According to geometric optics, light propagates in straight lines (in homogeneous media), reflects/refracts from surfaces according to Snell’s law: energy and tangential momentum are conserved. Thus, when reflecting from a hypersurface (which has codimension one) one gets the usual law of incident and reflected rays enclosing an equal angle to the normal to the surface. Indeed, conservation of ...

We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrödinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution, but does not alter the rapid oscillations. We consider initial states which are superpositions of slo...

This paper is devoted to a numerical simulation of the classical WKB system arising in geometric optics expansions. It contains the nonlinear eikonal equation and a linear conservation law whose coefficient can be discontinuous. We address the problem of treating it in such a way that superimposed signals can be reproduced by means of the kinetic formulation of “multibranch solutions,” original...

We propose and analyze a new way for obtaining an adiabatic geometric phase for light, via the sum-frequency-generation nonlinear process. The state of light is represented by the complex amplitudes at two different optical frequencies, coupled by the second order nonlinearity of the medium. The dynamics of this system is then shown to be equivalent to that of a spin-1/2 particle in a magnetic ...

The linear 2-dim irreducible representations of the dihedral groups (Dn) are interpreted as classical linear operators of geometrical optics. It is shown that the 2-dim irreducible representation of D4 is simply the refractive group described by Campbell [Optom. Vision Sci. 74, 381 (1997)]. The dihedral Fourier-inverse mechanism is introduced and shown to provide a systematic connection between...

where φ0 is real-valued. We are interested in the semiclassical limit ε → 0. The nonlinear Schrödinger equation (1) appears, for instance, in optics, and also as a model for Bose-Einstein condensates, with f(ρ) = ρ − 1, and the equation is termed Gross-Pitaevskii equation, or also with f(ρ) = ρ2 (see [13]). Some more complicated nonlinearities are also used especially in low dimensions, see [12...

In 1652 Huygens derived a formula specifying the rainbow angle for the primary bow (k = 1) in terms of the refractive index only. A generalization of this result for any k >or= 1 is outlined, along with an alternative representation. The details of the derivation can be found in (Adam, Mathematics Magazine, 2008, under review), but the results as stated may be of interest to the atmospheric opt...

We prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. For some perturbations of the data, we show that for very small times, we can neglect the Laplacian, and the mechanism is the same as for the corresponding ordinary differential equation. Our approach allows smaller perturbations of the data, where the instability occurs for times such that the p...

When thermal energies are weak, two-dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We present broad design principles to engineer the geometry of the underlying substrate so that a desired lamellar pattern can be obtained by self-assembly. Two dis...