Static Envelope Patterns in Composite Resonances Generated by Level Crossing in Optical Toroidal Microcavities

Abstract

We study level crossing in the optical whispering-gallery (WG) modes by using toroidal microcavities. Experimentally, we image the stationary envelope patterns of the composite optical modes that arise when WG modes of different wavelengths coincide in frequency. Numerically, we calculate crossings of levels that correspond with the observed degenerate modes, where our method takes into account the not perfectly transverse nature of their field polarizations. In addition, we analyze anticrossing with a large avoidance gap between modes of the same azimuthal number. Here we show experimentally and theoretically that optical resonances with a different number of wavelengths along the circumference can be tuned to cross in frequency. Modes of the same number of wavelengths, on the other hand, are shown to anticross with a large gap. Overall, diverse azimuthal and radial mode shapes are imaged, and modes with complex transverse shapes and polarizations are calculated. Using a fluorescent mode-mapping technique enabled imaging of noncircular mode patterns that signals crossing. This visual indication was hidden when we were using bare nonfluorescent cavities.Near the degeneracy region, modes exhibiting a different number of circumferential wavelengths are simultaneously excited with a single-frequency laser source to produce a standing interference envelope. Here, only modes circulating in one direction are examined. Level crossing of countercirculating modes is possible [10] but, in distinction to the relatively large spatial ''beats'' observed here, manifests itself as a fine standing-wave interference pattern with nodes lying just half an optical wavelength apart. Moreover, in experiments, countercirculating modes (of similar indices) are typically split (i.e., nondegenerate) [11][12][13] due to imperfect isotropy (broken axial symmetry). In our system, this split (10 7 Hz) is much smaller than the free spectral range (10 12 Hz); such a splitting and the type of anisotropy it originates from might therefore play only a minor role here in bringing levels close enough to cross.Optical resonators in general [14 -16] and dielectric whispering-gallery (WG) cavities [17,18] in particular have a set of discrete optical eigenfrequencies. For a geometry in which the wave equation is separable, such as a cylindrical or spherical resonator, each mode can be labeled by three integer indices. Normally, by convention, the resonance frequency increases monotonically with each mode index. In a dielectric ring, for example, a whispering-gallery mode with a fixed and small transverse mode index and a large azimuthal mode index of m 105, say, will have 105 wavelengths along the circumference and will resonate at an optical frequency higher than that of

Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion

“…The difference in FSR between different mode families is on the order of 1 GHz, resulting from different spatial mode structure within the toroid. Note that some mode positions are strongly affected by mode crossings with other families [20], leading to a locally increased dispersion. The evolution of the FSR of the mode with the highest optical Q-factor in this cavity is depicted in Fig.…”

Quantum-mechanical theory of optomechanical Brillouin cooling

“…In such resonators, the transverse (radial-polar) order of one mode can compensate for the frequency difference originating from the non-similar longitudinal (azimuthal) order of the other. This provides a pair of modes with different azimuthal wavevectors, but nearby frequencies, as experimentally observed via the resulting stationary interference pattern [5,6]. The energy flow in Brillouin anti-Stokes cooling [2] that is analyzed here, is opposite in respect to the Stokes excitation process [7][8][9].…”

“…This is because two optical resonances that have almost the same optical frequency but different propagation constants are needed in order to conseerve both the energy and momentum which are given to light by the acoustical phonon. One type of cavity that allows such optical-resonance pairs is whispering-gallery mode resonators [5,6]. In such resonators, the transverse (radial-polar) order of one mode can compensate for the frequency difference originating from the non-similar longitudinal (azimuthal) order of the other.…”

“…Here, q and Ω represent the wavevector and frequency of the acoustical mode. Conservation of both quantities is possible in the kind of experimental setup studied here due to high order transverse optical modes [5,6]. Taking into account the momentum and energy conservation, the relevant coupling term inĤ for this process will be:…”

Optomechanics with Surface-Acoustic-Wave Whispering-Gallery Modes

“…We show that a proper shaping of the resonator results in a significant reduction of the SAW WGM volume that further increases the process efficiency. To achieve phase matching between optical and acoustic waves we propose to use the interference patterns generated at the surface of the WGM resonator [18,19]. We have found that such patterns can be created even by WGMs having different polarizations, which significantly simplifies the control of the acousto-optical processes in resonators with birefringent host material.…”