Microspheres with Atomic-Scale Tolerances Generate Hyperdegeneracy

Degeneracies play a crucial rule in precise scientific measurements as well as in sensing applications. Spherical resonators have a high degree of degeneracy thanks to their highest symmetry; yet, fabricating perfect spheres is challenging because even a stem to hold the sphere breaks the symmetry. Here we fabricate a levitating spherical resonator that is evanescently coupled to a standard optical fiber. We characterize the resonators to exhibit an optical quality factor exceeding a billion, $10\mu \mathrm{m}$ radius, and sphericity to within less than 1 Å. Using our high quality and sphericity, we experimentally lift degeneracies of orders higher than 200, which we resolve with optical finesse exceeding 10 000 000. We then present our experimentally measured degenerate modes as well as their density of states next to our corresponding theoretical calculation. Our contactless photonic resonator is compatible with standard telecom fiber technology, exhibits the highest resonance enhancement as defined by (quality factor)/(mode volume), and the modes populating our cavity show the highest order of degeneracy reported in any system ever studied. This is in comparison with other settings that typically utilize the lowest-order twofold degeneracy.

Popular Summary

Spherical resonators, which trap circulating waves of light in a multitude of orientations, are useful for sensing. However, fabricating a perfectly spherical resonator is challenging because any point of contact that holds up the resonator will subtly deform the device and break the spherical symmetry needed to host multiple rings of light. Here, we fabricate a contactless, levitating optical resonator that is much closer to an ideal spherical shape than any other resonator of comparable size created so far.

Our resonators are $20\text{−}\mu \mathrm{m}$-wide droplets of silicone oil levitated by contactless optical tweezers—highly focused laser beams that support the droplets in midair. The droplet is coupled to an optical fiber, which pumps in light tuned to the resonances of the droplet. We measure these resonances by observing the transmission through the fiber. These measurements reveal more than 200 degenerate modes within the droplets (as opposed to other setups that support only two). The droplets themselves are spherical to within fabrication tolerances of roughly the size of an atom.

Except for the fundamental interest in experimentally demonstrating highest-order degeneracies, such hyperdegenerate devices might impact precise measurements, will support fundamental studies in light-matter and light-structure interactions, and serve in engineering applications.

Figure 1

Optical modes in spherical resonators. (a)-(d) High-degree degeneracy in a spherical resonator where a family of modes have the same propagation constant ${\beta }_m$ in the net direction of propagation (blue arrow), irrespective of their number of maxima between poles (here 1, 2, 3, and 5). (e),(f) Standing interference pattern generated by modes having the same resonance frequency, but with a different number of wavelengths along the equator ($m$). Red (blue) surfaces were calculated with Eq. (SM1) of the Supplemental Material [25] to represent the location where the electric field falls to half of its maximum (minimum). Green represents an equal intensity surface. The polarization is TE, the radial number is 1, the refractive index is 1.4, and the sphere’s circumference is 11 optical wavelengths.

Figure 2

Experimental setup for phase-matched coupling to a contactless sphere. A counterpropagating optical tweezer traps a silicone oil droplet. At the same time, a tapered-fiber coupler is brought nearby to the droplet. The transmission through the droplet resonator is monitored throughout the other side of the taper. The experiment is performed under the microscope while a monitor shows a micrograph of the droplet in which the bent tapered fiber is seen at the background, out of focus. The refractive index is 1.4 and the wavelength is 980 nm.

Figure 3

A photograph of the experiment. (a) The levitating microdroplet is visible to the naked eye via scattering of the tweezers’ green light. Micrographs of the experiment where we present top (b), side (c), and enlarged (d) views of the resonator-taper region. (e)–(j) Experimental results with 1–7 adjacent resonances (blue) together with a fit to a several-Lorentzians model (red). Resonances at (e)–(j) were measured at different spectral regions. The error bar represents the standard deviation. Orange lines represent regions of coresonating (partially overlapping) mode. $\lambda =980\mathrm{nm}$, $r=11.45\mu \mathrm{m}$ for (d), (h), and (i), and $13.5\mu \mathrm{m}$ for (f), (g), and (j), and $n=1.4$. Deviations of the measured transmission from the Lorentzian model relate to the nonperfectly uniform scanning rate of the tunable laser.

Figure 4

Density of modes. The spectral transmission of the levitating resonator (a),(b) indicates no resonances at the left-hand side of (b), followed by dense resonances [(b), middle], and then fewer resonances per frequency span [(b), right-hand side]. Panel (a) represents an enlargement of spectral regions that are marked by an arrow in (b). (c) The experimentally measured density of modes (blue) as a function of the laser frequency, next to a theoretical fit using Eq. (SM12) [25] (orange). $\lambda =980\mathrm{nm}$, $r=13.5\mu \mathrm{m}$, $n=1.4$, $l=113$, and degeneracy order is 227. Frequency error bars reflect standard deviation, and density-of-modes error bars reflect our estimation for fault counting of resonances. The irregular transmission background originates from fluctuations in the power of our pump laser.