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Design, Optimization and Implementation of Fresnel Domain Computer Generated Holograms (CGHs)
• Diffraction optical elements: reconstruct semi-arbitrary 2D or 3D optical fields• Numerical design: flexible encoding strategy high diffraction efficiency and uniformity • Avoid complications from conventional optical recording process• History: Detour (Brown, 1966), Kinoform (Lesem, 1969)• Applications: beam shaping, optical trapping, communications, 3D television, optical testing
- Pure phase: binary*, multi-level- Fabrication method: electron-beam lithography
2
Motivation• Increasing demand for smaller sized features, large working area semiconductor devices (e.g. LCD manufacture) need novel lithographic methods• CGHs promising candidates for replacing conventional 2D or 3D lithographic techniques• Key advantages:
Processing
In-line CGH Lithography Final Device
- Non-contact - Parallel exposure- High resolution- Large working area- 2D or 3D patterning
- Depth of focus control- Robust design- Standard fabrication- Simple optical setup- Cost effective
3
Problem Definition• Performance of CGHs depends primarily on optimization algorithm and fabrication method
• Previous work: X-ray (Jacobsen, 1992), UV (Wyrowski, 2001), EUV (Isoyan, 2006)
• Local search methods: inefficient, sensitive to initial point, get trapped at local minima
• Current multi-search schemes: optimize small size CGHsCGH Plane Reconstruction Plane
y
x
'y
'x
z
d
pix
pix'pix
'pix
sizeO
sizeO
sizeH
sizeH
Inverse Problem
11 1
iF F e
Encoding
11
idesF I e
Free parameter
Reconstruction Plane
22 2
iF F e
1FresnelO
Back-propagation
CGH Plane
22 2
iF F e
2 1F
2 H
HiH e
Desired Pattern
4
Problem Definition• Performance of CGHs depends primarily on optimization algorithm and fabrication method
• Previous work: X-ray (Jacobsen, 1992), UV (Wyrowski, 2001), EUV (Isoyan, 2006)
• Local search methods: inefficient, sensitive to initial point, get trapped at local minima
• Current multi-search schemes: optimize small size CGHsCGH Plane Reconstruction Plane
y
x
'y
'x
z
d
pix
pix'pix
'pix
sizeO
sizeO
sizeH
sizeH
Inverse Problem
11 1
iF F e
Decoding
HiH e
CGH Plane
22 2
iF F e
FresnelO
Forward-propagation
Reconstruction Plane
RiR R e
2| |
estI
Photoresist Exposure
Final Pattern
5
Reduced Complexity Hybrid Optimization Algorithm (RCHOA)
• Efficient optimization of Fresnel binary and multi-level phase CGHs
• Reduce problem complexity by introducing: Local Diffuser Phase Elements (LDPE) and Local Negative Power Elliptical Phase Elements (LNPEPE) masks
• Optimize reduced subset of variables
• Key features:- Multi-point parallel search- Robust: insensitive to initial points- Flexible choice of encoding signal- Reduced complexity
- Optical efficient results- Computationally efficient: GPU implementation
6
System GeometriesIn-Line Geometry* Off-Axis Geometry
TIR Geometry
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Local Diffuser Phase Elements Mask
• Maximize information transfer: amplitude (reconstruction plane) to phase (CGH plane)
• Step 1: decompose desired pattern into Nbp binary patterns
• Step 2: assign local diffuser phase element to each pattern
• Diffusivity of qth element controlled by: and
• LDPE mask:
Mask Decomposition
(q)
(q) (q) ( ) factorfactor shift
1 ev
2( )exp arg exp 2 Jinc
bpNq
LDPEq
FP q i i D R i A
(q) (q) (q)factor factor shift, D F
Binary functionRandom matrix
2(q)( ) factor
ev
q FA
3LDPE bpDOF N
• Reduced number of DOF:
Phase of Mask with Local Diffuser
200 250 300 350 400 450
200
250
300
350
400
450-3
-2
-1
0
1
2
3
Phase
Amplitudex
Binary Phase CGH
Multi-level
LDPE Mask
Desired Amplitude Mask
x (m)
y (
m)
-60 -40 -20 0 20 40 60
-60
-40
-20
0
20
40
60
0.5
1
1.5
2
2.5
3
3.5x (m)
y (
m)
-60 -40 -20 0 20 40 60
-60
-40
-20
0
20
40
60
-3
-2
-1
0
1
2
3
x (m)
y (
m)
-60 -40 -20 0 20 40 60
-60
-40
-20
0
20
40
60
-3
-2
-1
0
1
2
3x (m)
y (
m)
-60 -40 -20 0 20 40 60
-60
-40
-20
0
20
40
60
-3
-2
-1
0
1
2
3
Each element has different
diffusivity
Fresnel Back-Propagation
Reconstruction Plane CGH Plane
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Local Negative Power Elliptical Phase Elements Mask
• Maximize information transfer: amplitude (reconstruction plane) to phase (CGH plane)
• Step 1: decompose desired pattern into Nbp binary patterns
• Step 2: apply LNPEPE to each pattern• Controlled parameters:• LNPEPE mask:
2 2( ) ( )
(q) (q)(q) (q)
1 1 2
' '2( ) ( ) exp 'sin 'sin exp
bpq qN
c c
LNPEPE x yq
x x y yP q q i x y i
f f
(q) (q) (q) (q)1 2, , , x yf f
Binary function Truncation window
4LNPEPE bpDOF N
• Reduced number of DOF:
Binary pattern center coordinates
Phase
Amplitudex
LNPEPE Mask
Desired Amplitude Mask
Negative power elliptical phase
Fresnel Back-Propagation
Reconstruction Plane CGH Plane
9
Genetic Algorithms Block• Multi-point optimization scheme• Inspired in biological evolution: “survival of the fittest”• Reduced complexity allow optimizing large populations• Individual:
or
bp bp bp(N ) (N ) (N )(1) (1) (1)factor factor shift factor factor shift, , , , , ,kx D F D F
( ) ( ) ( ) ( )(1) (1) (1) (1)1 2 1 2, , , , , , , ,bp bp bp bpN N N N
k x y x yx f f f f
Global minimum
The MathWorksTM
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MER Block• Local search, iterative optimization method• Refine solution: fast convergence• Compare results with: diffracted field (DF) and simulated optically recorded hologram (SORH) encoding strategies
11
Error Metrics• Four considered error metrics• Choice of error metric is application dependent
2
2' 1 ' 1
1 N N
before est desx y
MSE I IN
Photoresist Contrast Curve
- Mean square error: bias estimator ( and )
2
2' 1 ' 1
1,
N N
desafterx y
MSE R IN
00
1
0 otherwise
DD D
R D
2
- Additional metrics: L1 (bias) and normalized cross-correlation (similarity), hybrid
dose
- Diffraction efficiency 22(0) (0) ( )
22 2 2( ) (0) (0)
4size size
size size size
dill d d
H Oub
dd d ill
O H H
f f f
f f f
:effInput power
Signal Power
Signal Power Inside Hsize
Amplitude Constraint
Effective efficiency inside pattern
(G. Zhou, et al., 2000)
12
Optimization Results
• Optimization example:
- binary phase CGH: resolution target - LDPE encoding strategy
Wavelength 532nm Elite Children 5
Working Distance 150μm Crossover Fraction 0.6
Pixel Size 200nm Generations 100
CGH Size 300μm Population Size 100
Object Window 180μm Iterations 400
Main Parameters:
Desired PatternOptimized LDPE MaskPhase Map: Optimized Binary CGHReconstruction from Multi-Level CGH at Photoresist Plane (Before Exposure)
Inte
nsity
Inte
nsity
27.18beforeMSE
79.46%eff
Reconstruction from Binary CGH at Photoresist Plane (Before Exposure)
Inte
nsity
Inte
nsity
147.31beforeMSE
35%eff
Convergence GA BlockConvergence MER Block
13
Optimization Results
• Optimization example:
- binary phase CGH: resolution target - LDPE encoding strategy
Comparison of Encoding Strategies After GAs Block: Multi-Level CGH
• Sensitivity Analysis: problem parameters (e.g. cross-over fraction, population size, etc.)
• Parallel implementation on graphic processing unit: speedup >180X
- GPU computational time: 4.47 hours
- CPU estimated time: 16.48 days!
14
Error Comparison: Binary CGH
Extending the Depth of Focus
• Extend DOF: tolerate potential axial misalignments during exposure process• Modify RCHOA to impose constraints at multiple planes •Regular DOF: 22NAeff
z
Multiple Plane Constraint
Extended DOF CGH
266nmz Extended: 2 z
15
CGH FabricationFabrication Process
• Fabricated using electron-beam lithography• Binary phase CGH• Resist: Hydrogen Silsesquioxane (HSQ)
Scanning Electron Microscope Image of Fabricated Sample
50μm
Fused Silica
Aluminum
HSQ
E-beam Patterning
Remove Aluminum & Develop HSQ
16
Characterization of Fabricated CGHs
• Implemented methods: evaluation algorithm*, optical characterization*, exposure test
• Evaluation algorithm: analyze fabricated CGH 2D error map (correct over/under dose)
Block Diagram of Evaluation Algorithm
Stitched Binarized Fabricated CGH 2D Error Map
17
Characterization of Fabricated CGHs
• Implemented methods: evaluation algorithm*, optical characterization*, exposure test
• Optical characterization: measure reconstructed intensity
Optical Setup: Coherent Illumination
Measured Reconstructed IntensitiesBinary CGH:
DF Encoding StrategyBinary CGH:
Diffuser Encoding Strategy-Fabricated CGHs not fully optimized- Eliminate speckle using partial coherence illumination
100μm
18
Sensitivity Analysis
• Estimate and assist in the correction of potential fabrication errors• Considered errors: e-beam over/under dose, proximity effect, uniform/nonuniform phase, stitching and positional errors
Dilation Test: Over Dose Error Stitching Error Analysis
MSE
Offset Distance