sensors Article

Monolithic Cylindrical Fused Silica Resonators with High Q Factors Yao Pan, Dongya Wang, Yanyan Wang, Jianping Liu, Suyong Wu, Tianliang Qu *, Kaiyong Yang and Hui Luo * College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, China; [email protected] (Y.P.); [email protected] (D.W.); [email protected] (Y.W.); [email protected] (J.L.); [email protected] (S.W.); [email protected] (K.Y.) * Correspondence: [email protected] (T.Q.); [email protected] (H.L.); Tel.: +86-731-8457-4730 (T.Q.); Fax: +86-731-8457-3747 (T.Q.) Academic Editor: Francesco De Leonardis Received: 9 May 2016; Accepted: 22 July 2016; Published: 28 July 2016

Abstract: The cylindrical resonator gyroscope (CRG) is a typical Coriolis vibratory gyroscope whose performance is determined by the Q factor and frequency mismatch of the cylindrical resonator. Enhancing the Q factor is crucial for improving the rate sensitivity and noise performance of the CRG. In this paper, for the first time, a monolithic cylindrical fused silica resonator with a Q factor approaching 8 ˆ 105 (ring-down time over 1 min) is reported. The resonator is made of fused silica with low internal friction and high isotropy, with a diameter of 25 mm and a center frequency of 3974.35 Hz. The structure of the resonator is first briefly introduced, and then the experimental non-contact characterization method is presented. In addition, the post-fabrication experimental procedure of Q factor improvement, including chemical and thermal treatment, is demonstrated. The Q factor improvement by both treatments is compared and the primary loss mechanism is analyzed. To the best of our knowledge, the work presented in this paper represents the highest reported Q factor for a cylindrical resonator. The proposed monolithic cylindrical fused silica resonator may enable high performance inertial sensing with standard manufacturing process and simple post-fabrication treatment. Keywords: cylindrical resonator; fused silica; Q factor; chemical etching; annealing

1. Introduction Coriolis vibratory gyroscopes (CVGs), especially those with axisymmetric shell resonators, are known for their outstanding capabilities of high accuracy, long durability, considerable reliability, low power consumption, short start-up time, high shock and acceleration resistance, etc. The well-known hemispherical resonator gyroscope (HRG) has claimed 30 million hours of continuous operation without a single mission failure [1]. Because this technology is complicated and rather expensive, researchers have tried to find acceptable compromise between cost and performance. As a simpler version, cylindrical resonator gyroscopes (CRGs) with metallic and piezoelectric cylindrical resonators have been intensively investigated [2–9]. The cylindrical resonator is the core component for CRGs with various applications where GPS may not be available, including inertial navigation systems, flight control systems, aircraft monitoring and maintenance, airborne platform stabilization, oil and gas platform stabilization, unmanned ground vehicles, north finding and automated guided vehicles, etc. There are tactical grade and industrial grade CRG products available on the market, such as Watson Industries Pro Gyro® series [10] and the Innalabs 2000 and 1000 series [11]. These gyroscopes are readily applicable to medium and low accuracy fields such as platform stabilization, industrial control systems, rail-tilt control systems, short-term navigation, etc. However, the intrinsic disadvantage

Sensors 2016, 16, 1185; doi:10.3390/s16081185

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of metallic and piezoelectric cylindrical resonators is their low quality (Q) factors, which limits the overall performance of the gyroscopes. With excitation and detection elements, the piezoceramic cup resonator has a sense Q of only ~52 [12]. Resonators made of metallic alloys typically have Q factors up to 104 ~4 ˆ 104 [7,13]. Notably, the Innalabs CVG25 resonators made from metallic alloys have shown a Q factor of 4 ˆ 104 , which is so far the highest Q factor reported for metallic cylindrical resonators [7]. The Q factor of the combined cylindrical resonator with a fused silica shell and a metal plate is only about 6250 [14]. Designed by STM Co./ATSU [15], cylindrical resonators made of high-purity single-crystalline sapphire have shown promising Q factors of more than 107 . However, the sapphire material has intrinsic anisotropy and high stress, which will induce frequency mismatch for resonators. It is also worth mentioning that, while various micro cylinder or hemispherical cylinder resonators have been reported recently with Q factors of 5 ˆ 105 ~1.2 ˆ 106 [16–18], these resonators have much higher mode frequency, resulting in much shorter ring-down time. Although some notable experimental results have been reported, these technologies are still premature for high-accuracy gyroscopic applications. The accuracy of CRGs operating in the force-to-rebalance mode is mainly determined by the Q factor and the natural frequency mismatch of the resonator [6]. Enhancing the Q factor is crucial for improving the rate sensitivity and noise performance of CRGs. Fused silica is an attractive material because of its intrinsically very low internal friction and high isotropy. Innalabs has predicted the Q factor of 25 mm-diameter resonators made from fused silica to be around 5 ˆ 105 ~7 ˆ 105 and the corresponding accuracy is a few of hundredths degree per hour in a temperature range of ´40 ˝ C to 50 ˝ C, approaching inertial grade. However, no experimental results have been reported on monolithic cylindrical fused silica resonators up to now. For the first time, we report a 25 mm-diameter monolithic cylindrical fused silica resonator with Q factor approaching ~8 ˆ 105 at a center frequency of 3974.35 Hz (the decay time is more than one minute) in moderate vacuum. In the following sections, the structure design of the resonators is first briefly presented. Then our non-contact experimental method to measure the Q factor of these resonators is demonstrated. Next the post-fabrication chemical and thermal treatment is presented and the Q improvement of six resonators is presented. Primary energy loss mechanism is analyzed and the corresponding solution is proposed. 2. Methods 2.1. Resonator Structure The structure of the monolithic cylindrical fused silica resonator is shown in Figure 1a. The resonator comprises a cylindrical resonant shell, a vibration-conducting shell, a bottom plate with eight holes equi-angularly distributed on, and a rigid stem on the bottom plate inside the cylindrical shell. The cylindrical resonator works at the n = 2 wineglass mode, with n referring to the number of circumferential node lines with respect to the axis of symmetry. For an ideal cylindrical resonator, the n = 2 wineglass mode actually includes two degenerate modes with the same natural frequencies and an angular interval of 45˝ , as depicted in Figure 1b, and they do not have a determined angular location around the axis of symmetry. These modes are often referred to as primary and secondary mode for excitation and detection mode, respectively. The finite element model was built using Ansys Multiphysics 14.0 software, as depicted in Figure 2a, and the simulated n = 2 wineglass mode was shown in Figure 2b. The cylindrical shell is the main vibrating body of the resonator, thus it demands extremely high machining precision. The vibration-conducting shell is considerably thinner than the cylindrical shell and the requirement on its machining precision is less stringent. The holes on the bottom plate ensure that the excitation and detection elements excite and detect the vibration mode of the resonator instead of membrane vibration of the bottom plate [19]. Furthermore, they may block the heat flow of the bottom plate while it deforms, thus decreasing the overall thermoelastic damping [20]. The rigid stem is used to fix the resonator, usually by gluing with epoxy or by indium bonding. In this

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design, the resonator can be excited and detected either by piezoelectric effect or by electrostatic force depending the electrode design. Sensors 2016, 16,on 1185 3 of 14 Sensors 2016, 16, 1185

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(a)(a)

(b) (b)

Figure 1.1. Schematic of: (a) the fused silica resonator structure; Figure 1. Schematic (a) themonolithic monolithiccylindrical cylindrical resonator structure; (b) (b) the nthe = n2n==22 Figure Schematic of:of: (a) the monolithic cylindricalfused fusedsilica silica resonator structure; (b)the wineglass mode of an ideal cylindrical resonator. wineglass mode idealcylindrical cylindricalresonator. resonator. wineglass mode of of anan ideal

(a)

(a)

(b)

(b)

Figure 2. The finite element model and n = 2 wineglass mode: (a) The sectional view of the finite

Figure 2. The finite element model and n = 2 wineglass mode: (a) The sectional view of the finite element model of the monolithic fused resonator withsectional Ansys Multiphysics 14.0; Figure 2. The finite element model andsilica n = cylindrical 2 wineglass mode: built (a) The view of the finite element model of the monolithic fused silica cylindrical resonator built with Ansys Multiphysics 14.0; (b) The simulated = 2 wineglass mode of the resonator.resonator built with Ansys Multiphysics 14.0; element model of the nmonolithic fused silica cylindrical (b) The simulated n = 2 wineglass mode of the resonator. (b) The simulated n = 2 wineglass mode of the resonator. It should be pointed out that, our design was inspired by the structure first proposed by It shouldetbe pointed out that, our we design was inspired by the structure first proposed by Chikovani [6,7,19], based on which fabricated the monolithic silica resonator, which by It should beal.pointed out that, our design was inspired by thefused structure first proposed Chikovani et al. [6,7,19], based on which we fabricated the monolithic fused silica resonator, which is different from the work reported by Xiang et al. [13,14,21], in which they investigated resonators is Chikovani et al. [6,7,19], based on which we fabricated the monolithic fused silica resonator, which made from of metallic alloys and resonators with fused silica shells gluedthey ontoinvestigated metallic disks, which willmade different the work reported by Xiang et al. [13,14,21], in which resonators is different from the work reported by Xiang et al. [13,14,21], in which they investigated resonators hardly result highresonators Q resonators forfused high-accuracy applications because of the material used of metallic alloysinand with silica shells glued onto metallic disks, which willand hardly made of metallic alloys and resonators with fused silica shells glued onto metallic disks, which will gluing procedure. In addition, our stem design was different from reported resonators for the result in high Q resonators for high-accuracy applications because of the material used and gluing hardly result in high Q precision resonators for high-accuracy applications because of the material used and simplicity better manufacturing. procedure. Inand addition, our stemofdesign was different from reported resonators for the simplicity and

gluing procedure. In addition, our stem design was different from reported resonators for the better precision of manufacturing. simplicity and better precision of manufacturing. 2.2. Manufacturing Process

2.2. Manufacturing A series ofProcess cylindrical fused silica (Suprasil-311) shell resonators of varied sizes were 2.2. Manufacturing Process manufactured. The manufacturing process was typical deep grinding consisting of four steps. First, A series of cylindrical fused silica (Suprasil-311) shell resonators of varied sizes were manufactured. the was roughly a preliminary cylinder and eight holes were formed, A fused seriessilica of block cylindrical fusedground silica to(Suprasil-311) shell resonators of varied sizes as were The manufacturing process was typical deep grinding consisting of four steps. First, the fused silica shown in Figure Then the inner and outer was finely ground,consisting as shown in 3b. The manufactured. The 3a. manufacturing process wassurface typical deep grinding ofFigure four steps. First, block roughly ground to a preliminary cylinder and eight holes were formed, as shown in Figure 3a. lastwas step wasblock to grind theroughly outer surface and to theabottom plate tocylinder the designed shown in Figure 3c. as the fused silica was ground preliminary and size, eightasholes were formed, Then the inner and outer surface was finely ground, as shown in Figure 3b. The last step was As the shell is very thin, a cylinder frock was used to avoid cracking. In this paper, six resonators to shown in Figure 3a. Then the inner and outer surface was finely ground, as shown in Figure 3b. The grind theconsidered. outer surface the bottom plateoftothe thesix designed size,were as shown in Figure As the were Theand overall dimensions resonators measured by a 3c. Werth last step was to grind the outer surface and the bottom plate to the designed size, as shown in Figure 3c. shell is very thin, a cylinder frock was used to avoid cracking. thisthe paper, sixthickness resonators Messtechnik GmbH (Giessen, Germany) measurement microscope,Inand bottom waswere As the shell is very thin, a cylinder frock was used to avoid cracking. In this paper, six resonators measured The by a micrometer. The resonator radius R is 12.5 mm and the diameterby ofathe bottom hole is considered. overall dimensions of the six resonators were measured Werth Messtechnik were3.5considered. The overall dimensions of resonators were measured by a Werth for resonator CR01 and CR02 and 5.5 the mm six forand the the restbottom resonators. The main GmbHmm (Giessen, Germany) measurement microscope, thickness was structural measured by Messtechnik GmbH (Giessen, Germany) measurement and the bottom thickness parameters wereresonator listed in Table 1, R where H,mm h and lmicroscope, stand for the andhole length of mm thewas a micrometer. The radius is 12.5 andL,the diameter ofthickness the bottom is 3.5 for measured byshell a micrometer. The resonator radius R respectively; is 12.5 mm and theddiameter of thickness the bottom hole is resonant and the vibration-conducting shell, h b and denote the of the resonator CR01 and CR02 and 5.5 mm for the rest resonators. The main structural parameters were 3.5 mm forplate resonator andof CR02 and 5.5 mm for the rest resonators. The of main structural bottom and theCR01 the bottom respectively; denotes the diameter the bottom listed in Table 1, wherediameter H, h and L, l stand hole, for the thicknessdand length of the resonant shell and hole. Due to our special support design, H, thehresonator be for finished with only and one length clamping, parameters were listed in Table 1, where and L, l can stand the thickness of the the vibration-conducting shell, respectively; hb and d denote the thickness of the bottom plate and resulting highthe manufacturing accuracy especially regarding cylinder roundness resonant resonant shellinand vibration-conducting shell, respectively; hb andrim d denote the and thickness of the the diameter of the bottom hole, respectively; d denotes the diameter of the bottom hole. Due to shellplate concentricity, which were within 3 μmhole, and 1respectively; μm, respectively. As a result these as-fabricated bottom and the diameter of the bottom d denotes the diameter of the bottom our special support design, the resonator can finished 0.031 with only one clamping, resulting in high havespecial small natural splitbe (typically without any special hole.resonators Due to our supportfrequency design, the resonator can beHz–0.555 finishedHz with only one clamping, resulting in high manufacturing accuracy especially regarding cylinder rim roundness and resonant shell concentricity, which were within 3 μm and 1 μm, respectively. As a result these as-fabricated resonators have small natural frequency split (typically 0.031 Hz–0.555 Hz without any special

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manufacturing accuracy especially regarding cylinder rim roundness and resonant shell concentricity, 4 of 14 which were within 3 µm and 1 µm, respectively. As a result these as-fabricated resonators have small natural frequency split which (typically 0.031crucially Hz–0.555important Hz without balancing which balancing procedure), is also toany thespecial performance of procedure), the gyroscope in is also crucially important to the performance of the gyroscope in addition to a high Q factor [22]. addition to a high Q factor [22]. Sensors 2016, 16, 1185

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balancing procedure), which is also crucially important to the performance of the gyroscope in addition to a high Q factor [22].

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(b)

(c)

(d)

Figure Figure 3. 3. Schematic Schematicof ofthe themanufacturing manufacturingprocess processof ofthe themonolithic monolithiccylindrical cylindricalfused fusedsilica silicaresonator: resonator: (a) (a) The The frock frock was was roughly roughly ground ground and and holes holes were were formed formed during during the the first first step; step; (b) (b) The The inner inner surface surface was finely finished during the second step; (c) The outer surface and the bottom plate were finished was finely finished (a) during the second (b) step; (c) The outer (c) surface and the bottom (d) plate were finished during during the the last last step; step; (d) (d) The The photo photo of of monolithic monolithic cylindrical cylindrical fused fused silica silica resonators. resonators. Figure 3. Schematic of the manufacturing process of the monolithic cylindrical fused silica resonator: (a) The frock was roughly ground and holes were formed during the first step; (b) The inner surface Table 1. geometric sizes of the considered resonators. Tableduring 1. Main Main geometric sizes thesurface considered was finely finished the second step; (c) The of outer and the resonators. bottom plate were finished during the last step; (d) The photo of monolithic cylindrical fused silica resonators.

Devices H L h l hb Devices H L h l hb Table 1. Main geometric sizes of the considered resonators. CR01 1.1730 9.8364 0.2643 8.1657 0.495 CR01 1.1730 9.8364 0.2643 8.1657 0.495 b 0.559d Devices H L h l h CR02 1.1730 9.8073 0.2762 8.2266 CR02 1.1730 9.8073 0.2762 8.2266 0.559 1.1730 9.8364 0.3405 0.2643 8.1657 3.45 CR03 0.6620 9.8067 8.0389 CR03 CR01 0.6620 9.8067 0.3405 8.03890.4950.950 0.950 CR02 0.2762 8.2266 0.5591.1055.43 CR04 0.7924 1.1730 9.9008 9.8073 0.3025 8.0124 CR04 0.7924 9.9008 0.3025 8.0124 1.105 CR03 0.3405 8.0389 0.9501.1005.52 CR05 0.8970 0.6620 10.28489.8067 0.3035 8.0401 CR05 CR04 0.8970 10.2848 0.3035 8.04011.1050.625 1.100 CR06 0.9965 10.0610 8.6200 0.7924 9.9008 0.2995 0.3025 8.0124 5.51 CR06 CR05 0.9965 0.8970 10.0610 10.2848

0.2995 0.3035

8.62001.1000.625 8.0401 5.53

d

d 3.45 3.45 5.43 5.43 5.52 5.52 5.51 5.51 5.53 5.53 5.53 5.53

CR06 0.9965 10.0610 0.2995 8.6200 0.625 5.53 2.3. Testing Apparatus 2.3. Testing Apparatus 2.3. Testing Apparatus The experimental setup is shown in Figure 4, which consists of our designed vacuum chamber and The experimental setup(Polytec, is shown in Figure consists of our designed vacuum The experimental setup is shown in Figure 4, which which consists of our designed vacuum chamber a laser Doppler vibrometer Irvine, CA, 4, USA). They were placed on an optical tablechamber to avoid and a laser Doppler vibrometer (Polytec, Irvine, CA, USA). They were placed on an optical tableand to and a laser Doppler vibrometer (Polytec, Irvine, CA, USA). They were placed on an optical table environmental vibrations. The rigid stem of the resonator is clamped with our designed to fixture avoid environmental vibrations. The rigid stem of the resonator is clamped with our designed fixture avoid environmental vibrations. The rigid stem ofepoxy the resonator is clamped our designed fixture together they are mounted on a rotary table with glue (EPO-TEK 301,with EPOXY TECHNOLOGY, and together they are mounted on a rotary table with epoxy glue (EPO-TEK 301, EPOXY and together they are mounted on a rotary table with epoxy glue (EPO-TEK 301, EPOXY Billerica, MA, USA). The resonator is excited by anis excited acoustic source itsand vibration detected by TECHNOLOGY, Billerica, MA, USA). The resonator by an acousticand source its vibration TECHNOLOGY, USA). The resonator is excited by an acoustic source and its vibration detected Billerica, byvibrometer, the laserMA, Doppler vibrometer, which enables non-contact characterization. Theresonator resonator the laser Doppler which enables non-contact characterization. The and the detected by the laser Doppler vibrometer, which enables non-contact characterization. The resonator and the acoustic source are placed in the vacuum chamber, where the air damping can be greatly acoustic source are placed in the vacuum chamber, where the air damping can be greatly reduced, reduced,source and the optical window on vacuum the vacuum chamberwhere allows the laserair Doppler vibrometry and placed in the chamber, damping can be greatly and the the acoustic optical window are on the vacuum chamber allows laser Doppler vibrometry characterization. characterization. This method measures the actual performance of the resonator itself, eliminating reduced, andmeasures the optical window on the vacuum chamberitself, allows laser Doppler vibrometry This method theand actual of the resonator eliminating electrical electrical damping otherperformance damping that may occur when driving and sensing elements were damping characterization. This method measures the actual performance of the resonator itself, eliminating attached onthat the resonator. and other damping may occur when driving and sensing elements were attached on the resonator. electrical damping and other damping that may occur when driving and sensing elements were attached on the resonator.

Figure 4. The experimental setup includes a Polytec laser Doppler vibrometer, a vacuum a chamber, an chamber, Figure 4. The experimental setup includes Polytec laser Doppler vibrometer, vacuum a rotary table. This This setup allows non-contact characterization of the resonator. an acousticacoustic sourcesource and and a rotary table. setup allows non-contact characterization of the resonator.

Figure 4. The experimental setup includes a Polytec laser Doppler vibrometer, a vacuum chamber, an acoustic source and a rotary table. This setup allows non-contact characterization of the resonator.

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It is known that, in practice, due to material anisotropy and manufacturing errors, the resonator has a natural frequency mismatch and a pair of principle axes of vibration. Moreover, in practice the Q factor of the resonator is not circumferentially homogeneous. The change of the excitation direction will further lead to the variation of the Q factor. However, with repetitive experiments we find that, the Q factor of the antinode is relatively stable. For example, to observe the stability of the antinode Q factor, we measured the Q factor variation by changing the excitation direction and measure the antinode Q factor of one resonator. The average value, standard derivation and relative variation were calculated, and the results are listed in Table 2. Table 2. The antinode Q factors of the resonator while changing the excitation angle. Average

9965.71

9728.33

STD

46.89

174.77

Variation

Angle (˝ )

Q Factor

0.47%

0 1 2 3 4 5 6

9896 9951 9921 10,024 10,001 10,004 9963

1.8%

0 5 10 15 20 25

9896 9951 9921 10,024 10,001 10,004

Results show that, a 6-degree change of the excitation direction leads to only 0.5% variation of the antinode Q factor, and a 25-degree change of the excitation direction results in no more than 2% variation of the antinode Q factor. Therefore, the Q factor testing procedure is carefully designed as follows. First the resonant frequency fr of the resonator is sought out by exciting the resonator with a sweeping signal in a relatively wide frequency range. Then the resonator is excited by a sinusoidal signal with its resonant frequency, during which the vibration pattern is recorded and the antinode is determined, as shown in Figure 5a. Then the resonator is again excited by a sweeping signal in a narrower frequency range, so as to have a better measurement resolution, and the vibration at the antinode is recorded. The Q factor of the resonator is then calculated by the Polytec software according to: Q“

∆f fr

(1)

where ∆f is the 3-dB bandwidth of the frequency response of the resonator, as shown in Figure 5b. The frequency resolution selected is 0.012 Hz. For much higher Q, the resonant characteristics are measured using the ring-down method, i.e., the shell is attenuated at its resonant frequency and the time constant τ for the vibration amplitude to decay by 1/e is measured. The resonant frequency is determined the same as in the frequency response method, and the sinusoidal signal with the resonant frequency is used to excite the resonator. When the wineglass mode vibration of the resonator is stable, the signal is shut down and the resonator is allowed to freely vibrate with damping. The data is recorded by the laser Doppler vibrometer, an appropriate length of which was acquired and fitted exponentially in MATLAB, as shown in Figure 5c. Q is then calculated by: Q “ π fr τ

(2)

9728.33

174.77

1.8%

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9951

10

9921

15

10,024

20

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25

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Specifically, the adata recorded is typical exponentially damped signal. We of have Results show that, 6-degree change of the excitation direction leadssinusoidal to only 0.5% variation calculated the frequency of the signal by fast Fourier transform, and found the difference between the antinode Q factor, and a 25-degree change of the excitation direction results in no more than 2% ´3 Hz. Therefore we use the calculated frequency and theTherefore, excitation the frequency on the order of 10 variation of the antinode Q factor. Q factorwas testing procedure is carefully designed as the frequency of the excitation signal during frequency asthe a good approximation follows. First the resonant frequency fr ofobtained the resonator is sought outsweeping by exciting resonator with a to calculate Q in the ring-down time method. We use cubic spline interpolation to derive the upper sweeping signal in a relatively wide frequency range. Then the resonator is excited by a sinusoidal envelope ofresonant the signal, and then during fit the envelope with exponential function by least nonlinear signal with its frequency, which the vibration pattern is recorded andsquare the antinode fitting. As shown in the enlarged picture in Figure 5, the decay curve has beating with a period is determined, as shown in Figure 5a. Then the resonator is again excited by a sweeping signal in a of ´ 3 roughlyfrequency 6 ˆ 10 s, whichsoisas caused by athe insufficient sampling frequencyand provided by the FFT mode narrower range, to have better measurement resolution, the vibration at the of the laser Doppler vibrometer. The error caused by this effect should be on the order of 10´3 s. antinode is recorded.

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Figure 5. Cont.

(c) Figure5.5. Exemplary Exemplary steps steps of of the theQQfactor factortest. test. (a) (a) First First the thevibration vibration pattern pattern was wasrecorded recordedand andthe the Figure antinode of n = 2 wineglass mode was selected to be the subsequent testing point; (b) The frequency antinode of n = 2 wineglass mode was selected to be the subsequent testing point; (b) The frequency responseof ofthe themonolithic monolithiccylindrical cylindricalfused fusedsilica silicaresonator resonatorwas wasmeasured measuredand andthe theQQfactor factorofofthe thenn==22 response wineglassmode modewas wascalculated calculatedby bythe thePolytec Polytecsoftware; software;(c) (c)The Thetime timesignal signalwas wasrecorded recordedand andthe thedata data wineglass was fitted in MATLAB to calculate the ring-down time of the n = 2 wineglass mode of the resonator. was fitted in MATLAB to calculate the ring-down time of the n = 2 wineglass mode of the resonator. Theenlarged enlargedpicture pictureshows showsthe thebeating beatingofofthe thesignal signaldue duetotothe thesampling samplingprocess. process. The

Q factor of the resonator is then calculated by the Polytec software according to: 2.4. QThe Factor Improvement

Δf Q is known to be affected by many different damping mechanisms, including surface loss Qsurface , Q= (1) material loss Qmaterial , air damping Qair and anchor floss Qanchor , etc. The total Q is limited by the r greatest damping as in [23]: where Δf is the 3-dB bandwidth of the frequency response of the resonator, as shown in Figure 5b. 1 is 0.012 Hz. 1 For much 1 1 1 The frequency resolution1selected are “ ` ` `higher Q,`the resonant characteristics (3) Q Q Q Q Q Q etc. air sur f ace material anchor measured using the ring-down method, i.e., the shell is attenuated at its resonant frequency and the time constant τ for the vibration amplitude to decay by 1/e is measured. The resonant frequency is The optimization of Q factor requires different damping mechanisms being individually determined the same as in the frequency response method, and the sinusoidal signal with the addressed. While air damping is the most dominant effect at atmospheric conditions, it can be resonant frequency is used to excite the resonator. When the wineglass mode vibration of the easily reduced by operating the resonator in vacuum. The proposed experimental apparatus enables resonator is stable, the signal is shut down and the resonator is allowed to freely vibrate with the measurement of Q in moderate vacuum (the pressure was ~10´2 Pa measured by an ionization damping. The data is recorded by the laser Doppler vibrometer, an appropriate length of which was acquired and fitted exponentially in MATLAB, as shown in Figure 5c. Q is then calculated by: Q=

πfr

(2)

Specifically, the data recorded is typical exponentially damped sinusoidal signal. We have

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gauge near the molecular pump, therefore in the vacuum chamber the pressure is higher, especially when the pump was shut down and there were organic chemicals inside. The pumping time was set to half an hour to ensure the same moderate vacuum conditions). Anchor loss is caused by energy transferring from resonator to substrate though the anchor, and is minimized by balancing the resonator and careful design of the fixture. To ensure the condition of the fixing, a torque wrench was used in the test and the fixing torque was set to 0.05 N¨ m. Surface loss is mainly caused by a defect layer formed during the grinding and polishing of the resonators. The surface of the fused silica resonator after mechanical grinding is subjected to subsurface damage (SSD), which refers to the residual fractured and deformed material in the near surface region. This layer of deformed and fractured material exhibits substantially different mechanical behavior, particularly a significantly higher internal friction. It was previously studied in optical fine machining that even by meticulously optical polishing, there would still be SSD remained. The surface loss can be effectively reduced by chemically removing the surface layer [24–27]. The material loss consists of several loss mechanisms, including thermal elastic damping and microscopic material defects, etc. The upper limit on the attainable quality factor is determined by material and geometric structure of the resonator [20]. However, the Q on the order of 107 is easily attainable for bulk materials of the high purity synthetic fused silica [28], thus in millimeter-diameter range other loss needs to be suppressed first before considering thermoelastic damping. Other material loss is usually minimized by using a high purity and isotropic material with low coefficient of thermal expansion. In addition, annealing can improve the Q factor through the relieving of internal stress and the removing of surface contaminants [29,30]. Post-fabrication annealing has been employed in various MEMS resonators [31,32], improving their Q for more than an order of magnitude, and more recently in micro shell resonators [33,34] whose Qs were also moderately improved. With the above considerations in mind, we performed both chemical and thermal treatments to improve the quality factor of the proposed monolithic cylindrical fused silica resonator. 3. Results 3.1. Chemical Etching The chemical etching procedure proposed in [25,26] was employed, and resonators CR01, CR05 and CR06 were chemically etched. For each etching round, the etching depth was measured by the Werth Messtechnik GmbH measurement microscope and the surface roughness was measured by a Taylor Hobson (Berwyn, PA, USA) profiler. Resonator CR01 was first etched in a solution of 10:1 in volume ratio 40 wt % NH4 F and 49 wt % HF in ambient temperature for two hours followed by three one-hour periods. The resonant frequency and Q factor for the n = 2 mode were measured five times continuously in each round and the relative errors were within 10 ppm and 2%, respectively. The results of the chemical etching process were listed in Table 3 and depicted in Figure 6. The parameters considered were the etching depth, the surface roughness, the resonant frequency and Q factor of the resonator. Results showed that the Q factor improves significantly, especially after the first etching round, when the Q factor improves by 78.7% and the resonant frequency decreased by 2.40%, while the etching depth is 16.9µm. Results also showed that the surface roughness deteriorates with each etching period. The surface roughness of resonator CR01 before and after chemical etching is depicted in Figure 7, which clearly shows the degradation of the resonator surface. The Q drop in the last experimental period suggests that further degradation of the surface roughness or other unknown factors might cause extra damping. According to the results measured in atmospheric condition, it seemed that the optimum etching depth should be around 33.3 µm, where the Q factor of resonator CR01 in air was improved by 92.9%. Later measurements in moderate vacuum suggested otherwise.

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etching depth should be around where thefactor Q factor of resonator CR01 inwas air was improved etching depth should be around 33.333.3 μm,μm, where the Q of resonator CR01 in air improved by 92.9%. Later measurements in moderate vacuum suggested otherwise. by Sensors 92.9%. Later measurements in moderate vacuum suggested otherwise. 2016, 16, 1185 8 of 14 Table 3. The etching depth, change of surface roughness resonant frequency, thefactor Q factor Table 3. The etching depth, change of surface roughness and and resonant frequency, and and the Q improvement of Resonator byof chemical etching. Table 3. The depth, change surface roughness and resonant frequency, and the Q factor improvement of etching Resonator CR01CR01 by chemical etching. improvement of Resonator CR01 by chemical etching. Time (hour) Depth Depth Ra (μm) fr (Hz) fr (Hz) Q factor Q factor Time (hour) (μm)(μm) Ra (μm) 0 0 Time (hour) 0 2 3 4 5

2

2

3

3

4

4

5

5

0 0 Depth (µm) 016.9 16.9 16.9 25.6 25.6 25.6 33.3 33.3 33.3 63.6 63.6 63.6

0.850 0.850 Ra (µm)

5556.1 5556.1 fr (Hz)

24102410 Q factor

1.874 1.874 0.850 1.874 2.0102.010 2.010 2.422 2.422 2.422 2.724 2.724 2.724

5423.0 5423.0 5556.1 5423.0 5401.5 5401.5 5401.5 5378.3 5378.3 5378.3 5353.3 5353.3 5353.3

430643062410 443044304306 4430 465046504650 423442344234

50005000

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Q factor Q factor Ra Ra

1.0 1.0

0.5 0.5 20002000 0 010 1020 2030 3040 4050 5060 6070 70

Etching Depth (m) Etching Depth (m)

Figure The surface roughness and factor change with etching depth resonator CR01. Figure 6. The surface roughness andand Q factor change with etching depth of resonator CR01. Figure 6.6.The surface roughness QQfactor change with etching depth ofofresonator CR01.

6 6 4 4 2 2 0 0 -2 -2 -4 -4

Ra (m) Ra (m) 0

6 6 4 4 2 2 0 0 -2 -2 -4 -4

Ra Ra

Ra (m) Ra (m)

Ra Ra

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4

4

Assessment Length (mm) Assessment Length (mm) (a) (a)

0

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Assessment Length (mm) Assessment Length (mm) (b) (b)

Figure roughness ofofCR01. (a) The surface roughness before chemical etching; (b) The Figure 7. Surface roughness of CR01. (a) The surface roughness plot plot before chemical etching; (b)etching; The Figure 7.7. Surface Surface roughness CR01. (a) The surface roughness plot before chemical surface roughness plot after chemical etching. surface roughness plot after chemical (b) The surface roughness plot afteretching. chemical etching.

To reduce air damping, resonator CR01 tested in moderate vacuum. To reduce the the air damping, the the resonator CR01 waswas thenthen tested in moderate vacuum. TheThe To reduce the air damping, the resonator CR01 was then tested in moderate vacuum. The ring-down signal exponential fit result plotted in Figure 8. The constant ring-down timetime signal andand the the exponential fit result werewere plotted in Figure 8. The timetime constant of of ring-down time signal and the exponential fit result were plotted in Figure 8. The time constant the resonator is 26.4745 s and the corresponding Q factor is 445,245, which 104.2 times greater the resonator is 26.4745 s and the corresponding Q factor is 445,245, which 104.2 times greater thanthan of the resonator is 26.4745 s and the corresponding Q factor is 445,245, which 104.2 times greater in Later air. Later experiments revealed thefactors Q factors of resonators the resonators in moderate vacuum before thatthat in air. experiments revealed thatthat the Q of the in moderate vacuum before than that in air. Later experiments revealed that the Q factors of the resonators in moderate vacuum 4, therefore the Q improvement here is mainly due to chemical etching were within chemical etching were within ~2 ×~210×4, 10 therefore the Q improvement here is mainly due to before chemical etching were within ~2 ˆ 104 , therefore the Q improvement here is mainly due to chemical etching. chemical etching. chemical etching.

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x 10 -4 1.5 x 10 1.5 1

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1

0.5 0.5 0

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τ=26.4745s τ=26.4745s

-1

-1.5 -1.5 15 15

20 20

25 25

30 30

Time (s)(s) Time

35 35

40 40

Figure 8. Ring-down time experiment of resonator CR01 in moderate vacuum shows τ =τ 26.4745 s at vacuum Figure 8. Ring-down time experiment of resonator CR01 in moderate vacuum shows τ = 26.4745 s at f =f 5353.29 Hz, giving a Q factor of 445,245. ==5353.29 5353.29 Hz, Hz, giving giving aa Q Q factor factor of of 445,245. 445,245.

3.2. 3.2.Annealing Annealing 3.2. Annealing We Weperformed performedannealing annealingonon onresonators resonatorsCR02, CR02,CR03 CR03and andCR04, CR04,which whichwere wereannealed annealedatat at1050 1050°C˝°C We performed annealing resonators CR02, CR03 and CR04, which were annealed 1050 C −2 for ananelectric vacuum furnace. All three resonators were annealed inin~10 vacuum −2Pa ´2 Pa for1212 12hhhin electricvacuum vacuumfurnace. furnace.All All three resonators were annealed ~10 Pa vacuum for ininan electric three resonators were annealed in ~10 vacuum to totoprotect the surface from being and surface defect during protect theresonator resonator fromcontaminated beingcontaminated contaminated andavoid avoid during protect the resonator surfacesurface from being and avoid surface defectsurface during defect annealing [30]. annealing [30]. The resonators were heated at a rate of 5 °C/min, then the furnace temperature was ˝ annealing [30]. The resonators were heated at a rate of 5 °C/min, then the furnace temperature was The resonators were heated at a rate of 5 C/min, then the furnace temperature was preserved at preserved atat1050 °C°Cfor the were cooled toto800 ofof1 1°C°C/min ˝ C for ˝ C at preserved 1050 for12 12h.two h.Then Then thetwo tworesonators resonators cooled 800°Cof °Cat1ata˝ C arate rate /mintono to 1050 12 h. Then the resonators were cooledwere to 800 a rate /min to ensure ensure nonoextra internal induced, bybyfree cooling totoroom ensure extra internalthermo-stress thermo-stress induced,followed followed free cooling roomtemperature. temperature. extra internal thermo-stress induced, followed by free cooling to room temperature. Q factors for n = 2 mode were tested and the results are listed in Table 4. All three of the Table ofofQQfactors mode of CR02, CR03 and before Table4.experienced 4.The Thecomparison comparison factorsand modefrequency frequency CR03 andCR04 CR04 beforeand and of Q. resonators a small increase inand resonant frequencyofasCR02, well as a notable improvement after annealing. after annealing. Their resonant frequencies were increased by 0.16%, 0.28% and 0.18%, respectively, while their Q factors Annealing After Annealing were increased by 88.1%, Before 51.2% and 61.4%, respectively. To reduce the airVariation damping, Before Annealing After Annealing Variation the resonator Device Device fr (Hz) Q factor fr (Hz) Q factor fr (Hz) QQ factor CR04 was then tested in moderate time signal and the exponential fr (Hz) vacuum. Q factorThe ring-down fr (Hz) Q factor fr (Hz) factor fit result CR02 5556.4 1896 5565.2 3567 0.16% 88.1% 5556.4 1896 Q factor 5565.2 3567 0.16% 3.8 88.1% were plotted inCR02 Figure 9. The corresponding was 37,015, which increases times than that in CR03 3456.4 4339 3466.2 6562 0.28% 51.2% 3456.4 4339 3466.2 0.28% air. ComparedCR03 with the Q improvement by chemical etching, 6562 which is 104.2 times, 51.2% the improvement CR04 4013.9 4784 4021.3 7725 0.18% 61.4% CR04 4013.9 4784 4021.3 7725 0.18% 61.4% brought by annealing is relatively small. QQfactors for were tested and the results listed ininTable 4.4.All three ofofthe factors forncomparison n= =2 2mode mode were tested resultsare areCR02, listed Table All three the Table 4. The of Q factors and and modethe frequency of CR03 and CR04 before and resonators experienced resonators experienceda asmall smallincrease increaseininresonant resonantfrequency frequencyasaswell wellasasa anotable notableimprovement improvementofof after annealing. Q.Q.Their Theirresonant resonantfrequencies frequencieswere wereincreased increasedbyby0.16%, 0.16%,0.28% 0.28%and and0.18%, 0.18%,respectively, respectively,while whiletheir theirQQ factors byby88.1%, 51.2% the Annealing Afterrespectively. Annealing factorswere wereincreased increasedBefore 88.1%, 51.2%and and61.4%, 61.4%, respectively.ToToreduce reduceVariation theairairdamping, damping,the the Device resonator CR04 was then tested in moderate vacuum. The ring-down time signal and the exponential resonator CR04 was then tested inQmoderate vacuum. time signal andQthe exponential fr (Hz) factor fr (Hz) The ring-down Q factor fr (Hz) factor fitfitresult plotted ininFigure corresponding QQfactor was which 3.8 times resultwere were Figure9.9.The The factor was37,015, 37,015, whichincreases increases CR02 plotted5556.4 1896 corresponding 5565.2 3567 0.16% 88.1% 3.8 times than with the QQimprovement by chemical etching, which times, the thanthat thatinCR03 inair. air.Compared Compared whichisis104.2 104.2 3456.4 with the 4339 improvement 3466.2 by chemical 6562 etching, 0.28% 51.2% times, the improvement brought by annealing is relatively small. CR04brought 4013.9 4784 4021.3 7725 0.18% 61.4% improvement by annealing is relatively small. -4

Velocity (mm/s) Velocity (mm/s)

x 10 -4 1 1 x 10

Data Data FitFit

0.50.5 00

-0.5 -0.5

τ=2.930s τ=2.930s

-1-1 2626 2828 3030 3232 3434 3636 Time (s)(s) Time Figure 9. Ring-down time experiment of resonator CR04 in moderate vacuum shows τ τ= 2.930 s at Figure 9. Ring-down time experiment of resonator CR04 in moderate vacuum shows τ = 2.930 s at f = 4031.02 Hz, giving a Q factor of 37,105. f ==4031.02 4031.02 Hz, Hz, giving giving aa Q Q factor factor of of 37,105. 37,105.

Enlightened Enlightenedbybyearlier earlierresults, results,we wefocused focusedour ourefforts effortsononchemical chemicaletching. etching.Resonators ResonatorsCR05 CR05and and CR06 were etched in a slightly different recipe with heating to accelerate the etching process. CR06 were etched in a slightly different recipe with heating to accelerate the etching process.Before Before

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Enlightened by earlier results, we focused our efforts on chemical etching. Resonators CR05 and CR06 were etched in a slightly different recipe with heating to accelerate the etching process. Before and after etching each resonator was tested in both atmospheric conditions and in moderate vacuum. The results were listed in Table 5, where fr and Q represent resonant frequency and Q factor in air while fr ’ and Q’ represent resonant frequency and Q factor in vacuum. The etching depths of resonator CR05 and CR06 were 91µm and 106 µm, respectively. Their Qs in air were 7647 and 9115, respectively, improved by 93.1% and 99.1% compared with initial Q factor in air. Their Q factors in vacuum were ~ 7 ˆ 105 and ~8 ˆ 105 , respectively, improvement of 37.8% and 47.7% compared10with Sensors 2016, 16, 1185 of 14 the initial Q factor in vacuum. The testing results in vacuum showed that, in contrast to earlier results, the etching is likely more than 100atmospheric µm. The ring-down and the fitting andoptimum after etching eachdepth resonator was tested in both conditionstime and signal in moderate vacuum. 5 at results were were depicted in in Figure CR06 show a Q factor approaching ~8 ˆ 10 The results listed Table10. 5, Notably, where fr resonator and Q represent resonant frequency and Q factor in air the center frequency of 3974.35Hz, which is more than thatfactor predicted by Innalabs while fr’ and Q’ represent resonant frequency and Q in vacuum. The [7]. etching depths of

resonator CR05 and CR06 were 91μm and 106 μm, respectively. Their Qs in air were 7647 and 9115, Table 5. Q factors of CR05 and CR06 before and after etching both in air and in moderate vacuum. respectively, improved by 93.1% and 99.1% compared with initial Q factor in air. Their Q factors in vacuum were ~7 × 105 and ~8 × 105, respectively, improvement of 37.8% and 47.7% compared with After Etching Before Etching Depth theDevice initial Q factor in vacuum. The testing results in vacuum showed that, in contrast to earlier results, (µm) (Hz) Q is likely fr’ (Hz)more Q’ fr ring-down (Hz) Qtime signal fr’ (Hz) Q’fitting the optimumfretching depth than 100 μm. The and the CR05 4814.7 4821.2 18,147 91CR06 4442.2 7647 approaching 4453.5 703,318 results were depicted in3960 Figure 10. Notably, resonator show a Q factor ~8 × 105 at CR06 4459.7 4577 4471.3 16,320 106 3964.1 9115 3974.5 794,148 the center frequency of 3974.35Hz, which is more than that predicted by Innalabs [7]. x 10

-4

Data Fit

0.5 0 -0.5 -1

τ=51.4742s 60

80

100

Time (s) (a)

4

Velocity (mm/s)

Velocity (mm/s)

1

-5

Data Fit

2 0 -2 -4

120

x 10

τ=64.529s 40

60

80

Time (s) (b)

100

120

Figure 10. 10.Ring-down Ring-down experiment of resonator CR05 CR06 in moderate vacuum. Figure timetime experiment of resonator CR05 and CR06and in moderate vacuum. (a) Resonator (a) Resonator CR05 shows τ = 51.4742 s at f = 4453.49 Hz, giving a Q factor of 720,178; (b) Resonator CR05 shows τ = 51.4742 s at f = 4453.49 Hz, giving a Q factor of 720,178; (b) Resonator CR06 shows shows s atHz, f = giving 3974.35a Hz, giving Q factor of 805,898. τCR06 = 64.529 s atτ f==64.529 3974.35 Q factor of a805,898. Table 5. Q factors of CR05 and CR06 before and after etching both in air and in moderate vacuum.

4. Discussion

Before Etching

Depth

After Etching

Results Device with chemical etching showed that our resonators experience dramatic Q factor (μm) fr (Hz) fr‘ (Hz) fr (Hz) fr‘ (Hz) Q Q’ Q Q’ improvements after surface treatment. This confirms that surface loss is indeed important, as suggested CR05 4814.7 3960 4821.2 18,147 91 4442.2 7647 4453.5 703,318 by Lunin [35], in macroscopic fused silica cylindrical resonators fabricated by grinding, in contrast CR06 4459.7 4577 4471.3 16,320 106 3964.1 9115 3974.5 794,148 with previous studies on bulk materials, which suggest that surface loss is not so significant [28,36]. This is likely due to the fact that our resonators have a large surface-to-volume ratio and their surfaces 4. Discussion are mostly ground curve surfaces. In addition, in contrast with the case with micro resonators [37], Results with chemical etching showed our resonators experiencethat dramatic Q factor the surface roughness alone does not have a clearthat relationship with Q, suggesting the micro-cracks improvements after surface treatment. This confirms that surface loss is indeed important, as which cannot be revealed by surface roughness are most likely the main source of surface loss. suggested the by improvement Lunin [Error! of Reference not found.], in ismacroscopic fused silica cylindrical Therefore Q factorssource by chemical etching likely due to the removing of the resonators fabricated by grinding, in contrast with previous studies on bulk materials, which suggest hydrated and contaminated surface layer suggested in [27]. that We compared surface is et al. [35] using notthe simple model so our results with thoseloss published by Lunin significant [28,36]. This is likely due to the fact that our resonators have a large surface-to-volume proposed in [28]. We used the same nomenclature as in [28], denoting total surface area as St , curved ratio and their are mostly ground curve surfaces. In addition, inratio contrast with thecylinder case with surface area as surfaces Sc , and total volume as V. The total-surface-to-volume of the bulk is micro resonators [37], the surface roughness alone does not have a clear relationship with Q, suggesting that the micro-cracks which cannot be revealed by surface roughness are most likely the main source of surface loss. Therefore the improvement of Q factors by chemical etching is likely due to the removing of the hydrated and contaminated surface layer suggested in [27]. We compared our results with those published by Lunin et al. [Error! Reference source not

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St /V~0.22 mm´1 , and the curved-surface to volume ratio is Sc /V~0.20 mm´1 . For our resonator, they are calculated as follows: Sc St „ 2.92 mm´1 , „ 2.46 mm´1 (4) V V From Equation (3), our cylindrical resonator has approximately 13.2 times more total surface and 12.3 times more curved surface relative to the bulk cylinder. Therefore, we would anticipate a maximum Q of 2.27 ˆ 105 before etching for our resonator, or 2.43 ˆ 105 if considering only the curved surface, and one would expect a maximum Q of 1.51 ˆ 106 after etching, or 1.62 ˆ 106 if considering only the curved surface. The fact that the Q factors of our resonators were below this value is likely due to the residual air damping and possible anchor loss. In addition, the mechanism of Q factor improvement due to annealing is commonly considered related with internal stress relieving [24,33], surface quality improving and material removing [16,33], etc. Various studies showed that annealing improves the Q factors of microresonators significantly [16,33,34], and also moderately improves Q factors of bulk fused silica material [30,36]. However, it does not, at least in our resonator structure, contribute to a marked Q factor improvement. For our specific resonator, Q factor improvement by chemical etching is one order of magnitude higher than that by annealing, which suggested the surface loss is the predominant loss after the reduction of air damping. Further studies are still necessary on understanding the mechanism of surface loss. It should be noted that, by “monolithic” we mean that our resonators are one-piece fused silica resonators, contrary to metallic cylindrical resonators or combined fused silica resonators as in literature [6,7,13,14]. We also acknowledge the fact that miniaturization is the future trend; however, our target at this stage is a macroscopic high-accuracy gyroscope for high-precision inertial applications. Although chemical and thermal treatments are widely used in literature to improve the resonator Q factor [25–27,30–35,37,38], no one has specially studied the effects of chemical and thermal treatments on monolithic cylindrical fused silica resonators. For example, in ref. [25], the distribution of subsurface damage in fused silica was studied systematically, specifically on the depth and morphology of subsurface fractures, aiming at correlating subsurface damage with abrasive size and load of the grinding process. However, the relationship between mechanical Q factor and the distribution of subsurface damage was not mentioned. In ref. [38], chemical etching is used to shape the LiNbQ3 disk resonator, while thermal treatment is used to improve the surface quality. However, the material, structure and application of the resonator are totally different from ours. Actually, the optimized process and effects of chemical and thermal treatments depend on the material and structural parameters of specific resonators, which is why there are many studies on chemical and thermal treatments to reduce the mechanical energy loss. We believe that, for our proposed resonator, it is of practical significance to study the suitable treatment procedures accordingly. As various literatures have indicated, with different materials and manufacturing processes, the effects of chemical and thermal treatments might be different [33,34,37] and optimum procedures need to be developed, in which we shall devote more effort. In conclusion, for our monolithic cylindrical fused silica resonators, the air damping and the surface loss are the two major losses in atmospheric conditions. The air damping limits Q factors to be below ~1 ˆ 104 even after the significant decreasing of surface loss by chemical etching. Therefore in practice resonators must work under appropriate pressure conditions. After the reduction of air damping by measuring the resonator in moderate vacuum, other losses limit Q to be within ~ 2 ˆ 104 . After the reduction of internal loss and air damping the remaining losses limit Q to be around ~ 4 ˆ 104 . After the reduction of surface loss and air damping, the Q factor approaches ~8 ˆ 105 . Therefore, with standard fabrication process and simple post-fabrication chemical treatment, the resonator Q is more than sufficient for medium accuracy gyroscope applications. However, our test apparatus limits the testing pressure to allow the energy transfer from acoustic source to the resonator. The real Q might still be higher than the present results suggest, therefore our future work shall include developing suitable excitation methods to excite resonators under higher

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vacuum. For even higher Q, functional anchor structure must be carefully designed to reduce the anchor loss. Furthermore, the Q factor may be further improved by understanding the mechanism of surface loss and internal-stress related loss, and by carefully designing combined procedure of chemical and thermal treatment. When all other losses are minimized, thermoelastic damping associated with geometric structure needs to be considered. We are also developing gyroscopes based on the proposed resonators with piezoelectric and electrostatic transducing. The sensibility and stability characteristics will be presented in the future. 5. Conclusions In this paper, the structure of the resonator is first briefly introduced, which is followed by the demonstration of the post-fabrication experimental characterization of the resonator Q factor. The experimental Q improvement is presented, including both chemical and thermal treatments. Results showed that the Q factor can be improved by more than an order of magnitude through chemical etching alone, while the Q improvement by annealing is small compared with chemical etching. Therefore for our monolithic cylindrical fused silica resonators, the surface loss is the major loss after the reduction of the air damping. In addition, we report the first 25-mm diameter monolithic cylindrical fused silica resonator with a Q factor approaching ~8 ˆ 105 at the frequency of 3974.35 Hz. The high Q factor is enabled by fused silica material with low internal friction and high isotropy, as well as a simple chemical etching process. To the best of our knowledge, the Q factor presented in this paper is the highest reported Q factor for a cylindrical resonator at 25 mm diameter. The monolithic cylindrical fused silica resonator may enable precision inertial sensing applications with standard manufacturing process and simple post-fabrication chemical treatment. Acknowledgments: This work was supported by the National Science Foundation of China (Grant No. 11304384, 61575220). Author Contributions: Y.P. and T.Q. conceived and devised the simulation and experiments, performed the measurements, analyzed and modeled the data, and prepared the manuscript. D.W. performed independent experiments and Y.W. assisted the experiments, both of whom helped with improvements of the measurement procedure. J.P.L. fabricated the resonator. S.W. directed the experiments and offered valuable advices on the overall improvement of the resonator. K.Y. and H.L. oversaw the research, provided advices and guidance and discussed the results at all stages of the research. All authors discussed the results and contributed to the refinement of the paper. Conflicts of Interest: The authors declare no conflict of interest.

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Monolithic Cylindrical Fused Silica Resonators with High Q Factors.

The cylindrical resonator gyroscope (CRG) is a typical Coriolis vibratory gyroscope whose performance is determined by the Q factor and frequency mism...
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