Ponderomotive dynamics of waves in quasiperiodically modulated media

Similarly to how charged particles experience time-averaged ponderomotive forces in high-frequency fields, linear waves also experience time-averaged refraction in modulated media. Here we propose a covariant variational theory of this ponderomotive effect on waves for a general nondissipative linear medium. Using the Weyl calculus, our formulation accommodates waves with temporal and spatial period comparable to that of the modulation (provided that parametric resonances are avoided). Our theory also shows that any wave is, in fact, a polarizable object that contributes to the linear dielectric tensor of the ambient medium. The dynamics of quantum particles is subsumed as a special case. As an illustration, ponderomotive Hamiltonians of quantum particles and photons are calculated within a number of models. We also explain a fundamental connection between these results and the well-known electrostatic dielectric tensor of quantum plasmas.

Figure 1

One-dimensional schematic of a PW (in red) propagating in a medium with a given dispersion operator affected by some MW. (a) Dynamics in the original variables. (b) Dynamics in the OC representation, in which the oscillations at the MW phase and its harmonics are removed.

Figure 2

(a) Comparison of the simulation results obtained by numerically integrating the full-wave Eq. (52) (solid fill) and the ray-tracing Eqs. (51) with $H_{\mathrm{eff}}$ taken from Eq. (57) (dashed) for a stationary MW. The initial wave packet is ${\mathrm{\Psi }}_0\left(x\right)={\left(2\pi {\eta }^2\right)}^{-1/4}exp\left[-{(x-\mu )}^2/\left(4{\eta }^2\right)\right]$, where $\mu =-50$ and $\eta =15$, and it is normalized such that $\int dx{\mathrm{\Psi }}_0^2\left(x\right)=1$. (The simulation is one dimensional, and $x$ denotes the spatial coordinate, unlike in the main text, where $x$ denotes the spacetime coordinate.) The initial conditions for the ray trajectory are $X\left(0\right)=-50$ and $P\left(0\right)=0$. (b) MW profile of the form $V\left(x\right)=0.15\mathrm{sech}(x/80)cos\left(x\right)$. Natural units are used such that $m=1, q=1, \hslash =1$, and $K=1$. At later times (not shown), diffraction effects become important, so the eikonal theory becomes inapplicable.