- Email: [email protected]
Normal Mode Theory Applied to Linear Arrays of Capacitive Micromachined Ultrasonic Transducers
Available online at www.sciencedirect.com
ScienceDirect Physics Procedia 70 (2015) 992 – 996
2015 International Congress on Ultrasonics, 2015 ICU Metz
Normal mode theory applied to linear arrays of capacitive Micromachined Ultrasonic Transducers Audren Boulmé*, Dominique Gross, Jacques Heller and Domnique Certon François Rabelais University, GREMAN UMR-CNRS 7347, 16 rue Pierre et Marie Curie, Tours, France.
Abstract
This paper describes the implementation of a new equivalent electroacoustic circuit for cMUT arrays. This circuit was obtained by using the normal mode theory. Theoretical simulations were also carried out from this new circuit to illustrate the impact of the trapped modes and some properties of the fundamental mode. © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of 2015 ICU Metz. Peer-review under responsibility of the Scientific Committee of ICU 2015 Keywords: cMUT; Subwavelength resonances; Baffle modes; Boundary Element Model.
1. Introduction In a previous work [1], an efficient and accurate modeling tool was presented to design linear arrays of cMUTs. Based on 2-D matrix representation, this model has been reformulated by applying the normal mode theory. From the modal decomposition, two categories of radiation mode were identified, one for which all cMUTs vibrate in phase (fundamental mode) and the others (baffle modes also called trapped modes [1]-[5]) where the columns of cells vibrate out of phase. Properties and description of these radiation modes are detailed in [1]. In this paper, the modal decomposition is reused to establish a new equivalent electroacoustic circuit of cMUTbased array element. The originality of this latter is that it contains one electrical port and as many acoustic ports as the number of eigenmodes. Consequently, the contribution of each mode on the element response can be easily distinguished and some strategies can be proposed to favour or cancel one mode. Here, this new electroacoustic circuit is used to analyze the impact of baffle modes on the basic performance of the element and to propose design guidelines focused on the optimization of the fundamental mode only.
* Corresponding author. Tel.:+33-247-424-000; fax: +33-247-424-937. E-mail address: [email protected]
1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015 doi:10.1016/j.phpro.2015.08.207
993
Audren Boulmé et al. / Physics Procedia 70 (2015) 992 – 996
2. cMUT array modeling 2.1. Matrix modeling cMUT arrays are modeled here using the 2-D matrix representation presented in [1]. Based on l-D periodic boundary conditions, this method consists of modeling cMUT array as a set of N columns, where each column is described by only four degrees of freedom: - the input electrical current Ii, - the applied electrical voltage Vi, - the spatial averaged particular velocity (mean velocity) vi, - the output acoustic force Fr,i. Assuming that each cMUT is a linear system, the previous parameters can be linked through the following matrix relationships: ሾܫሿ ൌ ሾܭ ሿ ሾܸሿ ሾܭ ሿ ሾݒሿ ሾܨ (1) ቐ ሿ ൌ ሾܭ ሿ ሾܸሿ െ ሾܭ ሿ ሾݒሿ ሾܨ ሿ ൌ ൣܩ௨ௗ ൧ ሾݒሿ [I], [V], [Fr] and [v] are vectors which respectively contain the N complex amplitudes of the electrical current (Ii), the applied voltage (Vi), the acoustic force (Fr,i) and the mean velocity (vi). [Kee], [Kem], [Kme] and [Kmm] are four matrices, which link respectively the electrical quantities ([Kee]), the acoustic and electrical quantities ([Kem] and [Kme]) and the acoustic quantities ([Kmm]). [Gfluid] is a boundary element matrix, so called radiation matrix, which models the fluid coupling between the N columns of cMUTs. More details on these matrices and on this model are given in [1] and [6]. 2.2. New electroacoustic equivalent circuit To implement the new electroacoustic circuit, the set of equations (1) was reorganized to express the input current vector [I] as a function of the input voltage vector [V] as follows: ሾܫሿ ൌ ݆ܥ ߱ ሾܸሿ ߶ ଶ ሾܻሿሾܸሿ
(2)
where C0 is the static capacitance of one cMUT and [Y] is the admittance matrix defined by: ሾܻሿ ൌ ቂሾܭ ሿ ൣܩ௨ௗ ൧ቃ
ିଵ
(3)
The admittance matrix [Y] is then broken down on the basis of its eigenmodes, made up of a set of N eigenvectors ([Ve,i]) and N eigenvalues (Oi) (i is an integer that selects each eigenmode). Relation (2) becomes now: ே
ሾܫሿ ൌ ݆ܥ ߱ሾܸሿ
߶ ଶ ߙ ߣ ൣܸǡ ൧
(4)
ୀଵ
with Di are the projection coefficient of the voltage vector [V] on the eigenvector basis. As already demonstrated in the previous study [1], the eigenvalues of the admittance matrix [Y] (Oi) are connected with the ones of the radiation matrix [Gfluid] (Ji) through the following equation: ͳ (5) ߣ ൌ ߛ ܼ௦ where Zms is the membrane mechanical impedance. From the relations (4) and (5), one can establish a new electroacoustic circuit of one cMUT array element (Fig. 1). The electrical port connects the input current vector [I] to the input voltage vector [V]. Each acoustic branch is made of: two series impedance, ܼ௦ and ߛ , one vector equivalent to an electrostatic force, ߙ ߶ൣܸǡ ൧, the mean velocity vector associated to the mode i, ൣݒǡ ൧.
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Fig. 1. Electroacoustic circuit of one element of a linear array deduced from the modal decomposition.
This new representation of an element of array points out several major interests to help any researchers or engineers to design a device. The output acoustic data are connected to the input electrical degrees of freedom through scalar relations rather than matrix relations as in the equation (1). The output acoustic vector ሾݒሿ is broken down as the sum of the N output acoustic eigenvectors: ሾݒሿ ൌ σே ୀଵൣݒǡ ൧. 3. Modal properties of the cMUT emitter response. In this part, the modal properties of the cMUT emitter response are illustrated by several numerical simulations. For this, an element composed of 4 columns of 25x25μm² square-shaped cMUTs was used. The metallization ratio of each cell was set to 50% and the column-to-column pitch (dx) to 40 μm. This element was design to vibrate around 10MHz in air. 3.1. Impact of baffle modes (mode 3) In this section, the impact of the baffle modes on the element performance is discussed through its electrical impedance (Fig. 2) and its pressure field (Fig. 3). Of course, for an element composed of 4 columns, 4 radiation modes can be excited. However, as already demonstrated in [1], since the same voltage is applied on the 4 columns of the element, only symmetrical modes contribute to the element response: the fundamental mode (mode 1) and one baffle mode (mode 3). The real part of the impedance was computed for different excitation conditions: a uniform excitation vector ([V]=[1111]) (solid lines Fig. 2.(a)), an excitation vector equal to the eigenvector of the mode 1 (dashed lines Fig. 2.(a)) and an excitation vector equal to the eigenvector of the mode 3 (Fig. 2.(b)). For the three bias voltages, simulations show that the shape of the electrical impedance is mainly driven by the mode 1, except around the cutoff frequency (§ 4MHz) caused by the mode 3. Consequently, it is clear that the performances of the element in term of sensitivity and bandwidth are mainly governed by the fundamental mode. More, it is interesting to notice that the mode 3 is highly impacted by the membranes “softening” effect, contrary to the mode 1. The directivity pattern of the element was computed at 5 MHz and 10 MHz for a uniform excitation and an excitation vector to select the fundamental mode only (Fig. 3). The excitation signal was one pulse centered at the desired frequency (5MHz or 10MHz). The elevation of the element was matched with the working frequency, i.e. 10mm at 5MHz and 3mm at 10MHz. The directivity pattern was computed for each element at its Fresnel distance using the DREAM Toolbox [7]. For the two working frequencies, a diminution of the angular directivity due to the baffle mode is clearly observable, especially for angles greater than 30°. This agrees with the fact that the mode 3 shows a lower angular directivity than the fundamental mode, since the columns of cMUT vibrate out of phase. To complete this study, the axial time domain response (at 1 mm from the array) was computed and compared for three different excitation configurations (Fig. 4): 1) the pressure emitted by one element excited with a uniform excitation vector, 2) the pressure emitted when the eigenmode of the fundamental mode is used as excitation vector and 3) the pressure emitted by a sub-aperture made of 5 elements excited with a uniform excitation vector. Again, two working frequencies were studied: 5 MHz and 10 MHz.
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Audren Boulmé et al. / Physics Procedia 70 (2015) 992 – 996
700
(a)
1/2 Vc
6000
3/4 Vc
5000 Real(Z),:
500 Real(Z),:
7000
1/4 Vc
600
400 300
2000 1000 5 7.5 10 Frequency (MHz)
12.5
1/2 Vc 3/4 Vc
3000
100 2.5
1/4 Vc
4000
200
0 0
(b)
0 2
15
3
4 5 Frequency (MHz)
6
Fig. 2. Real part of the electrical impedance for a uniform excitation ((a) solid lines), the mode 1 ((a) dashed lines) and the mode 3(b). In each case, the impedances curves are represented for three bias voltages.
5MHz 0 1
30
0.8
-30
0.6
60
Uniform Mode 1
30
-60
0.4
10MHz 0 1 0.8 0.6
60
90
-60
0.4
0.2
Uniform Mode 1
-30
0.2
90
-90
(a)
-90
(b)
Fig. 3. Directivity pattern computed at 5 MHz (a) and 10MHz (b) for a uniform excitation vector (blue solid lines) and an excitation vector proportional to the eigenvector of the mode 1 (red dashed lines). All computations were performed at the Fresnel distance of each element.
1.5
5MHz (a)
1 element uniform 1 element mode 1 5 elements uniform
1 0.5 0 -0.5 -1 -1.5 -2 0.5
1
1.5 2 Time (μs)
2.5
3
Normalized output pressure (A.U.)
Normalized output pressure (A.U.)
For the first case, undesirable oscillations are observed at the end of the impulse response for the two frequencies (Fig. 4). These oscillations, already reported in the literature [2], are caused by the baffle modes. For the two other excitation strategies, the suppression of the parasitic oscillations is clearly visible. Only the edge wave (from the element elevation) contribution remains at the end of the impulse response. For the first method, where each element was excited with a vector proportional to the eigenvector of the mode 1 (red dashed lines on Fig. 4), the oscillations were suppressed since only the mode 1 was selected. Although this strategy is highly efficient, it is difficult to implement experimentally since each column needs to be driven individually. For the third method (dotted black lines on Fig. 4), the principle consists to use simultaneously several elements (here 5). With this technique, even if the number of baffle modes is increased, the oscillations strongly decrease because their individual contributions are not constructive. This method can be assimilated to the plane-wave based beamforming strategies. 1.5 1
10MHz (b)
1 element uniform 1 element mode 1 5 elements uniform
0.5 0 -0.5 -1 -1.5 -2 0.5
1
1.5 2 Time (μs)
2.5
3
Fig. 4. Normalized output pressure computed at 5MHz (a) and 10MHz (b) for 1 element excited with a uniform vector (blue lines), 1 element excited according to the mode 1 (red dashed lines) and a sub-aperture made of 5 elements excited with a uniform excitation vector (black dotted lines).
3.2. Optimization of the fundamental mode Fig. 5.(a) shows the evolution of the pressure field spectrum (at 1 mm from the array) for four different ratios of metallization. The membrane thickness was tuned for each metallization ratio in order to keep the resonance frequency in air at the same value. We observe that the element bandwidth can be significantly increased when the metallization ratio is reduced. This result seems to show that the resonance frequency of the second membrane mode
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Audren Boulmé et al. / Physics Procedia 70 (2015) 992 – 996
(anti-symmetrical mode) is rejected toward high frequency when the electrostatic force is applied only on a small point at the center of the membrane. Finally, the pressure radiated by one element (at 1 mm from the array) was compared for two kinds of fluid: water and a silicone rubber (c0=1050m/s and U0= 1300kg/m3) (Fig. 5.(b)). When the silicone rubber substitutes water, a small gain of sensitivity and a significant reduction of the bandwidth are observed. It is clear that the silicone rubber impacts the element response and that the material used to encapsulate cMUT needs to be taken into account for the element design. To compensate for the silicone effects, the column-to-column pitch was tuned first. In the best case (dx=28μm, red curve), a gain of sensitivity was observed but without modifying the frequency bandwidth. For this, it was also necessary to match the membrane mechanical impedance (membranes thickness or size) in order to reject the mechanical cut-off frequency. With a thickness of 500 nm, the pressure spectrum was nearly the same than the one obtained with water (Fig. 5.(b)). 40 30
30
(a)
100% 75% 50% 35%
20 0 -10 -20
Silicone dx=40μm ; hm=450nm Silicone dx=28μm ; hm=450nm Silicone dx=28μm ; hm=500nm
0 -10
-30
-20
-40 -50 0
Water: dx=40μm ; hm=450nm
10 Abs(dB)
Abs(dB)
10
(b)
20
5
10 15 20 25 Frequency (MHz)
30
35
-30 0
5
10 15 20 25 Frequency (MHz)
30
35
Fig. 5. Impact of the metallization ratio (a) and the fluid (b) on the pressure field spectrum. On each figure, all curves are normalized by the maximum value, i.e. respectively 1533Pa/V and 1489Pa/V for the figures (a) and (b).
4. Conclusion In this paper, a new electroacoustic circuit, based on modal decomposition, was proposed. Theoretical simulations were then proposed to illustrate the impact of the baffle modes on the element performance. The fundamental radiation mode (mode 1) was also analyzed for different metallization ratios and different types of fluid. It was demonstrated that the element bandwidth is increased when the metallization ratio is reduced. This phenomenon seems to be mostly caused by a reduction of the electrostatic force aperture rather than by a mass modification. More, the impact of fluid properties on the element pressure spectrum was studied and we have shown that these modifications can be easily cancelled out by tuning the column-to-column pitch and the membrane thickness. Acknowledgements Work performed in the frame of TOURS 2015, project supported by the “Programme de l’économie numérique des Investissements d’Avenir”.
References [1] [2] [3] [4] [5] [6] [7]
A. Boulmé, D. Certon, N. Sénégond, C. Meynier, and F. Teston, "A strategy to predict and reduce baffle effects in linear array of cMUTs," in Proc.-IEEE Ultrason. Symp., 2010, pp. 1873-1876. A. Caronti, A. Savoia, G. Caliano, and M. Pappalardo, "Acoustic coupling in capacitive microfabricated ultrasonic transducers: modeling and experiments," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 52, pp. 2220-2234, 2005. S. W. Lani, K. G. Sabra, and F. L. Degertekin, "Modal and transient analysis of membrane acoustic metasurfaces," JAP., vol. 117, p. 045308, 2015. A. Atalar, H. Köymen, and H. O÷uz, "Rayleigh-bloch waves in CMUT arrays," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 61, pp. 2139-2148, 2014. A. Rønnekleiv, "CMUT array modeling through free acoustic CMUT modes and analysis of the fluid CMUT interface through Fourier transform methods," IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 52, pp. 2173-2184, 2005. C. Meynier, F. Teston, and D. Certon, "A multiscale model for array of capacitive micromachined ultrasonic transducers," J. Acoust. Soc. Am., vol. 128, pp. 2549-2561, 2010. F. Lingvall, B. Piwakowski, and G.-M. Zang. The DREAM Toolbox [Online]. Available: http://www.signal.uu.se/Toolbox/dream/
ScienceDirect Physics Procedia 70 (2015) 992 – 996
2015 International Congress on Ultrasonics, 2015 ICU Metz
Normal mode theory applied to linear arrays of capacitive Micromachined Ultrasonic Transducers Audren Boulmé*, Dominique Gross, Jacques Heller and Domnique Certon François Rabelais University, GREMAN UMR-CNRS 7347, 16 rue Pierre et Marie Curie, Tours, France.
Abstract
This paper describes the implementation of a new equivalent electroacoustic circuit for cMUT arrays. This circuit was obtained by using the normal mode theory. Theoretical simulations were also carried out from this new circuit to illustrate the impact of the trapped modes and some properties of the fundamental mode. © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of 2015 ICU Metz. Peer-review under responsibility of the Scientific Committee of ICU 2015 Keywords: cMUT; Subwavelength resonances; Baffle modes; Boundary Element Model.
1. Introduction In a previous work [1], an efficient and accurate modeling tool was presented to design linear arrays of cMUTs. Based on 2-D matrix representation, this model has been reformulated by applying the normal mode theory. From the modal decomposition, two categories of radiation mode were identified, one for which all cMUTs vibrate in phase (fundamental mode) and the others (baffle modes also called trapped modes [1]-[5]) where the columns of cells vibrate out of phase. Properties and description of these radiation modes are detailed in [1]. In this paper, the modal decomposition is reused to establish a new equivalent electroacoustic circuit of cMUTbased array element. The originality of this latter is that it contains one electrical port and as many acoustic ports as the number of eigenmodes. Consequently, the contribution of each mode on the element response can be easily distinguished and some strategies can be proposed to favour or cancel one mode. Here, this new electroacoustic circuit is used to analyze the impact of baffle modes on the basic performance of the element and to propose design guidelines focused on the optimization of the fundamental mode only.
* Corresponding author. Tel.:+33-247-424-000; fax: +33-247-424-937. E-mail address: [email protected]
1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Scientific Committee of ICU 2015 doi:10.1016/j.phpro.2015.08.207
993
Audren Boulmé et al. / Physics Procedia 70 (2015) 992 – 996
2. cMUT array modeling 2.1. Matrix modeling cMUT arrays are modeled here using the 2-D matrix representation presented in [1]. Based on l-D periodic boundary conditions, this method consists of modeling cMUT array as a set of N columns, where each column is described by only four degrees of freedom: - the input electrical current Ii, - the applied electrical voltage Vi, - the spatial averaged particular velocity (mean velocity) vi, - the output acoustic force Fr,i. Assuming that each cMUT is a linear system, the previous parameters can be linked through the following matrix relationships: ሾܫሿ ൌ ሾܭ ሿ ሾܸሿ ሾܭ ሿ ሾݒሿ ሾܨ (1) ቐ ሿ ൌ ሾܭ ሿ ሾܸሿ െ ሾܭ ሿ ሾݒሿ ሾܨ ሿ ൌ ൣܩ௨ௗ ൧ ሾݒሿ [I], [V], [Fr] and [v] are vectors which respectively contain the N complex amplitudes of the electrical current (Ii), the applied voltage (Vi), the acoustic force (Fr,i) and the mean velocity (vi). [Kee], [Kem], [Kme] and [Kmm] are four matrices, which link respectively the electrical quantities ([Kee]), the acoustic and electrical quantities ([Kem] and [Kme]) and the acoustic quantities ([Kmm]). [Gfluid] is a boundary element matrix, so called radiation matrix, which models the fluid coupling between the N columns of cMUTs. More details on these matrices and on this model are given in [1] and [6]. 2.2. New electroacoustic equivalent circuit To implement the new electroacoustic circuit, the set of equations (1) was reorganized to express the input current vector [I] as a function of the input voltage vector [V] as follows: ሾܫሿ ൌ ݆ܥ ߱ ሾܸሿ ߶ ଶ ሾܻሿሾܸሿ
(2)
where C0 is the static capacitance of one cMUT and [Y] is the admittance matrix defined by: ሾܻሿ ൌ ቂሾܭ ሿ ൣܩ௨ௗ ൧ቃ
ିଵ
(3)
The admittance matrix [Y] is then broken down on the basis of its eigenmodes, made up of a set of N eigenvectors ([Ve,i]) and N eigenvalues (Oi) (i is an integer that selects each eigenmode). Relation (2) becomes now: ே
ሾܫሿ ൌ ݆ܥ ߱ሾܸሿ
߶ ଶ ߙ ߣ ൣܸǡ ൧
(4)
ୀଵ
with Di are the projection coefficient of the voltage vector [V] on the eigenvector basis. As already demonstrated in the previous study [1], the eigenvalues of the admittance matrix [Y] (Oi) are connected with the ones of the radiation matrix [Gfluid] (Ji) through the following equation: ͳ (5) ߣ ൌ ߛ ܼ௦ where Zms is the membrane mechanical impedance. From the relations (4) and (5), one can establish a new electroacoustic circuit of one cMUT array element (Fig. 1). The electrical port connects the input current vector [I] to the input voltage vector [V]. Each acoustic branch is made of: two series impedance, ܼ௦ and ߛ , one vector equivalent to an electrostatic force, ߙ ߶ൣܸǡ ൧, the mean velocity vector associated to the mode i, ൣݒǡ ൧.
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Audren Boulmé et al. / Physics Procedia 70 (2015) 992 – 996
Fig. 1. Electroacoustic circuit of one element of a linear array deduced from the modal decomposition.
This new representation of an element of array points out several major interests to help any researchers or engineers to design a device. The output acoustic data are connected to the input electrical degrees of freedom through scalar relations rather than matrix relations as in the equation (1). The output acoustic vector ሾݒሿ is broken down as the sum of the N output acoustic eigenvectors: ሾݒሿ ൌ σே ୀଵൣݒǡ ൧. 3. Modal properties of the cMUT emitter response. In this part, the modal properties of the cMUT emitter response are illustrated by several numerical simulations. For this, an element composed of 4 columns of 25x25μm² square-shaped cMUTs was used. The metallization ratio of each cell was set to 50% and the column-to-column pitch (dx) to 40 μm. This element was design to vibrate around 10MHz in air. 3.1. Impact of baffle modes (mode 3) In this section, the impact of the baffle modes on the element performance is discussed through its electrical impedance (Fig. 2) and its pressure field (Fig. 3). Of course, for an element composed of 4 columns, 4 radiation modes can be excited. However, as already demonstrated in [1], since the same voltage is applied on the 4 columns of the element, only symmetrical modes contribute to the element response: the fundamental mode (mode 1) and one baffle mode (mode 3). The real part of the impedance was computed for different excitation conditions: a uniform excitation vector ([V]=[1111]) (solid lines Fig. 2.(a)), an excitation vector equal to the eigenvector of the mode 1 (dashed lines Fig. 2.(a)) and an excitation vector equal to the eigenvector of the mode 3 (Fig. 2.(b)). For the three bias voltages, simulations show that the shape of the electrical impedance is mainly driven by the mode 1, except around the cutoff frequency (§ 4MHz) caused by the mode 3. Consequently, it is clear that the performances of the element in term of sensitivity and bandwidth are mainly governed by the fundamental mode. More, it is interesting to notice that the mode 3 is highly impacted by the membranes “softening” effect, contrary to the mode 1. The directivity pattern of the element was computed at 5 MHz and 10 MHz for a uniform excitation and an excitation vector to select the fundamental mode only (Fig. 3). The excitation signal was one pulse centered at the desired frequency (5MHz or 10MHz). The elevation of the element was matched with the working frequency, i.e. 10mm at 5MHz and 3mm at 10MHz. The directivity pattern was computed for each element at its Fresnel distance using the DREAM Toolbox [7]. For the two working frequencies, a diminution of the angular directivity due to the baffle mode is clearly observable, especially for angles greater than 30°. This agrees with the fact that the mode 3 shows a lower angular directivity than the fundamental mode, since the columns of cMUT vibrate out of phase. To complete this study, the axial time domain response (at 1 mm from the array) was computed and compared for three different excitation configurations (Fig. 4): 1) the pressure emitted by one element excited with a uniform excitation vector, 2) the pressure emitted when the eigenmode of the fundamental mode is used as excitation vector and 3) the pressure emitted by a sub-aperture made of 5 elements excited with a uniform excitation vector. Again, two working frequencies were studied: 5 MHz and 10 MHz.
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Audren Boulmé et al. / Physics Procedia 70 (2015) 992 – 996
700
(a)
1/2 Vc
6000
3/4 Vc
5000 Real(Z),:
500 Real(Z),:
7000
1/4 Vc
600
400 300
2000 1000 5 7.5 10 Frequency (MHz)
12.5
1/2 Vc 3/4 Vc
3000
100 2.5
1/4 Vc
4000
200
0 0
(b)
0 2
15
3
4 5 Frequency (MHz)
6
Fig. 2. Real part of the electrical impedance for a uniform excitation ((a) solid lines), the mode 1 ((a) dashed lines) and the mode 3(b). In each case, the impedances curves are represented for three bias voltages.
5MHz 0 1
30
0.8
-30
0.6
60
Uniform Mode 1
30
-60
0.4
10MHz 0 1 0.8 0.6
60
90
-60
0.4
0.2
Uniform Mode 1
-30
0.2
90
-90
(a)
-90
(b)
Fig. 3. Directivity pattern computed at 5 MHz (a) and 10MHz (b) for a uniform excitation vector (blue solid lines) and an excitation vector proportional to the eigenvector of the mode 1 (red dashed lines). All computations were performed at the Fresnel distance of each element.
1.5
5MHz (a)
1 element uniform 1 element mode 1 5 elements uniform
1 0.5 0 -0.5 -1 -1.5 -2 0.5
1
1.5 2 Time (μs)
2.5
3
Normalized output pressure (A.U.)
Normalized output pressure (A.U.)
For the first case, undesirable oscillations are observed at the end of the impulse response for the two frequencies (Fig. 4). These oscillations, already reported in the literature [2], are caused by the baffle modes. For the two other excitation strategies, the suppression of the parasitic oscillations is clearly visible. Only the edge wave (from the element elevation) contribution remains at the end of the impulse response. For the first method, where each element was excited with a vector proportional to the eigenvector of the mode 1 (red dashed lines on Fig. 4), the oscillations were suppressed since only the mode 1 was selected. Although this strategy is highly efficient, it is difficult to implement experimentally since each column needs to be driven individually. For the third method (dotted black lines on Fig. 4), the principle consists to use simultaneously several elements (here 5). With this technique, even if the number of baffle modes is increased, the oscillations strongly decrease because their individual contributions are not constructive. This method can be assimilated to the plane-wave based beamforming strategies. 1.5 1
10MHz (b)
1 element uniform 1 element mode 1 5 elements uniform
0.5 0 -0.5 -1 -1.5 -2 0.5
1
1.5 2 Time (μs)
2.5
3
Fig. 4. Normalized output pressure computed at 5MHz (a) and 10MHz (b) for 1 element excited with a uniform vector (blue lines), 1 element excited according to the mode 1 (red dashed lines) and a sub-aperture made of 5 elements excited with a uniform excitation vector (black dotted lines).
3.2. Optimization of the fundamental mode Fig. 5.(a) shows the evolution of the pressure field spectrum (at 1 mm from the array) for four different ratios of metallization. The membrane thickness was tuned for each metallization ratio in order to keep the resonance frequency in air at the same value. We observe that the element bandwidth can be significantly increased when the metallization ratio is reduced. This result seems to show that the resonance frequency of the second membrane mode
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Audren Boulmé et al. / Physics Procedia 70 (2015) 992 – 996
(anti-symmetrical mode) is rejected toward high frequency when the electrostatic force is applied only on a small point at the center of the membrane. Finally, the pressure radiated by one element (at 1 mm from the array) was compared for two kinds of fluid: water and a silicone rubber (c0=1050m/s and U0= 1300kg/m3) (Fig. 5.(b)). When the silicone rubber substitutes water, a small gain of sensitivity and a significant reduction of the bandwidth are observed. It is clear that the silicone rubber impacts the element response and that the material used to encapsulate cMUT needs to be taken into account for the element design. To compensate for the silicone effects, the column-to-column pitch was tuned first. In the best case (dx=28μm, red curve), a gain of sensitivity was observed but without modifying the frequency bandwidth. For this, it was also necessary to match the membrane mechanical impedance (membranes thickness or size) in order to reject the mechanical cut-off frequency. With a thickness of 500 nm, the pressure spectrum was nearly the same than the one obtained with water (Fig. 5.(b)). 40 30
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Fig. 5. Impact of the metallization ratio (a) and the fluid (b) on the pressure field spectrum. On each figure, all curves are normalized by the maximum value, i.e. respectively 1533Pa/V and 1489Pa/V for the figures (a) and (b).
4. Conclusion In this paper, a new electroacoustic circuit, based on modal decomposition, was proposed. Theoretical simulations were then proposed to illustrate the impact of the baffle modes on the element performance. The fundamental radiation mode (mode 1) was also analyzed for different metallization ratios and different types of fluid. It was demonstrated that the element bandwidth is increased when the metallization ratio is reduced. This phenomenon seems to be mostly caused by a reduction of the electrostatic force aperture rather than by a mass modification. More, the impact of fluid properties on the element pressure spectrum was studied and we have shown that these modifications can be easily cancelled out by tuning the column-to-column pitch and the membrane thickness. Acknowledgements Work performed in the frame of TOURS 2015, project supported by the “Programme de l’économie numérique des Investissements d’Avenir”.
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