Connection of temporal coupled-mode-theory formalisms for a resonant optical system and its time-reversal conjugate

Temporal coupled-mode theory has been widely used to describe the physics of resonant optical systems. In general, an optical system can be constrained by energy conservation, time-reversal symmetry, and reciprocity. Most previous developments of temporal coupled-mode theory made use of all three constraints. In this paper, we consider separately the implication of each of these constraints on the parameters of temporal coupled-mode theory. For this purpose we made extensive use of the connection between a physical system and its time-reversal conjugate. This connection also indicates some of the nontrivial implications of the relation between the resonant properties of a physical system and its time-reversal conjugate. We validate these implications numerically by direct electromagnetic simulations of a guided resonance system. This work should enable the application of temporal coupled-mode theory to a wider range of resonant systems.

Figure 1

(a) Schematic of a resonator system described by a dielectric function $\mathbf{\epsilon }\left(\mathit{r}\right)$, coupling to multiple ports. (b) The time-reversal conjugate system with respect to (a) as described by a dielectric function ${\mathbf{\epsilon }}^{*}\left(\mathit{r}\right)$.

Figure 2

(a) Schematic of a unit cell of a dielectric grating that supports guided resonances. The relative permittivity of the grating (in blue) is 12. The relative permittivity of the bottom slab (in grey) is $2-i{\epsilon }_i$, where the imaginary part is varied. The periodicity is $l$, the grating total thickness $t=0.5l$, the thickness of top grating ridge $s=0.1l$, and the bottom slab thickness $b=0.2l$. To sample different resonant modes, the gap between the grating and the bottom slab $g$ is varied within 0–$0.2l$, and the width of the top grating ridge $w$ is varied within $0.2l$–$0.8l$. Only TM modes at $\mathrm{\Gamma }$ points are studied. For example, with parameters ${\epsilon }_i=0.5,g=0$, and $w=0.2l$, the electric field of a resonance near $\omega =0.52\times 2\pi c/l$ is shown in (b), and the transmission spectrum is shown in (c). (d) and (e) respectively show ${\tilde{\gamma }}_r-{\gamma }_r$ and ${\tilde{\gamma }}_i+{\gamma }_i$ as functions of ${\epsilon }_i$ for different resonant modes.