2.3 aberrations 2.5 depth of focus optical engineering
TRANSCRIPT
Universität Karlsruhe (TH)
Optical Engineering
Martina Gerken22.11.2007
4.2
Course outline1. Imaging optics
2.1 Light propagation across interfaces2.2 Photography and optical lenses2.3 Aberrations2.4 Plane parallel plates and reflective prisms 2.5 Depth of focus2.6 Magnifying glass and basic microscope2.7 Modern microscopes2.8 Beam expansion2.9 Telescopes2.10 Optic design
2. Optical sensors3. Optics in data storage4. Introduction to displays5. Fourier optics6. Diffractive optics and holograms7. Integrated optics8. Computerized imaging
4.3
Exercise: Selection of telescope
• Collect the advantages and disadvantages of the two telescopes!
• Decide as a group on one telescope!
4.4
Comparison of telescopes
• Refractor• Terrestrial observation
• Reflector• No spherical/chromatic aberrations• Mirror diameter larger, thus more
light• More compact• More expensive• Prettier• Aberrations due to diffraction on
second mirror
BRASKO 60700 Telescope Bresser Telescope Pluto 114/500
4.5
Important factors for telescope purchase• Two main tasks of telescope:
– Magnify small objects– Brighten faint objects
• Useful magnification limited by diffraction– Max. Magnification ≈ Diameter of objective in mm– Image may be magnified more, but image information is limited– Analog: Not more detail on newspaper image observable with magnifying
glass
• Aberrations and air turbulence ("Seeing") limit magnification further– Max. resolution of terrestrial telescope ≈ 1”– Hubble-Telescope ≈ 0.05” at visible wavelengths
• Larger aperture for brighter images!
• Quality of mount is also critical– Should not move more than 1 sec after touching
4.6
Course outline1. Imaging optics
2.1 Light propagation across interfaces2.2 Photography and optical lenses2.3 Aberrations2.4 Plane parallel plates and reflective prisms 2.5 Depth of focus2.6 Magnifying glass and basic microscope2.7 Modern microscopes2.8 Beam expansion2.9 Telescopes2.10 Optic design
• Analytical solution (optical quadrupole)• Matrix formulation• Sequential raytracing• Non-sequential raytracing• FDTD
4.7
Geometrical optics / Ray optics
ξ << Xλ << X
• Rays describe propagation of light in (most) optical systems sufficiently well, if dimensions X of objects and parts sufficiently larger than:
– Wavelength
– Coherence length
• Wave properties of light neglected
– No interference, diffraction, near field effects ...
• Geometrical optics may be derived from wave optics for wavelength approaching zero
• Direction of ray propagation is direction of wavevectors
4.8
βα
1n 2n1 2sin sinn nα β⋅ = ⋅
1 2n nα β⋅ = ⋅
Paraxial approximation / Gaussian optics• For propagation at small angles relative to optical axis simplified description of
optical parts is possible and analytical solutions exist.
4.9
Spherical surface in paraxial approximation• Approximated
• For rays parallel to optical axis
sn
rnnns−−′′
≈′
2θ1θ
O O’r
s s’
n n’
ϕ
A
S C
rnn
ns−′′
≈′
4.10
Thick lens with spherical surfaces• Calculation of optical power of thick lens• Optical power
– [D]=m-1=Diopter=dpt
fnD −=
d
n=1 nL
B1=G2B2=F
4.11
Thick lens: Focal point
d
n=1 nL
B1=G2B2=F
s1‘
11 1' r
nnsL
L
−≈
11' 1
112 −+−
=−−
=−=L
LL
L
L
nddnrndr
nndss
( )( )
( ) ( ) ( )[ ]11111
11'
12
12
1222
2 −+−−+−
=
+−−+−
=+−
=LLL
LL
LL
LLLLL ndrrnnddnrnr
ddnrnnn
rn
sn
rns
s2‘
s2
• Focal length is distance from principle point to focal point
4.12
Front principle plane• Front principle plane: Locus of intersection of diverging/converging bundle of
rays for parallel outgoing rays– Front principle point H: Intersection of front principle plane with optical axis
Source: Schröder/Treiber, Technische Optik
Positive (converging) system Negative (diverging) system
4.13
Rear principle plane• Rear principle plane: Locus of intersection of parallel ingoing rays with
diverging/converging outgoing rays– Rear principle point H’: Intersection of rear principle plane with optical axis
Source: Schröder/Treiber, Technische Optik
Positive (converging) system Negative (diverging) system
4.14
Thick lens: Focal length and optical power
f
n=1 nL
B1=G2B2=F
( ) ( ) ( )[ ]( ) ( ) ( )[ ]
21
12
12
21
2
21
1111
''
rrnndrrnnD
ndrrnnrrn
sssf
L
LLL
LLL
L
−+−−−=
−+−−==
H‘
4.15
Possible lens shapes
Source: Schröder/Treiber, Technische Optik
4.16
Special case: Plano-convex and Plano-concave lenses• For plane surface
• This results in
– Focal length independent of d!
Source: Schröder/Treiber, Technische Optik
11
−=
Lnrf
∞→2r
4.17
Special case: Spherical lens• For spherical surface
• This results in
Source: Schröder/Treiber, Technische Optik
21r
nnfL
L
−=
rdrrr 221 =−==
4.18
Special case: Thin lens• For negligible d
• This results in
• For thin lenses distance between principle planes may be neglected
( )( )
( )
L
L
L
L
nndHH
rrn
f
rrnrrf
1'
1111
1
21
12
21
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
−−=
( ) 121 rrnnd LL −<<−
4.19
Notation for imaging surfaces
Source: Schröder/Treiber, Technische Optik
4.20
Equivalent lens• System with several stages (e.g. objective with several lenses) may be
described by optical quadrupole (“Equivalent lens“)– Normally distance of individual systems measured between H1‘ und H2
– In case of microscopes tube length between focal points used
Source: Schröder/Treiber, Technische Optik
4.21
Course outline1. Imaging optics
2.1 Light propagation across interfaces2.2 Photography and optical lenses2.3 Aberrations2.4 Plane parallel plates and reflective prisms 2.5 Depth of focus2.6 Magnifying glass and basic microscope2.7 Modern microscopes2.8 Beam expansion2.9 Telescopes2.10 Optic design
• Analytical solution (optical quadrupole)• Matrix formulation• Sequential raytracing• Non-sequential raytracing• FDTD
4.22
2θ
1θ
z
x
1x 2x
A B 1i
i
xs
θ⎛ ⎞
= ⎜ ⎟⎝ ⎠
( )1 1,x θ ( )2 2,x θOptical system
2 1s s= ⋅12M
Matrix optics / ABCD-matrices
• Propagation of ray at point A completely described by distance x to optical axis and angle θ relative to optical axis
4.23
[ ]5 44' 3'4 12 1...s s= ⋅ ⋅ ⋅ ⋅M M M
2θ1θ
z
x
1x 2x 3x 4x
3θ
4θ
22'M 33'M 44'M12M 2'3M 3'4M
Eingang i Ausgangi
s s⎡ ⎤= ⋅⎢ ⎥⎣ ⎦∏M
Optical system with several components
• Effect of total system Output Input
4.24
2 1 1
2 1 1
10 1
x x Lx
θθ θ
= ⋅ + ⋅= ⋅ + ⋅
2 1 1
10 1 Freiraum
Ls s s⎛ ⎞= ⋅ = ⋅⎜ ⎟⎝ ⎠
M
2θ
1θ
z
x
1x 2x
A B
1. Matrix of translation
L
Translation
4.25
2 1 1
12 1 1
2
1 0
0
x xnxn
θ
θ θ
= ⋅ + ⋅
= ⋅ + ⋅
2 1 11
2
1 0
0 EbeneFläches s snn
⎛ ⎞⎜ ⎟= ⋅ = ⋅⎜ ⎟⎜ ⎟⎝ ⎠
M
2θ
1θ
z
x
1x 2x1x1n 2n
2. Refraction on plane interface
Plane interface
4.26
2 1 11 2 1
2 2
1 0
SphärischeFläches s sn n nn nρ
⎛ ⎞⎜ ⎟= ⋅ = ⋅−⎜ ⎟⎜ ⎟⎝ ⎠
M
2θ1θ
z
x
1x 2x1x ρ
1n 2n
0ρ >
3. Refraction on spherical interface• Snellius at interface• Angle depends on radius
– Convention for curvature and propagation
Spherical interface
4.27
Exercise: Thin lens• All important simple optical elements may be described using the tree
matrices (Translation, plane interface and spherical interface)– Thin lens (convex, concave)– Thick lens (convex, concave)– Mirror (plane, focusing)
• Derive the total matrix for a thin lens! • Derive the dependence of the focal length f and the interface curvatures (Lens
maker's equation for thin lens)!
• Derive the matrix for imaging with a convex lens!• Derive the imaging equation from the resulting matrix!
– Consider that x2 has to be angle-independent
4.28
( ) ( ) ( )2 2 2 1 1 1 2 1, ,SF FR SFs n n L n n sρ ρ= → ⋅ ⋅ → ⋅M M M
z
x 1n2n1n
f f
2h1h
Thick lens• Combination of two spherical interfaces and translation
– Focal length calculated from principal plane
L
SI SITr
4.29
12
A BC D⎛ ⎞
= ⎜ ⎟⎝ ⎠
M
112
2
det nAD BCn
= − =M
Matrix properties in Gaussian optics
• Because of Snellius law:
– Only 3 of the 4 matrix elements may be chosen independently
4.30
2 10A x Bθ= ⇒ =Focusing: 0 B
C D⎛ ⎞⎜ ⎟⎝ ⎠
2 10B x Ax= ⇒ =Optical imaging: 0A
C D⎛ ⎞⎜ ⎟⎝ ⎠
2 1Dθ θ=Deflection of parallel rays:
0A B
D⎛ ⎞⎜ ⎟⎝ ⎠
2 1Cxθ =Generation of parallel rays:
0A BC⎛ ⎞⎜ ⎟⎝ ⎠
Optical elements and their system matrices
4.31
Matrix optics for Gaussian beams and polarization• Matrix optics not limited to Gaussian optics, but may be used for Gaussian
beams as well. – ABCD-Matrices identical– Instead of ray vector s beam parameter q used:
• Polarization may be included as well in matrix algorithm– Jones-vector describes state of polarization– Jones-matrices describe optical elements
( )DCqBAqzq
++
=0
02
11w
iRq π
λ−=
4.32
Course outline1. Imaging optics
2.1 Light propagation across interfaces2.2 Photography and optical lenses2.3 Aberrations2.4 Plane parallel plates and reflective prisms 2.5 Depth of focus2.6 Magnifying glass and basic microscope2.7 Modern microscopes2.8 Beam expansion2.9 Telescopes2.10 Optic design
• Analytical solution (optical quadrupole)• Matrix formulation• Sequential raytracing• Non-sequential raytracing• FDTD
4.33
Sequential ray-tracing• Rays propagates through optical elements in sequential order• Used for design of imaging optical systems
– Microscope, telescope, camera ...
Source: http://optics.org/dl/2006/OLEJulAugPRODUCTGUIDE.pdf
4.34
Non-sequential ray-tracing• Rays split on surface and may pass optical elements several times• By propagation of Gaussian beams wave phenomena may be included• Used for design including stray light calculation
– Stray light calculation– Back lights, luminaires ...
Source: http://optics.org/dl/2006/OLEJulAugPRODUCTGUIDE.pdf
4.35
FDTD: Finite Difference Time Domain• Solution of Maxwell-equations using space and time grid• Used for micro- and nanosystems
– Integrated optics, photonic crystals, plasmonics ...
Source: http://optics.org/dl/2006/OLEJulAugPRODUCTGUIDE.pdf
4.36
Common optical design software
– At the university additionally COMSOL Multiphysics
Source: http://optics.org/dl/2006/OLEJulAugPRODUCTGUIDE.pdf
4.37
Course outline1. Imaging optics2. Optical sensors
2.1 Spectroscopy2.2 Material characterization2.3 Distance measurement2.4 Angle measurement2.5 Optical mouse
3. Optics in data storage4. Introduction to displays5. Fourier optics6. Diffractive optics and holograms7. Integrated optics8. Computerized imaging
4.38
SpectroscopySpectroscopy:
Frequency or wavelength analysis of light
Applications of spectroscopy– Material characterization– Sensors– Development of light sources
Example: Organic light emitting diode
inte
nsity
/ ar
b. u
nits
4.39
SpectroscopyOptical spectrum characteristic for chemical elements
– Known since appr. 1859 (Kirchhoff, Bunsen), e.g.:
– e.g.: Fraunhofer-lines in solar spectrum (1802, 1814) (Absorption spectra)
In general:
– Glowing solid or liquid objects have continuous spectra (e.g. thermal radiation, black body, Planck's law)
– Glowing gases or vapors have discontinuous spectra(Transition between discrete energy levels in atoms/molecules)
Source: Wikipedia and Gerthsen Physik
4.40
http://www.technoteam.de/
Spectrometer using absorptive filters
• Spectrally broad CCD-chip is illuminated through sequence of filters
4.41
Spectrometers using dispersive elements
• Goal: Spatial separation of light of different wavelengthse.g., on a screen or a CCD-chip
• Use dispersive elementMaterial or structure whose effect on light is wavelength-dependent (e.g. prisms, grating)
• Common types of spectrometers sorted by dispersive element– Prism spectrometer uses dispersion due to refraction – Grating spectrometer uses dispersion due to diffraction
• Common types of spectrometers sorted by functionality– Monochromator selects small wavelength interval– Spectrometer for observation of large wavelength interval– Spectrograph Spectrometer + CCD-camera / film etc.
4.42
Material dispersion• Refractive index depends on type of glass and on wavelength dependent
Source: http://en.wikipedia.org
4.43
Prism spectrometer – Refraction in prism
Minimum deflection angle for symmetric pass:
Dispersion:
Deflection angle:
Angular dispersion
Source: Wikipedia and „Optik Licht und Laser“ von Dieter Meschede
4.44
Prism spectrometer
Advantage: Unambiguous correlation between wavelength and position in image planeDisadvantage: Small dispersion and thus small resolution
Bild 16.3.1 ausNaumannSchröder
Basis of prism
Telescope
Source: „Bauelemente der Optik“ von Naumann und G. Schröder
4.45
Interference – special caseInterference: Superposition of waves with fixed phase relation.Interference is due to wave character of light and cannot be understood using ray
optics.
Example: Two monochromatic waves of identical polarization and identicalamplitude are superposed at position
Principle of superposition for fields.
Irradiance (intensity) at point is:
Spatially modulated interference pattern with minima and maxima
4.46
Interference – general caseGeneral case:
Superposition of fields:
Irradiance (intensity) obtained as temporal average of electrical field, i.e. square of field amplitude A
Interference term results as
Interference depends
on polarization
Only for complex field descriptionDivisor 2 from averaging
4.47
Constructive / destructive interferenceInterference term
Phase difference
Path difference
Constructive interference / Maximum of intensity for
Destructive Interference / Minimum of intensity for
Difference of opt. paths of two partial waves
Constant term
4.48
Grating principle• Interference of spherical waves exiting different slits• Each slit also causes diffraction
• Formula for N slits:
Interference term Diffraction term
Source: Optics lecture by PD Dr. Seifert, Universität Halle (http://www.physik.uni-halle.de/)
4.49
Source: Optics lecture by PD Dr. Seifert, Universität Halle (http://www.physik.uni-halle.de/)
Grating - intensity with angle• Main maxima for constructive interference of all wave parts
Example:N = 8,d = 2b
Diffraction of slit
1. order interference maximum
4.50
Grating equation• Maximum intensity at:
dm
mλθθ ±=− 0sinsin
Source: Wikipedia and Optics lecture by PD Dr. Seifert, Universität Halle (http://www.physik.uni-halle.de/)
Grating
Effective grating size
Image of light bulb through transmission
grating
4.51
Exercise: Grating period
• Determine the grating period of the given grating!
4.52
Grating ambiguity• Gratings are ambiguous!
Source: http://www.jobinyvon.com
4.53
Source: Optics lecture by PD Dr. Seifert, Universität Halle (http://www.physik.uni-halle.de/)
Grating – width of maximum• For more illuminated slits maximum is narrower
4.54
Source: Optics lecture by PD Dr. Seifert, Universität Halle (http://www.physik.uni-halle.de/)
Grating resolution• Two wavelengths are resolved if
they are at least separated by width of main maximum
Resolution in order m for grating for N illuminated slits
4.55
Types of gratings• Amplitude grating
Slits as origin of fundamental waves.
• Phase gratingModulation of optical path ! Modulation of phase
• Reflection gratingoften blazed
4.56
Blazed gratings• In Littrow configuration groove have such profiles that incident and diffracted
rays are in auto collimation (i.e., α = β). – Efficiency of grating increased
• In this case at the "blaze" wavelength λB we obtain:
Source: http://www.jobinyvon.com
dm Bλα
αβ
=
−=
sin2
4.57
Czerny-Turner monochromator• Light leaving the exit slit (G) contains entire image of entrance slit of selected
color plus parts of entrance slit images of nearby colors. • Rotation of dispersing element causes band of colors to move relative to exit
slit
Source: http://en.wikipedia.org
mdm
dd
θλθ
cos=
4.58
Échelle spectrographs • Prism for dispersion in one direction• Grating for dispersion in orthogonal direction
Source: http://www.andor.com
Spectrograph with standard prism
Spectrograph with Andorpatented prism
4.59
Resolution at 550nm
4 1
4 1
1,79260 , 60
2 104,5 10
12000
nb mmdn d nmd d nm
α
λ
δ λλλ
− −
− −
== = °
= ⋅
= ⋅
=∆
4 116 10
72000
d d nmα λλλ
− −= ⋅
=∆
Comparison grating / prism spectrometer• Reflection grating
– 1200 rules/mm– Width 60mm
• Prism– Glass SF11
4.60
Exercise: Spectroscope
• Build the spectroscope!
• Observe and sketch the spectrum of an incandescent source and a fluorescent lamp!– What influence does the entrance slit have?
4.61
Compilation of questions• When may Gaussian optics be used?• How are the principle planes of a thick lens defined?• What is a thin lens?• What are ABCD-matrices and what are they used for?• What is the difference between sequential and non-sequential raytracing?• How does a spectrometer using absorptive filters work?• Sketch a prism spectrometer!• What is constructive interference?• Sketch the outgoing intensity as a function of angle for a grating!• Why is the wavelength determination with a grating ambiguous?• What limits the resolution of a grating monochromator?• Sketch a grating spectrograph!