Traceable 2D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators
Mark Oxborrow
备注：
10 pages, 8 figures, submitted to IEEE Microwave Theory and Techniques. Last revision March 19 2007. This paper is a significantly abridged/revised version of quant-ph/0607156
This paper explains how a popular, commercially-available software package for solving partial-differential-equations (PDEs), as based on the finite-element method (FEM), can be configured to calculate, efficiently, the frequencies and fields of the whispering-gallery (WG) modes of axisymmetric dielectric resonators. The approach is traceable; it exploits the PDE-solver's ability to accept the definition of solutions to Maxwell's equations in so-called `weak form'. Associated expressions and methods for estimating a WG mode's volume, filling factor(s) and, in the case of closed(open) resonators, its wall(radiation) loss, are provided. As no transverse approximation is imposed, the approach remains accurate even for quasi-transverse magnetic/electric modes of low, finite azimuthal mode order. The approach's generality and utility are demonstrated by modeling several non-trivial structures: (i) two different optical microcavities [one toroidal made of silica, the other an AlGaAs microdisk]; (ii) a 3rd-order sapphire:air Bragg cavity; (iii) two different cryogenic sapphire WG-mode resonators; both (ii) and (iii) operate in the microwave X-band. By fitting one of (iii) to a set of measured resonance frequencies, the dielectric constants of sapphire at liquid-helium temperature have been estimated.
We have achieved low threshold lasing of self-assembled InAs/GaAs quantum dots coupled to the evanescent wave of the high-$Q$ whispering gallery modes of a silica microsphere. In spite of high temperature and $Q$-spoiling of whispering gallery modes due to diffusion and refraction on the high index semiconductor sample, room temperature lasing is obtained with less than 100 quantum dots. This result highlights the feasibility and interest of combining self-assembled quantum dots and microspheres in view of cavity-quantum electrodynamics experiments.
We theoretically study the properties of highly prolate shaped dielectric microresonators. Such resonators sustain whispering gallery modes that exhibit two spatially well separated regions with enhanced field strength. The field per photon on the resonator surface is significantly higher than e.g. for equatorial whispering gallery modes in microsphere resonators with a comparable mode volume. At the same time, the frequency spacing of these modes is much more favorable, so that a tuning range of several free spectral ranges should be attainable. We discuss the possible application of such res…
We propose an efficient scheme for the realization of quantum information transfer and entanglement with nitrogen-vacancy (NV) centers coupled to a high-Q whispering-gallery mode (WGM) microresonator. We show that, based on the effective dipole-dipole interaction between the NV centers mediated by the WGM, quantum information can be transferred between the NV centers through Raman transitions combined with laser fields. This protocol may open up promising possibilities for quantum communications with the solid state cavity QED system.
We demonstrate efficient upconversion of sub-THz radiation into the optical domain in a high-Q whispering gallery mode resonator with quadratic optical nonlinearity. The 5x10^{-3} power conversion efficiency of continuous wave 100GHz signal is achieved with only 16 mW of optical pump.
We present a quantum electrodynamical model involving nitrogen-vacancy centers coupled to a whispering-gallery mode cavity. Two schemes are considered to create W state and Bell state, respectively. One of the schemes makes use of the Raman transition with the cavity field virtually excited; The other enables the Bell state preparation and quantum information transfer by virtue of dark state evolution and adiabatic passage, which is tolerant to ambient noise and experimental parameter fluctuations. We justify our schemes by considering the experimental feasibility and challenge using currently…
We analyze the single-photon transport in a single-mode waveguide coupled to a whispering-gallery-type resonator interacting with a two-level atom. The single-photon transport properties such as the transmission and reflection amplitudes, as well as the resonator and the atom responses, are solved exactly via a real-space approach. The treatment includes the inter-mode backscattering between the two degenerate whispering gallery modes of the resonator, and the dissipations of the resonator and the atom. We also show that a generalized critical coupling condition, that the single-photon transmi…
We investigate the possibility of using dielectric microdisk resonators for the optical detection of single atoms trapped and cooled in magnetic microtraps near the surface of a substrate. The bound and evanescent fields of optical whispering gallery modes are calculated and the coupling to straight waveguides is investigated using finite-difference time domain solutions of Maxwell's equations. Results are compared with semi-analytical solutions based on coupled mode theory. We discuss atom detection efficiencies and the feasibility of non-destructive measurements in such a system depending on…
Conversion of microwave radiation into the optical range has been predicted to reach unity quantum efficiency in whispering gallery resonators made from an optically nonlinear crystal and supporting microwave and optical modes simultaneously. In this work we theoretically explore and experimentally demonstrate a resonator geometry that can provide the required phase matching for such a conversion at any desired frequency in the sub-THz range. We show that such a ring-shaped resonator not only allows for the phase matching, but also maximizes the overlap of the interacting fields. As a result, …
We theoretically propose a scheme to realize rotation sensing based on two coupled whispering-gallery-mode resonators with loss and gain. We consider that the active resonator with gain is rotated while the passive one with loss is stationary. The rotation will induce Sagnac effect and we show that the eigenfrequencies of the supermodes are sensitive to the Sagnac-Fizeau shift. Therefore, we can measure the average photon number in the steady state or the fluctuation spectrum of the output fields to detect the angular velocity of the rotation. We hope that our investigation will be useful in t…
Quasi-transverse-electric and -transverse-magnetic fundamental whispering gallery modes in a polymer-coated silica microtoroid are theoretically investigated and demonstrated to possess very high-quality factors. The existence of a nanometer-thickness layer not only evidently reduces the cavity mode volume but also draws the maximal electric field's position of the mode to the outside of the silica toroid, where single quantum dots or nanocrystals are located. Both effects result in a strongly enhanced coherent interaction between a single dipole (for example, a single defect center in a diamo…
We report on light emission from high-Q neodymium-implanted silica microtoroids. Following the description of the fabrication process of microtoroids, neodymium light emission is analysed. This emission is coupled to various cavity modes. Using evanescent wave coupling we achieve selective detection of Whispering Gallery Modes of a microtoroid.
We present experiments where a single subwavelength scatterer is used to examine and control the back-scattering induced coupling between counterpropagating high-Q modes of a microsphere resonator. Our measurements reveal the standing wave character of the resulting symmetric and antisymmetric eigenmodes, their unbalanced intensity distributions, and the coherent nature of their coupling. We discuss our findings and the underlying classical physics in the framework common to quantum optics and provide a particularly intuitive explanation of the central processes.
We have tuned the whispering gallery modes of a fused silica micro- resonator over nearly 1 nm at 800 nm, i.e.over half of a free spectral range or the equivalent of 10^6 linewidths of the resonator.This has been achieved by a new method based on the stretching of a two-stem microsphere.The devices described below will permit new Cavity-QED experiments with this ultra high .nesse optical resonator when it is desirable to optimise its coupling to emitters with given transition frequencies. The tuning capability demonstrated is compatible with both UHV and low temperature operation which should …
Squeezed vacuum states enable optical measurements below the quantum limit and hence are a valuable resource for applications in quantum metrology and also quantum communication. However, most available sources require high pump powers in the milliwatt range and large setups, which hinders real world applications. Furthermore, degenerate operation of such systems presents a challenge. Here, we use a compact crystalline whispering gallery mode resonator made of lithium niobate as a degenerate parametric oscillator. We demonstrate about 1.4 dB noise reduction below the shot noise level for only …
By direct application of the fluctuation-dissipation theorem, we numerically calculate the fundamental dimensional fluctuations of crystalline CaF2 whispering gallery resonators in the case of structural damping, and the limit that this noise imposes on the frequency stability of such resonators at both room and cryogenic temperatures. We analyze elasto-optic noise - the effect of Brownian dimensional fluctuation on frequency via the strain-dependence of the refractive index - a noise term that has so far not been considered for whispering-gallery resonators. We find that dimensional fluctuati…
Controlled tuning of the whispering gallery modes of GaN/InGaN μ-disk cavities is demonstrated. The whispering gallery mode (WGM) tuning is achieved at room temperature by immersing the μ-disks in water and irradiating with ultraviolet (UV) laser. The tuning rate can be controlled by varying the laser excitation power, with a nanometer precision accessible at low excitation power (~ several μW). The selective oxidation mechanism is proposed to explain the results and supported by theoretical analysis. The tuning of WGMs in GaN/InGaN μ-disk cavities may have important implication in cavity quan…
The microwave properties of a crystalline sapphire dielectric whispering gallery mode resonator have been measured at very low excitation strength (E/hf=1) and low temperatures (T = 30 mK). The measurements were sensitive enough to observe saturation due to a highly detuned electron spin resonance, which limited the loss tangent of the material to about 2e-8 measured at 13.868 and 13.259 GHz. Small power dependent frequency shifts were also measured which correspond to an added magnetic susceptibility of order 1e-9. This work shows that quantum limited microwave resonators with Q-factors > 1e8…
We report the direct observation of the electromagnetic-field distribution of whispering?gallery modes in silica microcavities (spheres and toroids). It is revealed by their excitation efficiency with a tapered fiber coupler swept along the meridian. The originality of this method lies in the use of the coupler itself for the near field mapping, eliminating the need of additional tools used in previous work. This method is successfully applied to microspheres and microtoroids
We introduce a microwave circuit architecture for quantum signal processing combining design principles borrowed from high-Q 3D resonators in the quantum regime and from planar structures fabricated with standard lithography. The resulting '2.5D' whispering-gallery mode resonators store 98% of their energy in vacuum. We have measured internal quality factors above 3 million at the single photon level and have used the device as a materials characterization platform to place an upper bound on the surface resistance of thin film aluminum of less than 250nOhms.
We demonstrate for the first time natural phase matching for optical frequency doubling in a high-Q whispering gallery mode resonator made of Lithium Niobate. A conversion efficiency of 9% is achieved at 30 micro Watt in-coupled continuous wave pump power. The observed saturation pump power of 3.2 mW is almost two orders of magnitude lower than the state-of-the-art. This suggests an application of our frequency doubler as a source of non-classical light requiring only a low-power pump, which easily can be quantum noise limited. Our theoretical analysis of the three-wave mixing in a whispering …
Stimulated Brillouin scattering recently allowed experimental excitation of surface acoustic resonances in micro-devices, enabling vibration at rates in the range of 50 MHz to 12 GHz. The experimental availability of such mechanical whispering gallery modes in photonic-MEMS raises questions on their structure and spectral distribution. Here we calculate the form and frequency of such vibrational surface whispering gallery modes, revealing diverse types of surface vibrations including longitudinal, transverse, and Rayleigh-type deformations. We parametrically investigate these various modes by …
We demonstrate a method to perform spectroscopy of near-infrared single photons without the need of dispersive elements. This method is based on a photon energy resolving transition edge sensor and is applied for the characterization of widely wavelength tunable narrow-band single photons emitted from a crystalline whispering gallery mode resonator. We measure the emission wavelength of the generated signal and idler photons with an uncertainty of up to 2 nm.
We perform an investigation into the properties of Pr3+:Y2SiO5 whispering gallery mode resonators as a first step towards achieving the strong coupling regime of cavity QED with rare-earth-ion doped crystals. Direct measurement of cavity QED parameters are made using photon echoes, giving good agreement with theoretical predictions. By comparing the ions at the surface of the resonator to those in the center it is determined that the physical process of making the resonator does not negatively affect the properties of the ions. Coupling between the ions and resonator is analyzed through the ob…
Current theoretical treatment of mode splitting and scattering loss resulting from sub-wavelength scatterers attached to the surface of high-quality-factor whispering-gallery-mode microresonators is not satisfactory. Different models have been proposed for two distinct scatterer regimes, i.e., a-few- and many-scatterers. In addition, many experimental results seem difficult to understand within the existing theoretical framework. Here we develop a unified approach that applies to an arbitrary number of scatterers, which reveals the applicable conditions and the limits of the existing theoretic…
We demonstrate trapping of a single 85Rb atom at a distance of 200 nm from the surface of a whispering-gallery-mode bottle microresonator. The atom is trapped in an optical potential, which is created by retroreflecting a red-detuned focused laser beam from the resonator surface. We counteract the trap-induced light shift of the atomic transition frequency by superposing a second laser beam with suitably chosen power and detuning. This allows us to observe a vacuum Rabi-splitting in the excitation spectrum of the coupled atom-resonator system. This first demonstration of stable and controlled …
Optical parametric downconversion has proven to be a valuable source of nonclassical light. The process is inherently able to produce twin beam correlations along with individual intensity squeezing of either parametric beam, when pumped far above threshold. Here, we present for the first time the direct observation of intensity squeezing of -1.2 dB of each of the individual parametric beams in parametric downconversion by use of a high quality whispering gallery mode disk resonator. In addition we observed twin beam quantum correlations of -2.7 dB with this cavity. Such resonators feature str…
Second order nonlinearities in whispering gallery mode resonators are highly investigated for their many applications such as wavelength converters, entangled photon sources and generation of frequency combs. In such systems, depending on the material under scrutiny, the derivation of quasi-phase matching equations can lead to the appearance of additional quanta in the selection rule on the azimuthal confinement order. Here, we demonstrate that these additional quanta show up due to the Berry phase experienced by the transverse spin angular momentum components of the whispering gallery modes d…
Whispering-gallery microcavity laser possesses ultralow threshold, whereas convenient free-space optical excitation and collection suffer from low efficiencies due to its rotational symmetry. Here we analytically study a three-dimensional microsphere coupled to a nano-sized scatterer in the framework of quantum optics. It is found that the scatterer is capable of coupling light in and out of the whispering-gallery modes (WGMs) without seriously degrading their high-Q properties, while the microsphere itself plays the role of a lens to focus the input beam on the scatterer and vice versa. Our a…
We propose a kind of plasmonic whispering gallery modes highly localized on the exterior surface of a metal-coated microresonator. This exterior (EX) surface mode possesses high quality factors at room temperature, and can be efficiently excited by a tapered fiber. The EX mode can couple to an interior (IN) mode and this coupling produces a strong anti-crossing behavior, which not only allows conversion of IN to EX modes, but also forms a long-lived anti-symmetric mode. As a potential application, the EX mode could be used for a biosensor with a sensitivity high up to 500 nm per refraction ind…
We report a highly efficient source of narrow-band photon pairs based on parametric down-conversion in a crystalline whispering gallery mode resonator. Remarkably, each photon of a pair is strictly emitted into a single spatial and temporal mode, as witnessed by Glaubers autocorrelation function. We explore the phase-matching conditions in spherical geometries, and determine the requirements of the single-mode operation. Understanding these conditions has allowed us to experimentally demonstrate a single-mode pair-detection rate of $0.97 \cdot 10^6$ pairs/s per mW pump power per 20 MHz bandwid…
We demonstrate an efficient optical guiding technique for coupling cold atoms in the near field of a planar nanophotonic circuit, and realize large atom-photon coupling to a whispering-gallery mode in a microring resonator with a single-atom cooperativity $C\gtrsim 8$. The guiding potential is created by diffracted light on a nanophotonic waveguide that smoothly connects to a dipole trap in the far field for atom guiding with subwavelength precision. We observe atom-induced transparency for light coupled to a microring, characterize the atom-photon coupling rate, extract guided atom flux, and …
We study the Rayleigh scattering induced by a diamond nanocrystal in a whispering-gallery-microcavity--waveguide coupling system, and find that it plays a significant role in the photon transportation. On one hand, this study provides a new insight into future solid-state cavity quantum electrodynamics toward strong coupling physics. On the other hand, benefitting from this Rayleigh scattering, novel photon transportation such as dipole induced transparency and strong photon antibunching can occur simultaneously. As potential applications, this system can function as high-efficiency photon tur…
We have theoretically studied the photon statistical properties in a nonlinear whispering-gallery-mode microresonator coupled with two nanoparticles. By tuning the relative position of two nanoparticles, the photon statistical features of the system can be modified remarkably. Interestingly, a controllable switching between unconventional and conventional photon blockade can be realized by manipulating the angular positions of two nanoparticles. We also investigate the influence of the Kerr effect on the second order correlation function and find that there is an optimal choice for the relativ…
Whispering gallery mode biosensors allow selective unlabelled detection of single proteins and, combined with quantum limited sensitivity, the possibility for noninvasive realtime observation of motor molecule motion. However, to date technical noise sources, most particularly low frequency laser noise, have constrained such applications. Here we introduce a new technique for whispering gallery mode sensing based on direct detection of back-scattered light. This experimentally straightforward technique is immune to frequency noise in principle, and further, acts to suppress thermorefractive no…
We investigate the single-photon transport problem in the system of a Whispering-Gallery mode microresonator directionally coupled with a two-level quantum emitter (QE). This QE-microresonator coupling system can usually be studied by cavity quantum electrodynamics and the single-photon transport methods. However, we find that if we treat a two-level QE as a single-photon phase-amplitude modulator, we can also deal with such systems using the transfer matrix method. Further, in theory, we prove that these three methods are equivalent. The corresponding relations of respective parameters among …
Whispering gallery modes (WGMs), circulating modes near the surface of a spheroidal material, have been known to exhibit high quality factors for both acoustic and electromagnetic waves. Here, we report an electro-optomechanical system, where the overlapping WGMs of acoustic and optical waves along the equator of a dielectric sphere strongly couple to each other. The triple-resonance phase-matching condition provides a large enhancement of the Brillouin scattering only in a single sideband, and conversion from the input radio-frequency signal exciting the acoustic mode to the output optical si…
In this paper we discuss the force exerted by the field of an optical cavity on a polarizable dipole. We show that the modification of the cavity modes due to interaction with the dipole significantly alters the properties of the force. In particular, all components of the force are found to be non-conservative, and cannot, therefore, be derived from a potential energy. We also suggest a simple generalization of the standard formulas for the optical force on the dipole, which reproduces the results of calculations based on the Maxwell stress tensor.
We have demonstrated a 165 micron oblate spheroidal microcavity with free spectral range 383.7 GHz (3.06nm), resonance bandwidth 25 MHz (Q ~ 10^7) at 1550nm, and finesse F > 10^4. The highly oblate spheroidal dielectric microcavity combines very high Q-factor, typical of microspheres, with vastly reduced number of excited whispering-gallery (WG) modes (by two orders of magnitude). The very large free spectral range in the novel microcavity - few hundred instead of few GigaHertz in typical microspheres - is desirable for applications in spectral analysis, narrow-linewidth optical and RF oscilla…
We have tuned the whispering gallery modes of a fused silica microresonator over nearly 1 nm at 800 nm, i.e. over 0.5 FSR or 10^6 linewidths of the resonator. This has been achieved by a new method based on the stretching of a two-stem microsphere. The devices described below will permit new Cavity-QED experiments with this high-Q optical resonator when it is desirable to optimize its coupling to emitters with given transition frequencies. The tuning capability demonstrated here is compatible with both UHV and low temperature operation, which should be useful for future experiments with laser …
Transmission electron microscopy is an excellent experimental tool to study the interaction of free electrons with nanoscale light fields. However, up to now, applying electron microscopy to quantum optical investigations was hampered by the lack of experimental platforms which allow a strong coupling between fast electrons and high-quality resonators. Here, as a first step, we demonstrate the broad-band excitation of optical whispering-gallery modes in silica microresonators by fast electrons. In the emitted coherent cathodoluminescence spectrum, a comb of equidistant peaks is observed, resul…
We present the design, fabrication and characterization of cubic (3C) silicon carbide microdisk resonators with high quality factor modes at visible and near infrared wavelengths (600 - 950 nm). Whispering gallery modes with quality factors as high as 2,300 and corresponding mode volumes V ~ 2 (λ/n)^3 are measured using laser scanning confocal microscopy at room temperature. We obtain excellent correspondence between transverse-magnetic (TM) and transverse-electric (TE) polarized resonances simulated using Finite Difference Time Domain (FDTD) method and those observed in experiment. These stru…
