# Design Method of Acoustic Metamaterials for Negative Refractive Index Acoustic Lenses Based on the Transmission-Line Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Distributed Transmission-Line Model and the Design Theory

_{0}, β, and l are the unit cell length, the characteristic impedance, the phase constant, and the line length, respectively, and βl means the electrical length. If we define the phase constant as $\beta =\omega \sqrt{\rho /K}$, we can apply the design theory of the model to the design of negative refractive index acoustic metamaterials, where ρ and K represent the mass density and the bulk modulus, respectively.

_{x}and k

_{y}show the wavenumbers in the x and y directions, respectively. Additionally, the formulas for the dispersion characteristics along the Γ-X path, that is represented with the red arrows ((k

_{x}= k and k

_{y}= 0 or k

_{y}= k and k

_{x}= 0)), and the refractive index (n) can be written as:

_{B}l

_{B}and β

_{NRI}l

_{NRI}) becomes:

_{B}) agrees with the absolute value of that of the negative refractive acoustic metamaterial (|n

_{NRI}|). In this case, we can solve the impedance matching problem if we choose those characteristic impedances (Z

_{0_B}and Z

_{0_NRI}) to the same value since those Bloch impedances (Z

_{Bloch_B}and Z

_{Bloch_NRI}) become the same value automatically from (7).

#### 2.2. Proposed Structure

_{Air}, c = c

_{Air}), and the wall surfaces in the acoustic waveguides are set to the sound hard boundary in the simulation with COMSOL of the following subsection. The boundary corresponds to the Neumann boundary condition that is the same as the perfect magnetic conductor for the TE incidence case in the analysis of electromagnetic waves, and the wall’s impedance is the set to infinity in this case. l

_{NRI}and l

_{B}are the waveguide lengths and Δd

_{NRI}and Δd

_{B}(=Δd

_{NRI}= l

_{B}) represent the unit cell lengths, and these correspond to the line length and the unit cell length of the model in Figure 1. w

_{NRI}and w

_{B}are the waveguide widths and these can be used to adjust the characteristic impedance for impedance matching between the NRI acoustic metamaterials and the background medium. Figure 3 shows the concept of an NRI acoustic lens. n

_{NRI}and n

_{B}are the refractive indices, θ

_{B_i}and θ

_{NRI_i}the incident angles, θ

_{B_r}and θ

_{NRI_r}the refractive angles, d

_{1}is the distance between the acoustic wave source and the focus, and d

_{2}is the distance between the focus and the refocus. In Figure 3, these parameters are assumed as |n

_{NRI}| = n

_{B}, θ

_{B_i}= |θ

_{NRI_r}| = θ

_{NRI_i}= |θ

_{B_r}|, and d

_{2}= d

_{1}in order to simplify the design. We determine the structural parameters in Figure 2 for constituting the NRI acoustic lens and the background medium of Figure 3 in the following.

#### 2.3. Acoustic NRI Lens Design

_{B}= Δd

_{B}= Δd

_{NRI}= 25 mm and f

_{NRI}= 2.5 kHz, respectively, for feasibility. Additionally, we defined the refractive indices as the quantities compared with the air, and obtained the refractive index of the background medium as n

_{B}=1.522 from (4) and determined that of the lens as n

_{NRI}= –1.522 from the condition of |n

_{NRI}| = n

_{B}. Then, we decided the waveguide length in Figure 2a to be l

_{NRI}= 113.5 mm by substituting f

_{NRI}and n

_{NRI}for (6) with m = 1. Moreover, we chose the waveguide widths as w

_{NRI}= 1.2 mm and w

_{B}= 1.2 mm for considering the feasibility. In this case, impedance matching is also considered and Z

_{0_B}= Z

_{0_NRI}and Z

_{Bloch_B}= Z

_{Bloch_NRI}are held based on (7). Incidentally, the mass density of the lens and the background medium become −4.038 × 10

^{2}and 4.038 × 10

^{2}kg/mm

^{3}and those bulk moduli become −2.092 × 10

^{−2}and 2.092 × 10

^{−2}GPa, respectively. These can be obtained from (4) and (7).

_{x}and k

_{y}= 0). Modes 1 and 1’ are the lowest modes and those phases only vary in the x direction. Modes 2 and 4 are known as flat band [6,14,35]. Modes 3 and 5 are identical higher modes and acoustic waves propagate in the x direction while those resonate with one wavelength in y direction. Additionally, the refractive indices become the negative and positive because the slopes are the negative and positive, respectively.

_{NRI}= 120.36 mm and l

_{B}= 26.47 mm, and we determined those with the small number of calculations by using the parametric sweep function of COMSOL. The frequency dispersion characteristics are drawn in Figure 4a with the orange and green dots, respectively. It can be seen from the figure that the frequency of the intersection of those dots agrees with the theoretical model. Incidentally, the NRI lens operates by exciting the mode 3 with the mode 1’. It has also been confirmed by the references of [6,8,9,14,15,35,36] that the higher modes operating as the lens are excited by the lower modes of other structures or materials.

## 3. Results and Discussion

_{NRI}× 5Δd

_{NRI}= 500 mm × 125 mm, and the lens is sandwiched between two background media. Additionally, the background medium and the lens are composed of the structures in Figure 5a,b, respectively. An acoustic wave source is placed at the position where is separated by d

_{1}/2 = d

_{2}/2 = 2.5Δd

_{NRI}= 62.5 mm (2.5 cell) from the lens. The nodes on all boundaries are set to the sound absorption boundary to perfectly suppress reflected waves. Under the conditions above, we calculated the complex sound pressure distributions in the acoustic waveguide by using COMSOL.

_{NRI}= 98.0 mm and the broken line represents the typical theoretical trajectory of the incident acoustic wave from the source. The intersection of the line in the area of the designed NRI acoustic lens represents the theoretical position of the focus and that in the area of the right-hand side background medium represents the theoretical position of the refocus.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Distributed transmission-line model for the design of NRI acoustic metamaterials. The red arrows show the Γ-X pass (k

_{x}= k and k

_{y}= 0 or k

_{y}= k and k

_{x}= 0) in the Brillouin Zone.

**Figure 2.**Unit cell structures. (

**a**) Proposed meander acoustic waveguide unit cell structure for NRI acoustic metamaterials; (

**b**) Straight acoustic waveguide unit cell structure for background media.

**Figure 3.**The concept of an NRI acoustic lens. |n

_{NRI}| = n

_{B}, θ

_{B_i}= |θ

_{NRI_r}| = θ

_{NRI_i}= |θ

_{B_r}|, d

_{2}= d

_{1}are assumed in this figure.

**Figure 4.**Calculated frequency dispersion characteristics for the NRI acoustic lens and the background medium (BM) at the Γ-X pass (k = k

_{x}and k

_{y}= 0), and modes of acoustic waves in the acoustic waveguide. (

**a**) Dispersion characteristics; (

**b**) Distributions of instantaneous values of the sound pressure of modes of 1–5 in (

**a**); (

**c**) Distributions of instantaneous values of the sound pressure of mode 1’ in (

**a**). The intersection of the solid line and the broken one in (

**a**) corresponds to the operating frequency of the NRI acoustic lens. The propagation direction of (

**b**) assumes x direction.

**Figure 5.**Modified unit cell structures. (

**a**) For the NRI acoustic lens (Δd

_{NRI}= 25 mm, w

_{NRI}= 1.2 mm, and l

_{NRI}= 120.36 mm); (

**b**) For the background medium (Δd

_{B}= 25 mm, w

_{B}= 1.2 mm, and l

_{B}= 26.47 mm). These refractive indices are set as n

_{NRI}= −1.522 and n

_{B}= 1.522, respectively.

**Figure 6.**The setup for the full-wave simulation for the designed NRI acoustic lens (Δd

_{B}= Δd

_{NRI}= 25 mm). The size of the background medium and the lens are 20Δd

_{NRI}× 5Δd

_{NRI}= 500 mm × 125 mm, and are composed of the structures in Figure 5a,b, respectively. An acoustic wave source is placed at the position where it is separated by d

_{1}/2 = d

_{2}/2 = 2.5Δd

_{NRI}= 62.5 mm from the lens.

**Figure 7.**Calculated complex sound pressure distributions at 2.5 kHz. The wavelength is λ = 3.92Δd

_{NRI}= 98.0 mm. The broken line represents an example of the theoretical trajectory of the incident acoustic wave from the source. The intersection of the line in the area of the designed NRI acoustic lens represents the theoretical position of the focus and that in the area of the right-hand side background medium represents the theoretical position of the refocus. (

**a**) Amplitude; (

**b**) Phase.

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**MDPI and ACS Style**

Takegami, I.; Nagayama, T.; Fukushima, S.; Watanabe, T. Design Method of Acoustic Metamaterials for Negative Refractive Index Acoustic Lenses Based on the Transmission-Line Theory. *Crystals* **2022**, *12*, 1655.
https://doi.org/10.3390/cryst12111655

**AMA Style**

Takegami I, Nagayama T, Fukushima S, Watanabe T. Design Method of Acoustic Metamaterials for Negative Refractive Index Acoustic Lenses Based on the Transmission-Line Theory. *Crystals*. 2022; 12(11):1655.
https://doi.org/10.3390/cryst12111655

**Chicago/Turabian Style**

Takegami, Ibuki, Tsutomu Nagayama, Seiji Fukushima, and Toshio Watanabe. 2022. "Design Method of Acoustic Metamaterials for Negative Refractive Index Acoustic Lenses Based on the Transmission-Line Theory" *Crystals* 12, no. 11: 1655.
https://doi.org/10.3390/cryst12111655