What will happen if two identical microspheres, with fundamental whispering gallery modes excited in each of them, become optically coupled? Conventional wisdom based on coupled-mode arguments says that two new modes, bonding and anti-bonding, with two split frequencies would be formed. In this paper we demonstrate, using exact multi-sphere Mie theory, that in reality an attempt to couple two fundamental modes of microspheres would result in a complex multi-resonance optical response with the field distribution significantly deviating from predictions of coupled-mode type theories.
We propose a series of conformal mappings for designing directional emission whispering gallery modes. The mappings transform circular cavity into quasi-polygonal transformation cavity. Anisotropic emission is demonstrated in the so-designed cavities.
A review of hollow core whispering gallery resonators (WGRs)is given. After a short introduction to the topic of whispering gallery resonators we provide a description of whispering gallery modes in hollow or liquid core WGRs. Next, whispering gallery mode (WGM) sensing mechanisms are outlined and some fabrication methods for microbubbles, microcapillaries and other tubular WGM devices are discussed. We then focus on the most common applications of hollow core WGRs, namely refractive index and temperature sensing, gas sensing, force sensing, biosensing, and lasing. The review highlights some o…
We simulate light propagation in perturbed whispering-gallery mode microcavities using a two-dimensional finite-difference beam prop- agation method in a cylindrical coordinate system. Optical properties of whispering-gallery microcavities perturbed by polystyrene nanobeads are investigated through this formulation. The light perturbation as well as quality factor degradation arising from cavity ellipticity are also studied.
We have observed for the first time a new photonic quantum ring emission of anti-whispering gallery modes from a negative mesa-type toroid cavity due to semiconductor photonic corrals.
In this paper concave surface scanning method based on x-ray scattering in whispering gallery effect is considered. The capabilities of this method are studied based on ray-tracing computer simulation. The dependence of the output x-ray beam on the RMS roughness and minimum detectable imperfection height estimations were obtained.
We report a theoretical study showing that rogue waves can emerge in whispering gallery mode resonators as the result of the chaotic interplay between Kerr nonlinearity and anomalous group-velocity dispersion. The nonlinear dynamics of the propagation of light in a whispering gallery-mode resonator is investigated using the Lugiato-Lefever equation, and we evidence a range of parameters where rare and extreme events associated with a non-gaussian statistics of the field maxima are observed.
We propose asymmetric resonant cavities with rational caustics and experimentally demonstrate interior whispering gallery modes in monolithic silicon mesoscopic microcavities. These microcavities demonstrate unique robustness of cavity quality factor (Q) against roughness Rayleigh scattering. Angle-resolved tapered fiber measurements and near-field images observe distinct resonant families and asymmetric emission from interior whispering gallery modes, which can be used for microcavity laser and CQED applications.
Stimulated Brillouin scattering with both pump and Stokes beams in resonance with whispering gallery modes of an ultra high Q CaF_2 resonator is demonstrated for the first time. The resonator is pumped with 1064 nm light and has a Brillouin lasing threshold of 3.5 microwatt. Potential applications include optical generation of microwaves and sensitive gyros.
We experimentally studied whispering-gallery modes(WGMs) and demonstrated resonance enhancement of optical forces evanescently exerted on dielectric microspheres. We showed that the resonant light pressure can be used for optical sorting of microparticles with extraordinary uniform resonant properties that is unachievable by conventional sorting techniques.
We present a first principle derivation of the coupled nonlinear Schrödinger equations that govern the interaction between two families of modes with different transverse profiles in a generic whispering-gallery-mode resonator. We find regions of modulational instability and the existence of trains of bright solitons both in the normal and in the anomalous dispersion regime.
Numerical evolutions of whispering gallery modes of both isotropic and anisotropic spheroidal resonators are presented and used to build analytical approximations of these modes. Such approximations are carried out mainly to have the possibility to have a manageable analytic formulas for the eigenmodes and eigenfrequencies of anisotropic resonators. A qualitative analysis of ellipsoidal anisotropic modes in term of superposition of spherical modes is also presented.
A surface acoustic wave (SAW) creates its own high-Q ultra-small volume whispering gallery modes (WGMs), different from usual bulk acoustic WGMs, in an optical dielectric WGM resonator. We show that it is possible to realize an externally controllable, efficient triply-resonant opto-mechanical interaction between two optical WGMs and the SAW WGM and to use such an interaction in various sensor applications.
Quasiclassical approach and geometric optics allow to describe rather accurately whispering gallery modes in convex axisymmetric bodies. Using this approach we obtain practical formulas for the calculation of eigenfrequencies and radiative Q-factors in dielectrical spheroid and compare them with the known solutions for the particular cases and with numerical calculations. We show how geometrical interpretation allows expansion of the method on arbitrary shaped axisymmetric bodies.
Different applications of crystalline whispering gallery mode resonators call for different properties of the resonator host material. We report on our recent study of resonators made out of sapphire, diamond, and quartz crystals and discuss possible applications of these resonators. In particular, we demonstrate Kerr frequency comb generation in sapphire microresonators.
Typical microresonators exhibit a large frequency spacing between resonances and a limited tunability. This impedes their use in a large class of applications which require a resonance of the microcavity to coincide with a predetermined frequency. Here, we experimentally overcome this limitation with highly prolate-shaped whispering-gallery-mode "bottle microresonators" fabricated from standard optical glass fibers. Our resonators combine an ultra-high quality factor of 360 million, a small mode volume, and near lossless fibre coupling, characteristic of whispering-gallery-mode resonators, wit…
An analytic solution for an uniaxial spherical resonator is presented using the method of Debye potentials. This serves as a starting point for the calculation of whispering gallery modes (WGM) in such a resonator. Suitable approximations for the radial functions are discussed in order to best characterize WGMs. The characteristic equation and its asymptotic expansion for the anisotropic case is also discussed, and an analytic formula with a precision of the order $O[ν^{-1}]$ is also given. Our careful treatment of both boundary conditions and asymptotic expansions makes the present work a par…
Whispering gallery modes are known for having orbital angular momentum, however the interplay of local spin density, orbital angular momentum, and the near-field interaction with quantum emitters is much less explored. Here, we study the spin-orbital interaction of a circularly polarized dipole with the whispering gallery modes (WGMs) of a spherical resonator. Using an exact dyadic Green's function approach, we show that the near-field interaction between the photonic spin of a circularly polarized dipole and the local spin density of whispering gallery modes gives rise to unidirectional behav…
We report on tuning the optical whispering gallery modes in a poly dimethyl siloxane-based (PDMS) microsphere resonator by more than a THz. The PDMS microsphere system consists of a solid spherical resonator directly formed with double stems on either side. The stems act like tie-rods for simple mechanical stretching of the microresonator over tens of microns, resulting in tuning of the whispering gallery modes by one free spectral range. Further investigations demonstrate that the whispering gallery mode shift has a higher sensitivity (0.13 nm/μN) to an applied force when the resonator is in …
We demonstrated the tuning of whispering gallery modes (WGMs) of a silica microsphere during optical levitation through the annealing process. We determined the annealing temperature from the power balance between the CO2 laser light heating and several cooling processes. Cooling caused by heat conduction through the surrounding air molecules is the dominant process. We achieved a blue shift of the WGMs as large as 1 \%, which was observed in the white-light scattering spectrum from the levitated microsphere.
We consider an optical fiber with nanoscale variation of the effective fiber radius supporting whispering gallery modes slowly propagating along the fiber, and reveal that the radius variation can be designed to support reflectionless propagation of these modes. We show that reflectionless modulations can realize control of transmission amplitude and temporal delay, while enabling close packing due to the absence of cross-talk, in contrast to conventional potentials.
We experimentally demonstrate the use of a novel liquid state, whispering-gallery-mode optical resonator as a highly sensitive humidity sensor. The optical resonator used consists of a droplet made of glycerol, a transparent liquid that enables high optical quality factor, doped with rhodamine 6G, which acts as fluorescent emitter. As glycerol is highly hygroscopic, the refractive index and radius of the droplet change with ambient humidity. This produces a shift on the whispering gallery modes wavelengths, which modulates the emission of rhodamine 6G. This easily-made device has an unpreceded…
We propose an original method to couple light into a whispering-gallery mode resonator. This method benefits from the mode selectivity and robustness of the prism-coupling along with the single-mode propagation of the fiber taper. It consists in a prism shaped crystal with a waveguide inscribed inside it. The waveguide is self-inscribed in-situ by beam self-trapping to allow an optimum coupling to a given resonator.
We developed a new method for fabricating crystalline whispering gallery mode cavities based on laser-heated pedestal growth. We fabricated sapphire cavities and obtained a Q factor of 16000 with a cavity whose diameter was about 240 um. We showed numerically that the cross-sectional shape of the cavity is sensitive to the cavity Q, and we controlled it successfully by changing the growth condition in the molten zone, without significantly degrading the crystal structure.
We develop a compact whispering-gallery-mode (WGM) sensing system by integrating multiple components, including a tunable laser, a temperature controller, a function generator, an oscilloscope, a photodiode detector, and a testing computer, into a phone-sized embedded system. We demonstrate a thermal sensing experiment by using this portable system. Such a system successfully eliminates bulky measurement equipment required for characterizing optical resonators and will open up new avenues for practical sensing applications by using ultra-high Q WGM resonators.
We demonstrate the efficient coupling (99.5%) of a silica whispering gallery mode microresonator directly with a silicon chip by using a silicon photonic crystal waveguide as a coupler. The efficient coupling is attributed to the small effective refractive index difference between the two devices. The large group index of the photonic crystal waveguide mode also contributes to the efficient coupling. A coupling Q of 2.68*10^6 is obtained, which allows us to achieve the critical coupling of a silica whispering gallery mode with an intrinsic Q of close to 10^7 with a Si chip.
We investigated the whispering gallery modes of cavities with a hexagonal cross-section. We found two different modes, namely perturbed and quasi-WGMs, of which the former exhibits the higher Q when the corner radius is large. We studied the dependence of Q on the curvature radius of the polygonal cavities and found that the coupling between the two modes determines the Q of the cavity. In addition we fabricated a cavity by employing laser heated pedestal growth and demonstrated a high Q.
Whispering gallery mode microdisk cavities fabricated by direct laser writing are studied using dark-field imaging and spectroscopy in the visible spectral range. {Dark-field imaging allows us to directly visualize the spatial intensity distribution of whispering gallery modes. We extract their azimuthal and radial mode indices from dark-field images, and find the axial mode number from the dispersion relation. The scattering spectrum obtained in the confocal arrangement provides information on the density of optical states in the resonator. The proposed technique is a simple non-invasive way …
The relative intensity noise (RIN) of a solid state whispering-gallery-mode class-B laser is studied both theoretically and experimentally under different pumping regimes. In particular, we show that harmonics of the spiking frequency are observed in the RIN spectrum. A rate equation model including Langevin forces and the nonlinear coupling between inverted ion and photon number fluctuations has been developed to reproduce the experimental results and to extract relevant physical parameters from the fitting of the RIN spectrum.
Using quasiclassical approach rather precise analytical approximations for the eigenfrequencies of whispering gallery modes in convex axisymmetric bodies may be found. We use the eikonal method to analyze the limits of precision of quasiclassical approximation using as a practical example spheroidal dielectric cavity. The series obtained for the calculation of eigenfrequencies is compared with the known series for dielectric sphere and with numerical calculations. We show how geometrical interpretation allows expansion of the method on arbitrary shaped axisymmetric bodies.
We propose and demonstrate a direct method to observe higher-order whispering-gallery modes in vertical cavity surface emitting lasers (VCSELs) at room temperature. Instead of introducing any defect mode, we show that suppression of lower-order cavity modes can be achieved by destroying vertical reflectors with a surface micro-structure. Up to the 23rd azimuthal order whispering-gallery mode confined laterally by the native oxide layers is observed in experiments through collecting near-field radiation patterns. Various vertical emission transverse modes are identified by the spectrum in exper…
We demonstrate a miniature self-injection locked DFB laser using resonant optical feedback from a high-Q crystalline whispering gallery mode resonator. The linewidth reduction factor is greater than 10,000, with resultant instantaneous linewidth less than 200 Hz. The minimal value of the Allan deviation for the laser frequency stability is 3x10^(-12) at the integration time of 20 us. The laser possesses excellent spectral purity and good long term stability.
We study whispering gallery modes in triple microdisks of equilateral and isosceles triangular configurations. The characteristic properties of resonant modes in three microdisks on vertices of an equilateral triangle are explained by discrete rotational symmetry of the triangle. The avoided crossings of resonant modes in three microdisks on vertices of an isosceles triangle are also studied in terms of a combination of single and coupled microdisks. Finally, we propose the matrix models which well explain the resonant modes in triple microdisks.
We show that electric field on the plane surface of truncated sphere or cylinders (so called Janus particles) have sharp resonances versus the depth of removed segment of a sphere or cylinder. These resonances are related to the excited whispering gallery waves caused by truncation. It is a new mechanism of the field localization. Optimization of this effect for cylinders permits to reach a super resolution in the line thickness what can be used for contacting optical lithography.
Chaotic whispering-gallery modes have significance both for optical applications and for our understanding of the interplay between wave phenomena and the classical ray limit in the presence of chaotic dynamics and openness. In strongly non-convex geometries, a theorem by Mather rules out the existence of invariant curves in phase space corresponding to rays circulating in whispering-gallery patterns, so that no corresponding modes of this type are expected. Here we discuss numerical computations of the electromagnetic fields in planar dielectric cavities that are strongly non-convex becauseth…
We outline a full-vectorial three-dimensional multi-mode matching technique in a cylindrical coordinate system that addresses the mutual coupling among multiple modes copropagating in a perturbed whispering-gallery-mode microcavity. In addition to its superior accuracy in respect to our previously implemented single-mode matching technique, this current technique is suitable for modelling waveguide-to-cavity coupling where the influence of multi-mode coupling is non-negligible. Using this methodology, a robust scheme for hybrid integration of a microcavity onto a silicon-on-insulator platform …
High-$Q$ whispering gallery mode microresonators are cornerstones to develop compact coherent sources like single frequency lasers or frequency combs. This work reports a detailed characterization of a whispering gallery mode microsphere in the near ultraviolet. Light coupling in the resonator is obtain thanks to a robust angle-polished fiber allowing to investigate the different coupling regimes. An intrinsic $Q_0$-factor of $2.2 \times 10^8$ and a finesse of $7.3 \times 10^4$ are reported at a wavelength of 420 nm. Physical mechanisms contributing to the $Q_0$-factor are discussed and routes…
Coupled-mode induced transparency is realized in a single microbubble whispering gallery mode resonator. Using aerostatic tuning, we find that the pressure induced shifting rates are different for different radial order modes. A finite element simulation considering both the strain and stress effects shows a GHz/bar difference and this is confirmed by experiments. A transparency spectrum is obtained when a first order mode shifts across a higher order mode through precise pressure tuning. The resulting lineshapes are fitted with the theory. This work lays a foundation for future applications i…
We show that when a microsphere is illuminated by an evanescent wave, the optical forces on- and off- whispering gallery mode (WGM) resonance can differ by several orders of magnitude. Such size selective force allows one to selectively manipulate the resonating particles, while leaving those particles at off-resonance untouched. As WGM resonances have very high-Q's, this kind of force could be deployed for size-selective manipulation with a very high accuracy (~1/Q), as well as simultaneous particle-sorting according to their size or resonant frequency.
We have studied magneto-optical effects in an optical whispering-gallery-mode resonator (WGMR) manufactured from a Faraday-rotator material with, to the best of our knowledge, the record quality factor ($Q = 1.45\times10^8$) achieved for such materials. We have experimentally measured the eigenfrequencies deviation amplitude under the application of an external magnetic field and demonstrated the polarization plane declination over the light path. An analytical model for arbitrary magnetic field geometries in magneto-optic birefringent WRMRs has been developed.
We have experimentally demonstrated an on-chip all-silk fibroin whispering gallery mode microresonator by using a simple molding and solution-casting technique. The quality factors of the fabricated silk protein microresonators are up to 10^5. A high-sensitivity thermal sensor was realized in this silk fibroin microtoroid with sensitivity of 1.17 nm/K, 8 times higher than previous WGM resonator based thermal sensors. This opens the way to fabricate biodegradable and biocompatible protein based microresonators on a flexible chip for biophotonics applications.
We report experimental demonstration of continuous-wave cascaded-harmonic generation and Raman lasing in a millimeter-scale lithium niobate whispering-gallery resonator pumped at a telecommunication-compatible infrared wavelength. Intensity enhancement through multiple recirculations in the whispering-gallery resonator and quasi phase-matching through a nonuniform crystal poling enable simultaneous cascaded-harmonic generation up to the fourth-harmonic accompanied by stimulated Raman, two-photon, three-photon, and four-photon Raman scattering corresponding the molecular vibrational wavenumbers…
A general model is presented for coupling of high-$Q$ whispering-gallery modes in optical microsphere resonators with coupler devices possessing discrete and continuous spectrum of propagating modes. By contrast to conventional high-Q optical cavities, in microspheres independence of high intrinsic quality-factor and controllable parameters of coupling via evanescent field offer variety of regimes earlier available in RF devices. The theory is applied to the earlier-reported data on different types of couplers to microsphere resonators and complemented by experimental demonstration of enhanced…
Directional coupling of light in nanophotonic circuits has recently attracted increasing interest, with numerous experimental realizations based on broken rotational or mirror symmetries of the light-matter system. The most prominent underlying effect is the spin-orbit interaction of light in subwavelength structures. Unfortunately, coupling of light to such structures is, in general, very inefficient. In this work, we experimentally demonstrate an order of magnitude enhancement of the directional coupling between two nanowaveguides by means of a whispering gallery microcavity. We also show th…
We report Raman lasing and the optical analog of electromagnetically-induced-transparency (EIT) in a whispering-gallerymode (WGM) microtoroid resonator embedded in a low refractive index polymer matrix together with a tapered fiber coupler. The Raman laser supports both single and multimode operation with low power thresholds. Observations of Fano and EIT-like pheonomena in a packaged microresonator will enable high resolution sensors and can be used in networks where slow light process is needed. These results open the way for portable, robust, and stable WGM microlasers and laserbased sensor…
An optical comb is shown to arise from a whispering gallery mode resonator pumped by two optical frequencies. Two externally excited modes couple due to Kerr nonlinearity to initially empty modes and give rise to new frequency components. This thresholdles process is much more efficient than the previously reported single-pump four-wave mixing. As a result, a few milliwatt pump is sufficient to generate strong secondary fields, that efficiently generate higher-order frequency components and so on, in a cascade process leading to an optical comb.
We report on the experimental demonstration of a tunable monolithic optical frequency comb generator. The device is based on the four-wave mixing in a crystalline calcium fluoride whispering gallery mode resonator. The frequency spacing of the comb is given by an integer number of the free spectral range of the resonator. We select the desired number by tuning the pumping laser frequency with respect to the corresponding resonator mode. We also observe interacting optical combs and high-frequency hyperparametric oscillation, depending on the experimental conditions. A potential application of …
We utilize a high quality calcium fluoride whispering-gallery-mode resonator to stabilize a simple erbium doped fiber ring laser with an emission frequency of 196 THz (wavelenght 1530 nm) to a linewidth below 650 Hz. This corresponds to a relative stability of 3.3 x 10^(-12) over 16 \mus. In order to characterize the linewidth we use two identical self-built lasers and a commercial laser to determine the individual lasing linewidth via the three-cornered hat method.
A fiber laser is stabilized using a Calcium Fluoride (CaF2) whispering-gallery-mode resonator. It is set up using a semiconductor optical amplifier as a gain medium. The resonator is critically coupled through prisms, and used as a filtering element to suppress the laser linewidth. Using the self-heterodyne beat technique the linewidth is determined to be 13 kHz. This implies an enhancement factor of 10^3 with respect to the passive cavity linewidth. The three-cornered hat method shows a stability of 10^(-11) after 10 μs.
Fiber based whispering-gallery-mode (WGM) microprobe, combining both the high optical field enhancement of the WGMs and the practicability of the fiber probe, is highly demanded in sensing and imaging. Here in this paper, we experimentally report the efficient far-field coupling of WGMs by scattering the focused laser beam through a nanotip. With the help of Purcell effect as well as the two-step focusing technique, a WGM excitation efficiency as high as 16.8% has been achieved. Both the input and output of the probe light propagate along the same fiber, which makes the whole coupling system a…
We perform numerical studies of the effect of sidewall imperfections on the resonant state broadening of the optical microdisk cavities for lasing applications. We demonstrate that even small edge roughness causes a drastic degradation of high-Q whispering gallery (WG) mode resonances reducing their Q-values by many orders of magnitude. At the same time, low-Q WG resonances are rather insensitive to the surface roughness. The results of numerical simulation obtained using the scattering matrix technique, are analyzed and explained in terms of wave reflection at a curved dielectric interface co…
We report the observation of stimulated Brillouin scattering and lasing at 1550~nm in barium fluoride (BaF$_2$) crystal. Brillouin lasing was achieved with ultra-high quality ($Q$) factor monolithic whispering gallery mode (WGM) mm-size disk resonators. Overmoded resonators were specifically used to provide cavity resonances for both the pump and all Brillouin Stokes waves. Single and multiple Brillouin Stokes radiations with frequency shift ranging from $8.2$ GHz up to $49$ GHz have been generated through cascaded Brillouin lasing. BaF$_2$ resonator-based Brillouin lasing can find potential a…
We demonstrate the excitation and detection of whispering gallery modes in optical microresonators using a "point-and-play", fiber-based, optical nano-antenna. The coupling mechanism is based on cavity-enhanced Rayleigh scattering. Collected spectra exhibit Lorentzian dips, Fano shapes, or Lorentzian peaks, with a coupling efficiency around 13\%. The spectra are characterized by the coupling gap, polarization, and fiber tip position. The coupling method is simple, low-cost and, most importantly, the \textit{Q}-factor can be maintained at $10^8$ over a wide coupling range, thereby making it sui…
We demonstrate that the surface second-harmonic generation can lead to the formation of nonlinear plasmonic whispering-gallery modes (WGMs) in microcavities made of metallic nanowires. Since these WGMs are excited by induced surface nonlinear dipoles, they can be generated even when they are not coupled to the radiation continuum. Consequently, the quality factor of these nonlinear modes can be as large as the theoretical limit imposed by the optical losses in the metal. Remarkably, our theoretical analysis shows that nonlinear plasmonic WGMs are characterized by fractional azimuthal modal num…
Mechanisms of whispering-gallery (WG) modes coupling in microdisk photonic molecules (PMs) with slight and significant size mismatch are numerically investigated. The results reveal two different scenarios of modes interaction depending on the degree of this mismatch and offer new insight into how PM parameters can be tuned to control and modify WG-modes wavelengths and Q-factors. From a practical point of view, these findings offer a way to fabricate PM microlaser structures that exhibit low thresholds and directional emission, and at the same time are more tolerant to fabrication errors than…
We study the optical interaction in a coupled dielectric microdisks by investigating the splitting of resonance positions of interacting whispering gallery modes (WGMs) and their pattern change, depending on the distance between the microdisks. It is shown that the interaction between the WGMs with odd parity about y-axis becomes appreciable at a distance less than a wavelength and causes directional emissions of the resulting interacting WGMs. The directionality of the interacting WGMs can be understood in terms of an effective boundary deformation in ray dynamical analysis. We also discuss a…
In limaçon-shaped gradient index dielectric cavities designed by conformal transformation optics, the variation of Q-factors and emission directionality of resonant modes was traced in their system parameter space. For these cavities, their boundary shapes and refractive index profiles are determined in each case by a chosen conformal mapping which is taken as a coordinate transformation. Through the numerical exploration, we found that bidirectionality factors of generic high-Q resonant modes are not directly proportional to their Q-factors. The optimal system parameters for the coexistence o…
We demonstrate sum-frequency generation in a nonlinear whispering gallery mode resonator between a telecom wavelength and the Rb D2 line, achieved through natural phase matching. Due to the strong optical field confinement and ultra high Q of the cavity, we achieve a 1000-fold enhancement in the conversion efficiency compared to existing waveguide-based devices. The experimental data are in agreement with the nonlinear dynamics and phase matching theory in the spherical geometry employed. The experimental and theoretical results point to a new platform to manipulate the color and quantum state…
We report on fabrication of whispering-gallery-mode microlasers in a Nd:glass chip by femtosecond laser three-dimensional (3D) micromachining. Main fabrication procedures include the fabrication of freestanding microdisks supported by thin pillars by femtosecond laser ablation of the glass substrate immersed in water, followed by CO2 laser annealing for surface smoothing. Lasing is observed at a pump threshold as low as ~69 μW at room temperature with a continuous-wave laser diode operating at 780nm. This technique allows for fabrication of microcavities of high quality factors in various diel…
Identifying the mode numbers in whispering-gallery mode resonators (WGMRs) is important for tailoring them to experimental needs. Here we report on a novel experimental mode analysis technique based on the combination of frequency analysis and far-field imaging for high mode numbers of large WGMRs. The radial mode numbers q and the angular mode numbers p=$\ell$-m are identified and labeled via far-field imaging. The polar mode numbers $\ell$ are determined unambiguously by fitting the frequency differences between individual whispering gallery modes (WGMs). This allows for the accurate determi…
We have stabilized an external cavity diode laser to a whispering gallery mode resonator formed by a protrusion of a single-crystal magnesiumdifluoride cylinder. The cylinder's compact dimensions (<1 cm^3) reduce the sensitivity to vibrations and simplify the stabilization of its temperature in a compact setup. In a comparison to an ultrastable laser used for precision metrology we determine a minimum Allan deviation of 20 Hz, corresponding to a relative Allan deviation of 6*10^-14, at an integration time of 100 ms. This level of instability is compatible with the limits imposed by fundamental…
