Acoustic coupling in capacitive microfabricated ... - IEEE Xplore

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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005

Acoustic Coupling in Capacitive Microfabricated Ultrasonic Transducers: Modeling and Experiments Alessandro Caronti, Member, IEEE, Alessandro Savoia, Student Member, IEEE, Giosuè Caliano, Member, IEEE, and Massimo Pappalardo, Member, IEEE Abstract—In the design of low-frequency transducer arrays for active sonar systems, the acoustic interactions that occur between the transducer elements have received much attention. Because of these interactions, the acoustic loading on each transducer depends on its position in the array, and the radiated acoustic power may vary considerably from one element to another. Capacitive microfabricated ultrasonic transducers (CMUT) are made of a twodimensional array of metallized micromembranes, all electrically connected in parallel, and driven into flexural motion by the electrostatic force produced by an applied voltage. The mechanical impedance of these membranes is typically much lower than the acoustic impedance of water. In our investigations of acoustic coupling in CMUTs, interaction effects between the membranes in immersion were observed, similar to those reported in sonar arrays. Because CMUTs have many promising applications in the field of medical ultrasound imaging, understanding of crosscoupling mechanisms and acoustic interaction effects is especially important for reducing cross-talk between array elements, which can produce artifacts and degrade image quality. In this paper, we report a finite-element study of acoustic interactions in CMUTs and experimental results obtained by laser interferometry measurements. The good agreement found between finite element modeling (FEM) results and optical displacement measurements demonstrates that acoustic interactions through the liquid represent a major source of cross coupling in CMUTs.

I. Introduction ecent imaging results obtained with capacitive microfabricated ultrasonic transducer (CMUT) probes, showing increased bandwidth and competitive image quality as compared to traditional piezoelectric transducers [1]–[3], have demonstrated the viability of the CMUT technology for diagnostic ultrasound imaging. Most of these results have been obtained using one-dimensional (1-D) linear and phased CMUT arrays and, more recently, curvilinear CMUTs [4]. However, one of the most promising applications of CMUT is the fabrication of 2-D ultrasonic phased arrays for real-time, 3-D medical imaging [5]. Cross-talk is a very important factor limiting the performance of phased arrays, resulting in reduced scanned sector and loss in the image resolution. In the earlier works on

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Manuscript received July 30, 2004; accepted February 23, 2005. The authors are with the Dipartimento di Ingegneria Elettronica, Universit` a Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy (e-mail: [email protected]).

cross-talk in immersion CMUT arrays, an interpretation of the main mechanism of coupling between the membranes was given, based on the propagation of surface acoustic waves at the fluid-silicon interface [6]–[8]. In this paper, a different interpretation is given based on the acoustic coupling, i.e., the loading effects of pressure waves propagated from a membrane on the surface of another membrane. In our previous investigations of cross-coupling mechanisms in CMUTs, acoustic interactions between the membranes were shown to have many important effects [9], [10]. Basically, they lead to a different acoustic loading on each membrane, depending on its position in the array element. As a consequence, when the membranes are driven in phase with a prescribed excitation voltage, the resulting average velocity of the membranes may be quite different from one another, and the pressure distribution over the surface of the array element may vary considerably from the ideal case of a piston transducer. Acoustic interaction effects are well-known in sonar transducer arrays, in which the mechanical load on some elements can become very low and even lead to a failure [11]. In the evaluation of the performance of sonar transducer arrays, the acoustic interaction effects often are investigated through the mutual radiation impedance. Mutual impedances have been calculated analytically or numerically by many authors for simple geometries and boundary conditions: circular pistons [12], circular flexural disks [13], and rectangular pistons [14] in an infinite rigid plane baffle; circular pistons on a rigid sphere [15]; and rectangular pistons on an infinite, rigid cylinder [16]. In most of these works, simplifications are made regarding the shape of the sources and the baffle that surrounds the radiating surfaces; in addition, transducers often are assumed to have uniform velocity profiles. More recently, Audoly [17] developed a computer model to calculate the mutual impedance matrix of densely packed sonar arrays with piston transducers, and investigated the influence of the size of the pistons on the pressure and velocity distributions on the array surface. Lee et al. [18] considered the acoustic interactions between 37 square pistons, mounted on a rigid infinite baffle, to calculate the acoustic radiation power of the transducer array. Lee et al. [19] calculated the mutual impedance of transducer arrays with planar circular pistons to determine the optimal arrangement of drivers for high radiation power efficiency.

c 2005 IEEE 0885–3010/$20.00

caronti et al.: fem study of acoustic interactions in cmut

2221 TABLE I Mechanical Properties of the Materials Used in Finite Element Simulations. Parameters

SiN

Si

Aluminum

Young’s modulus (GPa) Poisson’s ratio Density (kg/m3 )

180 0.25 2700

169 0.3 2330

67.5 0.35 2700

Fig. 1. Schematic cross section of a single CMUT cell.

The CMUT is made of a 2-D array of metallized membranes, electrically connected in parallel, and driven into flexural motion by the electrostatic force produced by an applied voltage. Except for dimensions and operating frequencies, CMUTs can be regarded as miniature sonar transducers. In this work, a finite-element study and experimental results of acoustic coupling in CMUTs are reported. The paper is organized as follows. In Section II, a finite element modeling (FEM) approach is used to evaluate the interaction effects between the membranes of an unbounded transducer, first, and of a typical CMUT array element with finite lateral size, after. The total radiation impedance of the element then is computed using FEM results. The calculation is carried out without evaluating the mutual radiation impedance of the membranes inside the elements, as described in the Appendix, in which definitions, concepts, and other points involved in numerical calculations are discussed. And, a discussion is provided on the use of Mason’s equivalent circuit for immersion CMUTs in transmission, including acoustic interactions. Section III reports the experimental results, obtained by laser interferometry measurements, that validate the results of the finite-element analysis and further enhance the understanding of the mechanisms of acoustic coupling in CMUTs. To explore several aspects of acoustic interactions, a number of experiments carried out on a variety of CMUTs with different size and number of elements are reported. The conclusions of this work are given in Section IV. II. Finite-Element Analysis of Acoustic Coupling in CMUTs The finite-element analysis (FEA) of acoustic coupling in CMUTs was performed using the commercial package ANSYS 8.1 (ANSYS Inc., Canonsburg, PA). Fig. 1 shows a cross section of the model of a single cell, including a plasma-enhanced chemical vapor deposition (PECVD) silicon nitride circular membrane of 20 µm radius and 0.5 µm thickness, with an aluminum top electrode of 13 µm radius and 0.25 µm thickness. A 0.6-µm thick silicon nitride layer is deposited over the top electrode, so that the total thickness of silicon nitride is 1.1 µm. The bottom electrode, which is assumed to be infinitesimally thin in the model, is covered with a 0.35-µm thick silicon nitride insulation layer and clamped at its bottom. The gap height is 0.35 µm. Note that the membrane is assumed without tensile stress, and it is supported along its rim on a 1-µm

wide silicon nitride post. In order to reduce computation time, the silicon substrate was not included in the FEM model. The material properties used in the simulations are listed in Table I. The eight-node, 3-D solid elements SOLID45 were used to model the structure, and acoustic elements FLUID30 were used to mesh the fluid medium. The fluid-structure interface (FSI) was flagged by surface loads to couple the structural motion and the acoustic pressure at the interface. A. Unbounded Transducer Acoustic interactions between the membranes of an unbounded CMUT transducer (having infinite lateral dimensions) were investigated by means of a FEM model, including only a single cell in contact with a fluid, with proper boundary conditions applied along the perimeter of the cell. This approach is based on the concept of image sources: the problem of a source confined between two parallel plane rigid boundaries is equivalent to an infinite array of image sources (mirror images of the original source) and no boundary [20]. Likewise, a CMUT membrane in contact with an infinitely long parallelepiped box of fluid with rigid walls is equivalent to a twofold infinity of identical image membranes, all lying in a plane transverse to the walls. A sketch of the unbounded CMUT, which is equivalent to the structure simulated by FEA, is shown in Fig. 2. The center-to-center distance between the membranes was set to 50 µm. To simulate an infinite extent of fluid in the z propagation direction, a fully absorbing layer (meshed with FLUID30 elements having unit boundary admittance) was placed at the top of the parallelepiped, thus avoiding reflection of outgoing pressure waves back on the transducer surface. In the analysis, the membranes were driven into harmonic motion by a 100 kPa uniform harmonic pressure applied over the metal electrodes. All electrostatic effects associated with the bias voltage [21], and the electrical excitation (including spring-softening [22]), were not taken into account in the simulations. A multifield analysis for solving coupled electrostatic, structural, and acoustic fields in CMUTs will be the object of a future paper. The response of the structure was characterized by means of the average displacement and pressure, as computed at the fluid-structure interface as a function of the excitation frequency. By symmetry, all membranes in the unbounded transducer have identical behavior. A plot of

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Fig. 2. A sketch of the unbounded CMUT transducer, made of an infinite matrix of circular membranes.

the average displacement and interface pressure of each membrane in a frequency range up to 20 MHz is shown in Fig. 3. A small resonance peak of displacement amplitude is observed at about 4 MHz, with a strong damping caused by the fluid loading. However, acoustic coupling between the membranes is responsible for efficient radiation into the fluid, as seen by the transmit pressure frequency response. The study of this unbounded transducer problem is of basic importance to understand the mechanism of acoustic interactions between CMUT membranes, and it will be regarded as a starting point for the analysis of the more realistic structure represented by an infinitely long CMUT element with finite lateral extent.

Fig. 3. Simulated membrane average displacement (top) and pressure (bottom) of an unbounded CMUT transducer in water.

B. Transducer Element of Finite Lateral Size One-dimensional CMUT arrays used for medical imaging applications typically consist of a number of long and narrow elements, including a few membranes per row and hundreds of membranes per column. As an example, Fig. 4 shows a portion of a fabricated, 64-element CMUT array, in which each element is 180-µm wide, 12-mm long, and includes 4 × 275 circular membranes of 40-µm diameter. In order to investigate the acoustic interactions between the micromembranes of a single CMUT array element, we modeled such a device by means of a 3-D FEM slice model. This slice model consists of two half-membranes placed onto a thin silicon substrate, and a fluid region in contact, bounded by absorbing acoustic elements FLUID130. By proper symmetry and boundary conditions applied along the edges of the structure, we were able to simulate an infinitely long element with four membranes per row, as

Fig. 4. Optical microscope image of a portion of a 64-element CMUT array.


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Fig. 6. Simulated membrane average displacement (top) and pressure (bottom) of the CMUT array element in water.

Fig. 5. An infinitely long CMUT array element with four membranes per row (top), and its reduced FEM slice model (bottom).

shown in Fig. 5. It should be noted that a simpler configuration of the element, with the membranes lined up on each row and column has been considered in place of the actual configuration of Fig. 4 to reduce model complexity and computation time. In the simulations, all membranes were driven in phase with a 100-kPa uniform excitation pressure applied over the electrodes. Without taking acoustic interactions into account, the velocity of each membrane within the array element would be the same. However, due to the finite lateral size of the transducer, acoustic interaction effects result in a different behavior of the membranes, depending on their position along each row of the element. By symmetry, the vibration of outer membranes differs from that of inner membranes, as shown by the frequency response of average displacement and transmit pressure of Fig. 6. For convenience, the membrane displacement and pressure of the equivalent unbounded transducer are plot-

ted in Fig. 6. Deviations are pronounced in a frequency region between 4 MHz and 6 MHz, that is, around the immersion resonance frequency of the membranes in which these are better coupled to the fluid. Because of the strong acoustic coupling in the region of the interaction effects, the membranes may vibrate out of phase at some frequencies. We also noticed a nonaxisymmetric deformed shape, with a nodal line inside, at about 13 MHz. Note that the average displacement and pressure over the area of the whole array element are smaller than the average values of the membranes only. The mechanism and entity of acoustic interactions depend on the mechanical properties of the membranes and their arrangement within the element, as well as on the acoustic impedance of fluid, as compared with the mechanical impedance of the membranes. To explore this latter aspect, we lowered the density of water while leaving the speed of sound unchanged in the simulations. The resulting average displacements and pressures, when the density is lowered by a factor of 2 and 10, are shown in Fig. 7. As can be seen, the coupling reduces with the fluid density, and the interactions shift at higher frequencies. The outer and inner membranes start to behave very much alike when

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Fig. 8. Effect of a change of fluid sound velocity, but with same acoustic impedance, on the average displacement.

Fig. 7. Effect of a change of fluid density on the average displacement (top) and transmitted pressure (bottom).

the density is lowered by 10, and the average displacement and pressure approach the behavior of the membrane in a low-impedance fluid such as air. The in-vacuum resonance of the membranes was calculated as 10.8 MHz. If the array element is operated in air, with a density much lower than that of water, the acoustic coupling, therefore, is negligibly small. Because the interaction effects are caused by the reaction pressure over the membranes of the propagating acoustic waves in the fluid, both fluid density and sound velocity can have a strong influence. This point is shown in Fig. 8, in which the displacement of the membranes is plotted for two combinations of ρ and c, giving the same acoustic impedance (ρc). In addition to the average transmit pressure and displacement, it is interesting to consider the radiated acoustic power per unit area, which was calculated by FEM according to (3) in the Appendix. The results are shown in Fig. 9. Because of the interactions, the acoustic radiation power of each membrane is different, especially between 4 MHz and 6 MHz, and the inner membranes may even absorb acoustic power from the outer ones at some frequencies. Analogous results were previously reported for

Fig. 9. Average acoustic intensity radiated in water by the membranes and the array element.

the planar transducer arrays that are used in sonar systems [11], [18]. For small acoustic pressures (linear acoustics), the total power radiated by the array element is simply the sum of the radiation power of each membrane. Of course, this total power is positive at any frequency. C. Radiation Impedance Calculations by FEM The transmitting characteristics of transducer arrays usually are evaluated using the familiar concepts of self and mutual radiation impedance. The mutual impedance describes the reaction force produced by the acoustic pressure generated by one transducer over the surface of another transducer. The total radiation impedance is a combination of self and mutual impedances. Because the object of the present analysis are the acoustic interactions between the membranes constituting a single array element, the mutual radiation impedance between adjacent


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Fig. 11. Comparison between the radiation impedance of a CMUT array element and that of a rectangular piston transducer on a rigid baffle.

Fig. 10. Radiation impedance of the membranes and the CMUT array element: real part (top) and imaginary part (bottom).

elements is not calculated. However, the total radiation impedance of the element is evaluated through those of the single membranes, as described in the Appendix. In this work, the radiation impedances are calculated using a definition that has a direct relationship with the average power radiated by the transducer (see Appendix), so that they act as a mean to compute a quantity with a precise physical meaning. Fig. 10 shows the total radiation impedance per unit area of outer and inner membranes, and that of the CMUT array element of Fig. 5, as a function of the element widthto-wavelength ratio (the width w is 200 µm). Deviations between the membranes still are observed. Note that inner membranes have negative radiation resistance close to w/λ = 0.7, in which they absorb acoustic power from outer membranes. The peaks in the radiation impedance curves correspond to changes in the vibration shape of the membranes, so that the surface motion of the whole array element has no fixed velocity distribution (in the meaning stated in the Appendix). Moreover, the numerical values of this radiation impedance depend on the reference velocity of (2), which has been assumed as the average velocity in the calculations [see (6)].

It should be noted that the radiation impedance curves have a familiar behavior when calculated for piston transducers, driven at a uniform and constant velocity. A comparison with the radiation impedance of a rectangular piston transducer on a rigid baffle [23], having the same width as the CMUT array element, is shown in Fig. 11. The radiation resistance Rr of the CMUT approximates that of a piston transducer, except for the frequencies in which acoustic interactions occur between the membranes, thereby reducing the radiated acoustic power. Differently, the radiation reactance Xr moves away from that of the piston, due to the fact that the CMUT membranes are not actuated at constant velocity, but with a prescribed excitation pressure; therefore, increasing portions of the membranes vibrate with opposite phases as the frequency increases, and the reactive acoustic power increases. The local peaks of the radiation reactance have a similar interpretation, but the acoustic energy is exchanged from outer to inner membranes. D. Implications of the Acoustic Interactions on the Equivalent Circuit Model by Mason There is an important implication of the acoustic interactions onto the classical electromechanical equivalent circuit by Mason [24]. As it is known, this two-port network makes use of a mechanical impedance, Zm , to describe the response of the transducer without fluid load (i.e., in vacuum); then the mechanical port is terminated with a radiation impedance, Zr , to describe the interaction of the transducer surface with a fluid in contact [Fig. 12(a)]. The basic assumption underlying this approach is that the vibration shape of the radiator is not affected by the fluid. However, this is not the case for the membranes of a CMUT. In [25], the validity of Mason’s equivalent circuit was discussed for a single membrane radiating in water. It was shown that the fluid load results in a shift of the membrane

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Fig. 14. Simulated transmit sensitivity of a CMUT array element: using a rectangular piston approximation (solid line) and including acoustic interaction effects (line with symbols).

Fig. 12. Equivalent circuits of an immersion CMUT in transmission. (a) Classical Mason’s circuit. (b) Modified circuit with mechanoacoustic impedance.

culation whenever the membranes profiles move away from a piston-like motion. Fig. 14 shows the simulated transmit sensitivity of the CMUT array element, as obtained with the Mason’s circuit using the radiation impedance of a rectangular piston transducer (solid line), and the FEM computation of mechanoacoustic impedance, taking acoustic interactions into account (line with open squares). Note that the rectangular piston CMUT, having the same area as the actual CMUT array element, features an improved sensitivity and larger transmit bandwidth. III. Experimental Results

Fig. 13. Mechanoacoustic impedance and Mason’s modeling results. All computations are made by FEM.

mechanical antiresonance to a lower frequency, so that the mechanical impedance and radiation impedance cannot be lumped separately. In this section, limitations in the use of Mason’s circuit are further investigated in the light of acoustic interactions. Fig. 13 shows a comparison between the direct addition of mechanical and radiation impedances, and a computation of the ratio of the driving force applied on the membranes to their average velocity. This ratio can be called a mechanoacoustic impedance, Zmr , [Fig. 12(b)], and it accounts for the acoustic interactions in transmit operation. The discrepancy between the results, especially in the regions of interaction effects, shows that the use of Mason’s equivalent circuit does not result in accurate cal-

The acoustic coupling between CMUT membranes was investigated by means of optical displacement measurements, using a Polytec MSV-300 laser scanning heterodyne vibrometer system (Polytec PI Inc., Auburn, MA), which can provide absolute measurements of displacement in the subnanometer range. The interferometric setup used to characterize the transducers, both in air-coupled and liquid-coupled operation, is depicted in Fig. 15. Displacement signals were detected by the optical probe in the middle of the membranes, visualized by a Tektronix TDS620B digital oscilloscope, (Tektronix Inc., Beaverton, OR), and stored using a computer-based data acquisition system. In order to investigate the various aspects of acoustic interactions, several experiments were carried out on a variety of CMUTs, with different size and number of elements. The results of these investigations are reported in the following subsections. A. 64-Element CMUT Array To validate the FEM model reported in Section II-B, optical displacement measurements were performed on some membranes of the CMUT array shown in Fig. 4. The device was immersed in a tank filled with oil, and one of


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Fig. 15. Interferometric setup used for CMUT characterization.

the elements was biased at 90 V and pulsed with a 10cycle sinusoidal burst using an HP 8116A function generator (Agilent Technologies, Englewood, CO). The temporal length of the burst was low enough to prevent standing waves from setting up inside the tank. The displacement amplitude in the midpoint of several membranes, located along some rows, then was acquired by sweeping the frequency from 1 to 10 MHz, so that the displacement spectrum of those membranes was reconstructed. A comparison between FEM simulations and experimental results of the center displacement of the membranes along one row of the device is shown in Fig. 16. The FEM model used for comparison has the same membrane arrangement as the actual device; it is shown in Fig. 17. The measured membranes were located in the middle of the element, for the condition of infinitely long element to be better approximated. Because the membranes located in a specular position with respect to the middle plane of the element have the same acoustic load, the simulated displacements of membranes 1,4 and 2,3 are identical. Indeed, a small discrepancy exists between the measured displacements, which can be attributed to unavoidable nonidealities, including fabrication tolerances of the geometry and the mechanical properties of the membranes, and finiteelement length. Nevertheless, the agreement between theory and experiments is quite satisfying, and demonstrates that acoustic interactions through the liquid are the main source of coupling in CMUTs. Note that, because the vibration of one membrane is affected by all the adjacent membranes, a good uniformity in their mechanical properties is required over several rows to have a good matching with simulation results. We have observed that even the failure of one membrane in a group of 10 can change the results in a nonnegligible way. To investigate the mechanism of acoustic coupling more deeply, the effect of a discontinuity of the solid-liquid interface also was studied [10]. For this purpose, another

Fig. 16. Center displacement of CMUT membranes in oil (as shown in the frame inside the bottom figure): FEM simulations (top), and optical displacement measurements (bottom).

fabricated 64-element CMUT array was used, including 11 additional test elements located at various distances from each other. A number of deep cuts of about 150 µm width, as deep as the wafer thickness, were made between the test elements along their full length, as shown in Fig. 18. One of the test elements was wire-bonded onto a circuit board, with an epoxy coating to provide electrical isolation in water operation. The other elements were not connected. The device then was immersed in a water tank, and the connected element was biased at 30 V and pulsed with a 5cycle, 1.5-MHz sinusoidal burst. The displacement signals of several membranes, both coupled through the cut and not, were detected by the optical probe; they are shown in Fig. 19. As can be seen, the cross-talk displacement is almost unaffected by the presence of the cut through the silicon wafer. The transmission delays between excited and coupled elements are consistent with the propagation velocity of sound in water. Note that the displacements of excited membranes are very much alike because acoustic interaction effects are negligible at 1.5 MHz, as it is shown in Fig. 16.

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Fig. 17. 3-D FEM model of one element of the CMUT array shown in Fig. 4.

Fig. 19. Membrane displacement waveforms detected in water on the elements shown in Fig. 18.

Fig. 18. A portion of some test elements (69 through 72) of a fabricated 64-element CMUT array, with a deep cut in between.

B. 6-Element CMUT Subarray Another device that was extensively characterized by optical interferometry measurements is a CMUT subarray consisting of six elements of 220 × 1900 µm2 surface area, each including 144 circular membranes arranged in a 36 × 4 honeycomb grid, as shown in Fig. 20. The membranes are 50 µm in diameter, and the neighboring elements are separated by nonmetallized membranes, so that the centerto-center distance (pitch) is 275 µm. Each row of the device includes 30 membranes. The CMUT array prototype was characterized both in air-coupled and liquid-coupled operation. In the experiments, attention was given to the acoustic interactions between the membranes when either one element or all array elements were excited simultaneously, generating ultrasound pulses. 1. Air-Coupled Operation: The resonant frequency of the majority of membranes in air was about 8 MHz; sev-

Fig. 20. Top view of the 6-element CMUT array prototype used in the experiments.


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Fig. 22. Displacement signals in air of a pulsed membrane (normal scale) and a coupled one (magnified by 10).

demonstrated. Therefore, because acoustic coupling is negligible in air, the displacement of nonactive membranes was attributed to the acoustic propagation in the silicon wafer. The average root mean square cross-talk displacement level in air was found below −30 dB for the first adjacent element in this device.

Fig. 21. Space-time displacement maps of a sample row of membranes (a) in air-coupled and (b) in oil-coupled operation.

eral membranes were found to resonate at higher frequencies, probably due to local variations in the parameters of the fabrication process [26]. Acoustic coupling was investigated by driving one of the subarray elements, biased at 80 V, with a 10-cycle, 8 MHz sinusoidal burst. For each row of the subarray, displacement waveforms of the 30 membranes along the row were collected. A space-time displacement map of a sample row of the device in air is shown in Fig. 21(a). Bright areas are associated with positive displacement cycles, dark areas represent negative displacement cycles. The oscillations of the four membranes of the excited element are clearly seen; however, the membranes of the adjacent elements are poorly coupled in air, as also shown in Fig. 22, comparing two representative waveforms of a pulsed membrane and one of the first neighbor. Note that a Bessel-shaped apodization function has been applied to the measured center displacements of the membranes to generate the maps of Fig. 21. From the observation of the propagation delays between the elements, no evidence of electrical coupling was

2. Liquid-Coupled Operation: The device was immersed in castor oil to provide electrical isolation of the wires bonding the CMUT to a circuit board. The acoustic impedance of the oil is very close to that of water, but the attenuation is much higher. In a first experiment, all the six elements of the CMUT were biased at 80 V and pulsed simultaneously with a fivecycle, 3.5-MHz sinusoidal burst. The displacement waveforms detected by the optical probe on the 30 membranes along a center row are shown in Fig. 23. The first signal is the center displacement of the excited membrane, the second one is the displacement echo produced by the emitted pressure wave, which is reflected at the oil-air interface back to the surface of the transducer. As we found by FEM harmonic analysis in Section IIB, because of the acoustic interactions the vibration of each membrane depends on the acoustic pressure caused by the adjacent membranes through the liquid. Thus, the displacement waveform changes with the position of the membrane in the array element. Although the displacement waveforms of excited membranes are quite distorted and different from one another, the displacement echoes in reception exhibit quite uniform amplitudes and no significant distortion. This behavior suggested that a recomposition effect occurs in the propagation of the transmitted pressure pulse. To confirm this result, the acoustic field produced by the measured displacement signals of all membranes in the transducer was simulated (the propagation model will be described in the next paragraph). Fig. 24 shows the transmitted pressure pulses simulated at several distances along the axis

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Fig. 24. Simulated axial pressure waveforms at (a) 5 µm, (b) 50 µm, (c) 500 µm, (d) 5 mm, distance from the transducer surface.

Fig. 23. Displacement waveforms detected in oil on the membranes of a center row when the six elements are driven in parallel. Dotted lines represent nonmetallized membranes.

of the array. Close to the CMUT surface, a strong acoustic interference exists; at a distance of 5 mm, which is in the far-field of the excited elements, the pressure pulse is nearly undistorted and well reproduces the driving electrical signal. It is worth noting that the received echo from the oilair interface yields displacement waveforms substantially undistorted for all membranes, which means that acoustic interaction effects are of minor importance in receiving operations. This aspect is the object of current investigations. Another experiment was carried out, in which only one of the array elements was pulsed with a five-cycle, 3.5 MHz sinusoidal burst. A map of displacement signals for a sample row in oil is depicted in Fig. 21(b), and the corresponding temporal displacement waveforms are shown in Fig. 25. The time delays of coupled membranes are clearly visible, which correspond to a propagation velocity estimated as 1250 ± 100 m/s. The strong attenuation of the signals with the propagation distance is mainly due to viscous losses in the layer of oil in contact with the silicon surface. 3. Pressure Field Simulations: To investigate the transient pressure field generated in oil by the transducer, the Field II program was used [27]. This is a versatile simulation tool for ultrasound systems, running under Matlab (MathWorks Inc., Natick, MA), that allows the user to divide transducer element apertures into a number of small

subapertures, so that any transducer geometry can be simulated. Time delays, as well as apodization, easily can be applied to each element. By using this program, a model was built that reproduces the actual configuration and geometry of the CMUT subarray, as shown in Fig. 26. Each membrane is treated as a physical element with its own excitation, and it is divided into 100 rectangular mathematical elements. A farfield approximation of the spatial impulse response is used by Field II for the rectangular elements. This makes it faster to calculate the response that, however, is accurate enough for our purpose. The accuracy of the envelope of emitted far-field pressure has been estimated at about 2– 5%, with a 100 MHz sampling frequency of the excitation signals. Pulsed ultrasonic fields transmitted in oil were simulated using the measured displacement waveforms of all membranes as inputs to the transducer model. To present the membrane profile in a more realistic way, a Besselshaped apodization function was applied to each displacement signal. A rigid baffle condition was used to compute the acoustic pressure by the Rayleigh integral formulation. Ultrasonic fields generated by exciting either one array element or the six elements simultaneously were investigated. When the six elements are pulsed simultaneously, the resulting simulated pressure wavefronts at a distance z = 5 mm, as a function of the lateral position along the array, are shown in Fig. 27 (top). A comparison with the field from an ideal transducer without membrane acoustic coupling also is shown (Fig. 27, bottom). Significantly, in the geometrical projection of the six-element aperture (between −0.8 mm and 0.8 mm) the far-field acoustic pressure of the actual CMUT array is in good agreement with that transmitted by the ideal transducer; this result is a generalization of Fig. 24(d) in which the far-field axial pressure is simulated. Fig. 28 compares the pressure wavefronts as a function of the axial distance generated by the actual transducer


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Fig. 25. Displacement signals detected in oil on the membranes of a center row when only one array element is pulsed.

Fig. 27. Simulated pressure wavefronts in the far-field when the six elements are pulsed simultaneously: using the measured displacement signals of the membranes (top), and assuming that all membranes are uncoupled (bottom).

Fig. 26. Field II model of the CMUT subarray.

(top) and by the ideal device with uncoupled membranes (bottom). It clearly shows the region of strong interference, close to the surface of the CMUT subarray, as well as the recomposition effect of the transmitted pressure in the farfield of the aperture.

IV. Conclusions We have presented the first comprehensive analysis of the acoustic interactions between the membranes of CMUT, based on finite-element modeling and experi-

ments. A FEM study of the interaction effects between the membranes of an unbounded CMUT transducer, and of a typical 1-D CMUT array designed for underwater imaging applications, has been reported. The results show that interaction effects can be considerable at some frequencies in the operational bandwidth of the transducer and, therefore, they must be considered in the analysis and design of CMUT arrays. These effects are similar to those found in sonar transducer arrays, resulting in different acoustic loading on the CMUT membranes, which then have different velocity and pressure distributions. The mismatch between membrane mechanical impedance and fluid radiation impedance has a major influence on the entity of acoustic interactions. For this reason, the acoustic coupling is found to be negligible in air, both by modeling and experiments. The total radiation impedance of a CMUT array element also is calculated by means of FEM results using a classical definition, without computing the mutual radia-

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teractions, including the effect on the transmitted pressure field, other experiments and simulations have been carried out, using several CMUTs with different size and number of elements. Our current studies are focused on the cross-coupling effects when the CMUT array is used as a probe in an ultrasound imaging system. Future design and fabrication of optimized CMUT arrays will benefit from the present study.

Appendix A Definitions of Self and Mutual Radiation Impedance In acoustics, the radiation impedance of a transducer in the presence of fluid loading is often defined as the ratio of a force to a velocity [28]. A more convenient definition, which was used in Section II for FEM computation, has been discussed by Foldy [29], who refers to a fixed velocity distribution transducer as a radiator whose vibrating surface has a fixed deflection profile. Indeed, all vibrating surfaces have many degrees of freedom, so that the fixed velocity distribution transducer is an idealization; however, this schematization is realized often in practice because it is desirable to excite in the transducer a dominant mode of vibration. If the normal velocity of the transducer surface is designated by v(r), the condition of fixed velocity distribution can be expressed as: v(r) = Vβ(r),

Fig. 28. Simulated pressure wavefronts as a function of the axial distance: (top) actual device and (bottom) ideal transducer without acoustic interactions.

tion impedance of the membranes inside the element. The radiation impedance then is compared with that of a rectangular piston transducer, showing that acoustic interaction effects manifest as peaks in the radiation impedance curves. Furthermore, the validity of Mason’s equivalent circuit is discussed. It is shown that, because the vibration shapes of the membranes are affected by the fluid loading, especially when acoustic interactions are stronger, the use of Mason’s circuit, with mechanical and radiation impedances separately lumped, does not result in accurate calculation. The acoustic coupling between CMUT membranes has been investigated by means of optical displacement measurements, using a laser scanning vibrometer system. A good agreement is found between FEM results and experiments, thus demonstrating that acoustic interactions through the liquid are the major source of cross coupling in CMUTs. To explore the various aspects of acoustic in-

(1)

where β(r) is a fixed function of the position (no matter how the motion is produced), which may be frequency dependent. All quantities are assumed to vary harmonically in time, and complex notation is used throughout this Appendix. Foldy’s definition [29] has the merit to establish a direct relationship with the radiated acoustic power; the selfradiation impedance of a fixed velocity distribution transducer with surface area S is given by [11]: 1 p(r)v∗ (r) dS = Rr + jXr , (2) Zr = VV∗ S where p(r) is the acoustic pressure on the transducer surface and V is a reference velocity, which can be taken either as the average normal velocity or as any nonzero value of the velocity (e.g., the maximum velocity), because the surface motion is always the same. The relation to the average radiated power, therefore, is: 1 1 1 Pr = Re p(r)v∗ (r) dS = Rr VV∗ = Rr |V|2 , 2 2 2 S (3) which is the most important property of the radiation impedance, as defined by Foldy [29]. Note that, for uniform velocity transducers, v(r) = V, the integral in (2) is


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the reaction force exerted by the fluid on the transducer surface, and the classical definition of radiation impedance is obtained. When assessing the performance of transducer arrays, radiation impedance calculations usually are done by means of mutual radiation impedances. In an array of N transducer elements with fixed velocity distribution, the total radiation impedance of the i-th element, Zri , is still calculated by (2), in which now p(r) is the sound pressure produced by all transducer elements over the surface of the i-th element. Using the superposition principle (small acoustic pressures), we have: Zri =

N j=1

where: Zji =

1 Vj Vi∗

Zji

Vj , Vi

pj (ri )vi∗ (ri ) dSi ,

(4)

(5)

Si

is the mutual radiation impedance between j-th and i-th elements when i = j, and the self-radiation impedance when i = j. The value of Zji is independent of the amplitudes of vibration of the elements, because pj is proportional to Vj and vi is proportional to Vi . The mutual radiation impedance only depends on the geometries (shape, size, separation, relative orientation) and the acoustical properties of the element surfaces, including the baffle effect. Expression (4) still applies to a single CMUT array element, when all its micromembranes are driven into harmonic motion with a given excitation. The total radiation impedance of the element may be computed through the mutual impedances of every pair of micromembranes in the element. This approach, which is the one used when performing calculations by analytical or numerical methods, is not the more straightforward when simulation results are available by FEA. In this work, the total radiation impedance of a single CMUT array element was computed by means of the total radiation impedances of individual micromembranes, without calculating their mutual radiation impedances. In fact, the FEM solution provides the velocity and pressure of each node at the fluid-structure interface. Because the surface normal velocity is negligibly small in the points on the silicon interface outside the membranes, the surface integral of (2), in which the integrand is the product of pressure and velocity, can be computed including only the contributions from the membranes. Thus, the total radiation impedance of the CMUT array element can be expressed as: N 1 1 ∗ 2 Zr = p(r)v (r) dS ≈ | vi | zri , | V |2 S | V |2 i=1 (6) where N is the number of membranes within the area S, vi is the average velocity of the i-th membrane, and zri is its total radiation impedance.

The magnitude of the average velocity of the totality of membranes over S is calculated as: N S1 vi , (7) × | V | = S i=1

where S1 is the surface area of the single membrane.

Acknowledgments The authors would like to thank V. Foglietti and E. Cianci for providing all the CMUT transducers used in the experiments; D. Fiasca and R. Carotenuto for their support in the experimental activity and helpful discussions; P. Gatta and C. Longo for their work in measurement equipment and automation; and anonymous reviewers for contributing suggestions that improved the quality of this manuscript. We wish to acknowledge the financial support of MUSTWIN project NMP2-CT-2003-505630.

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[12] R. L. Pritchard, “Mutual acoustic impedance between radiators in an infinite rigid plane,” J. Acoust. Soc. Amer., vol. 32, no. 6, pp. 730–737, 1960. [13] D. T. Porter, “Self and mutual radiation impedance and beam patterns for flexural disks in a rigid plane,” J. Acoust. Soc. Amer., vol. 36, no. 6, pp. 1154–1161, 1964. [14] E. M. Arase, “Mutual radiation impedance of square and rectangular pistons in a rigid infinite baffle,” J. Acoust. Soc. Amer., vol. 36, no. 8, pp. 1521–1525, 1964. [15] C. H. Sherman, “Mutual radiation impedance of sources on a sphere,” J. Acoust. Soc. Amer., vol. 31, no. 7, pp. 947–952, 1959. [16] J. E. Greenspon and C. H. Sherman, “Mutual radiation impedance and nearfield pressure for pistons on a cylinder,” J. Acoust. Soc. Amer., vol. 36, no. 1, pp. 149–153, 1964. [17] C. Audoly, “Some aspects of acoustic interactions in sonar transducer arrays,” J. Acoust. Soc. Amer., vol. 89, no. 3, pp. 1428– 1433, 1991. [18] J. Lee, I. Seo, and S. M. Han, “Radiation power estimation for sonar transducer arrays considering acoustic interaction,” Sens. Actuat. A, vol. 90, pp. 1–6, 2001. [19] H. Lee, J. Tak, W. Moon, and G. Lim, “Effects of mutual impedance on the radiation characteristics of transducer arrays,” J. Acoust. Soc. Amer., vol. 115, no. 2, pp. 666–679, 2004. [20] A. D. Pierce, Acoustics. An Introduction to Its Physical Principles and Applications. New York: American Institute of Physics, 1994, pp. 208–211. [21] A. Caronti, R. Carotenuto, G. Caliano, and M. Pappalardo, “The effects of membrane metallization in capacitive microfabricated ultrasonic transducers,” J. Acoust. Soc. Amer., vol. 115, no. 2, pp. 651–657, 2004. [22] F. V. Hunt, Electroacoustics. The Analysis of Transduction, and Its Historical Background. New York: American Institute of Physics, 1982. [23] D. S. Burnett and W. W. Soroka, “Tables of rectangular piston radiation impedance functions, with application to sound transmission loss through deep apertures,” J. Acoust. Soc. Amer., vol. 51, no. 5, pp. 1618–1623, 1972. [24] W. P. Mason, Electromechanical Transducers and Wave Filters. 2nd ed. Van Nostrand Company, 1943. [25] G. G. Yaralioglu, M. H. Badi, A. S. Ergun, and B. T. Khuri-Yakub, “Improved equivalent circuit and finite element method modeling of capacitive micromachined ultrasonic transducers,” in Proc. IEEE Ultrason. Symp., 2003, pp. 469–472. [26] A. Caronti, H. Majjad, S. Ballandras, G. Caliano, R. Carotenuto, A. Iula, V. Foglietti, and M. Pappalardo, “Vibration maps of capacitive micromachined ultrasonic transducers by laser interferometry,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 49, no. 3, pp. 289–292, 2002. [27] J. A. Jensen, “Field: A program for simulating ultrasound systems,” Med. Biol. Eng. Comput., vol. 34, no. 1, pp. 351–353, 1996. [28] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics. 4th ed. New York: Wiley, 2000. [29] L. L. Foldy, “Theory of passive linear electroacoustic transducers with fixed velocity distribution,” J. Acoust. Soc. Amer., vol. 21, no. 6, pp. 595–604, 1949.

Alessandro Caronti (M’00) was born in Roma, Italy on May 17, 1975. He received the M.S. and Ph.D. degrees in electronic engineering from the University Roma Tre of Roma, in 1999 and 2003, respectively. He is currently a contract researcher in the ACULAB group, at the Department of Electronic Engineering of the University Roma Tre. His main research areas are in the field of ultrasonic transducers, where he is experienced with FEM modeling, design, and characterization of capacitive micromachined ultrasonic transducers (CMUTs) for ultrasound imaging applications.

Alessandro Savoia was born in Edinburgh, Scotland on November 9, 1978. He received his M.S. degree in electronic engineering from the University of Roma “Roma Tre” in May 2003. He is studying towards a Ph.D. degree (Department of Electronic Engineering, Roma Tre). His main research interests are in the field of ultrasonic transducers for medical imaging applications. He is currently working on the design, the finite element modeling, and the characterization of capacitive micromachined ultrasonic transducers (cMUTs).

Giosu` e Caliano (M’00) was born in Salerno, Italy, in 1963. He received the Laurea degree in electronic engineering from the University of Salerno. After the degree, he served as postgraduate fellow at the Department of Electronics at the University of Salerno. His interests were in developing piezoelectric pressure sensors and measurement techniques for the characterization of piezoceramics. In 1995 he joined Pirelli-FOS as Industrial Automation Engineer. In this position he worked as design engineer for optical fibers production. Since 1997 he works in ACULAB, Department of Electronic Engineering of the University Roma Tre. He is currently engaged in the manufacturing, design and characterization of cMUT probes for medical applications and NDT evaluations. He has contributed several journal and conference papers.

Massimo Pappalardo (M’98) was born in Roma, Italy in 1942. He received the Dr. Sc. degree in electrical engineering from the University of Napoli, Italy, in 1967. He started his research activity in 1968 with a scholarship at the Istituto di Acustica C.N.R. in Roma, where he became Scientific Manager of the Department of Ultrasound and Acoustical Technology. He became professor of Biomedical Electronics at the University of Calabria in 1981. He joined the University of Salerno as professor of Electronics in 1986. Since 1995 he is full professor at the Department of Electronic Engineering of the University Roma Tre. He has worked in the field of ultrasonic transducers, piezoelectric devices, and echographic systems. He is currently mainly engaged in research on acoustical imaging for medical applications and non-destructive testing, and capacitive micromachined ultrasonic transducers. Massimo Pappalardo has published more than 100 papers in these fields, and he is an Associate Editor of the IEEE Trans. on UFFC.