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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

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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

• 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

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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

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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

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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

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)

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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

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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!

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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

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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

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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

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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

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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

MSE

Offset Distance