Selective frequency conversion with coupled time-modulated cavities

Time dependence in photonic structures leads to a number of intriguing and useful phenomena such as optical isolation, topological states of light, and frequency conversion. Here we present a mechanism to achieve efficient and selective frequency conversion using a system of two time-modulated cavities. This setup allows us to fine-tune the conversion process by controlling important parameters such as the intercavity coupling and the external excitation frequency. Both symmetric and asymmetric (up- or down-conversion) outputs can be targeted at will. We describe the processes extensively, with for example a leading role for the dynamic modes of the coupled system, the Floquet modes.

Figure 1

(a) Representation of the system of coupled resonances. (b) Possible implementation using two coupled arrays of graphene ribbons. Each cavity corresponds to a plasmonic resonance in the graphene array, where different ribbon widths ($D_1$ and $D_2$) provide different resonance frequencies. The ribbons are repeated in the horizontal direction creating two vertically offset gratings.

Figure 2

Blue lines: Floquet modes of the system for the degenerate case, where ${\omega }_{1,2}=2\pi \times {10}^{13}$ rad/s, $\mathrm{\Omega }=2\pi \times 2.5\times {10}^{11}$ rad/s, and $\delta =\mathrm{\Omega }/2$. Dashed red lines: Static resonance frequencies for the degenerate case $({\omega }_{\pm }={\omega }_{1,2}\pm \kappa )$. The four arrows near $\kappa =\mathrm{\Omega }/2$ represent the transitions of interest that we study in Sec. 4a while the two arrows near $\kappa =\mathrm{\Omega }$ represent the transitions that we study in Sec. 4b. An anticrossing appears at $\kappa =\mathrm{\Omega }/2$ where the static modes intersect.

Figure 3

Typical frequency comb produced by a time-modulated two-cavity setup. Here ${\omega }_1={\omega }_2$ and only the bright cavity is modulated. The parameters are such that no particular enhancement is achieved (${\omega }_0={\omega }_{1,2}=2\pi \times {10}^{12}$ rad/s, $\mathrm{\Omega }={\omega }_0/40$, $\kappa =\mathrm{\Omega }/2$, $\delta =\mathrm{\Omega }/4$, and $\gamma =2\times {10}^{11}$ rad/s).

Figure 4

Figures of merit ${\mathrm{\Gamma }}_{\pm }$ as a function of the incident frequency ${\omega }_0$. The frequency conversion is asymmetric at the anticrossing (when $\kappa =\mathrm{\Omega }/2$). When ${\omega }_0$ is equal to a Floquet mode frequency, the conversion efficiency is enhanced. The difference in conversion efficiency depends on the initial and final mode order: the coupling between source and Floquet modes is better for lower-order modes.

Figure 5

Comb produced by a set of parameters corresponding to the second peak of Fig. 4 (${\omega }_0=0.9890{\omega }_{1,2}$). The conversion to the upper sideband (${\omega }_0+\mathrm{\Omega }$) is more efficient than the conversion to the lower sideband (${\omega }_0-\mathrm{\Omega }$) and as a result the comb is asymmetric.

Figure 6

Figures of merit ${\mathrm{\Gamma }}_{\pm }$ for the four transitions highlighted in Fig. 2 around the band gap. The conversion efficiency is greater when the incident mode is a zeroth-order mode and at the band edge (for $\kappa =\mathrm{\Omega }/2$).

Figure 7

Figures of merit ${\mathrm{\Gamma }}_{\pm }$ as a function of the incident frequency ${\omega }_0$. In this case, $\delta =\mathrm{\Omega }/4$ and ${\omega }_1={\omega }_2$. At the first band crossing ($\kappa \approx \mathrm{\Omega }$), the frequency conversion is efficient and symmetric.

Figure 8

Comb produced by a set of parameters corresponding to the central peak of Fig. 7. The conversion to the upper sideband (${\omega }_0+\mathrm{\Omega }$) is nearly as efficient as the conversion to the lower sideband (${\omega }_0-\mathrm{\Omega }$).

Figure 9

Solid blue lines: Floquet modes of the system for the degenerate case (same as in Fig. 2). Dashed blue lines: Nondegenerate case (${\omega }_1\ne {\omega }_2$). We chose these two frequencies such that ${\omega }_1-{\omega }_2=2\mathrm{\Omega }/5$.

Figure 10

Figures of merit ${\mathrm{\Gamma }}_{\pm }$ as a function of the incident frequency ${\omega }_0$. In this case, $\delta =\mathrm{\Omega }$, ${\omega }_2-{\omega }_1=2\mathrm{\Omega }/5$, and $\kappa =\mathrm{\Omega }/2$ (around the anticrossing).

Figure 11

Figures of merit ${\mathrm{\Gamma }}_{\pm }$ as a function of the incident frequency ${\omega }_0$. In this case, $\delta =\mathrm{\Omega }$, ${\omega }_1-{\omega }_2=2\mathrm{\Omega }/5$, and $\kappa \approx \mathrm{\Omega }$ (around the second anticrossing).

Figure 12

Red circles are the results of the three-frequency model while the blue line is the full numerical solution. In (a) we imposed the constraint $b_{-}=b_{+}$ so the comb generated is perfectly symmetric. In (b) we imposed the constraint $b_0=b_{+}=0$, so the frequency conversion is asymmetric.

Figure 13

Solid lines: Numerical results for the eigenvalues of the Floquet matrix. Circles: Eq. (18) and with $\mathrm{\Omega }$ offset. The parameters used are the same as in Fig. 2.