Abstract

Pattern forming systems are usually ruled by partial differential equations with rotational (or, in general, elliptical) symmetry, so the spatial coupling is represented by the elliptical (Laplace) operator L_1,1 = ∂²/∂x² + ∂²/∂y², after trivial rescaling. This is the typical case of planar optical resonators, where L_1,1 follows from diffraction. It is possible however to change the situation into a hyperbolic case, represented by the D’Alembert operator L_1,−1 = ∂²/∂x² − ∂²/∂y². This requires that the diffraction coefficients in the resonator have opposite signs for two orthogonal transverse directions. The consequences are that (i) the rotational symmetry is broken, (ii) the pattern depends strongly on the underlying tilted wave and (iii) wavevectors with different magnitude can be excited simultaneously [1].

© 2007 IEEE