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.     

Compact, high-Q, zero temperature coefficient, TE011 sapphire-rutile microwave distributed Bragg reflector resonators

Some novel new resonator designs based on the distributed Bragg reflector are presented. The resonators implement a TE₀₁₁ resonance in a cylindrical sapphire dielectric, which is confined by the addition of rutile and sapphire dielectric reflectors at the end faces. Finite element calculations are utilized to optimize the dimensions to obtain the highest Q-factors and zero frequency-temperature coefficient for a resonator operating near 0°C. We show that a Q-factor of 70,000 and 65,000 can be achieved with and without the condition of zero frequency-temperature coefficients, respectively     

Cryogenic monolithic sapphire-rutile temperature compensated resonator oscillator

We have tested a new temperature-compensated sapphire resonator as frequency determining element for high-stability microwave oscillator. Temperature compensation has been obtained by coating the sapphire resonator with a thin rutile film. A 2-/spl mu/m rutile thickness is sufficient to reach turnover temperature higher than 40 K, and a 2/spl times/10/sup -12/ short-term frequency stability has been obtained.     

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.     

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.     

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     

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.     

Quantitative signal analysis in pulsed resonant photoacoustics

The pulsed excitation of acoustic resonances was studied by means of a high-Q photoacoustic resonator with different types of microphone. The signal strength of the first radial mode was calculated by the basic theory as well as by a modeling program, which takes into account the acoustic impedances of the resonator, the acoustic filter system, and the influence of the microphone coupling on the photoacoustic cavity. When the calculated signal strength is used, the high-Q system can be calibrated for trace-gas analysis without a certified gas mixture. The theoretical results were compared with measurements and show good agreement for different microphone configurations. From the measured pressure signal (in pascals per joule), the absorption coefficient of ethylene was calculated; it agreed within 10% with literature values. In addition, a Helmholtz configuration with a highly sensitive 1-in. (2.54-cm) microphone was realized. Although the Q factor was reduced, the sensitivity could be increased by the Helmholtz resonator in the case of pulsed experiments. A maximum sensitivity of the coupled system of 341 mV/Pa was achieved.     

Experimental verification of stress compensation in the SBTC-cut

The authors have demonstrated experimental verification of the stress compensation feature for the fast thickness shear mode of vibration of stress-compensated for B-mode and temperature-compensated for C-mode (SBTC)-cut quartz resonators. For the resonator design used in the cylindrical probe structure, the motional resistance for the B-mode of vibration was approximately 12% of that of the C-mode. The relatively large motional resistance for the C-mode of vibration of the SBTC-cut was found to be largely due to the lower piezoelectric coupling for the thickness excitation of this mode. In addition the proximity of the third overtone of the A-mode to the fifth overtone of the C-mode also contributed to the increase in the motional resistance. The authors have obtained experimental data on the temperature dependence of the planar stress coefficient and pressure dependence of the frequency-temperature characteristic for both the thickness-shear modes of the SBTC-cut. It is noted that such a doubly rotated cut can have applications in the design of either stable frequency sources or sensors for pressure and temperature measurements.     

Sensitivity and optimization of a high-Q sapphire dielectric motion-sensing transducer

A high-Q sapphire dielectric motion sensing transducer that operates at microwave frequencies has been developed. The device uses cylindrical whispering gallery modes of quality factor greater than 10 (5) at room temperature and greater than 10(8) at 4 K. The tuning coefficient of the transducer resonance frequency with respect to displacement was measured to be of the order of a few MHz/mum. An electromagnetic model that predicts the resonant frequency and tuning coefficient has been developed and was verified by experiment. We implemented the model to determine what aspect ratio and what dielectric mode is necessary to maximize the sensitivity. We found that the optimum mode type was a TM whispering gallery mode with azimuthal mode number of about 7 for a resonator of 3 cm in diameter. Also, we determined that the tuning coefficients were maximized by choosing an aspect ratio that has a large diameter with respect to the height. By implementing a microwave pump oscillator of SSB phase noise -125 dBc/Hz at 1 kHz; offset, we have measured a sensitivity of order 10 (-16) m/ radicalHz. We show that this can be improved with existing technology to 10(-18) m/ radicalHz, and that in the near future this may be further improved to 10(-19) m/ radicalHz.     

Analysis of temperature compensation in a plate thickness mode bulk acoustic wave resonator

We analyze temperature-induced frequency shift in a thickness mode bulk acoustic wave resonator with a layer of another material for temperature compensation. The perturbation integral by Tiersten is used to calculate frequency shifts in the resonator under a temperature change. It is shown that, with a proper design of the compensation layer, temperature sensitivity of the resonator can be reduced or made zero.     

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 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     

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.     

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.     

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.     

Designs of a microwave TE011 mode cavity for a space borne H-maser.

Method of Lines and Finite Element Analysis investigations have been performed to optimize parameters in a TE011 mode cavity resonator suitable for a spaceborne hydrogen maser. We report on designs that were explored to find a global maximum in the important design parameters for the microwave cavity used in a hydrogen maser. The criteria sought in this exercise were both the minimization of the total volume of the cavity and the maximization of the product of the z-component of the magnetic energy filling factor and the cavity TE011 mode Q-factor (Q.eta). Different configurations were studied. They were a sapphire tube in a copper cylinder, a sapphire tube in a copper cylinder with Bragg reflectors, and spherical copper cavities both empty and sapphire-lined on the inside cavity surface. At 320 K, the simulations resulted in an optimum product Q.eta = 4.9 x 10(4), with an inner cavity radius of 80 mm and unity aspect ratio. This represents a 54% improvement over an earlier design. The expected increase in the product Q . eta) with the inclusion of Bragg reflectors to the sapphire tube was not achieved. Moreover, the z-component of the magnetic energy filling factor was greatly reduced due to an increase in the radial magnetic field. The sapphire-lined spherical cavity showed no better performance than an equivalent-sized empty copper spherical cavity. For the empty cavity the simulations resulted in the product Q.eta = 4.4 x 10(4). The empty spherical cavity resonator is not suitable for the spaceborne hydrogen maser as the total volume in this case is 33% larger than that of the optimized sapphire tube resonator.     

Detailed performance modeling of a pulsed high-power single-frequency Ti:sapphire laser

Differential absorption lidar (DIAL) is a unique technique for profiling water vapor from the ground up to the lower stratosphere. For accurate measurements, the DIAL laser transmitter has to meet stringent requirements. These include high average power (up to 10 W) and high single-shot pulse energy, a spectral purity >99.9%, a frequency instability <60 MHz rms, and narrow spectral bandwidth (single-mode, <160 MHz). We describe extensive modeling efforts to optimize the resonator design of a Ti:sapphire ring laser in these respects. The simulations were made for the wavelength range of 820 nm, which is optimum for ground-based observations, and for both stable and unstable resonator configurations. The simulator consists of four modules: (1) a thermal module for determining the thermal lensing of the Brewster-cut Ti:sapphire crystal collinear pumped from both ends with a high-power, frequency-doubled Nd:YAG laser; (2) a module for calculating the in-cavity beam propagations for stable and unstable resonators; (3) a performance module for simulating the pumping efficiency and the laser pulse energy; and (4) a spectral module for simulating injection seeding and the spectral properties of the laser radiation including spectral impurity. Both a stable and an unstable Ti:sapphire laser resonator were designed for delivering an average power of 10 W at a pulse repetition frequency of 250 Hz with a pulse length of approximately 40 ns, satisfying all spectral requirements. Although the unstable resonator design is more complex to align and has a higher lasing threshold, it yields similar efficiency and higher spectral purity at higher overall mode volume, which is promising for long-term routine operations.     

Measurement and analysis of a microwave oscillator stabilized by a sapphire dielectric ring resonator for ultra-low noise

Phase-noise measurements are presented for a microwave oscillator whose frequency is stabilized by a whispering gallery mode sapphire ring resonator with Q of 2x10(5). The nature of the mode, which involves little metallic conduction, allows nearly full use of the very low dielectric loss in sapphire. Several mode families have been identified with good agreement with calculated frequency predictions. Waveguide coupling parameters have been characterized for the principal (lowest frequency) mode family, for n=5 to n =10 full waves around the perimeter. For a 5-cm wheel resonator in a 7.6-cm container, Q-values of above 10(5) were found at room temperature for all of the modes in this sequence. Coupling Q-values for the same modes ranged from 10(4) (n =5) to 10(5) (n=10) for a WR112 waveguide port at the center of the cylinder wall of the containing can. Phase noise measurements for a transistor oscillator locked to the n=10 (7.84-GHz) mode showed a 1/f(3) dependence for low offset frequencies, and a value of L(f)=-55 dB/Hz at an offset of 10 Hz from the carrier. The oscillator shows phase noise below the previously reported for any X-band source.     

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