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Optics 505 - James C. Wyant
Page 1
Optics 505 - James C. Wyant Part 9 Page 1 of 36
Part 9Phase Shifting Interferometry
• Classical Interferogram Analysis
• Phase Shifting Advantages• Phase Shifters• Algorithms• Removing Phase Ambiguities• Error Sources
• Classical Interferogram Analysis
• Phase Shifting Advantages• Phase Shifters• Algorithms• Removing Phase Ambiguities• Error Sources
Optics 505 - James C. Wyant Part 9 Page 2 of 36
Classical Analysis of Interferograms
• Classical Analysis• Measure positions of fringe
centers.• Deviations from straightness and
equal spacing gives aberration.
Surface Error = (λ/2) (∆/S)
Optics 505 - James C. Wyant
Page 2
Optics 505 - James C. Wyant Part 9 Page 3 of 36
Computer Analysis ofInterferograms
Largest ProblemGetting interferogram data into
computer
Solutions• Graphics Tablet• Scanner
• CCD Camera• Phase-Shifting Interferometry
Largest ProblemGetting interferogram data into
computer
Solutions• Graphics Tablet• Scanner
• CCD Camera• Phase-Shifting Interferometry
Optics 505 - James C. Wyant Part 9 Page 4 of 36
Advantages of Phase-ShiftingInterferometry
• 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
• 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
Optics 505 - James C. Wyant
Page 3
Optics 505 - James C. Wyant Part 9 Page 5 of 36
Phase-Shifting Interferometry
REF
TEST
INTERFEROGRAM
SOURCE1) MODULATE PHASE
2) RECORD MIN 3 FRAMES
3) CALCULATE OPD
CBA
0° 90° 180°
TAN-1 [
A - BC - B] OPD =
2πλ
Optics 505 - James C. Wyant Part 9 Page 6 of 36
Phase Shifting - Moving Mirror
Move λ/8
PZTPushingMirror
π/2PhaseShift
Optics 505 - James C. Wyant
Page 4
Optics 505 - James C. Wyant Part 9 Page 7 of 36
Phase Shifting - Diffraction Grating
+1 OrderDiffractionGrating
Move 1/4Period
π/2PhaseShift
Optics 505 - James C. Wyant Part 9 Page 8 of 36
Phase Shifting - Bragg Cell
+1 Order
BraggCell
fo
fo
fo+f
Frequencyf
0 Order
Optics 505 - James C. Wyant
Page 5
Optics 505 - James C. Wyant Part 9 Page 9 of 36
Phase Shifting - Tilted Glass Plate
π/2PhaseShift
Optics 505 - James C. Wyant Part 9 Page 10 of 36
Phase Shifting - Rotating Half-WavePlate
45° Rotation
CircularlyPolarized
Light
π/2PhaseShift
Half-Wave Plate
Optics 505 - James C. Wyant
Page 6
Optics 505 - James C. Wyant Part 9 Page 11 of 36
Four Step Method
I1(x,y) = I0 + I' cos [ φ (x,y)] φ (t) = 0
I2(x,y) = I0 - I' sin [ φ (x,y) ] φ (t) = π/2
I3(x,y) = I0 - I' cos [ φ (x,y) ] φ (t) = π
I4(x,y) = I0 + I' sin [ φ (x,y) ] φ (t) = 3π/2
I(x,y) = I0 + I' cos[φ(x,y)+ φ(t)]
Tan [φ(x,y)] = I4(x,y) -I2(x,y)
I1(x,y) -I3(x,y)
Optics 505 - James C. Wyant Part 9 Page 12 of 36
Relationship between Phase and Height
( ) ( ) ( )( ) ( )
( ) ( )yx
yxIyxI
yxIyxITanyx
,4
yx,ErrorHeight
,,
,,,
31
241
φπλ
φ
=
−−= −
Optics 505 - James C. Wyant
Page 7
Optics 505 - James C. Wyant Part 9 Page 13 of 36
Phase-Measurement Algorithms
Three Measurements
Four Measurements
HariharanFive Measurements
Carré Equation
φ = tan−1 I3 − I2I1 − I2[ ]
φ = tan−1 I4 − I2I1 − I3[ ]
φ = tan−1 2(I2 − I4)2I3 − I5 − I1[ ]
φ = tan−1 3(I2 − I3) − (I1 − I4)[ ] (I2 − I3) − (I1 − I4)[ ]( I2 + I3) − ( I1 + I4)
Optics 505 - James C. Wyant Part 9 Page 14 of 36
Phase-Measurement Algorithmfor N Intensity Measurements
αi = 2πiN for i = 1,...,N
Technique is also known as synchronous detection
N Measurements φ = -tan-1
Ii sinα ii=1
N
∑Ii cosαi
i=1
N
∑
Optics 505 - James C. Wyant
Page 8
Optics 505 - James C. Wyant Part 9 Page 15 of 36
Phase-Stepping PhaseMeasurement
MIR
RO
R P
OS
ITIO
N
A B C D
TIME
λ/2
λ/4
PHASE SHIFT
DE
TE
CT
ED
SIG
NA
L
1
0 π/2 π 3π/2 2πA B C D
Optics 505 - James C. Wyant Part 9 Page 16 of 36
Integrated-Bucket PhaseMeasurement
PHASE SHIFT
DE
TE
CT
ED
SIG
NA
L
0 π/2 π 3π/2 2πA B C D
1
MIR
RO
R P
OS
ITIO
N
A B C D
TIME
λ/2
λ/4
Optics 505 - James C. Wyant
Page 9
Optics 505 - James C. Wyant Part 9 Page 17 of 36
Integrating-Bucket and Phase-Stepping Interferometry
Measured irradiance given by
Integrating-Bucket ∆=αPhase-Stepping ∆=0
Ii = 1∆ Io 1+ γ o cosφ + αi (t)[ ]{ }dα(t)
αi −∆ /2
αi +∆ /2
∫= Io 1+ γ osinc ∆
2> @ cosφ + αi[ ]{ }
Optics 505 - James C. Wyant Part 9 Page 18 of 36
Phase Ambiguities
• If we know sign of Sin and sign of Cosine the Arc Tangent is calculated modulo 2π.
• Must correct for 2π ambiguities.
Optics 505 - James C. Wyant
Page 10
Optics 505 - James C. Wyant Part 9 Page 19 of 36
Typical Fringes For Spherical Surfaces
Fringes Phase map
Optics 505 - James C. Wyant Part 9 Page 20 of 36
Phase Ambiguities-Before Integration
2 π Phase Steps
Optics 505 - James C. Wyant
Page 11
Optics 505 - James C. Wyant Part 9 Page 21 of 36
Phase Ambiguities- After Integration
Phase Steps Removed
Optics 505 - James C. Wyant Part 9 Page 22 of 36
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
• 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
Optics 505 - James C. Wyant
Page 12
Optics 505 - James C. Wyant Part 9 Page 23 of 36
Error Sources
• Incorrect phase shift between data frames• Vibrations• Detector non-linearity• Stray reflections• Quantization errors • Frequency stability• Intensity fluctuations
Optics 505 - James C. Wyant Part 9 Page 24 of 36
Three π/2 Steps
�S 0 S
Phase measured
-0.04
-0.02
0
0.02
0.04
es
ah
Pr
or
rE
+s
na
id
aR
/
Peak �Valley Error +Radians / 0.0785398
Phase error due to 5% phase sh i ftcalibration error .
Optics 505 - James C. Wyant
Page 13
Optics 505 - James C. Wyant Part 9 Page 25 of 36
Four π/2 Steps
�S 0 S
Phase measured
-0.04
-0.02
0
0.02
0.04e
sa
hP
ro
rr
E+
sn
ai
da
R/
Peak �Valley Error +Radians / 0.0785302
Phase error due to 5% phase sh i ftcalibration error .
Optics 505 - James C. Wyant Part 9 Page 26 of 36
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.
Optics 505 - James C. Wyant
Page 14
Optics 505 - James C. Wyant Part 9 Page 27 of 36
Example of Algorithm Derivation
Averaging Technique4-FRAME (offset = 0) 4-FRAME (offset = π/2)
frames # 1,2,3,4 frames# 2,3,4,5
=2I -4 I
1I - 3I
N1D1
=2I -4 I
5I - 3I
N2D2
5-FRAME
I2-I+I
)I-I2(DDNN
=tan351
24
21
21 =++ϕ
Optics 505 - James C. Wyant Part 9 Page 28 of 36
Schwider-Hariharan Five π/2 Step Algorithm
�S 0 S
Phase measured
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
es
ah
Pr
or
rE
+s
na
id
aR
/
Peak �Valley Error +Radians / 0.0030859
Phase error due to 5% phase s hif tcalibration error .
Optics 505 - James C. Wyant
Page 15
Optics 505 - James C. Wyant Part 9 Page 29 of 36
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.
Optics 505 - James C. Wyant Part 9 Page 30 of 36
Error due to vibration for Schwider-Hariharan algorithm
0 0.2 0.4 0.6 0.8 1Frequency +in units of frame rate /
0
0.05
0.1
0.15
0.2
0.25
es
ah
Pr
or
rE
+s
na
id
aR
/
P�V phase error due to 0. 02 zero to pea kwaves of vibration
Optics 505 - James C. Wyant
Page 16
Optics 505 - James C. Wyant Part 9 Page 31 of 36
Best way to fix vibration problem
• Reduce vibration• Take data fast• Take all frames at once• Measure vibration and introduce
vibration 180 degrees out of phase to cancel vibration
Optics 505 - James C. Wyant Part 9 Page 32 of 36
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.
Optics 505 - James C. Wyant
Page 17
Optics 505 - James C. Wyant Part 9 Page 33 of 36
Schwider-Hariharan
�S 0 S
Phase measured
-0.002
-0.001
0
0.001
0.002
es
ah
Pr
or
rE
+s
na
id
aR
/Peak �Valley Error +Radians / 0.00480903
Phase error due to 1% detector non li near i tyof order 3.
Optics 505 - James C. Wyant Part 9 Page 34 of 36
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.
Optics 505 - James C. Wyant
Page 18
Optics 505 - James C. Wyant Part 9 Page 35 of 36
Error – Stray Reflections
Stray Beam
Test Beam
Measured Beam
Correct Phase
Phase Error
Optics 505 - James C. Wyant Part 9 Page 36 of 36
Phase-Shifting Interferometer
LASER
TEST SAMPLE
IMAGING LENS
PZT PUSHING MIRROR
BS
DETECTOR ARRAY
DIGITIZERPZT
CONTROLLER
COMPUTER