Bloch-Floquet waves in optical ring resonators

Modal coupling between frequency-degenerate resonances of an optical ring resonator is a commonly observed phenomenon that results in adverse mode splitting. Traditionally, this coupling is attributed to Rayleigh scattering of a propagating electromagnetic wave into its associated degenerate counterpropagating mode from small perturbations to the dielectric material of the resonator. We have chosen to reframe the problem of intracavity Rayleigh scattering by considering the optical ring resonator as an infinitely long, one-dimensional photonic crystal (PhC) that possesses a lattice constant equal to the perimeter of the ring. Through application of Bloch-Floquet theory, we show that modal coupling between degenerate resonances of a ring can effectively be described as the formation of photonic frequency bands in the dispersion relation of the resonator. We additionally demonstrate that the Bragg planes of the PhC lattice coincide with the phase matching conditions for constructive interference in the ring. Finally, we show that the magnitude of frequency splitting of a particular resonance is proportional to its associated coefficient in the Fourier expansion of the ring's periodic dielectric function. The fine control of the mode splitting that can be obtained in this way provides a straightforward way to obtain ring resonances with anomalous dispersion in arbitrary wavelength ranges.

Figure 1

A standard ring resonator of radius $R$ with point surface scatterers represented as an infinitely long dielectric waveguide. The high dielectric constant material is seen in black, while the low dielectric constant material is seen in white.

Figure 2

The photonic dispersion relation of a ring resonator in the reduced zone scheme (assuming a linear waveguide dispersion). The resonance of the ring occur wherever the phase matching boundary conditions (the blue vertical lines) intersect the photonic dispersion relation of the infinitely long waveguide. Phase matching conditions are separated by one reciprocal-lattice vector; consequently only the $m=0$ boundary condition represents a physically distinct state in the reduced zone scheme.

Figure 3

The dielectric function of a infinitely long SOI waveguide of width $w=450$ nm and effective dielectric constant ${\epsilon }_0=8.01$. The top image represents the Fourier expansion of the dielectric function of a waveguide with ${\kappa }_0=0.125$, ${\kappa }_{96}=0.025$, and ${\kappa }_{l\ne 96}=0$. The bottom image shows the geometry of an equivalent waveguide. For a $5\text{−}\mu \mathrm{m}$-radius ring, the waveguide possesses a dielectric function modulation of wavelength $\lambda =\frac{2\pi R}{96}=330$ nm and amplitude $\mathrm{\Delta }w=10$ nm.

Figure 4

The numerical prediction of Eq. (9) as compared to FDE and FDTD computational results for various values of ${\kappa }_{96}$. The dimensionless frequency of the FDE results has been scaled to the characteristic length unit for the system, the ring's perimeter $P$, while the FDTD results are given in THz.

Figure 5

The frequency spectrum of a $5\text{−}\mu \mathrm{m}$ 2D SOI ring resonator for (a) ${\kappa }_{96}=0$, (b) ${\kappa }_{96}=0.00375$, and (c) ${\kappa }_{96}=0.0075$. (d) The spectrum of a ring resonator with ${\kappa }_{90}=0.00235$ and ${\kappa }_{92}={\kappa }_{94}={\kappa }_{96}={\kappa }_{98}=0.0023$. In this case every other resonance in the spectrum possesses anomalous GVD.