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Sensing & Measurement

Toward whispering-gallery-mode disk resonators for metrological applications

Numerical simulation is used to characterize a novel method for efficiently inhibiting vibration- and acceleration-induced frequency fluctuations in whispering-gallery-mode disk resonators.
12 March 2012, SPIE Newsroom. DOI: 10.1117/2.1201202.004034

Optical resonators of millimeter size are characterized by significantly long photon lifetimes (>1μs), and accordingly by ultra-narrow resonance frequency linewidths. This feature is highly desirable in a wide range of applications, including frequency metrology, spectroscopy, sensing, and ultra-low phase noise microwave and terahertz wave generation. Whispering-gallery-mode (WGM) disk resonators with an ultra-high Q factor (a measure of resonator quality) are expected to play an increasing role in metrological applications. In these resonators, photons are trapped by total internal reflection into torus-like eigenmodes, similar to acoustic reflections ricocheting around the circular inside of a cathedral dome. For many applications, frequency stability in the resonator is critical. Fluctuations in either the refractive index or the cavity dimensions produce jitter and drift in the resonance frequency. Although the small size of typical WGM resonators limits their sensitivity to mechanical disturbances, both internal and environmentally induced fluctuations ultimately limit the noise floor for frequency stability. But while thermal noise limits in WGM resonators have been discussed before,1, 2 vibrations and accelerations can also deform the cavity enough to shift the eigenmode frequency.

To illustrate the impact of mechanical disturbances, consider the mere action of gravity on a WGM calcium fluoride disk resonator with a diameter of 5mm and thickness of 0.5mm. Under the influence of gravity, a disk resonator in a typical stem-mounting configuration—see Figure 1—undergoes a radius change of a few picometers (10× smaller than the diameter of the hydrogen atom). This microscopic change produces a resonance frequency shift as large as ∼200kHz for light with a wavelength of 1550nm. Metrology applications require a frequency precision on the order of 1Hz, which is the equivalent of a cavity radius stability better than ∼1fm, or roughly the diameter of a proton. Currently, the frequency stabilization of lasers locked to passive Fabry-Pérot optical cavities routinely reaches ∼10−15/Hz1/2.3, 4 Performance with WGM disk resonators is typically on the order of ∼10−12,5 but a thermal limit as low as ∼10−14 has already been reached.6To achieve a frequency stability comparable to Fabry-Pérot reference cavities, especially in practical devices, the problem of mechanical instability must be addressed.

Figure 1. An ultra-high-quality magnesium fluoride disk resonator.

A key challenge is therefore to design clever mounting architectures capable of reducing or eliminating environmental fluctuations of a mechanical nature. One interesting solution relies on the concept of ‘neutral mounting,’ in which the distribution of the mounting forces produces a null deformation of the optical cavity along a given direction.7The null force-to-displacement conversion prevents mechanical fluctuations from creating cavity-length deformations that would affect the resonance frequency. However, designing such a system is not straightforward given that a WGM disk resonator is subject to a number of stringent physical constraints. The most important requirement is that the rim, or lateral surface, of the disk must be free because any mechanical contact with that surface would preclude the propagation of whispering-gallery modes. The mounting scheme must also provide physical access to the edge of the disk by a fiber or prism for optical coupling.

Clamping a disk between two cylinders of unequal radii, when properly arranged, creates a radial displacement field that is null at the rim of the disk, regardless of the intensity of the mounting force. This finding is in fact foreshadowed by the Euler-Bernoulli beam theory, which states that when a beam is loaded (for example, bent downward), one portion is stretched while another is compressed. The plane of symmetry between those two portions, or a surface intersecting that plane, does not (to a first-order approximation) undergo any longitudinal deformation.

The same phenomenology occurs when a resonator is bent. Figure 2 shows the results of a finite-element numerical simulation of the displacement field corresponding to a clamped WGM disk resonator. This color-coded representation clearly demonstrates that the radial displacement is null near the plane of symmetry (the neutral plane), which is where the torus-like WGMs are physically located. Theoretical analysis indicates that the stress and strain fields are also null in this neutral plane. Hence, from an elastostatic point of view, the WGMs are subject to the same physical conditions as would exist in a load-free resonator. This finding holds over a wide range of mounting forces (see Figure 3).

Figure 2. Finite-element method simulation of the radial deformation field for a coaxially clamped disk resonator. Colors indicate the magnitude of radial deformation. The direction of this deformation has a turning point at z=65μm, which can be seen as the dark blue band around the outer edge of the resonator. This zero displacement of the radius ensures the stability of the resonant frequency reference. The inner and outer radii are 2 and 5mm, respectively, and the thickness of this magnesium fluoride disk is 0.5mm.

Figure 3. Simulation results showing that there is a location on the disk edge where the radial deformation is null for a wide range of mounting pressures (around z=65μm).

Our theoretical analysis, which has yet to be confirmed experimentally, suggests that a simple mounting strategy can reduce vibration- and acceleration-induced frequency fluctuations in WGM disk resonators. Successful implementation would make possible a small and robust frequency reference for use in portable and practical devices in non-laboratory environments. In future work, we will analyze mounting scheme optimization and temperature stabilization techniques as part of our ongoing research into the fundamental limits of these devices.

This research work was carried out at the Jet Propulsion Laboratory (JPL), California Institute of Technology (Caltech), under a contract with NASA. Yanne Chembo acknowledges the hospitality of the JPL during the completion of this research work.

Yanne K. Chembo
FEMTO-ST Institute
Besançon, France

Yanne Chembo is a French National Center of Scientific Research (CNRS) senior research scientist whose work focuses on microwave photonics and applications to communication and aerospace engineering. He is a European Research Council fellow and a former NASA postdoctoral fellow.

Lukas M. Baumgartel, Nan Yu
Pasadena, California

Lukas Baumgartel is a researcher at the JPL working towards a physics PhD from the University of Southern California. His research interests include microresonators, and he has worked with both piezoelectric acoustic transducers and crystalline whispering-gallery-mode resonators.

Nan Yu is a principal member of the technical staff at the JPL. He is also the group supervisor of the Quantum Sciences and Technology Group. His main research interests are frequency controls for metrology and sensor applications.

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