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Metrology and Sensing

Lecture 5: Interferometry I

2018-11-16

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

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

13 25.01.Metrology of aspheres

and freeformsAspheres, 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

Introduction

Interference

Types of interferometers

4

Interferometry

Basic idea:

- separation of a wave into two beams

(test and reference arm)

- every beam surpasses different paths

- superposition and interference of both beams

- analysis of the pattern

Different setups for:

- the beam splitting

- the superposition

- the referencing

Different path lengths

Difference equivalent of one fringe

Measurement of plates:

Haidinger fringes of equal inclination

Newton fringes of equal thickness

Ref: W. Osten

1 1 2 2 wn t n t N t

2wt

n

detector

beam

splitter

reference mirror

collimated

laser beam

surface

under

test

5

Classification of Interferometers

Division of amplitude: - Michelson interferometer

- Mach-Zehnder interferometer

- Sagnac interferometer

- Nomarski interferometer

- Talbot interferometer

- Point diffraction interferometer

Division of wavefront: - Young interferometer

- Rayleigh interferometer

Division of source: - Lloyds mirror

- Fresnel biprism

Ref: R. Kowarschik

6

Classification of Interferometers

Two-beam interferometers: - Michelson

- Twyman Green

- Sagnac

- Young

- Mach-Zehnder

- Rayleigh

- Fizeau

- Shearing

- Mireau

- Linnik

Multi-beam interferometers: - Fabry-Perot

- Lummer-Gehrke

Ref: R. Kowarschik

7

Localization of Fringes

Interference volume for a plate

Interference volume for a wedge

Ref: R. Kowarschik

volume of

interference

fringes

incident

light back side

reflectedfront side

reflected

volume of

interference

fringes

incident

light

back side

reflected

front side

reflected

8

Interference of Two Waves

Superposition of two plane waves:

1. Intensity

2. Phase difference

Spacing of fringes

Interference of two spherical waves

More complicated geometry

),,(cos2²²),,( 2121 zyxAAAAzyxI

rkkzyxzyxzyx

)(),,(),,(),,( 1212

Ref.: B. Dörband

2sin2

ns

9

Two Beam Interference

Interference of two point source spherical waves with perturbations

10

Two Beam Interference

Interference of two point source spherical waves

1. both wave are radiating outside

2. one incoming and one outgoing wave

11

Two Beam Interference

Interference of two plane waves under different directions

Fringe distance s1212

2

eenkks

12

Two Beam Interference

Interference of two plane waves

with finite spectral width w

1

0

))),,,(cos()()(2)²()²((1

),,( 2121

01

dzyxAAAAzyxI

13

Two Beam Interference

Interference of two spherical waves with finite bandwidth in x/z

Delay rotated cone of maximum contrast

bandwidth 20 nm bandwidth 60 nm bandwidth 100 nm

no

delay

delay

5 ms

14

Haidinger Fringes

Fringes of equal inclination:

Haidinger

Every inclination creates an individual delay in the plate

Two beam interference of two waves:

- propagation in the same direction

- same polarization

- phase difference smaller than axial length of coherence

Coherent superposition of waves

Difference of phase / path difference

Number of fringes

location of same phase

Conrtast

122121

2

21

cos2

IIII

EEI

122

s

sN

2

12

21

21

minmax

minmax2

II

II

II

IIK

Two Beam Interference

Two beam interference at a plane plate

- Fresnel fringes of equal thickness

- Haidinger fringes of equal inclination

Path difference

:

transparent

plane plate

detector

source

d

1

2

n

2sin2

2cos2 1

22

2

ndnds

Interference Fringes at a Plane Plate

17

Interference at a Plane-Parallel Plate

Multiple reflection superposition

Airy formulas

T: tranmittance

R: Reflectance

Ref: R. Kowarschik

n

n

n’ h’

r, t Reflection,

Transmission Coeff.

n n’

r’, t’ Reflection,

Transmission Coeff.

n’ n

’

Plane monochr.

wave

)(

22

2

)(

2sin4)1(

2sin4

ir I

RR

R

I

)(

22

2)(

2sin4)1(

it I

RR

TI

Multi beam interference

Intensity of pattern

Finesse determines the contrastd

n

1

2I( )

2m (2m+1) (2m+2)

R = 0.2

R = 0.6

R = 0.9

cos21

)1(2

2

RR

RIT

R

RF

1

2

2/1

Interference at a Plane Plate

Spectral filtering

Straylight suppression

Diameter adaptation

lensL1

lensL2

lensL3

lensL4

lensL5

LinseL6

CCD-camera

prism group

test surfaceM1

beam splitter

M1

stopB1

stopB2

stopB3

disrances1

distanceL1

distanceL2

distanceL3

distances2

D :2.5 mm D :

3.81 mm

D :10.0 mm D :

7.72 mm

D :3.81 x 9.49 mm

reference arm

straylight suppression and diameter adaptation

spectral filtering

detection

More complex Setup of an Interferometer

20

Real Interferometers

Ref: R. Kowarschik

21

Interferometers

Accuracy of interferometers

Ref: F. Hoeller

test surface

beamsplitter

reference surface

here: flat

illumination

to detector

path difference

mRrm

Test by Newton Fringes

Reference surface and test surface with nearly the same radii

Interference in the air gap

Reference flat or curved possible

Corresponds to Fizeau setup

with contact

Broad application in simple

optical shop test

Radii of fringes

22

Ref: W. Osten

spherical aberration coma

tilt astigmatism

Example Interferograms

Fizeau surface as part of the system work as reference

Fizeau surface near to test surface:

- large common path, insensitiv setup

- small cavity length

The test surface is imaged onto the detector

Fizeau Interferometer

detector

beam

splitter

collimator

plane test

surface

light

source

Fizeau

surface

stop

Fizeau Interferometer

Long common path, quite insensitive setup

Autocollimating Fizeau surface quite near to test surface, short cavity length

Imaging of test surface on detector

Straylight stop to bloc unwanted light

Curved test surface: auxiliary objective lens (aplanatic, double path)

Highest accuracy

detector

beam

splitter

collimatorconvex

surface

under test

light

source

Fizeau

surface

auxiliary lens

stop

no common path setup, sensitive

long distances, measurement of samples with small effects

beam

combiner

source

beam splitter

mirror

mirror

test arm

reference arm

detector

sample

Mach-Zehnder Interferometer

Test and reference arm separated:

setup sensitive

Both arms aligned:

fringes of equal inclination

Tilt in reference arm:

fringes of equal thickness

Setup corresponds to Twyman-Green-

interferometer

screen

reference mirror

laser source

compensator

plate

surface

under test

test beam

reference

beam

beam splitter

Michelson Interferometer

28

Michelson Interferometer

Visibility of fringes

Ref: R. Kowarschik

Haidinger Fringes Fizeau Fringes

B

S

S2’ S1’

M2

M1M2’

S

B

S2’

S1’

M2

M1

M2’

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

Straylight suppression in Twyman-

Green interferometer

Polarization of both arms by /4 plates

Analyzer in front of detector:

only signal light is passing

Optimization of azimuthal orientations

of the plates:

- reflectivity of test surface

- splitting of power in both arms

- largest contrast of interferogram

detector

lens

polarization

beam splitter

auxiliary

lens

surface under

test

reference mirror

collimated

laser beam

/ 2

plate

/ 4

plate

/ 4

plate

analyzer

Ri

1tan

RA tan

Suppression of Straylight by Polarization

Separation of wavefront:

self reference

Interferograms are looking completly different

Aperture reduced due to shear

Splitting and shift of wavefront:

- by thin plate

- by grating

d

Shearing Interferometer

source

shear

distance

Schematic drawing of sheared wavefronts

Typical interferogram

Shearing Interferometer

shear distance

wavefront W

x

Compact setup

Modified Mach-Zehnder setup with telescope

Radial Shearing Interferometer

beam splitterwavefront under

test

wavefront with

radial shear

lens

beam splitter

source

beamsplitter

mirror

mirror

test arm

reference arm

detector

telescope for

change of diameter

Separation of both arms by polarization

Shear principle

Used in microscopy for differential interference

contrast (DIC) pahse imaging

Nomarski Interferometer

condenser

object

Wollaston

prism

Wollaston

prism

compensator

objective

polarizer

analyzer

shear

distance

x

adjustment

phase

splitting

ratio

R1-R

Focussing onto a transparent plagte with pinhole

Pinhole creates a reference spherical wave

Optimization of contrast:

- size of pinhole

- numericalaperture

- transparency of the plate

Very stable setup

transparent plate

with pinhole

wavefront

under test reference

wavefront

Point Diffraction Interferometer

36

Point Diffraction Interferometer

Full setup according to Smartt

Beam splitting by stop / slit

Application:

measurement of inhomogeneities of refractive index

liquids must be in polishes glass cuvette

source detectorstop

test arm

reference arm

Rayleigh Interferometer

Jamin Interferometer

Köster Interferometer

source

test arm

reference arm

detector

source

test arm

reference arm

detector

Further Types of Interferometers

39

Fabry-Perot Interferometer

Setup of an etalon

Applications:

- spectral line resolution

- laser mode selection

Ref: R. Kowarschik

BFabry-Perot Etalon

Point source

h’

n’

40

Fabry-Perot Interferometer

Intensity

Finesse

Transmission

Contrast

Ref: R. Kowarschik

1

2

22

)(

)(

2sin

1

21

11

R

R

R

A

I

Ii

t

R

RF

1

2

max

)(

)(

11

R

A

I

Ii

t

p

2

22

min

)(

)(

max

)(

)(

41

1

1

F

R

R

I

I

I

I

C

i

t

i

t

41

Fabry-Perot Interferometer

Intrumental functions

Ref: R. Kowarschik

Properties )(W FRP

Absorption

Surface

imperfections

Finite range of

incidence

A

2

HF

FF

F

h 2

1

)(cos

1

0

AA

d

2

H

h

2

F )(cos

1

2

2

2

(sin)1(

41

1

d

R

R

R

T

d

mmm

m

),(0

)()( hfH

)cos(()( fF

Perfectly plane-

parallel plate

1R

1R

hR ,1

)(cos,1 R

42

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

zD

a

D

z1

z2

light source

screen with slits

distance D

detector

x

x

2

2

0 cos4)(z

xDIxI

D

zx 2

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

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

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)