Abstract
We aim to solve inverse problems in illumination optics by means of optimal control theory. This is done by first formulating geometric optics in terms of Liouville’s equation, which governs the evolution of light distributions on phase space. Choosing a metric that measures how close one distribution is to another, the formal Lagrange method can be applied. We show that this approach has great potential by a simple numerical example of an ideal lens.
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J. Chaves, Introduction to Non-imaging Optics (CRC Press, Boca Raton, 2008)
K.B. Wolf, Geometric Optics on Phase Space (Springer, Berlin, 2004)
V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1978)
A.J. Dragt, J.M. Finn, Lie series and invariant functions for analytic symplectic maps. J. Math. Phys. 17(12), 2215–2227 (1976)
A.S. Glassner, An Introduction to Ray Tracing (Academic, London, 1991)
B.S. van Lith, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, T.W. Tukker, A novel scheme for Liouville’s equation with a discontinuous Hamiltonian and applications to geometrical optics. J. Sci. Comput. 68, 739–771 (2016)
B.S. van Lith, Principles of computational illumination optics. PhD thesis, TU/e (2017)
F. Tröltzsch, J. Sprekels, Optimal Control of Partial Differential Equations: Theory, Methods and Applications (American Mathematical Society, Providence, RI, 2010)
M. Giaquinta, S. Hildebrandt, Calculus of Variations I (Springer, Berlin, 2004)
Acknowledgements
B.S. van Lith wishes to thank J.H.M. ten Thije Boonkkamp for presenting the work in Norway in his stead.
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Lith, B.S.v., Thije Boonkamp, J.H.M.t., IJzerman, W.L. (2019). Solving Inverse Illumination Problems with Liouville’s Equation. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_27
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DOI: https://doi.org/10.1007/978-3-319-96415-7_27
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