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Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates
Ultrasonics xxx (2014) xxx–xxx
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Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates Meng-Jhen Chiou a, Yu-Ching Lin b, Takahito Ono c, Masayoshi Esashi b, Sih-Ling Yeh a, Tsung-Tsong Wu a,⇑ a
Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan WPI-AIMR, Tohoku University, Sendai 980-0845, Japan c Department of Mechanical Engineering, Tohoku University, Sendai 980-0845, Japan b
a r t i c l e
i n f o
Article history: Received 19 February 2014 Received in revised form 10 May 2014 Accepted 10 May 2014 Available online xxxx Keywords: Phononic crystal Lamb wave Piezoelectric plate Focusing Waveguiding
a b s t r a c t This paper presents results on the numerical and experimental studies of focusing and waveguiding of the lowest anti-symmetric Lamb wave in micro-fabricated piezoelectric phononic plates. The phononic structure was based on an AT-cut quartz plate and consisted of a gradient-index phononic crystal (GRIN PC) lens and a linear phononic plate waveguide. The band structures of the square-latticed AT-cut quartz phononic crystal plates with different filling ratios were analyzed using the finite element method. The design of a GRIN PC plate lens which is attached with a linear phononic plate waveguide is proposed. In designing the waveguide, propagation modes in square-latticed PC plates with different waveguide widths were studied and the results were served for the experimental design. In the micro-fabrication, deep reactive ion etching (Deep-RIE) process with a laboratory-made etcher was utilized to fabricate both the GRIN PC plate lens and the linear phononic waveguide on an 80 lm thick AT-cut quartz plate. Interdigital transducers were fabricated directly on the quartz plate to generate the lowest anti-symmetric Lamb waves. A vibro-meter was used to detect the wave fields and the measured results on the focusing and waveguiding of the piezoelectric GRIN PC lens and waveguide are in good accordance with the numerical predictions. The results of this study may serve as a basis for developing an active micro plate lens and related devices. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The renewed properties, such as acoustic band gap [1–5], negative refraction [6,7] and local resonances [8–10], of acoustic waves in 2-D periodic structures called phononic crystals (PC) have been reported intensively in the literatures. In particular, due to the potential applications in high frequency Lamb wave devices and the readiness of micro-fabrication techniques, propagation of acoustic waves in 2-D PC plate structures have aroused much attention in physics as well as engineering communities over the last decade. In the literatures, the band gaps of Lamb waves in micro-fabricated silicon phononic plates have been demonstrated [11–15] and the related design and fabrication of Lamb wave resonators have also been reported [16–18]. By using the phononic band gap properties of Lamb waves in silicon thin plates, their results showed that the quality factors of the as-made Lamb wave resonators can be improved significantly and the devices’ size can be reduced. One of the important applications of phononic crystal ⇑ Corresponding author. Tel.: +886 2 33665891. E-mail address: [email protected] (T.-T. Wu).
is utilization of its complete band gap to form a linear waveguide by removing one or several layers of phononic crystals. Recently, numerical [19–22] as well as experimental [23] studies have been reported and shown that with a proper design, Lamb waves can be guided with little attenuation along the plate waveguide. A review on the propagation of Lamb waves in plates with phononic structure can be found in Ref. [24]. Regards to the focusing of bulk waves in PC structures, the gradient-index (GRIN) PC has been proposed [25,26]. Their results demonstrated that GRIN PC allows acoustic wave focusing over a wide range of operating frequencies and making it suitable for applications such as flat acoustic lenses and acoustic wave couplers. Recently, the concept of GRIN PC has been applied to the phononic plate structures and similar focusing capabilities were found. Wu et al. [27] numerically demonstrated focusing of the lowest anti-symmetric Lamb wave in a GRIN PC silicon plate and its application as a beam-width compressor for compressing Lamb wave into a stubbed phononic tungsten/silicon plate waveguide. Zhao et al. [28] employed a non-contact laser-ultrasonic technique to excite and detect flexural Lamb waves within a silicon GRIN PC plate and confirmed the focusing experimentally. However,
http://dx.doi.org/10.1016/j.ultras.2014.05.007 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
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realization of an active GRIN PC plate lens which is based on piezoelectric crystal remains necessary. In this paper, we demonstrate focusing and waveguiding of the lowest anti-symmetric (A0) Lamb waves in GRIN PC plates with/ without a PC waveguide. Toward the development of an active GRIN PC plate device, the piezoelectric AT-cut quartz was chosen as the plate material in this study. In the following, band structures of the square-latticed AT-cut quartz phononic crystal plates with different filling ratios are analyzed using the finite element method (FEM) firstly, then, the design of an AT-cut quartz based GRIN PC plate lens and the PC plate waveguide are followed. Finally, the micro-fabrication processes of the AT-cut quartz GRIN PC plate devices are described and the measured results of the focusing and guiding wave fields are discussed. Fig. 2. The band structure of a square-latticed quartz PC plate with lattice constant a and radius of the air hole r equal to 80 lm and 24 lm.
2. Band structure of an AT-cut quartz PC plate Before calculating the band structure of the PC plate, we examined the anisotropic characteristics of a pure AT-cut quartz plate with thickness (z-direction) equal to 80 lm. Fig. 1 shows the slowness curves of the three lowest plate modes of the pure AT-cut quartz plate on the x–y plane evaluated at 10 MHz, i.e., the antisymmetric Lamb mode (A0 mode, red line), the shear horizontal Lamb mode (SH0 mode, blue line) and the symmetric Lamb mode (S0 mode, green line). In the calculation, the x-axis of the AT-cut quartz plate was aligned with the X-axis of the crystal axes. The result in Fig. 1 showed that the slowness curves of the three lowest plate modes along the x and y-axis are slightly different; however, 90° symmetry exists. Fig. 2 shows the band structure of a squarelatticed quartz PC plate with lattice constant a and radius of the air hole r equal to 80 lm and 24 lm, respectively, and the corresponding filling fraction (ff) is 0.283. On considering the anisotropy of the AT-cut quartz, the irreducible Brillouin zone of the square-latticed PC plate are bounded by C, X, M and Y as shown in the inset of Fig. 2. In the figure, the displacement components of the Lamb modes (ux, uy, uz) are transformed into uL, uSH, uSV which are defined as
uL ¼ ux cos h þ uy sin h uSH ¼ ux sin h þ uy cos h
ð1Þ
uSV ¼ uz ; where uL and uSH are the components along and perpendicular to the wave vector direction, respectively and h is the angle between 1.0
A0 0.5
SH 0
ky(π/a)
S0 0.0
-0.5
-1.0 -1.0
-0.5
0.0
0.5
1.0
kx(π/a) Fig. 1. The slowness curves of the three lowest plate modes of the pure AT-cut quartz plate.
the wave vector and the x-axis. Further, we defined the ratio of polarization (ROP) as
RRR ROP ¼ RRR
ðu2x
u2p dV ; þ u2y þ u2z ÞdV
ð2Þ
where up represents uL, uSH, uSV. By assigning the green color to uL, blue to uSH, and red to uSV, and utilizing the ROP, the band structure of the square-latticed quartz PC plate was obtained as the color curves shown in Fig. 2. Along the CX section, uL, uSH, uSV are aligned with ux, uy, uz, respectively and the three lowest Lamb modes, A0 mode (red), SH0 mode (blue) and S0 mode (green) are similar to the case of a homogenous AT-cut quartz plate. However, the folded branch of the A0 mode becomes brown color (combination of red and green) which means both uL and uSV exist. Detail calculation of the band structure, one finds that there are level repulsions (anti-crossing) appeared (X1, X2, X3, X4) along the boundaries XM and MY [5]. In those regions, the polarizations of the wave displacements cannot be decoupled completely and therefore most of the colors of the bands are not pure green, red or blue. The band structure along the CY section is similar to that of the CX section except for a level repulsion appeared at X5. For later design of the GRIN PC plate lens, we calculated the band structures of the A0 mode along the CX direction with different filling fraction and the results are shown in Fig. 3(a). The results showed that the frequency band of the A0 mode drops and the group velocity decreases with the increase of the filling fraction. When the filling fraction is high (e.g. ff = 0.283) and the operation frequency greater than about 12 MHz, we note that the group velocity drops dramatically and vanishes at the X point. To evaluate the effect of anisotropy on the propagation of A0 mode in the square-latticed quartz PC plate, the equal frequency curves (EFCs) with different filling fractions evaluated at frequency equal to 10 MHz were calculated and the results are shown in Fig. 3(b). The results showed that the EFC becomes more circular (i.e., more isotropic) as the filling fraction increases. Fig. 3(c) shows the anisotropic ratio g of the A0 mode as a function of angle away from the CX direction (propagation direction) with different filling fractions. The anisotropic ratio was defined as the ratio of the wavenumber (kCX) along the CX-direction and those along the direction (k) which lies between the CX-direction and CY-direction, i.e., g = kCX/k. The results showed clearly that the anisotropic ratio decreases as the filling fraction increases from zero (pure AT-cut quartz plate) to ff = 0.283. The maximum anisotropic ratio is about 1.07 which appears on the curve of filling fraction equal to 0.126 (green line) and is about 7% difference as compared with the unity (i.e., CX-direction). Further, for the GRIN PC plate lens, the angles h of the focusing rays are less than 45° in general and therefore the anisotropic ratio can be down to less than 5% (Fig. 3(c)).
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(a)
level repulsions, the dispersion of the A0 mode is relatively simple and similar to that of a homogenous AT-cut quartz plate. For the A0 mode propagates along the CX direction, the group velocity drops dramatically around the X point, when the filling fraction is high. Thus the operation frequency of a GRIN PC plate lens should keep away from this region to have a better focusing. In addition, the maximum anisotropy ratio of the A0 mode is less than 5% which is reasonable to make the weakly anisotropic assumption in the following GRIN PC plate lens design.
ff=0 ff=0.126 15
ff=0.203 ff=0.250
Frequency (MHz)
ff=0.275 ff=0.283 10
3. Design of a GRIN PC plate lens 5
0
Γ
Shown in Fig. 4(a) is a schematic diagram of the GRIN PC plate lens on the AT-cut quartz plate with the input wave propagates along the x-direction. There are 41 rows of air holes along the y-direction arranged in square lattices with radii of the holes decreasing from a maximum of 24 lm (ff = 0.283) at the central line (y = 0) to a minimum of 16 lm (ff = 0.126) at the lens boundaries (y = ±20a). The lattice constant (a) and thickness (h) of the homogeneous plate are equal to 80 lm. Since the largest anisotropic ratio was estimated about 1.07 (ff = 0.126) along the
X
Wavevector (π/a)
(b)
1.0
(a) 0.5
ff=0
ky (π/a)
ff=0.126 ff=0.203 0.0
ff=0.250 ff=0.275 ff=0.283
-0.5
74a
(b)
-1.0 -1.0
-0.5
0.0
0.5
79a
1.0
kx (π/a)
y
(c) 1.10
x
Anisotropic ratio η
1.08
(c) 1.06 ff=0
1.04
ff=0.126 ff=0.203
1.02
ff=0.250 1.00
ff=0.275 ff=0.283
0.98
0
30
60
90
Angle θ (degree) Fig. 3. (a) Band structures of the A0 mode with different filling fraction. (b) Equal frequency curves with different filling fractions. (c) The anisotropic ratio g of the A0 mode as a function of angle away from the CX-direction with different filling fractions.
Based on the above analyses, although the band structure of the square-latticed quartz PC plate is complicated and contains several
Fig. 4. (a) Schematic diagram of the GRIN PC plate lens on an AT-cut quartz plate with the input wave propagates along the x-direction. (b) The focusing image of the lowest A0 mode in the designed GRIN PC quartz plate lens with a line force placed at x = 2a. (c) The amplitude distribution along the y-direction of the plate lens at x = 74a.
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boundaries of the lens, we assumed that the air/AT-cut quartz PC plate is weakly anisotropic, and approximated the refractive index of the A0 mode as the refractive index along the direction, n ¼ mCmX ,
where mCX is the group velocity along the CX direction and v is the referenced group velocity of the A0 mode of a homogeneous AT-cut quartz plate evaluated at the designed frequency 10 MHz. The refractive index profile in the form of a hyperbolic secant n = n0 sech(a y) was adopted, where n0 is the refractive index along the x-axis and a is the gradient coefficient [27]. Since the radii of the holes at y = 0 and y = ±20a are known, the gradient coefficient a were determined as 3.86 104 lm1. The radii of the holes along the y-direction can thus be found from the hyperbolic secant function shown above. Shown in Fig. 4(b) is the focusing image of the lowest A0 mode in the designed GRIN PC quartz plate lens with a line force placed at x = 2a. The result shows that there are several focusing regions appeared, the maximum one appeared at x = 79a with maximum amplitude 3.57 times and the second maximum one at x = 74a with maximum amplitude 3.53 times of that at the source. The focal distances of 79a or 74a are longer than the one predicted by the lens formula p/(2a), 51a, which is consistent with the observation given in reference [27]. In addition, unlike the sharp focusing of the decoupled shear vertical (SV) mode of BAW in the GRIN PC [25,26], the neck of the focus region is longer. The dotted ellipse in Fig. 4(b) shows the region with 90% of the maximum amplitude which starts at about 70a and ends at 88a. Shown in Fig. 4(c) is the amplitude distribution along the y-direction of the plate lens at x = 74a which shows the compressing of the wave beams and the full-width at half maximum (FWHM) is 4.5a. As compared with the aperture of the plate lens 40.5a, the compressing is about 11.1%. To investigate the bandwidth of the GRIN PC quartz plate lens, we set the operation frequency at 9 MHz and the results showed a similar focusing but the focusing regions appeared at longer distances away from the source, i.e., the maximum amplitude appeared at x = 81a while the amplitude amplification is 3.91 times. On the contrary, when the operation frequency was set at 11 MHz, the results showed that the maximum amplitude appeared at x = 73a with the amplitude amplification equal to 3.65 times. Based on the calculated results, it is found that slight deviation of the operation frequency from the designed one causes little variation on the amplitude amplifications, however, the changing of the focus distance have to be taken into consideration. Fig. 5 shows the focusing image when the operation frequency is increased to 12 MHz. Although a sharp focusing region still can be found at around x = 50a, the wave pattern becomes much more complicated. The reason may due to the sharp change of the group velocity of the A0 mode in this frequency range (see the CX section of Fig. 3(a)) which makes the distribution of the refraction index of the PC layers not so smooth.
Fig. 5. The focusing image with the operation frequency equal to 12 MHz.
Fig. 6. The schematic diagram of the GRIN PC plate lens which is connected with a PC plate waveguide.
4. GRIN PC plate lens with waveguide Shown in Fig. 6 is a schematic diagram of the GRIN PC Plate lens which is connected with a PC plate waveguide. To maximize the energy compressed into the waveguide, proper design of length of the GRIN PC plate lens, complete band gap and width of the PC plate waveguide are crucial. The design of the GRIN PC Plate lens is the same as the one shown in Fig. 4(a) except that the lens is terminated at x = 74a which is at the second maximum of the focusing regions. The reason for choosing the second instead of the maximum amplitude (x = 79a) is that their amplitudes are approximately the same; however, the former one makes the lens length shorter. To design a PC plate waveguide with operation frequency at 10 MHz, a square-latticed air/AT-cut quartz PC plate was chosen. The thickness of the plate waveguide is the same as that of the GRIN plate lens, the lattice constant is 200 lm and the radius of the hole is 96 lm. Numerical calculation on the band structure of the square-latticed air/AT-cut quartz PC plate was conducted and the result showed that a complete bandgap (8.7–10.9 MHz) formed around the operational frequency 10 MHz. To form a linear waveguide, one row of the periodic holes along the x-direction was removed to investigate the wave modes inside the waveguide with the waveguide width w = a. The supercell technique [19] was utilized to calculate the band structure of the square-latticed air/AT-cut quartz PC plate for wave propagates along the CX direction. The result shown in Fig. 7(a) shows that there are three propagating modes appeared within the band gap and only two of them (points 1 and 2) are intersected with the operation frequency 10 MHz. The mode shapes shown on the right part of Fig. 7(a) demonstrate that both of the two modes belong to the quasi-shear horizontal modes, i.e., the displacements are mostly on the x–y plane. Fig. 7(b) shows the propagating modes for the case of w = 3a where three of the intercepted points (points 2, 3, 4) belong to the quai-shear horizontal mode and one belong to the quasi-antisymmetric Lamb mode (point 1, quasi-A0 mode with displacement mostly on the x–z plane). For the cases of w = 2a and w = 4a, further simulation results showed that the number of the quasi-A0 mode are 2 and 4, respectively and those of the quasiSH0 mode are 2 and 1, respectively. Based on the above simulation results, we chose the case of w = 3a to make the propagation of the quasi-A0 mode in the waveguide simpler, i.e., only one quasi-A0 mode exists. Shown in Fig. 8(a) is the simulated image of the quasi-A0 mode in the designed GRIN PC quartz plate lens with PC plate waveguide. The result shows that the focused plate waves are compressed into the waveguide at x = 74a successfully. The maximum amplitude of the quasi-A0 mode in the waveguide is about 3.2 times of that at the source. Fig. 8(b) is the amplitude distribution along the transverse direction (y-axis) at x = 84a (middle of the waveguide and lies at the center of the neighboring two holes) which shows the FWHM is about the width of the waveguide, i.e., 3a. The two side peaks appeared at y = 18a and 23a are the local resonances between the side walls of the neighboring two holes; however, they decay very fast along the transverse direction.
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Fig. 7. The propagation modes inside the band gap for the case of (a) w = a and (b) w = 3a.
Fig. 8. (a) The simulated image of the quasi-A0 mode in the designed GRIN PC quartz plate lens with PC plate waveguide. (b) The amplitude distribution along the transverse direction (y-axis) at x = 84a.
5. Micro-fabrication of the GRIN PC plate lens with/without waveguide In the experimental part, double-side polished 80 lm thick ATcut quartz plates were utilized to fabricate the GRIN plate lens with and without PC waveguide. The geometrical parameters of both the plate lens with and without waveguide were the same as those
Fig. 9. The micro-fabrication process of the GRIN PC lens and waveguide.
used in the numerical simulations described in Sections 3 and 4. For the case of the plate lens with waveguide, the lens was terminated at x = 74a. We chose the case of w = 3a to insure that there is only one A0 mode exists in the waveguide.
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The micro-fabrication process is shown in Fig. 9. The upper and lower sides of the quartz plate were etched in a successive order in order to make the sidewall of the holes close to vertical. In the beginning, Au/Cr films were sputtered on both sides of the quartz plate. The Au film acts as a seed layer for the Ni electroplating and the Cr film as the adhesion layer between the Au film and quartz plate (Fig. 9(a)). An 8 lm thick photoresist (AZP 4620) was patterned on the front-side of the quartz plate to form the periodic cylinders, which defined the area for the Ni electroplating (Fig. 9(b)). It is noted that the double-side Au/Cr seed layers and the double-side Ni electrodeposition (Fig. 9(c)) were adopted to prevent the 80 lm thick quartz plate from bending and breaking due to the unbalanced intrinsic stress during the Ni electroplating. In the electroplating, the current density was 12 mA cm2, the electroplating bath was controlled at 55 °C and the Ni deposition rate was 0.1 lm min1. After the Ni electroplating, the photoresist was removed and the device was ready for the RIE process (Fig. 9(d)). Before the RIE process of the quartz plate, Xe gas was introduced to remove the Au/Cr layer which was not covered by the Ni mask. SF6 gas was used to etch the front side of the quartz plate with the RF stage temperature of 20 °C, gas pressure of 3 mTorr, power of 100 W, and a self-bias voltage of 390 V. The etching rate was 450 nm min1 under the aforementioned conditions. In order to fabricate the cylindrical air hole with a vertical sidewall, the etching process was divided into two parts. First, the etch depth of the front side was around 40 lm and the Ni mask was removed by using the nitric acid, as shown in Fig. 9(e). Then, the back-side of quartz plate was patterned again, followed by Ni electroplating and photoresist removal. By using the same etching parameters as those of the front side, the back side of the quartz plate was etched and the selectivity of the Ni mask to quartz plate was 20. Again, by using nitric acid, the Ni mask was removed and the PC gratings were accomplished (Fig. 9(f)–(h)). Finally, the IDTs were fabricated onto the left side of the GRIN PC plate lens to generate 10 MHz Lamb waves using the photolithography process (Fig. 9(i) and (j)). The gap between the IDT and the plate lens is 2a. The wavelength of the IDT, k, is 226 lm, the aperture is 15k and the number of electrode pairs is 20. Shown in Fig. 10 is the picture took at the junction of the GRIN PC plate lens and the linear PC waveguide.
6. Experimental results and discussions The wave fields (z-component of the displacement) in the GRIN plate lens with/without waveguide generated by the IDTs were detected using an ultra-high frequency vibrometer (UHF-120, Polytec). Fig. 11(a) shows the measured wave distribution of the GRIN PC plate lens under the IDT excitation at 10 MHz. The imaging result shows clearly the Lamb wave focusing in such a PC plate lens. The maximum amplitude appeared approximately at x = 69a which is about 3.4 times of that at the source x = 2a. The length of the neck region (90% of the maximum amplitude) is around 5a
Fig. 11. The measured wave distributions of (a) the GRIN PC plate lens and (b) the GRIN plate lens with a linear waveguide at 10 MHz.
and the FWHM is about 4a. In comparisons with the numerical predictions shown in the Section 3, where the maximum focusing amplitude is 3.53 times and FWHM is 4.5a, we find they are in good agreements. However, location of the maximum focusing amplitude (x = 69a) and length of the neck region (5a) are shorter than those in the numerical predictions, i.e., x = 74a and 18a. On examining the fabricated GRIN PC lens, we found that the radius of the largest hole is 25 lm instead of the designed one 24 lm, while that of the smallest hole is 15 lm instead of 16 lm. Possible reason for these deviations may come from the inaccuracy in the micro-fabrication of holes. In the etching process, smaller holes need more time than the larger holes to etch through. Therefore, as the smallest holes around the edges of the plate lens reached the design sizes, the largest hole along the centerline becomes bigger than the designed one and results an increase of the gradient coefficient a. This in turn may shorten the focal distance and length of the neck region according to the previous numerical simulations. It is noted that the deviation in the position and length of the neck region of the focusing may be minimized by fine tuning of the designed radii according to the experimental data. The measured wave field in the GRIN plate lens with a linear waveguide is shown in Fig. 11(b). The results show that the impinging of the focused Lamb waves into the waveguide and the maximum amplitude inside the waveguide is about 2.4 times of that at the source. The amplitude amplification is smaller than that predicted in the previous simulation, i.e., 3.2 times. Similarly, this inaccuracy may again due to the etching problem mentioned previously, i.e., the focused region in the GRIN plate lens move forward instead of just at x = 74a. 7. Summary and conclusions
Fig. 10. The picture at the junction of the GRIN PC plate lens and the linear PC waveguide.
In summary, we have conducted numerical and experimental studies of Lamb wave focusing in an AT-cut quartz GRIN PC plate lens without/with a linear PC waveguide. An approximated method for designing the GRIN PC plate lens was utilized to predict the focusing and waveguiding of the PC plate lens with PC waveguide. The micro-fabrication process of the AT-cut quartz GRIN PC lens and waveguide was developed. The measured focusing
Please cite this article in press as: M.-J. Chiou et al., Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.05.007
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and waveguiding images of the GRIN PC plate lens and waveguides are in good accordance with the numerical predictions. The deviation in the position and length of the neck region of the focusing due to the etching process may be minimized by fine tuning of the designed radii. Results of this study may serve as a basis for developing active GRIN PC plate devices based on a piezoelectric substrate. Acknowledgment The financial support from the National Science Council of Taiwan through the grant NSC 100-2221-E-002-104-MY2 is gratefully acknowledged. References [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composite, Phys. Rev. Lett. 71 (1993) 2022–2025. [2] R. Sainidou, B. Djafari-Rouhani, J.O. Vasseur, Surface acoustic waves infinite slabs of three-dimensional phononic crystals, Phys. Rev. B 77 (2008) 094304. [3] T.-T. Wu, Z.-G. Huang, S. Lin, Surface and bulk acoustic waves in two dimensional phononic crystal consisting of general anisotropic materials, Phys. Rev. B 69 (2004) 094301. [4] Z.-G. Huang, T.-T. Wu, Temperature effect on the bandgaps of surface and bulk acoustic waves in two-dimensional phononic crystals, IEEE, J. Ultrasonics, Ferroelectrics Freq. Control 52 (3) (2005) 365–370. [5] T.-T. Wu, Z.-G. Huang, Level repulsions of bulk acoustic waves in composite materials, Phys. Rev. B 70 (2004) 214304. [6] X.D. Zhang, Z.Y. Liu, Negative refraction of acoustic waves in 2D phononic crystals, Appl. Phys. Lett. 85 (2004) 341. [7] S.X. Yang, J.H. Page, Z.Y. Liu, M.L. Cowan, C.T. Chan, P. Sheng, Focusing of sound in a 3D phononic crystal, Phys. Rev. Lett. 93 (2004) 024301. [8] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally resonant sonic materials, Science 289 (2000) 1734. [9] G. Wang, X. Wen, J. Wen, L. Shao, Y. Liu, Two dimensional locally resonant materials with binary structures, Phys. Rev. Lett. 93 (2004) 154302. [10] J.C. Hsu, T.-T. Wu, Lamb waves in binary locally resonant phononic plates with two-dimensional lattices, Appl. Phys. Lett. 90 (2007) 201904. [11] R.H. Olsson III, I.F. El-Kady, M.F. Su, M.R. Tuck, J.G. Fleming, Microfabricated VHF acoustic crystals and waveguides, Sens. Actuat. A 145 (2008) 87–93.
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[12] S. Mohammadi, A.A. Eftekhar, A. Khelif, W.D. Hunt, A. Adibi, Evidence of large high frequency complete phononic band gaps in silicon phononic crystal plates, Appl. Phys. Lett. 92 (22) (2008) 221905. [13] N.-K. Kuo, C.-H. Zuo, G. Piazza, Microscale inverse acoustic band gap structure in aluminum nitride, Appl. Phys. Lett. 95 (9) (2009) 093501. [14] M.F. Su, R.H. Olsson III, Z.C. Leseman, I. El-Kady, Realization of phononic crystal operating at gigahertz frequencies, Appl. Phys. Lett. 96 (5) (2010) 05311. [15] Y.M. Soliman, M.F. Su, Z.C. Leseman, C.M. Reinke, I. El-Kady, R.H. Olsson, Phononic crystals operating in the gigahertz range with extremely wide band gaps, Appl. Phys. Lett. 97 (19) (2010) 193502. [16] S. Mohammadi, A.A. Eftekhar, W.D. Hunt, A. Adibi, High-Q micromechanical resonators in a two-dimensional phononic crystal slab, Appl. Phys. Lett. 94 (5) (2009) 051906. [17] C.-Y. Huang, J.-H. Sun, T.-T. Wu, A two-port ZnO/silicon Lamb wave resonator using phononic crystals, Appl. Phys. Lett. 97 (3) (2010) 031913. [18] N. Wang, F.-L. Hsiao, M. Palaniapan, C. Lee, Silicon two-dimensional phononic crystal resonators using alternate defects, Appl. Phys. Lett. 99 (2011) 234102. [19] J.-H. Sun, T.-T. Wu, Propagation of acoustic waves in phononic crystal plates and waveguides using a finite-difference time-domain method, Phys. Rev. B 76 (2007) 104304. [20] J.O. Vasseur, A.-C. Hladky-Hennion, B. Djafari-Rouhani, F. Duval, B. Dubus, Y. Pennec, P.A. Deymier, Waveguiding in two-dimensional piezoelectric phononic crystal plates, J. Appl. Phys. 101 (2007) 114904. [21] J.O. Vasseur, P.A. Deymier, B. Djafari-Rouhani, Y. Pennec, A.C. Hladky-Hennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates, Phys. Rev. B 77 (8) (2008) 085415. [22] Y. Pennec, B. Djafari-Rouhani, H. Larabi, J.O. Vasseur, A.C. Hladky-Hennion, Low-frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate, Phys. Rev. B 78 (10) (2008) 104105. [23] T.-T. Wu, Z.-G. Huang, T.-C. Tsai, T.-C. Wu, Evidence of complete band gap and resonances in a plate with periodic stubbed surface, Appl. Phys. Lett. 93 (11) (2008) 111902. [24] T.-T. Wu, J.-C. Hsu, J.H. Sun, Phononic plate waves, IEEE, J. Ultrasonics, Ferroelectrics Freq. Control 58 (10) (2011) 2146–2161. [25] S.-C.S. Lin, T.J. Huang, J.-H. Sun, T.-T. Wu, Gradient-Index phononic crystal, Phys. Rev. B 79 (2009) 094302. [26] S.-C.S. Lin, B.R. Tittmann, J.-H. Sun, T.-T. Wu, T.J. Huang, Acoustic beamwidth compressor using gradient-index phononic crystals, J. Phys. D: Appl. Phys. 42 (2009) 185502. [27] T.-T. Wu, Y.-T. Chen, J.-H. Sun, T.J. Huang, Focusing of the lowest antisymmetric Lamb wave in a gradient-index phononic crystal plate, Appl. Phys. Lett. 98 (2011) 171911. [28] J. Zhao, R. Marchal, B. Bonello, O. Boyko, Efficient focalization of antisymmetric Lamb waves in gradient-index phononic crystal plates, Appl. Phys. Lett. 101 (2012) 261905.
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Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates Meng-Jhen Chiou a, Yu-Ching Lin b, Takahito Ono c, Masayoshi Esashi b, Sih-Ling Yeh a, Tsung-Tsong Wu a,⇑ a
Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan WPI-AIMR, Tohoku University, Sendai 980-0845, Japan c Department of Mechanical Engineering, Tohoku University, Sendai 980-0845, Japan b
a r t i c l e
i n f o
Article history: Received 19 February 2014 Received in revised form 10 May 2014 Accepted 10 May 2014 Available online xxxx Keywords: Phononic crystal Lamb wave Piezoelectric plate Focusing Waveguiding
a b s t r a c t This paper presents results on the numerical and experimental studies of focusing and waveguiding of the lowest anti-symmetric Lamb wave in micro-fabricated piezoelectric phononic plates. The phononic structure was based on an AT-cut quartz plate and consisted of a gradient-index phononic crystal (GRIN PC) lens and a linear phononic plate waveguide. The band structures of the square-latticed AT-cut quartz phononic crystal plates with different filling ratios were analyzed using the finite element method. The design of a GRIN PC plate lens which is attached with a linear phononic plate waveguide is proposed. In designing the waveguide, propagation modes in square-latticed PC plates with different waveguide widths were studied and the results were served for the experimental design. In the micro-fabrication, deep reactive ion etching (Deep-RIE) process with a laboratory-made etcher was utilized to fabricate both the GRIN PC plate lens and the linear phononic waveguide on an 80 lm thick AT-cut quartz plate. Interdigital transducers were fabricated directly on the quartz plate to generate the lowest anti-symmetric Lamb waves. A vibro-meter was used to detect the wave fields and the measured results on the focusing and waveguiding of the piezoelectric GRIN PC lens and waveguide are in good accordance with the numerical predictions. The results of this study may serve as a basis for developing an active micro plate lens and related devices. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The renewed properties, such as acoustic band gap [1–5], negative refraction [6,7] and local resonances [8–10], of acoustic waves in 2-D periodic structures called phononic crystals (PC) have been reported intensively in the literatures. In particular, due to the potential applications in high frequency Lamb wave devices and the readiness of micro-fabrication techniques, propagation of acoustic waves in 2-D PC plate structures have aroused much attention in physics as well as engineering communities over the last decade. In the literatures, the band gaps of Lamb waves in micro-fabricated silicon phononic plates have been demonstrated [11–15] and the related design and fabrication of Lamb wave resonators have also been reported [16–18]. By using the phononic band gap properties of Lamb waves in silicon thin plates, their results showed that the quality factors of the as-made Lamb wave resonators can be improved significantly and the devices’ size can be reduced. One of the important applications of phononic crystal ⇑ Corresponding author. Tel.: +886 2 33665891. E-mail address: [email protected] (T.-T. Wu).
is utilization of its complete band gap to form a linear waveguide by removing one or several layers of phononic crystals. Recently, numerical [19–22] as well as experimental [23] studies have been reported and shown that with a proper design, Lamb waves can be guided with little attenuation along the plate waveguide. A review on the propagation of Lamb waves in plates with phononic structure can be found in Ref. [24]. Regards to the focusing of bulk waves in PC structures, the gradient-index (GRIN) PC has been proposed [25,26]. Their results demonstrated that GRIN PC allows acoustic wave focusing over a wide range of operating frequencies and making it suitable for applications such as flat acoustic lenses and acoustic wave couplers. Recently, the concept of GRIN PC has been applied to the phononic plate structures and similar focusing capabilities were found. Wu et al. [27] numerically demonstrated focusing of the lowest anti-symmetric Lamb wave in a GRIN PC silicon plate and its application as a beam-width compressor for compressing Lamb wave into a stubbed phononic tungsten/silicon plate waveguide. Zhao et al. [28] employed a non-contact laser-ultrasonic technique to excite and detect flexural Lamb waves within a silicon GRIN PC plate and confirmed the focusing experimentally. However,
http://dx.doi.org/10.1016/j.ultras.2014.05.007 0041-624X/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: M.-J. Chiou et al., Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.05.007
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realization of an active GRIN PC plate lens which is based on piezoelectric crystal remains necessary. In this paper, we demonstrate focusing and waveguiding of the lowest anti-symmetric (A0) Lamb waves in GRIN PC plates with/ without a PC waveguide. Toward the development of an active GRIN PC plate device, the piezoelectric AT-cut quartz was chosen as the plate material in this study. In the following, band structures of the square-latticed AT-cut quartz phononic crystal plates with different filling ratios are analyzed using the finite element method (FEM) firstly, then, the design of an AT-cut quartz based GRIN PC plate lens and the PC plate waveguide are followed. Finally, the micro-fabrication processes of the AT-cut quartz GRIN PC plate devices are described and the measured results of the focusing and guiding wave fields are discussed. Fig. 2. The band structure of a square-latticed quartz PC plate with lattice constant a and radius of the air hole r equal to 80 lm and 24 lm.
2. Band structure of an AT-cut quartz PC plate Before calculating the band structure of the PC plate, we examined the anisotropic characteristics of a pure AT-cut quartz plate with thickness (z-direction) equal to 80 lm. Fig. 1 shows the slowness curves of the three lowest plate modes of the pure AT-cut quartz plate on the x–y plane evaluated at 10 MHz, i.e., the antisymmetric Lamb mode (A0 mode, red line), the shear horizontal Lamb mode (SH0 mode, blue line) and the symmetric Lamb mode (S0 mode, green line). In the calculation, the x-axis of the AT-cut quartz plate was aligned with the X-axis of the crystal axes. The result in Fig. 1 showed that the slowness curves of the three lowest plate modes along the x and y-axis are slightly different; however, 90° symmetry exists. Fig. 2 shows the band structure of a squarelatticed quartz PC plate with lattice constant a and radius of the air hole r equal to 80 lm and 24 lm, respectively, and the corresponding filling fraction (ff) is 0.283. On considering the anisotropy of the AT-cut quartz, the irreducible Brillouin zone of the square-latticed PC plate are bounded by C, X, M and Y as shown in the inset of Fig. 2. In the figure, the displacement components of the Lamb modes (ux, uy, uz) are transformed into uL, uSH, uSV which are defined as
uL ¼ ux cos h þ uy sin h uSH ¼ ux sin h þ uy cos h
ð1Þ
uSV ¼ uz ; where uL and uSH are the components along and perpendicular to the wave vector direction, respectively and h is the angle between 1.0
A0 0.5
SH 0
ky(π/a)
S0 0.0
-0.5
-1.0 -1.0
-0.5
0.0
0.5
1.0
kx(π/a) Fig. 1. The slowness curves of the three lowest plate modes of the pure AT-cut quartz plate.
the wave vector and the x-axis. Further, we defined the ratio of polarization (ROP) as
RRR ROP ¼ RRR
ðu2x
u2p dV ; þ u2y þ u2z ÞdV
ð2Þ
where up represents uL, uSH, uSV. By assigning the green color to uL, blue to uSH, and red to uSV, and utilizing the ROP, the band structure of the square-latticed quartz PC plate was obtained as the color curves shown in Fig. 2. Along the CX section, uL, uSH, uSV are aligned with ux, uy, uz, respectively and the three lowest Lamb modes, A0 mode (red), SH0 mode (blue) and S0 mode (green) are similar to the case of a homogenous AT-cut quartz plate. However, the folded branch of the A0 mode becomes brown color (combination of red and green) which means both uL and uSV exist. Detail calculation of the band structure, one finds that there are level repulsions (anti-crossing) appeared (X1, X2, X3, X4) along the boundaries XM and MY [5]. In those regions, the polarizations of the wave displacements cannot be decoupled completely and therefore most of the colors of the bands are not pure green, red or blue. The band structure along the CY section is similar to that of the CX section except for a level repulsion appeared at X5. For later design of the GRIN PC plate lens, we calculated the band structures of the A0 mode along the CX direction with different filling fraction and the results are shown in Fig. 3(a). The results showed that the frequency band of the A0 mode drops and the group velocity decreases with the increase of the filling fraction. When the filling fraction is high (e.g. ff = 0.283) and the operation frequency greater than about 12 MHz, we note that the group velocity drops dramatically and vanishes at the X point. To evaluate the effect of anisotropy on the propagation of A0 mode in the square-latticed quartz PC plate, the equal frequency curves (EFCs) with different filling fractions evaluated at frequency equal to 10 MHz were calculated and the results are shown in Fig. 3(b). The results showed that the EFC becomes more circular (i.e., more isotropic) as the filling fraction increases. Fig. 3(c) shows the anisotropic ratio g of the A0 mode as a function of angle away from the CX direction (propagation direction) with different filling fractions. The anisotropic ratio was defined as the ratio of the wavenumber (kCX) along the CX-direction and those along the direction (k) which lies between the CX-direction and CY-direction, i.e., g = kCX/k. The results showed clearly that the anisotropic ratio decreases as the filling fraction increases from zero (pure AT-cut quartz plate) to ff = 0.283. The maximum anisotropic ratio is about 1.07 which appears on the curve of filling fraction equal to 0.126 (green line) and is about 7% difference as compared with the unity (i.e., CX-direction). Further, for the GRIN PC plate lens, the angles h of the focusing rays are less than 45° in general and therefore the anisotropic ratio can be down to less than 5% (Fig. 3(c)).
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(a)
level repulsions, the dispersion of the A0 mode is relatively simple and similar to that of a homogenous AT-cut quartz plate. For the A0 mode propagates along the CX direction, the group velocity drops dramatically around the X point, when the filling fraction is high. Thus the operation frequency of a GRIN PC plate lens should keep away from this region to have a better focusing. In addition, the maximum anisotropy ratio of the A0 mode is less than 5% which is reasonable to make the weakly anisotropic assumption in the following GRIN PC plate lens design.
ff=0 ff=0.126 15
ff=0.203 ff=0.250
Frequency (MHz)
ff=0.275 ff=0.283 10
3. Design of a GRIN PC plate lens 5
0
Γ
Shown in Fig. 4(a) is a schematic diagram of the GRIN PC plate lens on the AT-cut quartz plate with the input wave propagates along the x-direction. There are 41 rows of air holes along the y-direction arranged in square lattices with radii of the holes decreasing from a maximum of 24 lm (ff = 0.283) at the central line (y = 0) to a minimum of 16 lm (ff = 0.126) at the lens boundaries (y = ±20a). The lattice constant (a) and thickness (h) of the homogeneous plate are equal to 80 lm. Since the largest anisotropic ratio was estimated about 1.07 (ff = 0.126) along the
X
Wavevector (π/a)
(b)
1.0
(a) 0.5
ff=0
ky (π/a)
ff=0.126 ff=0.203 0.0
ff=0.250 ff=0.275 ff=0.283
-0.5
74a
(b)
-1.0 -1.0
-0.5
0.0
0.5
79a
1.0
kx (π/a)
y
(c) 1.10
x
Anisotropic ratio η
1.08
(c) 1.06 ff=0
1.04
ff=0.126 ff=0.203
1.02
ff=0.250 1.00
ff=0.275 ff=0.283
0.98
0
30
60
90
Angle θ (degree) Fig. 3. (a) Band structures of the A0 mode with different filling fraction. (b) Equal frequency curves with different filling fractions. (c) The anisotropic ratio g of the A0 mode as a function of angle away from the CX-direction with different filling fractions.
Based on the above analyses, although the band structure of the square-latticed quartz PC plate is complicated and contains several
Fig. 4. (a) Schematic diagram of the GRIN PC plate lens on an AT-cut quartz plate with the input wave propagates along the x-direction. (b) The focusing image of the lowest A0 mode in the designed GRIN PC quartz plate lens with a line force placed at x = 2a. (c) The amplitude distribution along the y-direction of the plate lens at x = 74a.
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boundaries of the lens, we assumed that the air/AT-cut quartz PC plate is weakly anisotropic, and approximated the refractive index of the A0 mode as the refractive index along the direction, n ¼ mCmX ,
where mCX is the group velocity along the CX direction and v is the referenced group velocity of the A0 mode of a homogeneous AT-cut quartz plate evaluated at the designed frequency 10 MHz. The refractive index profile in the form of a hyperbolic secant n = n0 sech(a y) was adopted, where n0 is the refractive index along the x-axis and a is the gradient coefficient [27]. Since the radii of the holes at y = 0 and y = ±20a are known, the gradient coefficient a were determined as 3.86 104 lm1. The radii of the holes along the y-direction can thus be found from the hyperbolic secant function shown above. Shown in Fig. 4(b) is the focusing image of the lowest A0 mode in the designed GRIN PC quartz plate lens with a line force placed at x = 2a. The result shows that there are several focusing regions appeared, the maximum one appeared at x = 79a with maximum amplitude 3.57 times and the second maximum one at x = 74a with maximum amplitude 3.53 times of that at the source. The focal distances of 79a or 74a are longer than the one predicted by the lens formula p/(2a), 51a, which is consistent with the observation given in reference [27]. In addition, unlike the sharp focusing of the decoupled shear vertical (SV) mode of BAW in the GRIN PC [25,26], the neck of the focus region is longer. The dotted ellipse in Fig. 4(b) shows the region with 90% of the maximum amplitude which starts at about 70a and ends at 88a. Shown in Fig. 4(c) is the amplitude distribution along the y-direction of the plate lens at x = 74a which shows the compressing of the wave beams and the full-width at half maximum (FWHM) is 4.5a. As compared with the aperture of the plate lens 40.5a, the compressing is about 11.1%. To investigate the bandwidth of the GRIN PC quartz plate lens, we set the operation frequency at 9 MHz and the results showed a similar focusing but the focusing regions appeared at longer distances away from the source, i.e., the maximum amplitude appeared at x = 81a while the amplitude amplification is 3.91 times. On the contrary, when the operation frequency was set at 11 MHz, the results showed that the maximum amplitude appeared at x = 73a with the amplitude amplification equal to 3.65 times. Based on the calculated results, it is found that slight deviation of the operation frequency from the designed one causes little variation on the amplitude amplifications, however, the changing of the focus distance have to be taken into consideration. Fig. 5 shows the focusing image when the operation frequency is increased to 12 MHz. Although a sharp focusing region still can be found at around x = 50a, the wave pattern becomes much more complicated. The reason may due to the sharp change of the group velocity of the A0 mode in this frequency range (see the CX section of Fig. 3(a)) which makes the distribution of the refraction index of the PC layers not so smooth.
Fig. 5. The focusing image with the operation frequency equal to 12 MHz.
Fig. 6. The schematic diagram of the GRIN PC plate lens which is connected with a PC plate waveguide.
4. GRIN PC plate lens with waveguide Shown in Fig. 6 is a schematic diagram of the GRIN PC Plate lens which is connected with a PC plate waveguide. To maximize the energy compressed into the waveguide, proper design of length of the GRIN PC plate lens, complete band gap and width of the PC plate waveguide are crucial. The design of the GRIN PC Plate lens is the same as the one shown in Fig. 4(a) except that the lens is terminated at x = 74a which is at the second maximum of the focusing regions. The reason for choosing the second instead of the maximum amplitude (x = 79a) is that their amplitudes are approximately the same; however, the former one makes the lens length shorter. To design a PC plate waveguide with operation frequency at 10 MHz, a square-latticed air/AT-cut quartz PC plate was chosen. The thickness of the plate waveguide is the same as that of the GRIN plate lens, the lattice constant is 200 lm and the radius of the hole is 96 lm. Numerical calculation on the band structure of the square-latticed air/AT-cut quartz PC plate was conducted and the result showed that a complete bandgap (8.7–10.9 MHz) formed around the operational frequency 10 MHz. To form a linear waveguide, one row of the periodic holes along the x-direction was removed to investigate the wave modes inside the waveguide with the waveguide width w = a. The supercell technique [19] was utilized to calculate the band structure of the square-latticed air/AT-cut quartz PC plate for wave propagates along the CX direction. The result shown in Fig. 7(a) shows that there are three propagating modes appeared within the band gap and only two of them (points 1 and 2) are intersected with the operation frequency 10 MHz. The mode shapes shown on the right part of Fig. 7(a) demonstrate that both of the two modes belong to the quasi-shear horizontal modes, i.e., the displacements are mostly on the x–y plane. Fig. 7(b) shows the propagating modes for the case of w = 3a where three of the intercepted points (points 2, 3, 4) belong to the quai-shear horizontal mode and one belong to the quasi-antisymmetric Lamb mode (point 1, quasi-A0 mode with displacement mostly on the x–z plane). For the cases of w = 2a and w = 4a, further simulation results showed that the number of the quasi-A0 mode are 2 and 4, respectively and those of the quasiSH0 mode are 2 and 1, respectively. Based on the above simulation results, we chose the case of w = 3a to make the propagation of the quasi-A0 mode in the waveguide simpler, i.e., only one quasi-A0 mode exists. Shown in Fig. 8(a) is the simulated image of the quasi-A0 mode in the designed GRIN PC quartz plate lens with PC plate waveguide. The result shows that the focused plate waves are compressed into the waveguide at x = 74a successfully. The maximum amplitude of the quasi-A0 mode in the waveguide is about 3.2 times of that at the source. Fig. 8(b) is the amplitude distribution along the transverse direction (y-axis) at x = 84a (middle of the waveguide and lies at the center of the neighboring two holes) which shows the FWHM is about the width of the waveguide, i.e., 3a. The two side peaks appeared at y = 18a and 23a are the local resonances between the side walls of the neighboring two holes; however, they decay very fast along the transverse direction.
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Fig. 7. The propagation modes inside the band gap for the case of (a) w = a and (b) w = 3a.
Fig. 8. (a) The simulated image of the quasi-A0 mode in the designed GRIN PC quartz plate lens with PC plate waveguide. (b) The amplitude distribution along the transverse direction (y-axis) at x = 84a.
5. Micro-fabrication of the GRIN PC plate lens with/without waveguide In the experimental part, double-side polished 80 lm thick ATcut quartz plates were utilized to fabricate the GRIN plate lens with and without PC waveguide. The geometrical parameters of both the plate lens with and without waveguide were the same as those
Fig. 9. The micro-fabrication process of the GRIN PC lens and waveguide.
used in the numerical simulations described in Sections 3 and 4. For the case of the plate lens with waveguide, the lens was terminated at x = 74a. We chose the case of w = 3a to insure that there is only one A0 mode exists in the waveguide.
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The micro-fabrication process is shown in Fig. 9. The upper and lower sides of the quartz plate were etched in a successive order in order to make the sidewall of the holes close to vertical. In the beginning, Au/Cr films were sputtered on both sides of the quartz plate. The Au film acts as a seed layer for the Ni electroplating and the Cr film as the adhesion layer between the Au film and quartz plate (Fig. 9(a)). An 8 lm thick photoresist (AZP 4620) was patterned on the front-side of the quartz plate to form the periodic cylinders, which defined the area for the Ni electroplating (Fig. 9(b)). It is noted that the double-side Au/Cr seed layers and the double-side Ni electrodeposition (Fig. 9(c)) were adopted to prevent the 80 lm thick quartz plate from bending and breaking due to the unbalanced intrinsic stress during the Ni electroplating. In the electroplating, the current density was 12 mA cm2, the electroplating bath was controlled at 55 °C and the Ni deposition rate was 0.1 lm min1. After the Ni electroplating, the photoresist was removed and the device was ready for the RIE process (Fig. 9(d)). Before the RIE process of the quartz plate, Xe gas was introduced to remove the Au/Cr layer which was not covered by the Ni mask. SF6 gas was used to etch the front side of the quartz plate with the RF stage temperature of 20 °C, gas pressure of 3 mTorr, power of 100 W, and a self-bias voltage of 390 V. The etching rate was 450 nm min1 under the aforementioned conditions. In order to fabricate the cylindrical air hole with a vertical sidewall, the etching process was divided into two parts. First, the etch depth of the front side was around 40 lm and the Ni mask was removed by using the nitric acid, as shown in Fig. 9(e). Then, the back-side of quartz plate was patterned again, followed by Ni electroplating and photoresist removal. By using the same etching parameters as those of the front side, the back side of the quartz plate was etched and the selectivity of the Ni mask to quartz plate was 20. Again, by using nitric acid, the Ni mask was removed and the PC gratings were accomplished (Fig. 9(f)–(h)). Finally, the IDTs were fabricated onto the left side of the GRIN PC plate lens to generate 10 MHz Lamb waves using the photolithography process (Fig. 9(i) and (j)). The gap between the IDT and the plate lens is 2a. The wavelength of the IDT, k, is 226 lm, the aperture is 15k and the number of electrode pairs is 20. Shown in Fig. 10 is the picture took at the junction of the GRIN PC plate lens and the linear PC waveguide.
6. Experimental results and discussions The wave fields (z-component of the displacement) in the GRIN plate lens with/without waveguide generated by the IDTs were detected using an ultra-high frequency vibrometer (UHF-120, Polytec). Fig. 11(a) shows the measured wave distribution of the GRIN PC plate lens under the IDT excitation at 10 MHz. The imaging result shows clearly the Lamb wave focusing in such a PC plate lens. The maximum amplitude appeared approximately at x = 69a which is about 3.4 times of that at the source x = 2a. The length of the neck region (90% of the maximum amplitude) is around 5a
Fig. 11. The measured wave distributions of (a) the GRIN PC plate lens and (b) the GRIN plate lens with a linear waveguide at 10 MHz.
and the FWHM is about 4a. In comparisons with the numerical predictions shown in the Section 3, where the maximum focusing amplitude is 3.53 times and FWHM is 4.5a, we find they are in good agreements. However, location of the maximum focusing amplitude (x = 69a) and length of the neck region (5a) are shorter than those in the numerical predictions, i.e., x = 74a and 18a. On examining the fabricated GRIN PC lens, we found that the radius of the largest hole is 25 lm instead of the designed one 24 lm, while that of the smallest hole is 15 lm instead of 16 lm. Possible reason for these deviations may come from the inaccuracy in the micro-fabrication of holes. In the etching process, smaller holes need more time than the larger holes to etch through. Therefore, as the smallest holes around the edges of the plate lens reached the design sizes, the largest hole along the centerline becomes bigger than the designed one and results an increase of the gradient coefficient a. This in turn may shorten the focal distance and length of the neck region according to the previous numerical simulations. It is noted that the deviation in the position and length of the neck region of the focusing may be minimized by fine tuning of the designed radii according to the experimental data. The measured wave field in the GRIN plate lens with a linear waveguide is shown in Fig. 11(b). The results show that the impinging of the focused Lamb waves into the waveguide and the maximum amplitude inside the waveguide is about 2.4 times of that at the source. The amplitude amplification is smaller than that predicted in the previous simulation, i.e., 3.2 times. Similarly, this inaccuracy may again due to the etching problem mentioned previously, i.e., the focused region in the GRIN plate lens move forward instead of just at x = 74a. 7. Summary and conclusions
Fig. 10. The picture at the junction of the GRIN PC plate lens and the linear PC waveguide.
In summary, we have conducted numerical and experimental studies of Lamb wave focusing in an AT-cut quartz GRIN PC plate lens without/with a linear PC waveguide. An approximated method for designing the GRIN PC plate lens was utilized to predict the focusing and waveguiding of the PC plate lens with PC waveguide. The micro-fabrication process of the AT-cut quartz GRIN PC lens and waveguide was developed. The measured focusing
Please cite this article in press as: M.-J. Chiou et al., Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.05.007
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Please cite this article in press as: M.-J. Chiou et al., Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates, Ultrasonics (2014), http://dx.doi.org/10.1016/j.ultras.2014.05.007