High-Q whispering gallery traveling wave resonators for oscillator frequency stabilization

Usually a frequency-stabilized standing wave resonator-oscillator incorporating a resonator as a frequency discriminator requires a circulator to separate the injected and reflected wave, A ferrite circulator is a noisy device and can limit the phase noise or frequency stability. Moreover, we show that the noise in a circulator varies, and detailed low noise measurements are necessary to choose an appropriate quiet circulator. Thus, by realizing a configuration that does not require a circulator, an improvement in performance and reliability can be obtained. A solution to this problem is to design a high-Q whispering gallery traveling wave (WGTW) resonator. This device naturally separates the injected and reflected wave in the same way as a ring cavity at optical frequencies, without degrading the frequency discrimination. Q-factor measurements of a WGTW sapphire resonator are presented, along with a derivation of critical parameters to maximize the frequency discrimination. New measurements of noise in ferrite circulators and isolators have also been made, which is followed with a discussion on oscillator design.     

High-Q whispering-gallery oscillations in a shielded spherical dielectric cavity

Results are presented of theoretical and experimental investigations of whispering-gallery oscillations in a shielded isotropic layered spherical dielectric cavity. Oscillations whose Q factor is two orders of magnitude higher than that of the oscillations of an open dielectric sphere are studied. Intermode interaction is observed between various cavity oscillation modes. Pis’ma Zh. Tekh. Fiz. 25, 20–25 (July 26, 1999)     

High-Q thermoelectric-stabilized sapphire microwave resonators forlow-noise applications

Two low-noise high-Q sapphire-loaded cavity (SLC) resonators, with unloaded Q values of 2×10⁵ and very low densities of spurious modes, have been constructed. They were designed to operate at 0°C with a center frequency of 10.000000 GHz. The cavity was cooled with a thermoelectric (TE) Peltier element, and in practice achieved the required center frequency near 1°C. The resonator has a measured frequency-temperature coefficient of -0.7 MHz/K, and a Q factor which is measured to be proportional to T^-2.5. An upper limit to the SLC residual phase noise of ℒ (100) Hz=-147 dBc/Hz, ℒ (1 kHz)=-155 dBc/Hz, and ℒ (10) kHz=-160 dBc/Hz has been measured. Also, we have created a free-running loop oscillator based on one of the SLC resonators, and measured a phase noise of ℒ(f)~-10-30log [f] dBc/Hz between f=10 /Hz and 25 kHz, using the other as a discriminator     

High-Q sapphire-rutile frequency-temperature compensated microwave dielectric resonators

A sapphiro-rutile composite resonator was constructed from a cylindrical sapphire monocrystal with two thin disks of monocrystal rutile held tightly against the ends. Because rutile exhibits low loss and an opposite temperature coefficient of permittivity to sapphire, it is an ideal material for compensating the frequency-temperature dependence of a sapphire resonator. Most of the electromagnetic modes in the composite structure exhibited turning points (or compensation points) in the frequency-temperature characteristic. The temperatures of compensation for the WG quasi TM modes were measured to be below 90 K with Q-factors of the order of a few million depending on the mode. For WG quasi TE modes, the temperatures of compensation were measured to be between 100 to 160 K with Q-factors of the order of a few hundreds of thousands, depending on the mode. The second derivatives of the compensation points were measured to be of the order 0.1 ppm/K(2 ), which agreed well with the predicted values.     

Micro-optical force sensor concept based on whispering gallery mode resonators

A micro-optical force sensor concept based on the morphology-dependent shifts of optical modes of dielectric microspheres is investigated. The optical resonances, commonly referred to as the whispering gallery modes (WGM), were excited by evanescently coupling light from a tunable diode laser using a tapered single-mode fiber. A compressive force applied to the sphere induces a change in both the shape and the index of refraction of the sphere leading to a shift in WGM. By tracking the shifts, the force magnitude is determined using solid silica as well as solid and hollow Polymethyl-methacrylate (PMMA) microsphere resonators. A measurement sensitivity as high as dlambda/dF=7.664 nm/N was demonstrated with a 960 mum hollow PMMA sphere.     

Transverse modes in one-port SAW resonators

The S(11) and equivalent S(21) frequency responses of a one-port surface acoustic wave (SAW) resonator with transverse modes derived from one-dimensional coupling-of-modes and transmission-matrix analysis. The two-dimensional nature of the problem is approximated by a summation of one-dimensional mode responses for each transverse mode. Comparison between theory and experimental data for a commercial 280-MHz one-port SAW resonator shows good agreement for the placement of transverse modes.     

Resonance shifts of transverse-electric whispering gallery modes in a spheroidal resonator

A variational method was applied to find the wave function of a transverse-electric whispering gallery mode (WGM) in a spheroidal resonator of a uniform refractive index (RI). It was found that the electric field is tangential to the resonator surface as in the sphere, up to the linear order of the ellipticity. Using the wave function, the resonance shift due to adsorption of a thin, uniform dielectric layer onto the surface and the shift by a uniform RI change in the surroundings were evaluated in the perturbation theory. The shift by the RI change is not affected by the ellipticity, but the shift by the layer adsorption now depends on the meridional order. However, the correction is not large unless the ellipticity is large and the meridional order is away from the one for the equatorial mode of WGM.     

Observation of whispering gallery modes in lasers with cut disk cavities

Whispering gallery modes (WGMs) have been observed in lasers with cut disk (half-and quarterdisk) cavities operating at room temperature in the 2.0–2.4 μm wavelength range. The possibility of WGM generation in a disk-sector cavity is theoretically studied. It is established that the intermode distance in a full-and half-disk cavity is the same, while that in the quarter-disk cavity is doubled. A new method for the radiation extraction from WGM lasers is proposed.     

Experimental investigation of eigenmodes of empty optical resonators with apertures

The spectra of an eigenmodal set of optical resonators with apertures are measured experimentally. Frequency shifts predicted by theory are confirmed for plane-parallel cavities. It is also shown that the local shape of the wave front at the aperture is significant for manifestation of these phenomena.     

Periodic shock waves in spherical resonators (survey)

This article surveys the literature on the problem of shock waves in spherical resonators. The published data is used to examine the feasibility of exciting shock waves in such resonators by means of a source of low-amplitude harmonic oscillations. A nonlinear wave equation is obtained to describe the propagation of unidimensional spherical waves in solids, liquids, and gases, as well as in bubbly liquids. A solution to the equation is constructed by the small-parameter method with the use of traveling-wave functions. Then, in solving boundary-value problems, linearized equations are integrated and the resonance frequencies at which the amplitudes of the oscillation increase without limit according to the linear solution are determined. Near these frequencies, the linear analysis is then refined by allowing for the nonlinear terms in the boundary-value problems. It is shown that an increase in the amplitude of the oscillations at resonance frequencies may lead to the formation of spherical periodic shock waves in the given resonators. An analogy is made between these waves and resonance shock waves excited in long unidimensional resonators.Translated from Problemy Prochnosti, No. 10, pp. 49–73, October, 1995.     

Optically tuned superconductive-dielectric resonators with whispering-gallery modes

An analytical method is presented for calculating the resonant frequency andQ-factor of a superconducting dielectric disk resonator operating in millimeter-wave regime with whispering-gallery mode. Resonant frequency shift due to the optical generation of quasi-particles in superconducting film is investigated as a function of photon flux. An optically tunable resonant frequency of about 500 MHz is estimated, and good agreement is found between numerical results and experimental ones.     

Excitation of ray oscillations in quasioptical dielectric cavities with whispering gallery modes

It is observed that ray oscillations may be excited together with whispering gallery modes in quasioptical dielectric cavities. It is shown that as the excitation conditions change, the fields of these oscillations gradually “split off” from the fields of the whispering gallery modes whose characteristics deteriorate. The angular spectrum of ray oscillations in dielectric cavities is determined. Pis’ma Zh. Tekh. Fiz. 23, 25–29 (August 12, 1997)     

Suppression of spurious lateral modes in thickness-excited FBAR resonators

Thin film bulk acoustic wave resonators (FBAR) utilize thickness-excited modes in which the resonant frequency is determined by the thickness of the structure and the wave velocity of the mode used. Unfortunately, other resonant modes also may be excited in the device. Some of these correspond to low-frequency, laterally-excited modes and, although a relatively small amount of the total energy is absorbed by these modes, their harmonics may produce an undesirable response around the fundamental resonance frequency of the desired thickness mode. This work explores various ways of suppressing the spurious effects caused by lateral-excited modes by studying their dependence of the electrode geometry. The origin of the lateral-excited modes is discussed in detail, and the results from a number of different electrode geometries are compared. A new elliptical electrode shape for suppression of spurious modes is developed and demonstrated.     

Study of the bending modes in circular quartz resonators

An experimental and theoretical study of bending modes in a partially electroded circular piezoelectric quartz (AT-cut) with free edge is presented. The quartz is excited by a voltage pulse applied on the electrodes, and its surface is scanned by a laser vibrometer that measures the out-of-plane displacements. The classical theory of bending of thin disks is used to describe the flexural modes at frequencies lower than the first thickness shear resonance (6 MHz). A fairly good agreement is found between experimental and theoretical results for the forced mode shapes and for the resonance frequencies. However, it appears that the two springs used to maintain the disk in position introduce extra clamping conditions. Several source shapes were studied, among which a collection of an arbitrary number of forces is particularly useful. The two-dimensional wavenumber representation shows the presence of anisotropy related to the crystallographic axes at higher frequencies, which is not predicted by the model. The experimental phase velocities are compared to those given by the classical theory of disks and to those of Lamb A(0) mode. This study confirms the correspondence at low frequencies between the A(0) mode and the bending eigenmodes of a disk with finite size.     

Selection of dark modes in resonators with conical reflectors

Selection of modes containing different dark regions was studied in resonators with conical reflectors. The possibility of selecting whole subgroups of such modes was shown in circularly symmetric resonators. To handle single-mode selection employing extra intracavity spatial filters, modified integral equations and a numerical method of their analysis are proposed. Usage of the filter symmetry reduces the size of the four-dimensional matrices corresponding to the equation kernels, and they are analyzed by algorithms for two-dimensional matrices with the best convergence. The optimum resonator parameters for effective selection of different dark modes are found.     

Characterization technique of optical whispering gallery mode resonators in the microwave frequency domain for optoelectronic oscillators

Optical Q factor measurements are performed on a whispering gallery mode (WGM) disk resonator using a microwave frequency domain approach instead of using an optical domain approach. An absence of hysteretic behavior and a better linearity are obtained when performing linewidth measurements by using a microwave modulation for scanning the resonances instead of the piezoelectric-based frequency tuning capability of the laser. The WGM resonator is then used to stabilize a microwave optoelectronic oscillator. The microwave output of this system generates a 12.48 GHz signal with -94 dBc/Hz phase noise at 10 kHz offset.     

Stokes resistance of a porous spherical particle in a spherical cavity

The boundary effect on the asymmetrical motion of a porous spherical particle in an eccentricspherical cavity is investigated in the quasi-steady limit under creeping flow conditions. The porous particletranslates and rotates in the viscous fluid, located within the spherical cavity, normal to the line connectingtheir centers. The fluid inside the porous particle is governed by the Brinkman equation. A tangential stressjump condition at the interface between the fluid and the porous particle is applied. A semi-analytical approachbased on a collocation technique is used. Due to the linearity of the present problem, the flow variables for theclear fluid region are constructed by superposing basic solutions of two problems: the first one is the regularsolution inside the cavity region in the absence of the porous particle where a first system of coordinates has itsorigin at the center of the cavity, while the second problem is the regular solution in the infinite region outsidethe spherical porous particle where a second coordinate system with its origin at the center of the porousparticle is used. Numerical results displaying the resistance coefficients acting on the particle are obtainedwith good convergence for various values of the physical parameters of the problem. The results are tabulatedand represented graphically. The findings demonstrate that the collocation results of the resistance coefficientsare in good agreement with the corresponding results for the impermeable solid particle.     

Calculating higher-order modes of one-dimensional unstable laser resonators

Abstract

Methods for calculating the mode profiles of unstable optical resonators are reviewed. Southwell's virtual source method (especially its inherent approximations and range of validity) is examined in depth, and ways of extending the celebrated Fox-Li method to handle higher-order modes are discussed.     

Detection of unlabeled oligonucleotide targets using whispering gallery modes in single, fluorescent microspheres

The development of an inexpensive, highly sensitive, whispering gallery mode (WGM) based biosensing system is demonstrated. The system comprises a silica microsphere functionalized with a fluorophore and a dense monolayer of single-strand oligonucleotides. The adsorption of the complementary strand causes spectral shifts in the emission spectrum of the microsphere that can be registered using a conventional optical microscope and CCD detector. The system is capable of detecting low-abundance, unlabeled-oligonucleotide targets. The spectral shifts can be used to monitor both the hybridization kinetics and the denaturation of the double-strand DNA at elevated temperatures. The hybridization appears to be completely reversible. Complementary oligonucleotide probes more than 30 bases in length are most readily detected while a complete assay takes only a few minutes.