Abstract
The nonlinear Schrödinger equation based on slowly varying approximation is usually applied to describe the pulse propagation in nonlinear waveguides. However, for the case of the front induced transitions (FITs), the pump effect is well described by the dielectric constant perturbation in space and time. Thus, a linear Schrödinger equation (LSE) can be used. Also, in waveguides with weak dispersion the spatial evolution of the pulse temporal profile is usually tracked. Such a formulation becomes impossible for optical systems for which the group index or higher dispersion terms diverge as is the case near the band edge of photonic crystals. For the description of FITs in such systems a linear Schrödinger equation can be used where temporal evolution of the pulse spatial profile is tracked instead of tracking the spatial evolution. This representation provides the same descriptive power and can easily deal with zero group velocities. Furthermore, the Schrödinger equation with temporal evolution can describe signal pulse reflection from both static and counter-propagating fronts, in contrast to the Schrödinger equation with spatial evolution which is bound to forward propagation. Here, we discuss the two approaches and apply the LSE with temporal evolution for systems close to the band edge where the group velocity vanishes by simulating intraband indirect photonic transitions. We also compare the numerical results with the theoretical predictions from the phase continuity criterion for complete transitions.
Highlights
In recent years, several theoretical predictions and experimental demonstrations were presented where the light propagating in guiding media is manipulated by a moving refractive index front [1,2,3,4,5]
In waveguides with weak dispersion the spatial evolution of the pulse temporal profile is usually tracked. Such a formulation becomes impossible for optical systems for which the group index or higher dispersion terms diverge as is the case near the band edge of photonic crystals
For the description of front induced transitions (FITs) in such systems a linear Schrödinger equation can be used where the temporal evolution of the pulse spatial profile is tracked instead of tracking the spatial evolution. This representation can deal with zero group velocities, as, for example, is the case at the band edge of photonic crystals
Summary
Several theoretical predictions and experimental demonstrations were presented where the light propagating in guiding media is manipulated by a moving refractive index front [1,2,3,4,5]. At the band edge the dielectric constant perturbation leads mainly to a frequency shift of the band diagram and not to a wavenumber shift [23] These non-unique dispersion functions frequently appear in periodic structures [23], such as photonic crystal waveguides [24], photonic crystal fibers [25], Bragg gratings [20,26,27]. In this equation, = ⁄ are the dispersion coefficients associated with the Taylor series expansion of the dispersion function at = 0, and , = In this case the temporal evolution of the envelope is a function of the spatial dispersion and the frequency shift of the dispersion relation with relative position with respect to the front. In the following we present several numerically integrated results for moving and static fronts and compare obtained results to theoretical predictions
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