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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
Physical division of amplitude
- three possibilities
Geometrical splitting by sub-apertures
Beam Splitting in Interferometers
a) Splitting by
coatings
o
e
b) Splitting by grating
orders
+1
-1
0
c) Splitting by
birefringence with
Polarization
By polarization
By interference
phase coupled light
sources
Q1
Q2 Q4Q3
polarization beam
splitterpolarization filter
phase coupled light
sources
Q1
Q2 Q4Q3
output
constructive
output
destructive
Superposition of Coherent Fields
By evanescent fields
By sub-aperture geometry
output
destructive
output
constructive
I1
I2I1 + I2
I = 0
prism coupling
device
Superposition of Coherent Fields
Geometrical superposition
Spectral superposition
sources
1
dichroitic
mirrors
2 3 4
Superposition of Incoherent Fields
10
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
11
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
12
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
13
Two Beam Interference
Interference of two point source spherical waves with perturbations
14
Two Beam Interference
Interference of two point source spherical waves
1. both wave are radiating outside
2. one incoming and one outgoing wave
15
Two Beam Interference
Interference of two plane waves under different directions
Fringe distance s1212
2
eenkks
Interference of a Double Pinhole
increasing separation D
decreasing
wavelength
Interference of a coherently illuminated double-pinhole setup
The observed pattern depends on the wavelength and the pinhole distance
17
Two Beam Interference
Interference of two plane waves
with finite spectral width w
1
0
))),,,(cos()()(2)²()²((1
),,( 2121
01
dzyxAAAAzyxI
18
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
19
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
22
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
25
Real Interferometers
Ref: R. Kowarschik
26
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
27
Ref: W. Osten
28
Newton Fringes
Movement of fringes
Determination of the OPD sign
Ref: B. Doerband
Autocollimation Principle
Spherical test surface:
- incoming and outgoing wavefront spherical
- concentric waves around center of curvature:
autocollimation
Aspherical test surface
auxiliary lensspherical test
surface
center of
curvature
wavefronts
spherical
auxiliary lens
aspherical test
surface
incoming wavefront
spherical
outcoming wavefront
aspherical
paraxial
center of
curvature
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
35
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
Grating Shearing Interferometer
Shearing interferometer with two identical Ronchi gratings with distance d
Self referencing system
Lateral shear offset d limizes transverse resolution
Interference by only the orders +1 and -1
Quite different interferogram pictures obtained
d
g g
+1
+1
-1
-1
(+1/+1)
(+1/-1)
(-1/+1)
(-1/-1)
orders
s
gdds
2sin2
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
41
Shearing Interferometer
Types of shear
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
43
Point Diffraction Interferometer
Full setup according to Smartt
44
Point Diffraction Interferometer
Setup integrated into Mach-Zehnder interferometer
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
47
Fabry-Perot Interferometer
Setup of an etalon
Applications:
- spectral line resolution
- laser mode selection
Ref: R. Kowarschik
BFabry-Perot Etalon
Point source
h’
n’
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
49
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
50
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
51
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)