no. 4
The near field refractor
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 4, pp. 655-684.

We present an abstract method in the setting of compact metric spaces which is applied to solve a number of problems in geometric optics. In particular, we solve the one source near field refraction problem. That is, we construct surfaces separating two homogeneous media with different refractive indices that refract radiation emanating from the origin into a target domain contained in an n-1 dimensional hypersurface. The input and output energy are prescribed. This implies the existence of lenses focusing radiation in a prescribed manner.

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     title = {The near field refractor},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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     doi = {10.1016/j.anihpc.2013.07.001},
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Gutiérrez, Cristian E.; Huang, Qingbo. The near field refractor. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 4, pp. 655-684. doi : 10.1016/j.anihpc.2013.07.001. http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.001/

[1] L.A. Caffarelli, C.E. Gutiérrez, Qingbo Huang, On the regularity of reflector antennas, Ann. of Math. 167 (2008), 299 -323 | MR | Zbl

[2] L.A. Caffarelli, Qingbo Huang, Reflector problem in 𝐑 n endowed with non-Euclidean norm, Arch. Ration. Mech. Anal. 193 no. 2 (2009), 445 -473 | MR | Zbl

[3] L.A. Caffarelli, V. Oliker, Weak solutions of one inverse problem in geometric optics, J. Math. Sci. 154 no. 1 (2008), 37 -46 | MR | Zbl

[4] C.E. Gutiérrez, Qingbo Huang, The near field refractor, Geometric Methods in PDE's, Conference for the 65th Birthday of E. Lanconelli, Lect. Notes Semin. Interdiscip. Mat. vol. 7 , Universitá degli Studi della Basilicata, Potenza, Italy (2008), 175 -188 | Zbl

[5] C.E. Gutiérrez, Qingbo Huang, The refractor problem in reshaping light beams, Arch. Ration. Mech. Anal. 193 no. 2 (2009), 423 -443 | MR | Zbl

[6] C.E. Gutiérrez, The Monge–Ampère Equation, Birkhäuser, Boston, MA (2001) | MR | Zbl

[7] S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field scattering data, Inverse Problems 13 (1997), 363 -373 | MR | Zbl

[8] A. Karakhanyan, Xu-Jia Wang, On the reflector shape design, J. Diff. Geom. 84 (2010), 561 -610 | MR | Zbl

[9] Xu-Jia Wang, On the design of a reflector antenna, Inverse Problems 12 (1996), 351 -375 | MR | Zbl

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