docOptimization of achromatic refractive beam shapers
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Optimization of achromatic refractive beam shapers
SPIE Optics + Photonics, Dan Diego, August 10th-14th 2008 Optimization of achromatic refractive laser beam shaping systems Frank Wyrowski, Friedrich-Schiller University, Jena, Germany Hagen Schimmel, LightTrans GmbH, Jena, Germany Introduction Reshaping of spatial coherent laser beams into intensity distribution with high homogeneity. Input field Beam Shaper Lens System Output field Introduction Phase of output field is not specified and can be used as design freedom. Conditions for Achromatization • Output intensity must be independent of wavelength • typically wavelength dependent – Single Lens – Free Space – Achromatic Lens System • typically wavelength dependent Conditions for Achromatization • Three different options for achromatization of beam shaping systems: – Achromatization of lens system and beam shaper – Compensation of chromatic effects of beam shaper by special design of lens system – Compensation of chromatic effects of lens system by special design of beam shaper We consider third choice for the most compact beam shaping situation. Compact Beam Shaping Setup Free Space Angular dispersion Must compensate angular dispersion Must be chromatic = Multiplexing Mapping Type Beam Shaper Elements • Beam shaper locally deflect light to reshape laser beams. • Local light deflection can be described by wave number vector • Local deflection directions Input field x x‘ Desired output field Achromatization of Compact Setup • Achromatization of local deflection directions: • Wavelength dependency of k-components must be identical. • With it follows condition on local deflection according to Diffractive Beam Shapers Free Space Light Deflection Diffractive Beam Shaper • Local light deflection of diffractive beam shapers can be described by grating equation. • 2D Grating equation • D local grating period, m diffraction order cannot be satisfied • Condition Design of Diffractive Beam Shaper • Different design methods published. • We use a two step design process. • Optimization of transmission function for one wavelength λ1. • Calculation of diffractive height profile from transmission phase. Monochromatic Design of Beam Shaper • Height profile of diffractive beam shaper • Maximum height h hmax x Example: Diffractive Beam Shaper Far Field Input field Diffractive Diameter 1 mm beam shaper Wavelength 540 nm Output field 5° half angle Chromatic Effects of Diffractive Beam Shapers Signal to Noise Ration in dB SNR of diffractive beam shaper depending on wavelength 50 45 40 35 30 25 20 15 10 5 450 500 550 Wavelength in nm 600 650 Refractive Beam Shapers Free Space Light Deflection Refractive Beam Shaper • Local light deflection of refractive beam shapers can be described by Eikonal equation. • It follows for thin element approximation Light Deflection Refractive Beam Shaper • Eikonal equation for thin element approximation • Condition for k to achromatize light deflection direction α, β • Fulfilled if • Fulfilled if material would not have dispersion Achromatization of Refractive Beam Shapers • Important result: Angular dispersion is “naturally” compensated if refractive beam shapers are used. • Since material dispersion is weak refractive beam shapers create output intensities with small chromatic effects. • Material dispersion is to be achromatized to improve result. • Compensation can be for example by using different materials. Achromatization of Refractive Beam Shapers h1(x,y) n0 h2(x,y) n1 hl-1 … hl nl-1 nl • Materials with different dispersion characteristics and multiple surfaces can be used to compensate chromatic effects. • Light propagation is assumed to be described by Thin Element Approximation. Method A: Design for 2 Wavelengths • Two step design process. • Optimization of a beam shaper transmission function per design wavelength for creation of same output. Optimization can be done for example by IFTA. • Calculation of several height profiles between different materials creating different transmission functions for the design wavelengths. • Solve problem exactly for two design wavelenghts. • Expect smooth behaviour between both wavelengths, because of nature of refractive shapers. Achromatization of Refractive Beam Shapers • Equation system with Q equations • Q design wavelengths and Q transmissions • L surfaces and L+1 materials Example: Refractive Beam Shaper Far Field Input field Fused Silica Diameter 1 mm Wavelengths 630 nm and 450 nm Schott/LASF40 Desired Output 5° half angle Surfaces Height Profile Height profile 1 Height profile 2 Wavelengths Dependency of Beam Shapers Signal to Noise Ration in dB SNR of different beam shaping systems depending on wavelength Diffractive Beam Shaper Refractive Beam Shaper Refractive Achromate 2 Design Wavelengths 60 55 50 45 40 35 30 25 20 15 10 5 450 500 550 Wavelength in nm 600 650 Method B: Compensation of Dispersion • Two step design process. • Optimization of transmission function for a central design wavelength. • Calculation of several height profiles between different materials to create the desired transmission and setting of phase derivative proportional 1/λ. • Summary: – Exact solution for one wavelength – Reduction of material dispersion for others Method B: Compensation of Dispersion • Calculation of phase for multiple surfaces and materials by thin element approximation. • It follows with difference (OPD) of an optical path Method B: Compensation of Dispersion • As derived for refractive beam shapers to achieve an achromatization of the condition must be fulfilled. • This means Method B: Compensation of Dispersion • The following equation system must be solved: • To solve this system typically two surfaces and three materials are required. Materials should be a low index, a high index and a surrounding material. Example: Refractive Beam Shaper Far Field Input field Fused Silica Diameter 1 mm Wavelengths 630 nm and 450nm Schott/LASF40 Desired Output 5° half angle Surfaces Height Profile Height profile 1 Height profile 2 Wavelengths Dependency Beam Shapers Signal to Noise Ration in dB SNR of different beam shaping systems depending on wavelength 60 Refractive Beam Shaper Refractive Achromate 2 Design Wavelengths Refractive Achromate Dispersion Compensation 55 50 45 40 35 30 450 500 550 Wavelength in nm 600 650 Multiplexing of Output Intensity • The design principle allowing the creation of different transmission functions for different design wavelengths can be also used for multiplexing of output intensity. • Beam shaper transmission functions can be calculated for different wavelengths that create different output intensities in the target plane. • Height profiles can be calculated from these transmission functions. Example – Wavelengths Multiplexing Far Field Input field Fused Silica Schott/LASF40 Diameter 1 mm Wavelength 630 nm and 450 nm Desired Output 5° half angle Surfaces Height Profile Height profile 1 Height profile 2 Wavelengths Sensitivity Output intensity 450 nm 540 nm 630 nm Summary • Achromatization of beam shaping systems means to generate within a wavelength range the same output intensity. • Achromatization of the output intensity requires typically a wavelength multiplexing feature of the beam shaping element. • Refractive beam shapers allow such kind of achromatization. • Two different methods were shown for achromatization which provide different optimization characteristics.
