Distortions of thickness shear mode shapes in plano-convex quartz resonators with mass perturbations

A perturbation technique for calculating the change in mode shape in a plano-convex thickness shear resonator blank due to off-center mass perturbations is developed. The perturbed shape is built as a linear combination of the familiar unperturbed modes and their anharmonics. The change in the location of the average center of the mode is calculated for different contours and different mass perturbation shapes and thicknesses. The results are used to predict changes in the acceleration sensitivity of two-point mounted 10-MHz, third SC-cut resonators when patches of metal film are deposited on the major surfaces off-center. These theoretical results are compared with published experimental results.     

Acceleration sensitivity of crystal resonators affected by the mass and location of electrodes

Predominant thickness-shear frequencies and modes of a crystal plate with electrodes of arbitrary shape and mass distribution are obtained by a finite-element method, based on Mindlin's first-order equations with platings. These frequencies and modes are used in a perturbation method for computing the acceleration sensitivity of crystal resonators with electrodes. Computations are made for a square AT-cut quartz plate that is supported by a four-point mount and coated with identical square and uniform electrodes on the upper and lower faces of the plate. To study the effect of uneven distribution of electrode mass, acceleration sensitivities are calculated when a small mass is added at various locations near the edges of the square electrodes. It is found that the percent increase of the acceleration sensitivity of the resonator with a small added mass to that of the resonator without added mass ranges from 3.8% to 541.7%, depending on the location of the small mass placed at the edges of the electrodes.     

Theory and design of piezoelectric resonators immune to acceleration: present state of the art

A typical low noise oscillator uses a crystal resonator as the frequency-determining element. An understanding of the fundamental nature of acceleration sensitivity in crystal oscillators resides primarily in understanding the behavior of the crystal resonator. The driving factor behind the acceleration-induced frequency shift is shown to be deformation of the resonator. The deformation drives two effects: an essentially linear change in the frequency-determining dimensions of the resonator and an essentially nonlinear effect of changing the velocity of the propagating wave. In this paper, the fundamental nature of acceleration sensitivity is reviewed and clarified, and attendant design guidance is developed for piezoelectric resonators. The basic properties of acceleration sensitivity and general design guidance are developed through the simple examples of “bulk acoustic wave (BAW) in a box” and “surface transverse wave (STW) in a box.” These examples serve to clarify a number of concepts, including the role of mode shape and the basic difference between the BAW and STW cases. The design equations clarify the functional dependencies of the acceleration sensitivities on the full range of crystal resonator design and fabrication parameters     

A micromachined vibration isolation system for reducing the vibration sensitivity of surface transverse wave resonators

A micromachined system has been developed for reducing the vibration sensitivity of surface transverse wave (STW) resonators. The isolation system consists of a support platform for mounting the STW resonator, four support arms, and a support rim. The entire isolation system measures 8 mm by 9 mm by 0.4 mm without the resonator mounted on the platform. The system acts as a passive vibration isolation system, decreasing the magnitude of high frequency (>1.2 kHz) vibrations. Finite element analysis is used to analyze the acceleration sensitivity of the mounted resonator. The isolation system is then modeled as a damped mass-spring system and the transmissibility of vibration from the support rim to the support platform is calculated. Multiplying the acceleration sensitivity of the resonator by the transmissibility results in the expected system vibration sensitivity. The isolation systems are fabricated using two sided bulk etching of (110) oriented silicon wafers. STW resonators were mounted on the isolation systems, and the isolated units were mounted on commercial hybrid oscillator substrates. Vibration sensitivity measurements were taken for vibrations with frequencies ranging from 100 Hz to 5 kHz. The measured data show that the system performs as expected with a low frequency (<500 Hz) vibration sensitivity of 1.8×10^-8/g and a high frequency roll off of 12 dB/octave     

Design and performance of diffractive optics for custom laser resonators

Diffractive optical elements are used as end mirrors and internal phase plates in an optical resonator. A single diffractive end mirror is used to produce an arbitrary real-mode profile, and two diffractive mirrors are used to produce complex profiles. Diffractive mirror feature size and phase quantization are shown to affect the shape of the fundamental mode, the fundamental-mode loss, and the discrimination against higher-order modes. Additional transparent phase plates are shown to enhance the modal discrimination of the resonator at the cost of reduced fabrication tolerances of the diffractive optics. A 10-cm-long diffractive resonator design is shown that supports an 8.5-mm-wide fundamental mode with a theoretical second-order mode discrimination of 25% and a negligible loss to the fundamental mode.     

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.     

An observation on quartz resonator aging

Aging and acceleration sensitivity of certain fabrication lots of quartz resonators show a trend that may arise from a common predominating mechanism, such as stresses arising from the mounting structure. The presence or absence of such a trend is discussed. As part of the work on bulk-acoustic-wave (BAW) resonators, a database of all known parameters of the resonators in question was assembled. Resonator processing and electrical parameters were examined for correlations with the acceleration sensitivity. A similar effort was being undertaken in regard to resonator aging, and possible correlations between the aging and acceleration sensitivities of the various resonators were examined. The results for two such fabrication lots are shown. The magnitude of the observed aging was typically in the 10(-10) per day range. The identity of the common contributor is unknown at the present time. It is likely that mounting related stresses are the common contributor.     

An estimate on the second-order normal acceleration sensitivity of a quartz resonator

An estimate is given for a second-order effect in the normal acceleration sensitivity of a perfectly symmetric Y-cut quartz resonator whose first-order normal acceleration sensitivity is zero. The order of the second-order frequency shift is 10-11 per g.     

复振型导数计算的可变移频值方法

实际结构系统由于存在多种不同性质的阻尼其动态特性很复杂,振型导数的计算也比较困难。采用模态加速和移频的思想发展了一种基于模态叠加的复振型导数计算方法。首先对控制方程进行移频处理,利用广义幂级数展开式获得模态迭代公式,并利用迭代结果与各阶振型表示复振型导数;然后把系统的广义动柔度矩阵表示为已知的低阶模态与截断的高阶模态之和,高阶模态部分采用多个矩阵多项式与一个广义幂级数的乘积表示,并利用系统的低阶模态和系统矩阵进行计算;各阶移频值表示为相应的移频系数与复特征值的乘积,它们仅与最低阶模态移频值的模和本阶模态的单位复特征值有关,而最低阶模态的移频系数通过精度分析获得。给出了合适的模态加速迭代次数。该方法仅需进行一次系统矩阵的分解就可获得高精度的多个复振型导数。算例表明方法正确、高效。     

Generation of a flat-top laser beam for gravitational wave detectors by means of a nonspherical Fabry-Perot resonator

We have tested a new kind of Fabry-Perot long-baseline optical resonator proposed to reduce the thermal noise sensitivity of gravitational wave interferometric detectors--the "mesa beam" cavity--whose flat top beam shape is achieved by means of an aspherical end mirror. We present the fundamental mode intensity pattern for this cavity and its distortion due to surface imperfections and tilt misalignments, and contrast the higher order mode patterns to the Gauss-Laguerre modes of a spherical mirror cavity. We discuss the effects of mirror tilts on cavity alignment and locking and present measurements of the mesa beam tilt sensitivity.     

Electromagnetic theory of an open resonator.

Particular higher-order sources give rise to electromagnetic Gaussian beams, which are linearly polarized and have their maximum in the propagation direction. For this dipolar beam the cross-sectional shape changes in the propagation direction. Nodal surfaces exist on which the tangential component of the electric field vanishes in the standing wave that is formed by the two oppositely directed dipolar, electromagnetic Gaussian beams. These surfaces are identified as the mirror shapes for an open resonator that supports this standing wave. For standing waves that have a particular cross-sectional shape at the waist the cross section of the beam near the mirror surfaces is circular. The resonant frequencies for the fundamental transverse mode of such a resonator have been determined as a function of the geometry and the axial mode number. By a perturbation technique the resonant frequency of an open resonator with spherical mirrors has been obtained. This result is valid in only the paraxial approximation. Illustrative numerical results are included.     

Designing for low acceleration sensitivity

Recent advances in acceleration sensitivity measurement and modeling are discussed, with an emphasis on what these advances indicate in terms of designing for low acceleration sensitivity. The design suggestions are separated into two parts, namely, those that use the crystal resonator as a mechanical vibrator and those that use the crystal oscillator as an electronic circuit. Such topics as symmetry considerations, metallization, mounting etc., in both bulk acoustic wave (BAW) and surface acoustic wave (SAW) devices are discussed for the former, while for the latter the equivalent circuit modeling of crystal resonators and other loop components is addressed.     

A study of the electromagnetic properties of a resonator for multichannel acceleration of light ions

We present an experimental study of a modified Interdigital-H-resonator which can be used for multichannel acceleration of light ions. The distribution of the electromagnetic fields is investigated with a perturbation method. Furthermore, measurements of the resonance frequency, resonator parallel resistance and quality factor are discussed. The measurements have been done on three types of accelerating structures, which have a different capacitance per unit length. The results are compared with an elementary model, in which the resonator is described in terms of discrete circuit elements. It is found that the resonance frequency can be calculated with good accuracy. For the other parameters scaling laws are derived.     

压力管道腐蚀灾变检测的应变敏度比法

基于结构振动模态测试理论和有限元方法提出了压力管道腐蚀检测的应变敏度比法。首先,通过位移模态及应变模态建立了腐蚀管道的定位检测判据。其次,制备不同缺陷的管道,应用应变敏度比法检测判断缺陷位置,结果表明,检测位置与实际损伤位置完全一致。最后,通过数值算例证实用前1阶模态分量数据检测的结果与实际缺陷基本一致。通过实测和数值算例证明了只测试损伤及未损伤结构的低阶模态参量,便可进行有效的检测,而低阶参量是相对简单易测,从而使该方法应用于工程实际更显便利。     

Comparison of resonator-originated and external beam shaping

The spatial shaping of laser beams is a subject of research in modern optics. Recently the introduction of diffractive elements in laser resonators has offered an alternative to external beam-shaping optics by mode shaping within the resonator. We describe the specification of the laser resonator mirrors to obtain by means of internal mode shaping a desired beam outside the resonator. Modal discrimination of the modified resonator and the mirror alignment sensitivity is discussed. Basic features of resonator-originated and external beam shaping are compared.     

Electrostatic transducers for micromechanical resonators: free space and solid dielectric

Three electrostatic transduction methods are analyzed for a micromechanical, longitudinal mode, beam resonator. The conventional parallel plate transducer placed at the location of maximum displacement is compared to two solid, dielectric transducers internal to the resonator. Although the solid dielectric offers higher permittivity than the free-space-filled transducers, the unfavorable locations of the internal transducers reduce or even remove the performance advantage of the higher permittivity.     

基于摄动法的多条裂纹欧拉梁特征模态分析

基于摄动理论推导了带多条开口裂纹的欧拉梁的特征模态参数的理论计算公式。采用最直接的方式将梁开口裂纹模拟成梁微段内的横截面折减并用δ函数表达了带开口裂纹的梁沿轴线的截面惯矩和线质量等物理参数。基于此,建立了裂纹梁动力微分方程,并采用一阶摄动理论推导得到了梁的模态频率和振型计算公式。简支梁及悬臂梁算例研究表明,该方法具有很好的精度,与有限元模拟结果及实验结果都能很好地吻合。并采用此方法分析了裂纹深度和位置对带多条开口裂纹梁的特征模态参数的影响。结果表明,裂纹对各阶模态频率虽然影响有限,但其引起的各阶频率变化有着明显的模式,可用于结构损伤定位;裂纹对模态振型影响不明显,但对模态曲率影响比较大,可用于结构损伤位置和程度的诊断。     

Iterative method for the design of a dual-phase-conjugation-mirror resonator with multiple apertures

Apertures have been used to select the low-order transverse modes in resonators. The additional diffraction losses result in a change in the transverse-mode structure, and the presence of apertures inside a resonator generally distorts the mode shape. The optimization of a multiple-aperture resonator demands an approach that differs from the conventional method in which the mode theory is used. We demonstrate an iterative design method to find optimal phase profiles for the reflector surfaces to build a resonator with multiple apertures to produce a lowest-order mode with much smaller diffraction loss and to satisfy the phase-conjugation conditions at the mirrors. The results are compared with conventional stable resonators, and we show that substantial improvement in round-trip loss and beam quality can also be obtained.     

Sensitivity of quartz oscillators to the environment: characterization methods and pitfalls

The advantages and disadvantages of characterizing a complete quartz crystal oscillator or characterizing only the quartz resonator by using a passive phase bridge are discussed. Measurements of temperature sensitivities, including quasistatic or dynamic thermal conditions, are presented. One important point is how to measure the real temperature of the device under test (quartz crystal, for instance) rather than the temperature of the probe. Methods for measuring acceleration and pressure sensitivities are presented, and spurious effects of temperature changes are considered. Problems are discussed in connection with the measurement of the sensitivity to magnetic fields, and to electric fields. Methods used for measuring these sensitivities and the many pitfalls that can be encountered are the focus of the investigation.