3 August 2000 SPIE 4095-04 1

Design of a Gradient-index Beam Shaping System Via a Genetic Algorithm Optimization Method

Neal C. Evans and David L. Shealy University of Alabama at Birmingham

Department of Physics, 1530 3rd Avenue South, CH310Birmingham, AL 35294-1170 USA

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Concepts and Context – Laser Beam Shaping

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Global Optimization

• General Concepts• Fitness Landscape

– Discrete and continuous variables

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Merit Function Topography• Merit Function

– Optical Design Parameters– Contributions from Individual Elements

ΡN

µ

M

M=(Irradiance Profile calculations on Output Surface) * (Radius of Output Surface) * (Wavefront Checks)

3 August 2000 SPIE 4095-04 5

Overview of Optimization Algorithms in Optics and Optical Design1

•Damped least-squares

•Expert Systems –ZEMAX

–SYNOPSYS

–Optics Toolbox

–CODE V

•Simulated Annealing

•Tabu Search

1. See D. Shafer, “Global Optimization in Optical Design,” Computers in Physics 8, 188-195 (1994).

•Proprietary Merit Functions

•Difficult to Scale

3 August 2000 SPIE 4095-04 6

Genetic Algorithms (GAs)• Useful Tool in Geometrical Optics?• Advantages

– Ease of implementation and broad applicability– Relative immunity to local extrema– Easily made parallel

• Drawbacks– Random starting designs– “This planet does not lack for life forms, but

there is a paucity of intelligent ones.2”2. Shafer,192.

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GA Concepts

• Biological Paradigm and Nomenclature• Expressing solutions in terms of binary

strings: genetic code

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GA Concepts

• Generating solutions• Populations and

individuals• Parents and children:

producing new populations

Init ialize GA byrandomly pickingnew individuals

Evaluate MeritFunct ion for eachindividual ingenerat ion

P erform genet ic operat ions(reproduct ion, mutat ions,cross-overs); produce newgenerat ion

Is the populat ionstagnant? (Micro-GAcheck)

Keep best individual andreplace the remainder withrandomly-selected individuals

Terminat ioncrit erion reached?

End

Y

N

Y

N

Step eligible for parallelizat ion

3 August 2000 SPIE 4095-04 9

Irradiance Profile Calculations

• A fast, efficient method is needed to calculate the irradiance profile on a specified Output Surface

• The Geometrical Optics form of the Energy Conservation Law is harnessed for these calculations:

( )2 2 20 ( ) 0n k e∇ + =r

0 0( ) ( ) exp ( )e e ik S = r r r

( )2 2S n∇ = 2 20 0 02 0e S e e S∇ ⋅∇ + ∇ =

( )f f f∇⋅ = ∇⋅ + ⋅∇v v v( ) 0I∇⋅ =a

1 1 2 2I dA I dA=

Source

1 1I dA

2 2I dA

3 August 2000 SPIE 4095-04 10

( ) ( )ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ,in in out out

I O

E da u dAσ ρ ρ ρ= ⋅ = Ρ Ρ ⋅ Ρ∫ ∫n v n v

( ) ( )11

cos( ) cos( )( ) .

cos( ) ( )

in outi i i ii

i i outi i i i

iu

i

ρ ρ ρ χσ ρ −

−

− Ρ = Ρ Ρ −Ρ 1 1 2 2I dA I dA=

3 August 2000 SPIE 4095-04 11

Beam Profile on Input Plane

( )2( ) expi iσ ρ αρ= −

( )σ ρ

,mmρ

3 August 2000 SPIE 4095-04 12

Merit Function Topography for Beam Shaper/Projector

ΡN

µ

M

uN

u ii

N

==∑1

1

( ).Ρ

2

1

1 { ( ) }N

ii

u uN

µ=

= Ρ −∑

21 exp 0.01{50 } ,NMµ

= − − Ρ

3 August 2000 SPIE 4095-04 13

Results for GA-optimized Beam Shaper/Projector System

• CODE V used to evaluate merit function• Simultaneous execution on 4 Sun Sparcs• Manually Tweaking the Merit Function

Lanscape• Total search time of 7 hours• “Best” solution found

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Zoomed Area (above)

Thin Lens Element Shaping

Element

Output Surface Input P lane

GA Solution for Beam Shaper/Projector System

3 August 2000 SPIE 4095-04 15

GA-determinedsurface

z

z h c hk c h

A hjj

j

( )( )

,=+ − +

+=∑

2

2 2 22

2

5

1 1 1

Surface Parameter Constraintc −10 20tok −10 0. toA4 − × ×−10 10 10 103 3. .toA6 − × ×−10 10 10 105 5. .toA8 − × ×−10 10 10 106 6. .toA10 − × ×−10 10 10 107 7. .to

Beam-shaping element

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Beam Profile on Output Surface

( )u Ρ

,mmΡ

( ) ( )11

cos( ) cos( )( ) .

cos( ) ( )

in outi i i ii

i i outi i i i

iu

i

ρ ρ ρ χσ ρ −

−

− Ρ = Ρ Ρ −Ρ

3 August 2000 SPIE 4095-04 17

Two-lens Beam Shaper Problem

W. Jiang, Shealy, Martin (1993)

Differential Equation Design Method: P. Rhodes and Shealy (1980)

3 August 2000 SPIE 4095-04 18

Lens 1

Lens 2

GA Solution to the Two-lens Shaper Problem

3 August 2000 SPIE 4095-04 19

Gradient-Index (GRIN) Shaper Problem*

21 1 1 1n a b z c z= + + 2

2 2 2 2n a b z c z= + +

*C. Wang and Shealy, “Design of gradient-index lens system for laser beam shaping,” Applied Optics 32, 4763-4769, 1993.

3 August 2000 SPIE 4095-04 20

GA Solution to GRIN:Part 1

Lens 2: LightPath G1SF, negat ive gradient

Lens 1: LightPath G1SF, posit ive gradient

Connector

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Free Form GRIN Laser Beam Shaping System

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Parameters Optimized for Free-form GA-designed GRIN ProblemParameter Description Type Limits[i]

1 Number of Elements Discrete 1-4 (integer)2 Radius of curvature of left surface of Element 1 Continuous -100 to 100 3 Radius of curvature of right surface of Element 1 Continuous -100 to 1004 Thickness of Element 1 Continuous 1 to 10 5 Distance between Element 1 and Element 2 Continuous 1 to 10 6 GRIN glass type for Element 1 Discrete 1-6 (integer)7 Positive or Negative GRIN for Element 1 Discrete 0 or 18 Radius of curvature of left surface of Element 2 Continuous -100 to 100 9 Radius of curvature of right surface of Element 2 Continuous -100 to 100 10 Thickness of Element 2 Continuous 1 to 10 11 Distance between Element 2 and Element 3 Continuous 1 to 10 12 GRIN glass type for Element 2 Discrete 1-6 (integer)13 Positive or Negative GRIN for Element 2 Discrete 0 or 114 Radius of curvature of left surface of Element 3 Continuous -100 to 100 15 Radius of curvature of right surface of Element 3 Continuous -100 to 100 16 Thickness of Element 3 Continuous 1 to 10 17 Distance between Element 3 and Element 4 Continuous 1 to 10 18 GRIN glass type for Element 3 Discrete 1-6 (integer)19 Positive or Negative GRIN for Element 3 Discrete 0 or 120 Radius of curvature of left surface of Element 4 Continuous -100 to 100 21 Radius of curvature of right surface of Element 4 Continuous -100 to 100 22 Thickness of Element 4 Continuous 1 to 10 23 Distance between Element 4 and Surface 10 (a dummy surface) Continuous 1 to 10 24 GRIN glass type for Element 4 Discrete 1-6 (integer)25 Positive or Negative GRIN for Element 4 Discrete 0 or 126 Distance from Surface 10 (a dummy surface) to the Output Plane Continuous 1 to 100

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Determining when a Solution is Found

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Free-form GA-designed GRIN Beam Shaper•Average evaluation time per generation: 7.80s. Total execution time: 26.8 hrs

•Integrating Output Profile over Output Surface yields 21.9 units; integrating Input Profile over Input Surface yields 21.7 units

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Free-form GA-designed GRIN Solution

Element 1

Element 2 Element 3

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Summary and Conclusions• Assumption: there are some problems in Geometrical

Optics that are intractable for Analytical or Conventional Optimization Techniques

• Proof-of-principle of GA optimization stage: the Beam Shaper/Projector and the Two-Lens Shaper

• Creative stage of GA optimization: the GRIN problems• Machine Learning Codes, Genetics Algorithms in

particular, are well-suited for these problems where discrete parameters are involved, or where the form of the solution is not known a priori

3 August 2000 SPIE 4095-04 27

Free Form 3-element GRIN