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Mathematical principles in photonic crystals. (English) Zbl 1217.78037

Introduction: Photonic crystals are nano-structures exhibiting the interesting behavior that light at certain frequencies cannot travel in the crystal, whereas at other frequencies it does travel and is scattered. This phenomenon is due to the periodic variation in the index of refraction of the crystal.
Such crystals also have the property that light at an allowed frequency may travel along a particular path, as a consequence of a small non-periodic variation (impurity) in the refractive index. The role of mathematics is crucial to the solution of several problems involving nano-structures. In particular, the design of photonic crystals is based on the identification of the refractive index of crystals when the allowed and forbidden frequencies are specified.
The purpose of this paper is to derive physically relevant mathematical results on the design of photonic crystals.
The outline of the paper is the following: in Section 2 we illustrate some physical properties, mention relevant applications and introduce the mathematical model that describes the propagation of light in photonic crystals. In Section 3 we give the basic mathematical results. In Section 4 we introduce the scattering matrix and the period map which allows us to characterize the propagation of the light in the crystal in terms of its behavior when propagating for only one period. In Section 5 we propose some new results on the design of photonic crystals and in Section 6 we draw some conclusions and mention some open problems.

MSC:

78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
78A60 Lasers, masers, optical bistability, nonlinear optics
34L25 Scattering theory, inverse scattering involving ordinary differential operators
78A40 Waves and radiation in optics and electromagnetic theory
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