The refractor problem with loss of energy

Gutiérrez, Cristian E.; Mawi, Henok

doi:10.1016/j.na.2012.11.024

Cited by 18 publications

(14 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Existence of refractors and uniqueness up to dilations is proved in [GH09] using mass transport techniques. This is also proved in [GM13] with a different method where a more general case that takes into account internal reflection is considered.…”

Section: Set Up Definitions and Statement Of Resultsmentioning

confidence: 82%

On the numerical solution of the far field refractor problem

2017

Self Cite

View full text Add to dashboard Cite

The far field refractor problem with a discrete target is solved with a numerical scheme that uses and simplify ideas from [CKO99]. A numerical implementation is carried out and examples are shown.From the definition, it is easy to see that refractors are Lipschitz continuous inDefinition 2.2. Given a refractor R = {ρ(x)x : x ∈Ω}, the refractor mapping of R is the multi-valued map defined for x o ∈Ω by N R (x o ) = {m ∈Ω * : E(m, b) supports R atρ(x o )x o f or some b > 0}.Given m o ∈Ω * the tracing mapping of R is defined bySuppose that we are given a nonnegative function g ∈ L 1 (Ω). We then recall the notion of refractor measure, see [GH09, Section 3.1]. * The physical problem considered is three dimensional; the mathematical extension to n dimensions is straightforward.

show abstract

Section: Set Up Definitions and Statement Of Resultsmentioning

confidence: 82%

On the numerical solution of the far field refractor problem

2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…Examples of other problems are that one can change the light source to be a parallel beam instead of a point source (see [37]) or consider multiple optical instruments instead of just one (see work of Glimm and Oliker [25] and Oliker [56]). Another interesting family of problems are models with nonperfect energy transmission, as studied by Gutiérrez and Mawi [30] and Gutiérrez and Sabra [31].…”

Section: Other Geometric Optics Problemsmentioning

confidence: 99%

Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations

Guillen

Kitagawa

2017

Comm Pure Appl Math

View full text Add to dashboard Cite

Abstract. The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Ampère type known as generated Jacobian equations. This class of equations, whose general existence theory has been recently developed by Trudinger, goes beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under minimal structural and regularity assumptions, covering situations analogous to that of costs satisfying the A3-weak condition introduced by Ma, Trudinger and Wang in optimal transport. These estimates are used to develop a C 1,α regularity theory for weak solutions of Aleksandrov type. The results are new even for all known near-field reflector/refractor models, including the point source and parallel beam reflectors and are applicable to problems in other areas of geometry, such as the generalized Minkowski problem.

show abstract

“…The formulation (2.2) is useful solve refraction problems for lens design, see [GH09], [GM13], [GS14], [GT13], and [DLGM17] for a numerical implementation.…”

Section: Introductionmentioning

confidence: 99%

General refraction problems with phase discontinuities on nonflat metasurfaces

2017

Self Cite

View full text Add to dashboard Cite

This paper provides a mathematical approach to study metasurfaces in nonflat geometries. Analytical conditions between the curvature of the surface and the set of refracted directions are introduced to guarantee the existence of phase discontinuities. The approach contains both the near and far field cases. A starting point is the formulation of a vector Snell's law in the presence of abrupt discontinuities on the interfaces.

show abstract

The refractor problem with loss of energy

Cited by 18 publications

References 7 publications

On the numerical solution of the far field refractor problem

On the numerical solution of the far field refractor problem

Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations

General refraction problems with phase discontinuities on nonflat metasurfaces

Contact Info

Product

Resources

About