### New iterative annealing and polishing increases the resonator’s finesse over the fundamental limit.

Resonators usually are characterized with two partially dependent values: finesse (ℱ) and quality factor (*Q*). The finesse of an empty Fabry-Perot (FP) resonator is defined solely by the quality of its mirrors and is calculated as

ℱ = π*R*^{1/2}/(1 – *R*).

The maximum up-to-date value of reflectivity *R* ≈ 1 – 1.6 × 10^{–6} is achieved with dielectric mirrors. An FP resonator made with the mirrors has finesse ℱ = 1.9 × 10^{6}. Further practical increase of the finesse of FP resonators is problematic because of the absorption and the scattering of light in the mirror material through fundamental limit on the reflection losses given by the internal material losses and by thermodynamic density fluctuations on the order of parts in 10^{9}. The quality factor of a resonator depends on both its finesse and its geometrical size. A one-dimensional FP resonator has *Q* = 2 ℱ *L*/λ, where *L* is the distance between the mirrors and λ is the wavelength. It is easy to see that the quality factor of the resonator is unlimited because *L* is unlimited. ℱ and *Q* are equally important.

In some cases, finesse is technically more valuable than the quality factor. For instance, buildup of the optical power inside the resonator, as well as the Purcell factor, is proportional to finesse. Sometimes, however, the quality factor is more valuable. For example, inverse threshold power of intracavity hyperparametric oscillation is proportional to *Q*^{2} and efficiency of parametric frequency mixing is proportional to *Q*^{3}. Therefore, it is important to know both the maximally achievable finesse and quality factor values of a resonator.

Whispering gallery mode (WGM) resonators are capable of achieving larger finesse compared to FP resonators. For instance, fused silica resonators with finesse 2.3 × 10^{6} and 2.8 × 10^{6} have been demonstrated. Crystalline WGM resonators reveal even larger finesse values, ℱ = 6.3 × 10^{6}, because of low attenuation of light in the transparent optical crystals. The larger values of ℱ and *Q* result in the enhancement of various nonlinear processes. Low- threshold Raman lasing, optomechanical oscillations, frequency doubling, and hyper-parametric oscillations based on these resonators have been recently demonstrated. Theory predicts a possibility of nearly 10^{14} room-temperature optical *Q*-factors of optical crystalline WGM resonators, which correspond to finesse levels higher than 10^{9}. Experiments have shown numbers a thousand times lower than that. The difference occurs due to media imperfections.

To substantially reduce the optical losses caused by the imperfections, a specific, multi-step, asymptotic processing of the resonator is implemented. The technique has been initially developed to reduce microwave absorption in dielectric resonators. One step of the process consists of mechanical polishing performed after high temperature annealing. Several steps repeat one after another to lead to significant reduction in optical attenuation and, as a result, to the increase of *Q*-factor as well as finesse of the resonator which demonstrates a CaF_{2} WGM resonator with ℱ >10^{7} and *Q*>10^{11}.

*This work was done by Lute Maleki of OE Waves and Anatoliy Savchenkov, Andrey Matsko, and Vladimir Iltchenko of Caltech for NASA’s Jet Propulsion Laboratory.*

*This invention is owned by NASA, and a patent application has been filed. Inquiries concerning nonexclusive or exclusive license for its commercial development should be addressed to the Patent Counsel, NASA Management Office–JPL. Refer to NPO-45053.*