ERROR ANALYSIS FOR CGH OPTICAL TESTING

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


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.


CGHs in Optical Interferometry

DIVERGERLENS


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.


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;


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.


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


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;


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.



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



• 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



• 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


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)


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)


Diffraction Wavefront Phase as a Function of Duty-cycle and Phase Depth

phase grating at m=0


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



m=2


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


Phase Grating Sample


Chrome-on-glass Grating(Top view)

Duty-cycle = 40%Spacing = 50 um

Duty-cycle = 50% Spacing = 50 um

20um gap

D = 40% D = 50%


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


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


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.


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)


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)


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)


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


(Sample Phase CGH)


+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)


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%


(Sample Chrome CGH)




Wavefront PhaseErrors

RMS Substrate Figure Error*(CGH Surface)

/10 (/5)*

Pattern Distortion 1 um /100

Chrome Thickness Error 2 nm 0


Root-Sum-Squared Errors : /100


Wavefront PhaseErrors

Pattern Distortion 1 um 0

Chrome Thickness Error 2 nm 0


Root-Sum-Squared Errors : 0

(Sample Chrome CGH)


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


Fizeau interferometer

Spherical reference

Test plate

(Sample Chrome CGH )



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


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.

ERROR ANALYSIS FOR CGH OPTICAL TESTING

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