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
1
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: - Non-contact
- Depth of focus control
- Robust design
- Parallel exposure
- Standard fabrication
- High resolution
- Large working area - Simple optical setup
- 2D or 3D patterning - Cost effective
In-line CGH Lithography
Final Device
Processing
2
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 CGHs
CGH Plane
x
Encoding
Reconstruction Plane
F1  I des ei1
Reconstruction Plane
H size
CGH Plane
Back-propagation
Free parameter
Desired Pattern
1
Fresnel
O

F2  F2 ei2
x'
Osize
Osize
H size
 pix
F2  1
2   H
H  eiH
y

y'
'
z
pix
 ' pix
 pix
d
F2  F2 ei2
F1  F1 ei1
Inverse Problem
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 CGHs
CGH Plane
x
Decoding
CGH Plane
H  eiH
Reconstruction Plane
H size
Reconstruction Plane
Forward-propagation
OFresnel 

R  R eiR
| |
x'
Osize
Osize
H size
 pix
2
y

y'
z
pix
 ' pix
 pix
I est
'
d
Photoresist
Exposure
Final Pattern
F2  F2 ei2
F1  F1 ei1
Inverse Problem
4
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
5
System Geometries
In-Line Geometry*
Off-Axis Geometry
TIR Geometry
6
Local Diffuser Phase Elements Mask
• Maximize information transfer: amplitude (reconstruction plane) to phase (CGH plane)
• Step 1: decompose desired pattern into Nbp binary patterns
Mask Decomposition
• Step 2: assign local diffuser phase element to each pattern
(q)
(q)
(q)
, Ffactor
• Diffusivity of qth element controlled by: Dfactor
and shift
• LDPE mask:
(q)


 2 Ffactor
 


(q)
(q)
(q)
   (q) exp i arg exp  i 2 Dfactor R  ishift   A Jinc 
 
q 1
 ev   




Nbp
PLDPE
Binary function
A( q )
Random matrix
(q)
 Ffactor

 

  ev 
CGH Plane
Reconstruction Plane
2
• Reduced number of DOF:
DOFLDPE  3Nbp
x
Fresnel
Back-Propagation
Each element
has different
diffusivity
7
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
Binary pattern
center coordinates
• Controlled parameters: f1(q) , f2(q) , x(q) ,  y(q)
• LNPEPE mask:


2

 


(q)
(q)
 x 'sin  x  y 'sin  y   exp i
   (q)(q) exp i


 

q 1



Nbp
PLNPEPE
Binary function
Truncation window
  x ' x ( q ) 2  y ' y ( q ) 2  
c
c




(q)
(q)

 
f1
f2

 
Reconstruction Plane
CGH Plane
• Reduced number of DOF:
DOFLNPEPE  4Nbp
x
Fresnel
Back-Propagation
Negative power
elliptical phase
8
Genetic Algorithms Block
• Multi-point optimization scheme
• Inspired in biological evolution:
“survival of the fittest”
• Reduced complexity allow
optimizing large populations
• Individual:
(1)
(1)
(1)
xk   Dfactor
, Ffactor
, shift
,
(N bp )
(Nbp )
(N )
, Dfactor
, Ffactor
, shiftbp 
or
xk   f1(1) , f 2(1) , x(1) , y(1) ,
( Nbp )
, f1
( Nbp )
, f2
, x
( Nbp )
, y
( Nbp )


Global
minimum
The MathWorksTM
9
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
10
Error Metrics
Photoresist Contrast Curve
• Four considered error metrics
• Choice of error metric is application dependent
- Mean square error: bias estimator (  and  2 )
MSEbefore
MSEafter
1
 2
N
1
 2
N
N
N
 Iest  Ides 
2
x '1 y '1
N
N
 R  I des  ,



x '1 y '1
2
dose
  D 
1 
R    D0 

0

inside pattern
otherwise
Amplitude
Constraint
- Diffraction efficiency
eff : Effective efficiency
D  D0
ub 
4  fill(0) f d(0)

 (d )
 fd

2

Osize


H size 
f d(0)
2
2
f


H size 
Signal Power
Inside Hsize
Signal
Power
(d ) 2
d
Osize
2
fill(0)
Input
power
2
H size
(G. Zhou, et al., 2000)
- Additional metrics: L1 (bias) and normalized cross-correlation (similarity), hybrid
11
Optimization Results
Main Parameters:
• 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
Intensity
Intensity
Convergence
GA
Block
Reconstruction
Reconstruction
fromfrom
Multi-Level
Binary
CGH
CGH
at
at
Photoresist
Photoresist
Plane
Plane
(Before
(Before Exposure)
Exposure)
Desired
Pattern
Phase
Map:
Optimized
Binary
CGH
Optimized
LDPE Mask
Convergence
MER
Block
MSEbefore
27.18
147.31
MSE
before 
Intensity
Intensity
effeff 79.46%
35%
12
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!
13
Extending the Depth of Focus
Multiple Plane Constraint
• Extend DOF: tolerate potential axial misalignments
during exposure process
• Modify RCHOA to impose constraints at multiple
planes
•Regular DOF: z 

2
2NAeff
Error Comparison: Binary CGH
Extended DOF CGH
z  266nm
Extended: 2  z
14
CGH Fabrication
Fabrication Process
• Fabricated using electron-beam lithography
• Binary phase CGH
• Resist: Hydrogen Silsesquioxane (HSQ)
Fused
Silica
Aluminum
HSQ
E-beam
Patterning
Remove
Aluminum &
Develop HSQ
Scanning Electron Microscope
Image of Fabricated Sample
50μm
15
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
16
Characterization of Fabricated CGHs
• Implemented methods: evaluation algorithm*, optical characterization*, exposure test
• Optical characterization: measure reconstructed intensity
Optical Setup: Coherent Illumination
Measured Reconstructed Intensities
-Fabricated CGHs not fully
optimized
Binary CGH:
DF Encoding Strategy
Binary CGH:
Diffuser Encoding Strategy
- Eliminate speckle using
partial coherence illumination
100μm
17
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
Stitching Error Analysis
MSE
Dilation Test: Over Dose Error
Offset Distance
18