www.iap.uni-jena.de

Metrology and Sensing

Lecture 6: Interferometry II

2017-11-23

Herbert Gross

Winter term 2017

2

Preliminary Schedule

No Date Subject Detailed Content

1 19.10. Introduction Introduction, optical measurements, shape measurements, errors,

definition of the meter, sampling theorem

2 26.10. Wave optics Basics, polarization, wave aberrations, PSF, OTF

3 02.11. Sensors Introduction, basic properties, CCDs, filtering, noise

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

5 16.11. Interferometry I Introduction, interference, types of interferometers, miscellaneous

6 23.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods

7 30.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods

8 07.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,

Scheimpflug setup

9 14.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology

10 21.12. Holography Introduction, holographic interferometry, applications, miscellaneous

11 11.01. Measurement of basic

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

12 18.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy

13 25.01. Metrology of aspheres

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

14 01.02. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous

15 08.02. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method

3

Content

Interferogram examples

Coherence

Young interferometer

Axial coherence

Interpretation of interferograms

Fringe evaluation methods

Wave aberrations in optical systems

Testing with Twyman-Green Interferometer

Short common path,

sensible setup

Two different operation

modes for reflection or

transmission

Always factor of 2 between

detected wave and

component under test

detector

objective

lens

beam

splitter 1. mode:

lens tested in transmission

auxiliary mirror for auto-

collimation

2. mode:

surface tested in reflection

auxiliary lens to generate

convergent beam

reference mirror

collimated

laser beam

stop

5

Interferometer

Ligo interferometer

Michelson type

Caltech, arm lengths 4 km

Measurement of gravitational waves

Nobel price 2017

Ref: S. Balmer

Interferograms of Primary Aberrations

Spherical aberration 1

-1 -0.5 0 +0.5 +1

Defocussing in

Astigmatism 1

Coma 1

6

7

Typical Interferometer Output

Digital output

Ref: R. Kowarschik

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. Drband

8

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

9

Interferometry

Color fringes of a broadband interfergram

Ref: B. Drband

10

11

Interferogram

Example Interferogram of a plate with step

Ref.: H. Naumann

Interferometric measurement with angles 0and 90 Layered structure confirmed

Interferometry 90- Rotation of Sample

axial

50mm

shadow image

transverse

12

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

13

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

]

Lichtquellengre [mm]size of lightsource in [mm]

rms of

field

difference

14

Shearing Interferograms

Typical shearing interferograms

of some simple aberrations

15

Coherence in Optics

Statistical effect in wave optic:

start phase of radiating light sources are only partially coupled

Partial coherence: no rigid coupling of the phase by superposition of waves

Constructive interference perturbed, contrast reduced

Mathematical description:

Averagedcorrelation between the field E at different locations and times:

Coherence function G

Reduction of coherence:

1. Separation of wave trains with finite spectral bandwidth D

2. Optical path differences for extended source areas

3. Time averaging by moved components

Limiting cases:

1. Coherence: rigid phase coupling, quasi monochromatic, wave trains of infinite length

2. Incoherence: no correlation, light source with independent radiating point like molecules

16

Coherence Function

Coherence function: Correlation

of statistical fields (complex)

for identical locations :

intensity

normalized: degree of coherence

In interferometric setup, the amount of describes the visibility V

Distinction:

1. spatial coherence, path length differences and transverse distance of points

2. time-related coherence due to spectral bandwidth and finite length of wave trains

ttrEtrErr ),(),(),,( 2

*

121

z

x

x1

x2

E(x2)

E(x1)Dx

r r r1 2

)()(

),,(),,()(

21

212112

rIrI

rrrr

)(),( rIrr

17

2

2

0 cos4)(z

xDIxI

D

zx 2

D

screen with

pinholes

detector

source

z2

region of interference

z2

x

D

Double Slit Experiment of Young

Young interference experiment:

Ideal case: point source with distance z1, ideal small pinholes with distance D

Interference on a screen in the distance z2 , intensity

Width of fringes

18

19

Young Interferometer

Division of the light from a source by two pinholes or two slits

Ref: R. Kowarschik

P1

S

A B

Q

P2

s1

s2

x

y

z D

a

D

z1

z2

light source

screen with slits

distance D

detector

x

Dx

Double Slit Experiment of Young

= 0 = 0.15 = 0.25 = 0.35 = 0.40 = 0.30

Partial coherent illumination of a double pinhole/double slit

Variation of the size of the source by coherence parameter

Decreasing contrast with growing

Example: pinhole diameter Dph = Dairy / distance of pinholes D = 4Dairy

20

Coherence Measurement with Young Experiment

Typical result of a double-slit experiment according to Young for an Excimer laser to

characterize the coherence

Decay of the contrast with slit distance: direct determination of the transverse coherence

length Lc

21

Typical contrast decay of Young double slit setup

Significant difference between the two orientations x/y

Perpendicular to slit window of laser cavity: sinc-type behaviour

Good agreement between Young contrast and Wigner measurement

0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000 1200

pinhole spacing

co

ntr

ast

Young-horizontal

Young_vertical

H-Gauss-fit after WDF

V-Gauss-fit after WDF

Excimer Laser: Laterale Coherence

22

D

ccl cc

Axial Coherence Length

starting

phase

in phase

2 time t

1

phase difference

180

Two plane waves with equal initial phase and differing wavelengths 1, 2

Idential phase after axial (longitudinal) coherence length

23

Temporal Coherence

t

U(t)

c

duration of a

single train

Damping of light emission:

wave train of finite length

Starting times of wave trains: statistical

24

tDD /1

t

tA

D

D

sin)(

deAtE ti2)()(

Temporal Coherence

I()

Radiation of a single atom:

Finite time Dt, wave train of finite length,

No periodic function, representation as Fourier integral

with spectral amplitude A()

Example rectangular spectral distribution

Finite time of duration: spectral broadening D,

schematic drawing of spectral width

25

Axial Coherence Length of Lightsources

Light source

lc

Incandescent lamp

2.5 m

Hg-high pressure lamp, line 546 nm

20 m

Hg-low pressure lamp, line 546 nm

6 cm

Kr-isotope lamp, line at 606 nm

70 cm

HeNe - laser with L = 1 m - resonator

20 cm

HeNe - laser, longitudinal monomode stabilized

5 m

26

Axial Coherence

Contrast of a 193 nm excimer laser for axial shear

Red line: Fourier transform of spectrum

contrast

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

-0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8

z-shift

in mm

measured

FFT-Data

27

Michelson-Interferometer

receiverfirst mirror from

source

signal

beam

reference

beam

beam

splitter

second

mirror

moving

overlap

lc

z z

relative

moving

I(z)

wave trains

with finite

length

Michelson interferometer: interference of finite size wave trains

Contrast of interference pattern allows to measure the axial coherence length/time

28

Young Experiment with broad Band Source

Realization with movable triple mirror

beam

splitter0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

-400 -300 -200 -100 0 100 200 300 400

x

contrast

laser

reference

mirror

movable

triple mirror

detector

scan

x

contrast

curve

interferogram

x

y

I(x,y)

29

d

xz

DD

2

Interpretation of Interferograms

D xd

Distance between fringes: d

Bending of fringes: Dx

Relation of surface error Dz

accross diameter

30

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

31

32

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

33

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 31

( , ) ( , ) ( , ) ( , ) ( , )4

S x y W x y W x y W x y W x y

Absolute Calibration of Interferometer

34

35

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

36

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

37

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

38

Fringe Tracking for Fringe Evaluation

Skeletonizing method

Ref: W. Osten

interferogram segmentation

improved

segment skeleton

phase map

39

Interferogram Examples

More complicated sample

Ref: Elta systems

40

Fourier Method of Fringe Evaluation

Intensity in interferogram

Substitution

gives

Fourier transform

interpretation:

A: low frequencies, background

C, C* : same information

Filtering with bandpass H(,):

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, background

- filtering of noise and asymmetrical contribution

Phase information

*( , ) ( , ) ( , ) ( , )I v A v C v C v

Im ( , )arctan

Re ( , )

C

C

j

| I(,) |

spatial

frequency

A(,)

C (,)* C (,)

filter-

function

H(,)

Fourier Method of Fringe Evaluation

41

42

Fourier Method of Fringe Evaluation

Fourier method

Ref: W. Osten

interferogram amplitude filtered amplitude

wrapped phase phase mapunwrapped phase

43

Sample Interferograms

Modulated surface

(left: ripple plate)

Test piece with constant

high frequency ripple

(right: Fourier spectrum)

Ref: B. Drband

44

Carrier Frequency Interferogram

Evaluation of data

With filtering step

Ref: B. Drband

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

45

46

Carrier Method of Fringe Evaluation

Ref: W. Osten

interferogram

interferogram

with carrier

amplitude

spectrum

spectrum filtered

and shifted

unwrapped

phaseunwrapped phase

47

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

48

Phase Shifting Method

Diversification of one parameter

Elimination and calibration by subtraction

Various technical 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

49

50

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

51

52

Phase Shifting Method for Fringe Evaluation

Ref: W. Osten

I2(90)

unwrapped

phase

I4(270)

I3(180) I1(0)

wrapped phase

53

Wave Aberrations in Optical Systems

Wave aberration in optical systems

Definition in exit pupil

Reference on chief ray and ideal sphere

Only for one object point and one wavelength

Raytrace into image plane

Backpropagation onto reference sphere

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

y'p

x'p

yp

xp x'

y'

z

yo

xo

object plane:

one point

one wavelength

entrance pupil

equidistant gridexit pupil

transferred grid

image planeoptical

system

surface of equal phase:

reference sphere

wave aberration

reference point

raytraceback to reference sphere

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 mnm

n j

j

j

54

Performance Description by Zernike Expansion

Vector of cj

linear sequence with runnin g index

Sorting by symmetry

0 1 2 3 4 -1 -2 -3 -4

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

circular

symmetric

m = 0cos terms

m > 0

sin terms

m < 0

cj

m

55

Changes of z-distance changes Zernikes

Relevant applications: 1. Human eye, iris pupil not accessible 2. Microscopic lens, exit pupil not accessible

Possible solution to determine the exact pupil phase front: 1. Calculation of Zernike changes by numerical propagation 2. Pupil transfer relay optical system

For a phase preserving transfer, a well corrected 4f-system is necessary A simple one-lens imaging generates a quadratic phase in the image plane

Zernike Coefficients in Different z-Planes

56

chief

ray

exit

pupil

rear

stopobject

plane

pupil

retina

fovea

cornea

iris

optical disc

blind spotcrystalline lens

lens capsule

anterior

chamber

posterior

chamber

vitreous

humor

temporal

nasal

final

plane

starting

plane

f1

f1

f2

f2

d'd

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

57

Zernike Calculation on distorted grids

Zernike Expansion of Local Deviations

Small Gaussian bump in

the topology of a surface

Spectrum of coefficients

for the last case

model

error

N = 36 N = 64 N = 100 N = 144 N = 225 N = 324 N = 625

original

Rms = 0.0237 0.0193 0.0149 0.0109 0.00624 0.00322 0.00047

PV = 0.378 0.307 0.235 0.170 0.0954 0.0475 0.0063

0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

58

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)( 249 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

Dc9 / c

9

Zernike Coefficients for Wrong Normalization

59