Direct Phase Measurement
Interferometry and Optical Testing
• Phase-Stepping and Phase-Shifting
–Basic Concept
–Phase Shifters
–Algorithms
–Phase-unwrapping
–Phase shifter calibration
–Errors
• Spatial synchronous and Fourier methods
• Solving vibration problems
• Multiple wavelength and vertical scanning
(Coherence Probe) techniques
Advantages of Phase-Shifting
Interferometry
• High measurement accuracy (>1/1000
fringe, fringe following only 1/10 fringe)
• Rapid measurement
• Good results with low contrast fringes
• Results independent of intensity variations
across pupil
• Phase obtained at fixed grid of points
• Easy to use with large solid-state detector
arrays
Phase Shifting Interferometry
CCD
Reference
Mirror
Beamsplitter
Sample
90o phase shift
between measurements
Phase Shifting - Moving Mirror
Move λ/8
π/2
Phase
Shift
PZT
Pushing
Mirror
Phase Shifting - Diffraction Grating
Diffraction
Grating
+1 Order
π/2
Phase
Shift
Move 1/4
Period
Phase Shifting - Bragg Cell
Bragg
Cell
fo+f
+1 Order
fo
fo
Frequency
f
0 Order
Phase Shifting - Rotating Half-Wave
Plate
Circularly
Polarized
Light
Half-Wave Plate
45° Rotation
π/2
Phase
Shift
Geometric Phase Shifter 1
λ/2 at
θo
Polarizer
at 45o
Input: Reference
and test beams
have orthogonal
linear polarization
at 0o and 90o.
λ/4 at
45o
λ/4 at
-45o
If the half-wave plate is rotated an angle θ the phase difference
between the test and reference beams changes by 4θ.
Geometric Phase Shifter 2
Input: Reference
and test beams
have orthogonal
linear polarization
at 0o and 90o.
λ/4 at
45o
Polarizer
at θo
If the polarizer is rotated an angle θ the phase difference between the
test and reference beams changes by 2θ.
Zeeman Laser
Two frequencies
ν and ν + Δν
Laser
Two frequencies have
orthogonal polarization
Frequency Shifting Source
Phase = (2 π/λ) (path difference) = (2 π/c) ν (path difference)
Laser
Phase shift = (2 π/c) (frequency shift) (path difference)
Four Step Method
phase shift
I(x,y) = Idc + Iac cos[φ(x,y)+ φ(t)]
measured object phase
I1(x,y) = Idc + Iac cos [φ (x,y)]
I2(x,y) = Idc - Iac sin [φ (x,y)]
I3(x,y) = Idc - Iac cos [φ (x,y)]
I4(x,y) = Idc + Iac sin [φ (x,y)]
Tan[φ (x, y )] =
φ (t) = 0
(0°)
= π/2 (90°)
=π
(180°)
= 3π/2 (270°)
I 4 (x, y ) − I 2 (x, y )
I1 (x, y ) − I 3 (x, y )
Relationship Between Phase Heights
⎡ I 4 ( x, y) − I 2 ( x, y) ⎤
⎥
⎣ I1 ( x, y) − I 3 ( x, y) ⎦
φ (x, y) = Tan-1 ⎢
Height ( x, y) =
λ
φ ( x, y)
4π
Phase-measurement Algorithms
[
]
[
]
Three Measurements
φ = tan−1 II3 −− II2
1
2
Four Measurements
φ = tan−1 II4 −− II2
1
3
Schwider-Hariharan
Five Measurements
φ = tan−1 2I −2 I −4 I
3
5
1
[2(I − I ) ]
Carré Equation
⎡ [3(I2 − I3 ) − (I1 − I4 )][(I2 − I3 ) − (I1 − I4 )] ⎤
φ = tan−1 ⎢
⎥
( I2 + I3 ) − ( I1 + I4 )
⎣
⎦
λ/2
λ/4
A
B
C
D
DETECTED SIGNAL
MIRROR POSITION
Phase-Stepping Phase Measurement
1
A
0
TIME
B
C D
π/2 π 3π/2 2π
PHASE SHIFT
λ/2
λ/4
A
B
C
D
DETECTED SIGNAL
MIRROR POSITION
Integrated-Bucket Phase
Measurement
1
A
0
TIME
B
C
D
π/2 π 3π/2 2π
PHASE SHIFT
Integrating-Bucket and PhaseStepping Interferometry
Measured irradiance given by
Ii = 1
Δ
α i +Δ/2
Io {1 + γ o cos[φ + αi (t)]}dα (t)
∫
α −Δ/2
i
{
}
[]
= Io 1 + γ osinc Δ
2 cos[φ + αi ]
Integrating-Bucket Δ=α
Phase-Stepping
Δ=0
Typical Fringes
For Spherical Surfaces
Fringes
Phase map
Phase Ambiguities
-Before Unwrapping
2 π Phase Steps
Phase Multiple Solutions
Phase
4π
2π
2π
X axis
Removing Phase Ambiguities
• Arctan Mod 2π (Mod 1 wave)
• Require adjacent pixels less than π
difference
(1/2 wave OPD)
• Trace path
• When phase jumps by > π
Add or subtract N2π
Adjust so < π
Phase Ambiguities - After Unwrapping
Phase Steps Removed
Phase-Shifting Interferometer
TEST SAMPLE
PZT
PUSHING
MIRROR
LASER
BS
IMAGING LENS
DETECTOR ARRAY
DIGITIZER
PZT
CONTROLLER
COMPUTER
Phase Shifter Calibration
Let the phase shifts be -2α, -α, 0, α, 2α
⎛ 1 I 5 ( x , y ) − I1 ( x , y ) ⎞
⎟⎟
I
x
y
I
x
y
2
(
,
)
−
(
,
)
2
⎝ 4
⎠
α = ArcCos⎜⎜
A limitation of this algorithm is that there are singularities for certain
values of the wavefront phase. To avoid errors, a few tilt fringes are
introduced into the interferogram and data points for which the numerator
or denominator is smaller than a threshold are eliminated.
It is often convenient to look at a histogram of the phase shifts. If the
histogram is wider than some preselected value it is known that there must
be problems with the system such as too much vibration present.
Error Sources
•
•
•
•
•
•
•
Incorrect phase shift between data frames
Vibrations
Detector non-linearity
Stray reflections
Quantization errors
Frequency stability
Intensity fluctuations
RMS Repeatability
Definition:
Take two frames of data and subtract
the two frames point by point to
determine rms of difference.
RMS surface height repeatability λ/1000.
(assuming good environment)
Averaging improves repeatability.
Repeatability generally limited by environment, rather than
interferometer.
Four π/2 Steps
Phase Error Compensating Techniques
Two data sets with π/2 phase shift.
• Calculate a phase for each set from algorithm, and then
average phases.
• Average Numerator and Denominator, and then
calculate phase.
Example of Algorithm Derivation
Averaging Technique
4-FRAME (offset = π/2)
4-FRAME (offset = 0)
frames # 1,2,3,4
frames# 2,3,4,5
I4 - I2 N1
=
I1 - I3 D1
I4 - I2 N2
=
I5 - I3 D2
5-FRAME
tan ϕ =
N1 + N2
2(I4 - I2 )
=
D1 + D 2 I1 + I5 - 2 I3
Schwider-Hariharan Five π/2
Step Algorithm
• Error of same double-frequency form as for
4-step algorithm, but magnitude of error
reduced by a factor of more than 25.
Error due to detector nonlinearity
• Generally CCD’s have extremely linear response
to irradiance
• Sometimes electronics between detector and
digitizing electronics introduce nonlinearity
• Detector nonlinearity not problem in well
designed system.
• Schwider-Hariharan algorithm has no error due to
2nd order nonlinearity. Small error due to 3rd
order.
Error due to Stray Reflections
• Stray reflections in laser source interferometers
introduce extraneous interference fringes.
• Stray reflections add to test beam to give a new
beam of some amplitude and phase.
• Difference between this resulting phase and phase
of test beam gives the phase error.
Error – Stray Reflections
Phase Error
Measured Beam
Stray Beam
Test Beam
Correct Phase
Quantization Error
rms phase error =
N
6
1
N = number of bits
3 2N
8
10
12
Phase error
9.0E-3 2.3E-3 5.6E-4 1.4E-4
Fringes
1.4E-3 3.6E-4 9.0E-5 2.2E-5
Surface error
(Angstroms)
4.54
1.14
0.28
0.07
Averaging further reduces error.
Reference: Chris Brophy, J. Opt. Soc. Am. A, 7, 537-541 (1990).
Spatial Synchronous and Fourier Methods.nb
1
5.6 Spatial Synchronous and Fourier Methods
Both techniques use a single interferogram having a large amount of tilt.
The interference signal is given by
irradiance#x_, y_' : iavg+1 J Cos#I#x, y' 2S f x'/
Spatial Synchronous
The interference signal is compared to reference sinusoidal and cosinusoidal signals.
The two reference signals are
rcos#x_, y_' : Cos#2S f x'
and
rsin#x_, y_' : Sin#2S f x'
Multiplying the reference signal times the irradiance signal gives sum and difference signals.
TrigReduce#irradiance#x, y' rcos#x, y''
1
cccc +2 iavg Cos#2 f S x' iavg J Cos#I#x, y'' iavg J Cos#4 f S x I#x, y''/
2
TrigReduce#irradiance#x, y' rsin#x, y''
1
cccc +2 iavg Sin#2 f S x' iavg J Sin#I#x, y'' iavg J Sin#4 f S x I#x, y''/
2
The low frequency second term in the two signals can be written as
s1
iavg J
ccccccccccccccccc Cos#I#x, y''
2
s2
iavg J
ccccccccccccccccc Sin#I#x, y''
2
Tan#I#x, y''
s2
cccccccccc
s1
The only effect of having the frequency of the reference signals slightly different from the average frequency of the
interference signal is to introduce tilt into the final calculated phase distribution.
Spatial Synchronous and Fourier Methods.nb
2
Fourier Method
The interference signal is Fourier transformed, spatially filtered, and the inverse Fourier transform of the
filtered signal is performed to yield the wavefront.
The Fourier analysis method is essentially identical to the spatial synchronous method. The irradiance can be
written as
irradiance#x, y'
iavg+1 J Cos#I#x, y' 2S f x'/
We can rewrite this as
irradiance#x, y'
1
1
iavg-1 cccc Æ2 Ç f S xÇ I#x,y ' J cccc Æ2 Ç f S xÇ I#x,y ' J1
2
2
We can take the Fourier transform of this irradiance signal and spatially filter to select the portion of the Fourier
transform around the spatial frequency f, and then take the inverse Fourier transform of this filtered signal to give
the wavefront.
FFT
FFT-1
fx
Fringes
Phase Map
Note that both the spatial synchronous method and the Fourier method require a large amount of tilt be introduced
to separate the orders. Since a spatially limited system is not band limited, the orders are never completely
separated and the resulting wavefront will always have some ringing at the edges. Also, the requirement for large
tilt always limits the accuracy of the measurement. The advantage of the techniques is that only a single
interferogram is required and vibration and turbulence cause less trouble than if multiple interferograms were
required.
Error Due to Vibration
• Probably the most serious impediment to
wider use of PSI is its sensitivity to
external vibrations.
• Vibrations cause incorrect phase shifts
between data frames.
• Error depends upon frequency of
vibration present as well as phase of
vibration relative to the phase shifting.
Best Way to Fix Vibration
Problem
•
•
•
•
•
•
•
Retrieve frames faster
Control environment
Common-path interferometers
Measure vibration and introduce vibration
180 degrees out of phase to cancel vibration
Grab all frames at once (Single Shot)
Carrier Frequency
Pixelated Array
1
Vibration Compensation Concept
• Example: Twyman-Green configuration
– Sense optical phase
– Feed back outout-ofof-phase signal to phase shifter
(Limited bandwidth)
PZT
DRIVER
PZT
Reference
Mirror
Vibration
BS
Single Mode Laser
Lens
Pupil Image Plane
Controller
Anti-Vibration Signal
“Point” Phase Sensor
Vibration Signal
Achieving High Speed Phase
Modulation
• Use polarization TwymanTwyman-Green configuration
• EOM changes relative phase between ‘S’ & ‘P’ components
– Can be very fast: 200 kHz - 1 GHz response
Reference
Mirror
S
EO
Modulator
QWP
P
Single Mode Laser
HWP
PBS
Pol
Diverger
Optics
Pupil Image Plane
Controller
& Driver
Test
Mirror
Point Phase Sensor
2
Results
Conclusions - Active Vibration
Cancellation Interferometer
System works amazingly well,
but it is rather complicated
and expensive.
3
Simultaneous PhaseMeasurement Interferometer
λ/2 Plate
Laser
Laser
λ/8 @ 0o
Reference
Beam
Test
Beam
Dielectric
B.S.
Linearly Polarized
180o
0o
270o
90o
Two fringe patterns
o
90 out of phase
Dielectric beamsplitter and phase shift upon
reflection for test and reference beams
Low - High
Low – High
&
High - Low
4
4D PhaseCam Operation
• TwymanTwyman-Green
– Two beams have
orthogonal
polarization
Diverger
• 4 Images formed
– Holographic
element
QWP
Test
Mirror
PBS
Laser
Optical Transfer CCD
& HOE
• Single Camera
– 1024 x 1024
– 2048 x 2048
• Polarization used
to produce 9090-deg
phase shifts
Holographic
Element
Phase Mask,
Polarizer &
Sensor Array
Four Phase Shifted
Interferograms on Detector
Effects of Vibration
Relative motion
between PhaseCam
and test object
Phase relationship is fixed
5
Testing a 0.5 meter diameter,
20 meter ROC mirror
0.5 m diameter mirror, 20 m ROC
5 nm rms repeatability (in air)
6
Phase Sweep at 408 Hz
0 deg
44 deg
88 deg
122 deg
168 deg
212 deg
266 deg
310 deg
360 deg
Hard Disk Platter Excited by PZT
at 408 Hz
7
Hard Disk Platter Excited by PZT
at 3069 Hz
Surface Subtraction
Subtraction in interferogram domain (diffuse and specular surfaces)
surfaces)
Stetson - 8 frame phasephase-difference
ΔZ =
⎛ [ D − B0 ][ A(t ) − C (t )] − [ A0 − C 0 ][ D(t ) − B(t )] ⎞
⎟
atan⎜⎜ 0
⎟
2
⎝ [ A(t ) − C (t )][ A0 − C 0 ] + [ D0 − B0 ][ D(t ) − B(t )] ⎠
λ
=
+
Unwrapping
A(t),B(t),C(t),D(t)
A(t),B(t),C(t),D(t)
A0,B0,C0,D0
• Not necessary to solve for random phase
8
Backplane Deformation Holder
Set Up for Backplane Deformation
Measurement
9
Backplane
Carbon Fiber Deformation
(Metrology for JWST Backplane)
10
Conclusions – Single Shot
Interferometer
•
•
•
•
Vibration insensitive, quantitative interferometer
Surface figure measurement (nm resolution)
Snap shot of surface height
Acquisition of “phase movies”
Still not perfect
Not easy to use multiple wavelength
or white light interferometry
N-Point Technique
(Carrier Frequency)
Phase shifting algorithms applied to consecutive
pixels thus requires calibrated tilt
4 pixels per fringe for 90 degree phase shift
11
Creating the Carrier Frequency
• Introduce tilt in reference beam
– Aberrations introduced due to beam
transmitting through interferometer off-axis
• Wollaston prism in output beam
– Requires reference and test beams having
orthogonal polarization
• Pixelated array in front of detector
– Special array must be fabricated
Use of Wollaston Prism to Produce
Carrier Fringes
Wollaston
Prism
Polarization
Interferometer
Detector
Polarizer
Reference and
test beams have
orthogonal
polarization
12
Pixelated Phase Sensor
• Compacted pixelated array placed
in front of detector
• Single frame acquisition
– High speed and high throughput
• Achromatic
– Works from blue to NIR
• True Common Path
– Can be used with white light
Use polarizer as phase shifter
test ref
LHC RHC
Circ. Pol. Beams (Δφ) + linear polarizer (α)
cos (Δφ + 2α)
PhasePhase-shift depends on polarizer angle
13
Array of oriented micropolarizers
α=45,
α=45, φ=90
α=0,
α=0, φ=0
Polarizer array
Matched to
detector array
pixels
Unit Cell
α=135,
α=135, φ=270
α=90,
α=90, φ=180
Electron micrograph of wire grid
polarizers
20 um
14
Array of phase-shift elements
unique to each pixel
R
Polarization
Interferometer
T
Sensor
Mask
B π/2
D
0 A
C
π
- π/2
Stacked
or
0 A B π/2
C D
- π/2
A
C
A
C
B A B
DC D
B
D
π
Circular
Phase error vs fringe tilt
Circular-4 and Stacked-4 Algorithm Error
0.016
RMS Error (waves)
0.014
Circular
0.012
Stacked
0.01
0.008
0.006
0.004
0.002
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
45 deg Tilt (waves/pixel)
15
Calculation of phase using 3 x 3
element array
Tan[θ5 ] =
2 ( I 2 + I8 − I 4 − I 6 )
− I1 − I 3 + 4 I 5 − I 7 − I 9
System Configuration
Source
Test Mirror
PBS
Camera
QWP
A
C
A
C
A
C
B
D
B
D
B
D
A
C
A
C
A
C
B
D
B
D
B
D
A
C
A
C
A
C
B
D
B
D
B
D
Pixelated Mask
QWP
Reference
Mirror
Parsing
Sensor
Array
PhasePhase-Shifted
Interferograms
16
Measurements
Every 4th pixel
Fringe pattern synthesized by
selecting every fourth pixel.
Phase map.
Measurement of 300 mm diameter,
2 meter ROC mirror
Mirror and interferometer on separate tables!
17
Measurement Results
Air burst
Phase w/reference subtraction
Heat Waves from Hot Coffee
OPD
Slope
18
Simultaneous Phase-Shifting Fizeau – Tilt Between
Reference and Test (Two Source Beams)
Reference
Wollaston
Prism
R
Laser
Imaging
Lens
T
T
Test
Surface
Stop
Polarization
Mask and
Detector
T and R have orthogonal polarization.
R reflected off reference and T reflected off test surface
are nearly parallel and give the interference fringes.
Simultaneous Phase-Shifting Fizeau – Tilt Between
Reference and Test
Reference
Half-Wave
Plate
R
Laser
Wollaston
Prism
Imaging
Lens
T
Stop
Polarization
Mask and
Detector
T
Test
Surface
R reflected off reference and T reflected off test
surface are tilted with respect to each other.
HalfHalf-wave plate makes polarization of reflected R
orthogonal to reflected T.
Wollaston prism make T and R nearly parallel.
19
Simultaneous Phase-Shifting Fizeau –
Short Coherence Length Source
Short
coherence
length
source
ΔL
ΔL
PBS
QWP
Imaging
Lens
Stop
Reference
Polarization
Mask and
Detector
Test
Surface
Interference pattern resulting from long path length
source beam reflected off reference and short path
length source beam reflected off test surface.
Test and reference beams have orthogonal polarization.
Fewer spurious fringes.
Simultaneous Phase-Shifting Fizeau – Short
Coherence Length Source – Testing Windows
Short
coherence
length
source
ΔL
ΔL
PBS
QWP
Imaging
Lens
Stop
Reference
Test
Surface
Polarization
Mask and
Detector
Testing window surfaces that are nearly parallel.
Test and reference beams have orthogonal polarization.
20
Interference Fringes Obtain
Testing a Thin Glass Plate
a) Long coherence
length source
b) Short coherence
length source
Pixelated Phase Sensor
Advantages
• Single frame acquisition
- High speed and high throughput
•Achromatic
– works from blue to NIR
• True Common Path
– Can be used with white light
• Fixed Carrier
– Faster processing and calib.
calib.
• Compact
– no bigger than camera
• Versatile
– can be interfaced to many interferometer
configurations
21
Multiple Wavelength and Vertical
Scanning Interferometry
• White Light Interferometry eliminates
ambiguities in heights present with
monochromatic interferometry
• Techniques old, but use of modern
electronics and computers enhance
capabilities and applications
How High is the Step?
Steps > λ/4 between adjacent detector pixels introduce
integer half-wavelength height ambiguities
Step > λ/4
Fringe Order ?
Interferograms
of Diffraction Grating
Red Light
White Light
Profile
Two Wavelength Measurement
• Measure Beat Frequency
• Long Effective Wavelength
1st Wavelength
2nd Wavelength
Beat - Equivalent
Wavelength
Two Wavelength Calculation
λ 1λ 2
λ1 − λ 2
Equivalent Wavelength
λ eq =
Equivalent Phase
ϕeq = ϕ1 −ϕ 2
No height ambiguities as long as height difference
between adjacent detector pixels < equivalent
wavelength / 4
Diffraction Grating Measurement
Single wavelength
(650 nm)
Equivalent wavelength
(10.1 microns)
3-D Two-Wavelength Measurement
(Equivalent Wavelength, 7 microns)
Two-Wavelength Measurement
of Step
Wavelength Correction
Compare
– Heights calculated using equivalent wavelength
– Heights calculated using single wavelength
λ eq heights
single λ heights
Add N x λ/2 to heights calculated using single wavelength so
difference < λ/4
Wavelength Correction Measurement
of Step
White Light Interference Fringes
• Fringes form bands of
contour of equal height
on the surface with
respect to the reference
surface.
• Fringe contrast will be
greatest at point of
equal path length or
“best focus.”
Principles of Vertical Scanning
Interferometry
• A difference between the reference and test
optical paths causes a difference in phase
• Best fringe contrast corresponds to zero
optical path difference
• Best focus corresponds to zero optical path
difference
Interference Microscope Diagram
Digitized Intensity
Data
Detector Array
Magnification
Selector
Beamsplitter
Illuminator
Light Source
Aperture
Stop
Controls amount
of light in
system. Keep it
fully open for
maximum light
throughput.
Translator
Microscope
Objective
Field
Stop
Controls the size of
field of view. Close it
down to position just
before clipping of the
image on the camera
occurs.
Mirau
Interferometer
Sample
Focus
Focus
Focus
Focus
Focus
Focus
Typical White Light Fringes for
Stepped Surfaces
Fringes
Phase map
Vertical Scanning Interferometer
VSI
White Light Interferograms
Focus Position A
Focus Position B
As the scan moves different areas of the part being measured come into
focus (have zero OPD or maximum contrast between fringes). A
determination of the point of maximum contrast and knowledge of the
motor position allows a reconstruction of the surface shape.
Intensity Signal Through Focus
I = I 0 [1 + γ (OPD ) cos (φ + α )]
where
Intensity
γ (OPD ) ≈
sin( OPD )
OPD
Coherence envelope
OPD
0
Typical Instrument
Light
Source
Aperture Stop
Camera
FOV
lenses
Field Stop
Stitched Measurement
VSI
mode
Step Measurement
Print Roller
Print Roller Measurement
Paper Measurement
Micromachined Silicon Measurement
Binary Optic Lens
Surface Stats:
RMS: 561.30 nm
PV: 2.37 um
493.7
um
um
1.189
400.0
0.500
300.0
0.000
200.0
-0.500
100.0
-1.185
0.0
0.0
um
100.0 200.0
375.5
Chatter Seen on Camshaft
um
3.759
Surface Stats:
Rq: 872.06 nm
Ra: 693.90 nm
Rt: 7.47 um
Terms Removed:
Cylinder & Tilt
2.500
1.000
-0.500
-2.000
-3.713
Heart Valve
X-Profile
um
469.3
384.24 nm
250.00
400.0
100.00
-50.00
300.0
-200.00
um
-343.18
0.00 50.00
200.0
150.00
250.00
350.00
419.90
Y-Profile
48.55 nm
100.0
0.00
um
0.0
-50.00
-100.00
0.0
100.0 200.0 300.0
419.9
-150.00
-219.10
0.00 50.00
150.00
Histogram
1849.79
350.00
um
469.34
Avg X PSD
Pts
86.76
1500.00
Data Statistics
Rt: 1.419 um
Ra: 87.391 nm
Rq: 113.942 nm
250.00
dB
80.00
1200.00
70.00
900.00
600.00
60.00
300.00
0.00
-69.39
nm
-30.00
10.00
69.39
52.46
0.00
1/mm
150.00
289.52
Pits in Metal
Surface Data
Mag: 10.0 X
Size: 248 X 239
Sampling: 1.70 um
Mode: VSI
um
12.707
10.000
Surface Stats:
5.000
Rq: 5.07 um
Ra: 3.44 um
Rt: 31.05 um
0.000
Terms Removed:
Tilt
-5.000
-10.000
-15.000
-18.339