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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
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Problem Definition• Performance of CGHs depends primarily on optimization algorithm and fabrication method
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
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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
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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
Diffuser Encoding Strategy-Fabricated CGHs not fully optimized- Eliminate speckle using partial coherence illumination
100μm
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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