Spatial solitons in periodic semiconductor-dielectric nano-structures

Abstract

Spatial solitons in periodic semiconductor-dielectric structures have been the subject of intense research over the past decade [1]. Recent progress with fabrication of nano-structures for photonics applications has stimulated research into light trapping and guiding on the sub-wavelength scale. For such nano-scale periodic structures the concept of evanescently coupled waveguides, largely used for analysis of various nonlinear phenomena in conventional periodic structures [1], is invalidated. In this work we develop theory and analyze existence and stability of spatial solitons in one-dimensional semiconductor-dielectric nano-structures: thin semiconductor layers embedded into a dielectric with much lower index of refraction. The analysis is done within full vector nonlinear stationary Maxwell equations where k = 2π/λ vac , c is speed of light in vacuum, e 0 is vacuum permittivity, λ vac is the wavelength of light in vacuum, for electric and magnetic fields it is assumed, e→,H→=1/2·E→,H→ exp (−ikct)+c.c., D→ is the displacement in SI units normalised to e 0 . Material parameters e = n2 and χ 3 are functions of the transverse coordinate x, at each interface they are assumed to vary sharply but continuously, so that semiconductor layers are described by the array of super-gaussian functions. Light propagates along z direction. In the considered 1D geometry, for both TE and TM polarizations we need to solve Maxwell equations for the electric field only. Soliton solutions are sought in the stationary form: {E x , E z }={f(x),ig(x)}exp(ikqz) (TM), E y = u(x)exp(ikqz) (TE), q is the relative change of the propagation constant with respect to its vacuum value. To characterize soliton solutions we use the power density P z = ∫ 〈S z 〉dx dxdefined through z-component of the time-averaged Poynting vector. Stability analysis of soliton solutions is done by linearizing Maxwell equations (1) for small perturbations {e→,h→} = →α(x)exp(ikλz) and solving the resulting eigenvalue problem.