Extraordinary sound transmission and collimation effect through a subwavelength slit with surface engineering of phase gradient

Abstract

Subwavelength slit diffraction, as a fundamental diffraction phenomenon, is of great significance in various fields. However, governed by the symmetry of coupling modes and diverse diffraction orders, designing a device that can independently achieve excellent extraordinary transmission performance and a desirable collimation effect remains a challenging task. Here, we propose a new paradigm for manipulating acoustic slit diffraction by engineering a reversed phase gradient on one side of the structural surface. For instance, under forward incidence, the extraordinary transmission efficiency can be significantly enhanced, where the phase gradient plays a key role in exciting unidirectional evanescent waves along the surface; under backward incidence, the beam collimation effect can be well maintained, as the phase gradient functions to suppress unwanted diffraction orders. We further experimentally demonstrate this novel phenomenon. Our work enriches the methods for manipulating acoustic single-slit diffraction and shows potential applications in sensors and holography.

Graphical abstract

Keywords

phase gradient / extraordinary transmission / beam collimation

1 Introduction

Single subwavelength metallic holes or slits play a crucial role in nano-optics, with their wave diffraction behaviors being core to the most advanced techniques, such as near-field scanning optical microscopy [1, 2], breaking the diffraction limit [3], molecular spectroscopy [4, 5], and laser devices [6, 7]. The underlying origin of subwavelength slit diffraction is also a physical cornerstone in many fundamental phenomena [8−13], including cavity resonances [14], waveguide modes [15, 16], surface plasmon polaritons (SPPs) [17], and spoof SPPs [18]. Typically, the most typical method previously used for controlling single-slit diffraction is based on periodic gratings, through which the incident light can be efficiently coupled to SPPs, thereby modifying the distribution of electromagnetic (EM) field at the metal surface [19−31]. Recently, it has been reported that the concept of phase gradient metasurfaces (PGMs) provides an alternative approach with significant advantages [32−36], enabling various effects such as directional beam control [37−39] and extraordinary optical transmission (EOT) [29, 40]. The mechanism benefits from the new degree of freedom of the phase gradient introduced by the well-designed metasurfaces, which can break the symmetry of wave coupling between the incident light and surface waves [42, 43], resulting in the unidirectional excitations of SPPs. Although a great deal of progress has been made in optics, very little work has been reported on the use of the concept of the phase gradient to manipulate the single-slit diffraction of acoustic waves. Unlike the electromagnetic wave, there is no cut-off wavelength for sound waves passing through subwavelength slits [44−48], and the physical mechanism of how the phase gradient manipulates slit diffraction in the acoustics still needs further investigation.

In this work, we investigate theoretically and experimentally how to manipulate the single-slit diffraction effect of sound waves with the additional degree of freedom of phase gradient. In the studies, the phase gradient is introduced by subwavelength air grooves with gradient depths (i.e., the PGMs), which are placed around the slit, thereby enabling the phase gradient to effectively guide and reshape sound waves. It is found that for forward incidence, i.e., the PGM is on the incident side, the extraordinary transmission efficiency can be significantly enhanced. For backward incidence, i.e., the PGM is on the transmission side, the beam collimation effect can be supported well without undesired diffraction orders. Compared with previous studies based on Fabry−Pérot (FP) resonances or periodic gratings, the energy utilization efficiency and modulation performance are greatly improved or enhanced. These results provide a versatile way to control acoustic slit transmission, offering potential applications in sound focusing, holographic and sensing elements.

2 Model and analysis

To illustrate our idea, Fig. 1 shows the proposed model, where the subwavelength slit is surrounded by well-designed phase gradient metagratings, i.e., PGMs. The PGMs on both sides are spatially symmetric with respect to the slit center, and the period of the supercell is p with m = 4 sub-cells, each sub-cell equipped with width w and subperiod a. To meet the conditions of the phase gradient, i.e., the reflection phase difference between two adjacent sub-cells should conform to \mathrm{\Delta }\phi =2\pi /m. Thus, the depth of well-designed air grooves is given as h_i=h_1+\left(\mathrm{i}-1\right)\mathrm{\Delta }h, where h₁ is the variable initial depth and \mathrm{\Delta }h=\pi /k_{0}m(k_0=2\pi /\lambda is the wave vector in free space). Hence, in this studied model, the additional momentum by the phase gradient along the two sides of the PGMs is: \mathit{\xi }=-\mathrm{d}\phi (x)/\mathrm{d}x=-2\pi /p=-\xi for x < 0 and \mathit{\xi }=+\mathrm{d}\phi (x)/\mathrm{d}x=+2\pi /p=+\xi for x > 0, respectively, where the “−” and “+” indicate that the direction of the phase gradient along the x direction.

For the case of forward incidence, the phase gradient enables an extraordinary acoustic transmission (EAT) effect, as shown in Fig. 1(a). As the incident wave enters the grooves, reflects at the bottom of each groove, and returns to the air-PGMs interface, the spatially accumulated reflection phase will be obtained along the surface. The phase gradient will provide a transverse wave vector to the incoming acoustic wave for efficient conversion from propagation waves (PWs) to evanescent waves (EWs). Different from common methods, such PGMs can result in a directional excitation of intrinsic EWs, because the phase gradient breaks the coupling symmetry, leading to the EW energy flow unidirectional toward the slit, and thereby effectively improving the incident energy coupling into the slit. Based on the generalized Snell’s law [32, 33], the wave vector along the x direction for both sides of the surface should satisfy the following relations:

(1)

where θ_in denotes the incident angle, G = 2π/p is the reciprocal lattice vector, and k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{L}} (k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{R}}) is the parallel wave vector of the EWs for the left (right) side. The integer v, where v = 0 is the lowest order here due to the phase gradient [32]. It can be clearly seen that the EWs exhibit opposite directions but toward the single-slit, thus enhancing the transmission efficiency. Further, the physical evolution process of excited directional EWs [29] can be visually described, as shown in Fig. 1(b). The phase gradient provided an additional momentum compensation, i.e., ξ, which breaks the coupling symmetry along the surface.

For the case of backward incidence, the phase gradient ensures a collimation effect, as shown in Fig. 1(c). As the incident waves pass through the subwavelength slit, its single-slit diffraction properties will be greatly modified by the PGMs surrounding the slit, producing phase gradient-dependent diffraction pattern in the far field. In particular, due to the phase gradient, the high-wave-vector components of the diffraction wave around the slit can be converted into propagating waves. According to the principle of reciprocity [49], the diffraction obeys the following process:

(2)

where θ_out denotes the diffraction angle to the free space, G = 2π/p is the reciprocal lattice vector, and k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{L}} (k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{R}}) is the wave vector of the EWs for the left (right) side. To obtain a collimated beam along the propagation direction (i.e., {\theta }_{\mathrm{o}\mathrm{u}\mathrm{t}}=0^{\circ }), the phase gradient should be opposite along the surface to satisfy the conditions of k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{L}}=\xi +\nu G and k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{R}}=\xi +\nu G on the left and right sides of the model. Typically, due to G = ξ, Eq. (2) can be re-expressed as k_0\sin {\theta }_{\mathrm{o}\mathrm{u}\mathrm{t}}=k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{L}}-n\xi (x < 0) and k_0\sin {\theta }_{\mathrm{o}\mathrm{u}\mathrm{t}}=n\xi -k_{\mathrm{E}\mathrm{W}\mathrm{s}}^{\mathrm{R}} (x > 0) with n = v + 1 [29]. Note that n = 1 (v = 0) is the lowest diffraction order of the PGMs. More importantly, this indicates that n = 1 is more easily coupled than other diffraction orders. Hence, the well-designed PGMs can freely select the needed diffraction orders by suppressing undesired diffraction to obtain a better collimation effect. Such processes of wave vector conversion from EWs to PWs in momentum space can be clearly described, as shown in Fig. 1(d).

3 Results and discussion

Figure 2(a) shows the experimental setup, where the sample [see Fig. 2(b)] is placed inside two acrylic plates to form a two-dimensional environment, and the sponges are surrounded all around to absorb noise and prevent scattering interference. The working wavelength is λ = 5 cm, corresponding to the frequency of 6860 Hz. The sample is fabricated by 3D-printing, where the selected material is resin, and the supercell contains m = 4 different depth air grooves to cover the 2π abrupt phase (see the enlarged box). The slight imperfections of the speaker array, residual noise, and fabrication errors are negligible in the present measurements. The relationship of the reflection (red line) and phase shift (blue line) for different depths of air grooves is calculated as shown in Fig. 2(c), where four different depths of air grooves are selected to obtain the discrete phase shift cover of 2π, and the measured results match well with the simulation results. The period of p = 4.8 cm, h₁ = 2.7 cm, h₂ = 3.325 cm, h₃ = 3.95 cm, and h₄ = 4.575 cm. The width of the single-slit is 1.2 cm; each sub-cell has a width w = 1 cm, and the sub-period is a = 1.2 cm; the thickness of the PGMs is 5.5 cm. The phase gradient introduced is \xi =1.04k_0>k_0.

For the case of forward incidence, Fig. 3(a) theoretically shows the transmission spectra for different phase gradient ξ and frequencies, where two modes result in EAT effect. The EAT marked the black dashed line is caused by the FP resonance of the single-slit, which is independent of the phase gradient. The mode marked by the green dashed line comes from the directional EWs excited by the well-designed phase gradient. In particular, at ξ = 1.04k₀ marked by the white dashed line, the corresponding results are replotted in Fig. 3(b), where two peaks appear at f₁ = 5190 Hz corresponding to FP resonances and f₂ = 6860 Hz corresponding to PGTMs. Compared with the single-slit alone (see the back curve), the introduced phase gradient leads to great enhancement of transmission not only at the position where the original FP resonance occurred, but also at the position of PGMs. Figure 3(c) shows the acoustic pressure field distribution at f₂, where the transmission side has obvious sound energy passing through the subwavelength slit, i.e., the EAT effect. To further confirm the pressure field at the transmission side, both the simulation results and experimental results are shown in Fig. 3(d), which are well-matched and confirm that the EAT effect occurred. Because the exit side does not have a phase gradient structure, the field distribution appears as cylindrical wave radiation. The amplitude distribution trends from simulation and experiment at y = −7.5 cm in Fig. 3(e) verify this observation. Further, Fig. 3(f) shows the calculated transmission of acoustic waves incident on the subwavelength slit for forward incidence, with different surface textures at the optimal groove depth. In detail, when the depth of the surface groove textures conforms to phase gradient, i.e., the model of “PGM”, ξ ≠ 0, when h₁ = 2.7 cm, p = 4.8 cm, the maximum transmission T_max = 0.67; but for the cases of surface groove texture without phase gradient, i.e., the models of “Grating_A” and “Grating_B”, ξ = 0, when h₁ = 3.01 cm, p = 4.8 cm (lattice match), T_max = 0.57; when h₁ = 1.65 cm, p = 1.2 cm (lattice mismatch), T_max = 0.25; and when the slit surrounding without surface groove texture, i.e., “Slab”, its T_max = 0.16. Thus, the scheme of a single-slit surrounded by PGMs has a better EAT effect than ordinary gratings.

For backward incidence, Fig. 4(a) shows the evolution of angular spectra for different frequencies. One can see that the phase gradient in the exit side can effectively control the wavefront directionally, with the diffraction energy mainly limited to the angle range from approximately −25° to 25°. Meanwhile, the side lobes are almost suppressed. Compared with the ordinary approach without PGMs [see Fig. 4(b)], although the collimation effect is also achieved, the energy is more dispersed. The reason is that the introduced phase gradient effectively suppresses the undesired diffraction orders [32, 49]. When the EWs mode is bound at the PGMs, it will be diffracted into free space via surface momentum matching engineering, and the radiation angles are determined by Eq. (2). We calculated the relationship between the diffraction angles and the ratio of p/λ, and the results are shown in Fig. 4(c). The lowest diffraction order in the PGM case is not n = 0, but n = 1, corresponding to v = 0 in Eq. (2), which is preferred to be diffracted. Hence, the diffraction is dominated by the first order, while the other diffraction orders are suppressed. Figures 4(d) and (e) show the far-field patterns at f₁ and f₂, respectively. Obviously, the PGMs greatly enhance the collimation effect in both cases. Finally, we calculated the transmission of the single slit while the PGM is on the transmission side, as shown in Fig. 4(f), where the PGM primarily plays a key role in reshaping the diffraction patterns rather than enhancing the efficiency.

Moreover, Fig. 5 illustrates the collimation performance at two frequencies, where Fig. 5(a) is for f₁ and Fig. 5(b) is for f₂. Thanks to the phase gradient having a certain bandwidth, the diffraction pattern in Fig. 5(a) still shows a clear directionality, but the acoustic energy gathers first and then gradually disperses as it evolves along the y-direction. While at the operating frequency f₂, the phase gradient can effectively suppress higher-order diffraction so that the energy intensity is mainly concentrated in the central area, i.e., the collimation effect. To verify these features, we further show the evolution process of the field intensity along the y-direction in Fig. 5(c), where the red line and blue line are acquired from x = 0 cm at f₁ and f₂, respectively. In detail, the trend of the red line appears to oscillate first and then stabilize, which means that it is affected by undesired diffraction orders. In comparison, the blue line gradually diminishes during the evolution process due to its primary contribution coming from a well-designed allowed diffraction order. Figure 5(d) shows the simulated and experimental results of the acoustic intensity at a distance of 40λ from the PGMs, where the red line is the working frequency at f₁, blue line is the working frequency at f₂. The experimental results and simulation results match well. Clearly, the introduced phase gradient, as shown by the blue line, produced better collimation performance.

4 Conclusions

In conclusion, we have theoretically analyzed and experimentally observed the EAT effect and collimation effect. These effects are believed to be attributed to the asymmetric coupling of the incident wave with the surface EWs or suppression of the undesired diffraction orders. The diffraction patterns are extremely dependent on the relative position between the incident wave and the phase gradient. Moreover, by calculating the performance of different corrugated metagratings, the “PGM” scheme has been proven to have significant advantages in enhancing transmission efficiency and directionality. Our work offers promising possibilities for effectively guiding and reshaping acoustic slit diffraction patterns, which may enable the design of high-performance acoustic elements.

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