Pressure dependence of resonance characteristics of the microcantilever fabricated from optical fiber

Vacuum/volume

47lnumbers 6-a/pages 475 to 47711996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0042-207X156 $15.00+.00

Pergamon PII: SOO42-207X(96)00146-7

Pressure dependence of resonance characteristics of the microcantilever fabricated from optical fiber H Kumazaki,” S habaa and K Hane,b”Department of Electrical Engineering, Motosu Gifu 501-04, Japan; bDepartment of Mechatronics and Precision Tohoku University Sendai 980, Japan

Gifu National College of Technology Engineering, Faculty of Engineering

A quartz core microcantilever was fabricated from an optical fiber tip and was vibrated photothermally by laser diode light through the optical fiber. The vibration of the cantilever is stable and complicated adjustment of the optical system is not necessary. Pressure dependence of the resonance curve was measured for cantilevers of various sizes, and evaluated using a “string-of-beads” model for the quartz oscillator. Copyright 0 1996 Elsevier Science Ltd. Key words: Optical vacuum gauge.

fiber, quartz core microstructure,

Introduction

Although the capacitance diaphragm gauge is widely used in low and medium vacuum regions, the use of this gauge is hazardous in corrosive or explosive gases, because of differential pressure applied to a thin diaphragm. As an alternative to the diaphragm gauge, an optical vacuum gauge with no electrodes is usable in explosive gas.’ This system can be utilized in a very small space since the control and measurement are performed remotely through a glass window. It can also be used in special environments such as strong magnetic fields. We have previously proposed a vacuum gauge using the pressure dependence of the vibration parameters of a cantilever vibrated photothermally.‘-’ Many mechanical microsensors using photothermal vibration have been proposed including the above-mentioned vacuum gauge.J ’ However. there are few practical examples because delicate adjustment of the optical system is required. Actually. it is difficult to irradiate the microcantilever through the glass window of a vacuum chamber with the near-infrared radiation of a laser diode. Recently. we developed a microcantilever fabricated from a quartz core by etching the clad layer of a fiber tip.h It was photothermally vibrated by the light transmitted through the optical fiber. Complicated adjustments of the optical system were not necessary and resonance frequency was determined with high reproducibility in this system. The quartz material of the cantilever is durable even in corrosive gases which damage capacitance manometers. Although vacuum sensing is possible using the resonance frequency, it is necessary to consider the temperature dependence. In contrast to the measurement using the resonance frequency. using the half-power width of the resonance curve is

photothermal

vibration,

photomechanical

sensor,

convenient because its temperature dependence is small. The resonance curves were measured for cantilevers of various sizes in the vacuum region from IO -‘to IO’ Pa. The effects of cantilever size on pressure sensing are evaluated in this paper using a “string-of-beads” model. Experimental

procedure

A schematic

view of the fabricated cantilever is shown in Figure I. The clad layer on the tip of the optical fiber (multimode SO/l 25 pm graded index type) was removed by chemical etching. The fiber tip was dipped in a solution composed of 50% hydrofluoric acid (HF), 40% ammonium fluoride (NH,F) and water (H,O) for etching. The volume ratio of NHJF: HF: Hz0 was 1.7:1:1.’ The diameter of the cantilever was adjusted by controlling the temperature and time of etching. As a result. the diameter of the

I:

Deposltlonof (‘r + Au

Figure 1. Schematic view of the microcantilecer of thr liber core. The clad layer was etched from the optical fiber and the cantilever was fixed with epoxy resin. 475

H Kumazaki

et al: Microcantilever

I

fabricated

from optical fibre

1

.

Experimental(fiber1) (D=49.0 u m.LGl S5mm) Experimental(fiber2) (D=31.6~ m.L=164mm) L-7 Experimental(fiber3) (D=41.0 u m.J_=3.12mm)

n

A

- - - - - Calculated(D=4l.Ou

to

.

m)

“;ci~~~,’

Figure 2. Block diagram of the experimental system for vacuum sensing.

J

clad increased gradually from the root of the cantilever. Cr and Au were deposited on one side of the core of the fiber tip to absorb the laser light for excitation of the vibration. Finally, the quartz was fixed with epoxy resin at a certain distance from the fiber tip. Figure 2 is a block diagram of the experimental system for vacuum sensing. The light from the laser diode was modulated at a frequency (~2OkHz) by a voltage-frequency converter. The quartz core cantilever was efficiently irradiated through the optical fiber. The light was absorbed at the Cr/Au-deposited side of the cantilever, but not at the other side. Thus, the cantilever was bent photothermally with the same period as the modulation frequency. The vibration was monitored with the light of a HeNe laser reflected from the Au layer through a slit. The light signal was transformed to an electrical signal by the displacement of the reflected light at the entrance of a photomultiplier and was detected using a lock-in amplifier. The resonance curves were obtained on an X-Y recorder to determine the resonance frequency and the half-power width. The cantilever was set in a vacuum chamber which was evacuated by a rotary pump and a diffusion pump. The pressure was measured with a diaphragm gauge and a mercury manometer.

10-l

IO’

lo3

,I..

“J

lo5

Pressure [Pa] Figure 3. Half power width of fibers l-3 as a function of air pressure

of the reciprocal of half-power width Af with the pressure, we discuss Ax Figure 3 shows Aj’of fibers l-3 as a function of air pressure. The experimental values are denoted by symbols. Aj monotonically decreased with decrease in the pressure from lo5 Pa. The following two points should be noted concerning this result. First, the rate of Aj’shift with air pressure differed for each cantilever. Second, the pressure at which Af became constant differed for each cantilever. Lines in Figure 3 show values calculated using the “string-ofbeads” model’ for the quartz oscillator. This model was adopted because it is difficult to obtain a precise solution for the drag force due to the surrounding gas for the cylindrical vibrator. The vibrator is modelled by a series of spheres of the same radius linked like a string of beads. It is assumed that all the spheres harmonically vibrate at the same frequency with different amplitudes. The one-dimensional equation of motion describing the forced vibration with damping in this model is given as

Results and discussion

(m +.fi)X +J;_i- + kx = Acosof,

The resonance curve of this cantilever was stable and reproducible, and the fluctuation of the resonance frequency was within + 1Hz. Resonance characteristics of three cantilevers with different sizes were measured as a function of air pressure. These cantilevers are referred to as fiber 1 (diameter D = 49.0 pm, length L = 1.55mm), fiber 2 (D = 31.6pm, L = 1.64mm) and fiber 3 (D = 41.0 pm, L = 3.12mm). The resonance frequencies of fibers I-3 monotonically increased about 1% with decrease in pressure from 10’ to 1 Pa. This characteristic is related to the temperature dependence of Young’s modulus of the quartz core. It is caused by the temperature rise of the cantilever due to the pressure dependence of the thermal conductivity in the medium vacuum region.* The resonance sharpnesses Q monotonically increased with decrease in the pressure from 10’ Pa and were saturated at around 10’ Pa, although the pressure at which Q saturated differed for each cantilever. The saturation was a result of the reduction of the shift of drag force due to surrounding gas in this vacuum region. Maximum values of Q were about 2010 (resonance frequency f; = 16.69 kHz, half-power width 4f = 8.3 Hz), 2350 (.f; = 11.54 kHz, dj’= 4.9 Hz), and 1500 (,I; = 3.44 kHz, Af= 2.3 Hz) for fiber 1, fiber 2 and fiber 3, respectively. Since the shift of Q with the pressure corresponds to the shift

where m is the mass of a sphere corresponding to the cantilever. .f’, andf,, respectively are the coefficient of a velocity term (damping constant) and the coefficient of an acceleration term (additional mass) of the drag force due to the surrounding gas. k is a spring constant. A and w are the amplitude and angular frequency of the forced vibration, respectively. By solving eqn (l), 4f‘in the model is derived as follows:‘”

476

Af=

(1)

Cf!

24m fh)

where C is a proportional constant. In the molecular flow region,

(3)

.j; = 0

(4)

In the viscous flow region, f, = &r/R+

3nR2J2ypw

(5) (6)

H Kumazaki et al:

Microcantilever

fabricated

from

optical

fibre

where R is the radius of the cantilever, p is the gas pressure, M the molecular weight, R,, the gas constant, T the absolute temperature. 4 and IJ are viscosity and density of the gas, and w is the angular frequency of the vibration. The effect of the emission of sound was neglected in eqn (5). since the value of the emission of sound at atmospheric pressure was more than seven orders of magnitude smaller than that of,/; in eqn (5). The proportional constant c’ in eqn (2) was determined by the condition that the calculated value agreed with the experimental value at IO’ Pa. Although their overall tendencies were similar, there was a difference of up to IO Hz between the experimental values and the calculated values. The difference in the molecular flow region was caused by the mechanical loss of the cantilever,f;,. Since it is assumed that the vibrations of the spheres are independent of each other. the influence off;) is ignored and is not included in eqn (2). The effect of f;,, however. exists in the actual cantilever and corresponds to the half-power width in the high vacuum region in Figure 3 (8.3 Hz, 4.9 Hz and 2.3 Hz for fibers I. 2 and 3. respectively). The difference in the viscous flow region was inherent in the model where the cylindrical core was regarded as a sphere. The ratios of the first term (Stokes’ term) and the second term (the term depending on frequency) of,/; in the viscous flow region (eqn (5)) of fibers I. 2 and 3 were 41:59, 56:44 and 65:35 at atmospheric pressure, respectively. The shift of 41’ is related to the accuracy of the pressure measurement. Theoretically, the value of df’iin the viscous flow region is determined by eqn (2) and eqn (5). Sincef; is very small compared with nl which is proportional to R’, 4f’is represented approximately as the sum of an R ‘-dependent term and an R ‘dependent term. If the resonance frequencies are the same and R decreases. the shift of Af’ with pressure increases, which was proved in this experiment where the value of the shift of Af’was largest for tiber 2 with the smallest R. The maximum rates of shift of Af’at around atmospheric pressure were 9.5 x IO-’ (Hz/Pa). I .4 x IO ’ (Hz, Pa) and 7.9 x IO -’ (Hz/Pa) for fibers I. 2 and 3, respectively. The above-mentioned rule did not apply to fiber 1 and fiber 3 because the resonance frequency ,/; of fiber 3 was about 20% of that of fiber 1. This experiment confirmed that a slender cantilever should be used to obtain better accuracy. The shift of Af’is actually affected by f;,. Since,f;, is added to the numerator of eqn (2) for the actual cantilever, a large f;) prevents a large shift of A/L The value of,/;, was more strongly affected by the state of the root of the cantilever than by the length and the radius of the core. The value 0f.f;) of fiber I was about 7.0 x 10 q(kg/s) and those of fiber 2 and fiber 3 were about 1.1x 10 ‘I (kg/s). These values were 34.2%, 11.6% and 10.3% of f; of the tibers 1. _7 and 3 at atmospheric pressure, respectively. The valur off,, also influences the lower limit of the pressure measurement. Figure 4 shows ,f;, obtained from this experiment and ,f; calculated using the above model in the molecular flow region. Since I, is proportional to the pressure and to R’ in this region as shoun in eqn (3). the values of./; at the same pressure increased with R ;IS shown in Figure 4. Since the shift of,/; & of fiber 3 was the largest. the shift of 4f’of fiber 3 continued as far as the lowest pressure, while Af’of fiber I and fiber 2 became constant at about IO’ Pa and about 3 x IO’ Pa, respectively. 41 of fiber 3 reached a constant value at about IO-’ Pa. The shift of ,f’,/f;,should be relatively large in order to extend the lower limit of the pressure measurement. The crucial point is to reduce the mechanical loss at the root of the cantilever. The actual relative sensitivity to pressure (shift off; :f;, ) would be higher in the case

Figure 4. Drag force of fibers I ~3 as a function of all- pressure. The symbol “f,,” represents mechanical loss of the cantllevrr and the value is determined from a resonance curve in high vacuum. The symbol “f,” represents the drag force by surrounding gas (molecular flow region) and the value is calculated using a “string-of-beads“ model in the molecular flow region.

of a slender and longer cantilever. A cantilever with higher Q will be advantageous not only at pressures higher than atmospheric pressure but also at lower pressures. Conclusions The resonance characteristics of microcantilevers of quartz core which were fabricated from optical fibers were measured. Pressure sensing using these resonance characteristics was possible in the region from IO’ Pa to 10” Pa. and may be possible to sense in a pressure region higher than the above-mentioned. It is necessary that a slender cantilever be used to obtain good accuracy and that the shift of f;,yf;, be relatively large in order to extend the measurement to lower pressures. Acknowledgements This work was supported in part by Grant\-in-Aid for General Scientific Research and Developmental Scientific Research frotn the Ministry of Education. Science and C’ulture of Japan. This work was also supported in part by Yasuda Corporation. References ‘S Ichimura. K Kokubun, H Shimlru and S Seklnc, .I C’tr~ 51 7&w/. A12, 1734 (1994). ‘S Inaba and K Hane. J V~‘rrc, Scr Tc~~/~ro/. A9. 2 13X ( IYY I ). ‘S Inaba and K Hane. Jpu J Appl Ph,v.r.32, 1001 (1993). ‘K Hane and S Hattori. Opt L~fr. 13, 550 (198X). ‘M V Andres. K W H Foulds and M J Tudor. ~‘/wrro,l LCII, 22, 1097 (1986). “H KumaLakl. S lnaba and K Hanc. ./ I ‘?I(.SW ./pr. 38. I76 (lYY5) (in Japanese). ‘T Pangaribuan. K Yamada. S Jlang. H Ohsaw;i and M Ohtsu. .I .1pl~/ P/II~S. 31, Ll302 (1992). “S Inaha. H Kumazaki and K Hane. ./D/I J Aod Plr~x. 34, 2018 (1995). ‘K Kokubun, M Hirata. H Murakami. Y ?oda and M Ono. L’wwnr. 34, 731 (1984). “‘A Watari. Kikui Shindo (Mechamcal Vibration). 3rd cd. Maw/en. Tokyo (1966) (in Japanese). 477