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 dimensional hypersurface. The input and output energy are prescribed. This implies the existence of lenses focusing radiation in a prescribed manner.
@article{AIHPC_2014__31_4_655_0, author = {Guti\'errez, Cristian E. and Huang, Qingbo}, title = {The near field refractor}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {655--684}, publisher = {Elsevier}, volume = {31}, number = {4}, year = {2014}, doi = {10.1016/j.anihpc.2013.07.001}, mrnumber = {3249808}, zbl = {1301.78001}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.001/} }
TY - JOUR AU - Gutiérrez, Cristian E. AU - Huang, Qingbo TI - The near field refractor JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 655 EP - 684 VL - 31 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.001/ DO - 10.1016/j.anihpc.2013.07.001 LA - en ID - AIHPC_2014__31_4_655_0 ER -
%0 Journal Article %A Gutiérrez, Cristian E. %A Huang, Qingbo %T The near field refractor %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 655-684 %V 31 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.001/ %R 10.1016/j.anihpc.2013.07.001 %G en %F AIHPC_2014__31_4_655_0
Gutiérrez, Cristian E.; Huang, Qingbo. The near field refractor. Annales de l'I.H.P. Analyse non linéaire, Volume 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] On the regularity of reflector antennas, Ann. of Math. 167 (2008), 299 -323 | MR | Zbl
, , ,[2] Reflector problem in endowed with non-Euclidean norm, Arch. Ration. Mech. Anal. 193 no. 2 (2009), 445 -473 | MR | Zbl
, ,[3] Weak solutions of one inverse problem in geometric optics, J. Math. Sci. 154 no. 1 (2008), 37 -46 | MR | Zbl
, ,[4] 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] The refractor problem in reshaping light beams, Arch. Ration. Mech. Anal. 193 no. 2 (2009), 423 -443 | MR | Zbl
, ,[6] The Monge–Ampère Equation, Birkhäuser, Boston, MA (2001) | MR | Zbl
,[7] Determination of reflector surfaces from near-field scattering data, Inverse Problems 13 (1997), 363 -373 | MR | Zbl
, ,[8] On the reflector shape design, J. Diff. Geom. 84 (2010), 561 -610 | MR | Zbl
, ,[9] On the design of a reflector antenna, Inverse Problems 12 (1996), 351 -375 | MR | Zbl
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