Coupled-mode theory for electromagnetic pulse propagation in dispersive media undergoing a spatiotemporal perturbation: Exact derivation, numerical validation, and peculiar wave mixing

Abstract

We present an extension of the canonical coupled mode theory of electromagnetic waves to the case of pulses and spatio-temporal perturbations in complex media. Unlike previous attempts to derive such a model, our approach involves no approximation, and does not impose any restriction on the spatio-temporal profile. Moreover, the effect of modal dispersion on mode evolution and on the coupling to other modes is fully taken into account. Thus, our approach can yield any required accuracy by retaining as many terms in the expansion as needed. It also avoids various artifacts of previous derivations by introducing the correct form of the solution. We then provide what is, to the best of our knowledge, the first ever validation of the coupled mode equations with exact numerical simulations, and demonstrate the wide range of possibilities enabled by spatio-temporal perturbations of pulses, including pulse shortening or broadening or more complex shaping. Our formulation is valid across the electromagnetic spectrum, and can be applied directly also to other wave systems. * Electronic address: sivanyon@bgu.ac.il

Femtosecond Transient Bragg Gratings

“…The theory of transient Bragg gratings is fully developed and described in the literature [87,88] starting from the wave equation. Here, we provide a short description of the theory starting from its derived coupled mode equation for transient grating.…”

Pulse propagation in the slow and stopped light regime

“…In order to eliminate the fast oscillations on the scale of the optical period and wavelength, the electric field is written as a convolution of the transverse field profile and a slowly-varying envelope [56], both written as sums of (strictly real) frequency components, i.e., E(ì r, t) = e iβ 0 z F t ω ∫ dt A(z, t )e i(ω−ω 0 )t…”

“…Otherwise (especially close to a zero group velocity point (ZGVP)), an exceeding number of dispersion terms (in the form of high-order time derivatives) has to be taken into account, because the various dispersion coefficients are proportional to the inverse of the group velocity (and its powers) and/or because of the slowly convergence of the series of higher-order derivatives of the propagation constant with respect to the frequency [54]. As a result, standard pulse propagation schemes (see e.g., [53,54]), as well as coupled mode models (see e.g., [55,56], which are all frequency-based expansions) become inefficient. In these cases, one is forced to resort to numerical solutions of the full Maxwell equations (e.g., using the Finite Differences Time Domain (FDTD) approach) which is far more demanding in terms of computational resources and run times [11-13, 37, 38, 57-59]; such an approach also hampers physical insights.…”

Instantaneous modulations in time-varying complex optical potentials

“…which is valid for an arbitrary shape of the time-dependent dielectric permittivity [41]. Here, Δ is the Laplace operator and c is the speed of light in vacuum (for the theory of non-stationary electromagnetism see, e.g., [42][43][44]). Without the modulation rate, i.e., when b=0, the standard linear dispersion relation k k c 0 w e = ( )ˆholds, where 0 1 e e e º + ( )˜is the permittivity at 0 t = and k is the unit vector in the direction of propagation.…”