Analysis and Design of Focused Interdigital Transducers

formance, it is essential to analyze the FIDTs with a variety of geometric shapes. However, among the existing studies concerning the diffraction of FIDTs, a detailed analysis and design of FIDTs is still in paucity. In this paper, we adopted the exact angular spectrum of plane wave theory (ASoW) to calculate the amplitude fields of FIDTs on Y-Z lithium nio-bate (LiNbO3) with the shape as a concentric circular arc

and the concentric wave surface. Based on the calculation results, we discussed the variations of the amplitude fields induced by changing number of pairs, degree of arc, and geometric focal length. In addition, the focusing properties of FIDTs on the (100)-oriented GaAs substrate were also analyzed and discussed. We also summarized the guiderules for designing a FIDT via four important factors. It is worth noting that the results of this study provide an important basis for designing various FIDTs to fit the desired appli-cations.

I. Introduction

T

he_{focused interdigital transducer (FIDT) has been} used widely in many applications recently due to its advantages of achieving high intensity and large beamwidth compression ratio. In signal processing, they were used as convolvers [1]–[4], time-Fourier transformers [5], and radio frequency (RF) channelizers [6]. As to optical communication, acousto-optic tunable filters (AOTF) with FIDTs can improve their performance effectively [7], [8]. In acousto-electric (AE) applications, FIDTs can generate high intensity acoustic field and enhance AE effect to ma-nipulate electron-hole pairs in GaAs quantum well [9]. An experimental study on the surface acoustic wave (SAW) band gap of two-dimensional (2-D) phononic structures in micrometer scale was proposed recently [10]. To realize a phononic crystal waveguide with a line defect (as shown in Fig. 1), a device that can provide SAW sources with high intensity and large beamwidth compression ratio is needed. The FIDT is very suitable for this application.

Manuscript received September 6, 2004; accepted December 20, 2004. The authors thank the financial support of this research from the National Science Council of R.O.C. through the grant NSC92-2212-E-002-001.

The authors are with the Institute of Applied Mechan-ics, National Taiwan University, Taipei, Taiwan (e-mail: daline@ndt.iam.ntu.edu.tw).

Fig. 1. Schematics of FIDTs and a 2-D phononic crystal waveguide.

The FIDT was first discussed by Kharusi and Farnell [11] in 1972. They proposed two sets of FIDTs with shapes that are a circular arc and wave surface, respectively. The results show that both of them are able to focus SAW and the FIDT with the shape as the wave surface has better focusing properties. In addition, they emphasized that the focusing properties of the FIDTs become worse when in-creasing number of pairs. Green and Kino [1], and Green

et al. [2] used Huygen’s principle to calculate the

ampli-tude field of a FIDT with circular-arc shape and predicted that the amplitude field has only one focal point. However, according to Fang and Zhang’s [12] calculation and exper-imental results in 1989, this amplitude field has no focal point, but a very long and narrow SAW beam in the main propagation direction. In 2003, de Lima et al. [13] studied the amplitude field of a FIDT on the (100)-oriented GaAs substrate. Although FIDT has been proposed over three decades, most of the literature simply involved the simple “curved” interdigital transducers (IDTs) and did not con-centrate on the design of FIDTs. In this paper, we present detailed analysis and design of a focused-type SAW trans-ducer by angular spectrum of plane wave theory.

II. Angular Spectrum of Plane Wave Theory

As shown in Fig. 2, we set the X-Y plane on the surface of a piezoelectric material and the IDT’s fingers are along the Y -axis. If the cut of the substrate is fixed, the wave vector ¯k can be expressed as a function of the propagation

angle φ. In addition, we assume that there are no disper-sion and propagation loss in the half-infinite substrate, so plane harmonic waves on this surface can be expressed

Fig. 2. Illustration of a conventional straight interdigital transducer.

as exp[−j(Xkx+ Y ky)], where kx and ky are the X and

Y components of wave vector k(φ), respectively.

Further-more, because the system is linear, the total amplitude distribution can be evaluated by a scalar field [14] as:

ψT(X, Y ) = N
i=1 1 2π  ∞ −∞ψi(ky) exp [−j {xikx(ky) + yky}] dky for X ≥ 0, (1) where Xi is equal to 0,−p, −2p, −3p, . . ., −Np and N is the finger number of IDT. ψi(ky) is the inverse Fourier transform of the acoustic source function ψi(Xi, Y) and is expressed as: ψi(ky) =  ∞ −∞ψi(X  i, Y) exp(jYky)dY. (2) The acoustic source function ψi(Xi, Y) is taken as:

ψi(Xi, Y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ A· (−1)|Xi
/p| |Y| < W 2 A 2 · (−1)| X_i
/p| |Y_{| =} W 2 0 |Y| > W 2 , (3)

where W is the aperture of IDT, A is an arbitrary con-stant, and p is the period of metal strips of IDT. Based on (1), (2), and (3), we can calculate the amplitude field of a conventional straight IDT on the surface of a half-infinite substrate.

III. FIDTs With the Shape as a Concentric Circular Arc

A FIDT with circular-arc shape [as shown in Fig. 3(a)] was used frequently because of its simplicity. However, when its finger number increases, the focusing properties

Fig. 3. FIDTs with the shape as (a) the conventional structure and (b) the concentric structure.

become unstable. In order to overcome this drawback, the concentric structure, as shown in Fig. 3(b), is adopted to be the shape of our FIDT. The amplitude field of a FIDT with the shape as a concentric circular arc can be cal-culated by using the ASoW theory as long as the acous-tic source function is modified by the equivalent aperture method. The equivalent aperture method, presented by Kharusi and Farnell [11], describes that a curved IDT can be approximated by a straight IDT with equiphase dis-tribution. Therefore, the new acoustic source function is given by:

ψi(Xi, Y) = ψi(Xi, Y)· exp[jk0· ∆(Xi)], (4) where ψi(Xi, Y) as shown in (3), is the acoustic source function of the straight IDT. ∆(Xi) is the path difference between the real aperture and the equivalent aperture of

ith _{finger of the curved IDT. According to (1), (2), and}

(4), the total amplitude field of a FIDT with the shape as a concentric circular arc can be calculated.

To design a FIDT, three design parameters must be considered: number of pairs Np, degrees of arc Da, and

Fig. 4. Total amplitude fields of FIDTs with the shape as a concentric circular arc. (a) Design 1-1, (b) design 1-2, (c) design 1-3, (d) design 2-1, (e) design 2-2, (f) design 2-3, (g) design 3-1, (h) design 3-2, (i) design 3-3.

TABLE I

FIDTs With a Variety of Designs. Number Degree Geo. focal Wave Design of pairs of arcs length length parameters (Np) Da) (fL) (λ) Design 1-1 5 pairs 40◦ 50 λ 58.13 µm Design 1-2 10 pairs 40◦ 50 λ 58.13 µm Design 1-3 20 pairs 40◦ 50 λ 58.13 µm Design 2-1 10 pairs 20◦ 50 λ 58.13 µm Design 2-2 10 pairs 40◦ 50 λ 58.13 µm Design 2-3 10 pairs 60◦ 50 λ 58.13 µm Design 3-1 10 pairs 40◦ 25 λ 58.13 µm Design 3-2 10 pairs 40◦ 50 λ 58.13 µm Design 3-3 10 pairs 40◦ 100 λ 58.13 µm

geometric focal length fL. Nine sets of different design pa-rameters are listed on Table I, where λ means the wave-length of SAW excited by a FIDT. In general, λ is equal to twice the period of the metal strips. In the following, we discuss one by one how the design parameters affect the amplitude fields. The substrate we used here is Y-Z LiNbO3.

A. Number of Pairs (Np)

The design parameters are listed in design 1-1 to 1-3 of Table I, and the corresponding simulated results are shown in Figs. 4(a)–(c). In Figs. 4(a)–(c), we observe that, because of the concentric structure, the total amplitude fields are quite similar no matter how many Np are. Fur-thermore, the total amplitude fields have a narrow, long, strong SAW beam, not a focal point. The results are con-sistent with Fang’s experimental results [12]. Besides, as shown in Fig. 5(a), the amplitude is proportional to Np.

The beamwidth compression ratio ηc is defined as:

ηc=

H Wc

, (5)

where H is the 3 dB transverse beamwidth and Wc is the equivalent aperture of the central finger of FIDT. Fig. 5(a) also shows that the beamwidth compression ration ηc does not change obviously when we change Np. In Fig. 5(c), we observe that the amplitude field has a peak at about 1.5 times of the geometric focal length (fL). It is worth noting that the amplitude maintains half of its maximum amplitude until about 4 times the fL.

Fig. 5. Cross sections of amplitude fields of FIDTs with the shape as a concentric circular arc. (a) and (b) show the X-cross sections with different number of pairs (design 1-1 to 1-3) and different degree of arcs (design 2-1 to 2-3), respectively. (c) and (d) show the cross sections along Y = 0 with respect to (a) and (b), respectively.

B. Degree of Arc (Da)

The design parameters are listed in design 1 to 2-3 of Table I, and the corresponding simulated results are shown in Figs. 4(d)–(f). The results show that, if we change the degree of arc (Da) of a FIDT, the total amplitude fields become much more different. This shortcoming can be avoided as long as Da is not too small [Figs. 4(e) and (f)]. Fig. 5(b) shows that, with the increasing of Da, the beamwidth compression a ratio become larger but the am-plitude remains almost constant. In Fig. 5(d), we find that the amplitude field has a peak at about 1.5 times the fL when Da is not too small. In contrary, when Da is too

small, the amplitude field has no obvious peak, but a cor-responding SAW beam that maintains half of its maximum until about four times the fL.

C. Geometric Focal Length (fL)

The design parameters are listed in design 3-1 to 3-3 of Table I, and the corresponding simulated results are shown in Figs. 4(g)–(i). As shown in Fig. 4(g), if the geometric focal length (fL) is too small, the total amplitude field becomes unstable. With the increasing of fL, as shown in Figs. 4(h) and (i), the total amplitude field becomes gradually identical. That is, fLshould not be too small in order to obtain stable focusing properties.

Fig. 6. Wave surface of Y-Z LiNbO3.

IV. FIDTs With the Shape as the Concentric Wave Surface

A FIDT with the shape as the wave surface was first proposed by Kharusi and Farnell [11]. They reported that its focusing ability is better than that of the FIDTs with the circular-arc shape. The wave surface is the locus of points tracked by the end of the energy velocity vector ve and is drawn from a fixed original point O as the propa-gation direction varies. For a plane wave, the projection of the energy velocity vector onto the propagation direction is equal to the magnitude of phase velocity vector, that is:

ve· η = v, (6) where n is the propagation direction vector and v is the magnitude of the phase velocity vector in n direction. In this paper, we adopted the effective permittivity approach to calculate the phase velocities in different propagation directions. By this approach, the ASoW theory can be extended to analyze a layered SAW device in the future [15], [16]. Based on (6), the wave surface of Y-Z LiNbO3is

obtained and shown in Fig. 6. The result shows that the curvature of a circular arc is somewhat smaller than that of the wave surface.

In this section, we also use the concentric structure as the shape of the FIDT due to the same reason stated in Section III. The total amplitude field of the FIDT with the shape as the concentric wave surface was calculated by us-ing the ASoW theory. But ∆(Xi) in (4) should be replaced with the path difference between the real aperture and the equivalent aperture of the ith _{finger of the FIDT. The}

lo-cus of the FIDT can be determined by the polar coordinate (ζNve(Φ), Φ), in which Φ represents the direction of the energy velocity and ζNve(Φ) is the corresponding radius.

ζN is a proportional constant and is defined as:

ζN= (fL+ N p)

ve(Φ)

ve(Φ0)

, (7)

Fig. 8(a), we see that, when adjusting Np, the amplitude is proportional to Np and the size of the focal area makes no obvious change. In addition, Fig. 8(c) shows that the maximum of the amplitude field is located at 0.9 times of fL. In other words, the real focal length is somewhat shorter than the geometric focal length.

B. Degree of Arc (Da)

The design parameters are listed in design 2-1 to 2-3 of Table I, and the corresponding simulated results are shown in Figs. 7(d)–(f). In Figs. 7(d)–(f), we observe that the total amplitude fields make significant difference when we change Da. This shortcoming can be avoided as long as Da is not too small [Figs. 7(e) and (f)]. As shown in Figs. 8(b) and (d), the energy of the generated SAW is highly focused on the focal area, and the real focal length appears at around 0.9 times the fL. With the increasing of Da, the size of the focal area become smaller. If Da is too small, the focal area will be unobvious. That is, if we want to focus the energy of the generated SAW at a point effectively, the degree of arc should not be too small.

C. Geometric Focal Length (fL)

The design parameters are listed in design 3-1 to 3-3 of Table I, and the corresponding simulated results are shown in Figs. 7(g)–(i). The results show that, with the increasing of fL, the size of the focal area becomes smaller. In other words, a larger fL lets the focal area approach to a focal point, but the size of the transducer also becomes larger.

V. Focusing Properties of GaAs Substrate

Two types of FIDTs discussed in Section III, including the shapes as a concentric circular arc and the concentric wave surface, were both on Y-Z LiNbO3. In this section,

we discuss the focusing properties of FIDTs on the (100)-oriented GaAs substrate.

For a FIDT with the shape as a concentric circular arc, the simulated results are shown in Fig. 9(a) and the cor-responding design parameters are listed in design 1-2 of

Fig. 7. Total amplitude fields of FIDTs with the shape as the concentric wave surface. (a) Design 1-1, (b) design 1-2, (c) design 1-3, (d) design 2-1, (e) design 2-2, (f) design 2-3, (g) design 3-1, (h) design 3-2, (i) design 3-3.

Table I. In Fig. 9(a), the shape of the total amplitude field is close to a focal point, not a SAW beam. In other words, a FIDT with a shape as a concentric circular arc cannot generate a SAW beam. The wave surface of GaAs and a circular arc are shown in Fig. 10. We observe that the wave surface is almost consistent with the circular arc. This is the reason why the amplitude field has a focal point, not a SAW beam. The results in Fig. 6 show that, if the cur-vature of a circular arc is a little lower than that of the wave surface, the amplitude field has a SAW beam. To verify this point of view, we chose a concentric elliptic arc, whose long axis is 1.8 times of the short axis, as the shape of the FIDT. The short axis of the concentric elliptic arc is along the main SAW propagation direction. The corre-sponding simulated results are shown in Fig. 9(b), and the design parameters are the same as that of Fig. 9(a). The results show that its amplitude field indeed has a narrow, long, strong SAW beam. In conclusion, no matter what the substrate is, we can generate a desired acoustic field that probably has a focal point or a beam, by adjusting the shape of a FIDT.

VI. Design of FIDTs

Through the above analyses, we summarized the guiderules for designing a FIDT with the desired focus-ing properties as follows:

A. The Focusing Property

The finger’s shape of a conventional FIDT (as shown in Fig. 1) is a circular arc. This leads to a poor focusing as the number of pairs increases. In order to overcome the drawback, the FIDT’s shape is suggested to be a concen-tric arc. As discussed in Section III, the amplitude fields of the FIDTs with different number of pairs are very sta-ble by using the concentric arc. In addition, the results of Figs. 4(g)–(i) and Figs. 7(g)–(i) show that the geometric focal lengths should not be too small in order to acquire better focusing properties.

Fig. 8. Cross sections of amplitude fields of FIDTs with the shape as the concentric wave surface. (a) and (b) show the X-cross sections of amplitude fields with different number of pairs (design 1-1 through 1-3) and different degree of arcs (design 2-1 through 2-3), respectively. (c) and (d) show the cross sections along Y = 0 with respect to (a) and (b), respectively.

B. Intensity

As seen in Section III, the amplitude of SAW is directly proportional to the number of pairs of a FIDT. Increasing the degree of arcs of a FIDT cannot enlarge the ampli-tude obviously. This is because the intensity of the SAW is defined as:

I(X, Y ) =|ψT(X, Y )| 2

, (8) and the intensity of the generated SAW is proportional to the square of the number of pairs of a FIDT.

C. Beamwidth Compression Ratio

In Section III, we find that the beamwidth compression ratio depends on the degree of arc of a FIDT. The bigger the degree of arc, the larger the compression ratio. Fig. 11 shows the comparisons of beamwidth compression ratio between a FIDT and a straight IDT on Y-Z LiNbO3. The

corresponding design parameters are listed in design 1-3 of Table I. As shown in Fig. 11, the intensity and compression ratio of a FIDT are much larger than those of a straight

IDT. In addition, the intensity and compression ratio of a FIDT with the shape as the concentric wave surface are bigger than those of a FIDT with the shape as a concentric circular arc.

D. Source Type

To excite a localized spot, a FIDT with the shape as the concentric wave surface is recommended. However, to generate a narrow, long, strong line SAW source, a FIDT with the shape as a concentric circular arc is not suitable due to the anisotropy of the substrate. It is suggested to adopt a FIDT with the shape as a concentric elliptic arc whose curvature is somewhat smaller than that of wave surface.

VII. Conclusions

In this paper, we systematically studied the focusing properties of FIDTs. The ASoW theory was adopted to calculate the amplitude fields of FIDTs with a variety of geometric shapes. Because the FIDT with the shape as a concentric arc has better focusing properties, the FIDTs

Fig. 9. Total amplitude fields of FIDTs on the (100)-oriented GaAs substrate. (a) FIDT with the shape as a concentric circular arc. (b) FIDT with the shape as a concentric elliptic arc.

Fig. 10. Wave surface of (100)-oriented GaAs.

with the shape as a concentric circular arc and the con-centric wave surface are analyzed, respectively. The results show that, for a FIDT with a concentric arc, the amplitude field is not sensitive to the number of pairs. The geomet-ric focal length should not be too small in order to obtain a better focusing property. To excite a localized spot, a FIDT with the shape as the concentric wave surface is sug-gested. However, to generate a narrow, long, strong line SAW source, a FIDT with somewhat smaller curvature of the transducer than the curvature of the wave surface is needed. In addition, we suggest that a FIDT with the shape as a concentric elliptic arc is suitable for this pur-pose. It is worth noting that the results of this study pro-vide an important basis for designing various FIDTs to fit the desired applications. In order to enhance the intensity of the generated amplitude fields, several design methods of conventional SAW devices, such as withdraw weighting,

single phase unidirectional transducers (SPUDT) . . . etc., also can be further applied to design FIDTs.

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plied Mechanics, National Taiwan University in 2004. He currently is on active military duty. His research interests are theoretical analysis and experiments of SAW devices.

Yung-Yu Chen received his B.S. degree

from National Chen-Kung University in 1995 and Ph.D. degrees from the Institute of Ap-plied Mechanics, National Taiwan Univer-sity in 2002. He is currently post doctor of the Institute of Applied Mechanics, National Taiwan University. His research interests are surface waves in layered anisotropic and/or piezoelectric materials, SAW devices and re-lated sensors.

Pei-Ling Liu was born in Taiwan in 1956.

She received the B.E. and M.S. degrees in civil engineering from National Taiwan University in 1979 and 1981, respectively, and the Ph.D. degree in civil engineering from the University of California, Berkeley in 1989. She joined the faculty of the Institute of Applied Mechan-ics, National Taiwan University in 1989. She is currently the director of the Institute. Her major research fields are nondestructive eval-uation of structures, structural reliability, and elastic waves. She is a member of the Ameri-can Society of Civil Engineers.