Information Optics and Photonics pp 3-11 | Cite as

# General Solution of Two-Dimensional Beam-Shaping with Two Surfaces

Chapter

First Online:

## Abstract

Beam shaping is a technique, by which a known input irradiance is transformed into a desired output irradiance by changing the local propagation vector of the wave front. Unlike one-dimensional beam-shaping, which leads to a simple differential equation which can be integrated in a straight forward manner, the two-dimensional beam shaping problem leads to a Monge–Ampere type equation, which is difficult to solve. In this paper, we generalize the problem to refractive and reflective systems and use the shifted-base-function approach to obtain a general solution.

## Keywords

Order Partial Derivative Beam Shaping Exit Plane Surface Gradient Entrance Plane
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- 1.L. A. Romero and F. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13, 751–760 (1996).ADSCrossRefGoogle Scholar
- 2.J. Hoffnagle and C. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488–5499 (2000).ADSCrossRefGoogle Scholar
- 3.F. Dickey and S. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, New York, 2000).CrossRefGoogle Scholar
- 4.J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).ADSCrossRefGoogle Scholar
- 5.D. Shealy and S. Chao, “Design and analysis of an elliptical Gaussian laser beam shaping system,” Proc. SPIE 4443, 24–35 (2001).ADSCrossRefGoogle Scholar
- 6.D. Shealy and S. Chao, “Geometric optics-based design of laser beam shapers,” Opt. Eng. 42, 1–16 (2003).Google Scholar
- 7.D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Grundlehren Math.Wiss. [Fund. Princ. Math. Sci.], Vol.
**224**(Springer-Verlag, Berlin, 1983).Google Scholar - 8.J. Benamou and Y. Brenier, “A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem,” Numer. Math. 84, 375–393 (2000).MathSciNetMATHCrossRefGoogle Scholar
- 9.E. J. Dean and R. Glowinski, Numerical analysis numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach, C. R. Acad. Sci. Paris, Ser. I 336, 779–784 (2003).Google Scholar
- 10.H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).MathSciNetADSCrossRefGoogle Scholar
- 11.V. I. Oliker and L. D. Prussner, “On the numerical solution of the equation \({\mathrm{z}}_{\mathrm{xx}}\ {\mathrm{z}}_{\mathrm{yy}} -{\mathrm{z}}_{\mathrm{xy}}^{2} = \mathrm{f}\) and its discretization. I,” Numer. Math. 54, 271–293 (1988).Google Scholar
- 12.V. I. Oliker and L. Prussner, “A new technique for synthesis of offset dual reflector systems,” In: 1994 Conference Proceeding, 10th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society), Vol. I, pp. 45–52 (1994).Google Scholar
- 13.V. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” In: Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, and F. Tomi, eds., pp. 191–222 (Springer-Verlag, Berlin, 2002).Google Scholar
- 14.T. Glimm and V. Oliker, “Optical design of two-reflector systems, the Monge–Kantorovich mass transfer problem and Fermat’s principle,” Indiana Univ. Math. J. 53, 1255–1278 (2004).MathSciNetMATHCrossRefGoogle Scholar
- 15.V. Oliker, “Optical design of two-mirror beam-shaping systems. Convex and non-convex solutions for symmetric and non-symmetric data,” Proc. SPIE 5876, 203–214 (2005).Google Scholar
- 16.K.-H. Brenner “Shifted base functions: an efficient and versatile new tool in optics,” J. Phys. Conf. Ser. 139, 012002, 11 pp, Workshop on Information Optics (WIO’08), 1–5August 2008 in Annecy (2008).Google Scholar
- 17.O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).ADSCrossRefGoogle Scholar
- 18.P. Scott and W. Southwell, “Reflective optics for irradiance redistribution of laser beams: design,” Appl. Opt. 20, 1606–1610 (1981).ADSCrossRefGoogle Scholar
- 19.P. H. Malyak, “Two-mirror unobscured optical system for reshaping irradiance distribution of a laser beam,” Appl. Opt. 31 4377–4383, (1992).ADSCrossRefGoogle Scholar
- 20.K. Nemoto, T. Fujii, N. Goto, H. Takino, T. Kobayashi, N. Shibata, K. Yamamura, and Y. Mori, “Laser beam intensity profile transformation with a fabricated mirror,” Appl. Opt. 36, 551–557 (1997).ADSCrossRefGoogle Scholar
- 21.D. L. Shealy, J. A. Hoffnagle, and K.-H. Brenner “Analytic beam shaping for flattened output irradiance profile,” SPIE Proceedings, Vol.
**6290**, No. 01, Annual Meeting of SPIE, Laser Beam Shaping Conference VII, San Diego (13–14 August 2006).Google Scholar - 22.R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern. Anal. Mach. Intell. 10(4), 439–451 (1988).MATHCrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media, LLC 2010