Scattering in Time-Varying Drude-Lorentz Models

TL;DR

The paper addresses how time-varying dispersion in Drude–Lorentz media affects wave scattering at a temporal interface. It develops four distinct models that realize the same initial and final permittivity but differ in which microscopic parameter is modulated, and uses a Laplace-transform approach to derive the four-frequency scattering coefficients, revealed to depend on model-specific boundary conditions and instantaneous current 'flashes' at the interface. Finite-difference time-domain simulations validate the analytical predictions and reveal limitations of abrupt plasma-frequency changes. The work emphasizes that precise physical implementation of time variation is crucial for accurate predictions in time-varying photonics and metamaterials, with potential experimental tests in optics and acoustics.

Abstract

Motivated by recent experiments, the theoretical study of wave propagation in time varying materials is of current interest. Although significant in nearly all such experiments, material dispersion is commonly neglected in theoretical studies. Yet, as we show here, understanding the precise microscopic model for the material dispersion is crucial for predicting experimental outcomes. Here we study the temporal scattering coefficients of four different time-varying Drude-Lorentz models, exploring how an incident continuous wave splits into forward and backward waves due to an abrupt change in plasma frequency. The differences in the predicted scattering are unique to time-varying media, and arise from the exact way in which the time variation appears in the various model parameters. We verify our results using a custom finite difference time domain algorithm, concluding with a discussion of the limitations that arise from using these models with an abrupt change in plasma frequency.

Figures (6)

• Figure 1: Time--varying Drude-Lorentz models: At each point $z$ within the medium an incident field couples to the harmonic motion of the polarization $\boldsymbol{P}$, with charge $q_e$, effective mass $m^{\star}$, resonant frequency $\omega_0$, and damping $\gamma$. The coupling into the motion is proportional to $\kappa_{\rm in}$, and the coupling back out into the field is proportional to $\kappa_{\rm out}$. For an abrupt change in the material parameters, we can achieve fixed initial and final permittivities, $\epsilon(\omega)$ through changing different combinations of these model parameters. In model 1, the time modulation occurs through the 'in--coupling', $\kappa_{\rm in}(t)$, in models 2 and 3 through the 'out--coupling', $\kappa_{\rm out}(t)$, and in model 4, through the effective mass, $m^{\ast}(t)$.
• Figure 2: Inverse Laplace Transform Contour: To perform the inverse Laplace transform in Eq. \ref{['eq:inverse_laplace_result']} we complete the integration contour with a semi circle in the right hand plane. We take $\Gamma<0$, so that the four poles are captured within the contour (contributing $-2\pi {\rm i}$ times each residue), equivalent to imposing a decay onto the four waves propagating within the material after the switch. The $\Gamma\to0$ limit is then taken, meaning that the contour must pass clockwise around each pole as we integrate along the imaginary $s$ axis.
• Figure 3: Evolution of the electric field in different time--varying Drude--Lorentz models: The electric field evaluated at a fixed point within the material for each of the four models described in Sec. \ref{['sec:four-models']}, with $\omega_-=2\times10^{13}(2\pi)$, $\omega_0=2\omega_-$, $\omega_{p-}=0.5\omega_-$, $\omega_{p+}=3\omega_-$ and $\gamma=0$. The dashed line represents the temporal boundary at $t=0$. Before the interface, the wave is taken as identical in all 4 models, after the interface the wave splits into 2 forward and 2 backward spatially propagating waves with 2 unique frequencies. The black dots represent the FDTD simulation whilst the red line represents the analytically derived solutions from the Laplace transform method.
• Figure 4: Forward and Backward Amplitudes $A_1^{\pm}$ for different time--varying Drude models ($\omega_{p-}=0.5\omega_-$, $\omega_0=0$): (a) forward $A_1^+$ and (b) backward $A_1^-$ scattering coefficients as a function of the final plasma frequency. All frequencies are given in terms of the initial frequency $\omega_-$, in a system with initial conditions given by $\omega_{p-}=0.5\omega_-$, $\omega_0=0$. The coloured lines represent the scattering coefficients for each of the different models. For this case with $\omega_0=0$, model 4 is identical to model 3 so we only show the results for model 3. The vertical dashed lines represent no switching ($\omega_{p-}=\omega_{p+}$) and the horizontal dashed lines show the amplitude of the initial wave. We see a clear trend in the order of the models for $\omega_{p+}>\omega_{p-}$ with the opposite being true for $\omega_{p+}<\omega_{p-}$. Most importantly, we see the predicted reciprocal, quadratic, and linear behaviours present in models 1,2 and 3 transformed by the initial conditions.
• Figure 5: Forward and backward scattering coefficients, $A_{1}^{\pm}$ and $A_{2}^{\pm}$ for different time--varying Drude--Lorentz models ($\omega_{p-}=0.5\omega_-$ and $\omega_0=0.7\omega_-$): As for Fig. \ref{['fig:no_res']}, with panels (a), (b), (c) and (d) showing the scattering coefficients $A_1^+$, $A_1^-$, $A_2^+$, $A_2^-$.