ERROR ANALYSIS FOR CGH OPTICAL TESTING
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Optical Sciences Center The University of Arizona
ERROR ANALYSIS FOR CGH OPTICAL TESTING
Yu-Chun Chang and James Burge
Optical Science Center
University of Arizona
Optical Sciences Center The University of Arizona
Applications of CGH in Optical Testing
• Optical interferometry measures shape differences between a reference and the test piece;
• Test pieces with complex surface profiles require reference surfaces with matched shapes or null lenses;
• Using CGHs to produce reference wavefronts eliminates the need of making expensive reference surfaces or null optics.
Optical Sciences Center The University of Arizona
CGHs in Optical Interferometry
DIVERGERLENS
Optical Sciences Center The University of Arizona
CGHs in Optical Interferometry
• Quality of the wavefront generated by CGHs affects the accuracy of interferometric measurements;
• Abilities to predict and analyze these phase errors are essential.
Optical Sciences Center The University of Arizona
CGH Fabrication Errors
• Traditional fabrication method is done through automated plotting and photographic reduction;
• Modern technique uses direct laser/electron beam writing;
• Fabrication uncertainties are mostly responsible for the degradation of the quality of CGHs;
Optical Sciences Center The University of Arizona
Sources of Errors CGH Fabrication
• A CGH may simply be treated as a set of complicated interference fringe patterns written onto a substrate material;
• CGH substrate figure errors;
• CGH pattern errors includes;– fringe position errors;
– fringe duty-cycle errors;
– fringe etching depth errors.
Optical Sciences Center The University of Arizona
Substrate Figure Errors
• Typical CGH substrate errors are low spatial frequency surface figure errors;
• Produce low spatial frequency wavefront aberrations in the diffracted wavefront.
CGH substrate
Transmitted wavefrontReflected wavefront
Incident wavefront
ss
(n = index of refraction)
(n-1)s
Optical Sciences Center The University of Arizona
Pattern Distortion
• The hologram used at mth order adds m waves per line;
• CGH pattern distortions produce wavefront phase error:
)y,x(S
)y,x(m)y,x(W
(x,y) = grating position error in direction perpendicular to the fringes;S(x,y) = localized fringe spacing;
Optical Sciences Center The University of Arizona
Binary Linear Grating Model
• Binary linear grating models are used to study grating duty-cycle and etching depth errors;
• Scalar diffraction theory is used for wavefront phase and amplitude calculations;
• Both phase gratings and chrome-on-glass gratings are studied;
• Analytical results are achieved.
Optical Sciences Center The University of Arizona
Binary Linear Grating Model
S
xcomb
S
1
b
xrect)AeA(A)x(u 0
i1o
• Output wavefront from a binary linear grating (normally incident plane wavefront):
bA1ei
A0
S
x
where A0 and A1 are amplitude functions and is phase depth
Optical Sciences Center The University of Arizona
Binary Linear Grating Model
• Diffraction wavefront function at Fraunhofer plane:
,...2,1m;)mD(csinD)sin(Ai)mD(csinDA)cos(A
0m;D)sin(AiDA)cos(AA
)(U
101
1010
where .z
'x
S
bD
Optical Sciences Center The University of Arizona
Binary Linear Grating Model
• Diffraction efficiency functions:
• Wavefront phase functions:
)cos()D1(DAA2DA)D1(A0m 1022
122
00m
)mD(csinD)cos(AA2AA,...2,1m 2210
21
200m
0m)cos(DA)D1(A
)sin(DAtan
10
1
,...2,1m)mD(csin)]cos(AA[
)mD(csin)sin(A)tan(
10
1
Optical Sciences Center The University of Arizona
Diffraction Efficiency for Zero (m=0) Diffraction Orders
0%
20%
40%
60%
80%
100%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
phase depth [waves]
effic
ien
cy D=10%
D=20%
D=30%
D=40%
D=50%
D=60%
D=70%
D=80%
D=90%
(Phase Grating)
Optical Sciences Center The University of Arizona
Diffraction Efficiency for Non-zero Diffraction Orders
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
duty-cycle
eff
icie
ncy
m=1
m=2
m=3
m=4
m=5
(Phase Grating)
Optical Sciences Center The University of Arizona
Diffraction Wavefront Phase as a Function of Duty-cycle and Phase Depth
phase grating at m=0
Optical Sciences Center The University of Arizona
Wavefront Phase vs. Etching Depthfor Non-zero Order Beams
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1phase depth [waves]
ph
ase
[w
av
es]
Duty-cycle: 0% - 100%
m=1
Duty-cycle: 50% - 100%
Duty-cycle: 0% - 50%
m=2
Optical Sciences Center The University of Arizona
Wavefront Phase vs. Duty-cycle for Non-zero Order Beams
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
ph
ase
[w
ave
s]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
ph
ase
[w
ave
s]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
ph
ase
[w
ave
s]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
ph
ase
[w
ave
s]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
ph
ase
[w
ave
s]
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
ph
ase
[w
ave
s]
m=4 m=5 m=6
m=1 m=2 m=3
Optical Sciences Center The University of Arizona
Phase Grating Sample
Optical Sciences Center The University of Arizona
Chrome-on-glass Grating(Top view)
Duty-cycle = 40%Spacing = 50 um
Duty-cycle = 50% Spacing = 50 um
20um gap
D = 40% D = 50%
Optical Sciences Center The University of Arizona
Interferograms Obtained at Different Diffraction Orders
-1
-0.5
0
0.5
1
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
-1
-0.5
0
0.5
1
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
(for chrome-on-glass grating)
m=0
Optical Sciences Center The University of Arizona
Wavefront Phase Sensitivity Functions
• Wavefront phase sensitivities to grating duty-cycle and phase depth.
cos)D1(DAA2)D1(ADA
cos)D1(DAADA
cos)D1(DAA2)D1(ADA
sinAA
D
:0m
1022
022
1
1022
10m
1022
022
1
100m
cosAA2AA
cosAAA
0D
:,...2,1m
1020
21
10210m
0m00 , for sinc(mD)=0
otherwise
Optical Sciences Center The University of Arizona
Wavefront Phase Sensitivity Functions
• Wavefront phase sensitivity functions provide an easy solution for CGH fabrication errors analysis;
• Applications of wavefront phase sensitivity functions in optical testing are given.
Optical Sciences Center The University of Arizona
CGH Errors Analysis Using Wavefront Sensitivity Functions
Fizeau interferometer
Phase type CGH Asphere Test
Piece
Spherical reference
Parameters Values
Grating Type Binary Phase Grating
Material Glass: n = 1.5
Operating Mode Transmission
Diffraction Order 1st order
Averaging Grating Period 40 um
Substrate Figure Errors /10 rms
Pattern Distortion 1 um
Grating Groove Depth 0.5 5%
Grating Duty-cycle 50% 2%
(Sample Phase CGH)
Optical Sciences Center The University of Arizona
Sources of Errors
• Wavefront errors come from:– Surface figure * (n-1)
– Pattern distortion/spacing
– Etch depth variation * sensitivity from diffraction analysis
– duty cycle variation * sensitivity from diffraction analysis
• RSS all terms give test error due to CGH
(Sample Phase CGH)
Optical Sciences Center The University of Arizona
Wavefront Phase Sensitivities to Grating Phase Depth Errors
-2
-1
0
1
2
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
d/d
(p
ha
se/
ph
ase
de
pth
)[w
ave
s/w
ave
s]
D=10% D=20%
D=30% D=40%
D=50% D=60%
D=70% D=80%
D=90%
grating phase depth [waves]
(Phase Grating at Zero-order Diffraction)
Optical Sciences Center The University of Arizona
Source of ErrorsFabricationTolerances
Wavefront PhaseErrors per Pass
RMS Substrate Figure Error(Front Surface)
/10 /20
RMS Substrate Figure Error(Back Surface)
/10 /20
Pattern Distortion 1 um /40
Grating Groove Depth Error 5% /80
Duty-cycle Error 2% 0
Root-Sum-Squared Errors : 0.076
CGH Errors Analysis Using Wavefront Sensitivity Functions
(Sample Phase CGH)
Optical Sciences Center The University of Arizona
+1 order
0 order
-1 order
incident
refraction
CONVEX ASPHERE
TEST PLATE
hologram ringpattern
REFERENCE BEAM:REFLECTED FROM HOLOGRAM
TEST BEAM:ZERO-ORDER THROUGH CGH,REFLECT FROM ASPHERE,BACK THROUGH CGH AT
ZERO ORDER
AT -1 ORDERST
CGH Errors Analysis Using Wavefront Sensitivity Function
(Sample Chrome CGH)
Optical Sciences Center The University of Arizona
Parameters Values
Grating Type Binary Chrome-on-glass Grating
Material (chrome) nchrome = 3.6-i4.4
Material (glass) nglass = 1.5
Reference Beam -1 reflected order (glass-cr)
Test Beam 0 transmitted order (glass-cr)
Averaging Grating Period 100 um
Substrate Figure Errors /10 rms
Pattern Distortion 1 um
Chrome Thickness 50 nm 2 nm
Grating Duty-cycle 20% 2%
CGH Errors Analysis Using Wavefront Sensitivity Function
(Sample Chrome CGH)
Optical Sciences Center The University of Arizona
CGH Errors Analysis Using Wavefront Sensitivity Functions
Source of ErrorsFabricationTolerances
Wavefront PhaseErrors
RMS Substrate Figure Error*(CGH Surface)
/10 (/5)*
Pattern Distortion 1 um /100
Chrome Thickness Error 2 nm 0
Duty-cycle Error 2% 0
Root-Sum-Squared Errors : /100
Source of ErrorsFabricationTolerances
Wavefront PhaseErrors
Pattern Distortion 1 um 0
Chrome Thickness Error 2 nm 0
Duty-cycle Error 2% 0
Root-Sum-Squared Errors : 0
(Sample Chrome CGH)
Optical Sciences Center The University of Arizona
Wavefront Phase Sensitivities Functions
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
duty-cycle
d/d
D [w
aves
/1%
duty
-cyc
le c
hang
e]
t=50nm
t=80nm
t=100nm
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%duty-cycle
d/d
t [w
ave
/nm
]
t=50nm
t=80nm
t=100nm
(Chrome-on-Glass Grating at Zero-order Diffraction)
Duty-cycle Errors Etching Depth Errors
Optical Sciences Center The University of Arizona
Fizeau interferometer
Spherical reference
Test plate
(Sample Chrome CGH )
CGH Errors Analysis Using Wavefront Sensitivity Function
Source of ErrorsFabricationTolerances
SensitivityWavefront Phase
Errors
Chrome Thickness Error 2 nm 0.0015815 /nm 0.003163
Duty-cycle Error 2% 0.001897 /1%duty-cycle 0.003794
Root-Sum-Squared Errors :
0115.0004938.0001.0 222
Optical Sciences Center The University of Arizona
Conclusions
• Wavefront phase deviations due to CGH fabrication errors are studied;
• Analytical solutions are obtained and verified with experimental results;
• Applications of wavefront sensitivity functions in optical testing are demonstrated;
• Wavefront sensitivity functions provide a direct and intuitive method for CGH error analysis and error budgeting.