Acoustic Negative Refraction and Planar Focusing Based on Purely Imaginary Metamaterials

Abstract

Acoustic purely imaginary metamaterials (PIMs) contain loss and gain uniformly distributed in space, but in different parameters. Therefore, the PIMs contain the elements of gain and loss simultaneously. As a result, some extraordinary wave modes may appear depending on whether gain or loss elements dominate. In this work, we theoretically and numerically investigate the general excitation conditions for acoustic lasing, coherent perfect absorption (CPA) and for their co-existence in the framework of acoustic PIMs. All-angle negative refraction and planar focusing are achieved by pairing two PIM slabs with conjugating parameters. The proposed structure provides an alternative basis for possible applications of acoustic perfect imaging.

1. Introduction

The diffraction of light in conventional media limits the sharpness of the image by the wavelength of light. Overcoming this barrier has inspired research on negative refraction, which possesses the ability to focus all the spatial Fourier components of a source, even the evanescent spatial spectrum [1]. However, negative refraction has not readily been found in natural materials. Thus, metamaterials exhibiting anomalous values have gained considerable interest in the last few years [1,2,3,4,5,6,7].

In particular, the exotic physical properties of parity-time (PT) symmetry materials have been widely studied [8,9,10,11,12,13,14]. PT symmetry in optics corresponds to a spatial distribution with complex refractive indices, i.e., systems that exhibit symmetric gains and losses by adjusting the refractive indices [9]. Many novel phenomena that cannot be achieved in traditional structures have been discovered. For example, PT-symmetry breaking [15,16,17,18], unidirectional invisibility [19,20,21,22], Bloch vibration [23,24] and planar perfect imaging system [12,25,26]. It is shown that by employing a pair of PT symmetric meta-surfaces, combined with a tailored nonlocal response, negative refraction and planar focusing have been realized. In this process, the illuminated half of the structure works as a coherent perfect absorber, fully absorbing the incident waves, while the other part operates in the mode of a coherent laser, reproducing the impinging waves on both sides [12,25].

Similar to PT-symmetric materials, purely imaginary metamaterials (PIMs) have been proposed by introducing the gain and loss acting as new freedoms. However, unlike the spatial distribution of loss and gain in PT-symmetric systems, the loss and gain of PIM are uniformly distributed in space but with different parameters. Thus, PIM is a homogeneous medium with a real refractive index. PIMs offer interesting underlying physics to explore and therefore they have been widely studied in the field of optics [7,27,28,29] and have also been introduced into the field of acoustics [30,31]. Interestingly, the coexistence of laser and anti-laser (coherent perfect absorption (CPA)) is also possible in PIM structures [27,30]. Inspired by the PT symmetry counterpart, there are some attempts to study the imaging phenomena in optics [28]. However, there are few works related to acoustic imaging based on PIM structures. In this work, we explore the anomalous scattering properties in the framework of acoustic PIM through analytical derivations and full-wave simulations. All-angle negative refraction and planar focusing have been demonstrated by using a PIM slab that supports the acoustic CPA mode and another PIM slab with conjugate parameters operating in the coherent acoustic laser mode. The proposed structure provides an alternative basis for the possible applications of acoustic amplification, acoustic absorption, and acoustic perfect imaging.

2. Scattering Properties of a PIM Slab

We first focus on the scattering properties of a purely imaginary medium (PIM) slab with a thickness of $d$ in air, as shown in Figure 1. The bulk compressibility and the mass density are given as ${\beta }_s=i{\beta }_r{\beta }_0$ and $\rho =-i{\rho }_r{\rho }_0$, respectively. Here, ${\beta }_0$ and ${\rho }_0$ are the corresponding parameters of air, and ${\beta }_r$, ${\rho }_r$ are real numbers. The density and velocity of air are ${\rho }_0=1.2{\mathrm{kg}/\mathrm{m}}^3$, and $c_0=343\mathrm{m}/\mathrm{s}$, respectively. When the imaginary part of the parameter is positive, it represents the gain, and when the imaginary part of the parameter is negative, it represents the loss. Although the PIMs contain both gain and loss elements, the phases of the bulk compressibility ${\beta }_s$ and the mass density $\rho$ are conjugates of each other, so the acoustic velocity $c=\pm \sqrt{{(\rho {\beta }_s)}^{-1}}$ and the refractive index $n=\pm \sqrt{\rho {\beta }_s/{\rho }_0{\beta }_0}$ are both real numbers.

Consider monochromatic acoustic waves with frequency ω and incident angle $\theta$. Under the $e^{i\omega t}$ convention, the acoustic fields $p$ satisfy the Helmholtz equation $(d^2/d{x}^2)p+{\omega }^2{\beta }_s\rho p=0$. The equation admits a solution

$$\begin{array}{l}\{\begin{array}{l}p(x,y)=p_{fl}{e}^{-j{k}_{x}x}{e}^{-k_{x}y}+p_{bl}{e}^{j{k}_{x}x}{e}^{-k_{x}y}, (x<0)\\ p(x,y)=p_{fr}{e}^{-j{k}_x(x-d)}{e}^{-k_{x}y}+p_{br}{e}^{j{k}_x(x-d)}{e}^{-k_{x}y},(x>d)\end{array}\end{array}$$

(1)

where, $k_y=k_0\sin \theta$ are the wave vector along the $y$ directions, $k_0=2\pi /\lambda$ is the wave vector in air, $k_x=\sqrt{k_0{}^2-k_y{}^2}$ and $k_x{}^{\prime }=\sqrt{n^{\prime 2}{k}_0{}^2-k_y{}^2}$ are the $x$ wave vector components in the air and PIM media, respectively. Due to the boundary conditions, $k_y$ remains constant in all layers. The coefficients $p_{fl}$, $p_{bl}$, $p_{fr}$ and $p_{br}$ are the complex amplitudes of the incoming or outgoing waves, with the subscript $f(b)$ for forward (backward) acoustic waves, and the subscript $l(r)$ for the left $x<0$ (right $x>d$).

In terms of the scattering matrix $S$, the relationship between the amplitudes of the incident and outgoing waves is

$$\left(\begin{array}{c}{p}_{bl}\\ p_{fr}\end{array}\right)=\left(\begin{array}{cc}r& t\\ t& r\end{array}\right)\left(\begin{array}{c}{p}_{fl}\\ p_{br}\end{array}\right),S=\left(\begin{array}{cc}{\mathrm{S}}_{11}& {\mathrm{S}}_{12}\\ {\mathrm{S}}_{21}& {\mathrm{S}}_{22}\end{array}\right)=\left(\begin{array}{cc}r& t\\ t& r\end{array}\right).$$

(2)

In the above equations, $r$ and $t$ represent the reflection and transmission coefficients of the slab, respectively. By calculating the elements of the scattering matrix, we obtain the transmittance and reflection coefficients as (see Appendix A)

$${r=\mathrm{S}}_{11}={\mathrm{S}}_{22}=\frac{({\eta }^2-1)2i\sin \varphi }{{(\eta +1)}^2{e}^{i\varphi }-{(\eta -1)}^2{e}^{-i\varphi }},{t=\mathrm{S}}_{12}={\mathrm{S}}_{21}=\frac{4\eta }{{(\eta +1)}^2{e}^{i\varphi }-{(\eta -1)}^2{e}^{-i\varphi }}.$$

(3)

Here, $\varphi =k_x{}^{\prime }d$ is the propagation phase in the slab, and $\eta ={\rho }^{\prime }{k}_x/{\rho }_0{k}_x{}^{\prime }$ is the relative impedance of the PIM-air interface. The equalities of ${\mathrm{S}}_{11}={\mathrm{S}}_{22}=r$ and ${\mathrm{S}}_{12}={\mathrm{S}}_{21}=t$ is observed due to the symmetry of the system.

3. Acoustic Laser Mode and CPA Mode

Since the material of PIM contains both gain and loss elements, some special wave modes, including acoustic laser and CPA, may appear depending on which element dominates. In the following, we explore the excitation conditions and mode characteristics of the CPA and laser modes of acoustic PIM by analyzing the scattering coefficients.

For an acoustic laser oscillator, the incident wave might induce quite intense transmission and reflection so strongly that the injected wave can be ignored. This means that the reflection and transmission coefficients tend to infinity. Thus, the condition for the acoustic laser mode can be obtained by setting the denominator of Equation (3) to zero, i.e., ${(\eta +1)}^2{e}^{i\varphi }-{(\eta -1)}^2{e}^{-i\varphi }=0$. Thus, the corresponding dispersion relations can be solved as

$$\{\begin{array}{l}\eta =-i\tan \frac{k_x{}^{\prime }}{2}d\left(\mathrm{even}\mathrm{mode}\right)\\ \eta =i\cot \frac{k_x{}^{\prime }}{2}d\left(\mathrm{odd}\mathrm{mode}\right)\end{array}$$

(4)

In contrast, for a perfect absorber, the in-phase coherent waves incident from both sides could be fully absorbed without any scattered waves. The boundary condition can be expressed as $p_{bl}=p_{fr}=0$. From Equation (2), we obtain $r\cdot p_{fl}+t\cdot p_{br}=0$ and $t\cdot p_{fl}+r\cdot p_{br}=0$. This requires either $p_{fl}=p_{br}$ (even mode) and finding the condition for $r+t=0$, or $p_{fl}=-p_{br}$ (odd mode) and finding the condition for $r-t=0$. Therefore, the condition for the CPA mode can be calculated as

$$\{\begin{array}{l}\eta =i\tan \frac{k_x{}^{\prime }}{2}d,\left(\mathrm{even}\mathrm{mode}\right)\\ \eta =-i\cot \frac{k_x{}^{\prime }}{2}d,\left(\mathrm{odd}\mathrm{mode}\right)\end{array}$$

(5)

The parity of the mode indicates whether the field distribution is symmetric or antisymmetric in the x-direction around the center of the PIM layer.

For a PIM slab, the scattering relations can be satisfied for both ${\beta }_r={\rho }_r$ and ${\beta }_r\ne {\rho }_r$. For the sake of simplicity, in the following we will use ${\beta }_r={\rho }_r$ as an example, such that $n=\sqrt{{\beta }_r{\rho }_r}={\beta }_r={\rho }_r$. Similar wave propagation still occurs in PIMs with ${\beta }_r\ne {\rho }_r$.

Based on Equation (4), the dispersion relationships between the wave vector $k_y$ and the refractive index $n$ of the acoustic laser modes is shown in Figure 2a, where the red solid and dashed curves correspond to the even and odd modes, respectively. Take $n={\beta }_r={\rho }_r=1.2$ as an example, the wave vectors in the $y$ direction wave vector for the even acoustic laser mode and odd acoustic laser mode are $0.42k_0$ and $0.84k_0$, respectively. To match the tangential momentum, the even and odd acoustic laser mode approximately to occur at the incident angles of ${25.12}^0$ and ${57.58}^0$, respectively.

To verify the analytical results, numerical simulations are performed using COMSOL Multiphysics software. A plane wave radiation boundary is imposed on the incident and transmitted boundaries, and a Floquet periodic boundary condition is applied in the y direction. The width of the PIM slab is set to $2\lambda$. Because a periodic boundary condition is set in the y-direction, the height of the PIM slab does not affect the acoustic field distribution, which is set to $4\lambda$ in the following simulations. For plane acoustic waves with an amplitude of 1.0 Pa incident from the left side with the corresponding incident angles, the results are displayed in Figure 2b,c. By observing the field patterns and the direction of the energy flow (black arrows) in Figure 2b,c, it can be found that quite intense outgoing waves are generated on both sides of the PIM slabs. In both cases, the total acoustic pressure fields are very strong with respect to the incidence, although the amplification is different (see color bars). The incoming acoustic waves oscillate within the slab to accumulate energy, and the additional energy is provided by the gain portion of the PIM, which is similar to the mechanism of the laser mode in a cavity system with a gain medium. The field distribution of Figure 2b is symmetric in the $x$ direction about the center of the PIM layer, corresponding to the symmetric mode, while the field distribution of Figure 2c exhibits an odd spatial distribution about the center of the PIM slab.

Based on Equation (5), the dispersion relation of CPA modes is shown in Figure 3a, and the blue solid and dashed curves correspond to the even CPA and odd CPA modes, respectively. For the case of $n=1.2$, the wave vector in the $y$-direction for the odd CPA mode and even CPA mode are $k_y=0.41k_0$ and $k_y=0.80k_0$, respectively.

To excite the odd CPA mode, two waves with opposite amplitudes of $p_{fl}=-p_{br}=1\mathrm{Pa}$ are incident from the left and right sides. The angle of incidence is ${24.27}^0$ to match the tangential wave vector of $0.41k_0$. The acoustic pressure field distributions are shown in Figure 3b. Likewise, when bilateral incident waves are of equal amplitude of $p_{fl}=p_{br}=1\mathrm{Pa}$ and with incident angle of $\theta ={53.35}^0$, which match the tangential wave vector of $0.80k_0$, an even CPA mode can be excited, as shown in Figure 3c. By observing the pressure field distributions in Figure 3b,c, it can be seen that the coherent incident acoustic waves are absorbed through a combination of interference and dissipation in the PIM slab.

4. Negative Refraction and Plane Focusing

It is worth pointing out that Equations (4) and (5) have the same form but with opposite signs. Therefore, if the conjugate operation is performed on the constitutive parameters, the acoustic laser mode can be converted to CPA mode and vice versa. In other words, if we obtain the CPA (acoustic laser) mode in a PIM slab with parameters ${\beta }_s=i{\beta }_r{\beta }_0$ and $\rho =-i{\rho }_r{\rho }_0$, then the acoustic laser (CPA) mode will be realized in a PIM* slab by conjugating the parameters, such that ${\beta }_s=-i{\beta }_r{\beta }_0$, $\rho =i{\rho }_r{\rho }_0$. The coexistence of CPA and acoustic laser modes offers the possibility of negative refraction.

As sketched in Figure 4, the structure devised consists of three blocks, in which the PIM slab and PIM* slab, both with a thickness of $d$, are separated by an air region of a width of $L$. The left PIM slab is designed to operate as a CPA, while the right PIM* slab with conjugate parameters is operated as an acoustic laser.

The thickness $d$ of both the PIM slab is $0.3\lambda$, and the length of the air gap $L$ is $12\lambda$. Based on the above theoretical derivation (see Appendix B), the absolute values of refraction coefficient $\left|r\right|$ and transmission coefficient $\left|t\right|$ (in dB) are related to the incident angle $\theta$ and the refraction index $n$, are shown in Figure 5a,b. It can be seen that there is a dark blue curve extending from coordinate [0, 0.8] to coordinate [90, 1.2] in Figure 5a. On this curve, the reflection coefficient is almost zero, while the corresponding transmission is approximately one, as shown in Figure 5b. To verify whether such a response is a negative refraction or a resonance phenomenon, the scattering coefficients of the PIM* are calculated as shown in Figure 5c,d. It can be seen that there is a dark red curve with the same shape. Since the absolute values of the refraction coefficients $\left|r\right|$ and transmission coefficients $\left|t\right|$ tend to infinity (refer to the color bar), this curve corresponds to acoustic laser modes of the PIM* slab. As can been seen from the black dotted and dash lines in Figure 6a, the corresponding overall transmission coefficients $\left|t\right|$ tend to unity, while the reflection coefficients $\left|r\right|$ are almost zero. Thus, such a response is consistent with negative refraction in the PIM-air-PIM* structure, with one side working as a coherent power sink and the other as a power amplifier, while the overall transmission is unitary.

5. All-Angle Negative Refraction and Planar Focusing

For illustration, we arbitrarily take the incident angles as $0^0$, ${30}^0$ and ${55}^0$. From the red curve ($n$ vs. $\theta$) of Figure 6a, it is found that the corresponding refractive indices are 0.84, 0.97 and 1.29, respectively. To verify the analytical results, numerical simulations are performed using COMSOL software. The Floquet periodic boundary condition is applied in the y direction. By launching plane acoustic waves from the left side with corresponding incident angles, pressure field distribution are shown in Figure 6b–d. In each picture, the white boxes with a thickness of $0.3\lambda$ on the left and right side represent the PIM slab and PIM* slab, respectively. The black arrows represent the Poynting vector field. It can be seen that negative refraction occurs at the boundary of both the PIM and PIM* slab, demonstrating the perfect negative behavior described in Figure 4.

In order to achieve planar focusing, negative refraction should be achieved for all angles of light emitted from the point source. Different longitudinal coordinates of the slab correspond to different incident angles. For example, a point source is placed to the left of the PIM slab, then the coordinate $y$ is relevant to an incident angle of $\theta =a\tan (y/f)$. Therefore, the refractive index of the PIM slab is inhomogeneous. The planar focusing performance of PIM-air-PIM* is calculated using Comsol software, and the material parameters are set to satisfy the dispersion relation between the refractive index and incidence angle, as shown in Figure 6a. The thickness $d$ is set to $0.3\lambda$, and the length of the air gap $L$ is set to $12\lambda$. The point source is located $6\lambda$ from the left PIM slab. Perfectly Matched Layers (PML) [32] boundary conditions are used at the perimeter of the structure to reduce reflections. As shown in Figure 7, the acoustic waves emitted from the point source are focused both in between the slabs and outside the right PIM* slab, respectively. The image resolution is diffraction-limited, as only the propagating wave components are considered.

6. Conclusions

In conclusion, we explore the excitation conditions and mode characteristics of the CPA and laser modes in a framework of acoustic purely imaginary materials (PIMs). Theoretical analysis and numerical simulations reveal some special scattering phenomena and whether gain or loss elements dominate in this process, since the PIM material contains both gain and lose elements. We further demonstrate that omnidirectional acoustic negative refraction and planar focusing can be achieved in the PIM-air-PIM* structures with tailored parameters. Such a finding may find important applications in amplification, acoustic absorption and acoustic perfect imaging.