Historical Perspective of Laser Beam Shaping David L. Shealy University of Alabama at Birmingham Department of Physics, 1530 3rd Avenue South, CH310 Birmingham, AL 35294-1170 USA 1
OUTLINE Note some current applications Overview of current practices Historical perspective of laser beam shaping using geometrical optics Rotationally symmetric systems 1 mirror, 1 lens, 2 plano-aspheric lenses Non-rotationally symmetric systems 2 mirror with no central obscuration Genetic algorithm optimization methods 3-element spherical surface GRIN system 2
http://www.pthmagazine.com/newsletters /2_11_02.html Optics and Fiber Optics February 11, 2002 BEAM-SHAPING TECHNOLOGY FOR OPTICAL PUMP LASERS Unique-m.o.d.e AG, a German laser company, has introduced a 975-nm fiber-coupled diode laser specifically designed for optically pumping erbium- and ytterbiumpumped solid-state lasers. 3
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Current Practices of 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 II, F.M. Dickey, S.C. Holswade, D.L. Shealy, eds., Proc. SPIE 4443, 2001. 6
Physical versus Geometrical Optics β = 2 2π ry f 0 0 λ = wavelength, r 0 = waist or radius of input beam, Y 0 = half-width of the desired output dimension f = focal length of the focusing optic, or the working distance from the λ optical system to the target plane Beam Shaping Guidelines: β < 4, Beam shaping will not produce acceptable results 4 < β < 32, Diffraction effects are significant β > 32, Geometrical optics methods should be adequate 7
Historical Perspective of Laser Beam Shaping Using Geometrical Optics Geometrical optics intensity law: ( I a) = 0 IdA= IdA 1 1 2 2 IdA 2 2 IdA 1 1 Source Constant optical path length condition: ( OPL ) = ( OPL ) 0 r 8
One Element Beam Shaping Systems Conservation of Energy: Scale: 0.58 IdA= IdA 1 1 2 2 Thin Lens Element Shaping Element Zoomed Area (above) Input Plane Output Surface 9
Two Element Beam Shaping Systems Conservation of Energy: r 2 R( r) =± Iin ( u) udu Iout 0 Constant OPL: ( OPL ) = ( OPL ) 0 r Equations of the optical surfaces: 10
Frieden, Appl. Opt. 4.11, 1400-1403, 1965: Lossless conversion of a plane wave to a plane wave of uniform irradiance. Conservation of Energy: 2 2 1 exp( r 2α ) ( ) =± R max 2 2 1 exp( r 2α ) max R r 1 2 Intensity shaping leads to OPL variation of 20λ Need to shape of output wavefront when phase is important Frieden requires rays to be parallel Z-axis Leads to OPL variation of λ/20 11
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. 12
Kreuzer, US patent 3,476,463, 1969. Conservation of Energy & Ray Trace Equations: zr 2 2( rr0) 1 e r+ R (, s S)sinθ Rmax 2 = 0 2( rmax r0 ) 1 e Constant OPL: ( ) ( ) d n 1 + R 1 ncosθ = 0 Mirror Surface Equations: ( ) = r dr ZR ( ) 2 0 ( ) 2 n 1 d ( n 1) + Rr ( ) = ( n 1) 1 2 R dr 2 0 ( ) 2 1 + n d Rr ( ) 13
Rhodes & Shealy, Appl. Opt. 19, 3545, 1980 Jiang, Shealy, Martin, Proc. SPIE 2000, 64, 1993 Energy Balance: Ray Trace Equation: Constant OPL: 14
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. 15
Non-Rotationally Symmetric System Cornwell, Non-projective transformations in optics, Ph.D. Dissertation, University of Miami, Coral Gables, 1980 and Proc. SPIE 294, 62-72, 1981. Differential Power I ( x, y) dxdy = I ( X, Y ) dxdy in out Conservation of Energy: Ein=Eout Magnifications of ray coordinates x ( u) du ( x( )) 1 x a mx( x) = C C x + A um u 1 2 0 X OPL condition Determine sag z(r) of first surface Determine inverse magnification Determine sag Z(R) of second surface 16
Two Mirror Beam Shaping System Shealy & Chao, Proc SPIE 4443-03, 2001 Input beam has 1-to-3 beam waist in x-to-y directions: I in 2 2 x y ( x, y) = exp 2 exp 2 x0 y 0 Iout = AX A const. 0 Y = 0 17
Genetic Algorithm Optimization Method N. C. Evans, D. L. Shealy, Proc. SPIE 4095, pp. 27-39, 2000. Can a spherical-surface GRIN beam shaping system be designed using catalog GRIN materials? Problem is well suited for Genetic Algorithms: discrete parameters Number of lens elements GRIN catalog number Continuous parameters: radii, thickness 18
M M Beam Shaping Merit Function Used in GA Optimization M Diameter Collimation = = M Uniformity exp exp 1 cos ( ) N 2 2 Q s( RTarget RN ) γ i i= 1 N N 1 1 Iout ( Ri) Iout R N i= 1 N k= 1 ( ) k R target = Output Beam Radius R N = Marginal Ray Height on Output Plane γ i = Angle i th Ray Make with the Optical Axis Q and s = Convergence Constants 19
Determining when a Solution is Found 20
3-Element GRIN Shaping System Element 1 Element 2 Element 3 21
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 have been used to design a 1 and 2-element beam shaping systems. Laser beam shaping merit function used with genetic algorithms to design a 3-element GRIN system with spherical surface lenses. 22