Microcavity morphology optimization

High spectral mode density of conventional optical cavities is detrimental to the generation of broad optical frequency combs and to other linear and nonlinear applications. In this work we optimize the morphology of high-$Q$ whispering gallery (WG) and Fabry-Perot (FP) cavities and find a set of parameters that allows treating them, essentially, as single-mode structures, thus removing limitations associated with a high density of cavity mode spectra. We show that both single-mode WGs and single-mode FP cavities have similar physical properties, in spite of their different loss mechanisms. The morphology optimization does not lead to a reduction of quality factors of modes belonging to the basic family. We study the parameter space numerically and find the region where the highest possible $Q$ factor of the cavity modes can be realized while just having a single bound state in the cavity. The value of the $Q$ factor is comparable with that achieved in conventional cavities. The proposed cavity structures will be beneficial for generation of octave spanning coherent frequency combs and will prevent undesirable effects of parametric instability in laser gravitational wave detectors.

Figure 1

Morphologies of the studied cavities: (a) single-mode WGM cavity; (b) single-mode FP cavity.

Figure 2

Spatial profile of the field intensity for the fundamental and the first dipole WGMs for different height of the protrusion: (a),(b) $x_0=2\mu \mathrm{m}$; (c),(d) $x_0=8\mu \mathrm{m}$; and (e),(f) $x_0=11\mu \mathrm{m}$. The intensity is illustrated by color density.

Figure 3

Dependence of the $Q$ factors of the fundamental and the first dipole as well as the first higher-order radial modes [quantization by $q$ number; see Eq. (7)] on the protrusion height for a WGM cavity (a) and a FP cavity (b). The numerical model is not accurate enough to predict $Q$ factors below ${10}^6$ and above ${10}^{10}$.

Figure 4

Dependence of the $Q$ factors of fundamental and first dipole mode versus the protrusion height. The simulation is performed for a WGM cavity formed by a protrusion with a square cross section with $10\mu \mathrm{m}$ width. The $Q$ factor of the modes drops below ${10}^6$ for height less than 1.5 $\mu \mathrm{m}$.

Figure 5

Variation of the $Q$ factor of the basic and first dipole modes of a single-mode WGM cavity with Gaussian shape versus thickness of the cavity, $h$. The Gaussian is characterized with $x_0=3\mu \mathrm{m}$ and $10\mu \mathrm{m}$ FWHM.

Figure 6

Spatial profile of the FP cavity mirrors and the field intensity for the fundamental (red solid lines) and the first dipole (blue dashed lines) modes for different heights of the mirrors: (a) $x_0=5\mu \text{m}$, (b) $x_0=8\mu \text{m}$, and (c) $x_0=14\mu \text{m}$. The intensity distribution of poorly confined modes have irregular spatial structure resulting from diffraction at the mirror edge. The arrows are used to directly show which scale (left or right) has to be used with each particular curve presented on the picture.

Figure 7

Dependence of the $Q$ factors of the fundamental, first axisymmetric, and first dipole modes on the protrusion height FP cavity mirror profiles controlled by parameters $\alpha ,\beta$ listed in Table 1.