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Metrology and Sensing
Lecture 6: Interferometry II
2016-11-22
Ralf Hambach / Herbert Gross
Winter term 2016
2
Preliminary Schedule
No Date Subject Detailed Content
1 18.10. Introduction Introduction, optical measurements, shape measurements, errors,
definition of the meter, sampling theorem
2 19.10. Wave optics (ACP) Basics, polarization, wave aberrations, PSF, OTF
3 01.11. Sensors Introduction, basic properties, CCDs, filtering, noise
4 08.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry
5 09.11. Interferometry I (ACP) Introduction, interference, types of interferometers, miscellaneous
6 22.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods
7 29.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods
8 06.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,
Scheimpflug setup
9 13.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology
10 20.12. Holography Introduction, holographic interferometry, applications, miscellaneous
11 03.01. Measurement of basic
system properties Bssic properties, knife edge, slit scan, MTF measurement
12 10.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy
13 17.01. Metrology of aspheres
and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms
14 24.01. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous
15 31.01. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method
3
Content
Interferogram examples
Interpretation of interferograms
Fringe evaluation methods
Interferograms of Primary Aberrations
Spherical aberration 1
-1 -0.5 0 +0.5 +1
Defocussing in
Astigmatism 1
Coma 1
4
Real Measured Interferogram
Problems in real world measurement:
Edge effects
Definition of boundary
Perturbation by coherent
stray light
Local surface error are not
well described by Zernike
expansion
Convolution with motion blur
Ref: B. Dörband
5
Interferogram - Definition of Boundary
Critical definition of the interferogram boundary and the Zernike normalization
radius in reality
6
Interferometry
Color fringes of a broadband interfergram
Ref: B. Dörband
Shearing Interferograms
Typical shearing interferograms
of some simple aberrations
d
xz
2
Interpretation of Interferograms
xd
Distance between fringes: d
Bending of fringes: x
Relation of surface error z
accross diameter
Intensity of fringes
I(x,y,t) intensity of fringes
V(x,y) contrast of pattern
W(x,y) phase function to be found
j(x,y,t) reference phase
Rs(x,y) multiplicative speckle noise
IR(x,y,t) additive noise
Tracing of fringes:
- time consuming method, interpolation, indexing of fringes, missing lines
Fourier method:
-wavelet method
- FFT Method
- gradient method
- fit of modal functions
Evaluation of Fringes
),,(),(),,(),(cos),(1),(),,( 0 tyxIyxRtyxyxWyxVyxItyxI RS j
11
Interferometry
General description of the measurement quantity:
superpostion of spatially modulated signal and noise
Io: basic intensity, source
T: transmission of the system, including speckle
j: phase, to be found
IN: noise, sensor, electronics, digitization
Signal processing, SNR improvement:
- filtering
- background subtraction
Ref: W. Osten
0( , ) ( , ) ( , ) cos ( , ) ( , )NI x y I x y T x y x y I x yj
original signal
filtered signal
background
processed signal
12
Interferometry
perfect interferogram
reduced contrast due
to background intensity
with speckle
with noise
Ref: W. Osten
Basic configuration
Test surface rotated by 180°
Cats eye configuration
Calibration
plane
mirror
1. Basic configuration
2. Surface rotated by 180°
3. Cats eye position
surface
under test
condenser
1 Re( , ) ( , ) ( , ) 2 ( , )f KondW x y W x y W x y S x y
2 Re( , ) ( , ) ( , ) 2 ( , )f KondW x y W x y W x y S x y
3 Re
( , ) ( , )( , ) ( , )
2
Kond Kondf
W x y W x yW x y W x y
1 2 3 3
1( , ) ( , ) ( , ) ( , ) ( , )
4S x y W x y W x y W x y W x y
Absolute Calibration of Interferometer
14
Fringe Evaluation
1. Fringe Tracking
2. Fourier-Transform Method
3. Spatial Phase Shifting
4. Phase Sampling Technique
5. Heterodyne Technique
6. Phase-Locking Method
7. Ellipse-Fitting Technique
Ref: R. Kowarschik
15
Evaluation of Fringe Pattern
Ref: R. Kowarschik
Static Methods Dynamic Methods
Fringe Tracking Phase Shifting Methods
Fourier-Transform Heterodyne Technique
Spatial-Carrier Frequency Phase-Locking Method
Spatial Phase Shifting
+ Only 1 interferogram Very variable + No specific components Accuracy better /100
- Difficult to automatize Calibration
- Accuracy below /100 Additional components
16
Evaluation of Fringe Pattern
Ref: R. Kowarschik
Static Methods Dynamic Methods
Fringe Tracking Phase Shifting Methods
Fourier-Transform Heterodyne Technique
Spatial-Carrier Frequency Phase-Locking Method
Spatial Phase Shifting
+ Only 1 interferogram Very variable + No specific components Accuracy better /100
- Difficult to automatize Calibration
- Accuracy below /100 Additional components
17
Fringe Tracking for Fringe Evaluation
Ref: R. Kowarschik
Fringe Tracking (fringe skeletonizing)
- Intensity distribution 1. Identification of local extrema
2. Fringe sampling points for interpolation
- determination of points with integer or half-integer order of interference
- absolute order has to be identified additionally
- relatively low accuracy of phase measurements
Processing:
- improvement of SNR by spatial and temporal filtering
- creation of the skeleton (segmentation)
- Improvement of the skeleton shape
- numbering the fringes
- reconstruction of the phase by interpolation
18
Fringe Tracking for Fringe Evaluation
Skeletonizing method
Ref: W. Osten
interferogram segmentation
improved
segment skeleton
phase map
Method of carrier frequency
- tilt creates carrier frequency
- essential signal: deviation from linearity
Evaluation in frequency space:
carrier frequency eliminated by filtering of the Fourtier method
Carrier Method of Fringe Evaluation
20
Fourier Method of Fringe Evaluation
Intensity in interferogram
Substitution
gives
Fourier transform
interpretation:
A: low frequencies, background
C, C* : same information
Filtering with bandpass:
elimination of A and C*:
Inverse Fourier transform
Pointwise calculation of phase
Unwrapping of the phase for 2
for a smooth surface
Ref: W. Osten
( , ) ( , ) ( , ) cos ( , )I x y a x y b x y x y
( , )1( , ) ( , )
2
i x yc x y b x y e
*( , ) ( , ) ( , ) ( , )I x y a x y c x y c x y
*J( , ) ( , ) ( , ) ( , )A C C
J( , ) ( , )C
( , )1( , ) ( , ) ( , ) ( , )
2
i x yI x y F J c x y b x y e
Im ( , )( , )
Re ( , )
c x yx y
c x y
Fourier method:
- representation in frequency domain
- A: noise
- filtering of noise and asymmetrical contribution
Phase information
),(),(),(),( * vuCvuCvuAvuI
),(Re
),(Imarctan
yxC
yxCj
| I(u,v) |
spatial
frequency
u
A(u,v)
C (u,v)* C (u,v)
filter-
function
H(u,v)
Fourier Method of Fringe Evaluation
22
Fourier Method of Fringe Evaluation
Fourier method
Ref: W. Osten
interferogram amplitude filtered amplitude
wrapped phase phase mapunwrapped phase
23
Carrier Method of Fringe Evaluation
Ref: W. Osten
interferogram
interferogram
with carrier
amplitude
spectrum
spectrum filtered
and shifted
unwrapped
phaseunwrapped phase
24
Carrier Method of Fringe Evaluation
Fourier spatial demodulation technique
Overlay of carrier frequency
Filtering of the spectrum: only one order
Inverse transform
Ref: G. Kaufmann
Interferogram Interferogram with carrier spectrum reconstructed phase
25
Phase Sampling
Diversification
Various possibilities for changes
Ref: R. Kowarschik
Phase shifting method TPMI
( temporal phase measuring interferometry )
- additional phase term a
- three different phases aj sequencially
measured (at least 3)
- elimination of phase values
background
contrast
- alternatively 4 frame method
- more phase values increase accuracy
aj ),(cos),(),(),( yxyxbyxayxI
3/2/13/2/1 cos aj baI
321231132
321231132
sinsinsin
coscoscosarctan
aaa
aaaj
IIIIII
IIIIII
2
3,,
2,0 4321
aa
aa
31
24arctanII
II
j
Phase Shifting Method of Fringe Evaluation
2 2
1 3 2 4
0
1
2C I I I I
I
1,4
1
4B j
j
I I
27
Phase Shifting
Errors of phase shifting, calibration:
- Nonlinearities of the detector
- Modulo 2
- Other systematic errors
- non-ideal reference surfaces
- aberrations of optical elements
- diffraction, ghosts
- digitization
- air turbulence
- mechanical vibrations
- detector noise
- frequency shift
Ref: R. Kowarschik
TPMI method variants
- 3-frame
- 4-frame
- 5-frame
Carre method:
- only phase differences essential
- higher accuracy
Comparison of accuracies:
larger number of frames is
more precise
PV-phase
error in
phase
error
a
0.05
10
in %
200-10-20
0.015-Frame
3- , 4-Frame
Carre
Phase Sifting Method
29
Phase Shifting Method for Fringe Evaluation
Ref: W. Osten
I2(90)
unwrapped
phase
I4(270)
I3(180) I1(0)
wrapped phase