Intermode reactive coupling induced by waveguide-resonator interaction

We report on a joint theoretical and experimental study of an integrated photonic device consisting of a single-mode waveguide vertically coupled to a disk-shaped microresonator. Starting from the general theory of open systems, we show how the presence of a neighboring waveguide induces a reactive intermode coupling in the resonator, analogous to an off-diagonal Lamb shift in atomic physics. Observable consequences of this coupling manifest as peculiar Fano line shapes in the waveguide transmission spectra. The theoretical predictions are validated by full vectorial three-dimensional finite-element numerical simulations and are confirmed by the experiments.

Figure 1

(a) A sketch of the microphotonic device. (b) The intensity profile of the first and second radial mode families (RMFs) of the resonator (top and middle panels) and of the waveguide mode (bottom panel). The (blue) curves show the cuts of the intensity profile, and the labels indicate the different materials. (c) (d) Results of ab initio numerical calculations for (c) the radiative decay rate ratio ${\Gamma }_{22}^{\mathrm{rad}}/{\Gamma }_{11}^{\mathrm{rad}}$ and (d) the frequency shifts of the first $({\Delta }_{11}$, top) and the second $({\Delta }_{22}$, bottom) radial family modes as functions of the waveguide position. The open circles indicate the waveguide position for the fabricated 40-$\mu \text{m}$-diameter resonator.

Figure 2

Analytically calculated transmission spectra in different regimes and for different detunings $\delta ={\omega }_2-{\omega }_1$. (a) Both modes are undercoupled, ${\Gamma }_{11,22}^{\mathrm{rad}}/{\gamma }_{1,2}^{\mathrm{nr}}=0.25$ and ${\Delta }_{12}=0$. (b) Both modes are overcoupled, ${\Gamma }_{11,22}^{\mathrm{rad}}/{\gamma }_{1,2}^{\mathrm{nr}}=4$ and ${\Delta }_{12}=0$. (c) Narrow (broad) mode is undercoupled (overcoupled), ${\Gamma }_{11}^{\mathrm{rad}}/{\gamma }_1^{\mathrm{nr}}=0.125$ $\left({\Gamma }_{22}^{\mathrm{rad}}/{\gamma }_2^{\mathrm{nr}}=2\right)$ and ${\Delta }_{12}/{\gamma }_2^{\mathrm{nr}}=0.6$. The ratio ${\gamma }_1^{\mathrm{nr}}/{\gamma }_2^{\mathrm{nr}}=0.1$ (a-b) and $0.16$ (c).

Figure 3

(a) Numerically calculated spectra for different detunings between the two radial family modes. The azimuthal mode order of the two resonances is reported in each graph. (b) Planar cuts of the intensity profile inside the resonator and inside the waveguide at the frequencies indicated as A, B, and C in (a). (c) The polygonal shape of the spatially oscillating mode profile is explained via an interference between the fields $E_1=E_1^{0}cos\left(\theta M_1\right)$ and $E_2=E_2^{0}cos\left(\theta M_2\right)$ of two modes with different azimuthal mode numbers $M_1$ and $M_2$. (d) This interference pattern is illustrated as a function of the azimuthal angle $\theta$ and the number $M_2$ of the second-order radial mode for a fixed $M_1=125$ of the first-order radial mode: as expected, the number of polygon vertices is determined by the difference $|M_2-M_1|$.

Figure 4

(Left) An optical photograph of the fabricated 40-$\mu \text{m}$-diameter microresonator and (a) the measured transmission spectrum as a function of the absolute incident frequency. The azimuthal mode numbers $M_1$ and $M_2$ are indicated next to the different first- and second-order radial modes. The two modal families have slightly different free spectral ranges of ${\mathrm{FSR}}_1\approx 1.236$ THz and ${\mathrm{FSR}}_2\approx 1.256$ THz. (b)–(g) Blow-ups of the regions marked in gray in (a). In each panel, the relative frequency is measured from the broader second family resonance. Red lines show fits to the spectra using the analytical model.

Figure 5

(a) Color-map plot merging six experimental transmission spectra of the $40\mu \text{m}$ resonator, shown in Figs. 4–4(g). (b) Color-map plot of the analytical prediction (3) for the transmittivity of a two-mode cavity using globally optimized parameters.

Figure 6

(a) Color map merging 21 experimental transmission spectra (indicated as S1–S21) for a $50\mu \text{m}$ resonator. On each row, the relative frequency is measured from the narrow mode frequency. (b) Analytic prediction for $T\left(\omega \right)$ using a three-mode extension of the model with optimized global parameters. (c) Selected examples of spectra. (d), (e) System parameters obtained by independently fitting each experimental spectrum with the analytical model.

Figure 7

Ratio of the decay rates and frequency shift of the resonator modes as obtained from finite-element simulations. In the left (a), (b) panels [the same plots as in Fig. 1], the different radial family modes are well detuned from each other and effectively independent. In the right (c),(d) panels, the considered modes are mixed by the off-diagonal reactive and dissipative coupling terms.