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Equation Chapter 1 Section 1
Design of ZnS/ZnSe Gradient-Index Lenses in the Mid-Wave Infrared and Design,
Fabrication, and Thermal Metrology of Polymer Radial Gradient-Index Lenses
by
James Anthony Corsetti
Submitted in Partial Fulfillment of the
Requirements for the Degree
Doctor of Philosophy
Supervised by Professor Duncan T. Moore
The Institute of Optics
Arts, Sciences, and Engineering
Edmund A. Hajim School of Engineering and Applied Sciences
University of Rochester
Rochester, New York
2017
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Dedication
This thesis is dedicated to my parents. I would not be where I am now without your
love and support.
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Table of Contents
Biographical Sketch .......................................................................................................... vii
Acknowledgments.............................................................................................................. ix
Abstract…........ ................................................................................................................. xii
Contributors and Funding Sources................................................................................... xiv
List of Tables… ................................................................................................................ xv
List of Figures.. ................................................................................................................ xvi
Chapter 1. Introduction to GRIN materials ................................................................. 1
Motivation ............................................................................................................ 1
Definition of GRIN Shape.................................................................................... 1
1.2.1 Axial GRIN ................................................................................................... 2
1.2.2 Radial GRIN ................................................................................................. 3
1.2.3 Spherical GRIN ............................................................................................. 4
GRIN Materials .................................................................................................... 5
1.3.1 Glass .............................................................................................................. 6
1.3.2 ZnS/ZnSe GRINs .......................................................................................... 6
1.3.3 Polymers ....................................................................................................... 7
Thermal Modeling ................................................................................................ 8
Thermal Metrology ............................................................................................ 10
Objective of Thesis............................................................................................. 12
Chapter 2. GRIN ZnS/ZnSe Design Studies ............................................................. 16
Background ........................................................................................................ 16
Color Correction using GRIN Materials ............................................................ 17
Design Study ...................................................................................................... 21
2.3.1 Spectral Considerations .............................................................................. 21
2.3.2 Material Selection ....................................................................................... 22
Singlet Studies .................................................................................................... 24
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2.4.1 Specifications .............................................................................................. 24
2.4.2 Homogeneous Designs................................................................................ 24
2.4.3 ZnS/ZnSe GRIN Design ............................................................................. 25
2.4.4 Weight Analysis .......................................................................................... 27
2.4.5 Alternative GRIN Material Designs ........................................................... 28
Objective lens studies ......................................................................................... 32
2.5.1 Background ................................................................................................. 32
2.5.2 System Specifications ................................................................................. 32
2.5.3 Design summary ......................................................................................... 33
2.5.4 Homogeneous and GRIN Comparison ....................................................... 34
Zoom lens designs .............................................................................................. 36
2.6.1 Preliminary Zoom Design ........................................................................... 36
2.6.2 5X Zoom Design ......................................................................................... 46
Conclusions ........................................................................................................ 54
Chapter 3. Copolymer GRIN Designs ...................................................................... 56
Introduction ........................................................................................................ 56
PMMA/polystyrene pairing................................................................................ 57
Color correction.................................................................................................. 59
Zoom designs ..................................................................................................... 61
3.4.1 2x zoom designs .......................................................................................... 62
3.4.2 GRIN Chromatic Macro ............................................................................. 65
10x zoom designs ............................................................................................... 69
Conclusions and future work ............................................................................. 73
Chapter 4. Fabrication of copolymer GRIN elements .............................................. 75
Background ........................................................................................................ 75
Rochester process ............................................................................................... 76
4.2.1 Monomer preparation.................................................................................. 76
4.2.2 Copolymerization ........................................................................................ 76
4.2.3 Initial samples ............................................................................................. 79
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4.2.4 Results ......................................................................................................... 81
2x zoom design using manufactured profile ...................................................... 85
Conclusions and future work ............................................................................. 88
Chapter 5. Athermalization of radial GRIN polymers .............................................. 90
Introduction ........................................................................................................ 90
Thermal Effects – Homogeneous ....................................................................... 91
Thermal effects - radial GRINs .......................................................................... 92
Athermalization .................................................................................................. 95
Polymers ............................................................................................................. 96
Validation and description of model .................................................................. 97
PMMA/polystyrene GRIN study ....................................................................... 99
Analytic modeling ............................................................................................ 102
Numerical modeling ......................................................................................... 104
Conclusions and future work ........................................................................ 107
Chapter 6. Thermal Interferometry ......................................................................... 109
Introduction ...................................................................................................... 109
Discussion of Instrument .................................................................................. 111
6.2.1 Previous Generation .................................................................................. 111
6.2.2 Updated System ........................................................................................ 112
Interferometric Measurements ......................................................................... 117
6.3.1 Beam Paths................................................................................................ 117
6.3.2 Data Acquisition ....................................................................................... 119
6.3.3 Athermalization of the test arm ................................................................. 122
Results .............................................................................................................. 124
6.4.1 Thermal measurement considerations ....................................................... 124
6.4.2 Steel Sample measurement ....................................................................... 125
6.4.3 ZrO2 Measurements .................................................................................. 127
6.4.4 CaF2 Measurement .................................................................................... 128
6.4.5 Zerodur Measurements ............................................................................. 130
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6.4.6 Sapphire measurement .............................................................................. 132
Polymer Measurements .................................................................................... 133
Conclusions and future work ........................................................................... 138
Chapter 7. Conclusions ........................................................................................... 139
Concluding remarks ......................................................................................... 139
Suggestions for future work ............................................................................. 142
References.….. … ........................................................................................................... 146
Appendix A. Lens listing for 5x MWIR zoom lens – homogeneous .......................... 153
Appendix B. Lens listing for 5x MWIR zoom lens – GRIN ...................................... 157
Appendix C. Lens listing for 2x visible zoom lens – homogeneous .......................... 160
Appendix D. Lens listing for 2x visible zoom lens – GRIN (optimized profile) ....... 163
Appendix E. CODEV® GRIN Chromatic macro........................................................ 166
Appendix F. Lens listing for 10x visible zoom lens – homogeneous ........................ 173
Appendix G. Lens listing for 10x visible zoom lens – GRIN ..................................... 176
Appendix H. Lens listing for 2x visible zoom lens – GRIN (JC018 profile) ............. 180
Appendix I. MATLAB code for identifying athermalized radial GRIN lenses ........ 183
Appendix J. MATLAB finite-element model (FEA) for modeling effect of temperature on radial GRIN elements ............................................................................ 186
Appendix K. Spectral data for thermal interferometer fused silica beamsplitter ........ 191
Appendix L. Thermal interferometer: data acquisition code (MATLAB) ................. 192
Appendix M. Thermal interferometer: data analysis code (MATLAB) ...................... 201
Appendix N. CTE and dn/dT for JC022 samples ....................................................... 216
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Biographical Sketch
James Anthony Corsetti was born in Rochester, New York and graduated from
Pittsford Sutherland High School in 2006. Thereafter, he attended the University of
Rochester in Rochester, New York. He graduated in 2010 with a Bachelor of Science
degree in optics. He received a Master of Science degree in optics in 2013. He pursued
research in gradient-index optics and optical design and metrology under the supervision
of Professor Duncan T. Moore.
List of Publications:
James A. Corsetti, William E. Green, Jonathan D. Ellis, Greg R. Schmidt, and Duncan T. Moore. “Simultaneous interferometric measurement of linear coefficient of thermal expansion and temperature-dependent refractive index coefficient of optical materials.” Appl. Opt. 55(29), 8145-8152 (2016) Rebecca E. Berman, James A. Corsetti, Keija Fang, Eryn Fennig, Peter McCarthy, Greg R. Schmidt, Anthony J. Visconti, Daniel J. L. Williams, Anthony J. Yee, Yang Zhao, Julie Bentley, Duncan T. Moore, and Craig Olson. “Optical design study of a VIS-SWIR 3X zoom lens,” Proc. SPIE 9580, Zoom Lenses V, 95800D (September 3, 2015); James A. Corsetti, Greg R. Schmidt, and Duncan T. Moore. “Axial and Lateral Color Correction in Zoom Lenses Utilizing Gradient-Index Copolymer Elements,” Proc. SPIE 9293, International Optical Design Conference 2014, 92930Y (Dec. 17, 2014). James A. Corsetti, Greg R. Schmidt, and D. T. Moore, "Axial and Lateral Color Correction in Zoom Lenses Utilizing Gradient-Index Copolymer Elements," in Classical
Optics 2014, OSA Technical Digest (online) (Optical Society of America, 2014), paper IW2A.2. James A. Corsetti, Greg R. Schmidt, and Duncan T. Moore. “Design and characterization of a copolymer radial gradient index zoom lens,” Proc. SPIE 9193, Novel Optical Systems Design and Optimization XVII, 91930U (Sep. 12, 2014). James A. Corsetti, Anthony J. Visconti, Kejia Fang, James A. Corsetti, Peter McCarthy, Greg R. Schmidt, and Duncan T. Moore. "Design, fabrication, and metrology of polymer
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gradient-index lenses for high-performance eyepieces", Proc. SPIE 8841, Current Developments in Lens Design and Optical Engineering XIV, 88411G (Sep. 28 2013). Anthony J. Visconti, James A. Corsetti, Kejia Fang, Peter McCarthy, Greg R. Schmidt, and Duncan T. Moore. "Eyepiece designs with radial and spherical polymer gradient-index optical elements", Opt. Eng. 52(11), 112102 (Aug. 2, 2013). Anthony J. Visconti, Kejia Fang, James A. Corsetti, Peter McCarthy, Greg R. Schmidt, and Duncan T. Moore. "Design and fabrication of a polymer gradient-index optical element for a high-performance eyepiece", Opt. Eng. 52(11), 112107 (Aug. 2, 2013). James A. Corsetti and Duncan T. Moore. "Color correction in the infrared using gradient-index materials", Opt. Eng. 52(11), 112109 (Jul 30, 2013). James A. Corsetti, Leo R. Gardner, and Duncan T. Moore. "Athermalization of polymer radial gradient-index singlets", Opt. Eng. 52(11), 112104 (Jul 25, 2013). James A. Corsetti and Duncan T. Moore, "Design of a ZnS/ZnSe Radial Gradient-Index Objective Lens in the Mid-Wave Infrared," Imaging and Applied Optics, ITu2E.3, (June 25, 2013). Peter McCarthy, James Corsetti, Duncan T. Moore, and Greg R. Schmidt. “Application of a Multiple Cavity Fabry-Perot Interferometer for Measuring the Thermal Expansion and Temperature Dependence of Refractive Index in New Gradient-Index Materials.” Optical Fabrication and Testing (OFT) Optical Testing II, OTu2D, (June 25, 2012).
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Acknowledgments
Thank you to all of the people that have helped me over the years:
To Duncan Moore, for giving me a chance to join your research group and for all
of your guidance and technical insights over the years. Thank you for being a great mentor
and teaching me so much about optical engineering and especially lens design. Thanks to
you, I am soon to begin my post-school career in optical design taking with me many of
the lessons I have learned from you during design group over the years.
To the members of my thesis committee: Jonathan Ellis, Julie Bentley, Thomas
Brown, and John Lambropoulos for all of your guidance and support of my research.
To the Defense Advanced Research Projects Agency (DARPA) for funding my
research throughout graduate school (Contract HR0011-10-C-0111).
To the members of the GRIN group, especially everyone who was a part of the M-
GRIN program with me and helped me to complete my thesis: Anthony Yee, Yang Zhao,
Eryn Fennig, Oscar Ta, Ben Feifke, and Ed White.
To Evelyn Sheffer and Lynn Doescher for always being willing to help and never
forgetting anyone’s birthday.
To Peter McCarthy and Kejia Fang for your technical advice and assistance and for
making the laboratory full of laughter.
A special thanks to Greg Schmidt for always having a smile and open door, as well
as an answer for every optics question I ever asked. I really appreciate all of the time you
spent teaching me about everything from polymer chemistry to interferometry to the
benefits of mantis shrimp.
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To the department staff for all of their help and support over the years, especially
Gayle Thompson, Noelene Votens, Kari Brick, Gina Kern, Lori Russell, Betsy Benedict,
and Lissa Cotter.
To Per Adamson, for always being willing to help me in the laboratory and giving
illuminating advice on lasers, lenses, and love.
To James Zavislan for the many conversations and for being my mentor during my
undergraduate studies and for helping me to always keep things in perspective.
To all of the friends I have made over the years in the department, including the
lunch table: Daniel Sidor, Coby Reimers, Eric Schiesser, Kyle Fuerschbach, and Dustin
Shipp, as well as the Baloneks (Hillary, Robert, and Greg but not Dan…just kidding Dan,
you too).
To Bill Green for your friendship and for the many hours we spent with our heads
jammed inside of the environmental chamber. I appreciate all of the time and effort you
put in for me and all of the jokes and stories and trips to Harry G’s and Dogtown.
To Daniel Savage, for your friendship and help and sense of humor since we began
in the department. I cannot believe it has been ten years since we were starry-eyed freshmen
tracing rays in ITS.
To Brandon Zimmerman, for your friendship and advice and for teaching me the
lasers course with Dan a few days before the prelim. A special thanks for setting Margaret
and I up together.
To Michael Kaiman for your friendship and rounds one through infinity. Here is to
many more great times and laugh-until-we-cry conversations and experiences.
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To Aaron Bauer and Anthony Visconti for being great housemates and better
friends and for all the days of jokes, playing nerd cards, football, WWE, swearing at
CODEV®, and hiking volcanoes that made graduate school so much more enjoyable.
To my family, for their love and guidance.
To my grandparents, Lillian and Anthony Provazza and Florence and Amato
Corsetti, for the love and support you have always given me.
To my brother Matthew and sister Julia for always knowing how to make me laugh
and keeping me sane. I love you and thank you for both being my best friends.
To my parents, Sandra and Jim, without whom I would never have been able to
finish this work. I am blessed to have you both as parents and will strive to bring up your
future grandchildren with the same amount of love and patience you have always shown
to me and my siblings. Words cannot express my gratitude, I love you both.
To my amazing fiancé Margaret, for always believing in me when I did not. Thank
you for your unwavering love and support each and every day through this journey. I hope
that I can now do as wonderful a job for you as you complete your thesis as you did for me
during mine. I cannot wait to begin our life together. Te amo por siempre mi amor!!!
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Abstract
Gradient-index (GRIN) materials are ones for which the index of refraction varies
as a function of spatial coordinate within an optical element. The radial GRIN is a specific
instance where the isoindicial surfaces, or surface of constant index of refraction, exist as
concentric cylinders centered upon the optical axis. The variation of the index of refraction
as a function of lens aperture yields a second source of optical power in the element with
the first coming from the lens’ surface curvatures. This fact, coupled with the chromatic
variation of the GRIN profile, provides the optical designer with additional degrees of
freedom as compared to a traditional homogeneous lens, most notably in the pursuit of
correcting chromatic aberration. This thesis explores a number of topics related to the
design, manufacture, and testing of radial GRIN elements.
Such elements are used in a series of design studies, the first on the application of
the crystalline ZnS/ZnSe GRIN material to the mid-wave infrared (MWIR) waveband
between 3 and 5 μm and the second to a copolymer GRIN of polymethyl methacrylate
(PMMA) and polystyrene over the visible spectrum. In both cases, GRIN singlets are seen
to act as achromats over their respective wavebands. A series of zoom lens design studies
are presented in which the GRIN designs consistently offer superior color correction and
imaging performance over homogeneous designs of the same number of elements.
Efforts to fabricate the PMMA/polystyrene radial GRIN are presented. For this
purpose, a centrifugal force method is employed whereby both MMA and styrene monomer
are rapidly rotated in a temperature-controlled environment. As copolymerization occurs,
the spinning of the sample causes the isoindicial surfaces to take on a cylindrical shape.
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Process challenges including monomer-to-polymer volume reduction and haze are both
presented along with a discussion of the fabricated radial samples. A profile manufactured
in this way is modeled as part of the aforementioned zoom lens studies in CODEV® to
determine the sensitivity of the design space to the GRIN profile shape.
When designing any optical system, it is important to know how that system will
behave with a change in temperature. In order to answer that, two key material parameters
are defined: (1) the coefficient of thermal expansion (CTE) which dictates how much a
material expands or contracts with a temperature change and (2) the temperature-dependent
refractive index (dn/dT) which determines how the index of refraction changes. A series of
computer models are presented for the purpose of determining how a radial GRIN element
is affected by a given temperature change. Analogous to it being possible to achromatize a
single radial GRIN element, modeling work shows that it is also possible to athermalize
such an element.
Finally, an interferometric system is presented for the purpose of measuring both
the CTE and dn/dT of a sample simultaneously. The system operates by tracking changes
in optical path difference between the sample and background as a function of temperature
in order to carry out these measurements. Results on a number of samples including steel,
ZrO2, CaF2, Zerodur, Sapphire, and a series of PMMA/polystyrene copolymers are
presented.
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Contributors and Funding Sources
The work was supported by a dissertation committee consisting of Professors
Duncan Moore (advisor), Julie Bentley, and Thomas Brown of The Institute of Optics, and
Professors Jonathan Ellis and John Lambropoulos of the Department of Mechanical
Engineering at the University of Rochester.
The work in this thesis was supported by funding from the Defense Advanced
Research Projects Agency (DARPA) Manufacturable Gradient-Index (M-GRIN) program
(Contract HR0011-10-C-0111).
The design and modeling work in this thesis was made possible with a student
license of CODEV® provided by Synopsys®.
The GRIN design work carried out in this thesis was done with the use of linear
composition model created by Dr. Peter McCarthy.
In Chapter 4, the centrifugal system for fabricating the radial GRIN samples was
put together by Dr. Greg Schmidt. In the same chapter, the Mach-Zehnder interferometer
used to measure those samples was assembled by Dr. Peter McCarthy.
In Chapter 6, the mechanical design of the thermal interferometer test arm,
beamsplitter mount, and support frame was done in conjunction with Dr. Jonathan Ellis
and Bill Green who also assisted in the assembly. The index of refraction measurements of
sample JC022 in the same chapter were carried out with the assistance of Dr. Anthony
Visconti.
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List of Tables
Table 2-1: Abbe numbers of GRIN materials over three infrared wavebands. ................ 20
Table 2-2: Comparison of the required element focal length and Δn values for various
GRIN materials in the MWIR for a system focal length of 50 mm (f/2) as calculated from
base and GRIN Abbe numbers of each material. The Δn values marked with a star
indicate that they are not physically realizable for the system specifications while
unmarked values are realizable. ........................................................................................ 29
Table 2-3: Homogeneous and GRIN Petzval-like objective designs ................................ 34
Table 2-4: 3x zoom lens design specifications ................................................................. 37
Table 2-5: Specification comparison between NEOS and GRIN 5x zoom lenses. .......... 48
Table 3-1: First order specifications of 2x zoom design ................................................... 62
Table 3-2: First order specifications of 10x zoom design ................................................. 70
Table 4-1: Summary of calculation of required monomer volumes for sample JC018
layers ................................................................................................................................. 82
Table 5-1: Material data for polymers used in thermal modeling studies. ....................... 97
Table 5-2: Effect of +40°C temperature change on EFL for five lenses in athermalization
study ................................................................................................................................ 103
Table 6-1: Summary of JC022 samples .......................................................................... 134
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List of Figures
Figure 1-1: Illustration of isoindicial surfaces for both an (a) axial and (b) radial GRIN.
Colors designate surfaces of constant index of refraction. ................................................. 3
Figure 1-2: Illustration of isoindicial surfaces for a spherical GRIN. ................................ 5
Figure 1-3: CTE measured as a function of temperature for a number of optical materials.
Figure adapted from [30] ................................................................................................. 10
Figure 1-4: (a) Illustration of the calculation of transverse ray aberration plots. (b)
Example of a transverse ray aberration plot showing transverse error as a function of
normalized pupil coordinate, both in the y-direction. This particular plot indicates the
presence of axial color. ..................................................................................................... 13
Figure 2-1: Atmospheric transmittance of the electromagnetic spectrum.
Figure adapted from [52]. ................................................................................................. 22
Figure 2-2: “Glass map” for a number of MWIR materials. Homogeneous materials are
shown as solid markers while GRIN materials are line markers. ..................................... 23
Figure 2-3: On-axis MTF performance comparison between homogeneous designs and
ZnS/ZnSe GRIN singlet. ................................................................................................... 25
Figure 2-4: Comparison of performance between the aspheric ZnSe singlet (top) and the
GRIN singlet (bottom). The transverse ray plots are shown in units of mm. Note change
in scale of transverse ray plots. ......................................................................................... 27
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Figure 2-5: On-axis MTF performance comparison between ZnS/ZnSe GRIN singlet of
different weights. ZnS/ZnSe (1) is 12.9g, ZnS/ZnSe (2) is 7.2g and ZnS/ZnSe (3) is 3.9g.
........................................................................................................................................... 28
Figure 2-6: Comparison of MTF performance between three MWIR GRIN singlets. ..... 30
Figure 2-7: Comparison of performance between three MWIR GRIN singlets From top to
bottom: ZnS/ZnSe, IG3/IG4, and IG2/IG3. ...................................................................... 31
Figure 2-8: Radial GRIN profiles for three MWIR GRIN singlets From top to bottom:
IG3/IG4 (Δn ~ 0.13), IG2/IG3(Δn ~ 0.13), and ZnS/ZnSe (Δn ~ 0.10) ........................... 31
Figure 2-9: (Left) Si-Si aspheric homogeneous design (middle) ZnS/ZnSe-Si GRIN
design (right) Si-Ge-Si homogeneous design. , scale of ±60 μm ..................................... 36
Figure 2-10: First order element layout for three zoom positions. From top to bottom:
EFL = 150 mm, 100 mm, and 50 mm. .............................................................................. 39
Figure 2-11: 3x zoom homogeneous lens drawing. From top to bottom: EFL = 150 mm,
100 mm, and 50 mm. ........................................................................................................ 40
Figure 2-12: 3x zoom homogeneous MTF plots. From left to right: EFL = 150 mm,
100 mm, and 50 mm. ........................................................................................................ 41
Figure 2-13: 3x zoom homogeneous transverse ray plots, scale of ±50 μm. From left to
right: EFL = 150 mm, 100 mm, and 50 mm. .................................................................... 41
Figure 2-14: Second lens located within surface sag of first lens for certain zoom
positions before adding user-defined constraints. ............................................................. 42
Figure 2-15: 3x zoom homogeneous aspheric MTF plots. From left to right: EFL =
150 mm, 100 mm, and 50 mm. ......................................................................................... 42
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Figure 2-16: 3x zoom homogeneous aspheric transverse ray plots, scale of ±50 μm. From
left to right: EFL = 150 mm, 100 mm, and 50 mm. .......................................................... 43
Figure 2-17: 3x zoom GRIN lens drawing (first and third elements are GRIN). From top
to bottom: EFL = 150 mm, 100 mm, and 50 mm. ............................................................ 44
Figure 2-18: 3x zoom GRIN MTF plots. From left to right: EFL = 150 mm, 100 mm, and
50 mm. .............................................................................................................................. 45
Figure 2-19: 3x zoom GRIN transverse ray plots. From left to right: EFL = 150 mm,
100 mm, and 50 mm. Note change in scale (now ±25 μm) compared to homogenous
designs............................................................................................................................... 46
Figure 2-20: 5x zoom homogenous lens drawing. From top to bottom: EFL = 250 mm,
100 mm, and 50 mm. ........................................................................................................ 49
Figure 2-21: 5x zoom homogenous lens MTF plots. From left to right: EFL = 250 mm,
100 mm, and 50 mm. ........................................................................................................ 50
Figure 2-22: 5x zoom homogenous lens transverse ray plots, scale of ±50 μm. From left
to right: EFL = 250 mm, 100 mm, and 50 mm. ................................................................ 50
Figure 2-23: 5x zoom GRIN lens drawing. From top to bottom: EFL = 250 mm, 100 mm,
and 50 mm......................................................................................................................... 51
Figure 2-24: 5x zoom GRIN lens MTF plots. From left to right: EFL = 250 mm, 100 mm,
and 50 mm......................................................................................................................... 52
Figure 2-25: 5x zoom GRIN lens transverse ray plots, scale of ±50 μm. From left to right:
EFL = 250 mm, 100 mm, and 50 mm. .............................................................................. 52
Figure 2-26: 5x GRIN zoom lens at 100 mm focal length zoom position ........................ 52
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Figure 2-27: Index profiles for the three radial GRIN elements in system (λ = 4µm)
plotted as a function of normalized radial coordinate (0 is the center of the lens) ........... 54
Figure 3-1: Dispersion plots of PMMA and PSTY .......................................................... 59
Figure 3-2: Lens drawings and ray aberration plots for singlet/doublet study. ................ 60
Figure 3-3: 2x zoom design lens layout ............................................................................ 63
Figure 3-4: Index profile of 2x zoom GRIN element ....................................................... 63
Figure 3-5: Ray aberration plots for 2x zoom design ....................................................... 64
Figure 3-6: MTF curves for 2x zoom design .................................................................... 65
Figure 3-7: N10 plotted as a function of chromatic coefficient for 2X zoom GRIN design
........................................................................................................................................... 68
Figure 3-8: Lateral color for both individual lens groups and system for both
homogeneous (left) and GRIN (right) 2x zoom designs (units of mm). ........................... 69
Figure 3-9: 10x zoom design lens layout .......................................................................... 70
Figure 3-10: Index profile of 10x zoom GRIN elements .................................................. 71
Figure 3-11: Ray aberration plots for 2x zoom design ..................................................... 72
Figure 3-12: MTF curves for 10x zoom designs .............................................................. 73
Figure 4-1: Layout of centrifugal radial GRIN setup (figure credit: Greg R. Schmidt) ... 78
Figure 4-2: Photograph of centrifugal radial GRIN setup ................................................ 78
Figure 4-3: Examples of radial GRIN samples: (a) a radial GRIN rod that is underfilled
leaving a central air pocket shown with a ruler for scale, (b) a radial GRIN rod with a
visible interface, and (c) a fully-filled radial GRIN rod. Both (b) and (c) are 14.4 mm in
diameter............................................................................................................................. 79
xx
Figure 4-4: Examples of fabricated radial GRIN samples. The left column shows the
interferograms of two approximately 0.6 mm-thick sections of samples and the right
column shows the index profiles through the center. ....................................................... 81
Figure 4-5: Illustration of copolymer layering process ..................................................... 82
Figure 4-6: GRIN profiles of various sections of radial sample JC018 ............................ 84
Figure 4-7: (a) Images and (b) CODEV® model of sample JC018. The sample has a
diameter of 14.4 mm ......................................................................................................... 85
Figure 4-8: (a) Meaured index profile and sixth-order fit for slice 2 of sample JC018. The
grayed-out area indicates the region greater than the aperture of the lens designed with
the fitted profile. (b) Comparison of GRIN profile shapes for 2X zoom lens design
between designed lens and fit of JC018, slice 2 profile. Note the change in aperture size
of the element between the two designs. .......................................................................... 86
Figure 4-9: Difference in index of refraction between the sixth-order fit to the
interferometrically-measured index profile data and the data itself. The accuracy of the
interferometric index measurements is ±2x10-5. ............................................................... 86
Figure 4-10: Ray aberration plots for homogeneous and both GRIN designs (fabricated
profile vs. designed profile) evaluated at the extreme zoom positions. ............................ 88
Figure 5-1: Effect of temperature increase on (a) a homogenous window and (b) a Wood
lens for materials with positive CTEs. Note that curvatures are induced in the radial
GRIN element. .................................................................................................................. 93
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Figure 5-2: Output from MATLAB athermalization model for radials GRINs composed
of DAP (on axis) and CR-39®. The solid black curve indicates athermalized solutions.
The dashed line indicates afocal lenses. ........................................................................... 99
Figure 5-3: Output from MATLAB athermalization model for radials GRIN lenses of
5 mm thickness, 10 mm diameter and a ΔT of +40°C. (a) Lenses composed of pure
polystyrene on axis and varying amounts of PMMA at the periphery. (b) Lenses
composed of pure PMMA on axis and varying amounts of polystyrene at the periphery.
......................................................................................................................................... 100
Figure 5-4: A zoomed in version of Figure 5-3b to see the singlets of interest to be
compared for degree of athermalization. The five white dots indicate the five lenses of
the same nominal focal length (50 mm) chosen for the design study. ............................ 101
Figure 5-5: Effect of thickness change on athermalized GRIN singlet solution space. All
plots are pure PMMA on axis and varying amounts of polystyrene at the periphery
(between 0 and 100% polystyrene). All lenses are biconvex with c1 = -c2 varying between
0 and 0.05 mm-1. ............................................................................................................. 102
Figure 5-6: Illustration of differential element model of lens......................................... 105
Figure 5-7: Effect of CTE discrepancy on surface deformation fit. Only using the c
coefficient in the fit for the HIRITM/DAP material pair results in a relatively large fitting
error. By introducing the k coefficient into the fitting algorithm, this error can be brought
down to a level consistent with that achieved fitting the PMMA/polystyrene pair to only
the curvature c. ................................................................................................................ 106
xxii
Figure 6-1. Photograph of the thermal interferometer setup. Note meter stick on chamber for
scale. ................................................................................................................................ 115
Figure 6-2: Model of the Twyman-Green interfereometer system as designed and built.
The system is designed to measure over a 2” aperture. The reference arm is located
outside of the chamber whereas the test arm enters the chamber through a port in the top
of the chamber. Only the test arm is subjected to the change in temperature. The
reference arm remains at the temperature of the room. The sample under test sits directly
upon the test mirror inside of the environmental chamber. ............................................ 117
Figure 6-3. Side view of the beam paths within interferometer. The sample is shown resting upon
the test mirror. The three different OPDs can be used to compute CTE and dn/dT from
background fluctuations. .................................................................................................... 118
Figure 6-4: Pixel intensity as a function of applied voltage from the piezo controller .. 120
Figure 6-5: Comparison between generated phase maps using different numbers of steps
in phase-shifting algorithm. Each column shows the results for a specific number of
steps. The top row shows the generated phase map and the bottom row a horizontal
cutthrough of the wrapped phase data as indicated in the 2D phase map. ...................... 121
Figure 6-6. (a) Coated CaF2 sample resting on mirror. The region bounded by the dashed-line
rectangle indicates (b) the associated computed wrapped phase map. .................................... 121
Figure 6-7. Photograph of the interferometer test arm inside the thermal chamber. The lengths of
the invar and aluminum rods were chosen to minimize the drift of the sample location as a
function of temperature. ..................................................................................................... 123
xxiii
Figure 6-8. Plots of the change in optical path difference as a function of temperature for two
measurements of background fringes. The differences in materials (invar versus aluminum and
steel) comprising the test arm make the background motion less sensitive to thermal fluctuations.
......................................................................................................................................... 124
Figure 6-9. Measured change in thickness of 20 mm-thick steel gauge block for ΔT = 20°C with
the difference between the fit and measured data plotted on the secondary axis. The measured
CTE for this gauge block was 10.65 x10-6/°C at 20°C. ......................................................... 126
Figure 6-10. Measurement of 20 mm-thick steel gauge block CTE along with the reference data
from the manufacturer and previously reported Okaji data[33]. Note that the samples measured
by Okaji are 100 mm-thick. ................................................................................................ 127
Figure 6-11: Steel (left) and ZrO2 (right) samples .......................................................... 128
Figure 6-12: Measurement of CTE of ZrO2 sample in five degree increments .............. 128
Figure 6-13. Comparison of CaF2 CTE measurements between Rochester and literature values
[110, 111]. ........................................................................................................................ 129
Figure 6-14. Comparison of CaF2 dn/dT measurements between Rochester and literature values
[110-112]. Note that Corning’s dn/dT measurement was carried out at 656 nm rather than
632.8 nm. .......................................................................................................................... 130
Figure 6-15: Comparison of Zerodur® CTE measurements between Rochester and
literature values from Schott. .......................................................................................... 131
Figure 6-16: Comparison of Zerodur® dn/dT measurements between Rochester and
literature values from Schott. No error bars were given in the literature. Note that the
Schott data is measured at a wavelength of λ = 656.3 nm .............................................. 132
xxiv
Figure 6-17: Results of measurement of CTE (left) and dn/dT (right) of sapphire sample.
......................................................................................................................................... 133
Figure 6-18: Index of refraction (λ=532 nm) and time to volume reduction as a function
of composition for PMMA/polystyrene copolymers. ..................................................... 135
Figure 6-19: CTE and dn/dT as a function of composition for PMMA/polystyrene
copolymers. The parameters are calculated over the full range between 5 and 35°C. The
solid red lines indicate the range of reported values for homogeneous PMMA and
polystyrene ...................................................................................................................... 137
1
Chapter 1. Introduction to GRIN materials
Motivation
In engineering, packaging represents a major hurdle to overcome in the design and
fabrication of a product. Constraints on both physical dimensions and weight are key
concerns that must be addressed. The field of optics is no different in this regard as there
is a constant push for systems to be smaller and lighter whether they be the camera lens in
a smart phone or the mirrors forming the optical train of a space telescope. For this reason,
it is of interest to further technologies that enable optical systems and devices to be made
more compactly while maintaining or improving overall performance. The aim of this
thesis is to explore the potential of one such technology: gradient-index (GRIN) optics [1].
Traditionally, optical systems are composed of a number of homogeneous elements
in which the individual index of refraction of each element is constant in space for a single
wavelength. By allowing the index of single elements to vary as a function of position,
new degrees of freedom are introduced into the design process. Through the use of such
optical elements, imaging performance can be improved while system size and weight may
be decreased. These materials for which the index of refraction varies as a function of
spatial coordinate are called GRIN materials.
Definition of GRIN Shape
GRIN elements are traditionally defined by a number of factors, chief among them
the profile shape and element material. While the index profile of a GRIN element can
theoretically take on any three-dimensional shape, there are a number of profile shapes that
2
are much more common and therefore warrant further discussion. The profile shapes are
defined using the element’s isoindicial surfaces. Isoindicial surfaces are physical surfaces
or contours within the lens that have a constant index of refraction. The added degrees of
freedom in the design process afforded by GRIN elements are useful for both
monochromatic and polychromatic aberration correction. This fact has prompted the
inclusion of GRIN elements in the design of a number of optical systems [2-9].
1.2.1 Axial GRIN
The simplest profile shape is that of the axial GRIN where the isoindicial surfaces
are planes perpendicular to the optical axis as shown in Figure 1-1. This profile can be
designed to perform an equivalent role to that of an asphere by correcting spherical
aberration [10]. To illustrate this, one can think of under-corrected spherical aberration as
being an excess of optical path at the periphery of a lens that causes the rays traversing the
edge of the lens to focus before paraxial focus. An asphere corrects this aberration by
shaping the surface geometry to reduce the physical thickness at the edge of the lens. As
optical path is the product of physical distance and index of refraction, it is also valid to do
the correction by instead reducing the index of refraction at the periphery. An axial GRIN
profile is defined to do exactly that within the surface sag of the element. Having the
isoindicial surfaces extend past the limits of the surface sag and into the bulk of the lens
does not help correct the spherical aberration (since all rays traveling through the lens
experience the same index profile, regardless of aperture position at that point) but can
occur due to manufacturing limitations.
3
Mathematically, an axial profile can be defined as a function of position along the
optical axis (traditionally defined as the z-direction in optics) as shown in
2 3 4
00 01 02 03 04( ) ...N z N N z N z N z N z= + + + + + (1-1)
where the term N00 is the base index of the material while the terms N0M (where M = 1,
2,…) define the coefficients for the higher order terms of the polynomial.
Figure 1-1: Illustration of isoindicial surfaces for both an (a) axial and (b) radial GRIN. Colors designate
surfaces of constant index of refraction.
1.2.2 Radial GRIN
The next step in geometric complexity is the radial GRIN where the isoindicial
surfaces exist as concentric cylindrical surfaces centered around the optical axis so that the
surfaces of constant index are now parallel to the optical axis. In such a gradient, the index
varies as a function of radial coordinate, making it possible to introduce optical power with
just the index profile shape (independent of the lens surface curvatures). The Wood lens
is the simplest example of a radial gradient where either positive or negative optical power
can be achieved with a plano-plano element [11]. The sign of the optical power coming
4
from the GRIN profile depends on the orientation of the gradient. A positive gradient
describes when the index is higher along the optical axis and then decreases towards the
periphery while the opposite is true of a negative profile. This is analogous to a positive-
power homogeneous lens having greater thickness and therefore greater optical path at the
center of the lens than at its edge. A radial gradient is defined mathematically by
2 4 6 800 10 20 30 40( ) ...N r N N r N r N r N r= + + + + + (1-2)
where N is the index of refraction at some point r, the radial distance measured outwards
from the optical axis (so that r = 0 corresponds to being along the optical axis) while NM0
are the various index coefficients forming the index profile polynomial. It should be noted
that unlike the axial GRIN, the variation of the profile with the aperture location causes the
GRIN element to contribute to the chromatic behavior of the lens as well [12]. The majority
of the work carried out in this thesis is centered upon the design, fabrication, and metrology
of radial GRIN elements and as such this geometry is discussed in much greater detail in
subsequent chapters.
1.2.3 Spherical GRIN
For spherical gradients, the isoindicial surfaces are concentric spheres centered
upon a point P, which is located somewhere along the optical axis as shown in Figure 1-2.
Point P is defined as being located a distance rG from the vertex of the first surface of the
lens. The value of rG is free to take on any value, including ones so that the curvature of
the GRIN profile matches the curvature of one of the lens surfaces, which can make
manufacturing the elements easier if they are coined. The spherical GRIN is defined
mathematically using a combination of both Equation 1-3 and Equation 1-4.
5 2 3 4
0 1 2 3 4( ) ...N N N N N Nρ ρ ρ ρ ρ= + + + + + (1-3)
222 )( Grzyx −++=ρ (1-4)
Because the index varies both axially and radially, a spherical GRIN can be thought
of as a combination of both an axial and a radial GRIN profile. Both the Maxwell fisheye
and Luneberg lenses are specific instances of the spherical GRIN, with both being ball
lenses that have symmetric index profiles (so that point P is located in the center of the
lens). Each is capable of perfect geometric imaging for certain conjugates; however, optical
applications are limited as the object and/or image are located on the surface of or within
the element [13]. Extensive work has been carried out by Visconti et al. on the design,
fabrication, and testing of spherical GRIN elements where point P is located outside of the
lens for use within an eyepiece design [14].
Figure 1-2: Illustration of isoindicial surfaces for a spherical GRIN.
GRIN Materials
It is possible to form a GRIN element from a variety of materials using different
processes. Depending upon the waveband of interest, a number of options are available
6
from the ultraviolet, through the visible, and into the infrared. This section provides a
summary of a number of these materials.
1.3.1 Glass
Much research has been devoted to the pursuit of glass GRIN elements that transmit in
the visible [15-17]. Ion-exchange is a commonly used technique to make GRIN glass where
a piece of homogeneous glass is submerged into a liquid salt bath. The diffusion process
occurs where the free ions within the bath exchange with those of the glass. Thus the
chemical composition of the glass piece is altered from the outside in as the ions penetrate
further into the center of the material. If the glass is a cylindrical shape, this process can
yield a radial GRIN element. This diffusion process is traditionally a long one for elements
of large diameters (greater than 20 mm), requiring times on the order of multiple weeks or
months to yield a desired profile. The time required to yield a quadratic profile scales with
the square of the diameter of the lens blank, limiting the practicality of this method to
smaller-diameter elements; however, the time can be reduced with proper modifications to
process controls such as temperature and salt bath composition as demonstrated by
Visconti et al. [18].
1.3.2 ZnS/ZnSe GRINs
Chemical vapor deposition (CVD) is a process whereby chemical reactions are
controlled within a chamber to cause thin layers of a material to be deposited, layer by
layer, upon a substrate. Homogeneous zinc sulfide (ZnS) and zinc selenide (ZnSe) are both
crystalline materials that can be grown through CVD. Both of these materials have a very
wide band of transmission from 0.43 to 14 μm for ZnS and from 0.55 to 17 μm ZnSe
7
respectively [19]. This very large waveband of transmission makes both of these materials
very intriguing from the perspective of optical design. It has been demonstrated that a
GRIN from the pairing of ZnS and ZnSe can be grown using CVD [20]. Because the gases
that are used in the CVD process are highly toxic, pursuit of the fabrication process for the
ZnS/ZnSe GRIN has never occurred at the University of Rochester. This material pairing
does exhibit some unique chromatic properties, namely a negative Abbe number in the
mid-wave infrared (MWIR) which makes it of special interest from a color-correction
standpoint [21]. The ramifications of this fact are explored in much greater detail in
Chapter 2, which describes a series of designs using this material pairing over the band
between 3 and 5 μm. Other MWIR-transmitting GRIN materials include the ceramics
aluminum oxynitride (ALON®) and spinel along with a number of chalcogenide glasses
such as the Schott ‘IRG’ materials [22, 23].
1.3.3 Polymers
Polymers offer an opportunity for optical designs of reduced cost and weight when
compared with other homogeneous and GRIN materials [9, 22, 24-26]. Optical polymers
are typically formed from liquid monomers in a process caused polymerization. In this
process, a catalyst is introduced to cause a chemical reaction among the individual
monomer molecules which then form long molecular chains and become a solid polymer.
By allowing two miscible monomers to come into contact, diffusion between the two
monomers can take place. As polymerization occurs, a copolymer of the two materials is
formed, resulting in the formation of a GRIN profile. The catalyst used to fuel the reaction
is typically either heat introduced through a controlled water or air system or light, often
8
ultraviolet, in a process known as photopolymerization [27]. Depending on the shape,
orientation, and motion of the vessel used to hold the monomers being copolymerized, it
is possible to form a copolymer GRIN element of the axial, radial, or spherical geometry.
The processes used to generate each of these geometries are discussed in greater detail in
this thesis with a special emphasis placed on the radial geometry, the focus of the design
and metrology studies carried out in this work.
Thermal Modeling
An important consideration in the design of any optical system is how that system will
behave when subjected to a certain temperature change [26, 28]. As materials, optical or
otherwise, are heated or cooled, they experience a change in physical dimension. The
amount that the material expands or contracts is dictated by a parameter known as the
coefficient of thermal expansion (CTE) which is represented mathematically by
' (1 )L L L Tα= + ∆ (1-5)
where α is the CTE and L and L’ respectively are the dimensions of the element before and
after a temperature change of ΔT.
As the physical dimensions of a lens change with temperature, so does its index of
refraction. The magnitude and sign of this effect are described by another physical
parameter, the temperature-dependent refractive index, dn/dT, in accordance with
'dn
n n TdT
= + ∆ (1-6)
where n and n’ are the indices of the material before and after the temperature change.
Equation 1-7 is the expression for the optical power of a thick lens
9 2
1 21 2
( 1)( 1)( )lens
n c c tn c c
nϕ
−= − − + (1-7)
where c1 and c2 are the curvatures of the lens while t is its thickness. The change in the
curvature of a lens surface is given by
'1
cc
Tα=
+ ∆ (1-8)
where c and c’ are the curvatures of the lens before and after the temperature change. From
Equation 1-7 in conjunction with Equation 1-5, 1-6, and 1-8, one can see how a lens
changing temperature affects its optical power. The vast majority of materials have a
positive-signed CTE, meaning that for a temperature increase (positive ΔT) the thickness
of the element increases, increasing the value of φlens while the values for the curvature
decrease, decreasing the value of φlens. While some materials do have negative dn/dT
values, most have positive values. The power of a lens is directly related to its index of
refraction and as such, the change to φlens from an index change due to temperature can be
of either sign, being dependent on the signs of both dn/dT and ΔT.
Up to this point in this manuscript, only the effect of temperature upon
homogeneous elements has been discussed. Thermal considerations are of equal, if not
greater concern, to GRIN systems as the variation of the material composition throughout
the element means that both CTE and dn/dT vary as a function of position. In Chapter 5
the possibility of designing a radial GRIN lens such that the contributions to the optical
power coming from changes to lens geometry and index profile counteract one another for
10
a certain temperature change, maintaining the nominal focal length of the element and
therefore athermalizing the lens is discussed [26, 29].
Thermal Metrology
As mentioned in the previous section, both CTE and dn/dT are key parameters for
describing the effects of temperature on an optical element. Before any such analysis can
be carried out, one must have a means to measure these values. It should be noted that often
both CTE and dn/dT are quoted as a single value; however, this is misleading as both of
these parameters vary with the temperature of reference. Figure 1-3 shows an example of
this, displaying the CTE for a number of optical materials as a function of temperature [30].
From the data it is apparent that the CTE on a single material can vary dramatically
depending on the temperature range it is measured over.
Figure 1-3: CTE measured as a function of temperature for a number of optical materials.
Figure adapted from [30]
11
CTE is traditionally measured mechanically through the use of a dilatometer [31].
To do this, a sample is placed in physical contact with the measuring instrument. As the
sample is heated and expands, it pushes against a probe which is able to record or determine
displacement as a function of applied temperature. One example is the capacitance
dilatometer where the sample is placed between two plates that together form a capacitor.
As the sample changes size with temperature, the distance between the plates does as well.
This change in the measured capacitance can be converted to a measurement of the
thickness change of the sample with temperature.
Traditionally, dn/dT is determined using a refractometer that is capable of
measuring absolute index of refraction. By measuring the absolute index of a sample at a
series of temperatures, dn/dT is calculated by taking the derivative of the measured data
with respect to temperature.
It is possible to measure CTE and dn/dT optically using interferometry [32-36]. A
means to measure both of these parameters optically and simultaneously is desirable for
three reasons: (1) increased accuracy from using a known wavelength of light as the
measuring scale, (2) avoidance of needing to contact both surfaces of a sample with the
instrument, and (3) reduced measurement time and complexity by carrying out both
measurements simultaneously. For this purpose a thermal interferometer is built, capable
of measuring both CTE and dn/dT simultaneously and with each arm of the interferometer
subject to different environmental conditions [37]. At the time of writing this thesis, this is
the first such interferometer of its kind. A number of other interferometric systems exist
but many measure only CTE or dn/dT alone. Of those that do measure both parameters
12
simultaneously, the vast majority do so under vacuum and using a Fabry-Perot or other
configuration so that both reference and test arms of the interferometer are exposed to the
same environment which simplifies the measurement process but increases the cost.
Chapter 6 of this thesis describes the design, construction, and capabilities of this system
in much greater detail along with a discussion of the aforementioned other systems.
Objective of Thesis
The overall purpose of this thesis is to explore a number of topics related to the
design, fabrication, and metrology of radial GRIN elements. The first objective of the work
is to illustrate a GRIN element’s ability to improve the imaging performance of a lens
system with special emphasis placed on the correction of chromatic aberration in different
wavebands using different materials. This is presented in a series of design studies on (1)
the ZnS/ZnSe GRIN over the mid-wave infrared spectrum between 3 and 5 μm and (2) a
copolymer formed between polymethyl methacrylate (PMMA) and polystyrene that
transmits over the visible. These design studies are discussed in Chapter 2 and Chapter 3
respectively.
One goal of this thesis is to present the great potential of materials with negative
GRIN Abbe numbers to improve imaging performance. The majority of this thesis
concentrates on copolymer GRIN materials, namely the PMMA/polystyrene GRIN which
has a positive Abbe number over the visible spectrum. Negative GRIN Abbe numbers in
copolymers are possible over this waveband using theoretical material pairings. A non-
copolymer material known to have a negative GRIN Abbe number: the ZnS/ZnSe GRIN
13
is chosen to illustrate the benefits of this type of material for achromatization in a series of
design studies carried out over the mid-wave infrared.
Transverse ray aberration plots are used throughout the thesis as a metric for
determining a lens’ imaging performance and diagnosing whether certain aberrations,
namely axial and/or lateral color, are present in the lens system. As depicted in Figure 1-4,
such plots show transverse ray error (the distance between where a real and an ideal ray
will intersect the image plane, labeled εy in the y direction) as a function of normalized
pupil coordinate ρy, the x-axis of the plot. Separate plots are necessary for each field point
and in the sagittal (x-z) and tangential (y-z) planes of the lens.
Figure 1-4: (a) Illustration of the calculation of transverse ray aberration plots. (b) Example of a transverse ray
aberration plot showing transverse error as a function of normalized pupil coordinate, both in the y-direction.
This particular plot indicates the presence of axial color.
A GRIN’s ability to correct color is founded on the fact that a single radial GRIN
element contains a second source of optical power and chromatic dispersion (in addition to
that which comes from the base index of refraction). As such, a GRIN element contains
additional degrees of freedom as compared to a homogenous singlet and can be shown to
correct axial color in a manner approaching that achieved by a traditional achromatic
doublet. The observations taken from studies of simple homogeneous and GRIN singlets
14
and doublets are extended to more complex designs with a special emphasis placed on that
of zoom lens systems. Monochromatic zoom lens designs have been explored in the past
[38, 39]. When originally published, the zoom lens studies in this thesis were the first of
their kind into the use of both materials in their respective full wavebands. The application
of GRIN elements to polychromatic zoom lens designs has since been explored in greater
detail [40, 41]. One purpose of this thesis is to demonstrate the potential of these radial
GRIN elements to improve the imaging performance of such zoom lens systems. A
software tool to assist in this analysis was developed in CODEV® optical design software
to quantify the contributions to both axial and lateral color coming from the GRIN
elements.
The second objective of this thesis is to present a series of experiments to generate
these radial GRIN elements in the laboratory. Because the gases necessary to grow layers
of ZnS and ZnSe are toxic, no attempts are made to fabricate that material at the University
of Rochester. An apparatus for making PMMA/polystyrene GRIN elements is used that
applies a centrifugal force to a continuously-rotating mixture of the two monomers. The
fast rotation causes the monomers to separate according to their relative densities but still
diffuse into one another during the copolymerization process to generate a radial GRIN
profile. Difficulties experienced during these experiments keeping the center of the final
samples clear due to the issue of the volume reduction of the material within the monomer
chamber as the liquid monomer because a solid copolymer are discussed in Chapter 4.
The third objective of this thesis is to demonstrate the potential to athermalize a
radial GRIN singlet by taking advantage of the additional degrees of freedom that also
15
make it possible to achcromatize such a singlet under different circumstances. A series of
algorithms are presented that make it possible to identify such solutions so that they may
be further modeled in CODEV® or other ray-trace software. Additional algorithms are
presented to act as a basic finite-element model that treats a GRIN element as a stack of
differential rectangles. As a temperature change is applied, the effect on each element is
calculated and combined to determine more accurately the resulting change in lens
geometry and index profile than is possible analytically. This is discussed in greater detail
in Chapter 5.
The fourth and final objective of the thesis is to demonstrate the ability of the
thermal interferometer to simultaneously measure both the CTE and dn/dT of a given
homogeneous or GRIN sample. Explanations are presented concerning the theory behind
the measurement along with a discussion of the methodology used to prepare the sample
and then acquire and analyze the data. The interferometer design is based on the goals of
reducing both the cost of the instrument itself as well as time required to make a
measurement by being able to determine both thermal parameters from a single run.
Measurements have been carried out between approximately -40 and +50 °C. A discussion
of the design, operation, and results of the system are presented in Chapter 6.
16
Chapter 2. GRIN ZnS/ZnSe Design Studies
Background
Gradient-index (GRIN) materials are ones for which the index of refraction varies
as a function of position within the optical element [42]. The added degrees of freedom in
the design process afforded by GRIN elements are useful for both monochromatic and
polychromatic aberration correction. As discussed in Chapter 1, the particular effect of a
GRIN in an optical system is largely dependent upon the shape of the index profile with
the three most common GRIN geometries being: axial, where the isoindicial surfaces
(contours of constant index) are planes perpendicular to the optical axis, (2) radial, where
the isoindicial surfaces are concentric cylindrical surfaces centered around the optical axis
so that the contours of constant index are now parallel to the optical axis, and (3) spherical,
where the isoindicial surfaces are concentric spheres centered upon some point along the
optical axis. Axial GRINs perform a role largely analogous to an asphere and are capable
of correcting multiple orders of spherical aberration [10]. Radial and spherical GRIN lenses
have a second source of optical power coming directly from the index profile. The simplest
example of this effect is the Wood lens: a plano-plano element that is capable of forming
an image with a radial GRIN profile [11]. Radial GRINs are defined mathematically by
2 4 600 10 20 30( ) ...N r N N r N r N r= + + + + (2.1)
where r is the distance measured outward from the optical axis. By allowing the quadratic
radial GRIN term to vary, a second source of refracting power is added to the lens system,
theoretically creating a doublet in a single element. Given a lens with a radial GRIN profile
17
of the form shown in Equation 2.1 and a positive value of N10, the total optical power,
including surface curvatures and index profile, can be calculated by
200 1 2 10 00 00 1 2
00
sinh( )(N 1)( ) cosh( ) [2 ( 1) ]GRIN
tc c t N N N c c
N
αϕ α
α= − − − − − (2.2)
10
00
2N
Nα = (2.3)
where N00 and N10 are the base and quadratic coefficients of the radial GRIN index profile,
c1 and c2 are the curvatures of the first and second surfaces of the lens, and t is the thickness
of the element [43]. If N10 is instead negative, the cosh and sinh functions in Equation 2.2
are replaced with the cos and sin functions respectively. Approximating Equation 2.2, an
expression for the power of just the GRIN profile (ignoring the contributions of the surface
curvatures so that c1 = c2 = 0) is given by
102GRIN N tϕ = − (2.4)
assuming the thickness (t) of the element to be relatively thin [44]. This work concentrates
on the application of the radial geometry to color correction; however, achromatization
using GRIN elements is also possible using the spherical geometry as shown in recent work
[45].
Color Correction using GRIN Materials
Dispersion is a property inherent to all refractive optical materials whereby the
index of refraction of that material varies as a function of wavelength. This fact is the cause
of chromatic aberrations in polychromatic optical systems. All-reflective designs do not
suffer from chromatic aberrations; however, many such systems are designed with an
18
obscuration leading to a significant degradation of imaging performance at the mid-spatial
frequencies [46]. Axial color is the aberration causing best focus to vary as a function of
wavelength. The standard solution to correct axial color is to replace a homogeneous
singlet with a doublet composed of two different materials with different Abbe numbers, a
value that dictates how dispersive an optical material is. In doing so, the differing
dispersions of the two materials are able to balance one another in order to bring two
wavelengths to the same focus. The textbook example of such an achromat in the visible
spectrum is to combine a positive-power element composed of a crown glass (low
dispersion) such as BK7 with a negative-power element composed of a flint glass (high
dispersion) such as SF2. For a wavelength band of 3 to 5 µm, an achromatic doublet can
be formed with silicon and germanium [47]. In order to determine the powers of each
element necessary to correct axial color based on first order optical properties, one must
solve a pair of equations:
1 2 systemϕ ϕ ϕ+ = and (2.5)
1 2
1 2
0ϕ ϕ
ν ν+ = (2.6)
where φ is optical power, ν is the Abbe number, and the subscripts 1 and 2 denote the first
and second lenses composing the doublet. The Abbe number of a homogenous material is
defined for a particular waveband using
1mid
short long
n
n nν
−=
− (2.7)
19
where nshort, nmid, and nlong are the indices of the material at the extremes and center of the
spectral range. In order to minimize the optical power required of each individual element
when correcting color, the ratio of the two Abbe numbers should be as large as possible.
In order to correct secondary axial color, three wavelengths are brought to the same focus.
This requires definition of an additional property known as the partial dispersion (P) of a
material. The partial dispersion is defined according to
mid long
short long
n nP
n n
−=
−. (2.8)
A method for determining the paraxial contributions to both axial and lateral color
for radial GRIN elements is present in the literature [48]. As the index profile of a radial
GRIN element provides a second source of optical power to a lens, it also provides a second
Abbe number, making color correction possible with a single optical element (discussed in
further detail in subsection “ZnS/ZnSe GRIN Design”) [49]. There are a number of
material combinations available that are capable of forming a GRIN profile in the infrared
between 1 and 5 µm, although not all of them have been manufactured at this time. These
include those from the ceramic aluminum oxynitride (ALON®)[50] as well as
combinations of both zinc sulfide (ZnS) and zinc selenide (ZnSe)[20] and the Schott
chalcogenide ‘IG’ glasses.
While the methodology of picking materials for color correction as well as
Equations 2.5 and 2.6 hold true for GRIN materials as well as homogeneous ones, there
are different equations to define the Abbe number and partial dispersion of a GRIN
20
material. Equation 2.9 and Equation 2.10 define the GRIN Abbe number and partial
dispersion
midGRIN
short long
n
n nυ
∆=
∆ − ∆ (2.9)
mid long
GRIN
short long
n nP
n n
∆ −∆=
∆ − ∆ (2.10)
where each of the three Δn terms correspond to the difference in index of refraction
between the two homogeneous materials composing the gradient at a given wavelength.
As in the case when dealing with homogeneous elements, when correcting primary color
the difference in Abbe number between the two materials should be as large as possible in
order to minimize individual element power. The Abbe numbers for a number of infrared
GRIN materials are shown for the waveband from 1 to 5 µm, 1 to 3 µm, and 3 to 5 µm in
Table 2-1.
Table 2-1: Abbe numbers of GRIN materials over three infrared wavebands.
Material ν (1 to 5µm) ν (1 to 3µm) ν (3 to 5µm)
ALON® 8.1 50.9 9.4
ZnS/ZnSe 14.0 11.6 -63.1
IG2/IG3 2.8 3.0 71.8
IG2/IG6 6.2 6.7 96.0
IG2/IG4 6.9 7.3 188.1
IG4/IG6 5.8 6.4 73.5
IG4/IG3 2.0 2.2 52.5
IG6/IG3 0.2 0.2 8.0
21
It is possible for materials which transmit over a wide waveband to act as crown
glasses in one portion of the spectrum and as flint glasses in another. A common example
of this is germanium, which is much more dispersive between 1 and 5 µm than it is between
8 and 12 µm. From Table 2-1, it is apparent that such behavior can be exhibited by GRIN
materials in the infrared as between 1 and 3 µm, ALON® is much less dispersive than it is
between 3 and 5 µm, while the opposite is true of the majority of the GRIN chalcogenides.
Of particular interest is ZnS/ZnSe, which has a negative Abbe number between 3 and 5µm.
This GRIN material has been used in the design of infrared systems [51]. This work
explores the potential of this material to color correct over this particular waveband.
Design Study
2.3.1 Spectral Considerations
The mid-wave infrared (MWIR) is the spectral band between 3 and 5 µm. As shown
in Figure 2-1 this region sits between the short-wave infrared (SWIR) and long-wave
infrared (LWIR) in the electromagnetic spectrum and is bound on either side by spectral
bands of very high absorption due to atmospheric gas/water vapor which make imaging
over those wavelength ranges very difficult [52]. The MWIR waveband is a source of
interest for both military and non-military application. Achromatization studies have
already been carried out over this waveband using homogeneous materials [47].
22
Figure 2-1: Atmospheric transmittance of the electromagnetic spectrum. Figure adapted from [52].
2.3.2 Material Selection
Figure 2-2 shows a plot of Abbe number versus refractive index for a number of
infrared materials. As mentioned before, a high-performance MWIR achromat is formed
from silicon and germanium. The reason for this is clearly illustrated in Figure 2-2 where
one can see the large discrepancy in Abbe number between the two materials. Both silicon
and germanium have much larger indices of refraction than their visible glass counterparts,
which is somewhat common of infrared materials as seen in Figure 2-2, and often very
beneficial in the design process.
Tra
nsm
itta
nce
[%
]
Wavelength [µm]
23
Figure 2-2: “Glass map” for a number of MWIR materials. Homogeneous materials are shown as solid markers
while GRIN materials are line markers.
From Figure 2-2, the largest difference between a homogeneous and GRIN Abbe
number is seen to be between ZnSe and the ZnS/ZnSe GRIN, suggesting this material
would be very useful for color correction. Homogeneous ZnS and ZnSe elements have
already been used as part of achromatized infrared systems [53]. Additionally, the fact that
the gradient’s Abbe number is negative allows the optical power of both the base material
(the curvatures) and the gradient to be positive while satisfying Equations 2.5 and 2.6.
GRIN materials, along with diffractive optics, are one of the only means available to
achieve a negative Abbe number. As mentioned previously, a lens with a positive-power
gradient has a lower index of refraction at the edge of the element than at the center. This
is useful from the perspective of aberration correction as such a GRIN profile assist in the
24
correction of the undercorrected spherical aberration inherent to the base homogeneous
element.
Singlet Studies
2.4.1 Specifications
Beginning with single elements, this design study is performed using a full field of
view (FFOV) of 1° over the entire MWIR spectrum between 3 and 5 µm. System
specifications also include an entrance pupil diameter (EPD) of 25 mm and an effective
focal length (EFL) of 50 mm (yielding an f/2 lens). It is in faster systems that the real
benefit of GRIN becomes most apparent. At higher f/#’s, diffraction-limited performance
is comparatively easier to achieve and the use of GRIN does not yield as significant of
performance improvement over the homogeneous. All designs are carried out using
CODEV® optical design software.
2.4.2 Homogeneous Designs
To compare with the GRIN ZnS/ZnSe singlet, homogeneous lenses of both
materials are designed. With its higher index of refraction and lower dispersion, the ZnSe
singlets always yield superior performance to that of the ZnS singlets and so the ZnS
designs are not shown. Homogeneous singlets of both silicon and germanium are also
designed. Silicon yields better performance than germanium which yields better
performance than ZnSe; however, these designs are not included as the intent is to directly
compare the GRIN lens to the homogeneous materials that compose it. For both ZnS and
ZnSe, the limiting aberrations are third order spherical aberration and primary axial color.
To counteract the effect of spherical aberration, the fourth and sixth order aspheric terms
25
on the front surface of the element are allowed to vary, yielding a lens dominated by axial
color. A silicon and germanium air-spaced doublet is also designed in order to compare the
GRIN performance to the homogeneous doublet standard for achromats in the MWIR.
Figure 2-3 shows the on-axis MTF performance comparison between the homogeneous
ZnSe lenses and the ZnS/ZnSe GRIN lens (the design of which is discussed in the following
section).
Figure 2-3: On-axis MTF performance comparison between homogeneous designs and ZnS/ZnSe
GRIN singlet.
2.4.3 ZnS/ZnSe GRIN Design
All GRIN designs in this study are carried out using a model which forces the index
coefficients to vary within the refractive index bounds and according to the actual
dispersions of the real materials [54]. The imaging performance of the lens is optimized,
allowing both the element’s surface curvatures and defocus, along with the N00, N10, N20
and N30 coefficients of the radial GRIN index polynomial, to vary. The axial performance
of the final design is shown in Figure 2-3 along with the three aforementioned
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150
Mo
du
lati
on
Spatial Frequency [cycles/mm]
Diffraction Limit
Si-Ge Doublet
ZnS/ZnSe GRIN
Aspheric ZnSe
ZnSe
26
homogeneous designs. It is apparent that the imaging performance has improved
dramatically from the aspheric singlet, yielding nearly diffraction-limited performance
with a single element. Most notably, Figure 2-4 shows that the addition of the GRIN profile
has acted to largely correct the axial color which is the limiting aberration of the aspheric
singlet. Now the lens is limited by a combination of spherochromatism and secondary
spectrum. While it is clear that the air-spaced doublet yields the best performance of the
designs, it is notable that the GRIN design is able to achieve nearly the same MTF
specifications with only a single element. Cutting down on element count is always
desirable in optical design in order to reduce both system weight and packaging size as
well as material cost (this is especially true in the infrared where materials such as
germanium are more expensive that standard BK7). Adding an asphere to the GRIN singlet
acts to correct some of the higher order spherical aberration but does not offer significant
improvement to the imaging performance.
27
Figure 2-4: Comparison of performance between the aspheric ZnSe singlet (top) and the GRIN singlet (bottom).
The transverse ray plots are shown in units of mm. Note change in scale of transverse ray plots.
2.4.4 Weight Analysis
Given Equation 2.4, in order for the GRIN profile to maintain a constant effective
focal length as the lens is made thinner, the magnitude of the coefficient N10 and therefore
the total change in index of refraction between the center and periphery of the lens (Δn)
must increase. To a certain degree, as long as the increased required Δn is consistent with
the real material bounds, the lens can be made thinner and therefore lighter without
sacrificing imaging performance. This is demonstrated by varying the center thickness of
the ZnS/ZnSe singlet between 2 and 5 mm and continually reoptimizing the imaging
performance, yielding designs of 12.9 g, 7.2 g, and 3.9 g. Figure 2-5 shows the MTF for
each of these singlets compared to that of the silicon and germanium doublet. Note that the
28
7.2 g design offers only a slight degradation in MTF performance when compared to the
12.9 g design; however, MTF performance drops off significantly for the 3.9 g design.
Though not included in the Figure 2-5, further decreasing the weight of the single element
GRIN design further degrades the MTF performance while increasing the weight beyond
12.9 g does not offer a significant improvement in imaging performance. The 7.2 g GRIN
design offers a 31% weight reduction when compared with the homogeneous doublet.
Figure 2-5: On-axis MTF performance comparison between ZnS/ZnSe GRIN singlet of different weights.
ZnS/ZnSe (1) is 12.9g, ZnS/ZnSe (2) is 7.2g and ZnS/ZnSe (3) is 3.9g.
2.4.5 Alternative GRIN Material Designs
To be thorough in the analysis, other GRIN materials are explored for their ability
to color correct over the MWIR. Table 2-2 shows the Abbe numbers for both the base
material and GRIN profile for each of these material combinations, as well as the required
focal lengths of each necessary to satisfy Equations 2.5 and 2.6 for a system focal length
of 50 mm. The final two columns of the table display the calculated Δn necessary for an
element thickness of 5 mm, as well as the maximum Δn allowed by that GRIN material at
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150
Mod
ula
tion
Spatial Frequency [cycles/mm]
Diffraction Limit
Si-Ge Doublet
ZnS/ZnSe (1)
ZnS/ZnSe (2)
ZnS/ZnSe (3)
29
a wavelength of 4 µm. From the table, it is apparent that the very short individual focal
lengths required of some of these GRIN materials for MWIR color correction make using
those materials infeasible for this design study. In addition to this, the necessary values of
Δn for the majority of the materials are inconsistent with the real index bounds of the two
materials comprising that GRIN pairing using the assumed thickness of 5 mm. Thus in this
case, high imaging performance cannot be expected from those GRIN materials even when
allowing the element thickness to be much greater than 5 mm as correcting spherical
aberration and primary color require opposite-signed GRIN profiles. For this second part
of the design study, the three best physically realizable GRIN material candidates (based
on requiring the longest element focal lengths) are explored. In order from best to worst
these are: the ZnS/ZnSe combination discussed in the previous section, the IG3/IG4
chalcogenide glass combination, and finally the IG2/IG3 combination.
Table 2-2: Comparison of the required element focal length and Δn values for various GRIN materials in the
MWIR for a system focal length of 50 mm (f/2) as calculated from base and GRIN Abbe numbers of each
material. The Δn values marked with a star indicate that they are not physically realizable for the system
specifications while unmarked values are realizable.
Material νbase νGRIN fbase [mm] fGRIN [mm] Δn |Δnmax| ALON® 8 9 -5 5 -3.16* 0.03 IG2/IG3 195 72 32 -86 0.18 0.29 IG2/IG4 195 188 2 -2 7.97* 0.11 IG2/IG6 195 96 25 -52 0.30* 0.28 IG3/IG4 195 53 37 -136 0.12 0.18 IG3/IG6 168 8 48 -1007 0.02* 0.01 IG4/IG6 195 74 31 -83 0.19* 0.17
ZnS/ZnSe 178 -63 68 191 -0.08 0.18
The two chalcogenide GRIN designs use the same linear composition model
described in the previous section for the ZnS/ZnSe GRIN. Again, the GRIN profile is
allowed to vary to give the best performance possible. The on-axis MTF plots for each
30
design are shown in Figure 2-6. The best results of the three are attained with the ZnS/ZnSe
GRIN, followed by the IG3/IG4 pair, and the worst with the IG2/IG3 combination. These
findings are consistent with the calculated focal length data presented in Table 2-2. The
chalcogenide GRINs are limited by axial color and higher order spherical aberration as the
GRIN profile is of the opposite sign as it was with the ZnS/ZnSe GRIN. Note that only the
ZnS/ZnSe GRIN corrects axial color over the MWIR as is evident from the transverse ray
aberration plots shown in Figure 2-7. A comparison in GRIN profile between the three
designs is presented in Figure 2-8. The ZnS/ZnSe GRIN is positive in sign so the profile is
consistently working towards correcting both spherical aberration and axial color
simultaneously, while the negative profiles of the chalcogenides enhance undercorrected
spherical aberration.
Figure 2-6: Comparison of MTF performance between three MWIR GRIN singlets.
0
0.2
0.4
0.6
0.8
1
0 30 60 90 120 150
Mod
ula
tion
Spatial Frequency [cycles/mm]
Diffraction Limit
Si-Ge Doublet
ZnS/ZnSe GRIN
IG3/IG4
IG2/IG3
31
Figure 2-7: Comparison of performance between three MWIR GRIN singlets From top to bottom: ZnS/ZnSe,
IG3/IG4, and IG2/IG3.
Figure 2-8: Radial GRIN profiles for three MWIR GRIN singlets From top to bottom: IG3/IG4 (Δn ~ 0.13),
IG2/IG3(Δn ~ 0.13), and ZnS/ZnSe (Δn ~ 0.10)
32
Objective lens studies
2.5.1 Background
The previous section demonstrated the potential of the ZnS/ZnSe GRIN material
for color correction. It is interesting to investigate the effect of this material on a more
complicated optical system. Traditionally, in the visible spectrum, the Petzval lens
describes a high performance objective configuration of relatively fast speed and narrow
field of view, limited by astigmatism [55]. In their most simple form, Petzval designs are
composed of two positive-power element groups spaced apart from one another by a large
distance. If the lens is composed of only two elements, both of them must be positive power
and the system suffers heavily from axial color. The abnormal dispersive properties of the
ZnS/ZnSe GRIN provide a unique opportunity for designing Petzval-like objectives over
the MWIR.
2.5.2 System Specifications
All designs are specified to be f/2 with an effective focal length (EFL) of 100 mm
(yielding an entrance pupil diameter of 50 mm) [56]. The lens system is designed to be
compatible with the FLIR SC8300 indium antimonide (InSb) infrared detector, which is
sensitive over the waveband of interest (λ = 3-5 μm) [57]. The diagonal dimension of this
detector is 21.8 mm. For the assumed EFL of 100 mm, an image plane of this size
corresponds to a half field of view (HFOV) of 6.22° (for an object located at infinity.) All
designs are carried out in CODE V® using five field points (0, 40, 70, 85, and 100% of the
HFOV) and five wavelengths (λ = 3.0, 3.5, 4.0, 4.5, and 5.0 μm). Field weighting is allowed
33
to vary between designs in order to achieve the best imaging performance while the five
wavelengths are equally weighted during optimization. Distortion is held to be 2% or less.
In order to minimize the required diameter of individual lens elements, the aperture
stop is constrained to be located at the first surface of the first lens element. This is
consistent with the standard Petzval configuration [55]. The majority of infrared imaging
systems contain a thin window for the purpose of protecting the detector array [58]. To
simulate this, all designs are modeled with a 0.5 mm thick window of germanium placed
2 mm in front of the image plane. The clearance between the final lens element and the
protective window is constrained to be greater than 5 mm. Finally, the overall system
length (from the vertex of the first lens surface to the detector plane) is held to be less than
150 mm.
2.5.3 Design summary
For this study, designs of two and three elements are carried out using the
specifications quoted in the previous section. In addition to the GRIN designs,
homogeneous designs with and without aspheres are included in the study. Each design is
optimized for best imaging performance. When appropriate, the CODE V® Glass Expert is
used in order to find the best infrared material combination possible. Optimizing in this
way yields solutions composed mainly of silicon and germanium due to these materials’
very high indices. The FLIR detector used in this study has a pixel pitch of 14 μm. This
corresponds to a Nyquist frequency of approximately 36 line pairs per millimeter (lp/mm).
For each design, the contrast values of the worst-performing field point are tabulated from
the modulation transfer function (MTF) plots the software generates. Table 2-3 summarizes
34
these results with the MTF plot evaluated at 9, 18, 27, and 36 lp/mm (25, 50, 75, and 100%
of the Nyquist frequency, respectively) for each design. Aspheric surfaces are allowed in
some of the designs shown in Table 2-3. Specifically, designs with a ‘yes’ in the ‘aspheric
surfaces’ column are allowed one aspheric surface to be placed at the best location available
for that particular design. The final row of the table displays the diffraction-limited MTF
contrast values for each spatial frequency of interest.
Table 2-3: Homogeneous and GRIN Petzval-like objective designs
Elements Aspheres GRINs 9 lp/mm 18 lp/mm 27 lp/mm 36 lp/mm
2 No 0 0.508 0.134 0.077 0.053
2 Yes 0 0.695 0.400 0.248 0.165
2 No 1 0.864 0.697 0.546 0.405
2 No 2 0.878 0.733 0.600 0.487
3 No 0 0.894 0.783 0.677 0.576
3 Yes 0 0.895 0.782 0.673 0.571
Diffraction Limit 0.908 0.817 0.727 0.639
2.5.4 Homogeneous and GRIN Comparison
From Table 2-3 it is apparent that the two-element homogeneous system has the
worst imaging performance of any of the designs shown, as one would expect. In this case,
the best design is found to be formed from two silicon elements. Approaching the Nyquist
frequency, the imaging performance becomes very poor and the system is limited by a
combination of axial color and spherical aberration on axis. To reduce spherical aberration,
an aspheric surface is added with best performance achieved by placing it on the front
surface of the first silicon element. The design is now limited purely by axial color as seen
in Figure 2-9 which shows the lens drawings and transverse ray plots for this and other
designs of interest. (Note that for figure clarity, these evaluations are shown for three field
35
points.) By removing the aspheric surface and introducing a third element into the system,
the first silicon element is split into a silicon-germanium air-spaced doublet as shown in
Figure 2-9. The negative germanium element is useful for helping to correct axial color
and improves the performance to nearly-diffraction-limited levels. Adding an aspheric
surface to the three-element system provides a significant boost in the axial performance
of the lens but only moderate improvement in the off-axis performance.
For comparison, the design space is explored using the ZnS/ZnSe GRIN material.
For a two-element configuration, the ZnS/ZnSe GRIN yields better results as the front
element of the system than as the back element. From Table 2-3, it is clear that a two-
element GRIN/homogeneous (silicon) configuration yields significantly improved results
over the aspheric two-element homogeneous configuration. From Figure 2-9, it is apparent
that replacing the homogeneous element with a GRIN one drastically improves axial color
and therefore imaging performance. It should be noted that the silicon-germanium-silicon
design is still superior to ZnS/ZnSe-silicon design from a performance standpoint which is
consistent with the finding of the ZnS/ZnSe singlet study [21]. Going one step forward and
replacing the remaining silicon element with a second GRIN one provides a minor increase
in improvement that would unlikely justify the increased difficulty in fabrication over a
homogeneous lens.
36
Figure 2-9: (Left) Si-Si aspheric homogeneous design (middle) ZnS/ZnSe-Si GRIN design (right) Si-Ge-Si
homogeneous design. , scale of ±60 μm
Zoom lens designs
2.6.1 Preliminary Zoom Design
From here, the ZnS/ZnSe GRIN material is applied to the design of a zoom lens
[59]. Some work has been carried out on the application of GRIN elements to zoom lens
designs; however very little work has carried out in particular on the MWIR spectrum. A
material with a negative GRIN Abbe number is a new design feauture as applied to the
design space of zoom lenses over the MWIR. A number of homogenous infrared zoom
lenses have been designed over the last few decades [60]. For this zoom lens study, the
lens is designed to be compatible with an f/4 FLIR Photon HRC camera [61]. This is a 640
x 512 InSb detector with a pixel pitch of 15 µm that is sensitive in the MWIR. This detector
needs to be cooled and as such the camera core includes a Dewar enclosure in addition to
JAC 03-Mar-13
MWIR Petzval 2E Aspheric.seq
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
-0.06
0.06
-0.06
0.06
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.06
0.06
-0.06
0.06
0.70 RELATIVE
FIELD HEIGHT
( 4.354 )O
-0.06
0.06
-0.06
0.06
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 6.220 )O
JAC 03-Mar-13
MWIR Petzval 1 GRIN.seq
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
-0.06
0.06
-0.06
0.06
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.06
0.06
-0.06
0.06
0.70 RELATIVE
FIELD HEIGHT
( 4.354 )O
-0.06
0.06
-0.06
0.06
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 6.220 )O
JAC 03-Mar-13
MWIR Petzval 3E Homogeneous.seq
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
-0.06
0.06
-0.06
0.06
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.06
0.06
-0.06
0.06
0.70 RELATIVE
FIELD HEIGHT
( 4.354 )O
-0.06
0.06
-0.06
0.06
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 6.220 )O
MWIR Petzval 3E Homogeneous.seq JAC 03-Mar-13
18.00 MM
MWIR Petzval 2E Aspheric.seq JAC 03-Mar-13
18.00 MM
MWIR Petzval 1 GRIN.seq JAC 03-Mar-13
18.00 MM
37
the focal plane array (FPA). In order to minimize the unwanted background infrared
radiation “seen” by the FPA, designs with cooled detectors contain a mechanical surface
known as a cold shield to limit the ray bundle incident upon the detector. When the cold
shield is the aperture stop, 100% cold shield efficiency occurs, resulting in a system cold
stop [62]. Thus, in this design study, the aperture stop is always located on a surface in
between the aforementioned Dewar window and the image plane.
Because the aperture stop is located after the zooming elements in the lens, the size
of the entrance pupil scales with the EFL for different zoom positions, resulting in a
constant f-number throughout zoom. The lens itself is designed to be a 3x zoom with an
EFL varying between 150 mm and 50 mm. For the given diagonal size of the detector, this
yields a full field of view that ranges between 4.7° and 14.0°. A summary of the design
specifications for the system is shown in Table 2-4. Lens length is specified to be the
distance between the first surface of the first element and the second surface of the last
element (element three in this case).
Table 2-4: 3x zoom lens design specifications
Parameter Specification
Aperture f/4 (at all zoom positions)
Wavelengths 3-5 µm
EFL 150-50 mm
FFOV 4.7-14.0°
Lens Length < 190 mm
Dewar Window 1 mm thick Ge window between lens and FPA
For this design, a three lens group zoom layout is chosen in which the first element
is stationary while the second and third elements are moveable. The motion of the second
38
lens is responsible for changing the focal length while the third lens moves maintain the
image position to be constant. By having two moving lens groups in a zoom system it is
possible to leave the image plane stationary over a number of zoom positions. Following
the method on three-element zoom lenses laid out in Warren J. Smith’s Modern Optical
Engineering, a first order solution to the design was carried out using
1 2
1( )A
R
R s sϕ
−=
+ , (2.11)
( 1)B A Rϕ ϕ= − + , (2.12)
( 1)( )3 1
AC
R R
R
ϕϕ
+ +Φ=
− , (2.13)
1
( 1)( 1)A
Rs
Rϕ
−=
+ , (2.14)
12
ss
R= and (2.15)
3 1'
( 1)R
lR R
−=
Φ + (2.16)
where φA, φB, and φC, are the individual powers of the first, second, and third lens elements
respectively, Φ is the power of the system at the “minimal shift” position (EFL = 100 mm),
R is the square root of the system magnification (3 in this case so R = 1.732), s1 and s2 are
the distances between the first and second and second and third lenses respectively, and l’
is the distance from the third lens to the image plane [63]. In this way, a thin lens solution
was generated for three zoom positions (EFL = 150 mm, 100 mm, and 50 mm) using
CODE V®’s zoom functionality as shown in Figure 2-10.
39
Figure 2-10: First order element layout for three zoom positions. From top to bottom: EFL =
150 mm, 100 mm, and 50 mm.
At this point, the design is carried out monochromatically (λ = 4 µm) using all
germanium elements (chosen for its very high refractive index). Note that the fourth
element is the Dewar window and that the aperture stop is located between the window and
the FPA in order to simulate the aforementioned cold stop. The system is held constant at
f/4 for all three zoom positions. From here, the distances between the first and second
lenses, the second and third lenses, and the third lens and the image plane, along with the
surface curvatures of the three lenses are optimized for best imaging performance holding
only EFL at each zoom position and an overall system length as constraints.
After creating the desired first-order lens layout, it is necessary to add thickness to
each element while moving the aperture stop farther away from the Dewar window. Also
at this point, the waveband is redefined to be the full 3-5 µm, prompting the need for
40
material variation within the design in order to correct chromatic aberrations. While it can
be possible to get away with designing using only germanium elements between 8 and
12 µm, the same is not true between 3 and 5 µm where the material is much more
dispersive. Using a combination of direct material variation and the CODE V® Glass
Expert, the best three-element homogeneous design using only spherical surfaces is found
to be silicon-germanium-silicon. The lens drawing is shown in Figure 2-11 along with the
MTF and transverse ray plots in Figure 2-12 and Figure 2-13 respectively. The MTF plots
are evaluated out to 33 cycles/mm, corresponding to the Nyquist frequency of the detector.
Figure 2-11: 3x zoom homogeneous lens drawing. From top to bottom: EFL = 150 mm, 100 mm,
and 50 mm.
41
Figure 2-12: 3x zoom homogeneous MTF plots. From left to right: EFL = 150 mm, 100 mm, and 50 mm.
Figure 2-13: 3x zoom homogeneous transverse ray plots, scale of ±50 μm. From left to right:
EFL = 150 mm, 100 mm, and 50 mm.
From the transverse ray plots, it is apparent that the system suffers from a number
of residual aberrations, most notably under-corrected spherical aberration on axis and
lateral color off axis. The inclusion of aspheric surfaces has become a common method of
correcting spherical aberration, especially for designs operating in the infrared since many
of the materials may be diamond turned. To that end, the lens is further optimized by
allowing each of the three elements to have one aspheric surface. It should be noted that at
this point in the design process, the second lens travels within the surface sag of the first
lens for certain zoom positions as shown in Figure 2-14. User-defined constraints are
written in CODE V® to prevent this from happing.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 150mm
DIFFRACTION MTF
JAC 28-May-13POSITION 1
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )1.64O
T
R1.0 FIELD ( )2.35O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 100mm
DIFFRACTION MTF
JAC 28-May-13POSITION 2
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )2.45O
T
R1.0 FIELD ( )3.52O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 50mm
DIFFRACTION MTF
JAC 28-May-13POSITION 3
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )4.90O
T
R1.0 FIELD ( )7.01O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.00000
JAC 28-May-13
EFL = 150mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 1
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 1.645 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 2.347 )O
JAC 28-May-13
EFL = 100mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 2
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 2.450 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 3.518 )O
JAC 28-May-13
EFL = 50mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 3
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 4.900 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 7.009 )O
42
The optimal location of the three aspheres are determined by a combination of
direct variation on the part of the user and CODEV®’s Asphere Expert. The MTF and
transverse ray plots for the three element aspheric design are shown in Figure 2-15 and
Figure 2-16 respectively (the lens drawing has been omitted as it looks essentially the same
as that for the all-spherical surfaces design). From the MTF plots, it is apparent that the
addition of the aspheres has improved the imaging performance at each zoom position as
expected. Looking at the transverse ray plots, the system is now limited by polychromatic
aberrations (both axial and lateral color). Normally, the lateral color could be corrected by
making the system symmetric around the aperture stop; however, the presence of the cold
stop prevents this from being an option as the aperture stop must be placed on the cold
shield.
Figure 2-14: Second lens located within surface sag of first lens for certain zoom positions
before adding user-defined constraints.
Figure 2-15: 3x zoom homogeneous aspheric MTF plots. From left to right: EFL = 150 mm,
100 mm, and 50 mm.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 150mm
DIFFRACTION MTF
JAC 28-May-13POSITION 1
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )1.64O
T
R1.0 FIELD ( )2.35O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 100mm
DIFFRACTION MTF
JAC 28-May-13POSITION 2
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )2.45O
T
R1.0 FIELD ( )3.52O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 50mm
DIFFRACTION MTF
JAC 28-May-13POSITION 3
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )4.90O
T
R1.0 FIELD ( )7.01O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.00000
43
Figure 2-16: 3x zoom homogeneous aspheric transverse ray plots, scale of ±50 μm. From left
to right: EFL = 150 mm, 100 mm, and 50 mm.
In order to help correct the residual polychromatic aberrations and improve imaging
performance, GRIN elements are introduced into the design process. Zoom lens systems
containing GRIN element have been designed in the past [38, 39]. To ensure the indices
and dispersions of each GRIN element remain consistent with the real materials composing
them (ZnS and ZnSe in this case) the same linear composition model mentioned before is
used in CODE V® [54]. Given the index bounds defined by the two materials, this model
outputs a series of constraints for each polychromatic GRIN coefficient to be used during
optimization.
JAC 28-May-13
EFL = 150mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 1
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 1.645 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 2.347 )O
JAC 28-May-13
EFL = 100mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 2
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 2.450 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 3.518 )O
JAC 28-May-13
EFL = 50mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 3
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 4.900 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 7.009 )O
44
Figure 2-17: 3x zoom GRIN lens drawing (first and third elements are GRIN). From top to
bottom: EFL = 150 mm, 100 mm, and 50 mm.
Design studies show that a significant increase in imaging performance both on and
off-axis is achievable with the introduction of one or more GRIN elements. A promising
design is found by allowing both the first and third elements to be ZnS/ZnSe GRIN lenses
while placing an asphere on the first surface of the second element (silicon). The layout of
this lens is shown in Figure 2-17. In this case, both GRIN elements are ZnSe on axis with
the first GRIN element (lens 1) having a total change of index between center and periphery
(Δn) of 0.081 while the second GRIN element (lens 3) has a Δn of about 0.048 (for the
reference wavelength of 4 µm). The maximum Δn possible for this material combination
is 0.18.
45
Figure 2-18 shows the MTF plots for this design while Figure 2-19 shows the
transverse ray plots. Compared to the homogeneous aspheric design, MTF contrast has
improved both on and off axis for all three zoom positions. The transverse ray plots show
that for the EFL = 150 mm and 100 mm zoom positions, axial color has essentially been
corrected so that the design is limited on axis by secondary spectrum at these two zoom
positions and by spherical aberration at the third. Off-axis, the design is limited most
notably by lateral color and coma, both of which would be curbed by system symmetry
around the stop or the addition of more elements. Note that the scale in Figure 2-19 is
different than that of the transverse ray plots for the homogenous designs in order to
highlight the limiting aberrations of the system.
Figure 2-18: 3x zoom GRIN MTF plots. From left to right: EFL = 150 mm, 100 mm, and 50 mm.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 150mm
DIFFRACTION MTF
JAC 29-May-13POSITION 1
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )1.64O
T
R1.0 FIELD ( )2.35O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 100mm
DIFFRACTION MTF
JAC 29-May-13POSITION 2
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )2.45O
T
R1.0 FIELD ( )3.52O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 50mm
DIFFRACTION MTF
JAC 29-May-13POSITION 3
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )4.90O
T
R1.0 FIELD ( )7.01O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.00000
46
Figure 2-19: 3x zoom GRIN transverse ray plots. From left to right: EFL = 150 mm, 100 mm,
and 50 mm. Note change in scale (now ±25 μm) compared to homogenous designs.
2.6.2 5X Zoom Design
While the preliminary design study demonstrates the usefulness of the ZnS/ZnSe
GRIN in zoom systems operating over the MWIR, it is interesting to see how a GRIN
system compares as the difficulty of the design increases. For this purpose, the designer
attempts to use the specifications of a 5x MWIR zoom lens available from New England
Optical Systems, Inc. (NEOS) as a goal [64]. The NEOS system is designed to be
compatible with a 256 x 256 sensor with a pixel pitch of 30 µm as compared to the 640 x
512 (15 µm pixel pitch) format being used for the GRIN lens study. Both lenses are
designed to have a 5x zoom with an EFL varying between 250 mm and 50 mm. Because
the detector size is different in each case while the focal lengths remain constant, the field
of view will be slightly different for the two systems with the GRIN system able to subtend
a greater field of view due to the image size being bigger. The field of view in each case is
determined by
tanh f θ= (2.17)
JAC 29-May-13
EFL = 150mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 1
-0.025
0.025
-0.025
0.025
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.025
0.025
-0.025
0.025
0.70 RELATIVE
FIELD HEIGHT
( 1.645 )O
-0.025
0.025
-0.025
0.025
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 2.347 )O
JAC 29-May-13
EFL = 100mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 2
-0.025
0.025
-0.025
0.025
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.025
0.025
-0.025
0.025
0.70 RELATIVE
FIELD HEIGHT
( 2.450 )O
-0.025
0.025
-0.025
0.025
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 3.518 )O
JAC 29-May-13
EFL = 50mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 3
-0.025
0.025
-0.025
0.025
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.025
0.025
-0.025
0.025
0.70 RELATIVE
FIELD HEIGHT
( 4.900 )O
-0.025
0.025
-0.025
0.025
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 7.009 )O
47
where h is the half-diagonal of the detector, f is the EFL, and θ is the half-field of view. It
is assumed that the image size on the detector is a constant through zoom positions. While
the NEOS system is specified to be f/4, it is unclear if this is for only the 50 mm EFL or
for every zoom position (as is the case with the GRIN design as discussed earlier). If the
aperture stop is located at the cold shield for the NEOS design as well, the system is f/4 at
every zoom position. Information on the NEOS lens packaging constraints, Dewar
window, distortion, and imaging performance are unavailable online but are shown in
Table 2-5 for the GRIN lens along with a summary of the other system specifications for
each design. The GRIN design is specified to meet at least 40% contrast at every field point
at every zoom position at 23 cycles/mm (70% of the Nyquist frequency). The lens is also
specified to be significantly shorter in length than the 3x zoom system to make it more
compact. Because the aperture stop is located in the back of the system, the front element
of the system is generally very large, prompting the addition of a constraint on the
maximum allowable lens diameter as well.
The 3x design discussed in the previous section is used as a starting point for this
design. The addition of a fourth (stationary) element group is necessary to proceed forward
with the design using the increased zoom ratio. The NEOS design is specified to be dual
field so that it can operate in either narrow or wide field of view mode. The GRIN designs
are carried out at three zoom positions (EFL = 50 mm, 100 mm, and 250 mm) to add a
medium field of view zoom capability.
48 Table 2-5: Specification comparison between NEOS and GRIN 5x zoom lenses.
Parameter NEOS Design GRIN Design
Aperture f/4 f/4 (at all zoom positions)
Wavelengths 3-5 µm 3-5 µm
EFL 250-50 mm 250-50 mm
Detector Format 256 x 256 640 x 512
Pixel Pitch 30 µm 15 µm
Nyquist Frequency 17 cycles/mm 33 cycles/mm
Detector Diagonal 10.9 mm 12.3 mm
FFOV 2.5-12.4° 2.8-14.0°
Lens Length Unavailable < 135 mm
Lens Diameter Unavailable < 85 mm
Dewar Window Unavailable 1 mm thick Ge window between lens
and FPA
Performance Unavailable MTF > 40% for all field points at
23 cycles/mm (70% Nyquist frequency)
Distortion Unavailable < 2.5%
Both homogeneous and GRIN designs are carried out for the 5x system in tandem
in order to determine the optimal locations for the GRIN elements in the system and to
evaluate their effect on improving imaging performance. The design is capped to have a
maximum of six elements with less than 2.5% distortion. Both CODEV®’s Asphere Expert
and Glass Expert are used in the design process. A drawing of the three zoom positions for
the best homogeneous design is shown in Figure 2-20. Even with one aspheric surface
allowed on each of the six homogeneous elements, it is not possible to reach the MTF
specifications shown in Table 2-5 for the given system configuration. From Figure 2-21, it
is apparent that at the two extreme zoom positions, namely the narrow field of view zoom,
the MTF curves are not close to the required 40% contrast at 23 cycles/mm for every field
point. Even by weighting these zoom positions more heavily in the optimization process
the performance requirement could not be met. The transverse ray plots for the
49
homogenous design are shown in Figure 2-22. From these, it is apparent that the system is
again limited by both axial and lateral color as the aspheric homogeneous 3x design is in
the previous section. This is encouraging as the potential for GRIN elements in the system
is investigated.
Figure 2-20: 5x zoom homogenous lens drawing. From top to bottom: EFL = 250 mm, 100 mm, and
50 mm.
25.00 MM
25.00 MM
25.00 MM
50
Figure 2-21: 5x zoom homogenous lens MTF plots. From left to right: EFL = 250 mm, 100 mm, and 50 mm.
Figure 2-22: 5x zoom homogenous lens transverse ray plots, scale of ±50 μm. From left to
right: EFL = 250 mm, 100 mm, and 50 mm.
Transitioning from a homogeneous to a GRIN design, the front element of the
system is first changed from silicon to the ZnS/ZnSe GRIN. Optimization from here shows
a significant improvement in imaging performance as the GRIN profile assists in the
correction of the polychromatic aberrations. From here, additional GRIN elements are
added to improve system performance while the materials of the remaining homogenous
elements along with the locations of the aspheric surfaces are systematically varied in an
iterative process to determine the optimal design configuration. Figure 2-23 shows a
drawing of the GRIN system at the three zoom positions while Figure 2-24 and Figure 2-25
show the MTF plots and transverse ray plots respectively for this lens. As is the case for
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 250mm
DIFFRACTION MTF
JAC 06-Jun-13POSITION 1
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )0.99 O
T
R1.0 FIELD ( )1.41 O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 100mm
DIFFRACTION MTF
JAC 06-Jun-13POSITION 2
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )2.46O
T
R1.0 FIELD ( )3.52O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 50mm
DIFFRACTION MTF
JAC 06-Jun-13POSITION 3
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )4.90O
T
R1.0 FIELD ( )7.01O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.00000
JAC 06-Jun-13
EFL = 250mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 1
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 0.986 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 1.409 )O
JAC 06-Jun-13
EFL = 100mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 2
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 2.462 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 3.518 )O
JAC 06-Jun-13
EFL = 50mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 3
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 4.900 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 7.009 )O
51
the homogeneous design, the MTF plots for the GRIN lens are evaluated out to
23 cycles/mm. From Figure 2-24 it is apparent that the GRIN design achieves the
performance requirement of 40% contrast at all fields at 23 cycles/mm. Of particular
interest is the comparison of the two sets of ray aberration curves between the homogenous
and GRIN designs (Figure 2-22 and Figure 2-25). Both sets of plots are shown on the same
scale to highlight the performance improvement achieved by adding the GRIN elements.
It is worth noting that while the homogeneous design is limited heavily by primary lateral
color, this aberration is largely corrected in the GRIN system where the transverse ray plots
show secondary lateral color instead.
Figure 2-23: 5x zoom GRIN lens drawing. From top to bottom: EFL = 250 mm, 100 mm, and
50 mm.
25.00 MM
25.00 MM
25.00 MM
52
Figure 2-24: 5x zoom GRIN lens MTF plots. From left to right: EFL = 250 mm, 100 mm, and 50 mm.
Figure 2-25: 5x zoom GRIN lens transverse ray plots, scale of ±50 μm. From left to right: EFL
= 250 mm, 100 mm, and 50 mm.
Figure 2-26: 5x GRIN zoom lens at 100 mm focal length zoom position
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 250mm
DIFFRACTION MTF
JAC 07-Jun-13POSITION 1
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )0.99 O
T
R1.0 FIELD ( )1.41 O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 100mm
DIFFRACTION MTF
JAC 07-Jun-13POSITION 2
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )2.46O
T
R1.0 FIELD ( )3.52O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.000001.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0
SPATIAL FREQUENCY (CYCLES/MM)
EFL = 50mm
DIFFRACTION MTF
JAC 07-Jun-13POSITION 3
DIFFRACTION LIMIT
AXIS
T
R0.7 FIELD ( )4.90 O
T
R1.0 FIELD ( )7.01 O
WAVELENGTH WEIGHT
5000.0 NM 1
4000.0 NM 1
3000.0 NM 1
DEFOCUSING 0.00000
JAC 07-Jun-13
EFL = 250mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 1
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 0.986 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 1.409 )O
JAC 07-Jun-13
EFL = 100mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 2
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 2.462 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 3.518 )O
JAC 07-Jun-13
EFL = 50mm
RAY ABERRATIONS ( MILLIMETERS )
5000.0000 NM
4000.0000 NM
3000.0000 NM
POSITION 3
-0.05
0.05
-0.05
0.05
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.05
0.05
-0.05
0.05
0.70 RELATIVE
FIELD HEIGHT
( 4.900 )O
-0.05
0.05
-0.05
0.05
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( 7.009 )O
53
Counting elements from the left to the right in Figure 2-26, lenses 1, 4, and 6 are
all ZnS/ZnSe GRIN. Optimization trials determined these locations to be the best
candidates for the GRIN profiles. Figure 2-27 shows plots of each of these three GRIN
profiles as a function of normalized aperture coordinate. The dotted lines indicate the index
bounds of homogenous ZnS and ZnSe at λ = 4 µm. Note that elements 1 and 4 have positive
optical power from both the surface curvatures as well as from the GRIN profile, while the
opposite is true of element 6 which has negative power coming from both sources. This
consistency in the signs of the power coming from each is the result of the negative Abbe
number of the GRIN profile. The trend is consistent with Equation 2-5 and Equation 2-6
which imply that when designing an achromat, the powers of both the surface curvatures
and GRIN profile may be the same sign as long as their Abbe numbers are of opposite sign.
The fact that CODE V® appears to be using the GRIN profile to achromatize single
elements within this design is consistent with a fundamental principle of zoom lens design,
namely that each lens groups should be individually achromatized (as opposed to balancing
system aberrations between lens groups) in order to yield high performance at each zoom
position [65].
54
Figure 2-27: Index profiles for the three radial GRIN elements in system (λ = 4µm) plotted as
a function of normalized radial coordinate (0 is the center of the lens)
There are a total of two aspheric surfaces in the system, one on the front surface of
lens 2 and one on the back surface of lens 5. While the design is only shown at the axial,
70% and full field point in this manuscript, it is checked at a number of intermediate field
points to ensure it meets performance specifications throughout the full field. All
specifications in Table 2-5 are met for the final GRIN design. CODEV® lens listings for
both the homogeneous and GRIN 5x zoom lenses are shown in 0 and Appendix B.
Conclusions
The ZnS/ZnSe GRIN has unique dispersion properties, a negative Abbe number, in
the MWIR that makes it an especially interesting material for the optical engineer designing
over this waveband. It is possible to both correct axial color and attain very high quality
imaging using only a single GRIN element. This work demonstrates the benefits of
2.25
2.30
2.35
2.40
2.45
0 0.2 0.4 0.6 0.8 1
Ref
ract
ive
Ind
ex
Normalized Lens Radius
Lens 1 Lens 4 Lens 6
55
ZnS/ZnSe GRIN elements in zoom systems operating over the MWIR. For a fixed element
count, GRIN aspheric designs are shown to offer superior imaging performance when
compared with homogeneous aspheric designs.
For the future it would be interesting to reattempt the design of the 5x system using
a different first order zoom layout. This design used the 3x system as a starting point and
resulted in a four element group layout with the first and third groups being positive power
and the second and fourth groups being negative power. Beginning the design with an
alternative power layout such as positive-negative-positive-positive or positive-negative-
negative-positive would likely result in different design forms than the one shown in this
paper and it would be useful to compare homogenous versus GRIN system performance
for each power combination.
It would also be worthwhile to explore the application of GRIN elements to more
complicated lenses with larger zoom ratios as there are a number of MWIR commercial
products having zoom ratios of 10x. Finally, there is a growing push towards developing
dual-band systems that are capable of imaging over multiple wavebands. It is interesting
to investigate how the addition of ZnS/Znse as well as other GRIN elements affect dual-
band mid/short wave infrared and mid/long wave infrared imagers.
56
Chapter 3. Copolymer GRIN Designs
Introduction
As mentioned in the introduction of this thesis, it is possible to form a GRIN profile
in a copolymer element. Polymers offer an opportunity for optical designs of reduced cost
and weight when compared with other homogeneous and GRIN materials. By allowing
two miscible monomers to come into contact, diffusion between the two monomers can
take place. As polymerization occurs, a copolymer of the two materials is formed, resulting
in the formation of a GRIN profile. Ohtsuka and Koike have performed extensive research
on a variety of copolymer GRIN systems, stemming mostly from the application of the
copolymers to GRIN fibers [24, 66, 67]. This group has also designed and fabricated much
larger radial polymer GRIN elements for the purpose of improving the human visual
system as elements for both eyeglasses and contacts lenses [68, 69]. Although the majority
of this work has concentrated on the radial geometry, Koike has worked on systems of both
the axial and spherical geometries [13, 70, 71].
Baer has pioneered a method of fabricating polymer GRIN elements from a
combination of polymethyl methacrylate (PMMA) and SAN17 by first forming sub
quarter-wavelength thick layers of the homogeneous materials and then combining them
in different ratios to form bulk material that has a refractive index dictated by the relative
composition of the two polymers [25]. Depending on the desired gradient-index profile,
individual layers of these various stock bulk materials are stacked and then thermoformed
at increased temperature and pressure in order to form a GRIN lens blank that can be coined
and diamond-turned into a lens.
57
In the Moore group at the University of Rochester, various studies have been
carried out on a variety of GRIN copolymer materials in a number of geometries. Gardner
investigated the application of the partial polymerization process for fabricating
combinations of DAP, HIRITM, and CR-39® in the radial geometry [26].
Prepolymerization essentially refers to the process of partially polymerizing a sample so
that the monomers begin to form long molecular chains but the full polymerization is not
completed. The net effect of this is to form a preform that can hold a desired shape
(cylindrical in the case of the radial) with a higher viscosity than the pure liquid monomer.
Once the preform is created, it is exposed to a second monomer or blend of monomers in
order to initiate material diffusion and generate an index profile. In order to simulate the
physiology of the insect eye, Schmidt modeled and fabricated tapered GRIN elements from
a variety of polymers through partial polymerization [72]. Photopolymerization is an
alternative means of fabricating GRIN polymer elements that uses light to catalyze the
chemical reaction [27].
PMMA/polystyrene pairing
Extensive work has been carried out regarding the copolymer formed between
(PMMA) and polystyrene. Both PMMA and polystyrene are desirable for use as they are
relatively cheap and readily available. Visconti has demonstrated the potential of this
material combination to greatly improve field aberrations and imaging performance in a
series of eyepiece studies [9]. While both radial and spherical geometry elements were
determined to be useful in the design process, the spherical geometry was chosen for further
pursuit due to greater maturity of the manufacturing process. Fang demonstrated the ability
58
to fabricate high optical quality axial-geometry GRIN rods in glass test tubes from PMMA
and polystyrene through a diffusion copolymerization process [73]. From there, these axial
geometry GRIN preforms were measured for index change and thermoformed to achieve
the desired profile slope [74]. From there, they were compressed in a spherical mold to
curve the planar isoindicial surfaces and yield a spherical GRIN. Efforts to generate radial-
geometry GRIN elements are discussed later in this chapter.
Note that the mathematics and color correction theory on radial GRIN elements
discussed in the previous chapter hold equally true for this material as for the ZnS/ZnSe
pairing As discussed before, the variation of the composition between two materials has an
Abbe number due to the refractive index changing throughout the material which makes it
possible for a GRIN singlet to act as a homogeneous doublet in an optical design [75].
Figure 3-1 displays the dispersion plots for both PMMA and polystyrene. This data is
measured on a Pulfrich refractometer and is used to calculate the base Abbe numbers of
each material as well as their GRIN Abbe number using Equation 2-7 and Equation 2-9
respectively. For the GRIN calculation, each Δnλ term corresponds to the difference in
index between PMMA and polystyrene at a given wavelength as shown in Figure 3-1. With
an Abbe number of 9, the GRIN profile is much more dispersive than either of the
homogeneous polymers that compose it (the Abbe numbers of PMMA and polystyrene are
57 and 30, respectively). Having two widely-spaced Abbe numbers is desirable from an
aberration and color-correction standpoint as it reduces the individual optical powers
required of each lens in the doublet. Thus, forming an imaging lens from this GRIN
59
copolymer demonstrates far superior chromatic properties when compared with a single
homogeneous PMMA or polystyrene element.
Figure 3-1: Dispersion plots of PMMA and PSTY
Color correction
Before progressing to the discussion of the zoom lens design, it is useful to show the
potential of these radial GRIN elements for color correction in simpler optical designs. For
this purpose, a series of singlets and doublets are designed at f/5 with a 0.5° half field of
view (HFOV) and an effective focal length (EFL) of 100 mm. Lens drawings and the on-
axis ray aberration plots for four of these designs are shown in Figure 3-2. As shown in
Figure 3-2, a homogeneous PMMA singlet (biconvex) is limited by a combination of axial
color and undercorrected spherical aberration. As expected, if an aspheric surface is added
to the front surface of that element, the spherical aberration is corrected but the design is
still limited heavily by axial color. If the aspheric singlet is replaced with a
PMMA/polystyrene copolymer radial GRIN singlet with spherical surfaces, it is apparent
60
that the imaging performance is vastly improved as the design is no longer limited by axial
color but rather by a combination of secondary spectrum and spherochromatism. Compared
to the standard BK7/SF2 homogeneous doublet pair, the GRIN singlet is inferior in terms
of performance; however, it is a vast improvement over either of the homogeneous singlets
and it is important to remember that in this case, a single copolymer element is being
compared to two glass elements.
Figure 3-2: Lens drawings and ray aberration plots for singlet/doublet study.
It is interesting to note how well the design results match the achromatic doublet
equations for a radial GRIN element. This particular GRIN element shown in Figure 3-2
has a thickness of t = 7.57 mm, first and second surface radii of curvature of R1 = 50.06 mm
and R2 = -219.72 mm respectively, base index of 1.5339 at a wavelength of 587.6 nm
(42.4% PMMA and 57.6% polystyrene), and quadratic index coefficient (from
Equation 1.2) of N10 = 0.000202 mm-2. Using the radii and refractive index in Equation 2.2,
the homogeneous contribution to the optical power is found to be φhom = 0.0131 mm-1.
61
Using the thickness and N10 term in Equation 2.4, the GRIN contribution to the optical
power is calculated to be φGRIN = -0.0031 mm-1. Note that the powers are of opposite sign
as is expected in traditional achromat theory when dealing with two positive-Abbe number
materials. Substituting these values into Equation 2.5, the system power is found to be
0.01 mm-1 which is consistent with the 100 mm focal length of the lens while substituting
them into Equation 2.6 (with a homogeneous Abbe number of 38.1 and a GRIN Abbe
number of 9.1) yields a sum of 3.2 x10-6 mm-1, very close to the expected value of zero for
a color-correcting doublet.
Zoom designs
Recent design studies including the one discussed in the previous chapter have
shown that GRIN elements are very useful for color correction in a variety of wavebands
including the visible, the dual band visible/short-wave infrared, and the mid-wave infrared.
Although a few GRIN zoom systems have been designed in the past, polychromatic designs
utilizing real materials have been very limited to this point [38, 39, 59]. It is the aim of this
thesis to see how much of a performance improvement these GRIN elements can offer to
homogeneous zoom system largely limited by chromatic aberrations. The copolymer
GRIN is used in the design of a pair of zoom lens “families” which varied in zoom ratio as
well as in the number of moving groups within the lens from family to family. In all cases,
GRIN designs are compared to homogeneous designs for a fixed total element count. All
designs are carried out using McCarthy’s aforementioned linear composition model in
CODEV® optical design software [76].
62
3.4.1 2x zoom designs
For this purpose, two sets of zoom systems, a 2x as well as a 10x system, are
compared [77-79]. The specifications for both of these designs are taken from a previous
paper by Youngworth et al. on zoom designs, in which a patent is referenced [80, 81]. The
first of these designs is a 2x zoom system specified at three zoom positions, the first order
specifications of which are shown in Table 3-1. While the specifications were taken from
the paper, the 2x zoom lenses are designed from scratch utilizing the same methodology
laid out in Chapter 2. From the lens drawings shown in Figure 3-3, it is clear that both the
homogeneous and GRIN designs have four elements with one copolymer GRIN element
present as the second element of the GRIN system. Both lenses contain aspheric surfaces
on the first surface of the front element and the second surface of the last element. During
the optimization of both designs, the homogeneous glasses are allowed to vary between
real materials for optimal imaging performance. Figure 3-4 shows a cut through of the
radial index profile present in the GRIN element of the 2x design. Based on the shape and
bounds of the profile, one can see that the element is essentially homogeneous polystyrene
along the optical axis and then utilizes about half of the available index change moving out
to the periphery of the lens. The distance from the first surface to the image plane is held
constant across zoom.
Table 3-1: First order specifications of 2x zoom design
Zoom 1 Zoom 2 Zoom 3
EFL [mm] 11.5 16.5 23
f/# 3.5 4.25 5
HFOV [°] 19.2 13.6 10
63
Figure 3-3: 2x zoom design lens layout
Figure 3-4: Index profile of 2x zoom GRIN element
1.48
1.50
1.52
1.54
1.56
1.58
1.60
-6 -4 -2 0 2 4 6
Lens Radius [mm]
Refr
acti
ve Index [λ=
58
7nm
]
PMMA
PSTY
64
Figure 3-5: Ray aberration plots for 2x zoom design
In order to evaluate system performance improvements in going from the
homogeneous to the GRIN design, Figure 3-5 shows the ray aberration plots across zoom
for the 2x zoom system on a scale of ±25 µm. From these plots it is apparent that the GRIN
design yields an improvement in polychromatic performance, namely by helping to reduce
the significant amount of lateral color very prominent in the extreme zoom positions of the
system. This improvement in chromatic behavior is due to the high dispersion of the GRIN
+25�m
-25�m
+25�m
-25�m
Tangential Sagittalhhhh = 1= 1= 1= 1
+25�m
-25�m
+25�m
-25�m
hhhh = 0= 0= 0= 0
+25�m
-25�m
+25�m
-25�m
Tangential Sagittalhhhh = 1= 1= 1= 1
+25�m
-25�m
+25�m
-25�m
hhhh = 0= 0= 0= 0
+25�m
-25�m
+25�m
-25�m
Tangential Sagittalhhhh = 1= 1= 1= 1
+25�m
-25�m
+25�m
-25�m
hhhh = 0= 0= 0= 0
+25�m
-25�m
+25�m
-25�m
Tangential Sagittalhhhh = 1= 1= 1= 1
+25�m
-25�m
+25�m
-25�m
hhhh = 0= 0= 0= 0
+25�m
-25�m
+25�m
-25�m
Tangential Sagittalhhhh = 1= 1= 1= 1
+25�m
-25�m
+25�m
-25�m
hhhh = 0= 0= 0= 0
+25�m
-25�m
+25�m
-25�m
Tangential Sagittalhhhh = 1= 1= 1= 1
+25�m
-25�m
+25�m
-25�m
hhhh = 0= 0= 0= 0
656.2725 NM656.3nm
Zoom
3:
Homogeneous GRINZoom
2:
Zoom
1:
656.2725 NM
587.5618 NM587.6nm486.1nm
587.5618 NM
486.1327 NM
587.6nm486.1nm
65
material. Figure 3-6 shows a comparison in modulation transfer function (MTF) curves
across zoom for the same 2x designs. These plots show the same performance improvement
in ray aberration curves for the two extreme zoom positions going from the homogeneous
to the GRIN design. CODEV® lens listings for both the homogeneous and GRIN 2x zoom
lenses are shown in Appendix C and Appendix D respectively.
Figure 3-6: MTF curves for 2x zoom design
3.4.2 GRIN Chromatic Macro
While ray aberration plots like those shown in Figure 3-5 are useful for looking at
the aberrations of a final optical system, they are not useful on either a surface-by-surface
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 7.7° (T)
F2: 7.7° (S)
F3: 13.4° (T)
F3: 13.4° (S)
F4: 16.3° (T)
F4: 16.3° (S)
F5: 19.2° (T)
F5: 19.2° (S)
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 5.5° (T)
F2: 5.5° (S)
F3: 9.5° (T)
F3: 9.5° (S)
F4: 11.6° (T)
F4: 11.6° (S)
F5: 13.6° (T)
F5: 13.6° (S)
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 3.9° (T)
F2: 3.9° (S)
F3: 6.9° (T)
F3: 6.9° (S)
F4: 8.4° (T)
F4: 8.4° (S)
F5: 9.9° (T)
F5: 9.9° (S)
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 7.7° (T)
F2: 7.7° (S)
F3: 13.4° (T)
F3: 13.4° (S)
F4: 16.3° (T)
F4: 16.3° (S)
F5: 19.2° (T)
F5: 19.2° (S)
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 5.5° (T)
F2: 5.5° (S)
F3: 9.5° (T)
F3: 9.5° (S)
F4: 11.6° (T)
F4: 11.6° (S)
F5: 13.6° (T)
F5: 13.6° (S)
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 3.9° (T)
F2: 3.9° (S)
F3: 6.9° (T)
F3: 6.9° (S)
F4: 8.4° (T)
F4: 8.4° (S)
F5: 9.9° (T)
F5: 9.9° (S)
Zoom
3:
Homogeneous GRIN
Zoom
2:
Zoom
1:
66
or an element-by-element basis for determining where in a system specific aberrations are
originating from. Seidel sums are a series of equations commonly used to tabulate the
contributions to third order and chromatic aberrations for each surface in an optical system,
making them a very useful tool for assessing where in a lens there is the most amount of a
certain aberration or where the balancing between aberrations is taking place. On its own,
CODEV® can calculate the Seidel sums for the third order surface contributions to the
aberrations as well as the transfer contributions to those aberrations which are the result of
the GRIN profile; however, it only does so for the monochromatic aberrations.
In order to calculate the contributions to both axial and lateral color for a radial
GRIN element, a macro is written in CODEV® based on a paper where the authors lay out
the methodology to calculate the polychromatic aberration contributions for GRIN
elements using Buchdahl notation [12]. The macro is included in Appendix E. While the
majority of the calculations are based on simple ray-trace values readily available from
CODEV® or other softwares, the definition of one specific parameter related to the GRIN
profile dispersion, ν11, warrants a more in-depth explanation.
When working with the Buchdahl model, the index of refraction is defined
according to
(3.1)
where Sharma defines n00λ0, ν01, and ν02 as the Buchdahl dispersion coefficients of the
optical material while ωλ is referred to as the chromatic coefficient and expressed
mathematically according to Equation 3.2
20 00 0 01 02( ) ...n nλ λ λ λω ν ω ν ω= + + +
67
0
0
( )
1 2.5( )
λ λω
λ λ
−=
+ − (3.2)
where λ is the wavelength (in micrometers) for which the chromatic coordinate is being
calculated and λ0 is defined by the user but taken to be the middle (reference) wavelength
(in microns). Thus the chromatic coefficient ω for wavelength λ0 is 0.
Based on the work of Moore and Sands, one can use Buchdahl notation to express
the index distribution for a radial GRIN as [82, 83]
20 1( , ) ( ) ( ) ...n r n n rλ λ λω ω ω= + + , (3.3)
where n0 and n1 are defined according to
20 00 0 01 02( ) ...n nλ λ λ λω ν ω ν ω= + + + (3.4)
21 10 0 11 12( ) ...n nλ λ λ λω ν ω ν ω= + + + (3.5)
where n0 and n1 are equivalent to the terms N00 and N10 from the standard equation for a
radial GRIN profile, specifically the base and quadratic coefficients. Given that fact, one
can determine v11 numerically by plotting the N10 values that CODEV® has optimized the
GRIN element to as a function of ωλ and finding the slope. Figure 3-7 shows an example
of this for the 2x zoom GRIN lens discussed in the previous section, with each N10 value
corresponding to a specific wavelength and therefore chromatic coordinate. In this case,
the relationship between N10 and ω is very linear and so it is sufficient to truncate
Equation 3.5 at only two terms though this may not always be the case. Sharma notes that
neglecting this second-order effect result in a loss of accuracy of less than 1% and considers
it a fair approximation [84].
68
Figure 3-7: N10 plotted as a function of chromatic coefficient for 2X zoom GRIN design
Using the aforementioned macro, the contributions to lateral color are calculated
across zoom for first the second element (the location of the GRIN element) and then for
the overall system for both the homogeneous and GRIN 2x zoom designs. The results of
these calculations for the second element and the system are summarized in Figure 3-8
(units of mm) along with those for the other lens groups that move together through zoom.
Note that the first group is just element one, the second group is just element two, and
group three is both elements three and four. Comparison of the plots show that at the
extreme zoom positions, lateral color is reduced in element two in going from a
homogeneous to a GRIN lens. Looking specifically at zoom position 1, the lateral color
has been reduced by a full order of magnitude with very significant decreases in the lateral
color at both of the other two zoom positions as well. The data shown in Figure 3-8 reflects
the improvements in system lateral color observed in the ray aberration plots from Figure
3-5. Again, the two extreme zoom positions show significant reduction of this aberration
while there is a minor degradation at the middle zoom although this is offset by the
69
improvements in imaging quality made at zoom 1 and zoom 3. Additionally, the GRIN
design has a better balance of the lateral chromatic aberration over the different zoom
positions and within groups.
Figure 3-8: Lateral color for both individual lens groups and system for both homogeneous (left) and GRIN
(right) 2x zoom designs (units of mm).
10x zoom designs
The second design is a 10x zoom system, the specifications of which are taken from
the same paper as the 2x designs [10, 77-79]. In this case, a patent lens from that paper is
used as the starting point [11]. The first order specifications for this system are shown in
Table 3-2. Figure 3-9 shows the lens layout of both the homogeneous and GRIN designs.
Each of these designs contain a total of nine elements with three moving groups. The
distance from the vertex of the first surface of the first element to the image plane is held
constant between zoom positions and distortion is constrained to be less than 3%. There
are two GRIN elements in this lens system as indicated in Figure 3-9 (elements six and
nine in the drawing) with plots of the GRIN profiles as a function of lens aperture shown
in Figure 3-10. From the figure it is apparent that both GRIN elements are utilizing the full
70
change in index available to them, an immediate sign that the GRIN profile is making a
significant difference in lens performance.
Table 3-2: First order specifications of 10x zoom design
Zoom 1 Zoom 2 Zoom 3
EFL [mm] 10 31 98
f/# 2 3.5 5
HFOV [°] 26.6 9.2 2.9
Figure 3-9: 10x zoom design lens layout
71
Figure 3-10: Index profile of 10x zoom GRIN elements
Figure 3-11 shows the ray aberration plots for the 10x zoom system on a scale of
±40 µm. From the figure it is clear that the homogeneous design is heavily limited by
chromatic aberration across the zoom range. At the short focal length zoom position the
design is limited by axial color while at the middle zoom position it is limited by lateral
color and at the final zoom position, it is limited by both axial and lateral color. Looking
at the corresponding GRIN design, each polychromatic aberration has been largely
corrected across zoom due to the presence of the two GRIN elements. Figure 3-12 shows
the MTF plots for both the homogeneous and GRIN systems across the zoom range. From
the plots, it is apparent that the addition of the GRIN elements has made the imaging
performance of the lens much more uniform across zoom with the performance vastly
improved at both the short and long focal length zoom positions. This enhancement is
largely due to the improved correction of both axial and lateral color in the GRIN system
over the homogenous as seen in Figure 3-11. As a reminder, the GRIN Abbe number of
1.48
1.5
1.52
1.54
1.56
1.58
1.6
-15 -10 -5 0 5 10 15
Ind
ex
of
Re
fra
ctio
n
Lens 6 Lens 9
Lens Radius [mm]
Refr
acti
ve In
dex [λ=
58
7nm
]
PMMA
PSTY
72
the visible spectrum is approximately 9. CODEV® lens listings for both the homogeneous
and GRIN 10x zoom lenses are shown in Appendix F and Appendix G respectively.
Figure 3-11: Ray aberration plots for 2x zoom design
Tangential Sagittal
h = 1 h = 1
-40�m
+40�m
-40�m
+40�m
h = 0 h = 0
-40�m
+40�m
-40�m
+40�m
Tangential Sagittal
h = 1 h = 1
-40�m
+40�m
-40�m
+40�m
h = 0 h = 0
-40�m
+40�m
-40�m
+40�m
Tangential Sagittal
h = 1 h = 1
-40�m
+40�m
-40�m
+40�m
h = 0 h = 0
-40�m
+40�m
-40�m
+40�m
Tangential Sagittal
h = 1 h = 1
-40�m
+40�m
-40�m
+40�m
h = 0 h = 0
-40�m
+40�m
-40�m
+40�m
Tangential Sagittal
h = 1 h = 1
-40�m
+40�m
-40�m
+40�m
h = 0 h = 0
-40�m
+40�m
-40�m
+40�m
Tangential Sagittal
h = 1 h = 1
-40�m
+40�m
-40�m
+40�m
h = 0 h = 0
-40�m
+40�m
-40�m
+40�m
656.2725 NM656.3nm
Zoom
3:
Homogeneous GRIN
Zoom
2:
Zoom
1:
656.2725 NM
587.5618 NM587.6nm486.1nm
587.5618 NM
486.1327 NM
587.6nm486.1nm
73
Figure 3-12: MTF curves for 10x zoom designs
Conclusions and future work
A series of design studies have been discussed utilizing a copolymer
PMMA/polystyrene radial GRIN. A number of simple homogeneous and GRIN singlets
and doublet designs were first compared to demonstrate the material’s ability to correct
color over the visible spectrum using a single element. This same type of element was then
applied to much more complex systems, first a 2x zoom lens followed by a 10x zoom lens.
In both cases the GRIN design was observed to have improved imaging performance over
its homogeneous counterpart of the same number of elements, largely due to the
improvements in the correction of color. A macro was written in CODEV® and MATLAB
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 10.6° (T)
F2: 10.6° (S)
F3: 18.6° (T)
F3: 18.6° (S)
F4: 22.6° (T)
F4: 22.6° (S)
F5: 26.6° (T)
F5: 26.6° (S)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 3.7° (T)
F2: 3.7° (S)
F3: 6.5° (T)
F3: 6.5° (S)
F4: 7.8° (T)
F4: 7.8° (S)
F5: 9.2° (T)
F5: 9.2° (S)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 1.2° (T)
F2: 1.2° (S)
F3: 2.0° (T)
F3: 2.0° (S)
F4: 2.5° (T)
F4: 2.5° (S)
F5: 2.9° (T)
F5: 2.9° (S)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 5.5° (T)
F2: 5.5° (S)
F3: 9.5° (T)
F3: 9.5° (S)
F4: 11.6° (T)
F4: 11.6° (S)
F5: 13.6° (T)
F5: 13.6° (S)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 5.5° (T)
F2: 5.5° (S)
F3: 9.5° (T)
F3: 9.5° (S)
F4: 11.6° (T)
F4: 11.6° (S)
F5: 13.6° (T)
F5: 13.6° (S)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100Cycles [mm-1]
Diff. Limit
F1: 0.0°
F2: 5.5° (T)
F2: 5.5° (S)
F3: 9.5° (T)
F3: 9.5° (S)
F4: 11.6° (T)
F4: 11.6° (S)
F5: 13.6° (T)
F5: 13.6° (S)
Zoom
3:
Homogeneous GRIN
Zoom
2:
Zoom
1:
74
for evaluating the surface and element contributions to both axial and lateral color for radial
GRIN elements utilizing Buchdahl notation. Results from this further support the GRIN
lens’ role in correcting color in individual elements and the design as a whole.
Going forward, it would be of interest to carry out further design studies utilizing
this material pairing with applications to more complicated systems. This could include
higher zoom ratios, greater fields of view, faster speeds, etc., all in an effort to further map
out the space of GRIN design and determine in what forms this specific GRIN copolymer
as well as GRIN materials in general improve performance. There are a number of other
miscible optical copolymers pairings that have been explored in the past and it would be
of great use to apply some of these to the design forms laid out in this chapter as well as
others to determine what pairing gives the highest-performance.
75
Chapter 4. Fabrication of copolymer GRIN elements
Background
Based on the results of the previous chapter, the potential for copolymer radial
GRIN elements to improve the imaging performance of a lens system is apparent. Attempts
to fabricate such elements in the laboratory are pursued using the polymerization process
first discussed in Section 1.3.3. As mentioned in Section 3.1, numerous methods for
copolymerization exist. The processes of both prepolymerization and photopolymerization
can be used to generate both positive and negative-signed gradients; however, they provide
limited maximum achievable index changes and are not compatible with many polymer
combinations. Because of these reasons, an alternative process is pursued: the centrifugal
force method.
In an attempt to achieve higher index changes, this thesis concentrates on the
fabrication of polymer radial GRIN elements through the use of centrifugal forces, based
on the methodology demonstrated by a number of other groups. Most simply, this process
involves rotating a vessel containing various monomers at 2,000 rpm during the fabrication
process so that as copolymerization occurs, diffusion between those monomers results in
an element with cylindrical isoindicial surfaces: a radial GRIN. Im has demonstrated index
changes between 0.002 and 0.015 in GRIN optical fibers formed from a PMMA and
polystyrene copolymer [85]. Cho has been able to use photopolymerization on the
monomers MMA and 2,2,3,3-tetrafluoropropyl methacrylate (TFPMA) to fabricate GRIN
76
rods [86]. Duijnhoven has been able to copolymerize 5 mm-diameter rods of MMA and
TFPMA to have an index change of up to 0.009 [87].
Rochester process
4.2.1 Monomer preparation
Initial attempts at sample fabrication are carried out without first filtering either of
the two monomers before copolymerization. These samples are found to be very hazy.
Based on this and the results of Fang and Schmidt, all subsequent samples are fabricated
using filtered monomer in order to remove any substances and impurities that would inhibit
the reactions [73]. To do this, the monomer is placed in a glass funnel above a glass tube
filled with fine cotton, molecular sieve, and Al2O3 powder. Over time, the monomer drips
through the filtration tube and into a clean glass beaker below. The unfiltered styrene
monomer has a yellowish hue which is not observed in the filtered styrene monomer.
During polymerization, a liquid monomer shrinks in size as it becomes a solid
polymer in a process known as volume reduction. A set of experiments in which test tubes
of liquid monomer were photographed once every five minutes during polymerization over
time determined that MMA monomer reduces to 81% of its initial volume after
polymerization to PMMA with that same metric being 87% going from styrene to
polystyrene.
4.2.2 Copolymerization
The Rochester process involves filling a temperature-controlled spinning monomer
chamber with varying amounts of two or more monomers to control the index profile. The
GRIN rod fills from the outside to the inside as the material pumped first into the spinning
77
chamber settles first to the outside of the chamber. Because MMA is heavier than styrene,
GRIN elements of that material pairing fabricated in this manner have more PMMA on the
outside of the aperture and more polystyrene on the inside. As a result of that, only
positive-power profiles are created in this way since PMMA has a lower refractive index
than polystyrene. Using a different combination of miscible polymers such as PMMA and
poly(benzyl methacrylate), a negative-signed GRIN profile may be fabricated as well.
A schematic of such a system is shown in Figure 4-1. In this system, two monomers
are stored in separate syringes and then pumped through both a passive mixing chamber
and a pre-heating chamber as controlled by a computer. As the monomers are stored cold,
this is meant to ensure that the liquid matches the required temperature of the reaction as it
enters the chamber. Based on the desired final index profile, the computer is used to dictate
the relative fill rates of the two monomers into the actual spinning chamber where
copolymerization occurs. The chamber is rotated at 2,000 rpm by a lathe while
continuously-circulating heated water falls onto the rotating chamber in order to hold it at
a set temperature during copolymerization. Figure 4-2 shows a photograph of the system
with one end of the monomer test tube mounted in the lathe. During the fabrication process,
the other side of monomer test tube is secured inside of a rotary bearing meant for
additional stability. The feed needle is carefully inserted into the spinning cap of the test
tube to drip in monomer. For clarity, the top of the water chamber is not shown. After being
copolymerized, the samples, now solid, are removed from the lathe while still within the
test tube and placed inside of a convection oven for post-cure. This process is meant to
finalize the reaction by using up any residual initiator molecules left in the samples. The
78
samples are heated from 65 °C to 95 °C over the course of five hours, held at 95 °C for one
hour and then brought down to 25 °C over an additional ten hours. Once the post-cure is
complete, the samples are separated from the test tube by carefully breaking and removing
the glass surrounding the now-solid copolymer.
Figure 4-1: Layout of centrifugal radial GRIN setup (figure credit: Greg R. Schmidt)
Figure 4-2: Photograph of centrifugal radial GRIN setup
79
4.2.3 Initial samples
Initial experiments with the system shown in Figure 4-1 and Figure 4-2 are carried
as a proof of principle in order to ensure that it is be possible to create radial GRINs in the
manner described. The first GRIN rods produced in the laboratory suffer from the presence
of a large central air pocket as shown in Figure 4-3. This is due to the fact that as the
monomer polymerizes from a liquid to a solid, it reduces in volume. This reduction in
volume must be accounted for when filling the syringe pumps; however, the filling must
be done slowly in order to ensure that the monomer chamber does not overfill with the
liquid monomer. If the filling is done too slowly, sections of the GRIN element
copolymerize too quickly resulting in the formation of interfaces. Figure 4-3 shows an
extreme example of this where a large air pocket was allowed to develop in an under-filled
sample.
Figure 4-3: Examples of radial GRIN samples: (a) a radial GRIN rod that is underfilled leaving
a central air pocket shown with a ruler for scale, (b) a radial GRIN rod with a visible interface,
and (c) a fully-filled radial GRIN rod. Both (b) and (c) are 14.4 mm in diameter.
Through constant monitoring of the system on the part of the operator, the central
air pocket is not able to grow large while ensuring that the monomer chamber does not
80
overflow. An example of a fully filled radial sample produced in this way with no interfaces
is shown in Figure 4-3. A typical experiment begins with a fill rate between 20 and
25 μL/min and ended with one below 1 μL/min.
Once a radial sample is produced, it is necessary to measure the index profile. For
this purpose, sections that are roughly 0.6 mm-thick are cut from the GRIN rod
perpendicular to the isoindicial surfaces and the section is placed in one arm of a phase-
shifting Mach-Zehnder interferometer for measurement. Single-pass interferometers like
Mach-Zehnders are often preferable for measuring GRIN profiles as the index change can
result in a large number of interference fringes that are difficult to resolve using a double-
pass interferometer. Double-pass interferometers are also undesirable as ray registration
errors can occur from having to transmit back through the GRIN material. Sample
measurements yield interferograms like those shown in Figure 4-4 with all measurements
carried out at a wavelength of 632.8 nm. The index profiles shown in Figure 4-4 are
measured from left to right through the center of each radial sample.
Note that alone, the Mach-Zehnder interferometer provides only relative index data
so that one only knows the relative change in index across the aperture. A measurement
from a refractometer or other instrument capable of measuring absolute index of refraction
is required to fully quantify the GRIN profile. Note that the samples produced from this
process are not truly radial GRIN elements but rather tapered gradients due to the index
having an axial-direction dependence. In this thesis, sections from these samples are
approximated to radial GRIN lenses. Tapered gradients are explored in detail by
Schmidt [72].
81
Figure 4-4: Examples of fabricated radial GRIN samples. The left column shows the
interferograms of two approximately 0.6 mm-thick sections of samples and the right column
shows the index profiles through the center.
4.2.4 Results
To reduce setup complexity, an alternative method of combining the monomers is
adopted in the fabrication process whereby MMA and styrene are premixed in varying
ratios and then pumped together from one single syringe into the spinning chamber. In
experiments, either three or four ratios are pumped in order to create one sample. For this
sample, three ratios of monomers are pumped into the spinning test tube. These are (in
order of pumping) (1) 75/25, (2) 50/50, and (3) 25/75% PMMA/polystyrene by volume of
each polymer. This is illustrated in Figure 4-5 and summarized in Table 4-1. The volumes
of each ratio are chosen so that the distance between the outer and inner diameter of each
calculated copolymer layer are close to one another. The calculated outer (r1) and inner
(r2) radii for each layer are also summarized in Table 4-1 along with the calculated total
solid copolymer volume of each layer and the corresponding calculated liquid monomer
82
volumes for both MMA and styrene. The inner diameter of the test tube used for JC018 is
14.4 cm while the length is 11 cm.
Figure 4-5: Illustration of copolymer layering process
Table 4-1: Summary of calculation of required monomer volumes for sample JC018 layers
Layer Thickness r1 r2 PMMA Polystyrene Copolymer
volume MMA
volume Styrene volume
Unit cm cm cm -- -- mL mL mL
1 0.25 0.72 0.47 0.75 0.25 10.28 9.54 2.95
2 0.25 0.47 0.22 0.50 0.50 5.96 3.69 3.43
3 0.22 0.22 0 0.25 0.75 1.67 0.52 1.44
Samples JC016 and JC017 were both carried out with the same layering recipe as
JC018; however, both suffered from process issues that were adjusted for JC018. JC016
was the last sample attempted using unfiltered monomer and the resulting haze in the
sample prompted the switch to using filtered monomer for all future samples as stated in
Section 4.2.1. Additionally, the final sample had a central hole (like that observed in Figure
4-3) in approximately one-third of the length of the sample. This is because even though
the test tube was completely filled when last observed, further volume reduction occurred
as the reaction finished. Because of this, both JC017 and JC018 were periodically
monitored by the operator during the filling of layer 2 and constantly monitored during the
83
filling of layer 3 to prevent a hole from appearing at any point once the test tube was first
filled with monomer. An additional volume of the layer 3 monomer mixture was pumped
into the test tube (between 0.5 and 1 mL) during this process.
After the post-cure, JC017 was found to have a series of air bubbles throughout its
length that were not present when the sample was removed from the lathe. This means that
the copolymerization reaction was not complete and that there was still a large number of
monomer molecules present in the sample. As a result of this, JC018 was given
significantly more time to copolymerize in the lathe setup than its predecessors. Each
sample run began at 60 °C and ended at 67 °C. Both JC016 and JC017 were polymerized
for an additional 20 hours after reaching 67 °C while that number was extended to 58 hours
for JC018.
Figure 4-6 shows an example of the profile measurements at various points in
sample (JC018). These profiles (namely slices 2 and 3) are much more quadratic and
therefore much more beneficial for imaging applications than those shown in Figure 4-4.
It is apparent from Figure 4-6 that there is a large axial component to these GRIN profiles
over the full length of the sample. This is due to the fact that the while spinning, the test
tube is tilted upwards so that the monomer may more easily enter the chamber and remain
there during the polymerization process.
84
Figure 4-6: GRIN profiles of various sections of radial sample JC018
Figure 4-7 shows photographs and a CODE V® model of one section of sample
JC018. To see how well the fabricated lenses are described by the design software, the back
focal length (BFL) of sample JC018 is measured and compared against the quantity
calculated by a CODE V® model. Rather than modeling the single element with the profile
varying axially and radially, the lens is modeled first with just the radial profile of slice 1
and then the profile of just slice 2, yielding calculated BFLs of 26.7 mm and 25.8 mm,
respectively. Direct measurements of this quantity is 26.6 ± 1.2 mm. The uncertainty is
the standard deviation of these measurements.
85
Figure 4-7: (a) Images and (b) CODEV® model of sample JC018. The sample has a diameter of 14.4 mm
2x zoom design using manufactured profile
As shown earlier, the 2x GRIN zoom design shown in Figure 3-3 contains a
positive-powered index profile, the sign of which is consistent with the aforementioned
centrifugal method. Figure 4-8 shows the optimized index profile of that element at a
wavelength of 632.8 nm. A question arose as to how sensitive this particular design form
is to the shape and depth of the index profile. The main issue is whether or not this non-
optimal index profile still provides improvements to image quality, most notably working
towards correcting the lateral color across the zoom range. To explore this, the measured
index profile of sample JC018, slice 2 is modeled in MATLAB. From here, the profile is
fitted to a sixth-order polynomial. The results of this curve fit are shown in Figure 4-8. The
fit is only carried out over the aperture of the modeled element as calculated by CODEV®
with the actual GRIN rod being of a larger diameter. The darkened areas of the plot indicate
the unused region of the manufactured GRIN rod larger than the diameter of the modeled
86
element. The difference in index of refraction between the sixth-order fit to the
interferometrically-measured index profile data and the data itself is shown in Figure 4-9.
Figure 4-8: (a) Meaured index profile and sixth-order fit for slice 2 of sample JC018. The grayed-out area
indicates the region greater than the aperture of the lens designed with the fitted profile. (b) Comparison of
GRIN profile shapes for 2X zoom lens design between designed lens and fit of JC018, slice 2 profile. Note the
change in aperture size of the element between the two designs.
-4 -3 -2 -1 0 1 2 3 4-6
-4
-2
0
2
4
6
Radius [mm]
Ind
ex o
f re
fra
ctio
n d
iffe
ren
ce
( x
10
-4)
Figure 4-9: Difference in index of refraction between the sixth-order fit to the interferometrically-measured
index profile data and the data itself. The accuracy of the interferometric index measurements is ±2x10-5.
While the relative index profile of the element is measured interferometrically, it is
necessary to perform an absolute index measurement somewhere on the sample to gain a
valid reference point for the GRIN profile. This is done using a Metricon 2010/M prism
-6 -4 -2 0 2 4 61.48
1.5
1.52
1.54
1.56
1.58
1.6
Radius [mm]In
de
x o
f re
fra
ctio
n (
λ =
632
.8n
m)
2x Design Profile
Fit to Fabricated Profile
-8 -6 -4 -2 0 2 4 6 80
0.005
0.01
0.015
0.02
0.025
0.03
Radius [mm]
∆n (
λ =
63
2.8
nm
)
Measured
ax6 + bx4 + cx2 + d fit
(a) (b)
PMMA
Polystyrene
87
coupler system at the edge of the sample. After calibration, this instrument has an absolute
index accuracy of ±0.0002.The fit to the measured index profile of sample JC018, slice 2
is shown in Figure 4-8 alongside the optimized profile. Looking at both profiles on the
same plot, one can see that the optimized profile is significantly steeper than the measured
one, while also trending heavily towards the higher index (greater amount of polystyrene)
side of the GRIN.
Using the 2x zoom design file as a starting point from section 3.4.1, the fit to the
measured index profile is entered into CODEV® and not allowed to vary. From there,
optimization trials are carried out allowing the thicknesses and curvatures of the elements
to vary around the frozen GRIN profile in order to recover imaging performance. The
performance results of this are shown by the aberration plots contained in Figure 4-10. This
figure compares these plots for three designs: (1) homogeneous, (2) GRIN (fabricated
profile), and (3) GRIN (designed profile) with all evaluated at the extreme zoom positions.
Generally in zoom lens systems (and reflected in both Figure 3-5 and Figure 3-11) the
shortest and longest effective focal length configurations exhibit the worst imaging
performance with the aberrations being better controlled for the intermediate zoom
configurations. Because of this fact and for figure clarity, the middle zoom position shown
in Figure 3-5 and Figure 3-11 is omitted from Figure 4-10.
It is important to note that while the GRIN design with the measured profile does
offer significantly better performance than its homogeneous counterpart, its performance
is not quite as good as that of the design using the optimized index profile. However, this
indicates that a GRIN design does not need to have the optimal profile in place in order to
88
benefit from the GRIN element’s chromatic properties. Note especially that the benefits
for lateral color correction of both GRIN designs over the homogeneous system for the
EFL = 23 mm zoom position are very similar. The optimized profile yields superior
imaging performance for the other extreme zoom position but, again, the measured profile
is a significant improvement over the homogenous. The CODEV® lens listings for GRIN
5x zoom lens with the JC018 profile is shown in Appendix H.
Figure 4-10: Ray aberration plots for homogeneous and both GRIN designs (fabricated profile vs. designed
profile) evaluated at the extreme zoom positions.
Conclusions and future work
A centrifugal-force method is presented for the purpose of manufacturing
copolymer PMMA/polystyrene radial GRIN elements. Liquid MMA and styrene monomer
89
is copolymerized in a test tube rotating at approximately 2000 rpm in order to accomplish
this. A procedure for monomer filtering and preparation, layering, and system operation is
presented along with a discussion of the manufactured samples. Note that although many
issues are resolved, a number still exist including residual haze in the sample (greatly
reduced from the first samples) stemming from the issue of volume reduction causing
liquid monomer to come in contact with largely-copolymerized material at the center of
the sample in the final steps of the manufacturing process. Sample JC018 is presented in
detail with photographs of the element along with index profile measurements showcasing
a desirable quadratic shape in some regions of the sample. One such profile is applied to
the 2x zoom copolymer lens design presented in Chapter 3 to partially determine how
sensitive a GRIN design’s ability to correct color is to a given optimal profile shape. As
desired and expected, the imaging performance for the lens with the manufactured profile
is better than that of the homogenous lens but not as good as the lens with the optimal index
profile.
Going forward it would be useful to manufacture a monomer chamber that is
capable of shrinking in size as the copolymerization process occurs in an attempt to
compensate for the sample reducing in size. It is expected this would reduce the amount of
haze by halting the aforementioned issue of the liquid monomer contacting largely-
copolymerized material. At the same time, it would be useful to refine the process to
achieve larger index changes and to reduce the tilt angle of the lathe to reduce the axial
component of the index profile. It would also be of interest to attempt fabrication of some
of the same miscible polymer combinations discussed in the future work of Chapter 3.
90
Chapter 5. Athermalization of radial GRIN polymers
Introduction
A fundamental concern in the design of any optical system is the issue of how that
system is affected by a change in temperature. Traditionally, athermalized systems are
specifically designed so that an increase or decrease in temperature has a minimal effect
upon the optical performance. In practice, one or more of three processes are employed in
order to achieve athermalization and avoid performance degradation: (1) allowing
individual lenses/lens groups or the sensor to move, (2) mounting the lenses using specific
materials and dimensions to cause favorable airspace changes, and (3) using the thermal
properties of optical elements to compensate for focus change [88]. These last two methods
are a passive, rather than active, means of athermalizing a lens. If a system is composed of
only a single homogeneous element, it is not possible to use differing optical and/or
mechanical materials to compensate for one another and the image plane must be translated
to recover imaging performance as the temperature is changed. Largely analogous to the
process of achromatizing a lens (correcting primary axial color), athermalizing a system
requires at least two elements of different homogeneous materials.
As discussed in the previous chapters on optical design, utilizing GRIN materials
introduces new degrees of freedom into the design process. These allow for improved
aberration correction and therefore imaging performance while making it possible to
replace a number of homogeneous lenses with fewer GRIN lenses in order to reduce system
size and weight [89]. At least two homogeneous elements are required either to athermalize
or to achromatize a lens system of a given focal length [90]. Because the index is varying
91
across the lens aperture, both radial and spherical GRIN lenses have optical power due to
the index profile, in addition to the ray bending due to the lens surfaces. Thus, it is possible
for a single GRIN element to replace a homogeneous doublet under certain circumstances,
one of which, as mentioned in earlier chapters, is color correction. The ability to
achromatize a single GRIN lens begs the question of whether or not it would be possible
to athermalize a single GRIN lens as well [26]. It should be noted that the work carried out
in this thesis applies only to monochromatic systems undergoing a uniform temperature
change.
Thermal Effects – Homogeneous
Two material parameters must be defined in order to understand how an optical
system is affected by temperature: (1) the linear coefficient of thermal expansion (CTE)
and (2) the temperature-dependent index of refraction (dn/dT). The CTE is a measure of
how much a material changes in dimension with a change in temperature (a positive CTE
corresponds to a dimensional expansion with an increase in temperature and vice versa)
while the dn/dT dictates how the index of refraction changes with a change in temperature
(a positive dn/dT value implies the index of refraction increases when the temperature is
increased and vice versa). Note that a discussion of how both of these quantities are
measured is presented in the following chapter. The changes in the axial thickness and
semi-diameter of a lens are calculated using
' (1 )t t Tα= + ⋅∆ (5.1)
0 0' (1 )r r Tα= + ⋅ ∆ (5.2)
92
where t’ and r0’ are the new thickness and semi-diameter respectively, t and r0 are the initial
thickness and semi-diameter respectively, ΔT is the change in temperature, and α is the
CTE [26].
For a positive CTE, the lens expands both radially and axially with an increase in
temperature, the net effect of which is to decrease the magnitude of the surface curvature.
For a given temperature change, the change in surface curvature is given by
'1
cc
Tα=
+ ∆ (5.3)
where c’ is the new surface curvature and c is the initial surface curvature. The change in
the index of refraction of a material is described by
00 00'dn
N N TdT
= + ∆
(5.4)
where N’00 is the index of refraction of a homogeneous material after a temperature change
while N00 is the initial index of refraction.
Thermal effects - radial GRINs
Due to their relative mathematical simplicity when compared with spherical
GRINS, radial GRINs are chosen as the focus of this athermalization study. In such a
gradient, the index varies as a function of radial coordinate, making it possible to introduce
optical power with just the index profile shape (independent of the lens surface curvatures).
The Wood lens is the simplest example of a radial gradient, where either positive or
negative optical power can be achieved with a plano-plano element depending on the sign
of the gradient (a positive gradient describes when the index is higher along the optical axis
93
and then falls off towards the periphery while the opposite is true of a negative profile)
[91]. The index of refraction of a radial gradient is defined mathematically by
2 400 10 20( ) ...N r N N r N r= + + + (5.5)
where N is index of refraction for a given r (the distance measured outward from the optical
axis) and Nx0 are the index polynomial coefficients for a radial GRIN profile.
Figure 5-1: Effect of temperature increase on (a) a homogenous window and (b) a Wood lens for materials with
positive CTEs. Note that curvatures are induced in the radial GRIN element.
As a GRIN material’s composition changes as a function of spatial coordinate, so
do CTE and dn/dT. In the case of the Wood lens, the CTE varies across the aperture of the
lens so that a temperature change induces curvatures in nominally-planar surfaces.
Assuming the values to all be positive, if the CTE along the optical axis is greater than the
CTE at the periphery of the lens, this curvature is positive (and vice versa). This fact is
shown in Figure 5-1 which illustrates the effect of temperature on the surface curvatures of
both a homogenous and radial GRIN window (Wood lens). Note that the homogenous lens
expands in both directions but remains a window. This effect on the surface curvatures is
94
also inherent to radial GRIN lenses with non-planar surfaces. Additionally, the fact that
dn/dT is a spatially-varying parameter in a GRIN element must also be taken into account.
This chapter assumes that the material composition is varying quadratically
between the optical axis (quantities designated by a subscript a) and the edge of the lens
(quantities designated by a subscript b). This means that the GRIN profile is described by
only the first two terms of the Equation 5.5 above (N20 and all higher order coefficients are
equal to 0). The change in the surface curvature of the lens is given by
20
2
(1 ) ( )'
(1 )
a b a
a
t Tc T
rc
T
α α α
α
∆+ ∆ − −
=+ ∆
(5.6)
where c’ is the curvature of the lens after a change in temperature and r0 is the semi-
diameter of the lens. Note that for a homogenous lens, the equation reduces to Equation 5.3.
The quadratic index coefficient (N10) is given by
10 20
b an nN
r
−= (5.7)
where na is the index at the center of the lens, nb is the index at the periphery of the lens.
Note that Equation 5.4 is still used to calculate the new base index of the profile. The
change in the quadratic coefficient of the index profile is given by
10 22
0
( )'
11 ( ( ))
3
b ab a
a b a
dn dnn n T
dT dTN
T rα α α
− + − ∆
=
+ + − ∆
(5.8)
95
where N’10 is the quadratic term after a change in temperature. Note that in Equation 5.8,
the denominator is equal to the square of the expanded radius lens after the temperature
change [26].
Athermalization
A radial GRIN element has two contributions to the overall optical power of the
singlet, with the first being that of the surface curvatures and the second being that of the
index profile. It is possible to design GRIN singlets so that as the temperature changes, the
change in optical power due to surface curvatures is opposite in sign to that of the index
profile so that the net change in the focal length of the lens is as close to zero as possible.
Equations 4.9a, 4.9b, 4.10, and 4.11 are used to determine the optical power of a radial
GRIN singlet
200
1 2 00 1 200
( 1) sinh( )( )( 1)cosh( ) sinh( )
N kc c N k b k cc t
N kφ
−= + − − − (5.9a)
200
1 2 00 1 200
( 1) sin( )( )( 1)cos( ) sin( )
N kc c N k b k cc t
N kφ
−= + − − − (5.9b)
00
btk
N= (5.10)
00 102b N N= (5.11)
where c1 and c2 are the curvatures of the first and second surfaces of the lens respectively
[43]. Equation 5.9a is used if N10 > 0 and Equation 5.9b is used if N10 < 0. If N10 = 0, both
equations for φ reduce to that for the power a homogeneous lens. Note that these equations
(and all equations in this chapter) are defined so that a positive curvature value always
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refers to a convex surface (on either surface). Using Equation 5.9a and Equation 5.9b for
φ, one can calculate the nominal optical power of a GRIN lens. For a given ΔT, one can
then determine the changes in the lens geometry and index profile using the equations
outlined in the previous section before finding the new optical power of the lens. The goal
is to find singlets where the difference in the optical power of the lens before and after the
temperature change is minimized.
Polymers
Thus far, this work has been focused on radial GRINs formed from a combination
of the optical polymers PMMA and polystyrene. Typically, polymers exhibit values of
CTE and dn/dT that are between one and two orders of magnitude greater than those of
optical glasses. For this reason, homogeneous plastic optical elements are much more
susceptible to a reduction in performance as a result of thermal effects than are most
homogeneous glass elements. To first order, the change in focal length of a lens (Δf) with
temperature is given by
/1 L H
dn dTf f T
nα α
∆ =− ∆ − +
− (5.12)
where f is the nominal focal length of the lens, αL, is the CTE of the lens material, and αH
is the CTE of the housing. Using Equation 5.12 and ignoring the contribution from the
housing (so that αH = 0) a BK7 singlet with a nominal focal length of 50 mm exhibits a
change in focal length of single microns over a temperature change of ΔT = +40°C [62].
Over the same temperature change, a polymer singlet of that same focal length has a focal
length change on the order of hundreds of microns. Table 5-1 summarizes the important
97
material parameters for these two polymers that were used in the athermalization model
found in literature [92].
Table 5-1: Material data for polymers used in thermal modeling studies.
Polymer Refractive Index [nD]
CTE [x10-5 / °C]
dn/dT [x10-5 / °C]
PMMA 1.4917 6.5 -8.5
Polystyrene 1.5903 6.3 -12.0
CR-39® 1.5016 10.38 -18.4
HIRITM 1.5594 13.51 -22.3
DAP 1.5728 8.29 -16.1
Validation and description of model
Initial studies into the possibility of athermalizing polymer GRIN lenses were done
by Leo Gardner in the late 1980s using three materials: CR-39®, HIRITM, and DAP (diallyl
phthalate) [26]. The relevant material parameters are summarized in Table 5-1 along with
those of PMMA and polystyrene. This first GRIN athermalization analysis by Gardner was
carried out using the method of double graphing [93]. New MATLAB code was written to
successfully reproduce Gardner’s double graphing plots used in the initial design studies
[26]. Because double graphing is a time consuming process, new code was written using
MATLAB, following the mathematical steps laid out in the previous section of this chapter.
This new model has been shown to find not only the same athermalized solutions the
double graphing method had, but many more and in only a small fraction of the time.
Before proceeding with the analysis of the PMMA/polystyrene GRIN pair, it was necessary
to confirm the MATLAB model with Gardner’s previous results regarding the other three
polymers.
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The contour plots output from the model show nominal total optical power for a
series of possible singlets as a function of both lens geometry (the x-axis) and GRIN profile
(the y-axis). An example is shown in Figure 5-2, displaying the MATLAB model output
for a lens that is 25 mm thick and 10 mm in diameter with pure DAP along the optical axis
of the lens and a GRIN profile that varies quadratically between 0% and 100% CR-39® at
the periphery. In MATLAB, each lens is modeled so that it is 100% of one material on axis
(DAP in Figure 5-2) while ranging between 0% and 100% of the second material (CR-39®)
at the periphery of the lens, thus forming the GRIN profile. If the y-axis of the contour plot
(the concentration of CR-39®) equals 0, the lens is homogeneous DAP and no GRIN profile
exists. If the y-axis is instead equal to 1, the lens is varying quadratically between pure
DAP along the optical axis and pure CR-39® at the edge of the lens. Due to the large number
of input parameters available in this model, lenses are modeled as either equi-convex or
equi-concave for simplicity so that a negative c1 value yields an equi-concave lens while a
positive c1 value yields an equi-convex lens as shown in Figure 5-2. A c1 value of 0
corresponds to a plano-plano element and therefore a Wood lens.
The contours of the plot represent the nominal optical power of a GRIN lens for a
given surface curvature and index profile combination as calculated using Equation 9a and
Equation 9b. The model also computes the new optical power of that same singlet after a
specified change in temperature (in the case of Figure 5-2, this is an increase of 40°C).
After the user sets a threshold acceptable change between these two values of power
(0.005% in the case of Figure 5-2), the model identifies potentially-athermalized solutions
with black marks. Those lenses which are afocal (yielding a net optical power of 0) have
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been identified with a dashed line in Figure 5-2. In Figure 5-2, the white dot designates the
combination of surface curvature and index profile that result in an athermalized solution
for this material pair as found by Gardner using the method of double graphing. The white
dot, Gardner’s solution, is shown to overlap right in the middle of the athermalized solution
space, showing agreement between the two methods. Testing the MATLAB model with
other material combinations of CR-39®, HIRITM, and DAP shows agreement between the
two methods.
Figure 5-2: Output from MATLAB athermalization model for radials GRINs composed of DAP (on axis) and
CR-39®. The solid black curve indicates athermalized solutions. The dashed line indicates afocal lenses.
PMMA/polystyrene GRIN study
With the model validated, it is further applied to the PMMA/polystyrene GRIN
combination. Figure 5-3 shows the MATLAB output for this material combination given a
lens thickness of 5 mm, lens diameter of 10 mm, and ΔT of +40°C. Figure 5-3a shows the
output for the case when there is pure polystyrene at the center of the lens and a variable
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amount of PMMA at the periphery, while the opposite is true of Figure 5-3b. The black
line indicates the solution space of lenses identified as athermalized by the model (note that
the criteria for the model to identify an athermalized lens in this case is for the optical
power of the lens to change by less than 0.005%). Note that the in each case, the
athermalized singlet solutions correspond to lenses where the power coming from the
surface curvatures is opposite that of the GRIN profile. For example, in Figure 5-3a, the
GRIN profile is always of positive-signed optical power since the refractive index of
polystyrene (at the center of the lens) is higher than of PMMA (at the edge of the lens). At
the same time, the solutions only exist when the surface curvatures are contributing
negative optical power to the lens. The opposite is the case of Figure 5-3b.
Figure 5-3: Output from MATLAB athermalization model for radials GRIN lenses of 5 mm thickness, 10 mm
diameter and a ΔT of +40°C. (a) Lenses composed of pure polystyrene on axis and varying amounts of PMMA at
the periphery. (b) Lenses composed of pure PMMA on axis and varying amounts of polystyrene at the
periphery.
From here, it is necessary to determine if the model is correctly identifying
athermalized lenses. Figure 5-4 indicates five singlets of the same nominal effective focal
length (50 mm) chosen for a design comparison. Note that Figure 5-4 uses the same system
parameters as those detailed at the beginning of this section for Figure 5-3 but with different
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axis limits in order to show the region of interest to the design study. In Figure 5-4, lens 1
is an athermalized singlet while the other four lenses are not. Of note is that lens 5 is a
homogeneous PMMA element. The model suggests that lenses which are closer to the
black curve experience a smaller change in EFL than those which are farther away from it.
Based on this, in Figure 5-4, one would expect lens 1 to have the least change in EFL with
a change in temperature while lens 5 would be expected to have the largest change.
Figure 5-4: A zoomed in version of Figure 5-3b to see the singlets of interest to be compared for degree of
athermalization. The five white dots indicate the five lenses of the same nominal focal length (50 mm) chosen for
the design study.
Figure 5-5 shows a series of athermalized lens solution spaces corresponding to the
same parameters as those discussed in the previous example and shown in Figure 5-4. All
lenses are biconvex with c1 = -c2 varying between 0 and 0.05 mm-1 while the lenses are
pure PMMA on axis with varying amounts of polystyrene at the periphery (between 0 and
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100% polystyrene). The nominal thicknesses of the lenses vary between 1 and 15 mm for
each of the six contour plots. Based on this series of plots, thicker lenses provide more
options for athermalization with a greater variety of nominal lens effective focal lengths
capable of being athermalized. This is likely due to the effects of the GRIN being more
pronounced in thicker elements. As the lens thickness is decreased, available solutions
converge more and more towards only those singlets of infinite focal length. This is
apparent from the plots as the black line designating solutions runs increasing parallel to
and overlaid upon the contour corresponding to 0 mm-1 nominal optical power.
Figure 5-5: Effect of thickness change on athermalized GRIN singlet solution space. All plots are pure PMMA
on axis and varying amounts of polystyrene at the periphery (between 0 and 100% polystyrene). All lenses are
biconvex with c1 = -c2 varying between 0 and 0.05 mm-1.
Analytic modeling
It is necessary to further validate the results of this study using other modeling tools.
To this end, CODE V® optical design software is used to model each of the five lenses
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before and after the temperature increase. Changes to the lens geometry and index profile
are calculated using the methods discussed earlier in this work. Once the specifications for
each lens are entered into CODE V®, it is possible to monitor the change in a number of
performance metrics over the +40°C temperature change. Because EFL is the only
parameter being directly controlled by the model, other metrics are not analyzed in this
study. The effective focal length values calculated by CODE V® are found to match those
values calculated by the MATLAB model.
The results of this study are summarized in Table 5-2 which tabulates both the
nominal and final EFLs of each of the five lenses. Note that the final column of the table
displays the percent change in EFL after the temperature increase. From Table 5-2, it is
apparent that the lens identified as athermalized by the model has exhibited a change in
EFL of less than one micron while the homogeneous lens 5 (predicted to be the least
athermalized of the group) has had an increase in EFL of almost one half-millimeter. This
is consistent with what is expected as the extra degrees of freedom inherent to the GRIN
singlet better enable it to maintain constant optical power over a temperature change than
a homogeneous element. Table 5-2 also confirms the predicted trend that lenses which are
farther away from the athermalized curve will exhibit a greater change in EFL for a given
change in temperature. The MATLAB code used in this section is available in Appendix I.
Table 5-2: Effect of +40°C temperature change on EFL for five lenses in athermalization study
Lens Nominal
EFL [mm] Final EFL
[mm] Change
[%]
1 50.000 50.000 0.000
2 50.000 50.117 0.234
3 50.000 50.235 0.470
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4 50.000 50.354 0.708
5 50.000 50.475 0.950
Numerical modeling
While the first-order analysis of the system is useful, it is not a complete model.
The equations laid out in this paper only apply to quadratic radial GRIN profiles and cannot
account for aspheric surfaces on the GRIN lens. In order to address more complicated radial
GRIN systems, a numerical model is developed in MATLAB which treats a GRIN lens as
a series of differential rectangular elements stacked up parallel to the isoindicial surfaces.
One can think of this as a relatively simple finite element model. Each of these differential
rectangles is modeled as a homogeneous element, the summed effect of which accounts for
the overall change in the lens with a change in temperature as shown in Figure 5-6.
Equation 5.1 and Equation 5.2 are used to calculate the change in the axial and radial
thickness of an individual element respectively, while Equation 5.4 is used to calculate the
change in refractive index of an element. Once the changes in geometry and index are
calculated for each of the individual elements (1,000 are used for the studies in this chapter)
the surface sag departure can be fit to an arbitrary number of aspheric coefficients using
Equation 5.13 (the equation for the sag departure of an aspheric surface)
24 6 8
4 6 82 2( ) ...
1 1 (1 )
csz s A s A s A s
k c s= + + + +
+ − + (5.13)
where z is the sag departure of the surface for a given radial distance s away from the
optical axis, c is the curvature of the surface, k is the conic constant, and An are the nth order
aspheric coefficients [94].
105
Figure 5-6: Illustration of differential element model of lens.
It is not always sufficient to model the change in lens geometry as a simple change
in surface curvature as in Equation 5.6. Just as a temperature change can induce surface
curvatures in a nominally plano-plano Wood lens, the differential CTE across the aperture
of a radial GRIN element can cause the surface to take on an aspheric shape, rather than a
purely spherical one. While the previous method is limited to describing only purely
quadratic radial GRIN profiles (the first two terms of Equation 5.5), using the numerical
model, the index profile can be fit to an arbitrary number of coefficients of the index
polynomial. Equation 5.5 is often truncated in this regard at the sixth-order term (N30) to
be consistent with CODE V®’s default radial GRIN definition. For quadratic profiles, the
lens thickness, diameter, surface curvatures, and index profile change in accordance with
the analytic method described in the previous section.
As a general rule, the need to model an optical surface with aspheric coefficients
increases as surface curvature increases and as the magnitude of the change in temperature
increases. As one would expect, an optical surface departs more from a sphere post-
temperature change as the discrepancy between the CTEs of the two materials increases.
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To illustrate this effect, two lenses are compared of the same dimensions (5 mm center
thickness, 10 mm clear aperature, 10 mm radius of curvature biconvex) with the first
varying quadratically between 100% PMMA on axis and 100% polystyrene at the lens edge
with the second varying quadratically between 100% HIRITM on axis and 100% DAP at
the lens edge. Given Table 5-1, the difference in material CTE for the HIRITM/DAP pair is
about 26 times greater than it is for the PMMA/polystyrene pair. It is expected that the
former material combination is more likely to require a fit beyond just the simple surface
curvature (c coefficient in Equation 5.13) than the latter. Subjecting both lenses to a +40 °C
temperature change, the surface fits are compared for each using only c and using a
combination of c and the conic coefficient k. Figure 5-7 shows a plot of the error between
the calculated sag of the surface after the temperature change and the fit to that data for the
four combinations of material pairs and fit variables. The MATLAB code used in this
section is available in Appendix J.
Figure 5-7: Effect of CTE discrepancy on surface deformation fit. Only using the c coefficient in the fit for the
HIRITM/DAP material pair results in a relatively large fitting error. By introducing the k coefficient into the
fitting algorithm, this error can be brought down to a level consistent with that achieved fitting the
PMMA/polystyrene pair to only the curvature c.
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Because the CTE’s of PMMA and polystyrene are so close to one another, it is
likely sufficient in most applications to model the surface deformation as simple change in
radius of curvature. In the case of the HIRITM/DAP; however, the difference between the
calculated and fitted surface deformation is on the order of a micron at the edge of the lens.
It is only by introducing the conic coefficient into the fitting algorithm that the error drops
down to the order of tens of nanometers. While useful in certain applications, Equation
5.13 may not always be the best choice to fit surface deformation to depending on the shape
of the GRIN profile being modeled and it is useful to explore alternatives in the future for
radial and other GRIN geometries.
Conclusions and future work
The spatially-varying refractive index profile of a GRIN singlet makes it possible
to maintain a constant EFL over a change in temperature. Work is successfully carried out
to reproduce the first-order results of a previous student and extend the space of possible
solutions. A preliminary first-order design study comparing an athermalized polymer
GRIN lens to a number of other singlets of the same EFL validates the MATLAB model’s
ability to identify athermalized GRIN lenses. This work in the study of thermal effects on
GRIN systems shows potential for the possibility of athermalizing a single optical element.
A second model is also developed to more accurately model the effects of temperature on
radial GRIN lenses. Unlike the first-order model, this simplified finite element model is
able to treat elements with radial GRIN profile terms higher than only the quadratic term
and also elements with aspheric surfaces.
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Practically speaking, when manufacturing polymer radial GRIN lenses such as
those discussed in this paper, it is necessary to account for the method of mounting in the
design process as the type of material used in the housing affects how the EFL changes
with temperature. Additionally, the method of mounting may cause stress in the element
so that the stress-induced birefringence of the element must also be monitored if such
elements are to be used as part of a real optical system.
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Chapter 6. Thermal Interferometry
Introduction
An important consideration in the design of any optical system is how that system
behaves when subjected to a temperature change. As the temperature of an optical system
is increased or decreased, both its physical dimensions as well as its index of refraction
change according to that material’s linear coefficient of thermal expansion (CTE) and
temperature-dependent refractive index coefficient (dn/dT) respectively. The dimensional
change due to the CTE, α, is
( )1' ,TL L α+ ∆= (6.1)
where L’ is the dimensional size after a change in temperature ΔT while L is the
dimensional size for ΔT = 0 From Equation 6.1, the additional length of the sample (ΔL) is
given by
.L L Tα∆ = ∆ (6.2)
The vast majority of materials have CTEs that are positive in sign such that an
increase in temperature corresponds to an increase in sample length. Most metals have CTE
values between 10 and 20 parts per million per degree Celsius (x10-6/°C) while most
plastics are about an order of magnitude greater [28]. NBK7, a standard crown optical
glass, has a nominal CTE of about 7.1 x10-6/°C while Zerodur®, a glass ceramic
characterized by its very low expansion, has a nominal CTE of 0 x10-6/°C between 0 and
50 °C [95]. It should be noted that although CTE is often quoted as a single number, the
dimensional change is not completely linear with temperature for most materials, meaning
110
the value of the instantaneous CTE is different over different temperatures ranges for the
same material.
The refractive index of a material after a temperature change is
0'00 0 ,
dnN T
dN
T
= + ∆
(6.3)
where N00 is the initial index of refraction, corresponding to ΔT = 0 [96]. Values of dn/dT vary
significantly from material to material in both magnitude as well as in sign. Most optical glasses
have dn/dT values between 0.1 and 5 x10-6/°C and are found to be either positive or negative in
sign while optical polymers such as PMMA or polystyrene are orders of magnitude greater with
dn/dT on the order of -60 to -100 x10-6/°C [92]. Just as the base index of refraction of an optical
material varies with wavelength, dn/dT values do as well. To fully characterize the behavior of the
index of refraction of an optical material, dn/dT must be measured for a number of wavelengths
and over different temperature ranges. In this paper, an instrument is presented for the purpose of
measuring both CTE and dn/dT.
Interferometric techniques for measuring CTE and dn/dT have previously been published.
A number of interferometer geometries have been reported for the purpose of measuring CTE
including the Fizeau, Michelson, and Fabry-Perot among others [34]. Okaji, et al. report
measurements on the CTE of steel and ceramic gauge blocks over the temperature range between
-10 and +60°C [33]. Kuriyama, et al. later developed a ring interferometer system and compared
their measurements of the same samples’ CTE values against those measured by Okaji [97]. The
Ultra Precision Interferometer has been developed by Schödel, et al. to measure CTE down to 7 K
(-266.15°C) using three different stabilized lasers (λ = 780, 633, and 532 nm) [35]. This system has
111
since been used to measure silicon carbide ceramics, silicon nitride ceramics, and single crystal
silicon between 7 K and 300 K [98]. Dupouy, et al. measured the dn/dT of fused silica between 25
and 70°C using a Fabry-Perot interferometer [99].
It is possible to interferometrically measure both the CTE and dn/dT of a sample
simultaneously. Measurements of both thermal quantities for Nd:YAG laser rods as well as for
YAG laser rods have been reported [100, 101]. With both groups using a Fabry-Perot
configuration, Jewell, et al. measured CTE and dn/dT of vitreous silica and heavy-metal fluoride
glass samples between 25 and 100°C while Kazasidis and Wittrock measured both quantities for
both praseodymium-doped yttrium lithium fluoride (Pr:YLF) crystals and fused silica between 20
and 80°C [102, 103]. It should be noted that the vast majority of these systems that measure both
properties simultaneously do so with both arms of the interferometer in the same environmental
conditions and often in vacuum to avoid the issue of the index of air changing with temperature.
This chapter explores a method of interferometrically measuring CTE and dn/dT simultaneously
with the two arms of the interferometer in different environments.
Discussion of Instrument
6.2.1 Previous Generation
To measure both the CTE and dn/dT of an optical material, an interferometric technique is
employed with one arm of an interferometer extending into an environmental chamber capable of
cycling temperature. The sample to be measured is located inside of the environmental chamber.
During measurement, the temperature is changed.
The previous generation of this technique at the University of Rochester was designed and
built by McLaughlin, et al. using a Fabry-Perot interferometer [32]. The system was used to make
112
measurements of both homogeneous and gradient-index (GRIN) samples at a single wavelength
(λ = 632.8 nm). Specifically, it was used to validate the results of a model meant to determine the
thermal properties of GRIN glass [104]. It should be noted that for a Fabry-Perot system, cavity
finesse is an issue as only certain wavelengths are supported by the etalon. As the work was
conducted in the 1980s before CCD and CMOS detectors were cheaply available, a galvonometer
scanning system was used with a photodiode to make measurements of the full aperture of the
sample.
In 2012, McCarthy, et al. rebuilt the system using the original Fabry-Perot cavity used by
McLaughlin but included a CCD array to simultaneously image the full region of interest [36]. This
version of the interferometer placed the Fabry-Perot inside a convection oven with a window in the
door. Collimated light was sent through a beamsplitter and into the chamber where it was reflected
90° by a fold mirror into the Fabry-Perot etalon containing the sample under test. One issue with
using this interferometer configuration is that because the fold mirror was inside the chamber, it
was subjected to dimensional changes as the temperature was varied, resulting in a time-varying
image shift on the detector. This is detrimental because GRIN optics require the fringes to be
recorded from the same spatial location. For this reason, it was necessary to design the system to
have no fold mirror within the environmental chamber going forward. Additionally, the new
system was designed so that the piezo controller of the reference arm would not be located within
the chamber, great reducing any environmental effects on its motion.
6.2.2 Updated System
For the next-generation system, it is desired for the interferometer to have the ability to
make measurements at multiple wavelengths between the visible and mid-wave infrared (MWIR).
113
This eliminates a Fabry-Perot configuration because the finesse changes with wavelength or
requires prohibitively-expensive coatings for the cavity surfaces. Thus, a Twyman-Green
configuration is chosen for the next-generation interferometer due to the aforementioned finesse
issue as well as because of the significantly-decreased component cost for building the
interferometer. The system is designed to have easily-swappable optical components (windows,
imaging optics, etc.) to make taking measurements in different wavebands easier. In the visible,
both a HeNe laser and an Ar+ laser are coupled into the interferometer, allowing measurements at
a series of four wavelengths: 457, 488, 514.5 and 632.8 nm. To enable measurement of samples in
the MWIR, a λ = 3.39 μm HeNe laser is integrated into the system. When measuring in the MWIR,
it is necessary to switch out the lenses, windows, and beamsplitter for optics composed of CaF2 that
transmit at the desired wavelength. While all measurements reported in this thesis are carried out
at a wavelength of 632.8 nm, the system is capable of making measurements at these other
wavelengths to quantify the dispersion of dn/dT for a material and will be the subject of future
work.
The original desire for this instrument is to be able to make measurements between -40 and
+80°C; however, the actual range is limited due to decreased fringe stability at temperatures greater
than about +40°C and less than 0°C. For measurement, an Espec BTX-475 environmental chamber
is being used that is capable of cycling between -70 and +180°C. The main disadvantage for the
Twyman-Green configuration as compared to the Fabry-Perot is that the test and reference arms
are subjected to different environments (while the Fabry-Perot cavity had previously been entirely
located within the thermal environment). While the reference arm of the interferometer is in
ambient room temperature, the test arm is located within the environmental chamber. Because of
114
this, vibrations from the environmental chamber and air index fluctuations are detrimental to the
stability of the fringes being observed if not properly addressed.
To reduce vibrations, the optics are mounted above the chamber on a breadboard. This
breadboard is supported by four vibration-isolating feet that are mounted on a metal frame built
around the chamber. The metal frame and breadboard do not contact the environmental chamber
anywhere to minimize the coupling of vibrations into the test arm. Figure 6-1 shows a photograph
of the system as it exists in the laboratory. Note the aforementioned support frame surrounding the
chamber which in turn supports the vibration-isolating feet that support the breadboard.
Atop the breadboard are mounted the two HeNe lasers next to one another while the Ar+
laser sits on a separate shelf. The vibrations generated by the Ar+ laser’s cooling fan make it
impractical to mount it on the breadboard, which, in turn, makes it difficult to direct the beam to
the rest of the optics through free space. For that reason and to ease alignment between
wavelengths, both the Ar+ and visible HeNe lasers are fiber coupled to the same output through a
two-color combiner. Using a color-correcting doublet, the spatially-filtered and diverging beam is
then collimated. Through a series of fold/flip mirrors the visible HeNe beam is also coaligned to
the MWIR HeNe beam through free space. This makes it easier to align the MWIR beam. As
mentioned before, the collimating lens and all subsequent refractive optics must be switched for
CaF2 elements when measuring at this wavelength. For measurements at all wavelengths, once the
beam is collimated, it is then directed through two fold mirrors that send the beam down through a
hole in the breadboard.
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Figure 6-1. Photograph of the thermal interferometer setup. Note meter stick on chamber for scale.
A four inch–diameter plate beamsplitter (fused silica for the visible and CaF2 for the
MWIR) is mounted directly underneath the hole in the breadboard. The beamsplitter mount is
attached to holes in the underside of the breadboard. Careful attention is paid to ensure that the
underside of the beamsplitter mount does not come in contact with the top of the chamber to avoid
the vibration-coupling issue. The reflection from the beamsplitter forms the reference arm of the
interferometer, parallel to the floor in the room. The fused silica beamsplitter was coated by
AccuCoat Inc. in Rochester, NY. The coatings for this beamsplitter are shown in Appendix K. The
reference mirror is mounted upon a piezo stage on the underside of the breadboard. This enables
phase-shifting during measurements. The beam that is transmitted downward through the
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beamsplitter and into the chamber forms the test arm of the interferometer, perpendicular to the
floor in the room. In order to thermally-isolate the environments of the room and the chamber from
one another, a series of anti-reflection-coated windows are mounted within the mechanics of the
test arm (to be discussed in greater detail in a later section, along with the sample under test). The
windows are mounted inside of removable lens tubes that can be switched in and out depending on
the waveband of interest. For the visible, six 3 mm-thick float glass windows are used while for the
MWIR, three 6 mm-thick CaF2 windows are used. Different windows are used because of the
antireflection coatings. For the visible, the windows are specified to have an average reflectance of
less than 0.5% between 425 and 700 nm. Finally, the test mirror is mounted below the windows
inside of the chamber and parallel to the floor. The sample sits directly atop the test mirror.
A rail is mounted to the underside of the breadboard using a pair of magnetic bases and
right-angle mounts. This rail holds the imaging lenses as well as the visible detector: a Point Grey
Flea3 CMOS camera. The inclusion of the rail greatly eases the process of attempting to align the
imaging optics in the space between the breadboard and chamber. While the rail is also used to
carry the MWIR imaging optics, the MWIR detector, a FLIR a6700sc, is too large to be mounted
to the rail and must be mounted using a right-angle mount attached to the breadboard itself. Figure
6-2 shows a computer model of the interferometer to illustrate the location of the input collimated
beam, beamsplitter, reference arm, test arm, and imaging optics in relation to one another and the
Espec environmental chamber. Again, note that the reference and test arms are subjected to
different environmental conditions.
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Figure 6-2: Model of the Twyman-Green interfereometer system as designed and built. The system is designed
to measure over a 2” aperture. The reference arm is located outside of the chamber whereas the test arm enters
the chamber through a port in the top of the chamber. Only the test arm is subjected to the change in
temperature. The reference arm remains at the temperature of the room. The sample under test sits directly
upon the test mirror inside of the environmental chamber.
Interferometric Measurements
6.3.1 Beam Paths
Because optical path length is the product of physical distance and index of refraction, a
method must be used whereby the user can distinguish the contribution of each to the optical path
difference (OPD) as the temperature is cycled. To make thermal measurements of a material using
this instrument, either two or three different sets of interferograms are analyzed concurrently. This
number is dependent upon the type of sample being measured with two sets of fringes being
necessary to measure the CTE of a reflective sample while three are required to measure both the
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CTE and dn/dT of a transmissive sample. These three possible beam paths are denoted by OPDx
and are illustrated in Figure 6-3.
Figure 6-3. Side view of the beam paths within interferometer. The sample is shown resting upon the test mirror. The three
different OPDs can be used to compute CTE and dn/dT from background fluctuations.
OPD1 is the difference in optical path between just the reference mirror and the test mirror,
often referred to as the “background.” OPD2 is the difference in optical path between the reference
mirror and the top of the sample. For a reflective sample this is readily accomplished; however for
a transmissive sample, a coating of some kind must be applied to the sample to make a region of
the sample’s aperture reflective. This is accomplished by coating half of the aperture with
approximately 50 nm of gold using a Denton sputter coater. OPD3 is the difference in optical path
between the reference mirror and the test mirror but having traveled through the sample as well.
These are represented by
( ) ( ) ( )1 2 ,air
T nO PD T z T= (6.4)
( ) ( ) ( ) ( )( )2 2 , a n da irT n T z T LP TO D = − (6.5)
( ) ( ) ( ) ( )( ) ( ) ( )3 2 2 ,sa i rT n T z T L T n T TO P D L= − + (6.6)
where z is the physical distance between the test mirror and some height above the sample, L is the
thickness of the sample, nair is the index of refraction of the air, and ns is the index of refraction of
the sample. All of these variables are a function of temperature, T. All of these interferograms are
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recorded simultaneously by imaging the collimated beam onto the CMOS detector. Because the
phase is tracked as a function of spatial coordinate, this technique can be used to see how CTE and
dn/dT vary across the aperture of a GRIN material. Often for an interferometric measurement, the
user is interested in observed OPD across a sample. For this instrument, the metric of interest is the
change in those OPDs as a function of time. The change in observed OPDx over time is denoted as
dOPDx.
6.3.2 Data Acquisition
To measure the thermal characteristics of a sample, both surfaces must be polished so that
the sample is optically flat with minimal wedge between surfaces. Before being placed inside of
the instrument, the samples and mirror are thoroughly cleaned to ensure better contact between
sample and mirror. In the current instrument, samples up to approximately 20 mm thick and 35 mm
in diameter can be measured. To accomplish phase-shifting in this interferometer, the reference
mirror is mounted on a Thorlabs NFL5DP20S piezo stage with a TPZ001 T-cube controller unit.
Phase maps are generated using the least-squares algorithm [105]. This method requires that the
piezo stage move through a full 2π cycle while images of the fringes are periodically recorded and
used to calculate the individual phase map. Calibration of the driving voltage required to achieve a
single cycle is carried out by recording the intensity of a series of pixels as the voltage is ramped
through a number of cycles. Figure 6-4 shows an example of this process with pixel intensity
plotted as a function of applied voltage from the piezo controller. The blue squares are the measured
data while the red dotted line is the sinusoidal fit to that measured data. The black circles in the plot
show the voltage values required to carry out the algorithm, in this case 25.
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Figure 6-4: Pixel intensity as a function of applied voltage from the piezo controller
Twenty-five steps are found to give the best mixture of speed of acquisition and accuracy. This
high number of steps is due to the vibration of the fringes observed in the system. Figure 6-5 shows
a comparison between phase maps generated using a varying number of phase steps. The four-step
algorithm is a common choice in literature, while an 11-step algorithm is used by McCarthy in the
fabrication of a number of Mach-Zehnder interferometers; however, these algorithms are not
sufficient for the thermal interferometer due to fringe stability [106]. This is apparent in Figure 6-5,
showing each full two dimensional phase map alongside a horizontal cut through of the data. The
wrapped phase data for the 25-step algorithm is significantly smoother than that of either the 4 or
11-step algorithms and is very similar to that of a 250-step algorithm plotted for reference.
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Figure 6-5: Comparison between generated phase maps using different numbers of steps in phase-shifting
algorithm. Each column shows the results for a specific number of steps. The top row shows the generated phase
map and the bottom row a horizontal cutthrough of the wrapped phase data as indicated in the 2D phase map.
As the temperature within the environmental chamber is varied and recorded, phase maps
are continuously recorded (requiring 0.8 seconds per phase map). Figure 6-6 shows an example of
a phase map side-by-side with the coated sample resting upon the test mirror. The MATLAB data
acquisition code is included in 0.
Figure 6-6. (a) Coated CaF2 sample resting on mirror. The region bounded by the dashed-line rectangle indicates (b) the
associated computed wrapped phase map.
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After temperature cycling is complete, the phase maps are unwrapped as a function of time
and therefore temperature. By unwrapping the three regions of interest, the change in piston for
OPD1, OPD2, and OPD3 is determined. It is necessary to compare the relative amounts of phase
accumulated by each of the three beam paths to determine the CTE and dn/dT of the material.
6.3.3 Athermalization of the test arm
Just as the optical sample being measured expands or contracts in accordance with its CTE,
so does the housing of the optical windows (shown in Figure 6-2) that connects the beamsplitter
to the test mirror. Because of this, the height of the test mirror changes with temperature. This
change in the height of the test mirror affects the measurements of all three beam paths’ change in
piston, as all three are affected by a global change in height of the test mirror. The change in height
of the test mirror, along with changes to the index of air within the chamber contribute a change in
optical path to OPD1. This change in the optical path difference is designated as dOPD1. In addition
to this, the index of refraction of the windows within the test arm changes as well in accordance
with the dn/dT of the windows. Minimizing the effects of these contributions is necessary to
increase the accuracy of this instrument. It should be noted that dOPD1 is also affected by the
changing index of air and changing physical length of the interferometer’s reference arm; however,
since this arm is outside of the chamber and shielded from temperature changes with insulation,
this effect should be minimal compared to that on the test arm.
This issue is more commonly known as thermal drift and is common to the majority of
opto-mechanical designs. A standard method of correcting this issue is passive athermalization
where sets of rods or spacers of different lengths and materials are integrated into the system
housing [107]. Their differing lengths and CTEs are meant to counteract one another to compensate
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for the image plane shifting with temperature. In this system, there are two sets of three rods
forming the mount of the test mirror: longer rods of low-CTE invar to form the forward direction
of the thermal circuit and shorter rods of higher-CTE aluminum to form the backward direction of
the thermal circuit. The mount for the test mirror rests on three adjustable stainless-steel screws, the
presence of which allows the user to change the amount of material in the backward direction of
the thermal circuit in the pursuit of locating the height of the test mirror for minimal change with
temperature. Figure 6-7 shows an image of the test arm extended down into the environmental
chamber.
Figure 6-7. Photograph of the interferometer test arm inside the thermal chamber. The lengths of the invar and aluminum
rods were chosen to minimize the drift of the sample location as a function of temperature.
It is important to note that all of the materials that compose the test arm must be
incorporated into the thermal drift model. Data for the CTE of the materials and the index of air as
a function of temperature are estimated based on NIST values [108, 109]. Figure 6-8 shows two
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measurements of the change in optical path length of the background (dOPD1) as a function of
temperature. It should be noted that each measurement is relative to its starting measurement (at
the colder temperature) which is set to be 0 μm. These optical path length changes take place over
the course of about 10-12 hours and are readily trackable.
Figure 6-8. Plots of the change in optical path difference as a function of temperature for two measurements of background
fringes. The differences in materials (invar versus aluminum and steel) comprising the test arm make the background motion
less sensitive to thermal fluctuations.
Results
6.4.1 Thermal measurement considerations
The environmental chamber introduces a noisy environment for the instrument from both
vibrations and thermal currents. As mentioned above, the optical instrument is mechanically
decoupled from the environmental chamber with a set of pneumatic active vibration isolating feet.
Further strategies are also implemented to mitigate vibrations, such as identifying the sources of
excitation, investigating the system's natural frequencies, and passively damping the test arm.
Vibrations are mainly an issue when making measurements below 0°C as the chamber’s fan and
compressor make generating high-quality phase maps difficult. Ultimately, for these lower
temperature ranges, the best method is to adopt a measurement procedure which allows for the
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chamber to be turned off during data collection. When the chamber is operating colder than
ambient, thermal currents are not an issue as the cold air remains inside the chamber. At elevated
temperatures, the hot air inside the chamber tends to rise out of the hole in the chamber. Special
attention is required to direct this turbulent air away from the instrument's optical path.
6.4.2 Steel Sample measurement
Several optical-grade samples are coated for use in the thermal interferometer including
steel, ZrO2, CaF2, sapphire, and Zerodur®; however, currently the only certified samples for their
thermal characteristics are a set of Mitutoyo steel and ZrO2 gauge blocks ranging in thickness from
1.2 mm to 20 mm. The steel gauge blocks are certified to be Grade 00 according to ASME B89.1.9-
2002 and are also NIST-traceable. The steel gauge blocks are all certified to have a CTE value of
10.8±0.5 x10-6/°C at a temperature of 20°C. As a preliminary measurement using the
interferometer, the 20 mm-thick steel gauge block is measured between 10 and 30°C (for a ΔT of
20°C). The changes in piston of the two paths of note in this case (dOPD1 and dOPD2) are tracked
throughout the measurement. These are used together along with the measured air temperature
within the chamber to determine the change in thickness of the steel sample.
It should be noted that the accuracy of the CTE measurement is directly related to the
accuracy of the measurement of the sample’s initial thickness. An n-percent error in measuring the
initial sample thickness will yield roughly an n-percent error in calculated CTE. Initial sample
thickness is measured using a micrometer accurate to ±1 μm. This contribution to the error is thus
very small for a 20 mm sample, in this case being approximately 0.005%.
The results of the steel measurement are shown in Figure 6-9 where the measured change
in thickness is plotted as a function of temperature along with the linear fit to the data. This linear
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fit is used to calculate the CTE of the material over the indicated temperature range. By utilizing a
rearranged Equation 6-2 to solve for α, L is the nominal thickness of the sample being measured
(20 mm) while the term ΔL/ΔT can be taken to be the slope of the linear fit to the data shown in
Figure 6-9. Using this, the linear CTE is found to be 10.65 x10-6/°C at 20°C which is well within
the error bars specified by the manufacturer at 20°C. This methodology is used in all cases to extract
CTE and dn/dT values from measured data. The MATLAB data analysis code is included in
Appendix M. The maximum deviation between the measured data and the linear fit is 47 nm while
the average value of the deviation is 11 nm.
Figure 6-9. Measured change in thickness of 20 mm-thick steel gauge block for ΔT = 20°C with the difference between the fit
and measured data plotted on the secondary axis. The measured CTE for this gauge block was 10.65 x10-6/°C at 20°C.
Figure 6-10 shows the results of a measurement of the steel gauge block over a larger
temperature range. In this case, the CTE is calculated and plotted over a series of ten-degree
increments. These data represent a single measurement run of the sample. The maximum value of
the error bars is ±0.1 x10-6/°C. This plot also shows Mitutoyo’s measurement of the nominal value
of the CTE along with the associated measurement error bars. Finally, Figure 6-10 displays the
results of Okaji’s measurements of four 100 mm-thick Mitutoyo steel gauge blocks.
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The largest source of error in a single measurement comes from variation between the
tracked dOPD1 and dOPD2 paths by different pixels on the detector. For this measurement, that
value is ±0.09 x10-6/°C. Also of note is that a thermal gradient exists within the chamber between
the top and bottom of the sample. Temperature is measured using a 12-channel Omega
RDXL12SD with a measurement resolution of 0.1˚C. For the 20 mm-thick steel sample this
difference in temperature is measured as high as 0.7˚C. Hardware improvements have reduced this
discrepancy to less than 0.1˚C over the course of a measurement. Comparing a measurement of
this same steel sample where the maximum thermal gradient is reduced from 0.7 to 0.1˚C
corresponds to a reduction in the measurement error bars (calculated purely due to this temperature
gradient) from ±0.25 to ±0.05 x10-6/°C.
Figure 6-10. Measurement of 20 mm-thick steel gauge block CTE along with the reference data from the manufacturer and
previously reported Okaji data[33]. Note that the samples measured by Okaji are 100 mm-thick.
6.4.3 ZrO2 Measurements
Zirconium dioxide (ZrO2) or zircon, is a ceramic noted for its high mechanical
strength and fracture toughness. An 18 mm-thick Mitutoyo gauge block of this material
has been acquired for CTE measurement. The gauge block is certified to have a CTE value of
9.3±0.5 x10-6/°C at a temperature of 20 °C. It appears white as shown in Figure 6-11 alongside
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a steel gauge block. It should be noted that the thermal interferometer is still capable of
measuring the CTE of this sample, despite the ZrO2 sample’s reduced reflectance as
compared to the steel samples. Figure 6-12 shows the results of measuring and calculating
the CTE of this sample in five degree increments. The measurement shows agreement with
the CTE certification.
Figure 6-11: Steel (left) and ZrO2 (right) samples
Figure 6-12: Measurement of CTE of ZrO2 sample in five degree increments
6.4.4 CaF2 Measurement
The interferometer is used to measure a CaF2 sample as well. Figure 6-13 and Figure 6-14
show the measurements of the CTE and the dn/dT of the sample respectively. Overlaid upon the
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CTE results are those of Cardinali, et al. as well as Corning [110, 111]. Figure 6-14 shows the
results from these same sources for dn/dT with additional results from NASA’s Cryogenic High-
Accuracy Refraction Measuring System (CHARMS) system [112]. It should be noted that for
dn/dT, Corning’s measurements are carried out at a wavelength of 656 nm, not 632.8 nm. The
plotted data is the result of five separate measurements of the sample. Uncertainty is based on the
variation between measurements rather than the uncertainty of any one trial. For the CTE
measurement, the maximum uncertainty is ±0.4 x10-6/°C while for dn/dT the maximum
uncertainty is ±0.9 x10-6/°C.
Figure 6-13. Comparison of CaF2 CTE measurements between Rochester and literature values [110, 111].
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Figure 6-14. Comparison of CaF2 dn/dT measurements between Rochester and literature values [110-112]. Note that
Corning’s dn/dT measurement was carried out at 656 nm rather than 632.8 nm.
6.4.5 Zerodur Measurements
Invented by Schott in 1968, Zerodur® is a glass ceramic valued for its near-zero CTE over
a wide temperature range. This property makes it very attractive in the fabrication of space-based
telescope systems among other applications requiring minimal change with temperature. A
25.4 mm-diameter, 14.1 mm-thick cylindrical Zerodur® sample having λ/20 surface flatness is
measured. The vendor states its CTE to be 0±0.1 x10-6/°C which is consistent with Schott’s listing
of Zerodur® Expansion Class 2 between 0 and +50 °C [113]. Figure 6-15 and Figure 6-16
respectively show the results of measuring the CTE and dn/dT of the Zerodur® sample
between -10 and +60 °C. In both cases the data is plotted against that reported by Schott. The
plotted data is the result of ten separate measurements of the sample. As for the CaF2 measurement,
uncertainty is based on the variation between measurements. For the CTE measurement, the
maximum uncertainty is ±0.3 x10-6/°C while for dn/dT the maximum uncertainty is ±0.4 x10-6/°C.
The red dotted lines in Figure 6-15 indicate the bounds of uncertainty in the measurement of the
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CTE of Expansion Class 2 Zerodur®. These bounds are consistent with the results of other groups.
Figure 6-16 shows data reported by Schott on the dn/dT of the material. No error bars are reported.
Note that the Schott measurement is carried out at a wavelength of 656.3 nm. The Rochester CTE
data appears to show a bias of approximately -0.1 x10-6/°C in the measurement range between 0
and 50°C. It is possible this bias exists in all aforementioned instrument measurements but is not
easily observable from the samples above since the quoted error bars are so much greater.
Additionally, the nominal CTE values are much greater than 0 x10-6/°C, yielding a significant
distinction between dOPD1 and dOPD2 and increasing the signal to noise ratio in the measurement.
Figure 6-15: Comparison of Zerodur® CTE measurements between Rochester and literature values from
Schott.
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Figure 6-16: Comparison of Zerodur® dn/dT measurements between Rochester and literature values from
Schott. No error bars were given in the literature. Note that the Schott data is measured at a wavelength of
λ = 656.3 nm
6.4.6 Sapphire measurement
A 5 mm-thick, 25.4 mm-diameter sapphire window (Thorlabs WG31050) is
acquired and coated with gold for the purpose of measurement. The CTE of the sample is
reported by the vendor to be 5.3 x10-6/°C [114]. Neither error bars nor a range of
temperatures of measurement is reported. The same reference stating the CTE also quotes
a dn/dT value of 13.1 x10-6/°C; however, this value is for a reference wavelength of
546 nm. After direct correspondence, the vendor’s best estimate of the dn/dT of the
material at the measurement wavelength of 632.8 nm is quoted as 12.6 x10-6/°C. The
results of measuring the CTE and dn/dT of the sapphire sample are shown in Figure 6-17
over the range between 10 and 40 °C with individual measurements between 10 and 20°C,
20 and 25°C, and 28 and 40°C.
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Figure 6-17: Results of measurement of CTE (left) and dn/dT (right) of sapphire sample.
Polymer Measurements
In the previous chapter, a model to determine how a radial GRIN element behaves
with a temperature change was developed. As mentioned, it was assumed in this model that
both the CTE and dn/dT of the element vary linearly with material composition. In order
to validate this assumption, a series of copolymer samples of varying ratios of PMMA and
polystyrene were fabricated and measured in the thermal interferometer. This series of 11
total samples is named JC022-X where X ranges between one and eleven.
The eleven samples of JC022 are copolymerized following the monomer
preparation method laid out by Fang and Schmidt in fabricating axial GRIN profiles [73].
Both the MMA and styrene liquid monomer volumes are separately filtered in order to
remove any substances and impurities that would inhibit the reactions. During
polymerization, a liquid monomer shrinks in size as it becomes a solid polymer in a process
known as volume reduction. A set of experiments in which test tubes of liquid monomer
are photographed once every five minutes during polymerization over time determines that
MMA monomer reduces to 81% of its initial volume after polymerization to PMMA with
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that same metric being 87% going from styrene to polystyrene. The two monomers are
measured out in quantities so that each of the eleven final samples are of a set ratio of
PMMA and polystyrene by volume in increments of 10 % and have a final volume of
approximately 10 mL. This is summarized in Table 6-1. This shows the calculated solid
volumes of both PMMA and polystyrene necessary for each sample. These values are used
to calculate the volumes of liquid MMA and styrene mixed together for each sample.
Table 6-1: Summary of JC022 samples
JC022-X Fraction
polystyrene Volume PMMA
Volume polystyrene
Polymer Volume
Volume MMA
Volume styrene
Monomer volume
1 0 10.02 0.00 10.02 12.40 0.00 12.40
2 0.1 8.97 1.00 9.97 11.10 1.15 12.25
3 0.2 8.00 2.00 10.00 9.90 2.30 12.20
4 0.3 7.03 3.01 10.04 8.70 3.46 12.16
5 0.4 5.98 3.99 9.97 7.40 4.58 11.98
6 0.5 5.01 5.01 10.02 6.20 5.76 11.96
7 0.6 4.04 6.06 10.10 5.00 6.97 11.97
8 0.7 2.99 6.98 9.97 3.70 8.02 11.72
9 0.8 2.02 8.08 10.10 2.50 9.29 11.79
10 0.9 0.97 8.73 9.70 1.20 10.03 11.23
11 1 0.00 10.00 10.00 0.00 11.49 11.49
Each samples’ monomer quantities are measured and mixed in separate test tubes.
The samples are copolymerized in the 14.4 mm inner diameter test tubes in a water bath
held at a constant temperature of 60 °C. Due to limited space, the samples are polymerized
in two batches with the odd-numbered ones first followed by the even-numbered samples
about 52 hours later. The time to copolymerization is measured by analyzing a series of
photographs taken once every five minutes. The results are summarized on the secondary
axis of Figure 6-18. Note that the results are largely linear with composition until the 90%
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polystyrene sample (JC022-10). It is of interest for future work to fabricate a series of
samples between JC022-10 and 11 in composition and investigate each of their
copolymerization times.
After being copolymerized, the samples, now solid, are placed inside of a
convection oven for post-cure. This process is meant to finalize the reaction by using up
any residual initiator molecules left in the samples. From here, the samples are removed
from the glass test tubes sectioned using an Isomert saw and polished by hand. The index
of refraction of each sample is measured at a wavelength of 532 nm using a Pulfrich
refractometer as shown in Figure 6-18. The index at this wavelength is shown to be linear
with composition. Each of the eleven samples is measured ten times on the refractometer
with the standard deviation of each set taken to be the measurement error for that sample
with values ranging between 0.00015 and 0.00077.
Figure 6-18: Index of refraction (λ=532 nm) and time to volume reduction as a function of composition for
PMMA/polystyrene copolymers.
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It is determined that the available hand polisher is insufficient for polishing the
copolymer samples to the surface figure and wedge for measurement in the thermal
interferometer. As such, they are sectioned to discs between 2.7 and 2.8 mm-thick and sent
to Syntec Optics in Rochester, NY to be diamond turned to a final thickness of 2.5 mm
with 5 μm parallelism, 2 fringes of surface figure, and between 70 and 100 Å surface
roughness. Once returned, the samples are given the gold coating mentioned in the
discussion of previous samples at the University of Rochester’s Nanosystems Center.
With their smaller diameters (14.4 mm), the samples are measured in pairs in the
thermal interferometer. The temperature of the chamber is programmed to ramp between
2.5 and 37.5°C linearly over the course of 13 hours. For these measurements, it is found
that at temperatures outside of this range, the fringes are unstable. At the lower
temperatures, the chamber’s compressor became active causing the fringes to shake and
leading to errors in the generated phase maps. The increased magnitude of fringe vibration
and thermal gradient introduced in the test arm while measuring at higher temperatures
also resulted in errors in the calculated phase maps.
The results of measuring all eleven samples over the full temperature range of 5 to
35 °C are shown in Figure 6-19 where both the measured CTE and the measured dn/dT are
plotted as a function of copolymer composition. Note that each data point is a separate
measurement of the same sample with the 0, 0.1, 0.4, 0.6, and 0.9 polystyrene samples each
having multiple measurements in an attempt to illustrate the repeatability of these particular
high-value CTE and dn/dT measurements. In Appendix N, the data for the two parameters
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as calculated over four different temperature ranges: between 5 and 15 °C, 15 and 25 °C,
25 and 35 °C, and finally the full 5 and 35 °C is summarized.
Figure 6-19: CTE and dn/dT as a function of composition for PMMA/polystyrene copolymers. The parameters
are calculated over the full range between 5 and 35°C. The solid red lines indicate the range of reported values
for homogeneous PMMA and polystyrene
Reported measurements of the CTE and dn/dT of both PMMA and polystyrene and
polymers in general vary dramatically between references. The CTE of PMMA are
reported to be between 50 and 90 x10-6/°C while that of polystyrene between 50 and
80 x10-6/°C with the majority of reports for both polymers between 60 and 75 x10-6/°C [92,
115-117]. The dn/dT of PMMA is reported to be between -85 and -105 x10-6/°C while that
of polystyrene is between -104 and -140 x10-6/°C. Measurements of both parameters using
the thermal interferometer fall within this reported range. These bounds are indicated in
Figure 6-19 for each of the two parameters and for each of the two homogenous polymers
with solid red lines.
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Conclusions and future work
An interferometer has been developed for the purpose of simultaneously measuring both the
CTE and dn/dT of optical materials. It has been demonstrated that these measurements can be
carried out using an interferometer where each arm is subjected to a different environment and
without the need to pull vacuum on the sample. In this regard, this interferometer is believed to be
the first of its kind. Preliminary measurements on the CTE and dn/dT of a number of samples show
agreement with literature values made by manufacturers and other research institutions. It would
be of use to extend this system to the measurement of additional materials in order to better quantify
its range of operation and accuracy.
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Chapter 7. Conclusions
Concluding remarks
The addition of gradient-index (GRIN) materials offers an opportunity to
significantly improve the optical performance of a lens system. Specifically, this thesis
focuses on GRIN elements of the radial geometry in which the isoindicial surfaces, the
surfaces of constant index of refraction, exist as a series of cylinders centered upon the
optical axis. For such an element, the index of refraction varies as a function of the semi-
aperture which in turn affects the chromatic properties of the lens. The GRIN profile
provides a second source of optical power to the lens with the first coming from the surface
curvatures. In addition to this, the profile variation with wavelength yields a second Abbe
number in a single element. These facts underlie the potential for a single radial GRIN
element to behave as a traditional homogeneous doublet by correcting chromatic
aberration.
The possibility to fabricate GRIN elements from a variety of materials including
optical and infrared chalcogenide glasses, ceramics, crystals, and polymers has been
discussed. Chapter 2 focused on the pairing of ZnS and ZnSe. The gases used in the
manufacturing process for this material are highly toxic and so the fabrication of this GRIN
has never been attempted at the University of Rochester. As a result, research efforts were
restricted to a series of design studies over the mid-wave infrared (MWIR) from 3-5 μm.
This range is of particular interest as over it, the GRIN Abbe number has a negative value.
Negative Abbe numbers have been shown to be useful in lens design as they can eliminate
the need for negative-powered elements under certain circumstances. Additionally, the
140
same-sign profile shape that works towards correcting color also works towards correcting
undercorrected spherical aberration. The ZnS/ZnSe GRIN was used in a series of design
studies comparing homogeneous and GRIN designs for simple singlets and doublets,
Petzval-like objective lenses, and finally, 3 and 5x zoom lenses. In all cases the GRIN
design showed significant imaging performance improvement over a homogeneous lens of
the same number of elements.
Chapter 3 catalogued a second series of design studies utilizing a material that could
be manufactured at the University of Rochester: a copolymer formed between polymethyl
methacrylate (PMMA) and polystyrene. The GRIN Abbe number is very dispersive over
this waveband with an Abbe number of approximately 9. Studies into singlet and doublet
designs and 2 and 10x zoom lens designs again show that the GRIN designs consistently
shows superior performance over their homogenous counterparts of the same number of
elements. Because CODEV® optical design software was unable to tabulate the Seidel
contributions to either axial or lateral color for GRIN materials as is, a macro was written
to perform these calculations using Buchdahl notation. Results supported the radial GRIN’s
ability to correct chromatic aberration.
Chapter 4 described attempts to manufacture the PMMA/polystyrene radial GRIN
material described in the previous chapter. Efforts were carried out using a centrifugal-
force method whereby both MMA and styrene monomer were copolymerized in a test tube
rotated at approximately 2000 rpm. The fabrication procedure as well as process challenges
such as monomer-to-polymer volume reduction and sample haze were described along with
the results of successfully-generated samples. The chapter concluded with a return to the
141
optical designs of Chapter 3 as a sixth-order profile manufactured in the laboratory was
applied to the 2x copolymer design. The study showed that while the manufactured profile
did not yield imaging performance as good as that of the optimized profile, it was better
than that of the analogous homogeneous design, largely because of the observed
corrections to chromatic aberrations.
The ability of a radial GRIN singlet to be achromatized raised the question of
whether or not it was possible to design such an element to instead be athermalized. This
issue was explored in Chapter 5, which introduces the reader to two material parameters:
the coefficient of thermal expansion (CTE) and temperature-dependent refractive index
(dn/dT). Respectively, these dictate how the geometry and index of refraction of an optical
element change with temperature. To expand the solution space first explored by a previous
student, a first-order model programmed in MATLAB was described for the purpose of
identifying athermalized radial GRIN singlets with spherical surfaces and quadratic
profiles. From here, a more advanced finite-element-like model was developed, capable of
treating higher-order radial profiles and more complete surface shapes such as aspheres.
The model showed that a radial GRIN element with nominally-spherical surfaces and a
large discrepancy in CTE as a function of space can become non-spherical with a change
in temperature.
Chapter 6 described the design, construction, and operation of an interferometer for
the purpose of simultaneous measurement both CTE and dn/dT. A discussion of other
institutions’ efforts to measure either or both of these parameters interferometrically was
presented. The Rochester instrument was a Twyman-Green interferometer with the test
142
arm extended into an environmental chamber and the piezo stage-mounted reference arm
outside of the chamber. By tracking the change in piston with temperature for the three
regions of interest (reflection off of the test mirror, reflection off of the top of the sample
under test, and reflection off of the test mirror after having transmitted through the samples)
one can calculate both CTE and dn/dT. The system was used to measure these parameters
as a function of temperature for a number of materials including steel, ZrO2, CaF2,
sapphire, Zerodur, and different PMMA and polystyrene copolymers. Results agreed with
those published in literature.
Suggestions for future work
Based on the promising results of the design studies, future work should be carried
out to apply both the ZnS/ZnSe and PMMA/polystyrene GRIN materials to additional
design forms. Of special note are traditional design forms limited by chromatic aberration
such as the zoom lens systems discussed in this thesis as well as certain eyepiece designs
such as those explored by Visconti et al. [40]. It is useful to more fully map out such
solution spaces, determining what degree of performance improvement can be gained as
compared to homogeneous designs of an equivalent element count. Specifically for the
zoom lens designs, it is of interest to expand the design studies to forms of higher zoom
ratios. Finally, it would be useful to investigate how such systems perform over an
expanded wavelength range as simultaneous imaging over multiple wavebands (the dual-
band mid-wave/long-wave infrared for example) becomes more prevalent.
Further research should be pursued on the optimal methods of manufacturing both
of these GRIN materials. As mentioned before, safety issues prevented any on-site attempts
143
at fabricating a ZnS/ZnSe GRIN element. The design studies in Chapter 2 indicate that
there is great benefit to be gained from such a material for a company or other institution
with the infrastructure necessary to create one. Even the highest-quality
PMMA/polystyrene radial GRIN samples manufactured in the laboratory still suffer from
readily-observable haze. Although steps were taken to mitigate this specific effect, this is
believed to still be the result of the volume reduction process as the incoming monomer
still comes into contact with largely-copolymerized material at the feed end of the test tube.
This could potentially be curbed by designing a deformable monomer chamber in such a
way so that as the liquid volume does solidify and shrink, the encapsulating vessel does as
well to compensate, preventing the need to ever add additional monomer. This could
perhaps be accomplished with a plunger that is able to slide further into the test tube as it
rotates while still maintaining a tight seal on the monomer within. Successful attempts to
manufacture profiles with greater index changes and specifically tuned to the requirements
of a given optical design would make these materials much more viable from a commercial
standpoint.
It would be of great use to fully validate the thermal model of radial GRIN elements.
To do this, one could manufacture a series of copolymer elements and fully characterize
their surface shapes, thicknesses, and index profiles. From here, one could test each
interferometrically as a function of temperature to determine how the focal length or other
parameters each change and determine if those results match those predicted by MATLAB
and CODEV®. Such work could be carried out by modifying the existing thermal
144
interferometer for measuring CTE and dn/dT by replacing the flat test mirror with an
element matched to the nominal power of the lens under test.
Research using the current system for measuring both CTE and dn/dT could be
carried out by applying its use to additional samples of interest. Of particular use would be
to attain some samples with certified measurements of both CTE and dn/dT in order to
cross–check the accuracy of the instrument with the results of another system. As is, only
the CTE measurements for the steel and ZrO2 gauge blocks contained any certification (and
those are only at a single temperature). The range of measurable temperatures using this
instrument is limited, with measurements lower than approximately +5 °C suffering from
the issue of the chamber compressor switching on leading to numerous discontinuities in
the data. At temperatures higher than approximately 40 °C, the thermal gradient within the
test arm of the system again led to sometimes unreliable data. To address this, a second
system built by Bill Green is being pursued, designed to heat and cool a much smaller
volume for the test arm and using a temperature unit much less prone to vibration than the
Espec chamber. In general, the signal-to-noise ratio of the system would be improved by
again pursing a Fabry-Perot design which has much more stable fringes as the test and
reference arms are not subjected to differing environments. Measuring in vacuum would
also improve the accuracy significantly by eliminating the issue of needing to account for
the temperature and therefore index of refraction of the air at the sample plane in the final
calculations.
Note that the change in the refractive index of a material with temperature is also
dependent on the stress on that sample. This is represented mathematically by
145
'dn dn d
n n TdT d dT
σ
σ
= +∆ +
(7.1)
where σ is the stress on the sample under test. Measurements of dn/dT taken with the
current thermal interferometer ignore the effect of stress on the sample being tested. Thus,
the quantity actually being measured is that shown in the brackets in Equation 7.1. Future
work should include a method to decouple these effects from one another to isolate
measurement of the dn/dT term from the second term related to the stress. Additionally,
the hysteresis of measurements of both CTE and dn/dT should be quantified for the
copolymer materials.
146
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153
Appendix A. Lens listing for 5x MWIR zoom lens – homogeneous
EFL = 250mm
RDY THI RMD GLA CCY THC GLC
> OBJ: INFINITY INFINITY 100 100
1: 97.18989 6.000000 SILICON_SPECIAL 0 0
SLB: "grin1"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :-.120788E-08 B :-.264738E-11 C :0.539877E-14 D :-.338892E-17
AC : 0 BC : 0 CC : 0 DC : 0
E :0.691409E-21 F :0.000000E+00 G :0.000000E+00 H :0.000000E+00
EC : 0 FC : 100 GC : 100 HC : 100
J :0.000000E+00
JC : 100
2: 121.80919 91.462979 0 0
3: 82.33401 5.000000 ZNS_SPECIAL 0 0
SLB: "e2"
4: 107.81032 1.235962 0 0
SPS QBF:
SCO/SCC
NRADIUS: 1.1835E+01 QB4: -2.8476E-01 QB6: -3.7952E-03
SCC C1: 0 SCC C3: 0 SCC C4: 0
QB8: 3.2673E-04 QB10: -9.8174E-04 QB12: -2.6453E-04
SCC C5: 0 SCC C6: 0 SCC C7: 0
QB14: 8.6756E-05
SCC C8: 0
ITR: FST
5: 119.71664 5.000000 GAAS_SPECIAL 0 0
SLB: "e3"
6: 46.33757 1.368567 0 0
SPS QBF:
SCO/SCC
NRADIUS: 1.0408E+01 QB4: 1.2765E-01 QB6: 3.2191E-03
SCC C1: 0 SCC C3: 0 SCC C4: 0
QB8: -9.7989E-05 QB10: 4.0243E-04 QB12: 9.4134E-05
SCC C5: 0 SCC C6: 0 SCC C7: 0
QB14: -1.5842E-05
SCC C8: 0
ITR: FST
7: 102.17313 3.000000 SILICON_SPECIAL 0 0
SLB: "grin4"
8: -806.85860 0.500634 0 0
SPS QBF:
SCO/SCC
NRADIUS: 1.7864E+01 QB4: 1.1990E-01 QB6: 2.6241E-02
SCC C1: 0 SCC C3: 0 SCC C4: 0
QB8: 1.6706E-03 QB10: -4.4422E-03 QB12: 4.1446E-03
SCC C5: 0 SCC C6: 0 SCC C7: 0
QB14: -2.6244E-03
SCC C8: 0
ITR: FST
154
9: 102.42023 6.984505 GAAS_SPECIAL 0 0
SLB: "e5"
10: 81.12093 2.545050 0 0
SPS QBF:
SCO/SCC
NRADIUS: 1.3684E+01 QB4: -1.1509E-01 QB6: -7.8829E-03
SCC C1: 0 SCC C3: 0 SCC C4: 0
QB8: 2.8088E-03 QB10: -1.3981E-03 QB12: 1.1912E-03
SCC C5: 0 SCC C6: 0 SCC C7: 0
QB14: 2.9247E-04
SCC C8: 0
ITR: FST
11: -58.03628 3.000000 BAF2_SPECIAL 0 0
SLB: "grin6"
SPS QBF:
SCO/SCC
NRADIUS: 1.3884E+01 QB4: 3.6366E-01 QB6: -1.2946E-01
SCC C1: 0 SCC C3: 0 SCC C4: 0
QB8: 8.9654E-03 QB10: 3.3594E-02 QB12: -2.4106E-02
SCC C5: 0 SCC C6: 0 SCC C7: 0
QB14: 7.3885E-03
SCC C8: 0
ITR: FST
12: -278.86647 3.000000 0 0
13: INFINITY 1.000000 GERMMW_SPECIAL 100 100
SLB: "dewar"
14: INFINITY 3.000000 100 0
STO: INFINITY 67.372289 100 PIM
IMG: INFINITY -0.070865 100 0
SPECIFICATION DATA
FNO 4.00000
DIM MM
WL 5000.00 4000.00 3000.00
REF 2
WTW 1 1 1
INI JAC
XAN 0.00000 0.00000 0.00000
YAN 0.00000 0.98595 1.40850
WTF 1.00000 1.00000 1.00000
VUX 0.00000 0.00091 0.00189
VLX 0.00000 0.00091 0.00189
VUY 0.00000 -0.00785 -0.00993
VLY 0.00000 0.01428 0.02747
POL N
PRIVATE CATALOG
PWL 5000.00 4000.00 3000.00
'grin1' 2.429527 2.433159 2.437579
GRC 0
URN 0.050000
URN C10 -0.6471E-04 -0.6402E-04 -0.6370E-04
GRC 0
URN C20 -0.7057E-08 -0.6981E-08 -0.6946E-08
GRC 0
URN C30 -0.1668E-11 -0.1650E-11 -0.1642E-11
GRC 0
155 'grin4' 2.396547 2.400530 2.405116
GRC 0
URN 0.050000
URN C10 -0.3722E-03 -0.3683E-03 -0.3664E-03
GRC 0
URN C20 -0.3417E-06 -0.3381E-06 -0.3364E-06
GRC 0
URN C30 -0.5509E-08 -0.5450E-08 -0.5423E-08
GRC 0
'grin6' 2.408828 2.412681 2.417204
GRC 0
URN 0.050000
URN C10 0.2352E-03 0.2327E-03 0.2315E-03
GRC 0
URN C20 -0.1020E-06 -0.1010E-06 -0.1004E-06
GRC 0
URN C30 -0.1162E-09 -0.1149E-09 -0.1144E-09
GRC 0
REFRACTIVE INDICES
GLASS CODE 5000.00 4000.00 3000.00
GERMMW_SPECIAL 4.015388 4.024610 4.044976
ZNS_SPECIAL 2.246097 2.251784 2.257187
GAAS_SPECIAL 3.301061 3.306776 3.316400
BAF2_SPECIAL 1.451022 1.456694 1.461146
SILICON_SPECIA 3.422272 3.425406 3.432338
SOLVES
PIM
No pickups defined in system
ZOOM DATA
ZOOM TITLE
POS 1 "EFL = 250mm"
POS 2 "EFL = 100mm"
POS 3 "EFL = 50mm"
POS 1 POS 2 POS 3
YAN F1 0.00000 0.00000 0.00000
YAN F2 0.98595 2.46228 4.90000
YAN F3 1.40850 3.51755 7.00877
WTF F2 1.00000 1.00000 1.00000
WTF F3 1.00000 1.00000 1.00000
VUY F1 -0.7940E-11 0.2947E-10 0.1029E-10
VLY F1 -0.7940E-11 0.2947E-10 0.1029E-10
VUY F2 -0.00785 -0.00134 0.00901
VLY F2 0.01428 0.02404 0.03194
VUY F3 -0.00993 0.00404 0.02514
VLY F3 0.02747 0.03666 0.05655
VUX F1 -0.7940E-11 0.1000E-09 0.1029E-10
VLX F1 -0.7940E-11 0.1000E-09 0.1029E-10
VUX F2 0.00091 0.00367 0.00675
VLX F2 0.00091 0.00367 0.00675
VUX F3 0.00189 0.00755 0.01401
VLX F3 0.00189 0.00755 0.01401
RSL DEF DEF DEF
156 THI S2 91.46298 40.14283 13.15711
THC S2 0 0 0
THI S6 1.36857 42.85671 78.26508
THC S6 0 0 0
THI S10 2.54505 12.37706 3.95440
THC S10 0 0 0
POS 1 POS 2 POS 3
INFINITE CONJUGATES
EFL 249.9978 99.9906 50.0046
BFL 67.3723 67.6088 67.3520
FFL -491.6330 26.3777 50.3676
FNO 4.0000 4.0000 4.0000
IMG DIS 67.3014 67.3014 67.3014
OAL 133.0977 133.0977 133.0977
PARAXIAL IMAGE
HT 6.1469 6.1464 6.1476
ANG 1.4085 3.5175 7.0088
ENTRANCE PUPIL
DIA 62.4994 24.9976 12.5011
THI 436.0316 174.2596 87.4929
EXIT PUPIL
DIA 16.8431 16.9022 16.8380
THI 0.0000 0.0000 0.0000
STO DIA 16.9989 17.0327 16.9615
157
Appendix B. Lens listing for 5x MWIR zoom lens – GRIN
EFL = 250mm
RDY THI RMD GLA CCY THC GLC
> OBJ: INFINITY INFINITY 100 100
1: 98.33239 13.000000 'grin1' 0 0
SLB: "grin1"
2: 122.25923 91.589556 0 0
3: 31.26944 5.000000 ZNS_SPECIAL 0 0
SLB: "e2"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.244036E-05 B :0.602579E-08 C :0.235799E-11 D :-.191889E-13
AC : 0 BC : 0 CC : 0 DC : 0
E :0.198364E-15 F :0.000000E+00 G :0.000000E+00 H :0.000000E+00
EC : 0 FC : 100 GC : 100 HC : 100
J :0.000000E+00
JC : 100
4: 109.16203 2.379366 0 0
5: -504.76764 5.000000 GAAS_SPECIAL 0 0
SLB: "e3"
6: 37.51429 1.305500 0 0
7: 70.09549 7.089514 'grin4' 0 0
SLB: "grin4"
8: -98.62930 0.500000 0 0
9: -9348.79936 3.000000 GAAS_SPECIAL 0 0
SLB: "e5"
10: 143.59910 2.466324 0 0
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.326453E-05 B :-.566373E-08 C :0.347352E-11 D :-.143962E-12
AC : 0 BC : 0 CC : 0 DC : 0
E :0.378435E-15 F :0.000000E+00 G :0.000000E+00 H :0.000000E+00
EC : 0 FC : 100 GC : 100 HC : 100
J :0.000000E+00
JC : 100
11: -33.38904 3.669740 'grin6' 0 0
SLB: "grin6"
12: -37.84814 3.000000 0 0
13: INFINITY 1.000000 GERMMW_SPECIAL 100 100
SLB: "dewar"
14: INFINITY 3.000000 100 0
STO: INFINITY 61.828250 100 PIM
IMG: INFINITY 0.359625 100 0
SPECIFICATION DATA
FNO 4.00000
DIM MM
WL 5000.00 4000.00 3000.00
REF 2
WTW 1 1 1
INI JAC
158 XAN 0.00000 0.00000 0.00000
YAN 0.00000 0.98595 1.40850
WTF 1.00000 1.50000 3.50000
VUX 0.00000 0.00109 0.00226
VLX 0.00000 0.00109 0.00226
VUY 0.00000 -0.00830 -0.00963
VLY 0.00000 0.01600 0.02777
POL N
APERTURE DATA/EDGE DEFINITIONS
CA
APERTURE data not specified for surface Obj thru 16
PRIVATE CATALOG
PWL 5000.00 4000.00 3000.00
'grin1' 2.429527 2.433159 2.437579
GRC 0
URN 0.050000
URN C10 -0.6526E-04 -0.6456E-04 -0.6424E-04
GRC 0
URN C20 -0.7219E-08 -0.7142E-08 -0.7106E-08
GRC 0
URN C30 -0.1880E-11 -0.1860E-11 -0.1850E-11
GRC 0
'grin6' 2.246386 2.251971 2.257308
GRC 0
URN 0.050000
URN C10 0.2619E-03 0.2591E-03 0.2578E-03
GRC 0
URN C20 -0.3111E-06 -0.3078E-06 -0.3062E-06
GRC 0
URN C30 0.2305E-10 0.2280E-10 0.2269E-10
GRC 0
'grin4' 2.429527 2.433159 2.437579
GRC 0
URN 0.050000
URN C10 -0.3817E-03 -0.3776E-03 -0.3757E-03
GRC 0
URN C20 -0.1140E-05 -0.1128E-05 -0.1122E-05
GRC 0
REFRACTIVE INDICES
GLASS CODE 5000.00 4000.00 3000.00
GERMMW_SPECIAL 4.015388 4.024610 4.044976
'grin1' 2.429527 2.433159 2.437579
URN 0.050000
URN C10 -0.6526E-04 -0.6456E-04 -0.6424E-04
URN C20 -0.7219E-08 -0.7142E-08 -0.7106E-08
URN C30 -0.1880E-11 -0.1860E-11 -0.1850E-11
ZNS_SPECIAL 2.246097 2.251784 2.257187
GAAS_SPECIAL 3.301061 3.306776 3.316400
'grin4' 2.429527 2.433159 2.437579
URN 0.050000
URN C10 -0.3817E-03 -0.3776E-03 -0.3757E-03
URN C20 -0.1140E-05 -0.1128E-05 -0.1122E-05
'grin6' 2.246386 2.251971 2.257308
URN 0.050000
URN C10 0.2619E-03 0.2591E-03 0.2578E-03
URN C20 -0.3111E-06 -0.3078E-06 -0.3062E-06
URN C30 0.2305E-10 0.2280E-10 0.2269E-10
159
SOLVES
PIM
No pickups defined in system
ZOOM DATA
ZOOM TITLE
POS 1 "EFL = 250mm"
POS 2 "EFL = 100mm"
POS 3 "EFL = 50mm"
POS 1 POS 2 POS 3
YAN F1 0.00000 0.00000 0.00000
YAN F2 0.98595 2.46228 4.90000
YAN F3 1.40850 3.51755 7.00877
WTF F2 1.50000 1.00000 1.00000
WTF F3 3.50000 1.00000 2.00000
VUY F1 0.3423E-11 0.2170E-11 0.1813E-12
VLY F1 0.3423E-11 0.2170E-11 0.1813E-12
VUY F2 -0.00830 0.00178 0.01486
VLY F2 0.01600 0.02606 0.03869
VUY F3 -0.00963 0.01108 0.03692
VLY F3 0.02777 0.03659 0.06274
VUX F1 0.3423E-11 0.2170E-11 0.1813E-12
VLX F1 0.3423E-11 0.2170E-11 0.1813E-12
VUX F2 0.00109 0.00475 0.00879
VLX F2 0.00109 0.00475 0.00879
VUX F3 0.00226 0.00967 0.01825
VLX F3 0.00226 0.00967 0.01825
RSL DEF DEF DEF
THI S2 91.58956 40.04167 13.21533
THC S2 0 0 0
THI S6 1.30550 43.11110 77.81068
THC S6 0 0 0
THI S10 2.46632 12.20861 4.33537
THC S10 0 0 0
POS 1 POS 2 POS 3
INFINITE CONJUGATES
EFL 250.0000 100.0000 50.0000
BFL 61.8282 62.2433 62.2544
FFL -506.0601 60.6080 78.6423
FNO 4.0000 4.0000 4.0000
IMG DIS 62.1879 62.1879 62.1879
OAL 142.0000 142.0000 142.0000
PARAXIAL IMAGE
HT 6.1470 6.1470 6.1470
ANG 1.4085 3.5175 7.0088
ENTRANCE PUPIL
DIA 62.5000 25.0000 12.5000
THI 504.8046 221.2678 118.8001
EXIT PUPIL
DIA 15.4571 15.5608 15.5636
THI 0.0000 0.0000 0.0000
STO DIA 15.6243 15.6931 15.6809
160
Appendix C. Lens listing for 2x visible zoom lens – homogeneous
EFL = 11.5mm
RDY THI RMD GLA CCY THC GLC
> OBJ: INFINITY INFINITY 100 100
1: 28.23444 2.049231 NFK5_SCHOTT 0 0
SLB: "e1"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.753363E-05 B :-.333147E-07 C :0.517518E-09 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
2: 19.60967 5.582847 0 0
3: -16.01143 2.014817 NFK5_SCHOTT 0 0
SLB: "e2"
4: -190.36272 20.211631 0 0
STO: INFINITY 0.100000 100 0
6: 6.64250 3.800000 NLAF3_SCHOTT 0 0
SLB: "e3"
7: -10.81224 0.100000 0 0
8: -10.21423 2.861179 SF10_SCHOTT 0 0
SLB: "e4"
9: 13.00755 13.280294 0 0
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.128935E-02 B :0.337402E-04 C :0.244789E-05 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
IMG: INFINITY 0.000000 100 100
SPECIFICATION DATA
EPD 3.28570
DIM MM
WL 656.27 587.56 486.13
REF 2
WTW 1 1 1
XAN 0.00000 0.00000 0.00000 0.00000 0.00000
YAN 0.00000 7.67200 13.42600 16.30300 19.18000
WTF 1.00000 1.00000 1.00000 1.00000 1.00000
VUX 0.00000 0.00511 0.01651 0.02498 0.03537
VLX 0.00000 0.00511 0.01651 0.02498 0.03537
VUY 0.00000 0.01058 0.04101 0.06522 0.09590
VLY 0.00000 0.02114 0.06352 0.09426 0.13153
POL N
PRIVATE CATALOG
PWL 656.27 587.56 486.13
'grin2' 1.540205 1.545253 1.555441
GRC 0
URN 0.050000
URN C10 -0.1653E-02 -0.1704E-02 -0.1839E-02
GRC 0
URN C20 0.1659E-04 0.1710E-04 0.1846E-04
GRC 0
161 URN C30 -0.4476E-07 -0.4613E-07 -0.4981E-07
GRC 0
'grin4' 1.487957 1.491402 1.497298
GRC 0
URN 0.050000
URN C10 0.3734E-02 0.3848E-02 0.4155E-02
GRC 0
URN C20 0.1907E-04 0.1966E-04 0.2122E-04
GRC 0
URN C30 -0.1984E-05 -0.2044E-05 -0.2207E-05
GRC 0
REFRACTIVE INDICES
GLASS CODE 656.27 587.56 486.13
NFK5_SCHOTT 1.485345 1.487490 1.492269
NLAF3_SCHOTT 1.712522 1.716998 1.727471
SF10_SCHOTT 1.720848 1.728250 1.746481
No solves defined in system
No pickups defined in system
ZOOM DATA
ZOOM TITLE
POS 1 "EFL = 11.5mm"
POS 2 "EFL = 16.5mm"
POS 3 "EFL = 23mm"
POS 1 POS 2 POS 3
EPD 3.28570 3.88235 4.60000
YAN F2 7.67200 5.45200 3.94800
YAN F3 13.42600 9.54100 6.90900
YAN F4 16.30300 11.58550 8.38950
YAN F5 19.18000 13.63000 9.87000
VUY F1 0.2487E-12 0.2706E-11 0.5369E-11
VLY F1 0.2487E-12 0.2706E-11 0.5369E-11
VUY F2 0.01058 -0.00045 -0.00018
VLY F2 0.02114 0.00096 -0.00247
VUY F3 0.04101 -0.00063 -0.00233
VLY F3 0.06352 0.00361 -0.00564
VUY F4 0.06522 -0.00023 -0.00403
VLY F4 0.09426 0.00530 -0.00766
VUY F5 0.09590 0.00062 -0.00609
VLY F5 0.13153 0.00640 -0.01007
VUX F1 0.2487E-12 0.2706E-11 0.5369E-11
VLX F1 0.2487E-12 0.2706E-11 0.5369E-11
VUX F2 0.00511 0.1402E-04 -0.00047
VLX F2 0.00511 0.1402E-04 -0.00047
VUX F3 0.01651 0.00028 -0.00141
VLX F3 0.01651 0.00028 -0.00141
VUX F4 0.02498 0.00060 -0.00204
VLX F4 0.02498 0.00060 -0.00204
VUX F5 0.03537 0.00102 -0.00277
VLX F5 0.03537 0.00102 -0.00277
THI S2 5.58285 14.45327 17.62899
THC S2 0 0 0
THI S4 20.21163 8.56197 1.85497
162 THC S4 0 0 0
THI S9 13.28029 16.05953 19.59081
THC S9 0 0 0
POS 1 POS 2 POS 3
INFINITE CONJUGATES
EFL 11.5000 16.5000 23.0000
BFL 13.3077 16.0958 19.5621
FFL 10.9788 7.4394 -2.9496
FNO 3.5000 4.2500 5.0000
IMG DIS 13.2803 16.0595 19.5908
OAL 36.7197 33.9405 30.4092
PARAXIAL IMAGE
HT 4.0002 4.0009 4.0017
ANG 19.1800 13.6300 9.8700
ENTRANCE PUPIL
DIA 3.2857 3.8824 4.6000
THI 18.9531 21.4928 20.2126
EXIT PUPIL
DIA 4.7384 4.5582 4.5678
THI -3.2768 -3.2768 -3.2768
STO DIA 5.8521 5.6337 5.6488
163
Appendix D. Lens listing for 2x visible zoom lens – GRIN (optimized profile)
EFL = 11.5mm
RDY THI RMD GLA CCY THC GLC
> OBJ: INFINITY INFINITY 100 100
1: 153.05250 2.002688 NFK5_SCHOTT 0 0
SLB: "e1"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.678809E-05 B :-.189994E-06 C :0.269889E-08 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
2: 28.83212 5.692019 0 0
3: -15.13370 2.332800 'grin2' 0 0
SLB: "grin2"
4: -117.97471 18.443687 0 0
STO: INFINITY 0.100000 100 0
6: 7.36119 3.270308 NLAK34_SCHOTT 0 0
SLB: "e3"
7: -15.26454 0.100000 0 0
8: -14.86794 3.800000 SF10_SCHOTT 0 0
SLB: "e4"
9: 15.61568 14.258498 0 0
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.967162E-03 B :0.246100E-04 C :0.101656E-05 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
IMG: INFINITY 0.000000 100 100
SPECIFICATION DATA
EPD 3.28570
DIM MM
WL 656.27 587.56 486.13
REF 2
WTW 1 1 1
XAN 0.00000 0.00000 0.00000 0.00000 0.00000
YAN 0.00000 7.67200 13.42600 16.30300 19.18000
WTF 1.00000 1.00000 1.00000 1.00000 1.00000
VUX 0.00000 0.00447 0.01490 0.02270 0.03199
VLX 0.00000 0.00447 0.01490 0.02270 0.03199
VUY 0.00000 0.00997 0.03622 0.05717 0.08287
VLY 0.00000 0.01935 0.06054 0.08742 0.11445
POL N
PRIVATE CATALOG
PWL 656.27 587.56 486.13
'grin2' 1.585265 1.591695 1.605584
GRC 0
URN 0.050000
URN C10 -0.2122E-02 -0.2187E-02 -0.2361E-02
GRC 0
URN C20 0.2187E-04 0.2254E-04 0.2433E-04
164 GRC 0
URN C30 -0.9741E-07 -0.1004E-06 -0.1084E-06
GRC 0
'grin4' 1.487957 1.491402 1.497298
GRC 0
URN 0.050000
URN C10 0.3734E-02 0.3848E-02 0.4155E-02
GRC 0
URN C20 0.1907E-04 0.1966E-04 0.2122E-04
GRC 0
URN C30 -0.1984E-05 -0.2044E-05 -0.2207E-05
GRC 0
REFRACTIVE INDICES
GLASS CODE 656.27 587.56 486.13
'grin2' 1.585265 1.591695 1.605584
URN 0.050000
URN C10 -0.2122E-02 -0.2187E-02 -0.2361E-02
URN C20 0.2187E-04 0.2254E-04 0.2433E-04
URN C30 -0.9741E-07 -0.1004E-06 -0.1084E-06
SF10_SCHOTT 1.720848 1.728250 1.746481
NFK5_SCHOTT 1.485345 1.487490 1.492269
NLAK34_SCHOTT 1.725090 1.729160 1.738469
No solves defined in system
No pickups defined in system
ZOOM DATA
ZOOM TITLE
POS 1 "EFL = 11.5mm"
POS 2 "EFL = 16.5mm"
POS 3 "EFL = 23mm"
POS 1 POS 2 POS 3
EPD 3.28570 3.88235 4.60000
YAN F2 7.67200 5.45200 3.94800
YAN F3 13.42600 9.54100 6.90900
YAN F4 16.30300 11.58550 8.38950
YAN F5 19.18000 13.63000 9.87000
VUY F1 -0.5433E-12 0.7268E-11 0.1783E-10
VLY F1 -0.5433E-12 0.7268E-11 0.1783E-10
VUY F2 0.00997 0.00113 0.00087
VLY F2 0.01935 0.00274 -0.00067
VUY F3 0.03622 0.00382 0.00124
VLY F3 0.06054 0.00911 -0.00050
VUY F4 0.05717 0.00628 0.00135
VLY F4 0.08742 0.01260 -0.00024
VUY F5 0.08287 0.00968 0.00146
VLY F5 0.11445 0.01407 -0.3744E-04
VUX F1 -0.5433E-12 0.7268E-11 0.1783E-10
VLX F1 -0.5433E-12 0.7268E-11 0.1783E-10
VUX F2 0.00447 0.00051 -0.2014E-04
VLX F2 0.00447 0.00051 -0.2014E-04
VUX F3 0.01490 0.00185 -0.1440E-04
VLX F3 0.01490 0.00185 -0.1440E-04
VUX F4 0.02270 0.00294 0.2143E-04
165 VLX F4 0.02270 0.00294 0.2143E-04
VUX F5 0.03199 0.00425 0.9015E-04
VLX F5 0.03199 0.00425 0.9015E-04
THI S2 5.69202 14.44287 16.71420
THC S2 0 0 0
THI S4 18.44369 6.38527 0.10000
THC S4 0 0 0
THI S9 14.25850 17.56606 21.58000
THC S9 0 0 0
POS 1 POS 2 POS 3
INFINITE CONJUGATES
EFL 11.5000 16.5000 23.0000
BFL 14.3105 17.6435 21.6265
FFL 9.1246 4.9032 -4.9621
FNO 3.5000 4.2500 5.0000
IMG DIS 14.2585 17.5661 21.5800
OAL 35.7415 32.4339 28.4200
PARAXIAL IMAGE
HT 4.0002 4.0009 4.0017
ANG 19.1800 13.6300 9.8700
ENTRANCE PUPIL
DIA 3.2857 3.8824 4.6000
THI 16.5254 17.7436 16.0420
EXIT PUPIL
DIA 5.1056 4.9888 5.0371
THI -3.5591 -3.5591 -3.5591
STO DIA 6.1108 5.9791 6.0431
166
Appendix E. CODEV® GRIN Chromatic macro
Instructions for running macro:
Ax_lat_GRIN_CODEV.seq manual (CODEV)
This document describes the operation of the CODE V sequence file ax_lat_GRIN_CODEV.seq. Note that all of the mathematics in the code are derived from: K. Siva Rama Krishna and Anurag Sharma, "Chromatic aberrations of radial gradient-index lenses. I. Theory," Appl. Opt. 35, 1032-1036 (1996). Please refer to that paper for more information on specific equations and definitions.
As of the writing of this document, CODEV® does not offer a means to calculate the polychromatic aberration coefficients of GRIN systems. This CODEV® macro provides a means to determine the axial and lateral color contributions from a quadratic radial GRIN element. Running ax_lat_GRIN_CODEV.seq on a GRIN surface in CODEV® calculates the third-order surface contributions to those aberrations (matching the values calculated by CODEV®’s built-in third-order aberration coefficient calculations (THO) in addition to the GRIN coefficients. To operate the macro, open ax_lat_GRIN_CODEV.seq in CODEV®. As shown in Figure A-1 this prompts user selection of both the surface number of the GRIN element as well as the zoom position (default to 1 if the system is not zoomed). Figure A-2 shows results from an example calculation, displaying axial and lateral color contributions for each surface and the GRIN as well as their sums. The constant ν11 is also displayed (discussed in Chapter 3).
Figure A-1: Dialog screen prompting user selection of surface number and zoom position
167
Figure A-2: Example output from chromatic macro
.seq file for GRIN Chromatic macro
!**********************************************************************
****
! Macro PLOTGRIN_polychrom
!
! Plots gradient index profile as a function of position at every
wavelength
!
! Usage:
! in PLOTGRIN surf# [type of GRIN]
!
!
! History: 2014_01_28 JAC Create
! Code was derived from: K. Siva Rama Krishna and Anurag Sharma,
"Chromatic aberrations of radial gradient-index lenses. I. Theory,"
Appl. Opt. 35, 1032-1036 (1996)
! See paper for more information
!**********************************************************************
****
! arg0 "Macro to plot gradient index profile as a function of
position."
!
! arg1 name "Surface number of GRIN element"
! arg1 type num
! arg1 default 1
! arg1 help "Surface number of GRIN material."
168 !
! arg2 name "Zoom Position"
! arg2 type num
! arg2 default 1
! arg2 help "Zoom position to plot."
!
!**********************************************************************
****
! Global variables
gbl num ^error ! Error flag for image evaluation
gbl str ^format ! format of value
gbl num ^ymax ! Maximum value on y-axis
gbl num ^ymin ! Minimum value on y-axis
! Local variables
lcl num ^surfnum
lcl num ^aray_y
lcl num ^lam_red
lcl num ^lam_green
lcl num ^lam_blue
lcl num ^w_red
lcl num ^w_blue
lcl num ^n_red_i
lcl num ^n_green_i
lcl num ^n_blue_i
lcl num ^V_i
lcl num ^v01_i
lcl num ^n_red_o
lcl num ^n_green_o
lcl num ^n_blue_o
lcl num ^V_o
lcl num ^v01_o
lcl num ^fa1
lcl num ^fa1_st
lcl num ^del_v_n
lcl num ^mu
lcl num ^ax
lcl num ^lat
lcl num ^C10
lcl num ^thi
lcl num ^alp
lcl num ^S
lcl num ^C
lcl num ^e
lcl num ^g0
lcl num ^g1
lcl num ^g2
lcl num ^v11
lcl num ^psi1
169 lcl num ^fa2
lcl num ^fa2_st
lcl num ^n10
lcl num ^ax_G
lcl num ^lat_G
! Buchdahl Coefficients
^lam_red == (wl w1)/1000
^lam_green == (wl w2)/1000
^lam_blue == (wl w3)/1000
^w_red == (^lam_red - ^lam_green)/(1+2.5*(^lam_red-^lam_green))
^w_blue == (^lam_blue - ^lam_green)/(1+2.5*(^lam_blue-^lam_green))
ver n
^surfnum == #1
^zoomnum == #2
! Define V and v01 for incident media
^n_red_i == (index(^surfnum-1,^zoomnum,1,1,0,0,0))
^n_green_i == (index(^surfnum-1,^zoomnum,2,1,0,0,0))
^n_blue_i == (index(^surfnum-1,^zoomnum,3,1,0,0,0))
if ^n_green_i = 1
^V_i == 0
^v01_i == 0
else
^V_i == (^n_green_i-1)/(^n_blue_i-^n_red_i)
^v01_i == (^n_green_i-1)/(^V_i*(^w_blue-^w_red))
end if
! Define V and v01 for refracting media
^n_red_o == (index(^surfnum,^zoomnum,1,1,0,0,0))
^n_green_o == (index(^surfnum,^zoomnum,2,1,0,0,0))
^n_blue_o == (index(^surfnum,^zoomnum,3,1,0,0,0))
if ^n_green_o = 1
^V_o == 0
^v01_o == 0
else
^V_o == (^n_green_o-1)/(^n_blue_o-^n_red_o)
^v01_o == (^n_green_o-1)/(^V_o*(^w_blue-^w_red))
end if
! Define V and v01 for third media
^n_red_3 == (index(^surfnum+1,^zoomnum,1,1,0,0,0))
^n_green_3 == (index(^surfnum+1,^zoomnum,2,1,0,0,0))
^n_blue_3 == (index(^surfnum+1,^zoomnum,3,1,0,0,0))
if ^n_green_3 = 1
^V_3 == 0
^v01_3 == 0
else
^V_3 == (^n_green_3-1)/(^n_blue_3-^n_red_3)
^v01_3 == (^n_green_3-1)/(^V_3*(^w_blue-^w_red))
end if
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Calculate change in v01/n00 for a surface
170 ^del_v_n == (^v01_o/^n_green_o) - (^v01_i/^n_green_i)
! Begin surface contribution calculation
^fa1 == ^n_green_i*(hmy z^zoomnum s^surfnum)*(imy z^zoomnum
s^surfnum)*^del_v_n/^n_green_i
^fa1_st == ^n_green_i*(hmy z^zoomnum s^surfnum)*(icy z^zoomnum
s^surfnum)*^del_v_n/^n_green_i
! Calculate surface coefficients
^mu == -1/(umy si z^zoomnum)
^ax == ^mu*^fa1*(^w_blue-^w_red)
^lat == ^mu*^fa1_st*(^w_blue-^w_red)
wri "Surface axial S1"
eva(-^ax)
wri "Surface lateral S1"
eva(-^lat)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Calculate change in v01/n00 for a surface
^del_v_n_S2 == (^v01_3/^n_green_3) - (^v01_o/^n_green_o)
! Begin surface contribution calculation
^faS2 == ^n_green_o*(hmy z^zoomnum s^surfnum+1)*(imy z^zoomnum
s^surfnum+1)*^del_v_n_S2/^n_green_o
^faS2_st == ^n_green_o*(hmy z^zoomnum s^surfnum+1)*(icy z^zoomnum
s^surfnum+1)*^del_v_n_S2/^n_green_o
! Calculate surface coefficients
^mu == -1/(umy si z^zoomnum)
^ax_S2 == ^mu*^faS2*(^w_blue-^w_red)
^lat_S2 == ^mu*^faS2_st*(^w_blue-^w_red)
wri "Surface axial S2"
eva(-^ax_S2)
wri "Surface lateral S2"
eva(-^lat_S2)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! End surface contribution calculation
! Begin tranfer contribution calculation
^n_green_axis == (index(^surfnum,^zoomnum,2,1,0,0,0))
^n_green_per == (index(^surfnum,^zoomnum,2,1,0,(hmy s^surfnum),0))
if ^n_green_axis - ^n_green_per = 0
^ax_G == ^mu*^fa2*(^w_blue-^w_red)
^lat_G == ^mu*^fa2_st*(^w_blue-^w_red)
else
171
^C10 == (GRN s^surfnum C10 w2 z^zoomnum)
^thi == (thi s^surfnum z^zoomnum)
^e == 2.718281828459046
!eva(^n_green_o)
^alp == sqrt(absf(2*^C10/^n_green_o))
!eva(^alp)
if ^C10 < 0
^C == cosf(^alp*^thi)
^S == sinf(^alp*^thi)/^alp
else if ^C10 > 0
^C == (^e**(^alp*^thi)+^e**(-^alp*^thi))/(2) !cosh
^S == (^e**(^alp*^thi)-^e**(-^alp*^thi))/(2*^alp) !sinh
else
^C == 1
^S == 0
end if
!eva(^C)
!eva(^S)
^g0 == 0.5*(^thi+^C*^S)
^g1 == ^S*^S
^g2 == (^thi - ^C*^S)/(2*^alp*^alp)
!eva(^g0)
!eva(^g1)
!eva(^g2)
^v11red == ((GRN s^surfnum C10 w1 z^zoomnum) - ^C10)/^w_red
^v11blue == ((GRN s^surfnum C10 w3 z^zoomnum) - ^C10)/^w_blue
!eva((^v11red+^v11blue)/2)
wri "v11"
^n10 == ^C10
^v11 == 0.00122889
!-0.00075901
eva(^v11)
!0.00132333
! 0.00028848
!-0.00075263
^psi1 == (^v11/^n10 - ^v01_o/^n_green_o)
!eva(^v01_o)
!wri "psi"
!eva(^psi1)
^fa2 == 2*^n10*^psi1*(^g0*(hmy z^zoomnum s^surfnum)*(hmy z^zoomnum
s^surfnum) + ^g1*(hmy z^zoomnum s^surfnum)*(umy z^zoomnum
s^surfnum+1)+^g2*(umy z^zoomnum s^surfnum+1)*(umy z^zoomnum
s^surfnum+1))
172 ^fa2_st == 2*^n10*^psi1*(^g0*(hmy z^zoomnum s^surfnum)*(hcy z^zoomnum
s^surfnum) + 0.5*^g1*((hmy z^zoomnum s^surfnum)*(ucy z^zoomnum
s^surfnum+1)+(hcy z^zoomnum s^surfnum)*(umy z^zoomnum s^surfnum+1)) +
^g2*(umy z^zoomnum s^surfnum+1)*(ucy z^zoomnum s^surfnum+1))
! Calculate GRIN coefficients
^mu == -1/(umy si z^zoomnum)
^ax_G == ^mu*^fa2*(^w_blue-^w_red)
^lat_G == ^mu*^fa2_st*(^w_blue-^w_red)
!eva(^fa2_st)
end if
wri "GRIN axial"
eva(^ax_G)
wri "GRIN lateral"
eva(^lat_G)
wri "Sum Axial"
eva(-^ax-^ax_S2+^ax_G)
wri "Sum Lat"
eva(-^lat-^lat_S2+^lat_G)
!wri "lF - lC"
!eva((^ax+^ax_S2-^ax_G)/(umy si z^zoomnum))
out y
173
Appendix F. Lens listing for 10x visible zoom lens – homogeneous
10x zoom, efl = 10mm
RDY THI RMD GLA CCY THC GLC
> OBJ: INFINITY INFINITY 100 100
1: 129.03733 3.000000 LLAH86_OHARA 0 0
SLB: "e1"
2: 64.13707 14.279409 SFPM2_OHARA 0 0
SLB: "e2"
3: -1037.40199 0.100000 0 0
4: 55.14020 8.371468 SFPM2_OHARA 0 0
SLB: "e3"
5: 126.73446 1.534938 0 0
6: 55.54215 2.500000 SFPM2_OHARA 0 0
SLB: "e4"
7: 15.74071 10.613936 0 0
8: -85.35226 1.000000 SFPM2_OHARA 0 0
SLB: "e45"
9: 24.24077 3.286086 0 0
10: 26.51113 3.240675 SLAL14_OHARA 0 0
SLB: "e5"
11: 84.38352 0.100000 0 0
12: 43.30257 67.331911 0 0
STO: INFINITY 0.100000 0 0
14: 27.73773 3.579720 SLAM3_OHARA 0 0
SLB: "e6"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :-.783204E-04 B :-.195374E-07 C :-.502482E-08 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
15: -76.61738 3.272434 0 0
16: 26.50625 4.000000 SFPL53_OHARA 0 0
SLB: "e7"
17: -14.69303 1.938504 0 0
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.656149E-04 B :-.642356E-06 C :0.348083E-08 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
18: 19.35413 2.000000 SNPH2_OHARA 0 0
SLB: "e8"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.499616E-05 B :0.801935E-07 C :-.364761E-08 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
19: 8.43398 15.750919 0 0
IMG: INFINITY 0.000000 100 100
SPECIFICATION DATA
EPD 5.00000
DIM MM
174 WL 656.27 587.56 486.13
REF 2
WTW 1 1 1
INI JAC
XAN 0.00000 0.00000 0.00000 0.00000 0.00000
YAN 0.00000 10.62600 18.59550 22.58025 26.56500
WTF 1.00000 1.00000 1.00000 1.00000 1.00000
VUX 0.00000 -0.00813 -0.02855 -0.04680 -0.07498
VLX 0.00000 -0.00813 -0.02855 -0.04680 -0.07498
VUY 0.00000 -0.03563 -0.11726 -0.19954 -0.34871
VLY 0.00000 -0.01609 -0.08467 -0.16298 -0.31624
POL N
PRIVATE CATALOG
PWL 656.27 587.56 486.13
'LV1' 1.844688 1.855040 1.880644
'LV2' 1.568498 1.570980 1.576506
'LV4' 1.455586 1.457200 1.460648
'LV6' 1.617322 1.620320 1.627108
'LV8' 1.695976 1.699790 1.708578
'LV10' 1.831307 1.842810 1.871441
'LV13' 1.568498 1.570980 1.576506
'LV14' 1.760862 1.766510 1.779977
'LV16' 1.568498 1.570980 1.576506
'LV18' 1.881698 1.888140 1.903476
'LV19' 1.602360 1.605200 1.611602
'LV21' 1.881698 1.888140 1.903476
'LV23' 1.515772 1.518250 1.523851
'grin5' 1.585273 1.591704 1.605594
GRC 0
URN 0.050000
URN C10 -0.8267E-03 -0.8521E-03 -0.9200E-03
GRC 0
URN C20 0.1732E-05 0.1785E-05 0.1928E-05
GRC 0
URN C30 -0.3209E-08 -0.3308E-08 -0.3571E-08
GRC 0
REFRACTIVE INDICES
GLASS CODE 656.27 587.56 486.13
SFPM2_OHARA 1.592555 1.595220 1.601342
LLAH86_OHARA 1.894221 1.902699 1.923335
SFPL53_OHARA 1.437333 1.438750 1.441954
SLAM3_OHARA 1.712528 1.717004 1.727488
SLAL14_OHARA 1.692974 1.696797 1.705521
SNPH2_OHARA 1.909158 1.922860 1.957994
No solves defined in system
No pickups defined in system
ZOOM DATA
ZOOM TITLE
POS 1 "10x zoom, efl = 10mm"
POS 2 "10x zoom, efl = 31mm"
POS 3 "10x zoom, efl = 98mm"
POS 1 POS 2 POS 3
175
EPD 5.00000 8.78130 19.66740
YAN F2 10.62600 3.69600 1.16440
YAN F3 18.59550 6.46800 2.03770
YAN F4 22.58025 7.85400 2.47435
YAN F5 26.56500 9.24000 2.91100
VUY F1 -0.1679E-12 0.7616E-11 -0.5662E-13
VLY F1 -0.1679E-12 0.7616E-11 -0.5662E-13
VUY F2 -0.03563 -0.00037 0.00739
VLY F2 -0.01609 0.00800 0.01010
VUY F3 -0.11726 0.00657 0.02455
VLY F3 -0.08467 0.02235 0.03510
VUY F4 -0.19954 0.01442 0.03825
VLY F4 -0.16298 0.03465 0.05550
VUY F5 -0.34871 0.02703 0.05642
VLY F5 -0.31624 0.05243 0.08204
VUX F1 -0.1679E-12 0.7616E-11 -0.5662E-13
VLX F1 -0.1679E-12 0.7616E-11 -0.5662E-13
VUX F2 -0.00813 0.00122 0.00276
VLX F2 -0.00813 0.00122 0.00276
VUX F3 -0.02855 0.00427 0.00908
VLX F3 -0.02855 0.00427 0.00908
VUX F4 -0.04680 0.00689 0.01403
VLX F4 -0.04680 0.00689 0.01403
VUX F5 -0.07498 0.01059 0.02042
VLX F5 -0.07498 0.01059 0.02042
THI S5 1.53494 40.36849 66.70971
THC S5 0 0 0
THI S12 67.33191 28.28023 3.15415
THC S12 0 0 0
THI S15 3.27243 2.74754 3.20177
THC S15 0 0 0
THI S17 1.93850 2.68153 1.01216
THC S17 0 0 0
POS 1 POS 2 POS 3
INFINITE CONJUGATES
EFL 10.0000 30.7345 98.3354
BFL 15.6816 15.7465 15.7032
FFL 53.4676 164.8288 135.8490
FNO 2.0000 3.5000 4.9999
IMG DIS 15.7509 15.7509 15.7509
OAL 130.2491 130.2491 130.2491
PARAXIAL IMAGE
HT 5.0000 4.9999 5.0004
ANG 26.5650 9.2400 2.9110
ENTRANCE PUPIL
DIA 5.0000 8.7813 19.6674
THI 57.7767 205.6110 557.7873
EXIT PUPIL
DIA 11.6033 6.6178 4.5836
THI -7.5250 -7.4158 -7.2145
STO DIA 14.4723 8.4359 5.5315
176
Appendix G. Lens listing for 10x visible zoom lens – GRIN
10x zoom, efl = 10mm
RDY THI RMD GLA CCY THC GLC
> OBJ: INFINITY INFINITY 100 100
1: 135.23819 3.000000 SLAH60_OHARA 0 0
SLB: "e1"
2: 49.60673 13.735028 SFPM2_OHARA 0 0
SLB: "e2"
3: -6547.29652 0.100000 0 0
4: 47.26300 8.273132 SFPM2_OHARA 0 0
SLB: "e3"
5: 119.88085 3.533455 0 0
6: 47.65881 2.500000 SLAH66_OHARA 0 0
SLB: "e4"
7: 15.76844 8.306959 0 0
8: -72.40898 1.000000 SLAL12_OHARA 0 0
SLB: "e45"
9: 42.74914 0.835482 0 0
10: 30.81078 5.478774 'grin5' 0 0
SLB: "grin5"
11: 51.50148 0.100000 0 0
12: 43.30257 63.254510 0 0
STO: INFINITY 0.100000 0 0
14: 24.36804 3.493146 LBSL7_OHARA 0 0
SLB: "e6"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :-.649574E-04 B :-.128590E-06 C :-.264636E-08 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
15: 95.43450 3.688683 0 0
16: 29.43545 4.000000 SFPM2_OHARA 0 0
SLB: "e7"
17: -23.86616 0.100000 0 0
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.789774E-05 B :-.287763E-06 C :0.127466E-08 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
18: 15.32186 4.577121 'grin8' 0 0
SLB: "grin8"
19: 9.99106 19.923709 0 0
IMG: INFINITY 0.000000 100 100
SPECIFICATION DATA
EPD 5.00000
DIM MM
WL 656.27 587.56 486.13
REF 2
WTW 1 1 1
INI JAC
XAN 0.00000 0.00000 0.00000 0.00000 0.00000
YAN 0.00000 10.62600 18.59550 22.58025 26.56500
177 WTF 1.00000 1.00000 1.00000 1.00000 1.00000
VUX 0.00000 -0.00736 -0.02575 -0.04147 -0.06357
VLX 0.00000 -0.00736 -0.02575 -0.04147 -0.06357
VUY 0.00000 -0.03181 -0.10167 -0.16644 -0.26656
VLY 0.00000 -0.01502 -0.07630 -0.13647 -0.22966
POL N
PRIVATE CATALOG
PWL 656.27 587.56 486.13
'LV1' 1.844688 1.855040 1.880644
'LV2' 1.568498 1.570980 1.576506
'LV4' 1.455586 1.457200 1.460648
'LV6' 1.617322 1.620320 1.627108
'LV8' 1.695976 1.699790 1.708578
'LV10' 1.831307 1.842810 1.871441
'LV13' 1.568498 1.570980 1.576506
'LV14' 1.760862 1.766510 1.779977
'LV16' 1.568498 1.570980 1.576506
'LV18' 1.881698 1.888140 1.903476
'LV19' 1.602360 1.605200 1.611602
'LV21' 1.881698 1.888140 1.903476
'LV23' 1.515772 1.518250 1.523851
'grin5' 1.585273 1.591704 1.605594
GRC 0
URN 0.050000
URN C10 -0.4979E-03 -0.5132E-03 -0.5541E-03
GRC 0
URN C20 -0.3724E-06 -0.3838E-06 -0.4144E-06
GRC 0
URN C30 -0.1136E-08 -0.1170E-08 -0.1264E-08
GRC 0
'grin8' 1.487957 1.491402 1.497298
GRC 0
URN 0.050000
URN C10 0.1299E-02 0.1339E-02 0.1446E-02
GRC 0
URN C20 0.1181E-05 0.1217E-05 0.1314E-05
GRC 0
URN C30 0.6002E-07 0.6186E-07 0.6679E-07
GRC 0
REFRACTIVE INDICES
GLASS CODE 656.27 587.56 486.13
SFPM2_OHARA 1.592555 1.595220 1.601342
'grin5' 1.585273 1.591704 1.605594
URN 0.050000
URN C10 -0.4979E-03 -0.5132E-03 -0.5541E-03
URN C20 -0.3724E-06 -0.3838E-06 -0.4144E-06
URN C30 -0.1136E-08 -0.1170E-08 -0.1264E-08
SLAH60_OHARA 1.827376 1.834000 1.849819
'grin8' 1.487957 1.491402 1.497298
URN 0.050000
URN C10 0.1299E-02 0.1339E-02 0.1446E-02
URN C20 0.1181E-05 0.1217E-05 0.1314E-05
URN C30 0.6002E-07 0.6186E-07 0.6679E-07
SLAH66_OHARA 1.767798 1.772499 1.783373
SLAL12_OHARA 1.674188 1.677900 1.686438
LBSL7_OHARA 1.513846 1.516330 1.521905
No solves defined in system
178
No pickups defined in system
ZOOM DATA
ZOOM TITLE
POS 1 "10x zoom, efl = 10mm"
POS 2 "10x zoom, efl = 31mm"
POS 3 "10x zoom, efl = 98mm"
POS 1 POS 2 POS 3
EPD 5.00000 8.78130 19.66740
YAN F2 10.62600 3.69600 1.16440
YAN F3 18.59550 6.46800 2.03770
YAN F4 22.58025 7.85400 2.47435
YAN F5 26.56500 9.24000 2.91100
VUY F1 0.1887E-12 0.1574E-10 0.5389E-11
VLY F1 0.1887E-12 0.1574E-10 0.5389E-11
VUY F2 -0.03181 0.00948 0.00615
VLY F2 -0.01502 0.01815 0.00887
VUY F3 -0.10167 0.03739 0.02040
VLY F3 -0.07630 0.05374 0.02725
VUY F4 -0.16644 0.06003 0.03097
VLY F4 -0.13647 0.08086 0.04084
VUY F5 -0.26656 0.08973 0.04412
VLY F5 -0.22966 0.11543 0.05758
VUX F1 0.1887E-12 0.1574E-10 0.5389E-11
VLX F1 0.1887E-12 0.1574E-10 0.5389E-11
VUX F2 -0.00736 0.00455 0.00245
VLX F2 -0.00736 0.00455 0.00245
VUX F3 -0.02575 0.01455 0.00768
VLX F3 -0.02575 0.01455 0.00768
VUX F4 -0.04147 0.02210 0.01150
VLX F4 -0.04147 0.02210 0.01150
VUX F5 -0.06357 0.03161 0.01621
VLX F5 -0.06357 0.03161 0.01621
THI S5 3.53346 38.93481 63.01677
THC S5 0 0 0
THI S12 63.25451 27.48391 0.41973
THC S12 0 0 0
THI S15 3.68868 2.24555 6.21845
THC S15 0 0 0
THI S17 0.10000 1.91238 0.92170
THC S17 0 0 0
POS 1 POS 2 POS 3
INFINITE CONJUGATES
EFL 10.0000 30.7348 98.3371
BFL 19.8896 19.9404 19.8712
FFL 53.1145 151.3719 101.8769
FNO 2.0000 3.5000 5.0000
IMG DIS 19.9237 19.9237 19.9237
OAL 126.0763 126.0763 126.0763
PARAXIAL IMAGE
HT 5.0000 5.0000 5.0005
ANG 26.5650 9.2400 2.9110
ENTRANCE PUPIL
DIA 5.0000 8.7813 19.6674
179 THI 56.4828 183.6983 397.3378
EXIT PUPIL
DIA 14.8445 8.3490 6.5458
THI -9.7994 -9.2812 -12.8580
STO DIA 15.0078 9.0403 6.1634
180
Appendix H. Lens listing for 2x visible zoom lens – GRIN (JC018 profile)
EFL = 11.5mm
RDY THI RMD GLA CCY THC GLC
> OBJ: INFINITY INFINITY 100 100
1: 22.79060 4.500000 NFK58_SCHOTT 0 0
SLB: "e1"
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.408898E-04 B :-.147288E-06 C :0.964058E-08 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
2: 8.95852 2.845927 0 0
3: -7.81877 3.310076 'grin44' 0 0
SLB: "grin2"
4: -16.63999 11.891328 0 0
STO: INFINITY 0.100000 100 0
6: 6.45838 4.244140 NSSK5_SCHOTT 0 0
SLB: "e3"
7: -10.14381 0.100000 0 0
8: -9.92368 2.562845 SF4_SCHOTT 0 0
SLB: "e4"
9: 28.82713 15.175441 0 0
ASP:
K : 0.000000 KC : 100
CUF: 0.000000 CCF: 100
A :0.115479E-02 B :0.195470E-04 C :0.244596E-05 D :0.000000E+00
AC : 0 BC : 0 CC : 0 DC : 100
IMG: INFINITY 0.000000 100 100
SPECIFICATION DATA
EPD 3.28570
DIM MM
WL 656.27 587.56 486.13
REF 2
WTW 1 1 1
XAN 0.00000 0.00000
YAN 0.00000 19.18000
WTF 1.00000 1.00000
VUX 0.00000 0.01526
VLX 0.00000 0.01526
VUY 0.00000 0.04201
VLY 0.00000 0.08325
POL N
PRIVATE CATALOG
PWL 656.27 587.56 486.13
'grin2' 1.612950 1.615235 1.633133
GRC 0
URN 0.050000
URN C10 -0.1835E-02 -0.1941E-02 -0.2076E-02
GRC 0
URN C20 0.3809E-05 0.6447E-05 0.6500E-05
GRC 0
URN C30 -0.9934E-06 -0.9796E-06 -0.9871E-06
GRC 0
'grin4' 1.487914 1.491360 1.497256
181 GRC 0
URN 0.050000
URN C10 0.3733E-02 0.3848E-02 0.4154E-02
GRC 0
URN C20 0.1906E-04 0.1964E-04 0.2121E-04
GRC 0
URN C30 -0.1984E-05 -0.2045E-05 -0.2208E-05
GRC 0
'grin78' 1.543635 1.548788 1.559258
GRC 100
URN 0.050000
URN C10 -0.5846E-03 -0.6026E-03 -0.6506E-03
GRC 100
URN C20 -0.1197E-04 -0.1234E-04 -0.1332E-04
GRC 100
URN C30 0.2873E-06 0.2961E-06 0.3197E-06
GRC 100
'grin44' 1.543542 1.548692 1.559154
GRC 100
URN 0.050000
URN C10 -0.5846E-03 -0.6026E-03 -0.6506E-03
GRC 100
URN C20 -0.1197E-04 -0.1234E-04 -0.1332E-04
GRC 100
URN C30 0.2873E-06 0.2961E-06 0.3197E-06
GRC 100
REFRACTIVE INDICES
GLASS CODE 656.27 587.56 486.13
'grin44' 1.543542 1.548692 1.559154
URN 0.050000
URN C10 -0.5846E-03 -0.6026E-03 -0.6506E-03
URN C20 -0.1197E-04 -0.1234E-04 -0.1332E-04
URN C30 0.2873E-06 0.2961E-06 0.3197E-06
NSSK5_SCHOTT 1.654554 1.658440 1.667494
SF4_SCHOTT 1.747298 1.755201 1.774681
NFK58_SCHOTT 1.454462 1.456000 1.459479
No solves defined in system
No pickups defined in system
ZOOM DATA
ZOOM TITLE
POS 1 "EFL = 11.5mm"
POS 2 "EFL = 16.5mm"
POS 3 "EFL = 23mm"
POS 1 POS 2 POS 3
EPD 3.28570 3.88235 4.60000
YAN F2 19.18000 13.63000 9.87000
VUY F1 0.4498E-11 0.1914E-10 -0.1616E-12
VLY F1 0.4498E-11 0.1914E-10 -0.1616E-12
VUY F2 0.04201 -0.00943 0.00400
VLY F2 0.08325 -0.01952 -0.01084
VUX F1 0.4498E-11 0.1914E-10 -0.1616E-12
VLX F1 0.4498E-11 0.1914E-10 -0.1616E-12
182 VUX F2 0.01526 -0.00444 -0.00123
VLX F2 0.01526 -0.00444 -0.00123
THI S2 2.84593 6.61588 5.49026
THC S2 0 0 0
THI S4 11.89133 4.12639 0.77794
THC S4 0 0 0
THI S9 15.17544 19.17043 23.64450
THC S9 0 0 0
POS 1 POS 2 POS 3
INFINITE CONJUGATES
EFL 11.5000 16.5000 23.0000
BFL 15.2222 19.2474 23.6728
FFL 6.1991 1.0795 -8.7233
FNO 3.5000 4.2500 5.0000
IMG DIS 15.1754 19.1704 23.6445
OAL 29.5543 25.5593 21.0853
PARAXIAL IMAGE
HT 4.0002 4.0009 4.0017
ANG 19.1800 13.6300 9.8700
ENTRANCE PUPIL
DIA 3.2857 3.8824 4.6000
THI 13.1851 12.9392 10.5965
EXIT PUPIL
DIA 5.4087 5.4014 5.4763
THI -3.7085 -3.7085 -3.7085
STO DIA 6.1572 6.1780 6.2720
183
Appendix I. MATLAB code for identifying athermalized radial GRIN lenses
clear all
close all
clc
tic;
% % Material 1 - PMMA - Pete
% n2_i = 1.4917; % base index
% dndT_2 = -13.4e-5; % dn/dT
% CTE_2 = 8.98e-5; % CTE
% % Material 2 - PS - KJah
% n1_i = 1.5903; % base index
% dndT_1 = -12e-5; % dn/dT
% CTE_1 = 5e-5; % CTE
% Material Specifications
% % Material 1 - CR-39
% n2_i = 1.5016; % base index
% dndT_2 = -18.4e-5; % dn/dT
% CTE_2 = 10.38e-5; % CTE
% % Material 2 - DAP
% n1_i = 1.5728; % base index
% dndT_1 = -16.1e-5; % dn/dT
% CTE_1 = 8.29e-5; % CTE
% Material 2 - PMMA -Leo
n1_i = 1.4917; % base index
dndT_1 = -8.5e-5; % dn/dT
CTE_1 = 6.5e-5; % CTE
% Material 1 - PS -Leo
n2_i = 1.5903; % base index
dndT_2 = -12e-5; % dn/dT
CTE_2 = 6.3e-5; % CTE
% %Materal HIRI
% n2_i = 1.5594; % base index
% dndT_2 = -22.3e-5; % dn/dT
% CTE_2 = 13.51e-5; % CTE
% System/lens specifications
r = 5; % lens radius
W = 5; % thickness
dT = 40; % change in temperature
inc = 1001; % vector increment
c_i_1 = linspace(0.01,0.05,inc);
c_sca = 1;
c_i_2 = c_sca.*c_i_1;
% GRIN calculation
comp_a = 1; % Percent of material a along the axis
comp_b = linspace(0,0.8,inc); % Definition of range of composition
profiles for periphery
[c_i_1 comp_b] = meshgrid(c_i_1,comp_b);
n_a = comp_a.*n1_i + (1 - comp_a).*n2_i; % Index at optical axis
184 n_b = (1 - comp_b).*n1_i + comp_b.*n2_i; % Index at edge of lens
dn = n_b - n_a; % Full change in index
% Calculation of constants at initial temperature
n10 = (n_b - n_a)./(r.^2);
n00 = n_a;
b = sqrt(abs(2.*n00.*n10));
k = b.*W./n00;
% Calculation
W_p = W.*(1 + CTE_1.*dT);
N00_p = n_a + dndT_1.*dT;
CTE_a = comp_a.*CTE_1 + (1 - comp_a).*CTE_2;
CTE_b = (1 - comp_b).*CTE_1 + comp_b.*CTE_2;
dCTE = CTE_b - CTE_a;
c_p_1 = ((c_i_1.*(1 + CTE_a.*dT))-(dCTE.*W.*dT./(r.*r)))./((1 +
CTE_a.*dT).^2); % assumes
c_p_2 = ((c_sca.*c_i_1.*(1 + CTE_a.*dT))-(dCTE.*W.*dT./(r.*r)))./((1 +
CTE_a.*dT).^2);
dndT_a = comp_a.*dndT_1 + (1 - comp_a).*dndT_2;
dndT_b = (1 - comp_b).*dndT_1 + comp_b.*dndT_2;
d_dndT = dndT_b - dndT_a;
% r = (1 + CTE_a.*dT).*r + dT
N10_p = (r.^-2).*(dn + d_dndT.*dT)./((1 + (CTE_a + dCTE./3).*dT).^2);
C1_p = c_p_1;
C2_p = -c_p_2;
b_p = sqrt(abs(2.*N00_p.*N10_p));
k_p = b_p.*W_p./N00_p;
phiG_i_pos = (c_i_1 + c_sca.*c_i_1).*(n00 - 1).*cosh(k) - b.*sinh(k) +
c_i_1.*(-c_sca.*c_i_1).*W.*(n00 - 1).*(n00 - 1).*sinh(k)./(k.*n00);
phiG_i_neg = (c_i_1 + c_sca.*c_i_1).*(n00 - 1).*cos(k) + b.*sin(k) +
c_i_1.*(-c_sca.*c_i_1).*W.*(n00 - 1).*(n00 - 1).*sin(k)./(k.*n00);
phiG_i_0 = (n00 - 1).*(c_i_1 + c_sca.*c_i_1 -
W.*c_i_1.*c_sca.*c_i_1.*(n00 - 1)./n00);
for xx = 1:size(n10,1);
if n10(xx,1) > 0
phiG_i(xx,:) = phiG_i_pos(xx,:);
elseif n10(xx,1) < 0
phiG_i(xx,:) = phiG_i_neg(xx,:);
else
phiG_i(xx,:) = phiG_i_0(xx,:);
end
end
phiG_f_pos = (C1_p - C2_p).*(N00_p - 1).*cosh(k_p) - b_p.*sinh(k_p) +
C1_p.*C2_p.*W_p.*(N00_p - 1).*(N00_p - 1).*sinh(k_p)./(k_p.*N00_p);
phiG_f_neg = (C1_p - C2_p).*(N00_p - 1).*cos(k_p) + b_p.*sin(k_p) +
C1_p.*C2_p.*W_p.*(N00_p - 1).*(N00_p - 1).*sin(k_p)./(k_p.*N00_p);
phiG_f_0 = (N00_p - 1).*(c_p_1 + c_sca.*c_p_1 -
W.*c_p_1.*c_sca.*c_p_1.*(N00_p - 1)./N00_p);
for yy = 1:size(N10_p,1);
if N10_p(yy,1) > 0
185 phiG_f(yy,:) = phiG_f_pos(yy,:);
elseif N10_p(yy,1) < 0
phiG_f(yy,:) = phiG_f_neg(yy,:);
else
phiG_f(yy,:) = phiG_f_0(yy,:);
end
end
% Find athermalized solutions
dphi = (phiG_f - phiG_i)./phiG_i;
phiG_i;
a = abs(dphi) <= .00005;
a = single(a);
a(a==0) = NaN;
% plot3(c_i_1,comp_b,a,'ko')
% Numerically find curvatures and GRIN profile of athermalized
solutions
format long
power = 0.02;
Q = abs(phiG_i - power) <= .00000000001;
Q = single(Q);
Q(Q==0) = NaN;
[row col] = find(Q==1);
for ijk = 1:length(row)
% check(ijk) = phiG_i(row(ijk),col(ijk));
yy = c_i_1';
C1 = yy(col(ijk))
row(ijk);
GRIN = comp_b(row(ijk))
end
% % % % % % % % % % % % % % % % % % % % % % % % % % % % %
figure(8)
hold on
plot3(c_i_1,comp_b,a,'k*')
plot3(c_i_1,comp_b,Q,'mo')
surf(c_i_1,comp_b,phiG_i)
ath_c = [0.02069038, 0.027251987, 0.03382204, 0.040384416, 0.046954];
ath_g = [0, 0.16386995, 0.3285438, 0.49362015, 0.659476];
% size(ath_c)
% size(ath_g)
plot3(ath_c,ath_g,[1,1,1,1,1],'o')
% set(gca,'zlim',[30 35])
% plot3(-.0138,0.45,1,'go') % CR-39/DAP solution
% plot3(.0093,0.5,1,'go') % DAP/CR-39 solution
shading interp
colorbar
% caxis([-.08 .08])
view(2)
toc;
% xlabel('Curvatures')
% ylabel('Composition of Mat. 2 at Edge')
% title('Base Power (\phi)')
186
Appendix J. MATLAB finite-element model (FEA) for modeling effect of temperature on radial GRIN elements
close all
clear all
clc
% Material 2 - PMMA - Precision Lens
n_PMMA = 1.4917; % base index
dndT_PMMA = -8.5e-5; % dn/dT
CTE_PMMA = 6.5e-5; % CTE
% Material 1 - PS - Precision Lens
n_PS = 1.5903; % base index
dndT_PS = -12e-5; % dn/dT
CTE_PS = 6.3e-5; % CTE
% % Material PS (nd) Greg
% n_PS = 1.591704087;
% dndT_PS = -12e-5; % dn/dT
% CTE_PS = 6.3e-5; % CTE
% % Material PMMA (nd) Greg
% n_PMMA = 1.491402003;
% dndT_PMMA = -12.5e-5; % dn/dT
% CTE_PMMA = 6.5e-5; % CTE
% 1.517214 1.50
% % Material BK7
% n_PS = 1.518522;
% dndT_PS = 15e-5; % dn/dT
% CTE_PS = 4.2e-6; % CTE
% % Material PMMA (nd) Greg
% n_PMMA = 1.51852200001;
% dndT_PMMA = 15e-5; % dn/dT
% CTE_PMMA = 4.2e-6; % CTE
% Material HIRI
n_PMMA = 1.5594;
dndT_PMMA = -22.3e-5; % dn/dT
CTE_PMMA = 13.51e-5; % CTE
% Material DAP
n_PS = 1.5728;
dndT_PS = -16.1e-5; % dn/dT
CTE_PS = 8.29e-5; % CTE
R1 = 10;%1./0.027251987; % Radius of curvature of surface 1
R2 = -10;%-1./0.027251987; % Radius of curvature of surface 2
CT = 5; % Center thickness of lens
CA = 10; % Clear aperture of lens
inc = 1001; % Increment for numerical calculations (should be an odd number)
mid = ceil(inc./2); % Defines the index of the middle of the inc vector
r = linspace(-CA./2,CA./2,inc); % radial coordinate, make sure it starts at r =
0 by making inc odd
dT = 40; % Change in temperature for thermal analysis
% Lines below calculate surface curvatures based on ROCs from above and
% surface sag departures referenced from the vertex of that lens surface
% (NB: sag is positive for a convex surface and vice versa)
187 C1 = 1./R1;
C2 = -1./R2;
Sag_S1 = (C1).*(r.^2)./(1 + sqrt(1 - (C1.*r).^2));
Sag_S2 = (C2).*(r.^2)./(1 + sqrt(1 - (C2.*r).^2));
figure(1)
plot(r,Sag_S1)
xlabel('r [mm])')
ylabel('Sag [mm]')
title('Nominal Sag')
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% This calculates half-slices of axial thickness of the lens as a function
% of distance from the axis
CT_S1 = CT./2 - Sag_S1;
CT_S2 = CT./2 - Sag_S2;
% The commands below plot the nominal lens shape
figure(2)
hold on
plot3(-CT_S1,r,linspace(1,1,inc)) % Surface 1
plot3(CT_S2,r,linspace(1,1,inc)) % Surface 2
plot3(linspace(-
CT_S1(inc),CT_S2(inc),inc),linspace(r(inc),r(inc),inc),linspace(1,1,inc)) % Top
of lens
plot3(linspace(-
CT_S1(1),CT_S2(1),inc),linspace(r(1),r(1),inc),linspace(1,1,inc)) % Bottom of
lens
ylim([-12 12])
axis equal
hold off
view(2)
xlabel('z [mm])')
ylabel('y [mm]')
title('Nominal Lens')
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% Here we define the material composition polynomial for our radial GRIN.
% Eq: n(r) = N00 + N10(r^2) + N20(r^4) + N30(r^6)
% This can be done in a number of ways. Below we assume a quadratic profile
% varying between some composition of PS on axis and some composition at
% the edge of the lens.
f_PS_a = 0;%0.78750833; % fraction of PS along optical axis
f_PS_e = 1;%0.16386995;%;0.6816;%0.85492616; % fraction of PS at lens' edge
N00 = n_PS.*f_PS_a + n_PMMA.*(1 - f_PS_a);
N10 = ((n_PS.*f_PS_e + n_PMMA.*(1 - f_PS_e))-(n_PS.*f_PS_a + n_PMMA.*(1 -
f_PS_a)))./((CA./2).^2);
% N10 = 0;
% N20 = ((n_PS.*f_PS_e + n_PMMA.*(1 - f_PS_e))-(n_PS.*f_PS_a + n_PMMA.*(1 -
f_PS_a)))./((CA./2).^4);
N20 = 0;
% N30 = ((n_PS.*f_PS_e + n_PMMA.*(1 - f_PS_e))-(n_PS.*f_PS_a + n_PMMA.*(1 -
f_PS_a)))./((CA./2).^6);
N30 = 0;
n = N00 + N10.*(r.^2) + N20.*(r.^4) + N30.*(r.^6); % Defines index polynomial
Q = (n - n_PMMA)./(n_PS - n_PMMA); % Defines concentration polynomial
CTE = Q.*CTE_PS + (1 - Q).*CTE_PMMA; % Defines CTE polynomial
188 dndT = Q.*dndT_PS + (1 - Q).*dndT_PMMA; % Defines dn/dT polynomial
CA_inc = CA./inc; % defines thickness of rectangular elements in radial
direction
CA_inc_p = CA_inc.*(1 + CTE.*dT);
CA_p = sum(CA_inc_p);
disp(['EPD = ' num2str(CA_p)])
r_p = linspace(-CA_p./2,CA_p./2,inc);
n_p = n + dndT.*dT;
x1 = r_p;
y1 = n_p;
format long
B0 = [N00 N10]; %
fh = @(B,x1) B(1) + B(2).*x1.^2;
ahat=nlinfit(x1,y1,fh,B0);
% B0 = ahat; B0(length(B0) + 1) = N20;
% fh = @(B,x1) B(1) + B(2).*x1.^2 + B(3).*x1.^4;
% ahat=nlinfit(x1,y1,fh,B0);
%
% B0 = ahat; B0(length(B0) + 1) = N30;
% fh = @(B,x1) B(1) + B(2).*x1.^2 + B(3).*x1.^4 + B(4).*x1.^6;
% ahat=nlinfit(x1,y1,fh,B0);
ahat
figure(3)
% plot(r,n)
hold on
% plot(r_p,n_p,'r')
% plot(r_p,fh(ahat,x1),'g')
plot(r_p,fh(ahat,x1)-n_p)
xlabel('r [mm]')
ylabel('n')
title('Index Polynomials')
% legend('Nominal','\Delta T','Fit')
% legend('\Delta T','Fit')
hold off
% Calculate the new surface curvature using Leo's thesis equation (p38)
% R_p = (((1+CTE00.*dT).^2)./(C1.*(1+CTE00.*dT)-0.5.*CT.*dT.*CTE10))^-1;
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% return
% This is the new center thickness of the lens after the temperature change
CT_F = CT.*(1 + CTE(mid).*dT);
disp(['CT = ' num2str(CT_F)])
% This calculates half-slices of axial thickness of the lens as a function
% of distance from the axis for the lens post temperature change
CT_S1_F = CT_S1.*(1 + CTE.*dT);
CT_S2_F = CT_S2.*(1 + CTE.*dT);
figure(4)
x = r_p;
y = (CT_S1_F(mid) - CT_S1_F); % Calculates the sag of the post-temp change lens
plot(r,y - Sag_S1) % Plots the difference in sag from post-temp to nominal
xlabel('r [mm])')
ylabel('\DeltaSag')
title('\DeltaSag = Sag(\DeltaT) - Sag(T0)')
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
189 % Below we fit the sag of the post-temp change lens (y) using the sag
% equation starting with just the surface curvature and then iteratively
% adding one aspheric term per step. You use the nominal surface curvature
% as the first guess for the curve fitting and then the results of one step
% as the guesses for the next step
% Fit with sag equation up to surface curvature (C)
A0 = C1; %
fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - A(1).^2.*x.^2));
bhat=nlinfit(x,y,fh,A0);
% % % % % % % Fit with sag equation up to conic coefficient (k)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2));
% bhat = nlinfit(x,y,fh,A0);
% % % %
% % % % % % % Fit with sag equation up to 4th order aspheric coefficient (A)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 6th order aspheric coefficient (B)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 8th order aspheric coefficient (C)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6 + A(5).*x.^8;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 10th order aspheric coefficient (D)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6 + A(5).*x.^8 + A(6).*x.^10;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 12th order aspheric coefficient (E)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6 + A(5).*x.^8 + A(6).*x.^10 + A(7).*x.^12;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 14th order aspheric coefficient (F)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6 + A(5).*x.^8 + A(6).*x.^10 + A(7).*x.^12 + A(8).*x.^14;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 16th order aspheric coefficient (G)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6 + A(5).*x.^8 + A(6).*x.^10 + A(7).*x.^12 + A(8).*x.^14
+ A(9).*x.^16;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 18th order aspheric coefficient (H)
190 % A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6 + A(5).*x.^8 + A(6).*x.^10 + A(7).*x.^12 + A(8).*x.^14
+ A(9).*x.^16 + A(10).*x.^18;
% bhat = nlinfit(x,y,fh,A0);
% %
% % % Fit with sag equation up to 20th order aspheric coefficient (I)
% A0 = bhat; A0(length(A0) + 1) = 0;
% fh = @(A,x) A(1).*x.^2./(1 + sqrt(1 - (1 + A(2)).*A(1).^2.*x.^2)) +
A(3).*x.^4 + A(4).*x.^6 + A(5).*x.^8 + A(6).*x.^10 + A(7).*x.^12 + A(8).*x.^14
+ A(9).*x.^16 + A(10).*x.^18 + A(11).*x.^20;
% bhat = nlinfit(x,y,fh,A0);
bhat
% close all
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% Below we plot (post temp change) the sag of the surface as calculated
% numerically by the model and also as fit to the aspheric sag equation
figure(5)
subplot(2,1,1)
hold on
plot(x,y,'g')%,'markersize',5,'color',[0,0,0]);
max(y)
xf = linspace(x(1), x(length(x)),inc);
% max(xf)
% min(xf)
plot(xf,fh(bhat,xf),'linewidth',1,'color',[1,0,0]);
hold off
xlabel('r [mm])')
ylabel('Sag [mm]')
legend('Raw Sag Data','Sag Equation Fit','location','Best')
% Here the error between the numerical model and the fit is plotted
subplot(2,1,2)
hold on
plot(r,y - fh(bhat,xf))
plot(r,0,'k')
xlabel('r [mm])')
ylabel('Raw - Fit')
title('Error in fit')
vv = (bhat(1))^-1
close all
figure(6)
hold on
err = y - fh(bhat,xf);
plot(r_p,err)
oo = err(inc);
plot(r_p,0,'k')
xlabel('r [mm])')
ylabel('Raw - Fit')
title('Error in fit')
figure(7)
hold on
xlim([min(r) max(r)])
plot(r,n)
plot(r,linspace(n_PMMA,n_PMMA,inc),'k')
plot(r,linspace(n_PS,n_PS,inc),'k')
191
Appendix K. Spectral data for thermal interferometer fused silica beamsplitter
Beamsplitter’s anitflection side coating:
Beamsplitter’s beamsplittering coating:
192
Appendix L. Thermal interferometer: data acquisition code (MATLAB)
This is the main set of code is used to acquire phase maps. Note that InitCamera, InitState, and DefineRegion are separate subfunctions also included in this Appendix.
%% Initial Setup % % This code will initialize the camera as well as the piezo stage.
If % the piezo stage does not turn on and crashes matlab instead, try % unplugging and replugging in the power cable clear all addpath(genpath('cC:\Data\ThermalInterferometerToolbox')) % This setting determines the kind of test. A value of 1 will ask for % information about the corrsponding region. FlagReflect = 0; FlagTransmit = 0; FlagGRIN = 0; % Do you want to look at the wedge in the sample? FlagWedge = 0;
%Important information about the test and sample waveLength = .6328; %microns pressure = 1000.*98.9259776400000; %pascals relativeHumidity = 26.5; %percent thickNess = 5000; %micronsc inDexSample = 1.4935; %Sample index at ambient temperature wavenumber = 2*pi/waveLength;
%Start up the camera! [src vidobj] = InitCamera(30);
%Start up the Stage! htrans = InitStage;
%% Crop the region that is being watched by the camera! % A figure will appear, drag and right click to select the region you
would % like to record interferograms for.
vidobj.ROIPosition = DefineRegion(vidobj,2); %% Single shot - Code to get phase shift data in first place (move
piezo, take pics) % This code records intensity values of an averaged region of the
camera. % This should trace out a sine wave for an applied linear voltage. This % data will be used in the calibration the instrument. pause(1) tic clear vlt_stps V_app V_ramp clbrtn_frms
for vlt_stps = 1:201; % Number of voltage
193 vlt_stps V_app = 4+.05.*vlt_stps./2; % applied voltage per step V_ramp(vlt_stps) = V_app; % generate voltage ramp htrans.SetVoltOutput(0,V_app); pause(.02) clbrtn_frms(:,:,vlt_stps) = getsnapshot(vidobj); end toc
%% Plot acquired linescan of a single pixel intensity vs. applied
voltage % % This code will give us our calibration of the piezo stage to ramp
voltage % through a single period. This is necessary for the least-squares
phase % shifting algorithm we are using (Malacara). This code takes the
intensity % values from the previous section of code, plots them, then fits them
to a % sine wave, and finally designates a series of voltage values (with
black % circles) for stepping as we carry out the phase shifting algorithm.
Note % that these voltage values should span a single period of the sine
wave.
clear abc pxl_R pxl_C pxl_int amp1 shftd_pxl_int; % clbrtn_frms = double(clbrtn_frms); pxl_R = 100; pxl_C = 100; sqr_sz = 2; for abc = 1:length(V_ramp); pxl_int(abc) =
mean(mean(clbrtn_frms(pxl_R:pxl_R+sqr_sz,pxl_C:pxl_C+sqr_sz,abc))); end;
amp1 = 0.5.*(max(pxl_int) - min(pxl_int)); shftd_pxl_int = pxl_int - amp1 - min(pxl_int);
close(figure(3)) % figure(3) % plot(V_ramp,shftd_pxl_int,'r.') grid on
% close(figure(4));figure(4); hold on; box on; xx_data = V_ramp; yy_data=shftd_pxl_int; % plot(xx_data, yy_data, '.b')
d = xx_data; Irrad = yy_data; dmax = d(find(Irrad == max(Irrad),1,'first'));
194
Ifft=ifft(Irrad-mean(Irrad)); fftmax = find(abs(Ifft(1:fix(length(d)/2))) ==
max(abs(Ifft(1:fix(length(d)/2)))),1,'first'); dfx=(1/abs(d(2)-d(1)))/length(d); fxguess = dfx*(fftmax-1)*2*pi; phiguess = -fxguess*dmax;
AA = amp1; BB = fxguess; CC = phiguess; DD = 0; plot(xx_data, AA .* cos(BB .* xx_data + CC) + DD, '-g') % <-- Adjust
these title('Manual Fit')
% Now feed the starting point to Matlab myfit = fittype('a*cos(b*x+c)+d', 'independent', 'x', 'dependent',
'y'); myopt = fitoptions('Method', 'NonlinearLeastSquares'); myopt.StartPoint = [AA BB CC DD]; % <-- Numbers from Line 9 [f, g] = fit(xx_data', yy_data', myfit, myopt); yfit = f.a .* cos(f.b .* xx_data + f.c) + f.d;
% Plot James' data and the regression result close(figure(12));figure(12) hold on; box on plot(xx_data, yy_data, '.b') plot(xx_data, yfit, '-r') title('Matlab Fit')
V_fit = linspace(xx_data(1),xx_data(length(xx_data)),10001); yy_fit = f.a .* cos(f.b .* V_fit + f.c) + f.d; [ee V_max] = max(yy_fit(1:round(length(yy_fit)./2))); V1 = V_fit(V_max); V2 = V1 + 2.*pi./f.b; hold on; box on; num_st =4; % plot(linspace(V1,V2,num_st),linspace(0,0,num_st),'go') % yyfit = f.a .* cos(f.b .* linspace(V1,V2,num_st) + f.c) + f.d; PSramp = linspace(V1,V2,num_st+1); PSramp = PSramp(2:num_st+1); plot(PSramp,f.a .* cos(f.b .* PSramp + f.c) + f.d,'ko') % max(PSramp)-min(PSramp) % % % V2-V1 grid on PSramp(end)-PSramp(1)
%% Acquire phasemaps % % cThis code carries out the phase shifting algorithm by first moving
the % piezo stage to the first specified value in the vector 'PSramp',
captures
195 % a snapshot of the interferogram, saves it within the matrix 'snap',
then % moves the piezo stage to the second value within PSramp, captures a % second image, etc. until the matrix 'csnap' is a 3D matrix with the % dimensions of the camera aperature x the number of steps speficied by % 'num_st' in the previous section of code (25 seems to work well).
Once % 'snap' is completed, the interferograms can be processed to generate
a % phasemap (variable 'phasetemp') using the Malacara algorithm. The % phasemaps can be calculated during a data run or afterwards from the % 'snap' matrics. Both options are available in this code. To only
capture % a series of interferograms (only the 'snap' matricses), use lines
191-195 % and comment out 198-215. To generate phasemaps use lines 198-215 and % comment out 191-195 clear allfrm_allstp snap FS = stoploop; xyz = 0; while ~FS.Stop() % tic %_1 Name the count for the phase map xyz = xyz + 1; if xyz < 10; cnt = ['00000' num2str(xyz)]; elseif (xyz > 9) && (xyz < 100) cnt = ['0000' num2str(xyz)]; elseif (xyz > 99) && (xyz < 1000) cnt = ['000' num2str(xyz)]; elseif (xyz > 999) && (xyz < 10000) cnt = ['00' num2str(xyz)]; elseif (xyz > 9999) && (xyz < 100000) cnt = ['0' num2str(xyz)]; else cnt = num2str(xyz); end
clk = clock; % seco = num2str(clk(6)./60); tmstmp = [cnt '_' num2str(clk(4)) 'h' num2str(clk(5)) 'm_'
num2str(clk(2)) '_' num2str(clk(3)) '_' num2str(clk(1))]; %_1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% xyz % frng_snp = getsnapshot(vidobj); % save([cnt '_snap'],'frng_snp','-v7.3') %_2 Main code for ijk = 1:length(PSramp) % Number of voltage steps htrans.SetVoltOutput(0,PSramp(ijk)); pause(.03) % for lmn = 1 % number of images you are averaging % onefrm_stp = getsnapshot(vidobj); % store a single frame % allfrm_stp(:,:,lmn)=onefrm_stp; % Frames to be averaged
196 % clear onefrm_stp % end % avfrm_stp=mean(allfrm_stp,3); snap(:,:,ijk) = getsnapshot(vidobj); % clear avfrm_stp end % % %
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Code for saving snapshots % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % save(tmstmp,'snap','-v7.3') % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Code for makeing phasemaps, variable 'phasetemp' % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % del = (1:length(PSramp)).*2.*pi./(length(PSramp)); I = double(snap);%(:,:,1:PSramp); N = 0; D = 0; for qrs = 1:(length(del)) N = N - I(:,:,qrs).*sin(del(qrs)); N=double(N); D = D + I(:,:,qrs).*cos(del(qrs)); D=double(D); end phasetemp = atan2(N,D);
save(tmstmp,'phasetemp','-v7.3') % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%
pause(.05) end
%% % % Code for examining a generated phasemaps. This will plot whatever
is % saved as variable 'phasetemp'. It will show an image of the phasemaps
as % well as both wrapped and unwrapped spatial cutthroughs of the
variable in % both the horizontal and vertical directions. These cuttrhoughs can be % specified by the user using commands hcut and vcut (lines 236-237) close(figure(3)) figure(3) subplot(2,4,[1:2 5:6]) % surf(phasetemp_fft) imagesc(phasetemp)
197 % ylim([0 550]) view(2) shading interp colorbar hold on vcut = 125; hcut = 50;%55; line([vcut vcut],[0 size(phasetemp,1)],[4 4],'LineWidth',1,'Color','k') line([0 size(phasetemp,2)],[hcut hcut],[4 4],'LineWidth',1,'Color','k') axis equal
subplot(2,4,3) plot((phasetemp(:,vcut)),'k.') title('Vertical') grid on xlim([0 size(phasetemp,1)]) subplot(2,4,7) plot((phasetemp(hcut,:)),'k.') title('Horizontal') grid on xlim([0 size(phasetemp,2)])
subplot(2,4,4) v_unwrap = unwrap(phasetemp(:,vcut)'); vymin = 1; vymax = size(phasetemp,1); vxdata = vymin:vymax; vydata = v_unwrap(vymin:vymax)./(2.*pi); myfit = fittype('a*x+b', 'independent', 'x', 'dependent', 'y'); myopt = fitoptions('Method', 'NonlinearLeastSquares'); myopt.StartPoint = [-1 20]; % [f, g] = fit(vxdata', vydata', myfit, myopt); plot(vxdata,vydata,'r.');hold
on;plot(vxdata,f.a.*vxdata+f.b,'LineWidth',2) xlabel('Pixel') ylabel('Fringes') grid on title('Vertical') xlim([0 10+size(phasetemp,1)])
subplot(2,4,8) h_unwrap = unwrap(phasetemp(hcut,:)); hymin = 1; hymax = size(phasetemp,2); hxdata = hymin:hymax; hydata = h_unwrap(hymin:hymax)./(2.*pi); myfit = fittype('a*x+b', 'independent', 'x', 'dependent', 'y'); myopt = fitoptions('Method', 'NonlinearLeastSquares'); myopt.StartPoint = [-1 20]; % [f, g] = fit(hxdata', hydata', myfit, myopt); plot(hxdata,hydata,'r.');hold
on;plot(hxdata,f.a.*hxdata+f.b,'LineWidth',2) xlabel('Pixel') ylabel('Fringes') grid on title('Horizontal') xlim([0 10+size(phasetemp,2)])
198
Function InitCamera:
function [src vidobj] = InitCamera(framerate,exposuretime) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % InitCamera v .1 Jan/14/2016 % Bill Green, James Corsetti % Description: Starts up the CMOS sensor used for capturing phasemaps
% src - a structure that contains video settings such as frame rate, % exposure time, etc.
% vidobj - a structure that references the video feed
% framerate - provided in FPS. This script will provide a default if
not % specified
% exposuretime - prodivded in seconds. A default is provided if not
specified. % This script will terminate if the exposure time is longer than the
time between frames. imaqreset
if nargin < 2, exposuretime = []; end if nargin < 1, framerate = []; end
% supply default parameters if isempty(framerate), framerate = 30; end if isempty(exposuretime), exposuretime = .015736; end
if exist('vidobj') delete(vidobj) clear vidobj end
vidobj=videoinput('pointgrey', 1, 'F7_Raw8_640x512_Mode1'); viewport = preview(vidobj); vidobj.ReturnedColorspace='grayscale'; src = getselectedsource(vidobj); vidobj.ROIPosition = [0 0 640 512];
disp('Please wait 10 seconds...')
pause(2) src.ExposureMode = 'Manual'; pause(2) maxfps = 120; src.FrameRatePercentageMode = 'Manual';
199 src.FrameRatePercentage = 100*framerate/maxfps; pause(2) src.GainMode = 'Manual'; pause(2) src.ShutterMode = 'Manual'; if exposuretime > 1/framerate exposuretime = 1/(1.1*framerate); disp('Exposure time exceeds time inbetween frames, shutterspeed
changed accordingly') end src.Shutter = exposuretime; pause(2) disp('Camera Initialized Successfully')
Function InitCamera:
function htrans = InitStage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % InitStage v .1 Jan/11/2016 % Bill Green, James Corsetti % Description: Starts up the Thorlabs piezometric stage
% htrans - a handle for the stage, used to set the voltage
disp('Initializing Piezometric stage...') close(figure(2)) fig = figure(2); % Define figure for stage control activex GUI % set(fig,'Position',[200 200 1100 400]);
htrans = actxcontrol('MGPIEZO.MGPiezoCtrl.1',[0 0 549 400],fig);%
Define control for translation stage set(htrans,'HWSerialNum',81834010);% Determine the serial number of the
translation stage driver htrans.StartCtrl;% Start control disp('Phase Shifting Stage Initialized Successfully.')
200
Function DefineRegion:
function [cropCoords] = DefineRegion(image,imagetype) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DefineRegion v .1 Jan/15/2016 % Bill Green, James Corsetti % Description: Grabs a screenshot from the camera, and allows you % to interactively select a region to subsample.
% image - The image that will be looked at in order to crop a region.
This can % either be the handle for the video feed, or a static image of the
phase % map. Designate this with the imagetype flag.
% imagetype - a flag that gets a screenshot differently depending on % whether you are pulling from a static image or the video feed. % imagetype = 1, use a static screenshot % imagetype = 2, use the video feed
% cropCoords - vector containing the information about the cropped % region: [xorigin yorigin width height]
switch imagetype case 1 imagetocrop = image;
case 2 imagetocrop = getsnapshot(image); end
fig = figure; [sampleRegion,cropCoords] = imcrop(imagetocrop); %imshow(sampleRegion); %crop = 'This is the region you have selected'; cropCoords = floor(cropCoords); close(fig) %j = text(.1*cropCoords(3),.1*cropCoords(4), crop,
'FontSize',14,'Color','r');
201
Appendix M. Thermal interferometer: data analysis code (MATLAB)
% Code for analyzing CTE and dn/dT clear timetab TAtab Ptab RHtab nairtab TStab timetab = excel_3_30_2017(:,1); Ptab = 1000.*98.9259776400000; RHtab = 26.5; TAtab = excel_3_30_2017(:,2); % air temperature TStab = (excel_3_30_2017(:,3) + excel_3_30_2017(:,4) +
excel_3_30_2017(:,5))./3; % sample temp nairtab = air_index_calc(TAtab,Ptab,RHtab,.6328); k = 2*pi/.6328; %% % Plot air and sample temp. vs time close(figure(154));figure(154) plot(excel_3_30_2017(:,1),TAtab,'m--') grid on; hold on plot(excel_3_30_2017(:,1),excel_3_30_2017(:,3),'r') plot(excel_3_30_2017(:,1),excel_3_30_2017(:,4),'g') plot(excel_3_30_2017(:,1),excel_3_30_2017(:,5),'b') plot(excel_3_30_2017(:,1),TStab,'k')
%% % Store all phasemaps in folder as variable D clear D vv fileloc = 'I:\2017_03_30_Syntec_5_&_7_p5_to35ramp_13hrs\PhaseMaps'; D = dir(fileloc); [vv,idx] = sort([D.datenum]); %% % Mark pixels in different regions to extract wrapped phase as a
function % of count close(figure(32)) figure(32); imagesc(phasetemp(:,:)); colorbar colormap summer axis equal hold on MKs = 6;
bg_ww_sft = 200; bg_ww = what_bg_pxls; ref_ww = what_ref_pxls; trans_ww = what_trans_pxls; regi4_ww = 1; reg55_ww =1;
% % % % % % % % % % % % % % % % % % Background region
202 % % % % % % % % % % % % % % % % % clear rows_bg cols_bg X_bg Y_bg X_bg_R Y_bg_R bg_cnt bg_pts
rows_bg=[50:5:130]; cols_bg=[132:5:192]'; [X_bg, Y_bg] = meshgrid(rows_bg,cols_bg); X_bg_R = reshape(X_bg,length(rows_bg)*length(cols_bg),1); Y_bg_R = reshape(Y_bg,length(rows_bg)*length(cols_bg),1);
for bg_cnt = 1:length(rows_bg)*length(cols_bg); bg_pts(bg_cnt,1) = X_bg_R(bg_cnt); bg_pts(bg_cnt,2) = Y_bg_R(bg_cnt); end
for bg_cnt2 = 1:size(bg_pts,1)
plot(bg_pts(bg_cnt2,2),bg_pts(bg_cnt2,1),'ro','MarkerSize',MKs,'MarkerE
dgeColor','k','MarkerFaceColor',[0 bg_cnt2./size(bg_pts,1) 0]) end
plot(bg_pts(bg_ww,2),bg_pts(bg_ww,1),'ro','MarkerSize',MKs,'MarkerEdgeC
olor','k','MarkerFaceColor',[1 0 0])
bg = [mean(bg_pts(bg_ww,1)) mean(bg_pts(bg_ww,2))];
text(bg(2)+5,bg(1),'BG') plot(bg(2),bg(1),'kx') plot(bg(2),bg(1),'ko')
% % % % % % % % % % % % % % % % % % % % % % % % % Reflection region (CTE of sample 1) % % % % % % % % % % % % % % % % % % % % % clear ref rows_ref cols_ref X_ref Y_ref X_ref_R Y_ref_R ref_cnt ref_pts
hold on rows_ref=[20:5:70]; cols_ref=[25:5:95]'; [X_ref, Y_ref] = meshgrid(rows_ref,cols_ref); X_ref_R = reshape(X_ref,length(rows_ref)*length(cols_ref),1); Y_ref_R = reshape(Y_ref,length(rows_ref)*length(cols_ref),1);
for ref_cnt = 1:length(rows_ref)*length(cols_ref); ref_pts(ref_cnt,1) = X_ref_R(ref_cnt); ref_pts(ref_cnt,2) = Y_ref_R(ref_cnt); end
for ref_cnt2 = 1:size(ref_pts,1)
203
plot(ref_pts(ref_cnt2,2),ref_pts(ref_cnt2,1),'ro','MarkerSize',MKs,'Mar
kerEdgeColor','k','MarkerFaceColor',[ref_cnt2./size(ref_pts,1) 0 1]) end
plot(ref_pts(ref_ww,2),ref_pts(ref_ww,1),'ro','MarkerSize',MKs,'MarkerE
dgeColor','k','MarkerFaceColor',[1 0 0]) ref = [mean(ref_pts(ref_ww,1)) mean(ref_pts(ref_ww,2))]; text(ref(2)+5,ref(1),'ref') plot(ref(2),ref(1),'kx') plot(ref(2),ref(1),'ko')
% % % % % % % % % % % % % % % % % % % % Transmission region (dn/dT of sample 1) % % % % % % % % % % % % % % % % % % % clear trans rows_trans cols_trans X_trans Y_trans X_trans_R Y_trans_R
trans_cnt trans_pts
hold on rows_trans=[92:5:142]; cols_trans=[25:5:95]'; [X_trans, Y_trans] = meshgrid(rows_trans,cols_trans); X_trans_R = reshape(X_trans,length(rows_trans)*length(cols_trans),1); Y_trans_R = reshape(Y_trans,length(rows_trans)*length(cols_trans),1);
for trans_cnt = 1:length(rows_trans)*length(cols_trans); trans_pts(trans_cnt,1) = X_trans_R(trans_cnt); trans_pts(trans_cnt,2) = Y_trans_R(trans_cnt); end
for trans_cnt2 = 1:size(trans_pts,1)
plot(trans_pts(trans_cnt2,2),trans_pts(trans_cnt2,1),'ro','MarkerSize',
MKs,'MarkerEdgeColor','k','MarkerFaceColor',[1-
trans_cnt2./size(trans_pts,1) 1-trans_cnt2./size(trans_pts,1) 1-
trans_cnt2./size(trans_pts,1)]) end
plot(trans_pts(trans_ww,2),trans_pts(trans_ww,1),'ro','MarkerSize',MKs,
'MarkerEdgeColor','k','MarkerFaceColor',[1 0 0]) trans = [mean(trans_pts(trans_ww,1)) mean(trans_pts(trans_ww,2))]; text(trans(2)+5,trans(1),'trans') plot(trans(2),trans(1),'kx') plot(trans(2),trans(1),'ko')
xlabel('Pixels') ylabel('Pixels')
% % % % % % % % % % % % % % % % % % % % % % % region 4 region (CTE of sample 2)
204 % % % % % % % % % % % % % % % % % % % % % % clear regi4 rows_regi4 cols_regi4 X_regi4 Y_regi4 X_regi4_R Y_regi4_R
regi4_cnt regi4_pts
hold on rows_regi4=[20:5:70]; cols_regi4=[230:5:300]'; [X_regi4, Y_regi4] = meshgrid(rows_regi4,cols_regi4); X_regi4_R = reshape(X_regi4,length(rows_regi4)*length(cols_regi4),1); Y_regi4_R = reshape(Y_regi4,length(rows_regi4)*length(cols_regi4),1);
for regi4_cnt = 1:length(rows_regi4)*length(cols_regi4); regi4_pts(regi4_cnt,1) = X_regi4_R(regi4_cnt); regi4_pts(regi4_cnt,2) = Y_regi4_R(regi4_cnt); end
for regi4_cnt2 = 1:size(regi4_pts,1)
plot(regi4_pts(regi4_cnt2,2),regi4_pts(regi4_cnt2,1),'ro','MarkerSize',
MKs,'MarkerEdgeColor','k','MarkerFaceColor',[0 0
regi4_cnt2./size(regi4_pts,1)]) end
plot(regi4_pts(regi4_ww,2),regi4_pts(regi4_ww,1),'ro','MarkerSize',MKs,
'MarkerEdgeColor','k','MarkerFaceColor',[1 0 0]) regi4 = [regi4_pts(regi4_ww,1) regi4_pts(regi4_ww,2)]; text(regi4(2)+5,regi4(1),'SS') plot(regi4(2),regi4(1),'kx') plot(regi4(2),regi4(1),'ko')
xlabel('Pixels') ylabel('Pixels')
% % % % % % % % % % % % % % % % % % % % % % reg55mission region (dn/dT of sample 2) % % % % % % % % % % % % % % % % % % % % % clear reg55 rows_reg55 cols_reg55 X_reg55 Y_reg55 X_reg55_R Y_reg55_R
reg55_cnt reg55_pts
hold on rows_reg55=[92:5:142]; cols_reg55=[230:5:300]'; [X_reg55, Y_reg55] = meshgrid(rows_reg55,cols_reg55); X_reg55_R = reshape(X_reg55,length(rows_reg55)*length(cols_reg55),1); Y_reg55_R = reshape(Y_reg55,length(rows_reg55)*length(cols_reg55),1);
for reg55_cnt = 1:length(rows_reg55)*length(cols_reg55); reg55_pts(reg55_cnt,1) = X_reg55_R(reg55_cnt); reg55_pts(reg55_cnt,2) = Y_reg55_R(reg55_cnt); end
205 for reg55_cnt2 = 1:size(reg55_pts,1)
plot(reg55_pts(reg55_cnt2,2),reg55_pts(reg55_cnt2,1),'ro','MarkerSize',
MKs,'MarkerEdgeColor','k','MarkerFaceColor',[1-
reg55_cnt2./size(reg55_pts,1) 1-reg55_cnt2./size(reg55_pts,1) 1-
reg55_cnt2./size(reg55_pts,1)]) end
plot(reg55_pts(reg55_ww,2),reg55_pts(reg55_ww,1),'ro','MarkerSize',MKs,
'MarkerEdgeColor','k','MarkerFaceColor',[1 0 0]) reg55 = [reg55_pts(reg55_ww,1) reg55_pts(reg55_ww,2)]; text(reg55(2)+5,reg55(1),'reg55') plot(reg55(2),reg55(1),'kx') plot(reg55(2),reg55(1),'ko')
xlabel('Pixels') ylabel('Pixels')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Background for fitting tilt clear backgroundrows backgroundcols backgroundrows = 60:120; backgroundcols = 132:192; line([backgroundcols(1)
backgroundcols(length(backgroundcols))],[backgroundrows(1)
backgroundrows(1)],'Color',[1 1 1],'LineWidth',2.5) line([backgroundcols(1)
backgroundcols(length(backgroundcols))],[backgroundrows(length(backgrou
ndrows)) backgroundrows(length(backgroundrows))],'Color',[1 1
1],'LineWidth',2.5) line([backgroundcols(1) backgroundcols(1)],[backgroundrows(1)
backgroundrows(length(backgroundrows))],'Color',[1 1
1],'LineWidth',2.5) line([backgroundcols(length(backgroundcols))
backgroundcols(length(backgroundcols))],[backgroundrows(1)
backgroundrows(length(backgroundrows))],'Color',[1 1
1],'LineWidth',2.5)
%% clear time TAmeas TSmeas RHmeas nair refphase bgphase transphase
regi4phase xtilt ytilt %% clear bgphase_unfil bgphase_fft refphase_unfil refphase_fft
transphase_unfil transphase_fft %% % f1 and f2 are the bounds of the phasemaps f1 =3; f2 =70403; tic for frame = f1:1:f2; frame PM = D(frame).name;
206 load(PM); clear phasetemp_unfil phasetemp_fft phasetemp_combo
% The 3 sections below extract different information from system and
should % be run separately from f1 to f2
% % % % This code calculates time, nair, temp as a function of phasemap % 1 time(frame-f1+1) = D(frame).datenum; % 1 TAmeas(frame-f1+1) = interp1(timetab,TAtab,time(frame-
f1+1));%,'cubic','extrap'); % 1 TSmeas(frame-f1+1) = interp1(timetab,TStab,time(frame-
f1+1));%,'cubic','extrap'); % 1 RHmeas(frame-f1+1) = RHtab; % 1 nair(frame-f1+1) = interp1(timetab,nairtab,time(frame-
f1+1));%,'cubic','extrap'); % % % %
% % % % This code is used to fit background tilt % 2 b = puma_ho(phasetemp(backgroundrows,backgroundcols),2); % 2 % Fit 2-D tilt to background data % 2 [x1,y1] = meshgrid(backgroundcols,backgroundrows); % 2 model_function0 = @(vars,r) vars(3) + vars(1)*r(:,1) +
vars(2)*r(:,2); % fit background tilt to a plane % 2 guess = [(b(round(size(b,1)/2),end)-
b(round(size(b,1)/2),1))/(backgroundcols(end)-backgroundcols(1))
(b(end,round(size(b,2)/2))-
b(1,round(size(b,2)/2)))/(backgroundrows(end)-backgroundrows(1))
mean(mean(b))]; % guess is taking endpoints and finding slope % 2 [varfit1,resid,J,Sigma] = nlinfit([x1(:)
y1(:)],b(:),model_function0,guess,statset('TolFun',1e-30,'TolX',1e-
20,'MaxIter',1000)); % 2 xtilt(frame-f1+1) = varfit1(1);%/(2*nair(frame-f1+1)*k); %
putting the tilt into units of waves % 2 ytilt(frame-f1+1) = varfit1(2);%/(2*nair(frame-f1+1)*k); % % % %
% % % % Extract wrapped phase vs. pixel for each region of interest % 3 for bgcnt = 1:size(bg_pts,1); % 3 bgphase(frame-
f1+1,bgcnt)=phasetemp(bg_pts(bgcnt,1),bg_pts(bgcnt,2)); % 3 end % 3 % 3 for refcnt = 1:size(ref_pts,1); % 3 refphase(frame-
f1+1,refcnt)=phasetemp(ref_pts(refcnt,1),ref_pts(refcnt,2)); % 3 end % 3 % 3 for transcnt = 1:size(trans_pts,1); % 3 transphase(frame-
f1+1,transcnt)=phasetemp(trans_pts(transcnt,1),trans_pts(transcnt,2)); % 3 end % 3
207 % 3 for regi4cnt = 1:size(regi4_pts,1); % 3 regi4phase(frame-
f1+1,regi4cnt)=phasetemp(regi4_pts(regi4cnt,1),regi4_pts(regi4cnt,2)); % 3 end % 3 % 3 for reg55cnt = 1:size(reg55_pts,1); % 3 reg55phase(frame-
f1+1,reg55cnt)=phasetemp(reg55_pts(reg55cnt,1),reg55_pts(reg55cnt,2)); % 3 end % % % %
end toc %% % scale tilt by wavenumber xtilt=(xtilt_interp./(2.*nair.*k)); ytilt=(ytilt_interp./(2.*nair.*k)); disp('done') %% % fit tilt of every tenth phasemap with spline. That gives an x and y
tilt % for each phasemap xtilt_interp =
interp1(1:10:70401,xtilt(1:10:70401),1:1:70401,'spline'); ytilt_interp =
interp1(1:10:70401,ytilt(1:10:70401),1:1:70401,'spline'); figure;plot(xtilt_interp,'b.'); hold on; plot(1:10:70401,xtilt,'ro') figure;plot(ytilt_interp,'b.'); hold on; plot(1:10:70401,ytilt,'ro')
%% % This code unwraps bgphase, refphase, transphase and stores them as % variables: bgphaseuw_shft, refphaseuw_shft, and transphaseuw_shft
% Unwrap background phase clear bgphaseuw bguw_cnt bgphaseuw = zeros(size(bgphase,1),size(bgphase,2)); for bguw_cnt = 1:size(bgphase,2); bgphaseuw(:,bguw_cnt) = unwrap(bgphase(:,bguw_cnt)); % plot(bgphaseuw(:,bguw_cnt),'Color',[0 1 0]);%%[0
bguw_cnt./size(bg_pts,1) 0]) % hold on end
% Unwrap reflection phase clear refphaseuw refuw_cnt refphaseuw = zeros(size(refphase,1),size(refphase,2)); for refuw_cnt = 1:size(refphase,2); refphaseuw(:,refuw_cnt) = unwrap(refphase(:,refuw_cnt));
208 % plot(refphaseuw(:,refuw_cnt),'Color',[1 0
1])%[refuw_cnt./size(ref_pts,1) 0 1]) % hold on end
% Unwrap transmission phase clear transphaseuw transuw_cnt transphaseuw = zeros(size(transphase,1),size(transphase,2)); for transuw_cnt = 1:size(transphase,2); transphaseuw(:,transuw_cnt) = unwrap(transphase(:,transuw_cnt)); % plot(transphaseuw(:,transuw_cnt),'Color',[0 0 0]);%[1-
transuw_cnt./size(trans_pts,1) 1-transuw_cnt./size(trans_pts,1) 1-
transuw_cnt./size(trans_pts,1)]) % hold on end
% Shifted background phase clear bgphaseuw_shft close(figure(40)) figure(40) subplot(2,2,1) shft = 1; aaa = shft; bbb = 72000;
clear bgphaseuw_shft bgphaseuw_shft = zeros(size(bgphaseuw,1),size(bgphaseuw,2)); for bguw_shft_cnt = 1:size(bgphaseuw,2); bgphaseuw_shft(:,bguw_shft_cnt) = bgphaseuw(:,bguw_shft_cnt)-
bgphaseuw(shft,bguw_shft_cnt); plot(bgphaseuw_shft(:,bguw_shft_cnt),'Color',[0
bguw_shft_cnt./size(bgphaseuw,2) 0]) hold on end plot(mean(bgphaseuw_shft,2),'k','LineWidth',2.5) grid on xlabel('Count,time') ylabel('Fringes') xlim([aaa bbb])
% Shifted reference region clear refphaseuw_shft subplot(2,2,2) clear refphaseuw_shft refphaseuw_shft = zeros(size(refphaseuw,1),size(refphaseuw,2)); for refuw_shft_cnt = 1:size(refphaseuw,2); refphaseuw_shft(:,refuw_shft_cnt) = refphaseuw(:,refuw_shft_cnt)-
refphaseuw(shft,refuw_shft_cnt);
plot(refphaseuw_shft(:,refuw_shft_cnt),'Color',[refuw_shft_cnt./size(re
fphaseuw,2) 0 1]) hold on end
209 grid on plot(mean(refphaseuw_shft,2),'k','LineWidth',2.5) xlabel('Count,time') ylabel('Fringes') xlim([aaa bbb])
% Shifted transmission region clear transphaseuw_shft subplot(2,2,3) clear transphaseuw_shft transphaseuw_shft = zeros(size(transphaseuw,1),size(transphaseuw,2)); for transuw_shft_cnt = 1:size(transphaseuw,2); transphaseuw_shft(:,transuw_shft_cnt) =
transphaseuw(:,transuw_shft_cnt)-transphaseuw(shft,transuw_shft_cnt); plot(transphaseuw_shft(:,transuw_shft_cnt),'Color',[1-
transuw_shft_cnt./size(transphaseuw,2) 1-
transuw_shft_cnt./size(transphaseuw,2) 1-
transuw_shft_cnt./size(transphaseuw,2)]) hold on end grid on xlabel('Count,time') ylabel('Fringes') xlim([aaa bbb]) clear bgphaseuw refphaseuw transphaseuw
%% % Checkderiv code. This identifies the "best behaved" pixels in terms
of % discontinuities in the unwrapped phase and stores those values as % what_bg_pxls, what_ref_pxls, and what_trans_pxls to be used in code
above % when speficying pixels of interest close(figure(57));figure(57); clear what_bg_pxls bg_diff_cnt chk_bg_diff_met chk_bg_diff clear what_ref_pxls ref_diff_cnt chk_ref_diff_met chk_ref_diff clear what_trans_pxls trans_diff_cnt chk_trans_diff_met chk_trans_diff
thrsh_bg=3.0138; thrsh_ref=3.081;%2.752;%2.96; thrsh_trans=2.98;%2.416; stopshft = 53870;%40000; ofst = 1;%11500;
subplot(3,1,1) bg_diff_cnt = 1; for pixels = 1:size(bg_pts,1); chk_bg_diff =
sort(abs(diff(bgphaseuw_shft(shft+ofst:stopshft,pixels))),'descend'); chk_bg_diff_met(pixels) = mean(chk_bg_diff(1:6)); if chk_bg_diff_met(pixels) < thrsh_bg what_bg_pxls(bg_diff_cnt) = pixels; bg_diff_cnt = bg_diff_cnt+1;
210 end clear chk_bg_diff end plot(chk_bg_diff_met,'go') hold on plot(what_bg_pxls,chk_bg_diff_met(what_bg_pxls),'gx') plot(what_bg_pxls,chk_bg_diff_met(what_bg_pxls),'ko')
subplot(3,1,2) ref_diff_cnt = 1; for pixels = 1:size(ref_pts,1); chk_ref_diff =
sort(abs(diff(refphaseuw_shft(shft+ofst:stopshft,pixels))),'descend'); chk_ref_diff_met(pixels) = mean(chk_ref_diff(1:6)); if chk_ref_diff_met(pixels) < thrsh_ref what_ref_pxls(ref_diff_cnt) = pixels; ref_diff_cnt = ref_diff_cnt+1; end clear chk_ref_diff end plot(chk_ref_diff_met,'mo') hold on plot(what_ref_pxls,chk_ref_diff_met(what_ref_pxls),'mx') plot(what_ref_pxls,chk_ref_diff_met(what_ref_pxls),'ko')
subplot(3,1,3) trans_diff_cnt = 1; for pixels = 1:size(trans_pts,1); chk_trans_diff =
sort(diff(transphaseuw_shft(shft:stopshft,pixels)),'descend'); chk_trans_diff_met(pixels) = mean(chk_trans_diff(1:6)); if chk_trans_diff_met(pixels) < thrsh_trans what_trans_pxls(trans_diff_cnt) = pixels; trans_diff_cnt = trans_diff_cnt+1; end clear chk_trans_diff end plot(chk_trans_diff_met,'ko') hold on plot(what_trans_pxls,chk_trans_diff_met(what_trans_pxls),'rx') plot(what_trans_pxls,chk_trans_diff_met(what_trans_pxls),'ro')
%% % The following section of code calculates the CTE and dn/dT of the
sample
clear startframe endframe % these values are defined earlier in the code. The unwrapped phase
values % of each region bgphaseuw1 = mean(bgphaseuw_shft(:,bg_ww),2); refphaseuw1 = mean(refphaseuw_shft(:,ref_ww),2); transphaseuw1 = mean(transphaseuw_shft(:,trans_ww),2);
211
dOPD1 = (bgphaseuw1)./k; dOPD2 = (refphaseuw1)./k; dOPD3 = (transphaseuw1)./k;
close(figure(54));figure(54);plot(dOPD1,'g');hold
on;plot(dOPD2,'m');plot(dOPD3,'k');grid on close(figure(55));figure(55);plot(TSmeas,dOPD1,'g');hold
on;plot(TSmeas,dOPD2,'m');grid on;plot(TSmeas,dOPD3,'k');grid on xlabel('Temp') ylabel('Change in OPD ({\mu}m)'); legend('Background','Sample','Transmission')
startframe=shft; endframe = length(TSmeas); dOPD1_bnd = dOPD1(startframe:endframe)'; dOPD2_bnd = dOPD2(startframe:endframe)'; dOPD3_bnd = dOPD3(startframe:endframe)';
nair_bnd = nair(startframe:endframe); xtilt_bnd = xtilt(startframe:endframe); ytilt_bnd = ytilt(startframe:endframe); TAmeas_bnd = TAmeas(startframe:endframe); TSmeas_bnd = TSmeas(startframe:endframe); time_bnd = time(startframe:endframe);
% base thickness (in microns) and index of sample t0=2495; n0=1.52869493454192;
% calculated thickness and index of the sample over time tsamp = (nair_bnd(1).*t0 + 0.5.*(dOPD1_bnd-dOPD2_bnd) -
1.*((nair_bnd(1).*xtilt_bnd(1) - nair_bnd.*xtilt_bnd).*(ref(2)-bg(2)) +
(nair_bnd(1).*ytilt_bnd(1) - nair_bnd.*ytilt_bnd).*(ref(1)-
bg(1))))./nair_bnd; nsamp = (n0.*t0 + .5.*(dOPD3_bnd-dOPD2_bnd) + 1.*((nair_bnd.*xtilt_bnd
- nair_bnd(1).*xtilt_bnd(1)).*(ref(2) - trans(2)) +
(nair_bnd.*ytilt_bnd - nair_bnd(1).*ytilt_bnd(1)).*(ref(1) -
trans(1))))./tsamp;
% The following code is used find the change in both thickness and % index and then to fix discontinuities in the data dt = tsamp - tsamp(1); dtA=931;dtB=923;dt(dtA:length(dt))=dt(dtA:length(dt))-(dt(dtA)-
dt(dtB)); dtA=1161;dtB=1151;dt(dtA:length(dt))=dt(dtA:length(dt))-(dt(dtA)-
dt(dtB)); dtA=1432;dtB=1421;dt(dtA:length(dt))=dt(dtA:length(dt))-(dt(dtA)-
dt(dtB));
dn = nsamp - nsamp(1);
212 dnA=931;dnB=927;dn(dnA:length(dn))=dn(dnA:length(dn))-(dn(dnA)-
dn(dnB)); dnA=1161;dnB=1151;dn(dnA:length(dn))=dn(dnA:length(dn))-(dn(dnA)-
dn(dnB)); dnA=1429;dnB=1416;dn(dnA:length(dn))=dn(dnA:length(dn))-(dn(dnA)-
dn(dnB));
close(figure(78)) figure(78)
subplot(1,2,1) % The following bounds are used to choose over range the CTE % and dn/dT values are calculated start2 = 1; end2 = 53870; xtilt_bnd=xtilt_bnd(start2:end2); ytilt_bnd=ytilt_bnd(start2:end2); dOPD1_bnd=dOPD1_bnd(start2:end2); dOPD2_bnd=dOPD2_bnd(start2:end2); dOPD3_bnd=dOPD3_bnd(start2:end2); TSmeas_bnd=TSmeas_bnd(start2:end2); TAmeas_bnd=TAmeas_bnd(start2:end2); dt=dt(start2:end2); dn=dn(start2:end2); nair_bnd=nair_bnd(start2:end2);
disp(['Temp range --> from ' num2str(TSmeas_bnd(1)) '',char(176) 'C to
' num2str(TSmeas_bnd(length(TSmeas_bnd))) '',char(176) ' C'])
% The following plots the results for CTE vs. temp and % carries out linear fit cnfd_val=.999; dt_FIT_CTE =
nlinfit((TSmeas_bnd),dt,@(b1,TSmeas_bnd)(b1(1).*TSmeas_bnd+b1(2)),[.06
-1.7]); % hold on CTE_slp = dt_FIT_CTE(1)./(t0); plot(TSmeas_bnd,dt,'bo','MarkerSize',2);xlabel(['Temp
(',char(176),'C)'],'FontSize',12);ylabel('Change in thickness
({\mu}m)','FontSize',12);title(['Sample: JC022-5 (t = '
sprintf('%.3f',t0./1000) 'mm), CTE = ' sprintf('%.2f',CTE_slp*(1e6))
'x10^{-6} [1/',char(176) 'C]'],'FontSize',12) hold on; %plot(TAmeas(eva_end),dt(eva_end),'ro'); xpy = linspace(min(TSmeas_bnd),max(TSmeas_bnd),201);
plot(xpy,dt_FIT_CTE(2)+dt_FIT_CTE(1).*xpy,'r','LineWidth',2) pub_CTE=t0*(7.5e-6); fit_diff = mean(dt_FIT_CTE(2)+dt_FIT_CTE(1).*xpy)-
mean(dt_FIT_CTE(2)+pub_CTE.*xpy); % %
plot(xpy,fit_diff+dt_FIT_CTE(2)+pub_CTE.*xpy,'g','LineWidth',2,'Color',
[0 .75 0]) % SS .375" pub_CTE_up = t0*(8e-6); pub_CTE_low = t0*(7e-6);
213 fit_diff1 = mean(dt_FIT_CTE(2)+dt_FIT_CTE(1).*xpy)-
mean(dt_FIT_CTE(2)+pub_CTE_up.*xpy); % % plot(xpy,fit_diff1+dt_FIT_CTE(2)+pub_CTE_up.*xpy,'g--
','LineWidth',2,'Color',[0 .75 0]) % SS .375" fit_diff1 = mean(dt_FIT_CTE(2)+dt_FIT_CTE(1).*xpy)-
mean(dt_FIT_CTE(2)+pub_CTE_low.*xpy); % % plot(xpy,fit_diff1+dt_FIT_CTE(2)+pub_CTE_low.*xpy,'g--
','LineWidth',2,'Color',[0 .75 0]) % SS .375" fitresult = fit(TSmeas_bnd',dt','poly1'); ci = confint(fitresult,cnfd_val); disp(['dL/dT CTE of JC022-5 = ' sprintf('%.3f',CTE_slp*(1e6)) 'x10^{-6}
[1/',char(176) 'C]'])% '']) % disp(['Error of CTE of JC022-5 = ' sprintf('%.2f',(CTE_slp-
ci(1,1)./t0)*(1e6)) 'x10^{-6} [1/',char(176) 'C]']) legend('Measured','Linear Fit');%,'Published') % xlim([-40 20]) grid on % ylim([-1 3]) CTE_delta=(1./t0).*((dt(end)-dt(1))./(TSmeas_bnd(end)-TSmeas_bnd(1))); disp(['DL/DT CTE of JC022-5 = ' sprintf('%.3f',CTE_delta*(1e6)) 'x10^{-
6} [1/',char(176) 'C]'])% ''])
% The following plots the results for dn/dT vs. temp and % carries out linear fit nsamp=n0; subplot(1,2,2) nsamp_FIT_dndT =
nlinfit((TSmeas_bnd),dn,@(b2,TSmeas_bnd)(b2(1).*TSmeas_bnd+b2(2)),[.06
-1.7]); dndT_slp = nsamp_FIT_dndT(1); plot(TSmeas_bnd,dn,'bo','MarkerSize',2);xlabel(['Temp
(',char(176),'C)'],'FontSize',12);ylabel('Change in index of
refraction','FontSize',12);title(['Sample: JC022-5 (n ='
sprintf('%.3f',n0) '), dn/dT = ' sprintf('%.2f',dndT_slp*(1e6))
'x10^{-6} [1/',char(176) 'C]'],'FontSize',12) hold on; %plot(TAmeas(eva_end),dt(eva_end),'ro'); xpy = linspace(min(TSmeas_bnd),max(TSmeas_bnd),201);
plot(xpy,nsamp_FIT_dndT(2)+nsamp_FIT_dndT(1).*xpy,'r','LineWidth',2) pub_dndT=14.3e-6; fit_diff = mean(nsamp_FIT_dndT(2)+nsamp_FIT_dndT(1).*xpy)-
mean(nsamp_FIT_dndT(2)+pub_dndT.*xpy); % %
plot(xpy,fit_diff+nsamp_FIT_dndT(2)+pub_dndT.*xpy,'g','LineWidth',2,'Co
lor',[0 .75 0]) % SS .375" pub_dndT_up = 14.8e-6; pub_dndT_low = 13.8e-6; fit_diff1 = mean(nsamp_FIT_dndT(2)+nsamp_FIT_dndT(1).*xpy)-
mean(nsamp_FIT_dndT(2)+pub_dndT_up.*xpy); % % plot(xpy,fit_diff1+nsamp_FIT_dndT(2)+pub_dndT_up.*xpy,'g--
','LineWidth',2,'Color',[0 .75 0]) % SS .375" fit_diff1 = mean(nsamp_FIT_dndT(2)+nsamp_FIT_dndT(1).*xpy)-
mean(nsamp_FIT_dndT(2)+pub_dndT_low.*xpy);
214 % % plot(xpy,fit_diff1+nsamp_FIT_dndT(2)+pub_dndT_low.*xpy,'g--
','LineWidth',2,'Color',[0 .75 0]) % SS .375" fitresult = fit(TSmeas_bnd',dn','poly1'); ci = confint(fitresult,cnfd_val); % % % % plot(xpy, ci(1,2)+ci(1,1).*xpy,'r--','LineWidth',2) % % % % plot(xpy, ci(2,2)+ci(2,1).*xpy,'r--','LineWidth',2) disp(['dn/dT of JC022-5 = ' sprintf('%.2f',dndT_slp*(1e6)) 'x10^{-6}
[1/',char(176) 'C]'])% '']) disp(['Error of dn/dT of JC022-5 = ' sprintf('%.2f',(dndT_slp-
ci(1,1))*(1e6)) 'x10^{-6} [1/',char(176) 'C]']) legend('Measured','Linear Fit'); grid on
%% % air_index_calc.m function % finds index of refraction as a function of temperature % methodology from Stone, J.A. and J.H. Zimmerman. % Index of Refraction of Air. % Available from:
http://emtoolbox.nist.gov/Wavelength/Documentation.asp.
function n_air = air_index_calc(varargin) % Usage is air_index_calc(t,p,RH,lam,xCO2) % t is temperature in Celcius, p is pressure in Pascals, RH is relative % humidity in percent (0 to 100) lam is wavelength in microns, and xCO2
is % carbon dioxide concentration (default 450). All input parameters are % required except for xCO2, which defaults to 450 if not entered. % To enter 40degC, 120kPa, 75% relative humidity, 633nm, default CO2: % air_index_calc(40,120000,75,.633,450) = 1.000299418310159
if size(varargin,2) < 4 || size(varargin,2) > 5 disp('Error: wrong number of input arguments') return end t = varargin{1}; p = varargin{2}; RH = varargin{3}; lam = varargin{4}; if size(varargin,2) == 4 xCO2 = 450; else xCO2 = varargin{5}; end
% A-I. Saturation Vapor Pressure K1 = 1.16705214528e3; K2 = -7.24213167032e5; K3 = -
1.70738469401e1; K4 = 1.20208247025e4; K5 = -3.23255503223e6; K6 =
1.49151086135e1; K7 = -4.82326573616e3; K8 = 4.05113405421e5; K9 = -
2.38555575678e-1; K10 = 6.50175348448e2;
215 T = t + 273.15; Q = T + K9./(T - K10); A = Q.^2 + K1.*Q + K2; B = K3.*(Q.^2) + K4.*Q + K5; C = K6.*(Q.^2) + K7.*Q + K8; X = -B + sqrt(B.^2 -4.*A.*C); psv = (10.^6).*((2.*C./X).^4);
% A-II. Determining Humidity alpha = 1.00062; beta = 3.14e-8; gamma = 5.60e-7; f = alpha + beta.*p + gamma.*t.*t; % enhancement factor f xv = (RH./100).*f.*psv./p; % xv is mole fraction
% A-III. Ciddor Calculation of Index of Refraction % part b w0 = 295.235; w1 = 2.6422; w2 = -0.03238; w3 = 0.004028; k0 = 238.0185; k1 = 5792105; k2 = 57.362; k3 = 167917; a0 = 1.58123e-6; a1 = -2.9331e-8; a2 = 1.1043e-10; b0 = 5.707e-6; b1 = -2.051e-8; c0 = 1.9898e-4; c1 = -2.376e-6; d = 1.83e-11; e = -0.765e-8; pR1 = 101325; TR1 = 288.15; Za = 0.9995922115; pvs = 0.00985938; R = 8.314472; Mv = 0.018015; % part c S = lam.^-2; % part d ras = (10^-8).*((k1./(k0 - S)) + (k3/(k2 - S))); rvs = (1.022e-8).*(w0 + w1.*S + w2.*S.*S + w3.*S.*S.*S); % part e Ma = 0.0289635 + (1.2011e-8).*(xCO2 - 400); raxs = ras.*(1 + (5.34e-7).*(xCO2 - 450)); % part f T = t + 273.15; Zm = 1 - (p./T).*((a0 + a1.*t + a2.*t.*t + (b0 + b1.*t).*xv + (c0 +
c1.*t).*xv.*xv)) + (d + e.*xv.*xv).*((p./T).^2); paxs = pR1.*Ma./(Za.*R.*TR1); pv = xv.*p.*Mv./(Zm.*R.*T); pa = (1 - xv).*p.*Ma./(Zm.*R.*T); % part g (index of refraction) n_air = 1 + (pa./paxs).*raxs + (pv./pvs).*rvs;