Wave Analysis and Homogenization of a Spatiotemporally Modulated Wire Medium

We develop a homogenization theory for a spatiotemporally modulated wire medium. We first solve for the modal waves that are supported by this composite medium, and we show peculiar properties such as extraordinary waves that propagate at frequencies below the cutoff frequency of the corresponding stationary medium. We explain how these unique solutions give rise to an extreme Fresnel drag that exists already with weak and slow spatiotemporal modulation. Next we derive the effective material permittivity that corresponds to each of the first few supported modes, and write the average fields and Poynting vector. Nonlocality, nonreciprocity, and anisotropy due to the spatiotemporal modulation are three inherent properties of this medium, and are clearly seen in the effective material parameters. Lastly, we validate that homogenization and spatiotemporal variation are not necessarily interchangeable operations. Indeed, in certain parameter regimes the homogenization should be performed directly on the spatiotemporally modulated composite medium, rather than the stationary medium being homogenized first and then the effect of the space-time modulation being introduced phenomenologically.

Figure 1

(a) Capacitively loaded wire. (b) The wire medium on a rectangular lattice with unit-cell dimensions $a$ along $x$ and $b$ along $y$. In this work we focus on waves that propagate transversely to the wires with $\overrightarrow{q}=q_x\hat{x}+q_y\hat{y}$.

Figure 2

(a) The time-modulated loaded wire. $L_0$ increases the self-inductance of the wire, and $C(t)$ introduces the time-modulated capacitance. These lumped elements are periodically loaded on the wire with periodicity $\mathrm{\Delta }$, thus creating the effective wire time-modulated susceptibility. (b) The wire medium. Each wire is modulated with the same temporal frequency $\mathrm{\Omega }$ but with a different phase that is determined by the modulation wave vector $\overrightarrow{\zeta }$. The propagation occurs with fundamental wave vector ${\overrightarrow{q}}_0$.

Figure 3

Dispersion relations as obtained by solutions No. $1\mathrm{–No.}6$ in Eqs. (31) and (32a). The fundamental wave number $q_0$ of each of the solutions is shown as a function of the frequency $\delta \omega /{\omega }_0$ for the four parameter cases that are tabulated in Table 1: (a) case I, (b) case II, (c) case III, and (d) case IV. Continuous lines represent the analytical results and circles represent the numerical results.

Figure 4

Dispersion diagrams for the fundamental solutions No. 1 and No. 2 and higher-order solutions No. 3–No. 6 for the parameters for case III in Table 1. (a) With $\xi =0$ the dispersion for the fundamental solutions $q_0^{(1,2)}$ is centered at the origin of the $q_x$-$q_y$ plane; however, its cutoff frequency is $\delta \omega /{\omega }_0\approx 0$. (b) In contrast, the higher-order solutions $q_0^{(3,4)}$ are propagating already at negative frequencies. These solutions are essentially nonreciprocal in light of their asymmetric dispersion. (c) is as (b) but for the additional solutions $q_0^{(5,6)}$. (d)–(f) are as (a)–(c) but with $\xi =\pi /4$.

Figure 5

Effective permittivity of a spatiotemporally modulated wire medium. Parameters for case II in Table 1. (a) Effective permittivity of the modal solutions when the propagation is collinear with the modulation. Circles represent no modulation (stationary wire media). Fundamental modal solutions No. 1,2 (continuous blue lines) are nearly identical to these of the stationary medium, and are reciprocal. The higher-order solutions No. 3,4 and No. 5,6 exhibit deviation with respect to the stationary-medium solutions and substantial nonreciprocally. (b) is as (a) but for propagation normal to the direction of modulation. The solutions are reciprocal in this case, and thus the strong anisotropy due to the spatiotemporal modulation clearly emerges from comparison of (a),(b). LTV, linear time variant.

Figure 6

Effective permittivity of the modal solutions for case III in Table 1. As in Fig. 5 except that that $\psi \sim 1$ implies a much-stronger effect by the modulation as evident from comparison with Fig. 5. LTV, linear time variant.

Figure 7

(a) Illustration of the structure. A $z$-polarized plane wave propagates along the $x$ axis and impinges on a finite lattice of spatiotemporally modulated wires. The array size is $N_x=201,N_y=141$ wires, with unit-cell parameters $a=0.07{\lambda }^0$ and $b=0.1{\lambda }^0$, so the overall sample shape is nearly a square. The remaining parameters are as given in cases III and IV in Table 1, and are used in the simulations for the dispersion diagrams of the infinite lattice. (b) For case III, at $\delta \omega =-0.15{\omega }^0$. The left panel shows a color map of the real part of the excited current wave, normalized by its maximal value. The strongly excited fundamental (stationary) solution is decaying rapidly, and the higher-order solutions are propagating. The right panel shows the discrete Fourier transform (implemented by FFT) carried over the field along $y=0$ reveals the excited spatial-frequency content, which nicely agrees with the dispersion calculations for the wave numbers of the wave that propagates along $x$. (c) As in (b) but for case IV in Table 1.

Figure 8

(a) Illustration of the structure. A current line source, with uniform amplitude and phase, oscillates at $\delta \omega =-0.15{\omega }^0$ and is located near the origin, along the $z$ axis. The current line excites a finite array with $N_x=201,N_y=141$ denoting the number of wires along $x$ and $y$, as in Fig. 7. The remaining parameters are as given in cases III and IV in Table 1, and are used in the simulations for the dispersion diagrams of the infinite lattice. (b) For case III, the excited current wave at the source frequency is shown, revealing substantial Fresnel drag, remarkably, already at the fundamental excitation harmonic with low modulation index and frequency. (c) As in (b) but for case IV in Table 1.

Figure 9

Effective permittivity of the modal solutions in a spatiotemporally modulated continuous plasmalike model. Results for case II in Table 2. This figure should be compared with Fig. 5, which is calculated by brute-force homogenization of the spatiotemporally modulated wire medium. (a) Effective permittivity when the propagation is collinear with the modulation. Circles represent no modulation (stationary plasma medium). Fundamental modal solutions No. 1,2 (continuous blue lines) are nearly identical to these of the stationary medium, and are reciprocal. The higher-order solutions No. 3,4 and No. 5,6 exhibit deviation with respect to the stationary-medium solutions and nonreciprocally. (b) is as (a) but for propagation normal to the direction of modulation. The solutions are reciprocal in this case, and thus the strong anisotropy due to the spatiotemporal modulation clearly emerges from comparison of (a),(b). A comparison with Fig. 5 demonstrates strong similarities between the results of the two homogenization approaches. LTV, linear time variant.

Figure 10

Effective permittivity in a spatiotemporally modulated continuous plasmalike model. As in Fig. 9 but calculated for case III in Table 2. These results should be compared with those in Fig. 6, exhibiting large differences between the two homogenization approaches in the case $\psi \sim 1$. LTV, linear time variant.

Figure 11

The equivalent circuit of a wire illuminated by an external field.