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Metrology and Sensing

Lecture 6: Interferometry II

2018-11-20

Herbert Gross

Winter term 2018

2

Schedule Optical Metrology and Sensing 2018

No Date Subject Detailed Content

1 16.10. IntroductionIntroduction, optical measurements, shape measurements, errors,

definition of the meter, sampling theorem

2 17.10. Wave optics Basics, polarization, wave aberrations, PSF, OTF

3 30.10. Sensors Introduction, basic properties, CCDs, filtering, noise

4 09.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry

5 13.11. Interferometry I Introduction, interference, types of interferometers, miscellaneous

6 20.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods

7 27.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods

8 28.11. Geometrical methodsTactile measurement, photogrammetry, triangulation, time of flight,

Scheimpflug setup

9 11.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology

10 18.12. Holography Introduction, holographic interferometry, applications, miscellaneous

11 08.01.Measurement of basic

system propertiesBssic properties, knife edge, slit scan, MTF measurement

12 15.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy

13 22.01.Metrology of aspheres

and freeformsAspheres, null lens tests, CGH method, freeforms, metrology of freeforms

14 29.01. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous

15 05.02. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method

3

Content

Interferogram examples

Interpretation of interferograms

Fringe evaluation methods

4

Interferometer

Basic idea:

- compare a reflected perturbed wave with a calibrated (ideal) reference wave

- the energy division determines the contrast

- the fringes of the superposed wave determines the phase differences

- integrating of the gradient field gives the desired wave

Important properties:

- coherent laser illumination

- small cavity length / large common path

- two beam or multiple beam interference

Interferograms of Primary Aberrations

Spherical aberration 1

-1 -0.5 0 +0.5 +1

Defocussing in

Astigmatism 1

Coma 1

5

6

Diffraction Limit

Strehl of 80%

Defocus, spherical, coma, astigmatism

Ref: R. Shack, lecture notes

Perfect lens, tilt and defocus

Spherical aberration with tilt and defocus

Coma, various azimuths

Coma with defocus

Interferograms of Primary Aberrations

6

Astigmatism with tilt in Petzval focus in various

azimuths

Astigmatism with tilt in sagittal focus n various

azimuths

Astigmatism with tilt in tangential focus n various

azimuths

Interferograms of Primary Aberrations

7

9

Multi-Wavelength Interferometer (MWLI)

Different wavelengths:

- large measurement interval

- unique evaluation of distance

Mathematical combination of interferometric signals

Large range of unambiguity

Better accuracy than single wavelength interferometer (better than +/- 50 nm)

Calculation

Ref: B. Fleck

1 2 2 1/ 2 ( ) / ( )

Interferometry

Color fringes of a broadband interfergram

Ref: B. Dörband

9

11

White Light Interferograms

Typical interferograms

monochromatic / white light

Additional information by different

wavelengths

Ref: B. Dörband

12

Interferograms

Examples

13

Sample Interferograms

Modulated surface

(left: ripple plate)

Test piece with constant

high frequency ripple

(right: Fourier spectrum)

Ref: B. Dörband

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

14

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

15

16

Interferogram Examples

More complicated sample

Ref: Elta systems

17

Interferogram

Example Interferogram of a plate with step

Ref.: H. Naumann

Coherent Superposition of Perturbations

Twyman-Green interferometer

Coherent defects on sample surface

(scratches, dots,...)

Very sensitive amplitude superpostion

Problems in fringe evaluation

Strong dependence on size of source:

relaxed problem for partial coherence

due to finite source size

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5 3 3.5

RM

S d

es D

iffe

renzf

eld

es [

nm

]

Lichtquellengröße [mm]size of lightsource in [mm]

rms of

field

difference

17

Coherent Superposition of Perturbations

Coherent defects on sample surface

(scratches, dots,...)

Superposition creates error in phase

Optimization of source size to suppress perturbations

without creates too large errors of the signal

18

Shearing Interferograms

Typical shearing interferograms

of some simple aberrations

19

d

xz

2

Interpretation of Interferograms

xd

Distance between fringes: d

Bending of fringes: x

Relation of surface error z

accross diameter

20

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

21

23

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

24

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

24

26

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

27

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

28

Phase Sampling

Diversification

Various possibilities for changes

Ref: R. Kowarschik

29

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

30

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 Fourier method

Equation: carrier frequency n

Carrier Method of Fringe Evaluation

without tilt same wave with tilt 20'

30

0( , ) ( , ) ( , ) cos ( , ) ( , )NI x y I x y T x y x y x I x yj n

Method of carrier frequency

- tilt creates carrier frequency

- creation by tilting the reference

- essential signal: deviation from linearity

Evaluation in frequency space:

- carrier frequency eliminated by filtering

of the Fourier method

- much better separation of low-frequency

components

Example:

water droplet as phase bump

Carrier Method of Fringe Evaluation

interferogram reconstructed phase

large

phase

bump

small

phase

bump

31

33

Carrier Method of Fringe Evaluation

Ref: W. Osten

interferogram

interferogram

with carrier

amplitude

spectrum

spectrum filtered

and shifted

unwrapped

phaseunwrapped phase

34

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

35

Carrier Frequency Interferogram

Evaluation of data

With filtering step

Ref: B. Dörband

36

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 Cn n n n

J( , ) ( , )Cn n

( , )1( , ) ( , ) ( , ) ( , )

2

i x yI x y F J c x y b x y e n

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

36

38

Fourier Method of Fringe Evaluation

Fourier method

Ref: W. Osten

interferogram amplitude filtered amplitude

wrapped phase phase mapunwrapped phase

Phase shifting method (PSI):

Gathering several interferograms by change of absolute phase

Realization:

- additional z-offset z

- shift of wavelength

Phase Shifting Method of Fringe Evaluation

38

reference offset /4 / +90° offset /2 / +180°

reference

black gray white

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

38

41

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

42

Phase Shifting Method for Fringe Evaluation

Ref: W. Osten

I2(90)

unwrapped

phase

I4(270)

I3(180) I1(0)

wrapped phase

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

40

Zernike Polynomials

+ 6

+ 7

- 8

m = + 8

0 5 8764321n =

cosj

sinj

+ 5

+ 4

+ 3

+ 2

+ 1

0

- 1

- 2

- 3

- 4

- 5

- 6

- 7

Expansion of wave aberration surface

into elementary functions / shapes

Zernike functions are defined in circular

coordinates r, j

Ordering of the Zernike polynomials by

indices:

n : radial

m : azimuthal, sin/cos

Mathematically orthonormal function

on unit circle for a constant weighting

function

Direct relation to primary aberration types

n

n

nm

m

nnm rZcrW ),(),( jj

01

0)(cos

0)(sin

)(),(

mfor

mform

mform

rRrZ m

n

m

n j

j

j

44

Deviation in the radius of normalization of the pupil size:

1. wrong coefficients

2. mixing of lower orders during fit-calculation, symmetry-dependent

Example primary spherical aberration:

polynomial:

Stretching factor of the radius

New Zernike expansion on basis of r

166)( 24

9 Z

r

14

24

44

2

949

23

)(13

)(1

Z

rZrZZ

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

c4

c1

c9 / c

9

Zernike Coefficients for Wrong Normalization

45

Conventional calculation of the

Zernikes:

equidistant grid in the entrance

pupil

Real systems:

Pupil aberrations and distorted

grid in the exit pupil

Deviating positions of phase

gives errors in the Zernike

calculation

Additional effect:

re-normalization of maximum

radius

46

Zernike Calculation on distorted grids

5% radial distorted grid with both signs:

+5%: pincushion

-5% barrel

Wavefront of only one selected Zernike is re-analyzed

on distorted re-normalized grid:

Effects of distortion:

- errors in the range of some percent

- cross-mixing to other zernikes of same symmetry

- neighboring Zernikes partly influenced until 20%

- larger effects of higher order Zernikes

- slightly larger effects for pincushion distortion

47

Zernike Calculation on distorted grids

48

Zonal Method for Wavefront Reconstruction

Primary data of measurement:

derivatives / gradients of the wave W(x,y)

Zonal reconstruction:

- derivatives by finite differences in every cell

of the pixel grid

Collecting all mesh values:

- matrix representation

- sparse bandwidth matrix B

Inversion of the linear system of equations

Problems:

- local integration suffers from problems in case of noise

- sequence of connecting the cells creates non-unique results in case of singular

points

x

y

j

Wj,1,1 W

j,2,1

Wj,1,2

x

y

Sj,x

Sj,y

SW W

xj x

j j

,

, , , ,

2 1 1 1

S

W W

yj y

j j

,

, , , ,

1 2 1 1

S B W

W B B B ST T

1

49

Modal Method of Wave Reconstruction

Expansion of the wave into orthonormal set of functions F

Analytical derivates used to describe the direct measurement values

System of equations to be solved

Solution by inversion of the the linear system

by SVD methods

Properties:

- modal function work as a low pass filter, more robust fitting

- eigenvalues indicate an unsufficient functional system

- unique solution, usually more accurate

W x y a F x yn n

n

( , ) ( , )

W

xa

F x y

xn

n

n ( , )

W

ya

F x y

yn

n

n ( , )

S A a

a A A A S

T T

1

50

Modal Method of Wave Reconstruction

Example 1:

Zernike expansion

Example 2:

Fourier modal functions

j

j

im

mn

m

nnm ea

rRcmi

W

,

jW

rc

d Rr

a

drenm

n

m

n m

im

,

xR

n

W

x

R

n

W

r

W

r'

' 'cos

sin

j

j

jy

R

n

W

y

R

n

W

r

W

r'

' 'sin

cos

j

j

j

ci n

ma

W

x

W

yR

r

ae r dr dnm n

m

a

im

( )sin cos

12

00

2

2

j

j j

j

Z m n

Nepq

i

Npm qn

( , )1

2

W m n a Z m npq pq

p q

( , ) ( , ),

N

q

N

p

qpWFeqpWFe

a

yN

ip

xN

iq

pq

22

22

sinsin4

),(ˆ1),(ˆ1