Analysis of the rutile-ring method of frequency-temperature compensating a high-Q whispering gallery sapphire resonator

The rutile-ring method of dielectrically frequency-temperature compensating a high-Q whispering gallery (WG) sapphire resonator is presented. Two and three-dimensional finite element (FE) analysis has been implemented to design and analyze the performance of such resonators, with excellent agreement between theory and experiment. A high-Q factor of 30 million at 13 GHz, and compensation temperature of 56 K was obtained. It is shown the frequency-temperature compensation can occur either because the rutile adds a small perturbation to the sapphire resonator or because of a mode interaction with a resonant mode in the rutile. The characteristics of both of these methods are described, and it is shown that for high frequency stability, it is best to compensate perturbatively     

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.     

Simple design rules for optimal design of dielectric temperature-compensated sapphire resonators

It has been shown that the use of two dielectric crystals with opposite temperature coefficient of permittivity allows the realization of a resonator with a zero temperature coefficient of frequency. By using sapphire and rutile materials, which have low-loss tangents, some compensated resonators with very high Q-factors have been realized. In this work we develop rules that greatly simplify the design of a dielectric-compensated resonator. We show that the optimum design for compensation at a specific temperature may be determined by simply selecting the aspect ratio of the sapphire resonator.     

Frequency-temperature compensation in Ti(3+) and Ti(4+ ) doped sapphire whispering gallery mode resonators

A new method of compensating the frequency-temperature dependence of high-and monolithic sapphire dielectric resonators near liquid nitrogen temperature is presented. This is achieved by doping monocrystalline sapphire with Ti(3+) ions. This technique offers significant advantages over other methods.     

Second generation 50 K dual-mode sapphire oscillator

Low-temperature, high-precision sapphire resonators exhibit a turning point in mode frequency-temperature dependence at around 10 K. This, along with sapphire's extremely low dielectric losses at microwave frequencies, results in oscillator fractional frequency stabilities on the order of 10(-15). At higher temperatures the lack of a turning point makes single-mode oscillators very sensitive to temperature fluctuations. By exciting two quasi-orthogonal whispering gallery (WG) modes in a single sapphire resonator, a turning point in the frequency-temperature dependence can be found in the beat frequency between the two modes. A temperature control technique based on mode frequency temperature dependence has been used to maintain the sapphire at this turning point and the fractional frequency instability of the beat frequency has been measured to be at a level of 4.3 X 10(-14) over 1 s, dropping to 3.5 X 10(-14) over 4 s integration time.     

Development of a high stability cryogenic sapphire dielectricresonator

The development of a new design of cryogenic sapphire dielectric resonator is reported. The temperature dependence of the resonant frequency and width is discussed. Preliminary results of a prototype oscillator referenced to this resonator are given     

Cooled, ultrahigh Q, sapphire dielectric resonators for low-noise, microwave signal generation

Ultra-high Q, X-band resonators, used in a frequency discriminator for stabilization of a low-noise signal generator, can provide a means of obtaining significant reduction in phase noise levels. Resonator unloaded Qs on the order of 500 K can be obtained in sapphire dielectric resonator (DR) operating on a low-order (i.e. TE(01)) mode at 77 K and employing high-temperature superconducting (HTS) films installed in the DR enclosure covers. Rigorous analysis for the determination of resonator frequency, modes, and unloaded Q have been carried out using mode matching techniques. Trade-off studies have been performed to select resonator dimensions for the optimum mode yielding highest unloaded Q and widest spurious mode separation. Field distributions within the resonator have been computed to enable practical excitation of the required mode. The results of both analysis and prototype device evaluation experiments are compared for resonators fabricated using enclosures consisting of conventional, metal sidewalls and covers employing HTS films as a function of cover conductivity.     

Conceptual Design of a High-Q, 3.4-GHz Thin Film Quartz Resonator

Theoretical analyses and designs of high-Q, quartz thin film resonators are presented. The resonators operate at an ultra-high frequency of 3.4 GHz for application to high-frequency timing devices such as cesium chip-scale atomic clocks. The frequency spectra for the 3.4-GHz thin film quartz resonators, which serve as design aids in selecting the resonator dimensions/configurations for simple electrodes, and ring electrode mesa designs are presented here for the first time. The thin film aluminum electrodes are found to play a major role in the resonators because the electrodes are onlyone third the thickness and mass of the active areas of the plate resonator. Hence, in addition to the material properties of quartz, the elastic, viscoelastic, and thermal properties of the electrodes are included in the models. The frequency-temperature behavior is obtained for the best resonator designs. To improve the frequency-temperature behavior of the resonators, new quartz cuts are proposed to compensate for the thermal stresses caused by the aluminum electrodes and the mounting supports. Frequency response analyses are performed to determine the Q-factor, motional resistance, capacitance ratio, and other figures of merit. The resonators have Q's of about 3800, resistance of about 1300 to 1400 ohms, and capacitance ratios of 1100 to 2800.     

分体圆柱谐振腔法用于金刚石膜微波介电性能测试的研究

针对金刚石膜微波介电损耗低、厚度薄带来的微波介电性能测试难点, 研制了一台分体圆柱谐振腔式微波介电性能测试装置。利用不同直径的蓝宝石单晶样品, 用上述装置对低损耗薄膜类样品微波介电性能的测试能力及样品直径对测试结果的影响进行了实验研究。在此基础上, 使用分体圆柱谐振腔式微波介电性能测试装置对微波等离子体化学气相沉积法和直流电弧等离子体喷射法制备的高品质金刚石膜在Ka波段的微波介电性能进行了测试比较。测试结果表明, 由Raman光谱、紫外-可见光谱等分析证明品质较优的微波等离子体化学气相沉积法制备的金刚石膜具有更高的微波介电性能, 其相对介电常数和微波介电损耗值均低于直流电弧等离子体喷射法制备的金刚石膜。     

TE013 Mode High Temperature Superconductor Millimeter Resonator

This paper describes a TE₀₁₃ mode dielectric resonator at 36.098 GHz. The dielectric resonator was formed by sandwiching a cylindrical piece of polished dielectric (sapphire) rod between two GdBa₂Cu₃O_7?x (GBCO) High Temperature Superconductor (HTS) thin films. The resonator was tested as a two-port device at a temperature of 77 K. A 62700 unload quality factor Q ₀ was obtained. The surface resistance (Rs) of the HTS film had been determined as 9.4 mΩ at 77 K and 36 GHz.     

Frequency-temperature behavior of flexural quartz resonators by means of Timoshenko's model

The frequency of a flexural resonator and its frequency-temperature behavior usually are computed by Bernoulli's classical approximation. This approach is valid for beams with a large length-over-thickness-ratio. For shorter beams, the effects of shear stress and rotary inertia may play a significant role for temperature-compensated resonators. These effects have been taken into account for isotropic beams. The aim of this paper is to discuss the extension of the shear coefficient in the case of an anisotropic material and to compute the frequency-temperature characteristic of an (XYt)theta cut resonator when the shear stress and the rotary inertia have been taken into account. Comparisons between the classical approximation and this treatment are given for quartz. Furthermore, the numerical predictions obtained by means of different sets of data available for thermal sensitivities of elastic coefficients are compared.     

Frequency-temperature behavior of spurious vibrations of rectangular AT-cut quartz plates

We evaluate the frequency-temperature behavior of spurious modes in a rectangular AT-cut quartz plate resonator based on three-dimensional linear equations. The elastic constants and three geometrical dimensions of the resonator are defined in terms of cubic polynomials of a temperature change. Assuming that the resonator holds its rectangular plate shape irrespective of temperature, we can determine the relationship between frequency and the dimensions of the resonator for a given temperature using the previous technique. We compare the calculated results with our own experimental data, and show that agreement between the calculated and observed data is excellent.     

Temperature-compensated sapphire resonator for ultra-stableoscillator capability at temperatures above 77 K

We report on the design and test of a whispering gallery sapphire resonator for which the dominant (WGH_n11) microwave mode family shows frequency-stable, compensated operation for temperatures above 77 K. The resonator makes possible a new ultra-stable oscillator (USO) capability that promises performance improvements over the best available crystal quartz oscillators in a compact cryogenic package. A mechanical compensation mechanism, enabled by the difference between copper and sapphire expansion coefficients, tunes the resonator to cancel the temperature variation of sapphire's dielectric constant. In experimental tests, the WGH₈₁₁ mode showed a frequency turnover temperature of 87 K in agreement with finite element calculations. Preliminary tests of oscillator operation show an Allan Deviation of frequency variation of 1.4-6×10^-12 for measuring times 1 s ⩽τ⩽100 s with unstabilized resonator housing temperature and a mode Q of 2×10⁶. We project a frequency stability 10^-14 for this resonator with stabilized housing temperature and with a mode Q of 10⁷     

Microwave oscillators incorporating cryogenic sapphire dielectricresonators

Progress is reported on efforts to develop a commercially-viable high purity X-band signal source incorporating a cryogenic sapphire dielectric resonator. The resonator design is of the whispering gallery type to take advantage of the excellent electromagnetic field confinement offered by this geometry. Complications resulting from the high spurious mode density of this type of resonator have been eliminated by developing a very accurate and complete mode analysis program which fully incorporates the dielectric anisotropies of the sapphire ring. This program allows the design of a window in the frequency domain where no unwanted modes exist, with accurate placement of the desired mode at the center of this region. Preliminary evaluation of the phase noise properties of simple oscillators incorporating these resonators have been performed. For example, in a dual-oscillator comparison of two oscillators operating near 13 GHz phase noise values of L(f)=-55 dBc/Hz, -145 dBc/Hz and -161 dBc/Hz were obtained for offset frequencies of 1 Hz, 1 kHz and 10 kHz, respectively     

TE013 Mode High Temperature Superconductor Millimeter Resonator

This paper describes a TE₀₁₃ mode dielectric resonator at 36.098 GHz. The dielectric resonator was formed by sandwiching a cylindrical piece of polished dielectric (sapphire) rod between two GdBa₂Cu₃O_7–x (GBCO) High Temperature Superconductor (HTS) thin films. The resonator was tested as a two-port device at a temperature of 77 K. A 62700 unload quality factor Q ₀ was obtained. The surface resistance (Rs) of the HTS film had been determined as 9.4 m at 77 K and 36 GHz.     

High Q-factor distributed bragg reflector resonators with reflectors of arbitrary thickness

The Bragg reflection technique improves the Q-factor of a resonator by reducing conductor and dielectric losses. This is achieved by designing a low-loss inner resonant region (usually free space) surrounded by an outer anti-resonant region made of distributed Bragg reflector layers. In this paper we develop a simple non-Maxwellian model and apply it to design three distinct cylindrical Bragg resonators based on the same set of single-crystal sapphire plates and rings by changing only the dimension of the cavity that supports the structure. To accomplish this, the simple model allows an arbitrary thickness for either the horizontal or the cylindrical dielectric reflectors by relaxing the condition that they must be lambda/4 thick. The model also allows for higher-order field variations in both the resonant and the anti-resonant regions. The resonators were constructed and experimental results were compared with the simple model and the rigorous method of lines analysis. For the fundamental mode, an unloaded Q-factor of 234,000 at 9.7 GHz was obtained. This is larger than that for a whispering gallery mode resonator. The resonator also exhibited a greatly reduced spurious mode density when compared to an overmoded whispering gallery mode resonator.     

Analysis of whispering-gallery superconducting dielectric resonator modes

The whispering-gallery (WG) modes of a superconducting dielectric resonator (SDR) based on a sapphire cylindrical dielectric resonator and a YBa₂Cu₃O_{7 –} shielding cylinder have been studied. A method for the determination of the resonant frequencies and the maximum quality factor of such modes is presented. Calculations have shown that most of the mode energy could be confined between the caustic surface of the WG modes provided the dimensions of the SDR are properly selected, and a magnitude of 10⁹ forQ of the SDR could be estimated. A phenomenal explanation is given to account for such outstanding microwave behavior.     

Whispering-gallery mode technique applied to the measurement of the dielectric properties of Langasite between 4 K and 300 K

We report new measurements of dielectric properties of Lanthanum gallium silicate (Langasite or LGS) conducted with the whispering-gallery mode technique at microwave frequencies and between 4.2 K and 300 K. The real part of the permittivity tensor of LGS presents two components having temperature coefficients of opposite sign. This unique property enables the design of a temperature compensated resonator that may be useful in building stable microwave oscillators or filters. We report also the first measurements of the two independent components of the imaginary part of the permittivity tensor. It appears LGS is a relatively high-loss dielectric material compared with sapphire or quartz.     

Microwave Power Dependence of Surface Resistance of High-Temperature Superconductor YBa2Cu3O y Films by Dielectric Resonator Method

The microwave power dependence of the surface resistance of YBa₂Cu₃O_y films of two different surface morphologies was measured using the dielectric resonator method. The dielectric resonator consisted of a high-quality sapphire rod sandwiched between two superconductor films. Measurements showed that the microwave power dependence of the surface resistance strongly depended on the surface morphology of the YBa₂Cu₃O_y film though the surface resistance of those films at low microwave power was comparable. In conclusion, the surface morphology is one of the reasons of the power dependence which is crucial for high power applications.     

Q-factor measurement of quasi-optical dielectric resonators under conditions of the whispering gallery mode degeneration removal

An approach that allows one to determine the unloaded quality factor of a quasi-optical dielectric resonator (QDR) under conditions of the resonance line splitting corresponding to twofold degenerative whispering gallery (WG) mode has been proposed. The resonator is represented by an equivalent network as two oscillatory circuits with a certain coupling between them. The approach allows one to determine the Q-factor of a single oscillatory circuit using three measured parameters, namely: 1) ratio between specific amplitudes of the resonator amplitude-frequency response (or its second derivative); 2) resonant frequency; and 3) frequency splitting of the response (or its respective second derivative). The proposed technique is illustrated by the measurements in the millimeter-wave range for sapphire quasi-optical resonators with conducting and high-T/sub c/ superconducting (HTS) film endplates.