Acoustic Focusing with Intensity Modulation Based on Sub-Wavelength Waveguide Array

Abstract

Acoustic focusing with intensity modulation plays an important role in biomedical and life sciences. In this work, we propose a new approach for simultaneous phase and amplitude manipulation in sub-wavelength coupled resonant units, which has not been reported so far. Based on the equivalent impedance and refractive index modulation induced by the change of geometry, arbitrary amplitude response from 0 to 1 and phase shift from 0 to 2π is realized. Thus, the acoustic focusing with intensity modulation can be achieved via waveguide array. Herein, the focal length can be adjusted by alternating the length of supercell, and the whole system can work in a broadband of 0.872f₀–1.075f₀. By introducing the coding method, the thermal viscosity loss is reduced, and the wavefront modulation can be more accurate. Compared with previous works, our approach has the advantages of simple design and broadband response, which may have promising applications in acoustic communication, non-destructive testing, and acoustic holography.

1. Introduction

Studies regarding metamaterial have become a hot topic in the last several years [1,2,3,4]. Among them, acoustic focusing is of great interest given its numerous promising applications such as biomedical imaging and non-destructive testing, among others. Traditionally, acoustic focusing can be achieved using phased array transducers, which has complex circuits and high costs. In recent years, the development of an acoustic artificial period structure, also known as acoustic metamaterial, has provided new means for the realization of acoustic focusing [5].

It has been demonstrated that acoustic gradient index (GRIN) lens [6,7,8] can be realized by gradually changing the size of phononic crystals. However, the size of GRIN lens is relatively large at low frequency due to the “wavelength-size” limitation in phononic crystals. To break this limitation, some researchers have proposed a new concept of metasurface (MS) in sub-wavelength scale, which can achieve wavefront manipulation using ultrathin structures. Although various previous works have realized acoustic focusing by taking advantage of MS with phase modulation (PM) [9,10,11,12,13,14,15,16,17,18], there are relatively few studies on the manipulation of focusing intensity. Essentially, the realization of acoustic focusing with intensity modulation in sub-wavelength scale needs to design an acoustic MS that can achieve both amplitude modulation (AM) and PM simultaneously. In 2017, Tian et al. [19] proposed a MS with decoupled AM and PM, which can manipulate the intensity of the transmitted waves. Nevertheless, it is based on the decomposition of the meta-cell into two units, one responsible for tuning the phase and the other serving as an amplitude modulator, which may increase the complexity of the structure. In 2018, Ghaffarivardavagh et al. [20] present a space-coiling metamaterial, enabling acoustic control with simultaneous AM and PM. However, they adopted four degrees of freedom including the width of the incident port, the thickness of coil’s wall, the number of coils, and the common radio of channel width to manipulate the wavefront, which may bring a limitation to the application. Therefore, it is important to explore a new kind of MS to achieve acoustic focusing with intensity modulation by utilizing the simple unit cell on a sub-wavelength scale.

In this work, we demonstrate the possibility of simultaneous AM and PM in sub-wavelength coupled resonant units, so as to achieve acoustic focusing with intensity modulation based on waveguide array. The units proposed in our work are composed of five resonators with different cross-sectional lengths. According to the effective medium method, we verify that the AM is due to the presence of the acoustic reactance term, and the PM owes to the shift of equivalent velocity. In addition, the whole acoustic system can work in a broadband of 0.872f₀–1.075 f₀. By introducing the coding method, the thermal viscosity loss could be reduced and the intensity modulation of focusing could be more accurate.

2. Design and Result

As shown in Figure 1a, the waveguide unit (L * H = 0.5λ * 0.1λ) proposed in this work is composed of five resonators, whose cross-sectional length varies with the gradient and can be regarded as a geometric sequence. L and H are the length and height of the waveguide unit. Following the reference [20], the structure featuring a gradient in channel spacing possesses complex acoustic impedance, which is a key to realizing simultaneous phase and amplitude modulation. Thus, we also design a gradient waveguide. If we define ‘c’ as the common ratio, the cross-sectional length and neck length of five resonators can be expressed as: a₁ = (1 − c)*L/(1 − c⁵), a₂ = ca₁, a₃ = c²a₁, a₄ = c³a₁, a₅ = c⁴a₁; b₁ = a₁/2, b₂ = cb₁, b₃ = c²b₁, b₄ = c³b₁, b₅ = c⁴b₁. It should be noted that different values of H₁ and H₂ lead to various effective acoustic velocities, so as to achieve different phase delays. Thus, the phase modulation can be realized by altering the values of H₁ and H₂. Thus, the heights of channel and resonators are adjustable and can be expressed as H₁ = H/w and H₂ = (H − H/w)/2, respectively. Where ‘w’ is a scale factor to control the height of H₁ and H₂. The waveguide unit is filled with air and the incident wavelength is 0.1 m. Of note, the wavefront modulation can be realized by changing the geometric parameters of the waveguide unit. Therefore, the number, length, or position of the resonators can be regarded as the degrees of freedom to manipulate the transmission waves. To simplify the design method, we only adopt common ratio ‘c’ and scale factor ‘w’ to manipulate the transmitted waves in this work. Herein, we fix the number of resonant unit (n = 5) and define the neck length as b_n = a_n/2.

To calculate the amplitude and phase response of the waveguide unit, finite element software “COMSOL Multiphysics” is adopted. The frequency domain in the pressure acoustic module is utilized, and the background of the medium is selected as air: ρ_air = 1.21 kg/m³, c_air = 343 m/s. The amplitude and phase responses can be obtained by “$\left|p_t\right|$” and “$-i\ln \left[p_t/\left|p_t\right|\right]$”, respectively, with $p_t$ being the acoustic pressure of the transmission wave. As shown in Figure 1b,c, although full PM can be achieved by manipulating the scale factor ‘w’, full PM and full AM are unable to be realized simultaneously by utilizing only one degree of freedom (‘c’ or ‘w’). Therefore, we have to add an extra degree of freedom and investigate the possibility of using both common ratio ‘c’ and scale factor ‘w’ to achieve complete wavefront modulation. As shown in Figure 1d,c, when the scale factor ‘w’ is not large enough (w < 18), the full PM and AM cannot be achieved even though two degrees of freedom are adopted here. With the increase of ‘w’, the range of phase and amplitude shift is expanded, that is to say, the ability of PM and AM could be improved with the decrease of channel width. In addition, the common ratio ‘c’ also has a great influence on wavefront modulation. For PM, it is hard to achieve 0–0.4π and 1.6π–2π without high-value of common ratio ‘c’; For AM, we have to increase the value of ‘c’ to obtain a lower transmittance (for example, the value of ‘c’ needs to exceed 5 if we want to obtain a transmittance around 0.1) with the altering of ‘w’. Therefore, full PM and AM could be realized independently when the value of ‘w’ and ‘c’ is large enough. In addition, as shown in Figure 1f, almost any combination of phase and amplitude can be realized accordingly by appropriately selecting geometric parameters. Each phase distribution in the range of 0–2π corresponds to an amplitude coverage of 0–1, which can be utilized to achieve both PM and AM simultaneously, so as to realize acoustic focusing with intensity modulation.

When the resonators shown in Figure 1a have the same cross-sectional length, that is c = 1 and a₁ = a₂ = a₃ = a₄ = a₅, the waveguide array becomes a traditional MS similar to the previous research. Considering a general metasurface with a thickness of L_m, equivalent acoustic impedance of Z_m and wave number of K_m placed in infinite space with acoustic impedance of Z, the following equation can be deduced [20]:

$$\theta =\arccos \left[Tcos\left(K_m{L}_m\right)\right]$$

(1)

where θ represents the transmitted phase. The complete wavefront modulation cannot be achieved under this condition owing to the existence of coupling between the transmission phase and amplitude shown in Equation (1). To avoid this situation, the presence of the acoustic reactance term is necessary [20].

The waveguide unit with several resonators proposed in our work, instead of having a constant resonator length, features a change in length at each step is governed by a constant common ratio, defined as: $c=\frac{a_n}{a_{n-1}}$. In which $a_n$ and $a_{n-1}$ are the resonator length of $n^{\mathrm{th} }$ and $n-1^{\mathrm{th} }$ resonator, respectively. By this definition, the waveguide can be well approximated as an exponential horn with a flare constant of $m=\frac{N}{L_{\mathrm{eff}}}\ln \left(c\right)$ for structures with $c>1$. In which N is the number of resonator and $L_{\mathrm{eff}}$ is the effective length of the waveguide. The acoustic transmission coefficient can be solved as

$$\begin{array}{ll}T=\left|T\right|{\mathrm{e}}^{i\theta }=& 4/(-\frac{kL}{\beta a_{\mathrm{out} }}+1+\frac{a_{\mathrm{in} }}{a_{\mathrm{out} }}-\frac{k^{a_{\mathrm{in} }}}{\beta L}\\ & +i\frac{\gamma }{\beta }\left(1-\frac{a_{ \mathrm{in} }}{a_{\mathrm{out} }}\right)){\mathrm{e}}^{\left(\gamma +i\beta \right)L_{\mathrm{eff} }+\left(\frac{kL}{\beta a_{\mathrm{out} }}+1+\frac{a_{\mathrm{in} }}{a_{\mathrm{out} }}+\frac{k{a}_{\mathrm{in} }}{\beta L}-i\frac{\gamma }{\beta }\left(1-\frac{a_{\mathrm{in} }}{a_{\mathrm{out} }}\right)\right){\mathrm{e}}^{\left(\gamma -i\beta \right)L_{\mathrm{eff} }}}\end{array}$$

with a_in and a_out are the resonator length for the first and last resonators, respectively, L is the length of the unit, k is wave number, and $\gamma$ and $\beta$ are defined as $\gamma =\frac{m}{2}$ and $\beta =\frac{1}{2}\sqrt{4k^2-m^2}$. Thus, it can be obtained that the transmission amplitude will be bounded by $\frac{2}{{\mathrm{c}}^{N/2}+{\mathrm{c}}^{-N/2}}\ge \left|T\right|\ge \frac{2}{{\mathrm{c}}^{N/2}+{\mathrm{c}}^{-N/2}}\left|\cos \left(\theta \right)\right|$. Therefore, waveguide unit featuring a gradient in resonator length has the ability to modulate the transmission amplitude.

Herein, the effective acoustic parameters could be obtained by calculating the reflection and transmission coefficients of the waveguide unit [21] (see Appendix A). As shown in Figure 2a, the effective acoustic velocity and mass density are different under various channel widths. Meanwhile, as illustrated in Figure 2b, the effective acoustic velocity and density depends on frequency as well. To verify the reliability of effective medium (EM), a data point with w = 6 is selected and the corresponding acoustic parameters are substituted into the EM at 3430 Hz. As shown in Figure 2c, the distribution of acoustic field in EM is almost the same as that in waveguide unit, which indicates that the proposed EM method has a good applicability. Note that the difference in the sound pressure between the resonator part and its effective medium can be neglected because our aim is to retrieve the equivalent properties of the waveguide unit, which only needs to ensure the agreement between the waveguide unit and its effective medium in transmission field and reflection field. In other words, the effective medium only needs to provide the same amplitude and phase of reflection and transmission waves, which can be verified in Figure 2c. Therefore, we could obtain the effective acoustic impedance with different values of ‘c’, as shown in Figure 3. For c = 1.0, or equally the traditional unit cell, the acoustic reactance term exists only at resonance coupling of the incident wavelength and the structure, which cannot reduce the amplitude sharply in a wide range. Compared with ‘c = 1′, the lower bound of transmittance expands gradually with the rising of ‘c’, and the full amplitude coverage (0–1) can be realized around c = 1.5. That is to say, as the value of ‘c’ increases, the amplitude of transmitted waves could be manipulated effectively because the complex acoustic impedance can be realized by the gradient change in cross-section of the resonators. Moreover, with the rising of ‘c’, the broad amplitude coverage could be achieved with a smaller value of ‘w’. For instance, in the case of c = 1.2 (Figure 3c), the transmission coefficient can be reduced to 0.3 only when w = 30. However, in the cases of c = 1.2–1.8 (Figure 3d–i), the transmission coefficient of 0.3 could be achieved in lower value of ‘w’. It is not hard to imagine when the value of ‘c’ is large enough, such as c = 10 (shown in Figure 1e), the wide range of AM may be realized by making use of different values of ‘w’ independently, which provides more degrees of freedom for complete wavefront modulation.

As mentioned in the work [22], the key to realize PM is to obtain the continuously gradient phase (CGP), which is equivalent to making the refractive index of the EM satisfy the following requirement:

$$n_i=1+\left(i-1\right){\lambda }_0/\left(pm\right)$$

(2)

where n_i represents the refractive index of the ith EM; λ₀ represents the incident wavelength; p represents the length of supercell composed of EM; m represents the number of rigid walls in a supercell. We could also consider each EM is filled with materials having different wave velocities:

$$c_i=c_{Air}/n_i$$

(3)

where c_i represents the velocity in the ith EM. According to Equation (3), the effective refractive index can be obtained by inversion of the effective velocity. Thus, it can be concluded that the phase shift is due to the change of effective velocity in waveguide unit. Different values of ‘w’ and ‘c’ lead to various acoustic velocities, so as to achieve a wide range of phase shift in individual parameter combination of ‘c’ and ‘w’.

Please note that the device proposed in reference [20] is a space-coiling metamaterial, through which the phase and amplitude modulations are able to be achieved simultaneously as well. However, they adopted four degrees of freedom including the width of incident port, the thickness of coil’s wall, the number of coils, and the common radio of channel width to manipulate the wavefront. As a comparison, only two degrees of freedom, common ratio ‘c’ and scale factor ‘w’, are needed in our work to obtain the simultaneous modulation of phase and amplitude for transmitted wavefront. Consequently, the approach we proposed is simpler than that in reference [20] because the parameter optimization is more convenient in this work, which greatly simplifies the design philosophy and is more suitable for practical applications. Moreover, the focusing intensity shown in Ref. [20] is fixed, and they did not mention the concept of intensity modulation of the focus, while it can be realized in our work, further promoting the application of the design.

Herein, as shown in Figure 4, five waveguide units with CGP in the range of 0.4π–2.0π in steps of 0.4π is designed. Analogous to the generalized Snell’s law in optics, if an acoustic wave with the frequency of f is incident from one medium to another, the generalized Snell’s law in acoustics can be expressed as follows:

$$k_t\sin {\theta }_t-k_i\sin {\theta }_i=d\varphi \left(x\right)/dx$$

(4)

where θ_i and θ_t represent the angle of incidence and refraction, respectively; k_i = 2πf/c_i, k_t = 2πf/c_t, c_i and c_t represent the acoustic velocity of the incident medium and the transmitted medium. When the plane wave is normally incident on the waveguide array (θ_i = 0), Equation (4) can be rewritten as:

$$k_t\sin {\theta }_t=d\varphi \left(x\right)/dx$$

(5)

As mentioned before, the EM of each unit have different wave velocities. Thus, the wave fronts inside different waveguide units accumulate different phase changes (ϕ_i) when they pass though the waveguide array. The phase difference between the ‘acoustic paths’ in neighboring waveguide units is a constant and is given by Δϕ = 2π/5 = 0.4π. Thus, the phase change between the neighboring supercells composed of waveguide units is always 2π, which is required to maintain a continuous and smooth wavefront pattern of the transmitted wave. We assume that the length of supercell is p, then we can use a constant transverse phase gradient (dϕ/dx = 2π/p) to describe the phase changes on the surface of the waveguide array composed of supercells. Such a transverse phase gradient gives rise to oblique transmitted waves due to the generalized Snell’s law of refraction in Equation (5), and the refraction angle can be manipulated by changing the parameter p:

$$\sin {\theta }_t=2\pi /k_{t}p={\lambda }_0/p$$

(6)

Thus, we could construct a PM structure by arraying waveguide units with CGP, and the acoustic focusing phenomenon can be achieved through constructive interference. Furthermore, according to

Table 1. The specific phase/amplitude value and corresponding parameters of c-w.

Wavefront Modulation (c-w)		Phase (φ/2π)
		0.2	0.4	0.6	0.8	1.0
Amplitude	0.2	0.81–29.5	1.46–20.0	1.7–19.5.0	1.76–13.0	3.7–11.6
	0.4	1.18–28.5	1.27–20.0	1.548–17.0	1.53–8.0	2.8–8.2
	0.6	0.87–28.0	1.15–22.0	1.43–16.0	1.4–13.0	2.1–7.2
	0.8	1.10–28.0	1.08–23.2	1.365–16.0	1.25–13.0	1.53–8.0
	1.0	1.10–5.0	1.10–2.0	1.10–18.5	1.10–14.0	1.10–9.0

, any single phase in a CGP corresponds to five continuously gradient amplitudes (CGA), which can manipulate the transmissivity of acoustic waves. Herein, finite element software (COMSOL Multiphysics) is used to calculate the acoustic intensity field of the structure. The frequency domain in pressure acoustic module is adopted, and the background of the medium is selected as air: ρ_air = 1.21 kg/m³, c_air = 343 m/s. The outside boundaries are set as plane wave radiation conditions, so as to avoid the unwanted reflections. As shown in Figure 5a,d, acoustic focusing phenomena with different intensities are obtained by altering the parameters of ‘c’ and ‘w’, which satisfies our purpose of manipulating the acoustic focusing intensity by utilizing only two degrees of freedom of the structure. In addition, as shown in Figure 5b, the focal length could be modulated by alternating the length of supercell, which provides a way to achieve long-distance acoustic beam collimation.

Another advantage of the waveguide unit shown in our work is its broadband in wavefront modulation. As shown in Figure 5c, acoustic focusing can be achieved within the band range of 1930 Hz–6430 Hz due to the coupling resonances given by five resonators in a waveguide unit. Of note, the focusing quality decreases in high frequency, which partly attributes to the non-strict linear response between the phase shift and the incident wavelength as shown in Figure 5e. On the other hand, the transmissivity of some waveguide units is reduced rapidly at high frequency, which also affects the quality of focus. Nevertheless, as shown in grey area in Figure 5e, it has a reasonably good performance when 0.872 ≤ f/f₀ ≤ 1.075 because the transmission coefficient is over 0.8 and the spanning range of phase shift expands to 80% of 2π within this band, which reveals that the acoustic system can work in a broadband.

When sound wave propagates in narrow buildings and geometric structures, thermal loss and viscous loss will lead to sound wave attenuation. Specifically, the loss occurs in the acoustic thermal boundary layer and viscous boundary layer near the wall. In order to establish a model that matches the actual situation, we need to consider this phenomenon and evaluate the impact of these losses on the thermoviscous acoustic system. To eliminate the thermal viscosity loss (TVL), we propose the coding method to construct the lens. Only two kinds of waveguide units with opposite phase responses are required, and the huge TVL caused by some harsh parameters of the units with narrow width can be avoided in this case. The key to realizing the acoustic focusing is to make the acoustic waves reaching the focus have the same phase, that is, constructive interference occurs (see Appendix B). Based on this theory, we can modulate the wavefront by arranging the position of two coding units with phase difference of π. To make acoustic waves converge at a certain point, the placement positions of two coding units shall meet the following requirements:

$$x_{i+1}=\sqrt{{\left({\lambda }_0/2+\sqrt{q^2+x_i^2}\right)}^2-q^2}$$

(7)

where x_i+1 and x_i represent the position of two coding units; q represents the focal length. Under this condition, as shown in Figure 6a, we only need two waveguide units with a phase difference of π, which provides an opportunity to improve the minimum size of the structure to reduce the TVL and facilitate sample preparation. Thus, we could select two coding units with low value of ‘w’ to achieve acoustic focusing. According to Equation (7), we set q = 0.6 m, λ = 0.1 m, x₀ = 0 m here. Then, the position of other coding units can be determined in the following order: x₁ = 0.25000 m, x₂ = 0.36056 m, x₃ = 0.45000 m, x₄ = 0.52915 m, x₅ = 0.60208 m, x₆ = 0.67082 m. To numerically calculate the lossless structure and lossy structure, we adopt the ‘Pressure acoustic module’ and ‘thermoviscous acoustic module’, respectively, in finite element software of COMOSL. As shown in Figure 6b, acoustic focusing is realized based on coding method using only two waveguide units with ‘w = 2′ and ‘w = 9.5′, respectively. By comparing the intensity of acoustic pressure in both lossy and lossless system shown in Figure 6c, it can be confirmed that the TVL is small and can be ignored here.

As another interesting application of the structure proposed in this work, the capability of wavefront shaping may be improved by means of the MS composed of waveguide units with simultaneous amplitude and phase modulation (APM) compared with traditional single-phase modulation (SPM). As an illustrative example, in the case of Airy beam shown in Figure 7, both APM and SPM have been considered. Figure 7a,b are the discrete amplitude and phase information for waveguide array. For Airy beam based on SPM, some unwanted waves along other directions are generated in the transmission field due to the parasitic diffraction [23]. While for Airy beam based on APM, perfect Airy beam is produced. The pressure fields shown in Figure 7c,d indicate that, compared with SPM method, the MS designed by APM could improve the beam quality.

3. Conclusions

In conclusion, we examined the possibility of simultaneous PM and AM in sub-wavelength coupled resonant units, so as to achieve acoustic focusing with intensity modulation based on waveguide array. By introducing a waveguide unit with variable cross-sections and channel widths, the complete wavefront modulation could be achieved utilizing only two degrees of freedom of the structure. The metalens proposed in our work has a broadband response and a low dissipation. Although dynamic manipulation of the wavefront is not realized in this work, it is expected to construct active metamaterial by utilizing the waveguide unit in the future. The results of complete wavefront modulation illustrated in our work are not only limited to acoustic focusing, but also can be extended to other phenomena, such as airy beam and splitting beam, which may have potential applications, including acoustic holography, communication, and ultrasonic medicine.