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Optical Design of Laser Beam Shaping Systems

David L. Shealy University of Alabama at Birmingham

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

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Outline of Presentation

• Overview of history and current practices• Geometrical methods for design• Applications:

• Two-plano-aspheric lens system• Two-mirror system with no central

obscuration• Three-element GRIN system

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Historical Background• Frieden, Appl. Opt. 4.11, 1400-1403, 1965: “Lossless

conversion of a plane wave to a plane wave of uniform irradiance.”

• Kreuzer, US patent 3,476,463, 1969: “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equi-phase surface.”

• Rhodes & Shealy, Appl. Opt. 19, 3545-3553, 1980:“Refractive optical systems for irradiance redistribution of collimated radiation – their design and analysis.”

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Contemporary Beam Shaping*•Process of redistributing the irradiance and phase

•Two functional categories:•Field Mapping•Beam Integrators

*Laser Beam Shaping: Theory and Techniques, F.M. Dickey & S.C. Holswade,eds., Mercel Dekker, 2000;Laser Beam Shaping, F.M. Dickey & S.C. Holswade,eds., Proc. SPIE 4095, 2000;Laser Beam Shaping, F.M. Dickey, S.C. Holswade, D.L. Shealy, eds., Proc. SPIE 4443, 2001.

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Field Mapping Beam Shaper

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Beam Integrators

D

d

F f

S

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Physical versus Geometrical Optics0 02 2

f r Yπβ

λ=

λ = wavelength, r0 = waist or radius of input beam, Y0= half-width of the desired output dimensionf = focal length of the focusing optic, or the working distance from the

optical system to the target plane

Beam Shaping Guidelines:β < 4, Beam shaping will not produce acceptable results4 < β < 32, Diffraction effects are significantβ > 32, Geometrical optics methods should be adequate

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Selected Chapter Titles:

•“Mathematical and Physical Theory of Lossless Beam Shaping,” L.A. Romero, F.M Dickey.

•“Gaussian Beam Shaping: Diffraction Theory and Design,” F.M. Dickey, S.C. Holswade.

•“Geometrical Methods,” D.L. Shealy

•“Optimization-based Techniques for Laser Shaping Optics,” N.C. Evans, D.L. Shealy.

•“Beam Shaping with Diffractive Diffusers,” D.R. Brown.

•“Multi-aperture Beam Integration Systems,” D.M. Brown, F.M. Dickey, L.S. Weichman.

•“Current Technology of Beam Profile Measurements,” C.B. Roundy.

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Overview of Geometrical Methods

• Geometrical optics intensity law:( ) 0I∇ =ai

1 1 2 2I dA I dA=

Source

1 1I dA

2 2I dA

• Constant optical path length condition:0( ) ( )rOPL OPL=

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Optical Design of Laser Beam Shaping Systems: Differential Equations vs. Global Optimization

• Geometrical methods leads to equations of the optical surfaces:

• Global Optimization with discrete & continuous variables:– Beam shaping merit function

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Applications of Geometrical Methods

• Two plano-aspheric lens system for shaping rotationally symmetric Gaussian beam.

• Two mirror system with no central obscuration for shaping elliptical Gaussian beam.

• Three-element GRIN system for shaping rotationally symmetric Gaussian beam.

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Two Lens Beam Shaping SystemJiang, Ph.D. Dissertation, UAB, 1993

Energy Balance:

Ray Trace Equation:

Constant OPL:

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Input and Output Beam Profiles

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Beam Shaping Applications

In a holographic projection processing system featured on 10 January 1999 issue of Applied Optics , a two-lens beam shaping optic increased the quality of micro-optical arrays.

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J.A. Hoffnagle & C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39.30, 5488-5499, 2000.

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Two Mirror Beam Shaping SystemShealy & Chao, Proc SPIE 4443-03, 2001

Input beam has 1-to-3 beam waist in x-to-y directions:Parameters:

0 0const.out X YI A A= =

2 2

0 0

( , ) exp 2 exp 2inx yI x yx y

= − −

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Sag of First Mirror

2 20

10; 5L h

l L h

= =

= +

00 0 max 0

00 max 0

max 0 max

1; ;2

3 ; 32

2

rr x x r

ry y r

X r Y

= = =

= =

= =

Parameters:

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Optical Performance Analysis*

• ZEMAX was used.

• Elliptical Gaussian input beam profile was modeled as user-defined surface.

• Two mirror surfaces were also modeled as user-defined surfaces.

*Shealy and Chao, Proc. SPIE 4443, 24-35, 2001: “Design and Analysis of an elliptical Gaussian laser beam shaping system.”

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Relative IlluminationInput Beam

Output Beam

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First Mirror Surface Analysis

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Second Mirror Surface Analysis

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Tolerance Analysis• In A, the first mirror was decentered 10% of its diameter about x-axis.• In B, both mirrors were decentered by 2.5% of their diameter about x- and y-axes and tilted about the three axes by 0.25 degrees.

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Conclusions for 2 Mirror System

• Designed a two-mirror system with no central obscuration for shaping a 3:1 elliptical Gaussian beam into a uniform output beam.

• ZEMAX used to do optical performance analysis: – First mirror has strong aspherical component along

direction of smaller waist (120µm for 6mm diameter mirror).

– Output beam profile is stable for decentering of less than 2.5% of mirror diameter and 0.25 degrees about coordinate axis.

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3-Element GRIN Beam Shaping SystemN. C. Evans, D. L. Shealy, Proc. SPIE 4095, pp. 27-39, 2000.

• Can a spherical-surface GRIN shaping system be designed using catalog GRIN materials?

• System would have practical applications.• Problem is well suited for Genetic Algorithms:

– discrete parameters • Number of lens elements • GRIN catalog number

– Continuous parameters: radii, thickness

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In itia lize G A by random ly p ick ing new ind iv iduals

E va luate M erit Function for each ind iv idual in generation

P erform genetic operation s (rep roduction , m utation s, cross-overs); p roduce new generation

Is th e popu lation stagnant? (M icro-G A check)

K eep best ind iv idual and rep lace the rem ainder w ith random ly-selected ind iv iduals

End

Y

N

Y

N

T erm ination criterion reached?

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Merit Function Used in GA Optimization

( )

( )

22

Target N1Diameter Collimation

Uniformityout out

1 1

exp exp 1 cos ( )

1 1( )

NQ

ii

N N

i ki k

s R RM MM

MI R I R

N N

γ=

= =

− − − − = =

−

∏

∑ ∑

Rtarget = Output Beam RadiusRN = Marginal Ray Height on Output Planeγi = Angle ith Ray Make with the Optical AxisQ and s = Convergence Constants

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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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3-Element GRIN Shaping System

Element 1

Element 2Element 3

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3-Element GRIN Shaping System•Average evaluation time for a 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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Summary and Conclusions• Geometrical methods for design of laser beam

shaping systems uses:– Conservation of energy within a bundle of rays,– Constant optical path length condition.

• Numerical and analytical techniques used to design a 2-plano-aspherical lens system and a 2-mirror system with no central obscuration.

• Laser beam shaping merit function used with genetic algorithms to design a 3-element GRIN system with spherical surface lenses.