Pathway interference in a loop array of three coupled microresonators

A system of three coupled toroidal microresonators arranged in a loop configuration is studied. This setup allows light entering the resonator setup from a tapered fiber to evolve along a variety of different pathways before leaving again through the fiber. In particular, the loop configuration of the resonators allows for an evolution which we term round-trip process, in which the light evolves from one resonator sequentially through all others back to the initial one. This process renders the optical properties of the system sensitive to the phases of all coupling and scattering constants in the system. We analyze the transmission and reflection spectra, and interpret them in terms of interference between the various possible evolution pathways through the resonator system. In particular, we focus on the phase dependence of the optical properties. Finally, we discuss possible applications for this phase sensitivity induced by the round-trip process, such as the measurement of the position of a nanoparticle close to one of the resonators and the measurement of changes in the refractive index between two resonators. Our analytical results for the applications are supported by proof-of-principle calculations based on the finite-difference time-domain solution of Maxwell’s equations in two dimensions on a grid.

Figure 1

The considered setup of three coupled resonators in loop configuration. The resonators are probed by a fiber coupled to one of the resonators. Due to the arrangement of the resonators, light can evolve in a round-trip process, e.g., from cavity 1 via cavities 2 and 3 back to cavity 1 without leaving the resonator array, which leads to rich interference effects.

Figure 2

Example pathways of photons passing through the resonator array. In (a), the photon enters only the first cavity, scatters inside cavity 1, and then leaves the cavity back into the fiber. In (b), the photon traverses two cavities, but scatters only in cavity 1. (c) shows a pathway through two cavities with scattering in both the cavities, (d) a path through all cavities with scattering in cavity 1 only, (e) a path through all cavities without scattering, and (f) an evolution through three cavities with scattering in cavities 2 and 3.

Figure 3

Transmission $T$ and reflection $R$ in the loop system. The parameters are ${\xi }_{12}={\xi }_{13}=30\gamma$ and ${\xi }_{23}=0$, such that scattering between cavities $1\leftrightarrow 2$ and $1\leftrightarrow 3$ are possible, but not between 2 and 3. Scattering occurs in all cavities with rates ${\xi }_{11}=30\gamma$ and ${\xi }_{22}={\xi }_{33}=20\gamma$. In (a) all phases are chosen zero, ${\varphi }_{\mathit{ij}}=0$. The dashed lines show corresponding results for transmission ($\mathrm{T̄}$) and reflection ($\mathrm{R̄}$) averaged over the phase angles ${\varphi }_{12}$ or ${\varphi }_{22}$. In (b), the phases are chosen as ${\varphi }_{12}=0.2\pi$, and all other ${\varphi }_{\mathit{ij}}=0$. In (c), ${\varphi }_{12}=-0.6\pi$ and ${\varphi }_{13}=0.4\pi$, and all other ${\varphi }_{\mathit{ij}}=0$.

Figure 4

Field configuration in resonator 1. [(a) and (c)] Scaled occupancies ${\mathcal{O}}_{a_1}=2\kappa \left|a_1\right|{}^2/\left|a_{\text{in}}\right|{}^2$ and ${\mathcal{O}}_{b_1}=2\kappa \left|b_1\right|{}^2/\left|a_{\text{in}}\right|{}^2$ of the two counterpropagating modes $a_1$ and $b_1$ inside cavity 1. [(b) and (d)] Phase ${\varphi }_a=\left|{\varphi }_{a_1}\right|$ of the counterclockwise propagating mode inside cavity 1 which is coupled out into the fiber in transmission direction. In (a) and (b), the parameters are chosen as in Fig. 3a, whereas in (c) and (d) they are as in Fig. 3b.

Figure 5

Eigenvalue analysis to interpret the phase dependence. (a) and (c) show the imaginary part of the eigenvalues ($\zeta$) of the matrix $\mathcal{M}$ describing the system dynamics, which correspond to the energy of the system’s dressed states. They are plotted against the phase angle ${\varphi }_{12}$. (b) and (d) show the power spectrum $\left|\mathrm{\zeta ̃}\right|{}^2$ of one eigenvalue as an example, revealing the periodicity of the eigenenergies in ${\varphi }_{12}$. The parameters in (a) and (b) are ${\xi }_{11}={\xi }_{13}=30\gamma$, ${\xi }_{22}={\xi }_{33}=20\gamma$, ${\xi }_{12}=30\gamma e^{i{\varphi }_{12}}$, and ${\xi }_{23}=0$, thus no round-trip process is possible. In (c) and (d) ${\xi }_{23}=15\gamma$, thus the round-trip process is possible.

Figure 6

Expansion of the transmission in orders of the round-trip process. For this, the transmission is expanded around 0 in $\left|{\xi }_{23}\right|/\gamma$. The parameters are ${\xi }_{11}=50\gamma$, ${\xi }_{22}=20e^{i{\varphi }_{22}}\gamma$, ${\xi }_{33}=20\gamma$, ${\xi }_{23}=10\gamma$, ${\xi }_{13}=30e^{i{\varphi }_{13}}\gamma$, and all ${\varphi }_{\mathit{ij}}=0$. (a) Zero-order contribution (${\xi }_{23}=0$), (b) first-order contribution $c_1$, (c) second-order contribution $c_2$, and (d) complete transmission for ${\xi }_{23}=3\gamma$.

Figure 7

[(a),(c),(e)] Reflection R (red) and [(b),(d),(f)] transmission T (blue) spectra for parameters which allow for the round-trip process. The parameters are the same as in Fig. 6 except for ${\xi }_{23}=5\gamma$. In (a) and (b), all phases are zero (${\varphi }_{\mathit{ij}}=0$). In (c) and (d), ${\varphi }_{22}=1.6\pi$ and ${\varphi }_{13}=0.4\pi$, and all other phases are chosen zero. In (e) and (f), ${\varphi }_{22}=1.6\pi$, ${\varphi }_{13}=-0.4\pi$, and all others ${\varphi }_{\mathit{ij}}=0$.

Figure 8

Transmission T (blue) and reflection R (red) spectra for parameters which allow for the round-trip process. The spectra are plotted against the phase ${\varphi }_{13}$. The other parameters are chosen as in Fig. 7 with $\Delta =-19.5\gamma$.

Figure 9

Setups for the two example applications. (a) shows the measurement of the position of a particle at position ${\mathrm{R⃗}}_p$ close to resonator 2. (b) shows the measurement of changes in the index of refraction between resonators 2 and 3 induced, e.g., by slabs of different refractive indices.

Figure 10

Transmission $T,\mathrm{T̄}$ and reflection $R,\mathrm{R̄}$ for parameters as in Fig. 5. In (a), the solid lines $T,R$ show the case with phases ${\varphi }_{\mathit{ij}}=0$; the dashed lines $\mathrm{T̄},\mathrm{R̄}$ show results averaged over ${\varphi }_{22}$. In (b), the solid lines $T,R$ show the case with phases ${\varphi }_{\mathit{ij}}=0$; the dashed lines $\mathrm{T̄},\mathrm{R̄}$ show results averaged over ${\varphi }_{12}$.

Figure 11

Configuration of the field component $H_z$ obtained from the FDTD simulation for three resonators in loop configuration without particle or slab. The image shows one snapshot in time. The excitation wavelength is $\lambda =577.21$ nm. See text for further explanation.

Figure 12

Transmission spectrum obtained from FDTD simulations for three resonators in loop configuration without particle or slab.

Figure 13

Transmission spectrum obtained from FDTD simulations for three resonators in loop configuration with a particle placed close to the top left resonator. The different curves show particle positions (a) $\theta ={90}^{°}$, (b) $\theta ={95}^{°}$, and (c) $\theta ={180}^{°}$, whereas (d) shows the result for the case without nearby particles. Dots indicate the actual results of the calculation, and the lines connect the dots to guide the eye.

Figure 14

Transmission spectrum obtained from FDTD simulations for three resonators in loop configuration with a slab placed between the two rightmost cavities [see Fig. 9b]. The curves show two spectral lines at different wavelengths. In the direction of the arrow the different curves show results for slabs with $\epsilon =4.0$, $\epsilon =4.001$, $\epsilon =4.01$, $\epsilon =1$ (without slab), $\epsilon =4.02$, $\epsilon =4.05$, and $\epsilon =4.1$.