Abstract

Acoustic metamaterials have been proposed for numerous applications including subwavelength imaging, impedance matching, and lensing. Yet, their application in compressive sensing and imaging has not been fully investigated. When metamaterials are used as resonators at certain frequencies, they can generate random radiation patterns in the transmitted waves from the transducers and received waves from a target. Compressive sensing favors such randomness inasmuch as it can increase incoherence by decreasing the amount of mutual information between any two different measurements. This study aims at assessing whether the use of resonating metamaterial unit cells in a single-layered non-optimized array between a number of ultrasound transceivers and targets can improve the sensing capacity, point-spread function of the sensing array (their beam focusing ability), and imaging performance in point-like target detection. The theoretical results are promising and can open the way for more efficient metamaterial designs with the aim of enhancing ultrasound imaging with lower number of transceivers compared to the regular systems.

1 Introduction

Inspired by the theoretical formulation and experimental realization of left-handed electromagnetic metamaterials [13], the existence of double-negative acoustic metamaterials were shown in Ref. [4]. The principles of acoustic metamaterials were founded on similar concepts discussed in electromagnetics by Veselago [5]. The introduction of acoustic metamaterials has paved the way for diverse applications such as cloaking [68], focusing and lensing [914], elastic wave manipulation [15,16], subwavelength imaging [17,18], sound absorption [1921], canceling out aberrating layers [22], and impedance matching [23,24]. More practically, real-life applications associated with these new opportunities vary from anti-earthquake infrastructures for buildings [25,26] to penetrating bones via ultrasound waves [22].

Particularly, to enhance imaging capability, Zhu et al. proposed a holey-structured metamaterial to exceed the limit that diffraction sets on the image resolution [18]. The structure included 1600 holes made into a rigid brass block and it caused Fabry–Pérot resonances to be formed and successfully transmitted to the other side. Through simulation and experimental results, it was shown that details down to 2% of the exciting wavelength could be restored in the image from an object. Previously, a metamaterial hyperlens [13] and a single-negative metamaterial [17] were utilized to achieve resolutions of 14.7% and 13% of the used wavelength, respectively.

In this work, aiming at improving the sensing capacity, the idea of acoustic metamaterials has been combined with the concept of compressive sensing by a randomization process that enables collecting more information from the imaging domain (ID). Compressive sensing theory states that using nonadaptive linear projections, one can reconstruct a signal by samples taken at a much lower rate than what Shannon–Nyquist theorem requires [27]. One way to take advantage of compressive sensing is to create random measurements, that is, to randomize the outgoing signal from the transmitter and the incoming signals to the receivers [28]. This randomization can effectively enhance sensing mechanisms [29], which may seem unexpected at first glance. When looked into the matter more closely, however, the counter-intuitiveness of the role of randomizing the measurements in enhancing the sensing capacity can be resolved: random signals minimize the amount of mutual information between two successive measurements and thus they maximize the information that can be transferred from the imaging domain to the receivers. Utilizing metamaterials that exhibit resonant behavior at certain frequencies is one way to randomize the measurements and has been adopted in this work.

To the knowledge of the authors, the use of ultrasound metamaterials has not yet been reported for megahertz frequency imaging. This could be due to the challenges in fabricating such fine structures, since obtaining resonant frequencies in the megahertz domain requires features at micrometer scale. However, the use of metamaterials to create an ultrasound matching layer [30] and to manipulate acoustic waves [31] has been experimentally and theoretically studied, respectively, at megahertz frequencies. In the first reference, features of sizes down to 122 μm were fabricated, and in the second one, employing micro gas bubbles to alter ultrasound waves were introduced. Here, the use of metamaterial resonators in increasing the sensing capacity, enhancing beam-forming ability or point-spread function (PSF) of the sensing array, and improving the imaging performance is investigated, in a theoretical, proof-of-concept fashion.

2 Metamaterial Design and Analysis

This section first describes the design of a resonant metamaterial unit cell to merely have at least a resonance in the working frequency band of the sensing system, without any optimization or deterministic approach. Next, the design of a randomly distributed line-array of two different sizes of the designed unit cell will be explained, which will be used in an imaging application in the upcoming section. The exact size of the unit cell, scaling factor, number of the unit cells in the array, and the distribution method are arbitrary to make the process as random as possible.

2.1 Metamaterial Unit Cell.

Metamaterial unit cells that are composed of a combination of stiff and soft materials have been studied as single-frequency and multi-frequency local resonators [32]. Ao and Chan introduced a metamaterial with alternating layers of soft and stiff materials to create local resonances [33], and Zhu et al. demonstrated that specific arrangements of such materials, again in multi-layered structures, can generate multi-resonant behavior in a single cell [32]. The same idea has been applied here and the metamaterial Ao et al. introduced is scaled down and tuned to give a single resonance between 1 and 7 MHz. This is possible since if a metamaterial unit dimensions are scaled down by a factor of σ, its resonant frequency is increased by the same factor, i.e., fnew=σfold, where fnew and fold denote the new and old resonant frequency, respectively. As shown in Fig. 1, the square unit cell is made up of a steel central part coated by layers of rubber and aluminum, in order, in a water background. The radii of the three layers are ral = 0.25a, rrub = 0.2a, and rst = 0.15a, from outside to the center, with a being the square unit cell side. The numerical simulations were carried out in comsol multiphysics, and the reflection (R) and transmission (T) coefficients were calculated on the left and right boundaries of the simulation domain according to the following formulae, respectively:
R(f)=e2kb(f)d10h(Pt(x,y,f)Pi(f))dy0hPi(x,yf)dy|x=L
(1)
T(f)=e2kb(f)d20hPt(x,y,f)dy0hPi(x,y,f)dy|x=L
(2)
where Pt and Pi denote the total and incidence pressure fields, respectively; h is the height of the simulation domain; kb is the background wavenumber; f is the guided frequency; and d1, d2, and L are geometric dimensions and are defined in Fig. 1. In the configuration shown in the figure, h = a = 20 μm, d1 = d2 = 60 μm, and L = 70 μm. The properties of each material, as used in the simulations, are given in Table 1. It is clear from the transmission and reflection curves that the metamaterial unit cell is working in a band-pass mode, reflecting most of the incoming waves and only passing the wave through almost completely at the resonant frequency, that is 4.5 MHz.
Fig. 1
The single unit cell design and performance: (a) simulation configuration, (b) the total pressure field at two different frequencies, and (c) reflection and transmission coefficients in dB versus frequency in Hz. The frequencies for which the total pressure field is plotted are shown with black dots.
Fig. 1
The single unit cell design and performance: (a) simulation configuration, (b) the total pressure field at two different frequencies, and (c) reflection and transmission coefficients in dB versus frequency in Hz. The frequencies for which the total pressure field is plotted are shown with black dots.
Close modal
Table 1

The acoustic properties of the materials used in the metamaterial simulations

MaterialDensity (kg/m3)Sound speed (m/s)Reference
Water10001490[34]
Rubber100055[32]
Steel77005050[32]
Aluminum27306800[32]
MaterialDensity (kg/m3)Sound speed (m/s)Reference
Water10001490[34]
Rubber100055[32]
Steel77005050[32]
Aluminum27306800[32]
The obtained reflection and transmission coefficients can be used to retrieve the effective properties—density and bulk modulus—of the metamaterial unit cell, provided that the wavelength of the exciting waves is at least four times bigger than the metamaterial unit cell size, i.e., λ>4a. This condition is known as the homogeneity condition [35]. Therefore, one can set the conditions on the size of a unit cell and its contents in this study as follows: a<λmin/4(=53μm), 0 < ral < a/2, 0 < rrub < ral, 0 < rst < rrub, with λmin being the minimum wavelength used in imaging. Now, the approach that was introduced by Fokin et al. for locally resonant acoustic metamaterials [36] is used to obtain the effective parameters. When R(f) and T(f) are available, the effective refractive index n and impedance ξ in the case of normal incidence can be found as [36]
n(f)=±cos1([1(R2T2)]/(2T))+2πqkbd
(3)
ξ(f)=±(1+R2)T2(1R2)T2
(4)
where q = 0, ±1, ±2, ±3, … is the branch number of the inverse cosine function and d is the unit cell thickness in the direction of the waves. To remove the ambiguities that arise in the signs and the branch number in the above formulations, Fokin et al. set two conditions: (i) physical realization of the solution, i.e., Re(ξ)>0, with Re(·) showing the real part operator on a complex number, and (ii) the use of minimum metamaterial thickness for which q = 0. Their approach is summarized schematically in Fig. 2. Another method to resolve the mentioned ambiguities is introduced by Szabó et al. [37], using Kramers–Kronig relationship between the real and imaginary parts of an analytical complex function such as the refractive index.
Fig. 2
Finding metamaterial effective properties using the method suggested by Fokin et al. [36]
Fig. 2
Finding metamaterial effective properties using the method suggested by Fokin et al. [36]
Close modal

The retrieval results are shown in Fig. 3 for the designed unit cell, using both Eqs. (3) and (4) and the algorithm outlined in Fig. 2. In adopting the mentioned equations, the positive signs have been used. As can be seen, the algorithm has modified the signs of the parameters at certain frequency intervals. Furthermore, the resonant behavior observed in transmission and reflection curves also appears in the effective parameters plots, exhibiting an extremum at 4.5 MHz. This property will be employed to create the desired randomness in the compressive sensing approach, as described in the next section. It should be noted that inasmuch as the density and bulk modulus take negative values, the unit cell is of double-negative type.

Fig. 3
The effective properties of the unit cell shown in Fig. 1: (a) the real and imaginary parts of the retrieved complex density and (b) the real and imaginary parts of the retrieved complex bulk modulus. The dashed lines show the properties predicted by Eqs. (3) and (4), using the positive signs, titled as basic properties. The solid lines are obtained by applying the method described in Fig. 2.
Fig. 3
The effective properties of the unit cell shown in Fig. 1: (a) the real and imaginary parts of the retrieved complex density and (b) the real and imaginary parts of the retrieved complex bulk modulus. The dashed lines show the properties predicted by Eqs. (3) and (4), using the positive signs, titled as basic properties. The solid lines are obtained by applying the method described in Fig. 2.
Close modal

2.2 Metamaterial Line Array.

Scaling the designed metamaterial will create unit cells of different resonant frequencies. If the original metamaterial unit has a dimension of r0 and a resonant frequency of f0, scaling its dimensions by a factor σ creates a metamaterial unit of dimension σr0 and resonant frequency f0/σ. As an example, upscaling the multi-layered circle inside the designed cell by a factor of 1.2 yields a unit cell of resonant frequency 3.75 MHz. A combination of such unit cells in an array can create multiple resonant frequencies. Placing the resultant array in the way of the waves coming from the acoustic sources will randomize the wave fields that reach a target by altering their patterns at different frequencies.

In this study, two types of unit cells, one having a resonant frequency of 4.5 MHz and the other having a resonance at 3.75 MHz, are arranged randomly in a line array and placed between the sources and the targets. As mentioned, the size of the unit cells and their contents are selected so that they give one or more resonances in the operating frequency range. Moreover, the unit cells are placed side by side, without any space between them, so that they cover the entire width of the domain. It is important to note that in this work, to take advantage of the favored randomness in compressive sensing, no optimization will be done in the design of the individual unit cells or the array. Rather, unit cells with arbitrary resonances in the frequency band and their random distribution will be relied on to make the sensing matrix less coherent. Regarding the behavior of the array, the following remarks are noteworthy:

  • Using these cells together in a two-element array will not necessarily create isolated resonant frequencies precisely at the predicted values as shown in Fig. 4.

  • When the waves are not perpendicular to the metamaterial unit cell—as in the case of waves coming off a point source and reaching at the boundary of unit cells that are not directly situated below the source—the behavior of the metamaterial might change significantly. When the incident fields are not normal to the unit cell boundary, another set of formulations are required to restore the effective properties of the metamaterial cell [38]. Although these effects cannot be easily controlled, in this case, they are advantageous in increasing the randomness of the measurements.

Fig. 4
Unit cell designs: (a) the characteristics of an array of two unit cells with disparate individual resonant frequencies in the 1–7 MHz span, (b) the transmission–reflection (T–R) coefficient plots versus frequency are shown for the combination of two unit cells of different sizes, and (c) the pressure fields associated with the resonant frequencies are shown on the top. Each unit cell exhibits resonance at its associated resonant frequency.
Fig. 4
Unit cell designs: (a) the characteristics of an array of two unit cells with disparate individual resonant frequencies in the 1–7 MHz span, (b) the transmission–reflection (T–R) coefficient plots versus frequency are shown for the combination of two unit cells of different sizes, and (c) the pressure fields associated with the resonant frequencies are shown on the top. Each unit cell exhibits resonance at its associated resonant frequency.
Close modal

As shown in Fig. 4, the upscaling of the base unit cell has created a transmissive response at 6.05 MHz. This can be viewed as if the TR plot has been contracted along the frequency axis and has brought in a new resonant frequency that was outside the frequency band for the base metamaterial. The interaction between the unit cells has also generated other extermums on the plot at 2.755 MHz and 4.35 MHz that are not of reflective or transmissive nature.

3 Ultrasound Imaging Using Metamaterials

To assess the performance of a sensing system in the presence of resonant metamaterials as measurement randomizers, an imaging simulation is carried out with an array of ultrasound transceivers and a number of targets. In this section, the simulation domain and its parameters are introduced. Furthermore, the imaging algorithm that is utilized to retrieve the image of the targets is established.

3.1 Forward Modeling and Simulation Setup.

The geometry shown in Fig. 5 is employed for the forward model simulations. Assuming time-harmonic pressure fields, the frequency-domain acoustic pressure equation that is numerically solved in the entire domain by the finite element method solver comsol multiphysics is as follows [39]:
ρ(r)(1ρ(r)P(r,ω))+k2(r,ω)P(r,ω)=F(r,ω)
(5)
in which ω=2πf is the angular frequency with f denoting the frequency, ρ(r) is the density of the medium, k(r,ω)=ω/c is the wave-number with c being the speed of sound, P(r,ω) is the pressure field, and F(r,ω) is the acoustic source term. The source term can be defined in various ways. If it is a monopole, and the root-mean-square (RMS) of power per unit length, that is Prms, is used as excitation, then [40]
F(r,ω)=2eiϕs2ρ(r)ωPrmsδ(rrs)
(6)
where ϕs and rs are the phase and the location of the source, in order, and δ(rrs) is the shifted Dirac delta function. The dependency of F(r,ω) on frequency is due to the two-dimensional (2D) modeling; however, in 3D modeling, this dependency does not exist [40]. Thus, in the 2D case, for the exciting source term to have a magnitude of one and no phase terms, the parameters must be chosen as Prms(ω)=1/(8ρ(r)ω) and ϕs=0.
Fig. 5
Imaging simulation domain: (a) the configuration of layers of coupling, intermediate, and background media, alongside with the location of the sources and targets are shown. One layer of metamaterials is placed between the sources and the tumors on top of the intermediate medium. The number of unit cells used is a function of the width of the domain. (b) The pseudo-random distribution of the basic and upscaled unit cells is shown versus the index of the metamaterial from left to right to enable the regeneration of the results by other researchers. No. 1 corresponds to the basic metamaterial cell and No. 2 corresponds to the upscaled metamaterial cell.
Fig. 5
Imaging simulation domain: (a) the configuration of layers of coupling, intermediate, and background media, alongside with the location of the sources and targets are shown. One layer of metamaterials is placed between the sources and the tumors on top of the intermediate medium. The number of unit cells used is a function of the width of the domain. (b) The pseudo-random distribution of the basic and upscaled unit cells is shown versus the index of the metamaterial from left to right to enable the regeneration of the results by other researchers. No. 1 corresponds to the basic metamaterial cell and No. 2 corresponds to the upscaled metamaterial cell.
Close modal

The definition of the geometric parameters of the domain is given in Table 2, alongside with the numerical values used in this case study. Since this study is closely related with the idea of multimodal early detection of breast cancer using a mechatronic system [41,42], the domain was set up accordingly, including an acrylic interface medium between the coupling liquid and the imaging domain.

Table 2

The geometric parameters of the imaging simulation domain

ParameterDefinitionNumerical value (mm)
wDDomain width6
hDDomain height6
tcCoupling medium height2
tiIntermediate medium height2
tbBackground medium height2
diCircular target diameter0.2
(xs,1, ys,1)Source 1 location(−0.25, 2.9)
(xs,2, ys,2)Source 2 location(0.25, 2.9)
(xt,1, yt,1)Target 1 location(−0.6, 2)
(xt,2, yt,2)Target 2 location(0.8, 2)
ParameterDefinitionNumerical value (mm)
wDDomain width6
hDDomain height6
tcCoupling medium height2
tiIntermediate medium height2
tbBackground medium height2
diCircular target diameter0.2
(xs,1, ys,1)Source 1 location(−0.25, 2.9)
(xs,2, ys,2)Source 2 location(0.25, 2.9)
(xt,1, yt,1)Target 1 location(−0.6, 2)
(xt,2, yt,2)Target 2 location(0.8, 2)

From the top of the figure to the bottom, respectively, three layers of different materials are used: (i) water in which the acoustic transceivers are placed, (ii) acrylic sheet which is assumed to be the bottom of the water container or the breast compression paddle, (iii) background medium (fibroglandular breast tissue) that encompasses a number of targets (invasive ductile carcinoma (IDC) tumors). The properties—density and speed of sound—of each of these materials, except water whose properties were given in Table 1, are listed in Table 3. The longitudinal speed of sound is given for an IDC mass in Ref. [43], but to compute the tumor’s density, the data given in Ref. [44] on the mean values of elasticity (Eidc), sound shear velocity (csw,idc), and the relationship between the two can be used. Having Eidc = 140.7 kPa and csw,idc = 6.7 m/s, the density can be calculated as ρidc=Eidc/3csw,idc2=1077kg/m3 [44]. The geometry as well as the simulation were set up using the COMSOL Multiphysics with matlab module. The simulation was set up to scan the medium by a total number of 61 frequencies in the band 1–7 MHz.

Table 3

The acoustic properties of the materials used in the imaging simulations

DomainDensity (kg/m3)Sound speed (m/s)References
Intermediate12002730[45,46]
Background10351487[47,48]
Targets10771549[43,44]
DomainDensity (kg/m3)Sound speed (m/s)References
Intermediate12002730[45,46]
Background10351487[47,48]
Targets10771549[43,44]

3.2 Inverse Modeling and Imaging Algorithm.

Similar to the electromagnetics problem, pressure waves satisfy the Helmholtz equation in a homogeneous medium [49]. Since the density difference between the targets and the background medium, or the density gradient in the whole imaging domain, is small in the case study considered here, with approximation, Helmholtz equation can be used to formulate the problem even when the targets are present. Thus, the pressure fields obtained by the forward computations using comsol multiphysics full-field simulations can be reasonably assumed to satisfy the Helmholtz equation if the mentioned conditions are met. If the background and total fields are denoted by Pb(r,ω) and Pt(r,ω) for the cases without and with targets, respectively, the Helmholtz equations in the ID can be written as [50]
(2+kb2(r,ω))Pb(r,ω)=F(r,ω),rID
(7)
(2+kt2(r,ω))Pt(r,ω)=F(r,ω),rID
(8)
where kb(r,ω) and kt(r,ω) denote the wave-number for the cases without and with the targets, respectively. The solution to the background field problem can be obtained by superposition as [50]
Pb(r,ω)=Gb(r,r,ω)F(r,ω)dr
(9)
in which Gb(r,r,ω) is the solution of the background pressure when the exciting source is an impulse function δ(rr) located at r′. If the scattered field is defined as Ps(r,ω)=Pt(r,ω)Pb(r,ω), then Eq. (5) can be rewritten as
2(Pt(r,ω)Ps(r,ω))+kb2(r,ω)(Pt(r,ω)Ps(r,ω))=F(r,ω)
(10)
Next, by adding and subtracting the term kt2(r,ω)Pt(r,ω), the above equation turns into
2Pt(r,ω)+kt2(r,ω)Pt(r,ω)kt2(r,ω)Pt(r,ω)2Ps(r,ω)+kb2(r,ω)(Pt(r,ω)Ps(r,ω))=F(r,ω)
(11)
Referring to Eq. (8), it can be seen that the first two terms on the left-hand side of Eq. (11) cancel with F(r,ω) and the following equation results in:
(2+kb2(r,ω))Ps(r,ω)=(kb2(r,ω)kt2(r,ω))Pt(r,ω)
(12)
which is similar to Eq. (5), except the exciting source H(r,ω)=(kb2(r,ω)kt2(r,ω))Pt(r,ω) and the subscript s that stands for the scattered field. Thus, the solution to (12) can be found by (9), after making the relevant changes in the subscript and the source term as follows:
Ps(r,ω)=Gb(r,r,ω)H(r,ω)dr=Gb(r,r,ω)kb2(r,ω)X(r,ω)Pt(r,ω)dr
(13)
where X(r,ω)=(1/cb2(r,ω)1/ct2(r,ω))/(1/cb2(r,ω))=1(cb(r,ω)/ct(r,ω))2 is the contrast variable and the relationships kb(r,ω)=ω/cb and kt(r,ω)=ω/ct are used, with cb(r,ω) and ct(r,ω) being the speed of sound inside the domain in the absence and presence of the targets, respectively.
The introduced parameter X(r,ω) is the unknown of the imaging problem. Given that the sound velocity does not change much with frequency [51] in the used frequency band, the approximation X(r,ω)X(r) can be made. From the numbers given in Table 2, X(r) takes a relatively small value of 7.47% inside the tumor and it is equal to zero everywhere else in the imaging domain. To determine X(r) from (13), the value of the total pressure Pt(r,ω) and scattered pressure field Ps(r,ω) are required at each point of the imaging domain. This is not feasible in most cases as the number of transceivers that can be placed in the space is limited. Yet, since the contrast value is small in the imaging domain, the total field can be estimated with the background field using the first-order Born approximation [52,53], which yields the following equation:
Ps(r,ω)Gb(r,r,ω)kb2(r,ω)X(r)Pb(r,ω)dr
(14)
Now, Pb(r,ω) can be computed with different methods such as numerical computation and Ps(r,ω) can be obtained by measuring the total field at the position of the receivers and subtracting the result from the simulated background field. Equation (14) can be transformed into matrix form for a defined number of pixels n in the imaging domain and a particular number of measurements m, assuming noiseless simulations:
An×mx^m=bn×1
(15)
where A is the sensing matrix, x^ is the unknown vector—the discretized and vectorized version of X(r)—and b is the measurement vector. In other words, x^ is the column-wise stacking of the values of the contrast parameter, in 2D space, at each pixel in the imaging domain. As mentioned, due to the limitation on the number of measurements, the number of unknowns is much larger than that of the known variables and thereby Eq. (15) is underdetermined, having infinitely many solutions. Consequently, a regularization method is needed to find an optimized solution for the linear system.
In this study, with the assumption that the solution is sparse, the following convex norm-1 regularization approach was employed to find vector x^ [54]:
minimizex^1,subjecttoAx^b2<δ
(16)
where x^1 is the norm-1 of vector x^, i.e., Σi=1m|xi^|, with xi^ being the ith element of x^; and δ is an upper limit for the residual error Ax^b2, with ℓ2 denoting the norm-2. NESTA—short for Nesterov’s method—can be used to solve the above minimization problem. The code that implements this algorithm is available online2 and it requires the input of the norm-1 regularization parameter μ to optimize the solution.

3.3 Sensing Capacity and Point-Spread Function.

Apart from imaging that was mentioned earlier, two additional criteria, useful in comparing the performance of the system with and without the use of metamaterials, are discussed. One is the sensing capacity that can be defined for nmin = min (n, m) orthogonal parallel channels as [28,55]
C=Σi=1nminlog2(1+Piλi2n0)
(17)
where the ratio Pi/n0 is the signal-to-noise ratio (SNR) of the ith channel, with Pi and n0 being the power of the signal and the noise, respectively, and λi is the ith nonzero singular value of the sensing matrix that can be obtained from its singular value decomposition [28]
A=Um×mΞm×nVn×n*
(18)
in which U = (u1, …, um), V = (v1, …, vn), and Ξ=diag(λ1,,λnmin). The other criterion is the beam focusing ability of the system or its PSF that can be established using the phase compensation focusing method as follows [56,57]:
BF(r|rp)=i=1Ntk=1NfPb,ik(r,fk)ejϕp,ik
(19)
where Nt is the number of transmitters, Nf is the number of frequencies, Pb,ik (r, fk) is the background fields due to transmitter i at frequency fk at location r, and ϕp,ik is the phase of the background field at the location of the focus point rp = (xp, yp), that is Pb,ik(rp,fk). Using this method, the acoustic beams can be focused at rp, giving the highest intensity at the location of the point and fading intensity as one moves from the point. The quality of focusing is contingent upon various factors such as the number of frequencies, number of transmitters, and, as will be seen, the use of metamaterials.

4 Results

The results of the simulations are presented in this section and the impact of adding metamaterials between the transceivers and the targets is assessed. The sensing matrix was computed from the background pressure for each pixel of the imaging domain, at each frequency and for each transceiver. Having the sensing matrix and background pressure, sensing capacity C and beam focusing BF(r|rp) are obtained immediately using Eqs. (17) and (19), for both cases where the metamaterials were present or absent. The beams were focused at the location of the tumors, as given in Fig. 5. The results are demonstrated in Figs. 6 and 7. It is clear that the addition of metamaterials has enhanced the phase-compensation focusing by narrowing down the region of the highest intensity around the focus points. Furthermore, it is evident that the metamaterials have increased the sensing capacity of the system, particularly at SNRs more than 20 dB.

Fig. 6
Beam focusing performance, comparison between the case where the metamaterials are absent versus the case where they are present. The use of metamaterial has enhanced the focusing ability by shrinking the area of high intensity around the focus point.
Fig. 6
Beam focusing performance, comparison between the case where the metamaterials are absent versus the case where they are present. The use of metamaterial has enhanced the focusing ability by shrinking the area of high intensity around the focus point.
Close modal
Fig. 7
Comparison between the sensing performance of the system in the absence and presence of metamaterials: (a) the sensing capacity C versus the SNR for the cases with and without metamaterials. The sensing capacity in the presence of the metamaterial line starts to surpass the case where the metamaterial line is absent after around an SNR of 20 dB. The difference becomes more significant as the SNR increases. (b) The amplitude of the singular values of the sensing matrix.
Fig. 7
Comparison between the sensing performance of the system in the absence and presence of metamaterials: (a) the sensing capacity C versus the SNR for the cases with and without metamaterials. The sensing capacity in the presence of the metamaterial line starts to surpass the case where the metamaterial line is absent after around an SNR of 20 dB. The difference becomes more significant as the SNR increases. (b) The amplitude of the singular values of the sensing matrix.
Close modal

The imaging results, obtained by NESTA, are shown in Fig. 8. These images are optimized by varying the NESTA algorithm parameters aimed at fewer artifacts and better localization. Also, they have been both saturated to only include the solution that is within the range 0 to −7 dB, for better demonstration of the contrast between the tumor and the background. It is clear that in the absence of the metamaterial line, the targets are not detected; rather, only their whereabouts in the y-direction has been revealed. On the other hand, the addition of the metamaterial line has enabled the system to detect the location of the targets—(−0.6, −2) mm and (0.8, −2) mm—both in x- and y-direction, with acceptable accuracy.

Fig. 8
Imaging performance without and with metamaterials. In the absence of the metamaterials, the targets are not distinguishable in cross-range; however, the metamaterial line has enabled both targets to appear in the retrieved image.
Fig. 8
Imaging performance without and with metamaterials. In the absence of the metamaterials, the targets are not distinguishable in cross-range; however, the metamaterial line has enabled both targets to appear in the retrieved image.
Close modal

5 Conclusions

In this study, the use of metamaterials for increasing the sensing capacity, focusing ability, and imaging performance of ultrasound waves was discussed. Since compressive sensing technique in imaging, which enables signal retrieval at much lower rates than the Nyquist frequency, favors randomness, a metamaterial unit cell was designed at two different sizes to create random patterns in the waves at different frequencies. It was shown by three criteria—sensing capacity, beam focusing, and imaging using NESTA—that the addition of metamaterials between the transceivers and the targets can meaningfully enhance the performance of the sensing system. The design of the mostly reflective metamaterial in this work was rather preliminary and theoretical, to only exhibit transmissive resonant behavior at certain frequencies. What is more, the unit cell and array design were intentionally non-optimized to show the power of random processes in compressive sensing, which could lead to desirable results without any optimization. In future studies, it should be seen whether different metamaterial designs with different configurations will be able to improve the imaging even more (for a study of different metamaterial types and the impact of various random distributions of the unit cells, refer to [58]). Moreover, the fabrication aspects of such small, multi-layer metamaterials should be addressed.

Footnote

Acknowledgment

This work has been partially funded by the Department of Energy (Award DE-SC0017614) and the NSF CAREER Program (Award No. 1653671).

Nomenclature

     
  • x =

    discretized contrast variable or the unknown vector

  •  
  • C =

    sensing capacity

  •  
  • A =

    sensing matrix

  •  
  • k(r, ω) =

    wavenumber as a function of location vector r and angular frequency ω

  •  
  • n(f) =

    refractive index as a function of frequency

  •  
  • BF =

    beam focusing

  •  
  • F(r,ω) =

    ultrasound source term as a function of location vector r and angular frequency ω

  •  
  • Gb(r,r,ω) =

    Green’s function as a function of transmitter location vector r′, location vector at each point in the domain r, and angular frequency ω

  •  
  • Pb(r,ω) =

    background pressure field as a function of location vector r and angular frequency ω

  •  
  • Ps(r,ω) =

    scattered pressure field as a function of location vector r and angular frequency ω

  •  
  • Pt(r,ω) =

    total pressure field as a function of location vector r and angular frequency ω

  •  
  • R(f) =

    reflection coefficient as a function of frequency

  •  
  • T(f) =

    transmission coefficient as a function of frequency

  •  
  • X(r,ω) =

    contrast variable a function of location vector r and angular frequency ω

  •  
  • ξ(f) =

    impedance as a function of frequency

References

1.
Pendry
,
J. B.
,
2000
, “
Negative Refraction Makes a Perfect Lens
,”
Phys. Rev. Lett.
,
85
(
18
), p.
3966
. 10.1103/PhysRevLett.85.3966
2.
Smith
,
D. R.
,
Vier
,
D.
,
Kroll
,
N.
, and
Schultz
,
S.
,
2000
, “
Direct Calculation of Permeability and Permittivity for a Left-Handed Metamaterial
,”
Appl. Phys. Lett.
,
77
(
14
), pp.
2246
2248
. 10.1063/1.1314884
3.
Shelby
,
R. A.
,
Smith
,
D. R.
, and
Schultz
,
S.
,
2001
, “
Experimental Verification of a Negative Index of Refraction
,”
Science
,
292
(
5514
), pp.
77
79
. 10.1126/science.1058847
4.
Li
,
J.
, and
Chan
,
C.
,
2004
, “
Double-Negative Acoustic Metamaterial
,”
Phys. Rev. E
,
70
(
5
), p.
055602
. 10.1103/PhysRevE.70.055602
5.
Veselago
,
V. G.
,
1968
, “
The Electrodynamics of Substances With Simultaneously Negative Values of ε and μ
,”
Sov. Phys. Usp.
,
10
(
4
), p.
509
. 10.1070/PU1968v010n04ABEH003699
6.
Cummer
,
S. A.
, and
Schurig
,
D.
,
2007
, “
One Path to Acoustic Cloaking
,”
New J. Phys.
,
9
(
3
), p.
45
. 10.1088/1367-2630/9/3/045
7.
Chen
,
H.
, and
Chan
,
C.
,
2007
, “
Acoustic Cloaking in Three Dimensions Using Acoustic Metamaterials
,”
Appl. Phys. Lett.
,
91
(
18
), p.
183518
. 10.1063/1.2803315
8.
Torrent
,
D.
, and
Sánchez-Dehesa
,
J.
,
2008
, “
Acoustic Cloaking in Two Dimensions: A Feasible Approach
,”
New J. Phys.
,
10
(
6
), p.
063015
. 10.1088/1367-2630/10/6/063015
9.
Ambati
,
M.
,
Fang
,
N.
,
Sun
,
C.
, and
Zhang
,
X.
,
2007
, “
Surface Resonant States and Superlensing in Acoustic Metamaterials
,”
Phys. Rev. B
,
75
(
19
), p.
195447
. 10.1103/PhysRevB.75.195447
10.
Zhang
,
S.
,
Xia
,
C.
, and
Fang
,
N.
,
2011
, “
Broadband Acoustic Cloak for Ultrasound Waves
,”
Phys. Rev. Lett.
,
106
(
2
), p.
024301
. 10.1103/PhysRevLett.106.024301
11.
Guenneau
,
S.
,
Movchan
,
A.
,
Pétursson
,
G.
, and
Ramakrishna
,
S. A.
,
2007
, “
Acoustic Metamaterials for Sound Focusing and Confinement
,”
New J. Phys.
,
9
(
11
), p.
399
. 10.1088/1367-2630/9/11/399
12.
Zhang
,
S.
,
Yin
,
L.
, and
Fang
,
N.
,
2009
, “
Focusing Ultrasound With an Acoustic Metamaterial Network
,”
Phys. Rev. Lett.
,
102
(
19
), p.
194301
. 10.1103/PhysRevLett.102.194301
13.
Li
,
J.
,
Fok
,
L.
,
Yin
,
X.
,
Bartal
,
G.
, and
Zhang
,
X.
,
2009
, “
Experimental Demonstration of an Acoustic Magnifying Hyperlens
,”
Nat. Mater.
,
8
(
12
), pp.
931
934
. 10.1038/nmat2561
14.
Zhou
,
X.
, and
Hu
,
G.
,
2011
, “
Superlensing Effect of an Anisotropic Metamaterial Slab With Near-Zero Dynamic Mass
,”
Appl. Phys. Lett.
,
98
(
26
), p.
263510
. 10.1063/1.3607277
15.
Zhu
,
R.
,
Chen
,
Y.
,
Wang
,
Y.
,
Hu
,
G.
, and
Huang
,
G.
,
2016
, “
A Single-Phase Elastic Hyperbolic Metamaterial With Anisotropic Mass Density
,”
J. Acoust. Soc. Am.
,
139
(
6
), pp.
3303
3310
. 10.1121/1.4950728
16.
Zhu
,
R.
,
Yasuda
,
H.
,
Huang
,
G.
, and
Yang
,
J.
,
2018
, “
Kirigami-Based Elastic Metamaterials With Anisotropic Mass Density for Subwavelength Flexural Wave Control
,”
Sci. Rep.
,
8
(
1
), p.
483
. 10.1038/s41598-017-18864-z
17.
Deng
,
K.
,
Ding
,
Y.
,
He
,
Z.
,
Zhao
,
H.
,
Shi
,
J.
, and
Liu
,
Z.
,
2009
, “
Theoretical Study of Subwavelength Imaging by Acoustic Metamaterial Slabs
,”
J. Appl. Phys.
,
105
(
12
), p.
124909
. 10.1063/1.3153976
18.
Zhu
,
J.
,
Christensen
,
J.
,
Jung
,
J.
,
Martin-Moreno
,
L.
,
Yin
,
X.
,
Fok
,
L.
,
Zhang
,
X.
, and
Garcia-Vidal
,
F.
,
2011
, “
A Holey-Structured Metamaterial for Acoustic Deep-Subwavelength Imaging
,”
Nat. Phys.
,
7
(
1
), pp.
52
55
. 10.1038/nphys1804
19.
Yang
,
Z.
,
Dai
,
H.
,
Chan
,
N.
,
Ma
,
G.
, and
Sheng
,
P.
,
2010
, “
Acoustic Metamaterial Panels for Sound Attenuation in the 50–1000 Hz Regime
,”
Appl. Phys. Lett.
,
96
(
4
), p.
041906
. 10.1063/1.3299007
20.
Mei
,
J.
,
Ma
,
G.
,
Yang
,
M.
,
Yang
,
Z.
,
Wen
,
W.
, and
Sheng
,
P.
,
2012
, “
Dark Acoustic Metamaterials as Super Absorbers for Low-Frequency Sound
,”
Nat. Commun.
,
3
(
1
), pp.
756
756
. 10.1038/ncomms1758
21.
Cummer
,
S. A.
,
Christensen
,
J.
, and
Alù
,
A.
,
2016
, “
Controlling Sound With Acoustic Metamaterials
,”
Nat. Rev. Mater.
,
1
(
3
), p.
16001
. 10.1038/natrevmats.2016.1
22.
Shen
,
C.
,
Xu
,
J.
,
Fang
,
N. X.
, and
Jing
,
Y.
,
2014
, “
Anisotropic Complementary Acoustic Metamaterial for Canceling Out Aberrating Layers
,”
Phys. Rev. X
,
4
(
4
), p.
041033
.
23.
D’Aguanno
,
G.
,
Le
,
K.
,
Trimm
,
R.
,
Alu
,
A.
,
Mattiucci
,
N.
,
Mathias
,
A.
,
Aközbek
,
N.
, and
Bloemer
,
M.
,
2012
, “
Broadband Metamaterial for Nonresonant Matching of Acoustic Waves
,”
Sci. Rep.
,
2
, p.
340
. 10.1038/srep00340
24.
Xie
,
Y.
,
Konneker
,
A.
,
Popa
,
B.-I.
, and
Cummer
,
S. A.
,
2013
, “
Tapered Labyrinthine Acoustic Metamaterials for Broadband Impedance Matching
,”
Appl. Phys. Lett.
,
103
(
20
), p.
201906
. 10.1063/1.4831770
25.
Farhat
,
M.
,
Guenneau
,
S.
, and
Enoch
,
S.
,
2009
, “
Ultrabroadband Elastic Cloaking in Thin Plates
,”
Phys. Rev. Lett.
,
103
(
2
), p.
024301
. 10.1103/PhysRevLett.103.024301
26.
Miniaci
,
M.
,
Krushynska
,
A.
,
Bosia
,
F.
, and
Pugno
,
N. M.
,
2016
, “
Large Scale Mechanical Metamaterials as Seismic Shields
,”
New J. Phys.
,
18
(
8
), p.
083041
. 10.1088/1367-2630/18/8/083041
27.
Baraniuk
,
R. G.
,
2007
, “
Compressive Sensing [Lecture Notes]
,”
IEEE Signal Process. Mag.
,
24
(
4
), pp.
118
121
. 10.1109/MSP.2007.4286571
28.
Lorenzo
,
J. A. M.
,
Juesas
,
J. H.
, and
Blackwell
,
W.
,
2016
, “
A Single-Transceiver Compressive Reflector Antenna for High-Sensing-Capacity Imaging
,”
IEEE Antennas Wirel. Propag. Lett.
,
15
(
1
), pp.
968
971
. 10.1109/LAWP.2015.2487319
29.
Candès
,
E. J.
, and
Wakin
,
M. B.
,
2008
, “
An Introduction to Compressive Sampling
,”
IEEE Signal Process. Mag.
,
25
(
2
), pp.
21
30
. 10.1109/MSP.2007.914731
30.
Li
,
Z.
,
Yang
,
D.-Q.
,
Liu
,
S.-L.
,
Yu
,
S.-Y.
,
Lu
,
M.-H.
,
Zhu
,
J.
,
Zhang
,
S.-T.
,
Zhu
,
M.-W.
,
Guo
,
X.-S.
,
Wu
,
H.-D.
,
Wang
,
X. L.
, and
Chen
,
Y. F.
,
2017
, “
Broadband Gradient Impedance Matching Using an Acoustic Metamaterial for Ultrasonic Transducers
,”
Sci. Rep.
,
7
(
1
), p.
42863
. 10.1038/srep42863
31.
Vanhille
,
C.
,
2017
, “
Two-Dimensional Numerical Simulations of Ultrasound in Liquids With Gas Bubble Agglomerates: Examples of Bubbly-Liquid-Type Acoustic Metamaterials (Blamms)
,”
Sensors
,
17
(
1
), p.
173
. 10.3390/s17010173
32.
Zhu
,
R.
,
Huang
,
G.
, and
Hu
,
G.
,
2012
, “
Effective Dynamic Properties and Multi-Resonant Design of Acoustic Metamaterials
,”
ASME J. Vib. Acoust.
,
134
(
3
), p.
031006
. 10.1115/1.4005825
33.
Ao
,
X.
, and
Chan
,
C.
,
2008
, “
Far-Field Image Magnification for Acoustic Waves Using Anisotropic Acoustic Metamaterials
,”
Phys. Rev. E
,
77
(
2
), p.
025601
. 10.1103/PhysRevE.77.025601
34.
Medwin
,
H.
,
1975
, “
Speed of Sound in Water: A Simple Equation for Realistic Parameters
,”
J. Acoust. Soc. Am.
,
58
(
6
), pp.
1318
1319
. 10.1121/1.380790
35.
Caloz
,
C.
, and
Itoh
,
T.
,
2005
,
Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications
,
John Wiley & Sons
,
New York
.
36.
Fokin
,
V.
,
Ambati
,
M.
,
Sun
,
C.
, and
Zhang
,
X.
,
2007
, “
Method for Retrieving Effective Properties of Locally Resonant Acoustic Metamaterials
,”
Phys. Rev. B
,
76
(
14
), p.
144302
. 10.1103/PhysRevB.76.144302
37.
Szabó
,
Z.
,
Park
,
G.-H.
,
Hedge
,
R.
, and
Li
,
E.-P.
,
2010
, “
A Unique Extraction of Metamaterial Parameters Based on Kramers–Kronig Relationship
,”
IEEE Trans. Microwave Theory Tech.
,
58
(
10
), pp.
2646
2653
. 10.1109/TMTT.2010.2065310
38.
Park
,
J. H.
, and
Kim
,
Y. Y.
,
2016
, “
Characterization of Anisotropic Acoustic Metamaterials
,”
INTER-NOISE and NOISE-CON Congress and Conference Proceedings
,
Hamburg, Germany
,
Aug. 20–24
, Vol.
253
,
Institute of Noise Control Engineering
, pp.
1468
1475
.
39.
Jensen
,
F. B.
,
Kuperman
,
W. A.
,
Porter
,
M. B.
, and
Schmidt
,
H.
,
2000
,
Computational Ocean Acoustics
,
Springer Science & Business Media
,
New York
.
40.
COMSOL Inc.
,
2013
,
Acoustics Module User’s Guide
,
COMSOL Inc.
,
Burlington, MA
.
41.
Dagheyan
,
A. G.
,
2016
, “
A Near-Field Radar Mechatronics System for Early Detection of Breast Cancer
,”
Master's Thesis
,
Northeastern University
,
Boston
.
42.
Dagheyan
,
A. G.
,
Molaei
,
A.
,
Obermeier
,
R.
, and
Martinez-Lorenzo
,
J.
,
2016
, “
Preliminary Imaging Results and Sar Analysis of a Microwave Imaging System for Early Breast Cancer Detection
,”
2016 IEEE 38th Annual International Conference of the Engineering in Medicine and Biology Society (EMBC)
,
Orlando, FL
,
Aug. 16–20
,
IEEE
,
New York
, pp.
1066
1069
.
43.
Li
,
C.
,
Duric
,
N.
,
Littrup
,
P.
, and
Huang
,
L.
,
2009
, “
In Vivo Breast Sound-Speed Imaging With Ultrasound Tomography
,”
Ultrasound Med. Biol.
,
35
(
10
), pp.
1615
1628
. 10.1016/j.ultrasmedbio.2009.05.011
44.
Youk
,
J. H.
,
Son
,
E. J.
,
Park
,
A. Y.
, and
Kim
,
J.-A.
,
2014
, “
Shear-Wave Elastography for Breast Masses: Local Shear Wave Speed (m/sec) Versus Young Modulus (kPa)
,”
Ultrasonography
,
33
(
1
), p.
34
. 10.14366/usg.13005
45.
Carlson
,
J.
,
Van Deventer
,
J.
,
Scolan
,
A.
, and
Carlander
,
C.
,
2003
, “
Frequency and Temperature Dependence of Acoustic Properties of Polymers Used in Pulse-Echo Systems
,”
2003 IEEE Symposium on Ultrasonics, Vol. 1, IEEE
,
Honolulu, HI
,
Oct. 5–8
,
New York
, pp.
885
888
.
46.
Martin
,
J.
,
2006
,
Materials for Engineering
, 3rd ed.,
Woodhead Publishing
,
Cambridge, UK
.
47.
Wittek
,
A.
,
Joldes
,
G.
,
Nielsen
,
P. M.
,
Doyle
,
B. J.
, and
Miller
,
K.
,
2017
,
Computational Biomechanics for Medicine: From Algorithms to Models and Applications
,
Springer
,
New York
.
48.
Malik
,
B.
,
Klock
,
J.
,
Wiskin
,
J.
, and
Lenox
,
M.
,
2016
, “
Objective Breast Tissue Image Classification Using Quantitative Transmission Ultrasound Tomography
,”
Sci. Rep.
,
6
(
1
), p.
38857
. 10.1038/srep38857
49.
Levine
,
H.
, and
Schwinger
,
J.
,
1948
, “
On the Theory of Diffraction by an Aperture in an Infinite Plane Screen. I
,”
Phys. Rev.
,
74
(
8
), p.
958
. 10.1103/PhysRev.74.958
50.
Obermeier
,
R.
,
2016
, “
Compressed Sensing Algorithms for Electromagnetic Imaging Applications
,”
Ph.D. thesis
,
Northeastern University
,
Boston
.
51.
Zimmer
,
M.
,
Bibee
,
L.
, and
Richardson
,
M.
,
2005
, “
Frequency Dependent Sound Speed and Attenuation Measurements in Seafloor Sands From 1 to 400 KHz
,”
Tech. Rep
.,
Naval Research Lab Stennis Space Center MS Marine Geosciences Div
.
52.
Obermeier
,
R.
,
Juesas
,
J. H.
, and
Martinez-Lorenzo
,
J. A.
,
2015
, “
Imaging Breast Cancer in a Hybrid DBT/NRI System Using Compressive Sensing
,”
2015 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting
,
Vancouver, BC, Canada
,
July 19–25
,
IEEE
,
New York
, pp.
392
393
.
53.
Poli
,
L.
,
Oliveri
,
G.
, and
Massa
,
A.
,
2012
, “
Microwave Imaging Within the First-Order Born Approximation by Means of the Contrast-Field Bayesian Compressive Sensing
,”
IEEE Trans. Antennas Propag.
,
60
(
6
), pp.
2865
2879
. 10.1109/TAP.2012.2194676
54.
Beck
,
A.
, and
Teboulle
,
M.
,
2009
, “
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
,”
SIAM J. Imag. Sci.
,
2
(
1
), pp.
183
202
. 10.1137/080716542
55.
Tse
,
D.
, and
Viswanath
,
P.
,
2005
,
Fundamentals of Wireless Communication
,
Cambridge University Press
,
Cambridge
.
56.
Ahmed
,
S. S.
,
Schiessl
,
A.
, and
Schmidt
,
L.-P.
,
2009
, “
Multistatic mm-Wave Imaging With Planar 2D-Arrays
,”
2009 German Microwave Conference, IEEE
,
Munich, Germany
,
Mar. 16–18
,
IEEE
,
New York
, pp.
1
4
.
57.
Molaei
,
A.
,
Juesas
,
J. H.
, and
Lorenzo
,
J. A. M.
,
2017
,
Antenna Arrays and Beam-Formation
,
M.
Shbat
, ed.,
InTech
,
London, UK
, pp.
31
51
.
58.
Dagheyan
,
A. G.
,
2018
, “
High-Sensing-Capacity, Bimodal Mechatronic Imaging System for Early Detection of Breast Cancer
,”
Doctoral dissertation
,
Northeastern University
,
Boston
.