Unusual electromagnetic modes in space-time-modulated dispersion-engineered media

We report on electromagnetic modes in space-time-modulated dispersion-engineered media. These modes exhibit unusual dispersion relation, field profile, and scattering properties. They are generated by coupled codirectional space-time harmonic pairs and occur in space-time periodic media whose constituent materials exhibit specific dispersion. Excitation of a slab of such a medium with subluminal modulation results in periodic transfer of energy between the incident frequency and a frequency shifted by a multiple of the modulation frequency, whereas superluminal modulation generates exponentially growing anharmonics. These modes may find applications in optical mixers, terahertz sources, and other optical devices.

Figure 1

Comparison of dispersion diagrams of space-time-modulated periodic media composed of nondispersive and dispersive background materials for $M\to 0$. (a) Case of nondispersive background material. (b) Case of strongly dispersive background material. The magenta arrow shows the modulation (or space-time period) vector (${\mathbf{p}}_{\text{m}}$). Here, unessentially, the modulation for both cases is subluminal ($v_{\text{m}}={\omega }_{\text{m}}/{\beta }_{\text{m}}<v_{\text{b}}$). For clarity, the two space-time harmonics $n=0$ and $n=-1$ are emphasized.

Figure 2

(a) Dispersion diagram of a subluminally space-time-modulated lossless Lorentz medium for $M\to 0$. The magenta arrow represents the modulation vector. The three space-time harmonics $n=0$ (green), $n=+1$ (red), and $n=-1$ (blue) are emphasized. The dashed lines represent the edges of the Brillouin zone. (b) Dispersion diagram zoomed in at the crossing circled in Fig. 2 for the finite modulation depth $M=0.01$ and $f_{\text{per}}(.)=cos(.)$. The solid curves represent the real parts of the dispersion. The imaginary parts are zero.

Figure 3

Field profiles for (a) the additive mode ($E^{\text{A}}$), corresponding to the cyan point in Fig. 2, and 2 the subtractive mode ($E^{\text{S}}$), corresponding to the purple point in Fig. 2, and 2 their superposition ($E^{\text{A}}+E^{\text{S}}$).

Figure 4

(a) Dispersion curves of a superluminal space-time-modulated Lorentz medium for $M\to 0$. The magenta arrow represents the modulation vector. The three space-time harmonics $n=0$ (green), $n=+1$ (red), $n=-1$ (blue) are emphasized. The dashed line represents the edge of the Brillouin zone. (b) Dispersion diagram zoomed in at the crossing circled in Fig. 4 for the finite modulation depth $M=0.01$ and $f_{\text{per}}(.)=cos(.)$. The solid and dashed curves represent the real and imaginary parts of the propagation constant, respectively.

Figure 5

Field profiles for (a) the growing mode ($E^{\text{G}}$), corresponding to the cyan point in Fig. 4, and 4 the decaying mode ($E^{\text{D}}$), corresponding to the purple point in Fig. 4, and 4 their superposition ($E^{\text{G}}+E^{\text{D}}$), where $\beta =\text{Re}\left({\beta }_G\right)=\text{Re}\left({\beta }_D\right)$, $\alpha =\text{Im}\left({\beta }_G\right)=-\text{Im}\left({\beta }_D\right)$.

Figure 6

Transmission power in dB scale, $T=20{log}_{10}|{\mathbf{E}}^{\text{t}}|/|{\mathbf{E}}^{\text{i}}|$, through a slab of the medium in Fig. 2 excited by a plane wave versus its length $L$. The magnitude of different transmitted harmonics at ${\omega }_0+n{\omega }_{\text{m}}$ are plotted separately.

Figure 7

Transmission power in dB scale, $T=20{log}_{10}|{\mathbf{E}}^{\text{t}}|/|{\mathbf{E}}^{\text{i}}|$, from a slab of the medium in Fig. 4 excited by a plane wave, versus its length $L$. The magnitude of different transmitted harmonics at ${\omega }_0+n{\omega }_{\text{m}}$ are plotted separately.

Figure 8

Different scenarios for intercoupling between space-time harmonics. Top row: nondispersive systems supporting only forward-backward or contradirectional coupling. Bottom row: dispersive systems supporting forward-forward or codirectional space-time harmonic coupling. The left and right columns describe subluminal and superluminal space-time-modulated media, respectively. The dashed black curves represent the background (uncoupled) space-time harmonics. The solid blue curves represent the real parts of the dispersion relations. The red and green curves represent the imaginary parts of the complex propagation constants and complex frequencies, respectively.

Figure 9

Magnitude and phase of the harmonics for the additive and subtractive modes in Fig. 2. (a) Magnitude of the harmonics of the additive and subtractive modes. For clarity the dashed lines are shifted by 0.1 units. (b) The phase difference between the harmonics of the additive and subtractive modes.

Figure 10

Magnitude and phase of the harmonics for the growing and decaying modes in Fig. 4. (a) Magnitude of the harmonics of the growing and decaying modes. For clarity the dashed lines are shifted by 0.1 units. (b) The phase difference between the harmonics of the growing and decaying modes.

Figure 11

A slab of space-time-modulated dispersion-engineered medium is sandwiched between two nondispersive media and excited by a forward-propagating wave ${\mathbf{E}}^i$. ${\mathbf{E}}^r$ and ${\mathbf{E}}^t$ represent reflected and transmitted fields.