Cecilia Casarini

Ben Tiller

James F C Windmill

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Keywords:

This paper presents a 3D printing technique based on stereolithography and direct light processing for the fabrication of low resonance frequency thin plates suitable for acoustic metamaterials applications. It was possible to achieve a better resolution with respect to other 3D printing methods such as fusion deposition modeling and to obtain plates with a thickness of 70 μm. The plates were characterized using three different methods: laser Doppler vibrometer supported by modal analysis, impedance tube measurements backed by a transfer matrix model and nanoindentation. All results are in good agreement. The physical parameters retrieved through the characterization methods can be used for future designs and integrated into finite element analysis to better predict the noise impact of these materials. Thanks to the small radius and thickness of the plates presented in this paper and to their low resonance frequency, it is suggested that they could be arranged in various configurations and used as unit cells in acoustic metamaterials applications for noise attenuation in smallscale electroacoustic devices.

I. INTRODUCTION

ecent advances in additive manufacturing techniques have now made it possible to fabricate thin membranes and plates that are essential for the study of several acoustic applicationsfor example, microphones and loudspeaker prototyping and acoustic metamaterials [START_REF] Tiller | Piezoelectric microphone via a digital light processing 3D printing process[END_REF]- [START_REF] Casarini | Enhancing the Sound Absorption of Small-Scale 3-D Printed Acoustic Metamaterials Based on Helmholtz Resonators[END_REF]. In laboratories, these membranes are usually obtained through a procedure where a piece of material is cut and glued on a rigid frame or support. On the other hand, industrial fabrication typically relies on other techniques -such as clean room manufacturing -that have high maintenance costs. In comparison with these established production processes, additive manufacturing provides with a cheap and reliable fabrication approach, which additionally offers the opportunity to choose the material for the membrane. Furthermore, 3D printing software allows to create abstract shapes and to integrate the product inside other structures, for example by 3D printing a membrane inside a Helmholtz resonator. Haque et al. [START_REF] Haque | Inkjet-Printed Membrane for a Capacitive Acoustic Sensor: Development and Characterization Using Laser Vibrometer[END_REF] fabricated a thin silver layer on a purchased Mylar film using a inkjet 2D printer, that deposits drops of liquid material on pre-existing substrate. Previous research on 3D printing membranes relied on fusion deposition modeling [START_REF] Leblanc | Three-dimensional-printed membranetype acoustic metamaterial for low frequency sound attenuation[END_REF], which deposits a filament layer by layer through a heated extruder nozzle head. Stereolithography via direct light processing has recently been used to 3D print thin layers with thickness down to 35 μm [START_REF] Tiller | Piezoelectric microphone via a digital light processing 3D printing process[END_REF], thanks to new techniques that rely on absorbers that allow to control shapes and thickness of the materials in a finer way with respect to fusion deposition modelling, that has a minimum thickness of 0.1 mm. The advantages of these new stereolithography techniques can be found not only in the practical utility of fabricating a thin layer, but also in the opportunity to finely control its material. Moreover, multilayer plates made of different materials can be fabricated, paving the way of manufacturing a sensor or even an entire device in one print [START_REF] Tiller | Piezoelectric microphone via a digital light processing 3D printing process[END_REF], [START_REF] Haque | Inkjet-Printed Membrane for a Capacitive Acoustic Sensor: Development and Characterization Using Laser Vibrometer[END_REF]. In particular, when using polymers it is possible to choose the desired physical properties of the plate, given the vast range of different options available within the polymers Engineering Department, University of Strathclyde, Glasgow, Scotland, UK. email: cecilia.casarini@strath.ac.uk.

Vicent Romero-García and Jean-Philippe Groby are members of the Laboratoire d'Acoustique de l'Université du Maine (LAUM), Le Mans, France. Fundamental Frequency realm [START_REF] Tiller | Piezoelectric microphone via a digital light processing 3D printing process[END_REF], [START_REF] Domingo-Roca | 3Dprinting polymer-based permanent magnets[END_REF]. In this work, polymer-based thin low-frequency circular acoustic plates are fabricated using stereolithography and experimentally characterized via three different techniques. Although the fabrication technique described in this paper could be useful for many different applications based on membranes and plates, e.g. in microphones manufacturing [START_REF] Tiller | Piezoelectric microphone via a digital light processing 3D printing process[END_REF] and non-destructive evaluation [START_REF] Zhu | Pipe Organ' Inspired Air-Coupled Ultrasonic Transducers With Broader Bandwidth[END_REF], the design presented here is specifically tailored to noise control through acoustic metamaterials. Plate-type acoustic metamaterials, originally proposed by Yang et al. [START_REF] Yang | Membranetype acoustic metamaterial with negative dynamic mass[END_REF]- [START_REF] Yang | Acoustic metamaterial panels for sound attenuation in the 50-1000 Hz regime[END_REF], present a new and promising solution to the problem of low-frequency noise attenuation. Their small thickness and light weight could make them more desirable, for certain applications, than traditional materials used for noise control. Plates with or without a mass attached present a region of negative effective density with high sound transmission loss between the first and the second resonance, when backed with a rigid cavity, while plates suspended on an open supportas the ones presented in this papershow this behavior in the frequency band preceding the first mode [START_REF] Lee | Acoustic metamaterial with negative density[END_REF]- [START_REF] Sui | A lightweight yet sound-proof honeycomb acoustic metamaterial[END_REF]. Thin membranes and plates can be used as unit cells in acoustic metamaterials presenting different configurations. For example Yang et al. [START_REF] Yang | Acoustic metamaterial panels for sound attenuation in the 50-1000 Hz regime[END_REF] arranged them in in a panel, Sui et al. [START_REF] Sui | A lightweight yet sound-proof honeycomb acoustic metamaterial[END_REF] sandwiched the membranes between honeycomb panels and Naify et al. [START_REF] Naify | Scaling of membrane-type locally resonant acoustic metamaterial arrays[END_REF] built stacked membranes arrays. The thin plate presented in this paper could be arranged in any of the configurations described above. Furthermore, the fabrication technique presented here provides the opportunity to directly 3D print and integrate the plates inside other complex unit cells such as Helmholtz resonators and pipes to broaden their band gap. Being able to control the thickness, radius and material properties of the plates through additive manufacturing allows to choose the best combination of parameters to match a specific frequency range to be attenuated, while respecting the dimension constraints of a specific application. For example, in this paper the goal is to fabricate thin plates that could be included in small-scale devices, such as headphones and smart speakers, and at the same time having a low resonance frequency. Therefore, a polymer that had a low Young's modulus with respect to other materials used in previous work [START_REF] Casarini | Enhancing the Sound Absorption of Small-Scale 3-D Printed Acoustic Metamaterials Based on Helmholtz Resonators[END_REF] was selected. The 3D printing technique here developed made it possible to fabricate layers 70 μm thick. The chosen materials allowed to obtain a small-scale thin plate with a low resonance frequency.

Characterization is a crucial step in the production process to predict the acoustic behavior of a device at the design stage. This is particularly true for 3D printing applications, where the physical quantities of the materials change throughout the polymerization process. A good estimate of properties such as Young's modulus, Poisson's ratio, density and loss factor is very important to design more complicated devices that include these plates and to predict their acoustic impact before fabrication using finite element analysis or other modeling techniques. In the following sections the fabrication process used to 3D print the plates is first described in detail, then, three different characterization methods are introduced. Laser Doppler vibrometer is used to show that the 3D printed layer presents the typical modes of vibration found in circular plates and membranes. Impedance tube measurements assessed by the transfer matrix modeling approach are employed to retrieve the main physical parameters such as Young's modulus, density, Poisson's ratio and loss factor, and to obtain the acoustic coefficients, namely absorption, transmission and reflection. Finally, nanoindentation together with scanning electron microscopy give further insight on the properties of the plate. Future work and potential industrial applications are outlined in the Conclusion section.

II. MATERIALS AND METHODS

A. Plates Fabrication

This section firstly presents the design choices in terms of geometry and materials and secondly describes in detail the thin plate fabrication steps. An Asiga Pico Plus 27 (Sidney, Australia) with a pixel image resolution of 27 by 27 μm, and a minimum build layer thickness of 1 μm was used to 3D print the samples. A circular shape was selected for the plate, since the analytical model of circular membranes and thin plates is well known and also due to previous research based on cylindrical Helmholtz resonators [START_REF] Casarini | Enhancing the Sound Absorption of Small-Scale 3-D Printed Acoustic Metamaterials Based on Helmholtz Resonators[END_REF] that could be integrated with thin plates in future work. A hollow cylinder 1 mm in height and with 1 mm thick walls was also designed to support the plate. The 3D printer build plate size is 33.49 x 20.99 x 76 mm and the outer diameters of the support and plate were set at 20 mm, which is the maximum diameter. Therefore, the effective diameter of the membrane was 15 mm. There are two main reasons for selecting the maximum possible diameter. Firstly, in order to characterize the plate with an impedance tube, the diameter should be as large as possible, given the standard dimension of the tube, in this case having a 30 mm diameter. Secondly, since the resonance of a thin plate is inversely proportional to the square of its radius, a larger diameter would result in a lower frequency, as required for noise applications. As shown in Figure 1, two different samples were fabricated. The first sample had as main component polyethylene glycol diacrylate 250 (PEGDA 250) and the second polyethylene glycol diacrylate 700 (PEGDA 700). The numbers 250 and 700 represent the molecular weights (MW) of the two different types of PEGDA. In particular, PEGDA 700 was preferred for this work as it is characterized by a lower Young's modulus than PEGDA 250 and hence a lower resonance frequency, that could match and attenuate lower frequencies. BaTiO3 nanopowder with a 500 nm diameter average particle size, purchased from US-NANO, was added to the resin with a 50% weight with respect to PEGDA when fabricating the plate. BaTiO3 main function was to scatter the light in a uniform way on the horizontal plane, to obtain a flat plate, since previous attempts that did not include BaTiO3 resulted in a curved plate. It is important to notice that by adding BaTiO3 to the resin, a piezoelectric material can be obtained [START_REF] Tiller | Piezoelectric microphone via a digital light processing 3D printing process[END_REF] and it could be used as such in future implementation of this work. Irgacure 819 (I819) was added to both mixtures as a photoinitiator with a 1% weight with respect to PEGDA and 0.1 % Sudan I (SI) was added as absorber to better control the amount of light that could contribute to the polymerization of the resin. These chemicals were purchased from Sigma Aldrich. While the resin described above was used to print the thin plate, the support material was made of PEGDA 250, 0.1% Sudan I and 1% Irgacure 819. The exposure time to light contributes to determining the thickness of the membrane [START_REF] Gong | Optical Approach to Resin Formulation for 3D Printed Microfluidics[END_REF] and different exposure times between 0.5 second and 9 seconds where tried for different materials as reported in [START_REF] Casarini | Enhancing the Sound Absorption of Small-Scale 3-D Printed Acoustic Metamaterials Based on Helmholtz Resonators[END_REF]. Since the Asiga 3D printer software does not allow to change the exposure time to light or other parameters while the print has started, the chosen exposure time of 5 seconds was of a duration that allowed to print both the materials of the support and the materials of the membrane. The first step in the fabrication process, as shown in Figure 2, consisted in mixing the chemicals to obtain the resin for the thin plate and the resin for the support. The two resins were then independently mixed for 2 minutes using Thinky ARE 250 planetary mixer (INTERTRONICS) and then sonicated for 15 minutes. The CAD design was then uploaded into the Asiga Composer 3D Printing Software and automatically sliced into 101 layers, each 10 μm thick. The Layers 1 to 100 corresponded to the support and layer 101 to the thin plate. The resin for the support was then poured into the build tray and the 3D printing process was started. The support and the plate were printed in one piece, by pausing the 3D printer in correspondence to the layer before the plate, i.e. layer 100, leaving the 3D printed PEGDA part attached to the build platform to keep the right calibration. After cleaning the part on the build platform with isopropyl alcohol (IPA), the liquid resin in the build tray was changed to the resin of the thin plate and the last layer was fabricated. It is important to notice that, while the chosen layer thickness is 10 μm, when printing on a void the final thickness is determined by the intensity of light, the absorption coefficient of the fluid and the exposure time to light [START_REF] Gong | Optical Approach to Resin Formulation for 3D Printed Microfluidics[END_REF]. With commercial resins, even when the build layer thickness was set to 10 μm or less, the plates resulted to be 1 mm thick. As shown in Figure 3, when adding SI to the resin to reduce the amount of light, it was instead possible to finely control the resolution and to obtain a thickness down to 35 μm [START_REF] Tiller | Piezoelectric microphone via a digital light processing 3D printing process[END_REF], [START_REF] Casarini | Enhancing the Sound Absorption of Small-Scale 3-D Printed Acoustic Metamaterials Based on Helmholtz Resonators[END_REF]. In this case, the fabricated plate had a thickness of 70 μm, which was measured using a digital micrometer. Panel (a) of Figure 3 shows the 3D printing process of the support first 100 layers. When the last layer is fabricated (panel (b)), the part of fluid that polymerizes underneath the air cavity expands beyond the expected position in the z axis, resulting in a thicker membrane. When SI is added to the resin (panel (c)), the polymerization process can be finely controlled, and the last layer corresponds to the selected thickness.

B. Characterization Using Laser Doppler Vibrometer

C. Transfer Matrix Model and Characterization Using Impedance Tube

To acoustically characterize the plates, a 30 mm in diameter impedance tube (CTTM, Le Mans, France) represented in Figure 5 and shown in Figure 6 (a) was used to experimentally measure the transmission, reflection and absorption coefficients. A brass sample holder shown in Figure 6 (b) and(c) to be inserted into the tube was also fabricated. The sample holder consists of two hollow cylinders, the first one having a cavity where the plate sample can be fit. Four joints extrude from the second cylinder and secure the connection with the first cylinder that presents four matched hollows. The plate is inserted into the first cylinder and then the second cylinder is connected and simply secured. The system was modeled through a transfer matrix approach [START_REF] Song | A transfer-matrix approach for estimating the characteristic impedance and wave numbers of limp and rigid porous materials[END_REF]. The parameters assumed in the model were then optimized by fitting the measurements results into the model [START_REF] Geslain | Acoustic characterization of silica aerogel clamped plates for perfect absorption[END_REF].

As shown in Figure 5, the system to be modeled consisted of the plate with thickness ℎ, the sample holder modeled as two air cylinders of length 𝑙 1 and 𝑙 2 on both sides of the plate and their end corrections. The sample holder is modeled as a circular tube and its wave number 𝑘 𝑡 and the impedance 𝑍 𝑡 are derived as in [START_REF] Stinson | The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape[END_REF]. Therefore, the transfer matrix 𝑇 = ( 𝑇 11 𝑇 12 𝑇 21 𝑇 22 ), as discussed in [START_REF] Song | A transfer-matrix approach for estimating the characteristic impedance and wave numbers of limp and rigid porous materials[END_REF], [START_REF] Jiménez | Iridescent Perfect Absorption in Critically-Coupled Acoustic Metamaterials Using the Transfer Matrix Method[END_REF], is equal to:

𝑇 = 𝑀 𝑒𝑛𝑑𝑐𝑜𝑟𝑟 𝑀 𝑡𝑢𝑏𝑒1 𝑀 𝑚𝑒𝑚 𝑀 𝑡𝑢𝑏𝑒2 𝑀 𝑒𝑛𝑑𝑐𝑜𝑟𝑟 , (1)
where 𝑀 𝑒𝑛𝑑𝑐𝑜𝑟𝑟 is the transfer matrix of the end correction 𝛥𝑙 [START_REF] Merkel | Control of acoustic absorption in one-dimensional scattering by resonant scatterers[END_REF]:

𝑀 𝑒𝑛𝑑𝑐𝑜𝑟𝑟 = ( 1 ⅈ𝑍 𝑡 𝑘 𝑡 𝛥𝑙 0 1 ). ( 2
)
𝑀 𝑡𝑢𝑏𝑒𝑗 is the transfer matrix of the sample holder on the left (j = 1) and right (j = 2) sides of the plate [START_REF] Groby | The use of slow waves to design simple sound absorbing materials[END_REF]: ).

𝑀
(3)

𝑀 𝑚𝑒𝑚 = ( 1 𝑍 𝑚 0 1 ), (4)
where 𝑍 𝑚 is the acoustic impedance of the thin plate [START_REF] Bongard | Acoustic transmission line metamaterial with negative/zero/positive refractive index[END_REF]:

𝑍 𝑚 = -ⅈ𝜔𝑚 ⋅
𝑘 𝑚 is derived in [START_REF] Bongard | Acoustic transmission line metamaterial with negative/zero/positive refractive index[END_REF] 𝑎 is the radius of the plate, 𝐽 𝑛 and 𝐼 𝑛 are respectively regular and modified Bessel's functions of the first kind of order n and 𝑚 is the mass of the plate. The wave number of the plate 𝑘 𝑚 is equal to [START_REF] Bongard | Acoustic transmission line metamaterial with negative/zero/positive refractive index[END_REF]:

𝑘 𝑚 2 = 2𝜋𝑓√ 𝜌′′ 𝑚 12(1-𝑣 2 ) 𝐸ℎ 3 , (6)
where 𝜌 𝑚 ′′ = 𝜌 𝑚 ℎ is the surface mass density of the plate with mass density 𝜌 𝑚 = 𝑚 𝑆ℎ , where 𝑚 is the mass of the plate and 𝑆 its cross-section. The resonance frequency 𝑓 0 is given in terms of the the Young's Modulus 𝐸, Poisson's Ratio 𝜈 and loss factor 𝜂 and the geometry by [START_REF] Morse | Vibration and Sound[END_REF]:

𝑓 0 = 0.4694 ℎ 𝑎 2 √ 𝐸 𝜌 𝑚 (1-𝑣 2 ) . ( 7
)
Transmission, reflection and absorption coefficients can be retrieved from the total transfer matrix 𝑇 according to [START_REF] Mechel | Formulas of Acoustics[END_REF], [START_REF] Blackstock | Fundamentals of physical acoustics[END_REF]. The impedance tube used to measure the acoustic coefficients had four microphones, an anechoic termination on one side and a loudspeaker on the other termination, as illustrated in Figure 6 (a). A sine sweep between 20 Hz and 2500 Hz was used as input [START_REF] Niskanen | Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements[END_REF]. Each sample was measured 10 times to assess the repeatability of the measurements. The reflection, transmission and absorption coefficients were retrieved from the measurements [START_REF] Song | A transfer-matrix approach for estimating the characteristic impedance and wave numbers of limp and rigid porous materials[END_REF], [START_REF] Blackstock | Fundamentals of physical acoustics[END_REF]. Then, the physical quantities, chosen to initialize the model, namely density, Poisson's ratio loss factor and Young's modulus, were optimized to fit the experiments.

D. Characterization Using Nanoindentation

In order to compare different characterization methods and to determine which one is most suitable for this kind of samples, a 3D nanoindenter was also used to retrieve the Young's modulus of the plate. The 3D printed plate was cut and just a small sample of it was used during this process. A Berkovich tip made of single crystalline diamond was used for nanoindentation and a IBIS software retrieved the Young's Modulus of the materials. The values of the reduced elastic modulus were determined using the method of Oliver and Pharr [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF]. The Young's modulus was then obtained from the reduced Elastic modulus assuming a Poisson's ratio of 0.35. To better understand if this characterization process was compatible with the 3D printed plate, a scanning electron microscope was used to obtain a detailed image of the sample and to analyze the distribution of BaTiO3 and PEGDA particles.

III. RESULTS

A. Laser Doppler Vibrometer Results

The measurements obtained using laser Doppler vibrometer method showed that the 3D printed samples presented in this paper behave as thin plates. Two models exist in the literature to predict the modes of vibration of circular thin layers. The model of a membrane considers a thin layer only dependent on tension, while the model of the plate is only dependent on stiffness. Furthermore, a membrane resonance frequency is inversely proportional to its thickness, while a plate is directly proportional to it [START_REF] Morse | Vibration and Sound[END_REF]. While fabricating the samples, it become clear that a thicker sample had a higher resonance frequency and therefore a thin plate model was chosen (see Eq. 7). As shown in Figure 7a for PEGDA 250 and in Figure 7b for PEGDA 700, it can be clearly seen that the plates present several modes of vibration, both radial and circular modes. It is possible to predict the frequency of the modes of vibration of a circular plate as reported in [START_REF] Morse | Vibration and Sound[END_REF], where modes (1,1) and (0,2) are predicted to be respectively 2.091 and 3.909 times higher than mode (0,1). The 3D printed plates present all the modes of vibrations predicted by [START_REF] Morse | Vibration and Sound[END_REF] and their frequencies are in good agreement with those predicted in the literature.

Table I shows the average resonance frequency for the two types of PEGDA samples. These values were useful while modeling the plates with the transfer matrix method, as they were used as initial guess to initiate the optimization procedure to retrieve the other parameters.

B. Impedance Tube Results

The transmission and reflection coefficients were first retrieved through impedance tube measurements and are represented respectively by the solid magenta and green lines in Figure 8 for PEGDA 250 (a) and 700 (b). The same coefficients together with absorption were then retrieved using the transfer matrix model and initialized using the values given by laser Doppler vibrometer for the resonance frequency and by vendors data sheets and literature values for density, Young's modulus, Poisson's ratio and loss factor. To minimize the difference between transmission, reflection and absorption coefficients obtained experimentally with those obtained analytically, the values of density, Young's modulus, Poisson's ratio and loss factor assumed at the modeling stage where changed iteratively during an optimization process, until the modeled coefficients graph in Figure 8 fit into the graph representing the experimental coefficients. Therefore, the density, Young's modulus, Poisson's ratio and loss factor parameters retrieved in this section result from the process of minimization of the difference between experimental coefficients obtained through measurements in the impedance tube and theoretical transmission, reflection and absorption coefficients modeled with the transfer matrix approach. Through this inverse method it was therefore possible to obtain the values of Young's modulus, density, Poisson's ratio, loss factor and fundamental frequency listed in Table II. The differences between the values of fundamental frequency given by laser vibrometer and the impedance tube approach might be due to the differences in samples preparation. In the first case, the plate is in the free field and its support is glued to a piece of glass. Hence, a tiny volume of air is created between the plate and the support. In the second case, there is no enclosed volume of air underneath the plate, which is in a tube instead than the free field. The values retrieved through LDV measurements should therefore be used just as initial values for our model and to show the presence of the modes in the plates. Figure 8 shows that the plates fabricated in this work would be suitable for acoustic metamaterials applications. In fact, kinds of samples show a high reflection coefficient hence low transmission before the first mode of resonance, which corresponds to a region of negative effective density [START_REF] Sui | A lightweight yet sound-proof honeycomb acoustic metamaterial[END_REF], [START_REF] Hu | Investigation of the negative-mass behaviors occurring below a cut-off frequency[END_REF], [START_REF] Varanasi | The low frequency performance of metamaterial barriers based on cellular structures[END_REF].

C. Nanoindentation Results

Nanoindentation measurements were gathered on both top and bottom sides of the samples. By looking at Figure 9, it can be seen that the particles distribution is different on the two sides of the plate. The bottom side tends to contain more BaTiO3 particles than the top side. Therefore, two different values for the Young's Modulus of both PEGDA 250 and PEGDA 700 samples were retrieved, as outlined in Table III. This nonhomogeneous distribution of particles in the plate can possibly explain the different values of Young's modulus given by nanoindentation and retrieved through impedance tube measurements. Since the tip used for nanoindentation is small in comparison to the dimension of the particles of PEGDA and BaTiO3 in the plate, it can be inferred that, for this specific type of plates, impedance tube measurements are more reliable than nanoindentation. Moreover, the sample required for nanoindentation must be small and cut from the support, hence possibly eliminating the small stress that the plate could have received while being attached to the support during the 3D printing process. This sample preparation process could cause a modification of the physical properties of the plate. Therefore, in future designs containing these plates, the results from impedance tube measurements will be preferred when modeling the system. Properties of the materials used to 3D print the plates as retrieved by using the transfer matrix approach and optimization with the impedance tube measurements.

IV. CONCLUSION

Thin flat plates were successfully 3D printed using innovative stereolithography techniques based on direct light processing. Thanks to these new techniques it was possible to carefully choose suitable materials and engineer a thin plate that could be used to attenuate low frequencies for example in road traffic, engine or aircraft noise. Recently this goal has been achieved by exploiting active noise cancellation in headphones, and in the future acoustic metamaterials could be integrated in electroacoustic devices or inside cars and aircrafts to achieve a similar noise attenuation. Other frequency ranges can be attenuated by using different materials or thicker plates. Three different characterization techniques were used, namely laser Doppler vibrometer, nanoindentation and impedance tube with transfer matrix model to retrieve the physical quantities and parameters related to the plate under test. In particular, impedance tube measurements confirmed to be a suitable and complete technique for this kind of materials. Laser Doppler vibrometer was also very useful to quickly identify the vibration modes and it could be employed to retrieve the physical parameters if used in conjunction with a simulation technique such as finite element analysis. Nanoindentation was less preferred, in that the tip is very small and could randomly sample a part of the plate where BaTiO3 is not present. The obtained physical parameters will be integrated in future designs where the plates will be incorporated in more complicated structures. The fabrication technique presented in this paper could be scaled to cover different frequency ranges and for larger structures, for example to attenuate noise in the field of aerospace and other transportation industries. Nevertheless, this specific design would be more suitable for small-scale applications, such as headphones, smart speakers and other electroacoustic devices.

  A Polytec 3D scanning laser Doppler vibrometer (LDV) was used to perform the first characterization of the modes and fundamental resonance of the plates under test. Each sample was fixed to a glass support, and a loudspeaker (ESS Heil Air-Motion Transformer) was used to create a sweep between 100 and 20000 Hz. The vibration of the plate was mapped by 289 scanning points. The Polytec software was used to analyze the results and plot the vibration modes of the plate under test.

Fig. 1 .

 1 Fig. 1. 3D printed plates front (left) and back (right).

Fig. 3 .

 3 Fig.3. The first 100 layers of the sample, i.e. the empty cylindrical support, are first 3D printed thanks to the projection of the UV light according to the selected geometry and to the consequent polymerization of the liquid material contained in the build tray (a). When no SUDAN I is added to the liquid resin, the last layer, i.e., the plate, polymerizes beyond its defined thickness, expanding in the void inside the support (b). Adding SUDAN I allows to better control the amount of light absorbed by the resin and to 3D print a plate with the correct thickness (c).

Fig. 2 .

 2 Fig. 2. Diagram representing the 3D printing process step by step.

  𝑡𝑢𝑏𝑒𝑗 = ( 𝑐𝑜𝑠(𝑘 𝑡 𝑙 𝑗 ) ⅈ𝑍 𝑡 𝑠𝑖𝑛(𝑘 𝑡 𝑙 𝑗 ) 𝑖𝑠𝑖𝑛(𝑘 𝑡 𝑙 𝑗 ) 𝑍 𝑡 𝑐𝑜𝑠(𝑘 𝑡 𝑙 𝑗 )

𝐼 1 (

 1 𝑘 𝑚 𝑎)𝐽 0 (𝑘 𝑚 𝑎)+(𝐽 1 𝑘 𝑚 𝑎)𝐼 0 (𝑘 𝑚 𝑎) 𝐼 1 (𝑘 𝑚 𝑎)𝐽 2 (𝑘 𝑚 𝑎)-𝐽 1 (𝑘 𝑚 𝑎)𝐼 2 (𝑘 𝑚 𝑎) ,

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Experimental setup used during data acquisition with a Polytec laser Doppler vibrometer (LDV). The vibration modes of the sample were excited acoustically though a sine sweep.

Figure 7 .

 7 Figure 7. Modes (0,1), (1,1), (0,2), (0,3) for the PEGDA 250 sample (a). Modes (0,1), (1,1), (0,2), (1,2) for the PEGDA 700 sample (b).

Fig. 8 .

 8 Fig. 8. Transmission, reflection and absorption coefficients represented by the dotted lines correspond to the parameters retrieved with the optimized transfer matrix model. The solid lines show reflection and transmission coefficients measured through the impedance tube and normalized to the input signal. PEGDA 250 values are shown in graph (a) and PEGDA 700 in graph (b).

Fig. 9 .

 9 Fig. 9. Cross-section of the PEGDA 250 sample used for nanoindentation and obtained with scanning electron microscope. It can be noticed that the top and bottom side contain different amounts of BaTiO3 and PEGDA particles.

TABLE I MATERIAL

 I

		PROPERTIES -LDV
	Symbol Quantity	PEGDA 250	PEGDA 700
	𝑓 0 (0,1) Fundamental	940	400
		frequency (Hz)
	(1,1)	First radial mode	1500	1070
		(Hz)
	(0,2)	First circular mode	2300	1530
		(Hz)
	Values of fundamental frequency and first circular and radial modes for samples
	of PEGDA 250 and PEGDA 700 having a 15 mm diameter.

TABLE III

 III 's modulus of PEGDA 250 and PEGDA 700 obtained through nanoindentation measurements. Both top and bottom of the samples have been measured because of non-homogeneous distribution of BaTiO3 in the sample, as highlighted by scanning electron microscopy.

		MATERIAL PROPERTIES -NANOINDENTATION
	Symbol Quantity	PEGDA 250	PEGDA 700
	𝐸	Young's Modulus	0.30	0.062
		(GPa) Top
	𝐸	Young's Modulus	0.12	0.059
		(GPa) Bottom

Young

ACKNOWLEDGMENT The authors would like to thank Dr. Milovan Cardona for his help in performing nanoindentation and in using the scanning electron microscope at the University of Strathclyde Biomedical Engineering Department. This article is based upon work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology) and by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. [615030].

This research is funded by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. [615030] and by COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology).

Vicente Romero-García is CNRS researcher at the Acoustic Laboratory of Le Mans University (LAUM, UMR CNRS 6613) since 2014. His research interests cover a wide range of subjects in wave physics. In particular wave propagation in periodic media as phononic and sonic crystals, random media, complex structures as well as acoustic metamaterials, and metasurfaces.

Jean-Philippe


I Herrero-Durá

A Cebrecos

V Romero-García

L M Garcia-Raffi

Matrix formulation in Acoustics: the Transfer Matrix Method

Matrices are introduced in mathematical subjects related to vector spaces and linear Algebra, being disconnected with their applications in other fields of Science and Engineering studies. The transmission of this knowledge is done, in many occasions, in a purely theoretical manner and posing problems to students that are sometimes abstract, lacking application to real world. In this paper we present an application of matrix formalism in Acoustics: The Transfer Matrix Method. It is a simple method widely used in transmission problems. For a system composed of a waveguide and an aluminum plate, subjected to the pressure of an acoustic wave, we can establish a simple connection between the matrix formalism, the use of vectors and the physical magnitudes

Introduction

Matrix formalism is part of the basic mathematical education of scientists and technologists The term "matrix"was introduce by Sylvester but was a collaborator of him, Calley (1821-1895), who used a form of matrices in the work "On the Theory of Linear Transformations"using the term "matrix"in its modern mathematical sense [START_REF] Keiner | A History of Abstract Algebra[END_REF]. Usually, matrices are introduced in mathematical subjects related to vector spaces and linear algebra, being disconnected form their applications in other fields of science. Of course, matrices are used in the classroom in many subjects but sometimes students are not able to establish the correct connection between certain mathematical objects and its application. In many cases, classical applications such as those relating to mechanics, especially those relating to diagonalization and its relationship with the determination of principal moments and axes of inertia of a body are the only ones treated in the math classroom. The other typical example is related to the matrix formulation of quantum mechanics, but its interpretation is much more complex for students.

Acoustics, one of the disciplines of classical physics, is often absent in the general physics syllabus. However, it is a discipline that provides facts of physics that in many occasions can be checked by the student in a simple way, using very simple setups. The theoretical description of acoustics can be complex involving the resolution of Navier-Stokes equations, but in many cases acoustic phenomena can be described using a simple mathematical setting. This is the case of the Transfer Matrix Method (TMM). This method is a matrix formalism used in many field of physics (optics, acoustics, quantum mechanics) to analyze wave propagation in a 1-D setting (see for example Markoš and P., Soukoulis, C.M., 2008). This method gives teachers the possibility to pose simple problems to students in the field of physics, where diagonalization, eigenvectors and eigenvalues have a simple interpretation. The example proposed in this article starts with a simple problem, easily understandable for students with basic knowledge in physics, and follows with building up of the mathematical formalism based on matrices. After that, the resolution of the problem is connected with finding eigenvalues and eigenvectors of a matrix, giving a physical meaning to them. Under the perspective of didactics, the activity proposed can be considered a modeling eliciting activity [START_REF] Lesh | Foundations of a models and modeling perspective on mathematics teaching, learning, and problem-solving[END_REF].

First, we introduce the TMM, that has a transversal character from a teaching perspective, as the same formalism can be applied at different levels in science and technology, covering from a basic course on Physics to a more specialized one. The system under analysis is a transmission line (a waveguide) with an obstacle (a clamped plate). TMM relates the acoustic pressure and the particle velocity in the left side of the obstacle to those on the right side. Transmission and reflection coefficients will be defined from the transfer matrix formalism. Next, we will introduce the scattering matrix, which describes the outgoing waves form the obstacle in terms of the ingoing waves and is directly related to the transfer matrix. Eigenvalues and eigenvectors of this scattering matrix will be calculated and interpreted in terms of the physical behavior of the system. Finally, an experimental activity in acoustics will be proposed in order to bring mathematics to the laboratory. All the concepts of matrix algebra that appear in this paper can be found in several textbooks (see for example, Lay, D. C., 1997 or Strang, G., 2016).

Transfer matrix method

We start considering a square aluminum clamped plate placed into a tube (see Figure 1), in other words, a resonant inhomogeneity in a waveguide. We consider that the thickness of the plate, h, is much smaller than the wavelength of the incoming waves, λ. Imagine we want to transmit a wave through it, for example an acoustic wave. We place a sound source (loudspeaker) at the beginning of the tube that emits a signal containing several frequencies that will be our range of study, always with frequencies lower than the cutoff frequency of the waveguide, which correspond to the range of frequencies where only plane waves can propagate. Thus, the system can be considered one dimensional. The incident plane wave impinges the plate, inducing the vibration of it. As a result of this interaction, one part of the energy is reflected back and other part is transmitted through the plate. If no losses are considered, by energy conservation, all the energy that is not reflected is transmitted. If losses are considered, part of the incident energy is reflected, part absorbed by heating the plate, and part is transmitted. The acoustic process can be characterized by the state vector s constructed from the sound pressure, p, that is related with the force applied to the medium and the normal particle velocity, v = v e x , with e x the unitary vector along the x-direction. The particle velocity is related with the reaction of the medium to the application of this force, i.e., related with the opposition of the medium to be perturbed or acoustic impedance. This state vector is in R 2 , and can be represented as,

s = p v . (1)
The TMM relates the sound pressure, p, and normal acoustic particle velocity, v, between the two faces of a system extending from x = 0 to the thickness x = h by means of the transfer matrix T, defined as follows

p v x=0 = T p v x=h , (2)
In other words, in the math language at classroom, the matrix T is a linear mapping.

We consider waves having a wavelength much longer than the diameter of the tube. In this case only plane waves are excited in the tube and this can be considered as a 1-D (x-axis) problem (see Figure 2(a)) where the reflected and transmitted waves can be easily related to the incident wave through the reflection and transmission coefficients. Under this conditions, we are only interested in the first resonant mode of the clamped plate, i.e., the "piston mode"as can be seen in Figure 2 The acoustic behavior of the clamped plate can be characterized by its acoustic impedance, Z p , which is a classical resonant function expressed as follows,

Z p = iΓω ω 2 0 -ω 2 + iσω -1 , (3)
where Γ and σ are the admittance parameter and the dissipation term, respectively. ω 0 is the frequency of the first resonant mode of the clamped 4 plate that can be obtained from the following approximated equations,

C p = 3.73 10 -4 a 6 D , M p = 2.06ρh a 2 , (4)
ω 0 = 1 M p C p , (5)
where ρ the density of the plate, and D = Eh 3 /12(1 -ν 2 ), the bending rigidity, with E the Young's modulus and ν the Poison's ratio; a represents the side of the square clamped plate. In this work we will use a = 0.1 m, E = 60 GPa, ρ = 2700 kg/m 3 , ν = 0.3 and h = 0.5 mm, Γ = 0.5 m/Pa s -2 and σ (s -1 ) will vary depending on the analyzed case (σ = 0 represents the lossless case and σ = 0 the lossy case). Notice that in this work we will use the temporal dependence e iωt . On the other hand, we can consider that the clamped plate is a locally resonant elements in series in a main waveguide. As h << λ, we consider the plate as punctual resonator in the x direction. Defining the pressure at both sides, p x=0 , p x=h , and the corresponding particle velocity v x=0 , v x=h , the effect of the plate on the acoustic field is twofold: (i) the velocity is the same on both sides due to continuity boundary conditions, i.e., v d = v u ; (ii) there is a pressure drop between each side due to the presence of the plate, and we can relate it at both sides of the punctual resonator using the impedance of the resonator as

Z p = ∆p v = p x=0 -p x=h v x=h . (6)
Then, the equations relating pressure and velocity at both sides are

p x=0 = p x=h + Z p v x=h , (7)
v x=0 = v x=h . (8)
Relating the state vectors (p x=0 , v x=0 ) and (p x=h , v x=h ), the transfer matrix, can be obtained as

T p = 1 Z p 0 1 , (9)
where Z p is the acoustic impedance of the clamped plate given by Eq. 3. This matrix represents the transfer matrix of the clamped plate. But, at the beginning of the section we talk about the scattering coefficients. Hence, which are the reflection and transmission coefficients of our system?

Reflection and Transmission coefficients

We consider here the general case of a scattering problem by an slab of material of thickness L and represented by a transfer matrix T . Then, applying Eq. 2 we obtain:

p(0) = T 11 p(h) + T 12 v(h), (10)
v(0) = T 21 p(h) + T 22 v(h).
Due to the geometry of the considered system, the scattering is symmetrical, meaning that the reflection coefficients obtained from both sides are identical. This condition implies that the elements of the transfer matrix must fulfill this equation T 11 = T 22 . Moreover, we are in the linear regime and the scattering process should be reciprocal, meaning that transmission is identical in both senses. This implies that the determinant of the transfer matrix is unity, T 11 T 22 -T 12 T 21 = 1. Notice that Eq. 9 fulfills both conditions because a clamped plate in the linear regime creates a reciprocal and symmetric 1D scattering problem. For more details, we refer to reference (Song, B.H and Bolton, J.S., (1999)).

Under these conditions, for a wave of amplitude one traveling from left to the right, the pressure in the left-hand side of the plate will be the result of the incident pressure plus the reflected one, that is:

p(0) = 1 + R,
and for the velocity we have to subtract the part corresponding to the reflected wave that travels in the opposite sense than the incident wave and divide by the impedance of the air in the tube, that is:

v(0) = 1 -R Z 0 .
In the same manner, the wave in the right part will be the transmitted one:

p(h) = T e ıkh ,
where we have taken into account the phase e ıkh corresponding to the plane wave traveling to the right part e ıkx evaluated at x = h. For the velocity

v(h) = T e ıkh Z 0 .
Combining these equations with Equation 10 we obtain:

1 + R = T e ıkh (T 11 + T 12 Z 0 ), ( 11
) 1 -R = T e ıkh (Z 0 T 21 + T 22 ),
and then adding both equations and solving for T

T = 2e -ıkh T 11 + T 12 Z 0 + Z 0 T 21 + T 22 . ( 12
)
Dividing equations in Eq. 11, we obtain

1 + R 1 -R = T 11 + T 12 Z 0 Z 0 T 21 + T 22 ,
from which we can solve

R = T 11 + T 12 Z 0 -Z 0 T 21 -T 22 T 11 + T 12 Z 0 + Z 0 T 21 + T 22 . ( 13
)
Now, we can particularize Eqs. 12 and 13 with the transfer matrix of the clamped plate, given by Eq. 9, then simple equations for the reflection and transmission coefficients can be obtained:

T = 2e -ikh (2 + Z p /Z 0 ) 2 (2 + Z p /Z 0 ) , (14)
R = Z p /Z 0 2 + Z p /Z 0 . ( 15
)
We notice that in the lossless case |R|+|T | = 1. In the lossy case, |R|+|T | = 1, and an absorption coefficient should be defined as

α = 1 -|R| 2 -|T | 2 .

Scattering of waves by a clamped plate

Figure 3 represents the acoustic impedance and the scattering coefficients of both the lossless and the lossy cases. In the lossless case, we can observe that at the resonant frequency ω/ω 0 = 1, the condition Im(Z p )=0 with Re(Z p ) = 0 is fulfilled (see Fig. 3(a)), which represents the resonant condition. In fact, if we look at the scattering coefficients, Fig. 3(b), we can see that at this particular frequency, the transmission is enhanced producing perfect transmission of waves through the plate with R = 0. However as soon as the losses are introduced in the plate, at the resonant frequency, the real part of the impedance is no longer zero (see Fig. 3(c)). This dramatically impacts the reflection and transmission coefficients. Figure 3(d) shows the enhancement of the absorption at the resonant frequency as a consequence of the presence of losses in the plate.

Scattering matrix: eigenvalues and eigenvectors

As we have mentioned above, when the acoustic wave is transmitted along the tube, it arrives to the plate, makes it to vibrate and some part of the wave is transmitted and some part is reflected. This process is intimately related with the fact that the plate scatters the wave: is the scattering problem. This interaction is in general too complex to be easily described, but in many cases it is sufficient to know the wave function in the left and the right part of the resonant inhomogeneity.

The waves in the left and the right-hand side of the plate can be defined in terms of plane waves, which in mathematical terms are simple complex exponentials (see Figure 4):

p + L (x) = Ae -ıkx p + R (x) = Ce -ıkx , (16)
p - L (x) = Be ıkx p - R (x) = De ıkx . ( 17
) (18)
Then the total pressures at both sides of the structure are given by:

p(x) = Ae -ıkx + Be +ıkx for x < x p , (19)
p(x) = Ce -ıkx + De +ıkx for x > x p + h, (20)
being x p the position of the plate.

In general, the relationship between incoming and outgoing amplitudes remains:

C B = S A D = T R R T C B ( 21
)
where T and R are the transmission and reflection coefficients calculated above.

As it has been mentioned above, this system is symmetric and reciprocal, and the resulting scattering matrix is symmetric and diagonalizable (moreover it is unitary SS † = I where † is the conjugated transposed operation and I is the identity matrix, and det(S) = 1).

By using the matrix diagonalization, it is possible to obtain the eigenvalues and eigenvectors, which as we will see have relevance from a physical point of view.

Eigenvalues

A scalar λ is called an eigenvalue of A if there is a non-trivial solution x of A x = λ x; where x is called eigenvector corresponding to λ (Lay, D.C. 1997 or Strang, G. 2016). In order to obtain the eigenvalues of a matrix, it is necessary to solve the following equation (S -λI) x = 0, or calculate the Kernel of the mapping, Ker (S -λI).

Obtaining a solution implies that the determinant should be equal to zero, that is:

|S -λI| = 0, ( 22
)
where S is the scattering matrix, λ is the eigenvalue that we want to determine and I is the identity matrix.

Equation 22 results in a second-degree polynomial which solutions are the two eigenvalues of the system (a problem in R 2 )

(T -λ) 2 -R 2 = 0, ( 23
)
λ 1,2 = T ∓ R. ( 24
)
Figure 5 shows the dependence of the eigenvalues of the scattering matrix with frequency. Two cases are represented: the first one without losses (σ = 0) (Figs. 5(a)) and the second one with losses (σ = 416) s -1 (Figs. 5(b)). In the lossless case the eigenvalues take the same value at the resonance frequency of the system λ 1,2 = 1 as a consequence of the energy conservation, i.e., |R| 2 + |T | 2 = 1. In fact, at the resonant frequency, |R| = 0 and |T | = 1, then λ 1 = λ 2 = 1. In the lossy case, the situation is a little bit more complicated, as losses induce absorption of energy. In this case, we can observe that for the chosen case, R = T = 0.5 (see Fig. 3(d)). As a consequence, we have one of the eigenvalues that has a zero value at the resonant frequency and the other one that is 1.

We can evaluate the dependence of the scattering coefficients and the eigenvalues of the scattering matrix on the losses of the system, σ. Figure 6(a) represents the absorption of the system at the resonance frequency as the losses increase. We can observe that the absorption cannot pass the threshold of α(ω 0 ) = 0.5. In fact, this particular point correspond to the analyzed case in Fig. 5(b). This particular situation is due to the fact that only one of the two eigenvalues of the scattering matrix is zero at the resonant frequency, meaning that only half of the energy can be efficiently absorbed. In such kind of problems, with single resonators, only one mode can be excited in the plate, and only one of the two eigenvalues can be used to dissipate the energy; the other one remains almost invariable. In this work, this situation corresponds to the case with σ = 416 s -1 (see Fig. 6(c)). If we have smaller or bigger values of losses than this value, the efficiency in absorption is reduced, as shown in Figs. 6(b) and (d) respectively.

Eigenvectors

As in any other mathematical problem of diagonalization of a symmetric matrix, we are interested also in calculating the eigenvectors. Both, eigenvalues and eigenvectors are identified with resonances of the system. The eigenvector v 1 corresponding to the eigenvalue λ 1 must satisfy the equation A v 1 = λ 1 v 1 for v 1 unknown (Lay, D.C. 1997 or Strang, G. 2016). In fact we are dealing with obtaining a base element of the corresponding subspace associated to the eigenvalue λ 1 . The equation of this subspace is given by:

(S -λ 1 I) v 1 = 0, ( 25
)
where S is the scattering matrix, λ 1 is one of the two eigenvalues calculated above and v 1 is the eigenvector to be determined.

Solving the previous equation for λ 1 and λ 2 ,i.e., the two previous eigenvalues calculated, one gets the two eigenvectors of the system (one corresponding to each eigenvalue).

v 1 = [R, -R] , v 2 = [R, R] . (26)
At this point, due to the fact that we are dealing with a symmetric matrix, we can show for example that, if λ 1 = λ 2 eigenvectors corresponding to different eigenvalues are ortogonal. This can be easily checked using the standard euclidean inner product:

v 1 • v 2 = [R, -R] [R, R] t = R 2 -R 2 = 0, ( 27
)
where the symbol t represent the matrix operation transpose.

4 Proposal of action: How to connect with maths in the acoustics lab

Considering the information given along this work, it is possible to present a proposal of a practical lesson to be applied in the first course of engineering and science degrees. This lecture consists of solving the proposed problem by using the analytical tools and comparing the results obtained with experimental measurements. The students, having a previous background in mathematics, especially regarding to matrix diagonalization, complex exponentials and calculation of eigenvalues and eigenvectors, can check and compare the results of the model with experimental measurements. Also a basic knowledge on acoustics is necessary at a level of basic physics syllabus. The fundamentals of acoustics and TMM can be given to the students in a previous session. Once this step has been completed, the practical case that has been previously developed in this article will be explained and the students will have to obtain its analytical solution by using the given tools. The activity will be completed with a lab session in which students will perform experimental measurements with the device shown in Fig. 7. These measurements are performed by using the transfer-function method, detailed in the standard ISO 10534-2:2002. Figure 7 shows a picture with some elements necessary for the experimental measurement.

Conclusions

In this article we present the development of the formalism corresponding to the Transfer Matrix Method. This method gives the opportunity to use elements of matrix algebra that appear in math subjects in the first course of practically all degrees in science and technology in a real problem. The activity in the math classroom can be translated to the laboratory where, without the need of a sophisticated experimental setup, students can compare the predictions of a mathematical model with experimental results, and increase their understanding of the involved physical phenomena. This gives the opportunity not only to put mathematics in context, but also organize a multi-disciplinary activity where mathematical and physical concepts go together. In summary, the activity should contribute not only to a better understanding of the physical phenomena and their interpretation in terms of the analytical results, but also it should also anchor the mathematic knowledge via its relation with real problems.

Figure 1 :

 1 Figure 1: 3-D scheme of the problem in transmission. Dark blue represents the incident plane wave generated by the source, medium blue represents the sound wave that is reflected by the plate and soft blue represents the transmitted wave. The plate is represented in grey.

Figure 2 :

 2 Figure 2: (a) 1-D scheme of the problem in transmission. (b) First mode (piston mode) of the plate.

Figure 3 :Figure 4 :

 34 Figure 3: Analysis of the acoustic impedance of the clamped plate and the scattering coefficients. (a) and (b) represent the case without losses (σ = 0) and (c) and (d) represent the case with losses (σ = 416 s -1 ). (a) and (b) represent the imaginary (blue continuous line) and the real (red dashed line) part of the normalized impedance of the clamped plate. (b) and (c) represent the absolute value of the scattering coefficients, i.e., reflection (blue continuous line), transmission (red dashed line) and absorption (black dashdotted line). In all the cases the dependence is shown with respect to the normalized frequency with respect to the resonant frequency of the plate, ω 0 .

Figure 5 :Figure 6 :

 56 Figure 5: Analysis of the eigenvalues of the system. (a) represents the case without losses (σ = 0) and (b) represents the case with losses (σ = 416 s -1 ). (a) [(b)] represents the absolute value of the both eigenvalues of the system, |T + R| (blue continuous line) and |T -R| (blue dashed line) for the lossless [lossy] case. In all the cases the dependence is shown with respect to the normalized frequency with respect to the resonant frequency of the plate, ω 0 .

Figure 7 :

 7 Figure 7: Experimental set up for measurements.


M Malléjac

Aurélien Merkel

J Sánchez-Dehesa

J Christensen

V Tournat

J.-P Groby

V Romero-Garcia

Zero-phase propagation in

published or not. The documents may come

Acoustic and elastic metamaterials have attracted considerable attention in the last decades [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF][START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF][START_REF] Yang | Sound Absorption Structures: From Porous Media to Acoustic Metamaterials[END_REF][START_REF] Deymier | Acoustic Metamaterials and Phononic Crystals[END_REF] , certainly fuelled by the possibilities of tailoring their wave-dispersion properties and making previously unexpected behaviors real. Actually, a plethora of effective dynamic mass density values can be achieved, from negative [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Sheng | Locally resonant sonic materials[END_REF][START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF][START_REF] Park | Amplification of acoustic evanescent waves using metamaterial slabs[END_REF] to positive mass density, thus presenting a zero mass density at specific frequency. At the frequency at which the material mass density vanishes, the supported acoustic waves present a phase velocity tending to infinity in the lossless case, leading to an effective quasi-static field distribution [START_REF] Alù | Transmission-line analysis of ε-near-zero-filled narrow channels[END_REF][START_REF] Edwards | Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide[END_REF][START_REF] Park | Giant acoustic concentration by extraordinary transmission in zero-mass metamaterials[END_REF] , or equivalently to an infinite wavelength. In light of these features, some outstanding effects and applications were initially proposed in electromagnetics and optics by using the epsilon near zero metamaterials. Radiation patterning and lensing [START_REF] Alù | Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern[END_REF] or energy super-squeezing and tunneling via narrow ducts are examples of these possibilities [START_REF] Silveirinha | Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using ε-Near-Zero Materials[END_REF][START_REF] Silveirinha | Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials[END_REF][15][START_REF] Edwards | Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects[END_REF] . In acoustics, extraordinary properties have been numerically predicted based on DNZ lossless metamaterials, like extraordinary sound transmission through ultranarrow channels [START_REF] Fleury | Extraordinary sound transmission through density-near-zero ultranarrow channels[END_REF] , unity transmission through sharp bends and perfect power dividers [START_REF] Graciá-Salgado | Negative mass density and ρ -near-zero quasi-two-dimensional metamaterials: Design and applications[END_REF] . The geometric mismatch between a large cross-sectional air-filled waveguide and this ultranarrow channel is perfectly compensated by the extremely low density of the material occupying the channel, thus realizing a perfect impedance matching and an extraordinary tunneling via a supercoupling effect. This effect is accompanied by a large and uniform field enhancement along the channel and a zero-phase propagation of the wave front from the input to the output of the waveguide, implying a quasi-static wave propagation due to the extremely low value of the density. It is worth noting here that the mass density should not be zero, since it would imply an infinite impedance mismatch, but its value needs to be in the near zero region in order to compensate the cross section drop. Therefore, in the ultranarrow channels with DNZ metamaterials, the impedance matching and the wave tunneling do not depend neither on the channel length nor on the presence of bends, twists, and even absorbing sections, along the channel. Supercoupling is thus ideal for long-distant waveguide coupling with a high transmittance and no phase delay [START_REF] Fleury | Extraordinary sound transmission through density-near-zero ultranarrow channels[END_REF][START_REF] Alù | Coaxial-to-waveguide matching with ε-near-zero ultranarrow channels and bends[END_REF] , light concentration and harvesting [START_REF] Silveirinha | Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using ε-Near-Zero Materials[END_REF][START_REF] Silveirinha | Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials[END_REF][START_REF] Alù | Light squeezing through arbitrarily shaped plasmonic channels and sharp bends[END_REF] , sensing [START_REF] Alù | Dielectric sensing in ε-near-zero narrow waveguide channels[END_REF] , filtering [START_REF] Alù | Transmission-line analysis of ε-near-zero-filled narrow channels[END_REF] and nonlinear applications [START_REF] Powell | Nonlinear control of tunneling through an epsilon-near-zero channel[END_REF][START_REF] Argyropoulos | Boosting optical nonlinearities in ε-near-zero plasmonic channels[END_REF] . In the absence of huge cross-sectional change, the impedance matching should be achieved when the bulk modulus is in addition nearly infinite [START_REF] Ziolkowski | Propagation in and scattering from a matched metamaterial having a zero index of refraction[END_REF] .

A DNZ material can be made using different systems including plate-type acoustic metamaterials [START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF][START_REF] Fleury | Extraordinary sound transmission through density-near-zero ultranarrow channels[END_REF][START_REF] Bongard | Acoustic transmission line metamaterial with negative/zero/positive refractive index[END_REF][START_REF] Huang | Membrane-and plate-type acoustic metamaterials[END_REF] or periodic distributions of structured cylindrical scatterers embedded in two-dimensional waveguides [START_REF] Graciá-Salgado | Negative mass density and ρ -near-zero quasi-two-dimensional metamaterials: Design and applications[END_REF] .

On the one hand, plate type acoustic metamaterials 2,3 are of great practical interest thanks to their huge tunability. In particular, structured arrays of such elastic elements allow to break the density law and thus make them good candidates for achieving DNZ metamaterials [START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF][START_REF] Fleury | Extraordinary sound transmission through density-near-zero ultranarrow channels[END_REF][START_REF] Bongard | Acoustic transmission line metamaterial with negative/zero/positive refractive index[END_REF] . However, wave energy losses, not avoidable in this kind of systems [START_REF] Ma | Acoustic metasurface with hybrid resonances[END_REF][START_REF] Romero-García | Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators[END_REF][START_REF] Henriquez | Viscothermal losses in double-negative acoustic metamaterials[END_REF] , can dramatically alter its dispersion properties and as a consequence drastically affect the DNZ metamaterial behavior. When plate-type materials are considered, the viscoelasticity of the plate material in the ultranarrow tube influences the DNZ transmission peak amplitude [START_REF] Fleury | Extraordinary sound transmission through density-near-zero ultranarrow channels[END_REF] . Low damping viscoelastic plate material has thus to be selected. On the other hand, when periodic distributions of structured cylindrical scatterers embedded in a two-dimensional waveguide are considered, impedance matching is fulfilled in the double negativity regime, which is reached over a narrow frequency band. However, these systems are not able to maintain the extraordinary features associated with double negativity because of the viscothermal losses [START_REF] Henriquez | Viscothermal losses in double-negative acoustic metamaterials[END_REF] which are inherent to the system geometry.

In this work, we show that zero phase propagation can be achieved in a Plate-type Acoustic Metamaterial (PAM) and can be a promising tool for directive acoustic emission or wavefront shaping devices. The unit cell of length L unit forming the PAM is shown in Fig. 1 and is composed of one plate of thickness h = 102 μm and radius R = 1.615 cm which is surrounded on both side by two cylindrical cavities of length L gap /2 giving L unit = L gap + h. The inner and outer radius of the cylinders, which are made of aluminum, are R i = 1.5 cm and R o = 1.615, respectively. The plate is clamped between the two aluminum cylinders using 500 μm thick annular Teflon rings in order to enhance the reproducibility of the clamped condition. N unit cells are then hold together in a sample holder and an uniform pressure is applied on the sample by screws at the extremities.

The elastic parameters characterizing the plates are extracted from the experimental results (see supplementary material S.III). These parameters are the complex Young's Modulus E = E 0 (1 + iβ ), that contains the elastic part E 0 = 4.6 GPa and the loss factor β = 0.13 to account for the viscoelastic losses, the Poisson's ratio ν = 0.4 and the mass density ρ = 1400 kg.m -3 . The plastic shim composing the plates is chosen for its low loss factor. The resonance frequency of one single plate is experimentally measured at f m = 438 Hz which gives in the lossless case f m = 423 Hz.

In the cylindrical cavities surrounding the plates, the viscothermal losses are accounted for by considering a complex wavenumber k 0 (ω) and an impedance Z 0 (ω) [START_REF] Zwikker | Sound absorbing materials[END_REF][START_REF] Stinson | The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape[END_REF] , where ω is the cyclic frequency.

The theoretical effective dynamic mass density ρ(ω) of an infinite lossless PAM are first analyzed and paralleled up with the transmission and reflection coefficients of a PAM of finite size composed of N = 1, 3, 6 and 9 unit cells as shown in Fig. 2. The effective mass density is found with ρ(ω) = Z(ω)/c(ω) where c(ω) = ω/k(ω) is the effective sound velocity and k(ω) and Z(ω) are the effective wavenumber and impedance of the PAM, respectively. The effective frequency dependent wavenumber k(ω) and impedance Z(ω) of the PAM are obtained from the transfer matrix of a single unit cell as follows

k(ω)L unit = cos -1 [(T 11 + T 22 )/2] , (1)
Z(ω) = (T 12 /T 21 ) 1 2 , (2)
where T i j are the i j-th element of the 2×2 transfer matrix which is obtained theoretically (see supplementary material S.I) and can be extracted from the experimental and numerical scattering coefficients.

We define three characteristic frequencies.

The first one is the resonance frequency of the plate, f m , where the PAM impedance matches that of the air. At f m , perfect transmission is achieved, because of the impedance matching condition, with a phase shift related to the length L = NL unit of the material. The phase equals the one produced in an air-filled cavity of same length. The second frequency of interest is that at which the dynamic mass density reaches the zero value f ρ 0 . At this particular frequency, zero-phase delay propagation and constant wavefield within the PAM are expected, but the PAM is not impedance matched with the surrounding medium. The amplitude of the transmission coefficient is thus not unitary and more importantly depends on the number N of unit cells. The phase shift at f ρ 0 is weaker than that at f m (when the system is impedance matched), but even more importantly it depends on N. Therefore, no propagation without phase change is supported by the system at f ρ 0 .

The third important frequency is that at which the phase of the transmission coefficient is exactly zero f φ =0 . At this frequency, the effective density is negative and equal to ρ(ω) = -ρ 0 κ 0 /κ(ω)

(see supplementary material S.V). The negative effective density regime (grey mapped areas in Fig. 2(c)) corresponds to a stop band for an infinite system. Although f φ =0 lies in the negative mass range, the transmission remains considerable due to the small size of the considered PAM.

The amplitude of the transmission coefficient also depends on the number of the unit cells composing the system, but the zero phase propagation is operated independently of the number of unit cells. Having zero density, zero phase and unitary transmission at the exact same frequency would also imply to have an infinite bulk modulus (see supplementary material S.V), which is not the case in the PAM.

We now analyze the limits of zero phase transmission in the presence of losses when N ≤ 6.

These numbers of unit cell constitute a good balance between finite size of the system and the transmission coefficient amplitude value in the lossless case at f φ =0 (|T | ≥ 0.9). Both viscothermal losses in the waveguide and viscoelasticity of the plate material are accounted for. First, we analyze the dependence of the zero phase frequency, f φ =0 , on the viscoelastic losses which are the most important loss source in usual PAM. The zero phase frequency decreases with the increase of the loss factor (see Fig. 3(a)), thus entering more and more in the PAM stop band. However, the change in frequency is less than 10% with respect to the lossless case. Figures 3(b) and(c) respectively depict the amplitude and the phase of the transmission coefficient of the N = 6 PAM for different values of the loss factor. The transmitted amplitude is reduced because of the losses, but remains reasonable for the application of zero phase propagation in realistic situations. Moreover in Fig. 3(d) we represent the dependence of the phase of the transmission coefficient on the number of unit cells considered in the finite length PAM. The variation of the phase remains lower than 8% at f φ =0 in the cases N ≤ 6.

In order to validate the previous results, we followed a twofold procedure focusing the anal- ysis on the case of N = 6 PAM. The first procedure consists in performing full-wave numerical simulations in a 2D-axisymmetric configuration using FEM. The system is insonified by a plane incident wave from left to right and the plates are constituted of viscoelastic material of above mentioned properties. A plane wave radiation condition is applied at the waveguide end boundary to avoid spurious reflections. The second procedure is the measurement of the scattering parameters and the recovery of the effective dynamic mass density of the system using the experimental set-up shown in Fig. 1(d). Details on the retrieval procedure can be found in Ref. [32] and in the supplementary material S.II.

Red dashed line and symbols in 3(b-d) shows respectively the full numerical and experimental validation of the analytical predictions for the N = 6 PAM. at f m = 439 Hz due to the quasi impedance matching condition. The phase of the transmission coefficient is shown in Fig. 3(c-d). A zero phase propagation is measured at f φ =0 = 389 Hz, in agreement with the predictions, thus experimentally confirming the feasibility of zero-phase propagation. A good agreement is observed between the measurements, the analytical, and the numerical results. agative regime, the real part of the effective dynamic mass density is positive while its imaginary counterpart is negative, thus fulfilling the causality principle. In contrast, both real and imaginary parts of the density are negative in the forbidden band gap. Moreover, the real and imaginary parts of the density are of the same order of magnitude over the frequency range of interest, contrary to the bulk modulus, the imaginary part of which is much smaller than its real part (see supplementary material Fig. S1). Thus, most of the losses can be attributed to the complex effective mass density. A fairly good agreement is found between the measurements, the numerical, and the analytical predictions. The visible discrepancies are attributed to a residual variability on the unit cells, due either to the clamping condition or to the intrinsic properties of the plate. This variability mainly affects the reflection coefficient (see supplementary material S.IV) and therefore the effective density.

As an example of application, this peculiar zero phase propagation property is used to create a subwavelength dipole source by considering two waveguides: the first waveguide is filled with a DNZ PAM and the other one is a coiled-up Fabry-Perot resonator (FPR) occupied by a porous material in order to compensate the amplitude decrease due to the losses of the PAM in the first waveguide. A dipole source can be approximated by two out-of-phase monopoles of equal flow rates, resulting into two symmetric lobes in the polar directivity pattern. By designing the length of this FPR such as its first resonance coincides with the zero-phase frequency of the PAM, the acoustic fields at both waveguide boundaries are out-of-phase thus fulfilling the design condition for an acoustic dipole.

Figure 5 shows a 2D full-wave simulation of such a device. Using the previously characterized plastic shim material, a L = 6 cm periodic arrangement of 6 square plates of width w = 2 cm and thickness h = 102 μm is tested. The zero-phase frequency of the plate lattice is f φ =0 = 488 Hz.

The 2 cm wide FPR waveguide length is thus L FP = c 0 /(2 f φ =0 ) ≈ 35 cm long to resonate at that frequency. In the simulation, the FPR is coiled-up to reduce the device total volume (total width: 19.7 cm and total height 6 cm). To reproduce the sound attenuation of the PAM, the Fabry-Perot waveguide is filled with a porous medium (porosity 0.96, flow resistivity 2847 Pa.s.m -2 , viscous length 273 μm, thermal length 672 μm, tortuosity 1.07, and Biot frequency f c = 334 Hz) [START_REF] Groby | Total absorption peak by use of a rigid frame porous layer backed by a rigid multi-irregularities grating[END_REF][START_REF] Allard | Propagation of sound in porous media: modelling sound absorbing materials[END_REF][START_REF] De Ryck | Acoustic wave propagation in a macroscopically inhomogeneous porous medium saturated by a fluid[END_REF] lying in the intertial regime at f φ =0 . Both waveguides are excited by a plane wave pressure field at the top boundary and radiate in a semi infinite domain. An out-of-phase equal mean flow rate is found at each outer boundary, leading to a two lobes directivity pattern (purple line) and thus evidencing the efficiency of the device as a subwavelength dipole (L x ≈ λ /4 for the width and L y ≈ λ /12 for the height).

In this work, we have reported the impact of the viscothermal and viscoelastic losses on the zero phase propagation regime of a PAM. We have shown that the zero phase propagation appears at frequencies in the negative mass density regime. In this regime, corresponding to the band gap, the amplitude of the transmission coefficient depends on the number of the unit cells in the system. In contrast, the phase of the transmission coefficient remains constant in the lossless case.

Therefore a compromise between the number of unit cells and the variation of the amplitude should be reached. In this case, we consider systems made of N ≤ 6, leading to a lossless transmission amplitude of |T | ≥ 0.9. Once the losses are introduced in the system, a weak dependence of both the zero phase frequency, f φ =0 , and the phase of the transmission coefficient, φ , on both the number of unit cells and the amount of losses is observed. In both cases, the variation is less than 10%. These analytical results have been numerically reproduced by full-wave simulations and experimentally validated by measuring the scattering parameters of a PAM made of N = 6 plates as well as the effective mass density. The agreement between the analytical predictions, the numerical simulations and the experimental results are found very good. The results shown in this work pave the way to design devices based on PAM with zero phase propagation. As an example, a subwavelength acoustic dipole has been designed numerically.

SUPPLEMENTARY MATERIAL

See the supplementary material for details on the transfer matrix analytical model (S.I); the experimental reconstruction of the scattering parameters (S.II); the plate characterization process (S.III); the measured scattering parameters and effective dynamic mass density of a 6 PAM (S.IV); and the theoretical reason for the frequency offset between the zero-phase, zero-density, and unit transmission in the case of a non-infinite effective bulk modulus system (S.V).

FIG. 1 .

 1 FIG. 1. (a) Considered plate-based metamaterial sketch, (b, c) Unit cell, (d) photography of an exploded view of the unit cell composed of a thin elastic plate embedded between two Teflon and aluminum rings, and (e) photography of the experimental setup.

  FIG. (Color online) Effective mass density and scattering properties of the PAM in the lossless case. (a) Effective dynamic mass density in terms of function as obtained from Eqs. (1) and (2). (b) and (c) represent the amplitude of the transmission [left vertical axis of (b)] and reflection [right vertical axis of (b)] coefficients as well as the normalized phase of the transmission coefficient for a finite PAM made of N = 1, 3, 6 and 9 plates respectively.. Vertical blue, gray and green lines in (a), (b) and (c) represent the frequencies for the zero phase, f φ =0 , zero mass density, f ρ=0 , and impedance matching, f m , frequencies respectively. The grey mapped area delimits the zero-frequency band gap of an infinite system.

Figure 3 (FIG. 3 .

 33 FIG. 3. (Color online) Analysis of the zero phase frequency and the scattering properties of the PAM considering the viscoelastic and viscothermal losses. (a) Dependence of the zero phase frequency on the viscoelastic losses of the system. (b) and (c) represent the amplitude of the transmission coefficient as well as its normalized phase for a finite PAM made of N = 6 plates depending on the losses. (d) represents the dependence of the phase of the transmission coefficient with the number of unit cells in the PAM for the case with β = 0.13 (experimental losses). Continuous lines, dashed lines and symbols in (b-d) represent respectively the experimental, numerical and analytical results. Horizontal and vertical blue lines in (a-d) represent the frequency for the zero phase PAM experimentally analyzed in this work.

Figure 4 FIG. 4 .

 44 Figure4depicts the real and imaginary parts of the effective dynamic mass density. In the prop-

FIG. 5 .

 5 FIG. 5. (Color online) 2D Full-wave simulation of a subwavelength dipole device built of a coiled-up Fabry-Perot waveguide of length L FP ≈ 35 cm and a 6 plates PAM. The y component of the velocity field is shown as well as the normalized directivity polar plot of the dipole (purple line).

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, "Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies," Physical Review Letters 100,1-4 (2008).

ACKNOWLEDGMENTS

This article is based upon work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology). This work has been funded by the Metaroom Project No. ANR-18-CE08-0021, co-funded by ANR and RCG. J. Christensen acknowledges the support from the MINECO through a Ramón y Cajal grant (Grant No. RYC-2015-17156). J. Sánchez-Dehesa acknowledges the support by the Ministerio de Economía y Competitividad of the Spanish government and the European Union Fondo Europeo de Desarrollo Regional (FEDER) through Project No. TEC2014-53088-C3-1-R.


P.-J Nacher

S Kumaragamage

G Tastevin

C P Bidinosti

A Fast MOSFET RF Switch

published or not. The documents may come

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Introduction

TRansmit Array Spatial Encoding (TRASE) is a MRI method using repeated pulses of radiofrequency (RF) B 1 fields from different phase-gradient transmit coils instead of the standard imaging gradients of B 0 to manage spatial encoding. 1,2 An evaluation of the benefits and limitations of TRASE MRI at low field (a few mT) is under way using hyperpolarised gas samples. 3,4 Fast switching between two or more transmit coils with sufficient isolation and negligible wait time between pulses is required to jointly reduce the total RF encoding time and the RF peak power. PIN diodes are often used for coil switching,2 but bias control with RF isolation involves response times (several RF periods) which are too slow for the targeted low-frequency application.

Methods

Switching times (delays and jitter) have been inferred from datasheet or measured for different mechanical switches (MEMS, reed or standard relays). Bidirectional MOSFET switch prototypes, with floating and optically isolated gate control, have been designed and wired (Figs. 1 and2). Electrical tests have been performed using different RF power amplifiers (a 250-W-peak, BT00250-Alpha A model, Tomco Technologies, and a 100-W-CW in-house linear amplifier), different loads (a 50-Ω resistive load or tuned and untuned transmit coils), and different frequencies (in the range DC -200 kHz). An Apollo Tecmag console (low-frequency LF1 model), allowing for 100 ns time resolution of sequence events, was used to manage RF pulses and TTL signals to control switches. RF currents were recorded using a 4-channels digital oscilloscope by monitoring the voltage drop across a 1-Ω resistor inserted on the ground return line at the output of the RF amplifier. This is the peer reviewed version of the abstract n° 2874 which can be found in final form in: Proc. of the Intl. Soc. Mag. Reson. Med. 26 (2018) 1699 Abstract 2874 Figure 1: Schematic drawing of the MOSFET switches used for low-frequency TRASE MRI. Two of the three identical channels are displayed. On the right-hand side, independent DC-DC converters (Conv.) are used as floating 12-V power supplies of the gate control circuits. On the left-hand side, optocouplers (OC) are used to isolate the TTL control inputs from the power MOSFETS QA and QB which act as bipolar switches between terminals SA and SB.

Figure 2: Photograph of a prototype MOSFET switch with 3 independent channels (see Fig. 1). The tested components are: IK0512SA or ISE1212A (Conv.), HCPL2201 (OC), and IRF840 or IRF730 (QA, QB).

Results

Switching times of mechanical relays range from 10 µs (for MR-compatible MEMS [START_REF] Spence | [END_REF]) to several ms. Fast reed relays are widely available and economical, can be operated in the vicinity of low-field MR systems, and have close/release times of 0.15/0.1 ms for typical carry currents of 1 A, while miniature relays with 4-A carry currents have typical switching times of 1 ms. The MOSFET switch of Fig. 2 (with IRF 840, up to 32 A) has switching times below 1 µs. It has a low ON impedance (about 1 Ohm) and a frequency-dependent OFF impedance, consistent with a 500 pF capacitance. This usually provides efficient RF current switching, both on a resistive load (Fig. 3) and on transmit coils (various situations have been evaluated and will be reported). A small RF interference due to the operation of DC-DC converters is sometimes observed, but a suitable choice of the component eliminates the problem (different components operate at different switching frequencies).

Figure 3: Current driven in a 50-Ohm resistive load through the MOSFET switch by a 50-V-peak-topeak voltage (QA, QB: IRF840). The control TTL signal is applied at t=0 for 18 RF periods (frequency: 83.682 kHz, period: 12.950µs). In the OFF state, the current is reduced by a factor 50, and corresponds to an impedance of the gate Z=3.3 kOhm, i.e. a capacitance C=580 pF.

Discussion/Conclusion

Although magnetically actuated relays provide an excellent switching efficiency and can be operated in low-field MR systems, their slow response times, on the same order as the duration of individual B1 pulses in TRASE sequences (typically 10-20 RF periods around 100 kHz [START_REF] Pj Nacher | Proc ESMRMB15[END_REF]), introduce significant wait times during encoding pulse trains. This may result in prohibitive diffusion-induced resolution loss in gas samples, and acceleration through the use of shorter RF pulses is impractical due to Bloch-Siegert shifts and to the influence of concomitant RF fields [6]. MOSFET-based switches eliminate wait times and RF pulse currents can be switched to different transmit coils with sufficient isolation ratio and no delay between pulses. A choice of the MOSFET components can be made to achieve a suitable compromise between reducing the OFF-state capacitance and managing the required RF voltage and current. x 10

Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, Paris, France

Department of Physics, University of Winnipeg, Winnipeg, MB, Canada

Rady Faculty of Health Sciences, College of Medicine, University of Manitoba, Winnipeg, MB, Canada


Christopher Bidinosti

Pierre-Jean Nacher

Geneviève Tastevin

Unconventional trajectories on the Bloch Sphere: A closer look at the effects and consequences of the breakdown of the rotating wave approximation

TRASE MRI uses rapid π-pulses of phase gradient fields, and in general requires as many as two distinct phase-gradient coils per encoding direction. This tends to restrict one to large amplitude, linear B fields, which in low B field leads to a breakdown of the rotating wave approximation. We have studied this regime both numerically and experimentally. Our results show a rich behavior involving a complex interplay of the Bloch-Siegert shift, the B start and stop phase, and B amplitude transients.

Introduction

TRansmit Array Spatial Encoding (TRASE) is a relatively new MRI method for which spatial encoding is achieved through the application of phase gradients of the B field -rather than magnitude gradients of the B field -and in general requires as many as two distinct phase-gradient coils per encoding direction. Moreover, traversal through k-space with TRASE requires a train of π-pulses generated by alternating phase-gradient coils. As a result, there is very strong motivation here to employ linear rather than rotating B fields (lest the already large number of rf coils should further double) and to understand the full details of rapid π-pulse generation with such fields. This is particularly important for our ongoing evaluation of the benefits and limitations of TRASE at low field, where the magnitude of the linear B field may not be negligibly small compared to B leading to a breakdown of the rotating wave approximation (RWA). Here we explore the unconventional trajectories on the Bloch sphere arising from the breakdown of the RWA. Our work reveals a complex interplay of Bloch-Siegert shift, B start or end phase, and B amplitude transients that must be taken into account to achieve the expected trajectory terminus.

Methods

We studied the effects and consequences of the breakdown of the RWA through analytical and numerical solutions of the Bloch equations. For convenience, we express the magnitude of the B field as the parameter N = B /B , which is the number of cycles of the oscillating field required to achieve a π-radian rotation. As such, the continuous wave Bloch-Siegert shift becomes ∆B = B /(4 N ) . For all simulations presented here, we fixed the B frequency to its nominal value ω , applied rectangular pulses of linear field B cos(ω t + φ) along the x-axis in the laboratory frame, and managed any de-tunings through a field shift δB away from B . Experimental studies were performed in B fields of a few mT or less, using both water and hyperpolarized He gas samples.

Results

Numerical simulations are shown in Figures 1 through 4; experimental results are in Figure 5. Figure 1 highlights the general effects of the breakdown of the RWA due to a strong counter-rotating B field: the trajectory undergoes cycloid motion and a perfect π-pulse cannot be achieved without de-tuning the field. Every half-period, the trajectory without/with de-tuning lands on that due to a rotating B field with/without de-tuning of the same magnitude. Figure 2 highlights the fact that the specific details of the cycloid trajectory, and in particular its terminus, depends strongly on the phase of the B field. Starting and stopping pulses at the cusps in the trajectory (ie. zero field and zero coil current, occurring every half-period) is an important strategy for eliminating further complications due to B amplitude transients in real coils. As shown in Figure 3, a perfect π-pulse can always be achieved by de-tuning the field by δB = β ∆B , where to lowest order the prefactor β depends only on the phase φ. In Figure 4, we show the effect of the breakdown of the RWA on a π-pulse intended to rotate transverse magnetization about the x'-axis in the rotating frame. Again, the appropriate de-tuning of the static field will achieve the desired rotation. In Figure 5, we present a small compilation of experimental results achieved to date. In particular, we show data that confirms the 1/ N behavior of the Bloch-Siegert shift, as well as the magnitude and change of sign of the prefactor β(φ). The Bloch-Siegert shift is determined experimentally as the de-tuning δB required to achieve a perfect π-pulse. We also present a comparison of the performance a composite π-pulse versus a simple rectangular π-pulse. Arbitrary pulse shapes and durations have also been investigated.

Discussion and conclusion

We have made extensive numerical and experimental studies of the breakdown of the RWA due to a strong counter-rotating B field. Our results show a rich behavior involving the Bloch-Siegert shift, B start and stop phase, and B amplitude transient that is of fundamental interest in magnetic resonance. It is of great practical importance for low-field MR applications, in particular TRASE MRI where one is more or less restricted to using large amplitude, linear B fields. Achieving the expected trajectory terminus in this regime requires an appropriate phase-dependent de-tuning of the field. Using pulse durations and sequence timings that are half-integer multiples of the B period is also beneficial.

1. JC Sharp, et al. Magn Reson Med. 2010;63:151 and NMR Biomed. 2013;26: 1602. 2. Q Deng, et al. Magn Reson Imaging. 2013;31:891. 3. PJ Nacher, et al. Proc ESMRMB 15, Magn Reson Mater Phy. 2015:28 Suppl.1;S64. 4. S Kumaragamage, et al. Proc ESMRMB 16, Magn Reson Mater Phy. 2016:29 Suppl.

1;S34. 1 2 2 1 2 1 0 1 1 1,2 1 0 1 3,4,5 1 0 6 1 7 1 8 1 π 1 0 6 cw 0 π 2 1 0 1 0 0 0 3 1 1 1 1 cw π 2 1 1 1 1 1 5. PJ

FiguresFigure1:

  Figures

Figure 2 :

 2 Figure 2: Trajectories near the beginning (left) and end (right) of a π-pulse with N = 9 and δB=0. With no Bloch-Siegert shift, a perfect π-pulse can only be achieved with φ = 30 . The circles indicate time intervals equal to half-periods of the B field from the start of the pulse. For φ = 90 these occur at the cusps in the trajectory, which occur when the net B field is zero.

Figure 3 :

 3 Figure 3: Trajectories near the end (left) of a π-pulse with N = 9 and δB = β ∆B . With the appropriate de-tuning, a perfect π-pulse can be achieved for any φ. Plot of the analytic form of β for the case where N = n/2, where n is an integer. Simulations for N = 1 and 3 reveal a higher order dependence of β on the magnitude of B .

Figure 4 :

 4 Figure 4: A π-pulse starting with unit magnetization in the transverse plane: N = 9 and φ = 0 . Left: On-resonance. The terminus overshoots or undershoots, depending on the original phase of the magnetization φ (0). Final M values, shown in the plot, are up to +/-0.08. Right: Off-resonance (B = B -δB, with δB/B = -1/(4N ) ). Final M values are reduced to below 1ppt for the nominal shift (β=-1, points, red dashed line). Using a more precise value of β to account for higher order effects reduces M to the ppm level (points).

Figure 5 :

 5 Figure 5: Some experimental results. Left: Bloch-Siegert shift versus N for φ = 90 with data (points) and theory (line). Center: Bloch-Siegert shift versus φ for various N with data (points) and theory (lines). Right: Flip angle error (degrees from 180 ) versus de-tuning for CORPSE composite pulse and an N =12 rectangular pulse. The former remains insensitive to de-tuning despite the departure from the RWA.

of the Intl. Soc. Mag. Reson. Med. 26 (2018) 2701

  This is the peer reviewed version of the abstract n° 4309 (Poster 2701) which can be found in final form in: Proc.

Nacher, et al. This conference. 6. F Bloch and A Siegert. Physical Review. 1940;57:522. 7. D MacLaughlin. Rev Sci Instrum. 1970;41:1202. 8. M Mehring and J Waugh. Rev Sci Instrum. 1972;43:649.