Controlling the flow of waves and guiding them to directions and locations of interest has long been an important area of research. Potential applications include imaging, communication, energy harvesting, audio systems, etc. Among the solutions for wave controlling problems, metamaterials demonstrate promising performance by achieving effective material properties that does not exist in nature with engineered subwavelength resonating structures in both electromagnetics Zheludev and Kivshar (2012) and acoustics (Cummer et al., 2016; Ma and Sheng, 2016). As the 2D realizations of metamaterials, metasurfaces utilize the Huygens principle to discretize the surface into subwavelength unit cells, each having designed phase and/or amplitude responses (Yu et al., 2011; Sun et al., 2012; Kildishev et al., 2013; Chen et al., 2016; Glybovski et al., 2016). In acoustics, most commonly explored metasurfaces are phase gradient metasurfaces (Li et al., 2014; Tang et al., 2014; Xie et al., 2014; Zhu et al., 2016; Li et al., 2018; Zhu et al., 2018). They are composed of phase shifters that can generate discrete values of phase shifts in the range of (0, 2π). Those phase shifters are arranged in the metasurface to form the designed phase gradient for wavefront manipulation (Zhao et al., 2013; Li et al., 2015; Wang et al., 2016), acoustic absorption (Mei et al., 2012; Ma et al., 2014; Li et al., 2016), cloaking (Zhang et al., 2013; Esfahlani et al., 2016), hologram (Xie et al., 2016), etc.

There is a common property in the acoustic phase shifters that are previously designed: the phase shift values provided by the phase shifters are frequency dependent. For example, in the work Xie et al. (2014) and Liang and Li (2012), the phase shifters are formed by curved waveguide channels that generate phase delays proportional to the total wave path lengths. For a phase shifter with a fixed length, the time delay after passing through the waveguide channel is a constant as the frequency changes, which means the phase shift is a function of frequency. Most of the previously designed phase shifters belong to this type. In this paper, we will introduce a new type of phase shifters for acoustic reflected waves in which the phase shift values do not change with frequency. Our work is inspired by the coupled transmission line design in electrical circuits (Quirarte and Starski, 1993; Guo et al., 2006; Zhang et al., 2011). Here we extend the idea from electrical circuits into acoustics with the help of acoustic transmission line theory (Beranek, 1954; Zhang et al., 2009). The designed phase shifters are then applied to metasurfaces for manipulation of acoustic reflected waves.

In the previous acoustic metasurface designs, the resolution of spatial discretization is usually larger than five unit cells per period in order to form smooth phase gradient. Such dense arrangement of unit cells limits the lateral resolution of the phase gradient. The complexity in design and fabrication of phase shifters also increases with the increasing resolution of phase response discretization. Here we propose a new way to achieve acoustic wave control using only two discrete phase shift values, 0 and π.

The rest of the paper is organized as the following: Section 2 describes the theoretical design and simulation of a broadband differential phase shifter with π phase difference between its two components. Section 3 describes the design of acoustic reflected wave metasurfaces for wave steering and focusing based on the broadband differential phase shifter designed in section 2. Section 4 summarizes the results and provide further discussion about the advantages and applications of the phase shifter and metasurface designs.

First of all, we will demonstrate the design of a broadband differential phase shifter that generates π phase difference between its two components. Those two components will be the key elements for the binary metasurfaces we will introduce later in this work.

In Zhang et al. (2011), the authors presented the design of a Schiffman differential phase shifter for microwave circuits, which consists of two components. The output phase of each individual component is frequency-dependent. However, the output phase difference between the two components is nearly constant over a broadband frequency range. Similar as in Zhang et al. (2011), we consider a differential phase shifter for reflected acoustic wave, in which the reflected wave phase difference between two components is constant over a broadband frequency range. Here, one component is a straight acoustic waveguide with width d and length l ₀ (Figure 1A), and the other component is a straight acoustic waveguide (width d and length l ₁) with two identical Helmholtz Resonator (HR) side branches (neck r ₁, h ₁ and cavity r ₂, h ₂) (Figure 1B). We call the first component “tube” and the other component “resonator”.

The effective circuit model for the tube is a delay line of length 2l ₀, as shown in Figure 1A. The factor “2” is due to the fact that wave is reflected at the bottom of the tube and the total travel path length is twice the length of the tube. Assume P _in is the incident acoustic pressure, and P out e j φ 0 is the reflected acoustic pressure. We have: φ 0 = − 2 k l 0 . (1) The delay line generates a phase shift −2kl ₀, where k = 2πf/c is the wavenumber in air.

As shown in Figure 1B, the effective circuit model for the resonator is a delay line with length 2l ₁ that generates phase shift value of −2kl ₁, and four side branches that are identical. Each side branch is composed of a capacitor and an inductor. The characteristic impedance of a tube with width d is Z c = c ρ d , where ρ is the density of the material in the tube and c is the sound velocity inside the tube. The impedance of the side branch is Z s = 1 j ω C + j ω L . Here, C = r 2 h 2 ρ c 2 is the effective capacitance of the cavity in the HR and L = ρ h eq r 1 is the effective inductance of the neck of the HR, where h _eq is the modified neck length that is determined by the real neck length h ₁ and the end loading. We will determine h _eq according to simulation of the resonator. The effective acoustic impedance at the location of side branch can be written as the parallel of Z _c and Z _s , which is Z = 1 j ω C + j ω L Z c Z c + 1 j ω C + j ω L . (2) For the resonator component, We also assume P _in as the incident pressure, P out ′ e j φ 1 as the reflected pressure field. We have: φ 1 = − 2 k l 1 + 4 ∠ 2 Z Z c + Z . (3) Here 2 Z Z c + Z is the transmission coefficient T at the location of the HR when the wave passes by one of the HR side branches.

The differential phase shift between the tube and the resonator is Δ φ = φ 1 − φ 0 = − 2 k ( l 1 − l 0 ) + 4 ∠ 2 Z Z c + Z . (4) By tuning l ₀, l ₁ and the geometric parameters that determines the value of Z _c and Z, different phase difference value between the tube and the resonator can be achieved. In our design, the π phase difference is achieved with the following parameters: l ₀ = 50 mm, l ₁ = 46 mm, d = 5 mm, r ₁ = 1 mm, h _eq = 1.595 mm, r ₂ = 20 mm, h ₂ = 5 mm. (Assume the background material in the tubes is air (c = 343 m/s, ρ = 1.29 kg/m³)). Here, the value of h _eq is determined by matching the helmholtz resonance ( f = 1 2 π L C = 4340 Hz) of the theoretical calculation with the simulation.

The comparisons between theoretical calculation and FEM frequency domain simulation are shown in Figure 2. FEM frequency domain simulation is implemented in COMSOL 5.5. The background material and the geometrical parameters are set as the same values in the theoretical calculation, except for l ₁. In theoretical calculation, l ₁ is 46 mm, while in simulation, l ₁ is fine-tuned to 45 mm to generate π phase difference. In Figure 2A, we plotted the theoretical and simulated phase responses (φ ₀ and φ ₁) of the tube component and the resonator component. We can see that the theoretical calculation and simulation give the same phase response for the tube component. For the resonator component, the theoretical calculation and the simulation are nearly the same in the frequency range from 10 KHz to 16 KHz (the orange shaded region in Figure 2A), which is between the two resonance peaks at f = 8,620 Hz and f = 17.6 KHz found in simulation. Those two resonance peaks correspond to the first and second resonances of the cavity with r ₂ = 20 mm and h ₂ = 5 mm in the Helmholtz resonator. The difference between the theoretical and simulated resonator phase responses (the red curve and the black curve in Figure 2A) is due to the fact that the theoretical calculation neglects higher order resonances of the Helmholtz resonator cavity and the neck, and also neglects the small reflections at the entrance of each Helmholtz resonator side branch in the resonator component. In the frequency range from 10 KHz to 16 kHz, the phase differences (Δφ) between the tube and the resonator from both theoretical calculation and simulation are around π, as plotted in Figure 2C. We also plot the theoretical and simulated amplitude responses of both the resonator component ( P out ′ / P in ) and tube component (P _out/P _in) in Figure 2B. The phase shifters are used in a reflection setting. If there are no loss in the system, we expect all the components to have the amplitude response of one. However, when we calculate the theoretical amplitude response of the resonator, we only consider the transmitted energy at each Helmholtz resonator side branch and neglects the reflected part. As a result, the theoretical amplitude response of the resonator component is smaller than one in the range from 10 KHz to 16 KHz.

The tube component and the resonator component in this π differential phase shifter will be used as the 0 and π phase modulators respectively to construct binary metasurfaces for reflected wave steering and focusing, which are two of the basic functions in wave control.

In summary, we developed a method to design broadband differential phase shifters. Nearly constant phase difference between the two components in a phase shifter is generated over a range of frequencies. Using this method, we designed a broadband π differential phase shifter using a tube and a resonator. The nearly constant π phase difference between the tube and the resonator is preserved in frequency range from 10 KHz to 16 KHz. The tube and the resonator are further applied to the design of binary acoustic metasurfaces for reflected wave steering and focusing.

There are two main advantages of the metasurface design based on the broadband differential phase shifters. Firstly, we propose to achieve acoustic wave control using only two discrete phase shift values, 0 and π. Thus the metasurfaces we designed are much easier to fabricate and assemble, and the spatial phase variation frequency can be much larger than previous designs. Second, our phase shifter design is broadband. The phase shift values do not change with frequency. We can first fabricate the tubes and resonators as individual elements, and then tune the reflected wave angle (in wave steering function), focal location (in wave focusing function), and the working frequencies that those functions are achieved by simply changing the locations of those individual elements in the metasurface. We don’t need to re-fabricate the individual phase shifter elements for different frequencies.

The phase shifters can be fabricated with 3D printing techniques or template-based molding process. For applications in air, most of the solid materials can be used for the fabrication, such as metals, plastics, polymers, etc. The phase shifters serve as the basic building elements for metasurfaces that can be applied to in-door sound shaping, acoustic imaging, communication and energy harvesting applications.