Coupled-resonator-induced transparency

We demonstrate that a cancellation of absorption occurs on resonance for two (or any even number of) coupled optical resonators, due to mode splitting and classical destructive interference, particularly when the resonator finesse is large and the loss in the resonator farthest from the excitation waveguide is small. The linewidth and group velocity of a collection of such coupled-resonator structures may be decreased by using larger resonators of equal size, by using larger resonators of unequal size where the optical path length of the larger resonator is an integer multiple of that of the smaller one, or by using a larger number of resonators per structure. We explore the analogy between these effects and electromagnetically-induced transparency in an atomic system.

Figure 1

Typical energy level diagram for the observation of EIT.

Figure 2

Illustration of $N$ coupled ring resonators. The numbering scheme for the rings and electric fields is shown.

Figure 3

(a) Absorptance vs single-pass phase shift for two coupled ring resonators with $a_1=0.9999$, $a_2=0.88$, $r_1=0.999$, and $r_2=0.9$. The resonance is split, analogous to the Autler-Townes splitting and destructive interference that occurs in three-level atomic systems, such that transparency rather than absorption occurs for ${\varphi }_1={\varphi }_2=0$. (b) Effective phase shift for a single ring resonator $\left({\varphi }_1^{\left(\text{eff}\right)}\right)$ and for two coupled ring resonators $\left({\varphi }_2^{\left(\text{eff}\right)}\right)$. Whereas anomalous dispersion occurs on resonance for (a), normal dispersion occurs for (b). (c) The slope of the effective phase shift vs the single-pass phase shift for two coupled rings. This quantity is proportional to the difference between the group index and the phase index. The light is slowed by a factor of about $100\rho Z$ on resonance.

Figure 4

Relative splitting $\Delta \omega ∕\gamma$ plotted vs $r_1$ for various values of $r_2{a}_2$ (solid curve) and $\eta$ (dashed curve). The dotted curve represents the approximation of small resonator losses for $r_2{a}_2=0.9$.

Figure 5

(a) Absorptance vs single-pass phase shift for two unequal coupled ring resonators such that ${\varphi }_1=4{\varphi }_2$, where $a_1=0.9999$, $a_2=0.88$, $r_1=0.999$, and $r_2=0.9$. The CRIT linewidth is 4 times smaller than for the case of equal-sized rings $\left({\varphi }_1={\varphi }_2\right)$. (b) The group index on resonance is 4 times larger than for the case of equal-sized rings $\left({\varphi }_1={\varphi }_2\right)$.