Modal Parameter Tuning of an Axisymmetric Resonator via Mass Perturbation

Abstract

Abstract-This paper reports the permanent frequency mismatch reduction of the primary wineglass modes in a planar axisymmetric resonator by strategic mass loading. The resonator consists of a set of concentric rings that are affixed to neighboring rings by a staggered system of spokes. The outer layers of spokes are targets for mass deposition. The paper develops modified ring equations that guide the mass perturbation process and despite the fact that the deposited mass and deposition locations are quantized, it is possible to systematically reduce the frequency difference of the wineglass modes to effective degeneracy such that two modes cannot be distinguished in a frequency response plot. Results on five resonators are reported with nominal wineglass modes near 14 kHz, quality factors of 50 k, and frequency mismatches exceeding 30 Hz in some cases but with post-perturbation mismatches smaller than 80 mHz. Furthermore, it is also shown that the quality factors remain unchanged.

Tailored Etch Profiles for Wafer-Level Frequency Tuning of Axisymmetric Resonators

“…The innermost large spoke layer is designated as the first layer (i = 1) and the outermost spoke layer is designated as the fourth layer (i = 4). The resonator is a modification of the design described in [3]. The reader is referred to this paper for information on the resonator fabrication, electrode layout, test procedure, and the process for extracting pertinent modal parameters from empirical frequency response data.…”

“…In this report the resonator is fabricated without the gold film on, or the reservoirs in, the large spokes. Consequently, these areas are targets for further etching in order to selectively remove material from the resonator instead of adding material as was pursued in [3]. It will be shown that material removal at the large spokes locations does not modify the stiffness properties of the resonator and so models that readily predict the changes in the resonator modal properties can be developed by assuming only the mass distribution is perturbed.…”

“…Small fabrication errors, however, cause the modal frequencies to detune, thereby reducing the signal-to-noise ratio that can be achieved with the associated CVG (see [1] for the analysis of a disk resonator). With the successful development of modematched micro-scale disk resonators [2], [3], [4], quadruple mass resonators [5], hemispherical [6], [7], [8], [9], [10], [11], [12], and hemitoroidal [13], [14] resonators, it has become imperative to create post-fabrication corrective procedures so that the pairs of modal frequencies can be brought back to degeneracy. This paper reports the development and application of a wafer-level targeted SiDRIE process for eliminating the modal frequency differences in a planar axisymmetric silicon resonator.…”

“…Laser ablation of a ring was also reported in [15] but it degraded the quality factors. The planar resonator design in this paper was also used for modal frequency mismatch reduction by mass-loading the resonator and the resonator frequency models introduced in [3] are further refined in the present paper. One advantage of the present approach over the mass deposition in [3] is the fact that the etch simply removes silicon and does not introduce metals or other material that may not be compatible with downstream processing and packaging.…”

“…II-B and II-C develop a semi-analytical model of the resonator's n = 2 modal frequency difference as a function of mass perturbations. A global sensitivity parameter is also defined and unifies several aspects of the model developed in [3]. Sec.…”