Laser-Incident Angle Optimization for Measuring In-Plane Mode Shapes in Piezoelectric Resonators with Polished Surfaces Yasuaki WATANABE, Shigeyoshi GOKA and Hitoshi SEKIMOTO
Toru YAMAMOTO Graduate School of Science and Engineering,
Graduate School of Engineering, Tokyo Metropolitan University Tokyo 1920397, JAPAN
[email protected]
Tokyo Institute of Technology Tokyo 1528552, JAPAN
[email protected] mirror placed on the opposite side of the laser. In the developed system, the mirror was tilted so that the reflected beam from the device returned to the surface of the device. The incident angle of the laser was set to 16 degrees from the sample surface, that is 74 degrees from the normal, and the azimuth is parallel to the x axis of the resonator under test. This value was selected to ensure sufficient scatter intensity on the surface of the device and corresponded to the incident angle described in ref. 5. The angle of the mirror was adjusted to prevent damage to the LD by the returning beam.
Abstract— Based on an optical reflection model in polished multilayered surfaces, reflectance and transmittance for the incident angle of the laser beam have been analyzed. The analysis demonstrates that the laser incident angle used in the previous experiment is optimum for measuring in-plane mode shapes in polished resonators with gold and chrome electrodes.
I.
INTRODUCTION
The laser speckle method using a CCD camera and burstdevice-excitation enables very fast visualization of mode shapes in piezoelectric devices[1-3]. We have previously reported that this method can be applied to the device with polished-metalized surfaces by choosing an optimum incident angle of the laser beam[4]. We used a laser incident angle set to 16 degrees from the device surface to maximize detection sensitivity. However, the theoretical background for the optimum incident angle was not assessed. We analyzed the reflectance and transmittance of the metalized surface which causes laser diffusion in polished resonator surface, and validated the optimum laser incident angle. II.
Figures 3 (a) and (b) highlight the results for (1, 1, 1) and (1, 1, 3) modes [4]. The numbers in the parentheses indicate overtone, the wave number along the x axis, and the wave number along the z’ axis. The black dots and white bright lines seen are effects of the dust and the actual edge of the mesa portion. We can also see from these figures that inplane vibrational displacement is trapped at the center of the mesa portion.
Mesa portion
EXPERIMENTAL METHOD AND RESULTS[4]
We previously reported that the mode shapes in piezoelectric resonators with polished surface can be measured by laser speckle method [4]. We used a bi-mesa shaped piezoelectric resonator and this can be seen in Fig. 1. We used an 8.3 MHz bi-mesa shaped AT-cut quartz resonator with polished surfaces with partial electrodes. The electrodes were made of gold, 150 nm thick, and had a chromium substratum about 5 nm thick.
z’
Figure 2 illustrates the optical system used for measuring in-plane mode shapes of polished piezoelectric devices. A collimated beam from the laser diode (LD) illuminates the polished and electroded sample surface and is then reflected off the surface. Polarization of the laser beam was parallel to the sample surface. The reflected beam illuminates the
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x
Fig. 1. Bi-mesa shaped rectangular AT-cut quartz crystal resonator with partial electrodes.
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pattern 3) is at a maximum, then the reflection generated by pattern 1) is at a minimum and vice versa.
Image Trigger
Transfer
PC
Figure 4 outlines a structure of the multilayered resonator surface. Laser incident angle dependency on reflections and transmission of the polished surface was analyzed using Fresnel’s reflectance and transmittance.
CCD Cam.
Low Freq. Oscillator
Mirror
Signal
LD
Generator
AM Modulation
ρ0
Collimator
SWR Bridge
Coaxial
Resonator
Switch
Under Test
d1
GP -IB
τ0
N0
(Air)
N1
(gold)
N2
(chrome)
N3
(crystal)
ρ1
Network Analyzer
d2
ρ2
τ1
Fig. 4. Experimental setup of the full-field in-plane motion
τ2
visualization system. Fig. 2. Experimental setup of full-field in-plane motion visualization system.
Fig. 4. Reflection and transmission model of multilayered device surface.
IV.
ANALYSIS
We analyzed the reflectance and transmittance of the metalized device surfaces by applying the Fresnel coefficients of p-polarized beam to each interface. (a)
(b)
Refractive indexes ηxp of the materials are given by
Fig. 3. Experimental results for mesa-shaped resonator with polished surface. (a) fundamental thickness shear (1,1,1) mode, and (b) nearby inharmonic (1,1,3) mode.
III.
η xp = N x cos θ x ,
where N is the refractive index of the medium, θ is the incident angle of the beam, and x indicates the layer.
OPTICAL REFLECTION MODEL FOR MULTILAYERED SURFACES
The Fresnel coefficients of the p-polarized beam at the interfaces can be written as
In general, optical reflections on an object are classified into the following four patterns [6]. 1)
Direct reflection.
ρ xp =
2) Reflected over twice on a surface. (Surface roughness is larger than wavelength of the incident light.) 3)
Penetrated object surface and reflected in the object.
4)
Micromorphological reflection.
(1)
and
τ xp =
Patterns 2), 3) and 4) are categorized as diffusion reflections. Pattern 4) is generally very small and can be therefore ignored. Pattern 2) is unrelated to polished objects. Therefore, the diffusion reflections on polished surfaces are almost generated by pattern 3. When a reflection caused by
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η x +1, p − η xp η x +1, p + η xp 2η xp
η x +1, p + η xp
.
(2)
(3)
Equations (2) and (3) show the reflectance and transmittance for the interfaces. The overall Fresnel coefficients due to layer 1 and 2 (metals) can be written as
ρ and
' 0p
ρ 1 p + ρ 2 p e − i 2 ∆ 2 − i 2 ∆1 e = ρ0 p + −i 2 ∆ 2 1 e ρ ρ + 1p 2 p
τ τ e −i 2 ∆ τ 0' p = τ 0 p 1 p 2 p −i 2 ∆ e −i∆ 1 + ρ1 p ρ 2 p e 2
1
2
ρ + ρ 2 p e − i 2 ∆ 2 −i 2 ∆1 1 + ρ 0 p 1 p e −i 2 ∆ 2 1 e + ρ ρ 1p 2 p
ρ + ρ 2 p e − i 2 ∆ 2 − i 2 ∆1 1 + ρ 0 p 1 p e −i 2 ∆ 2 1 ρ ρ e + 1 p 2 p ,
where ∆1 and ∆2 are phase shifts of the laser in medium 1 and 2, and can be written as
and
∆1 =
2π
λ
N 1d 1 cos θ 1
(6)
N 2 d 2 cos θ 2 ,
(7)
(4)
(5)
compared to reflectance, the diffusion reflection was at a maximum when the direct reflection was at a minimum. In the experiments [4], the incident angle of the laser was set to 16 degrees from the sample surface, that is 74 degrees from the normal. This value was slightly higher than the analysis shown in Fig. 5.
∆2 =
2π
λ
where λ is wavelength of the laser, d1 and d2 are thicknesses of medium 1 and 2. Therefore, net reflectance and transmittance can be expressed as 2
R p = ρ 0' p and
Tp =
Re(η 3 p )
Re(η 0 p )
τ 0' p
(8)
2
.
(9) Fig. 5. Reflection characteristics of multilayered device surface.
V.
RESULTS When the diffusion reflection is at a maximum at 70 degrees, then the detection sensitivity should also be at a maximum at this incident angle. In the laser speckle system, however, the detection sensitivity for in-plane vibration abruptly decreased below 70 degrees[5]. Therefore, the optimum incident angle should be greater than 70 degrees. This approach is consistent with the incident angle used in the experiments.
Reflectance and transmittance of the polished multilayered surface for the incident laser beam were analyzed using the above equations. Figure 5 highlights the analysis of the reflectance-incident angle characteristics for the metalized surface of the resonator. According to the actual mesa-shaped resonator, the top electrode of the resonator was gold, 150-nm thick, the substratum was chromium, 5-nm thick, and the substrate was quartz crystal. The wavelength of the laser diode was 630-nm.
VI.
It is assumed from the figure that the direct reflection was at a minimum at around 70 degrees from the normal.
CONCLUSIONS
Based on the optical reflection model in multilayered polished-devices, reflectance and transmittance were analyzed for the resonator used in the laser speckle method.
Figure 6 shows the analysis of the transmittance-incident angle characteristics. It is naturally understood that when reflectance is at a minimum then transmittance is at a maximum. Since transmittance was relatively small
The analysis showed that the maximum diffusionreflection is obtained when the beam angle is at 70 degrees
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REFERENCES
for the sample device, and the laser incident angle used in the experiments was validated.
[1]
[2]
[3]
[4]
[5]
[6]
Fig. 6. Transmission characteristics of multilayered device surface.
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Y. Watanabe, Y. Shikama, S. Goka, T. Sato, H. Sekimoto, ”Mode Shape Measurement of Piezoelectric Resonators Using Image Processing Technique,” Japanese. Journal of Appl. Phys. Vol. 40, pp. 3572-3574, 2001. Y. Watanabe, T. Tominaga, T. Sato, S. Goka, H. Sekimoto, “Visualization of Mode Patterns of Piezoelectric Resonators using Correlation Filter,” Japanese. Journal of Appl. Phys., Vol.41, Pt.1, No.5B, pp. 3313-3315, 2002. Y. Watanabe, T. Sato, S. Goka and H. Sekimoto, “Non-Scanning Means for Determining Vibrational Distribution in BAW and SAW Devices,” in Proc. IEEE Ultrasonic Symp., 2002, pp. 928-931 Y. Watanabe, S. Goka, T. Sato, H. Sekimoto, “Nonscanning Measurements for Determining In-Plane Mode Shapes in Piezoelectric Devices with Polished Surfaces,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 51, no. 5,pp. 491 – 495, may 2004. Y. Watanabe, T. Sato, S. goka and H. Sekimoto, “Measurement of InPlane and Out-of-Plane Mode Shapes in Piezoelectric Devices using Laser Speckle Interferometry,” Acoust. Sci. & Tech. Vol.23, No.5 pp.284-285, (2002-08) M. Kobiyama, “Fundamental theory on optical membranes,” Optonics, 2003 (in Japanese)
