Anomalous diffractions of water waves in parity-controlled metagratings

Abstract

Owing to the formal analogy between their governing equations, concepts from electromagnetic wave physics have been successfully extended to water-wave systems. Here, we propose and experimentally demonstrate the water-wave metagratings (WMGs) capable of wavefront modulation based on the generalized Snell’s law. These WMGs generate anomalous diffraction, including both retroreflection and negative refraction, by engineering the integer parity of supercells. As a proof of concept, we realize broadband water-wave focusing using WMGs. This work opens new avenues for compact and tunable water-wave devices, with potential applications in wave-energy harvesting and marine engineering.

Graphical abstract

Keywords

water wave manipulations / metagratings / wave diffractions

1 Introduction

The exploration and manipulation of water waves hold significant scientific and practical importance across multiple disciplines, such as fluid mechanics and analog physics [1−10], sustainable energy harvesting [11, 12], and marine engineering [13]. Water surface waves have attracted growing attention since their governing wave equation is comparable to Maxwell’s equations. This similarity enables the direct application of photonic concepts to hydrodynamic systems [1]. Drawing inspiration from electromagnetic metamaterials and metasurfaces [14−25], researchers have designed a variety of functional water wave devices. For example, these include invisibility cloaks [6, 26], water wave diodes [27], water-wave metamaterial shifters [28], concentrators [2] and anisotropic media [29]. These intriguing water-wave devices can be designed via effective medium theory and transformation optics [1]. Recently, inspired by photonic crystals, diverse devices composed of periodic cylinder arrays have been developed to enable the demonstration of water wave refraction and focusing [30, 31], and fast water waves [4], with potential applications in enhanced energy capture. Despite these advances, most current designs rely on bulky two-dimensional (2D) architectures, limiting their integration and scalability. Moreover, significant propagation losses persist, arising from parasitic scattering at the liquid–solid interface and impedance mismatches at device boundaries [32−34]. These losses remain major obstacles to practical implementation. Motivated by analogies with spoof surface plasmons in structured surfaces [35], the unidirectional propagation of water wave polaritons at the interface of a one-dimensional (1D) groove array is revealed [36]. While such 1D-array structures offer promising avenues for compact water wave manipulation [37−39], experimental demonstrations of flexible, arbitrary control using simple geometries are still lacking, although this method is highly desirable in this field.

In this work, we propose and experimentally demonstrate water-wave metagratings (WMGs) that precisely control the wave diffractions via engineered gradient water depths of WMGs [Fig. 1(a)]. By tailoring the phase distributions at the interface, we leverage the generalized Snell’s law (GSL) to manipulate the transmitted and reflected wavefronts of water waves arbitrarily. Crucially, we reveal that the integer parity of unit-cell numbers per supercell dictates whether the WMGs operate in transmission or reflection mode. We achieve the intriguing retroflection and negative refraction in odd-parity WMGs by simply inverse the incident angle. As a proof of concept, we demonstrate broadband water-wave focusing, showcasing the WMGs’ potential for wave-energy harvesting and coastal protection

2 Method

The GSL enables arbitrary control of the wave diffractions by engineering the spatial-dependent phase shift of meta-gratings [40, 41]. Inspired by acoustic metagratings [42, 43], we design WMGs composed of periodic supercells for water wavefront modulations. Unlike conventional acoustic metagratings, which use acoustic hard boundaries in each unit, we propose constructing WMGs by exploiting the water depth of the unit cells to reduce phase distortion caused by liquid‒solid coupling. This is achieved by using the bump structure to control the water depth (see Fig. S1 in Supplementary Note 1), and the overall structure is immersed in a constant water depth of h_0 [see Fig. 1(b)]. The phase speed (v_p) of water waves in 2D scenarios is given by

(1)v_p=\sqrt{{\displaystyle \frac{g}{k_0}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(k_{0}h\right)},

where g is gravitational acceleration and where k_0=\frac{2\pi }{{\lambda }_0} and {\lambda }_0 are the wavenumber and wavelength of water waves in the background region, respectively. In the WMG region, the gap width of the unit cells is much smaller than the wavelength (i.e., a_0\ll {\lambda }_0); thus, we assume that only the fundamental mode of water waves can be supported inside these unit cells. Borrowing from the concept of electromagnetic waves, we define the effective refractive index of a water wave in WMG unit cells related to that of the background as

(2)\beta \left(x\right)={\displaystyle \frac{v_{p0}}{v_{pW}}}=\sqrt{{\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(k_0{h}_0\right)}{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(k_W{h}_W\left(x\right)\right)}}},

where h_W\left(x\right) is the spatially dependent water depth in the WMG region and k_W=\beta \left(x\right)k_0 is the wavenumber of water waves in the WMG region. v_{p0} and v_{pW} represent the phase speed of water waves in the background region and WMGs region, respectively. The background water depth is set by default as h_0=15 mm. To derive the analytical geometrical structures of WMGs, we first design the effective refractive index of each unit cell based on the desired phase modulation, such as 2π/3 or 4π/3. In particular, the constant-index profile is transformed to a linearly gradient-index design. The continuous index around the interface of WMGs can considerably reduce wave reflections since the interface impedance is matched (see Supplementary Note 1). To further reduce the propagation loss, the thickness (d) of WMGs should be as small as possible, while a smaller d can result in the excitation of higher-order modes in the unit cell due to nonadiabatic mode evolution. Consequently, there is a compromise between the insertion loss and the adiabatic evolution of water-wave modes. Here, we carefully choose d={\lambda }_0 to balance these two effects. To steer the outgoing wave, the transmitted phase across a supercell covers a complete range of 2\pi, such as 0, \frac{2\pi }{3},\mathrm{a}\mathrm{n}\mathrm{d}\frac{4\pi }{3}. In each supercell, the maximum index of the j-th unit cell is given as {\beta }_{m,j}=1+2\left(j-1\right){\lambda }_0/(md), and the transmitted phase can be modulated by adjusting {\beta }_{m,j} of each unit cell. Then, according to Eq. (2), the analytical height profile of each unit cell can be obtained:

(3)h_W^{\prime }\left(x\right)=h_0-{\displaystyle \frac{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left({\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(k_0{h}_0\right)}{{\beta \left(x\right)}^2}}\right)}{\beta (x)k_0}}.

The typical bump structures of unit cells in a supercell are shown in Fig. 1(b).

The diffraction behaviour of our designed WMGs follows a similar principle to that of acoustic metagratings [41] (see details in Supplementary Note 2). Before the water wave departs from WMGs, it oscillates back and forth within each unit cell, as shown in Fig. 1(c). Let the number of single path traversals be defined as L. Generally, the relationship between L and diffraction order n is [42]

(4)L=m+n.

Then the diffraction would be transmission/reflection when L is odd/even. Furthermore, for the case of n=\pm 1, we can reverse the transmission or reflection of diffraction order by adjusting the parity of the number of slots m in each supercell. In other words, the diffraction order is correlated with the anomalous reflection (refraction) when m is odd (even).

Without loss of generality, we focus on exploring the wavefront modulations via two types of WMGs, m=3 and m=4 (see details in Supplementary Note 3). We utilize COMSOL MULTIPHYSICS to conduct simulations to assess the effectiveness of the designed WMGs in controlling water waves. For m=3, when the incident angle is {\theta }_{in}=-30°, the water wave is retroreflected exactly. The WMG is operated in reflection mode, and incident waves are manipulated to produce a diffraction order of n=1 termed anomalous reflection (R_1), as shown in panel (i) of Fig. 2(a). When the incident angle of water waves is {\theta }_{in}=30°, strong anomalous refraction with a diffraction order of n=-1 is obtained (T_{-1}), as shown in panel (ii) of Fig. 2(a). The diffraction properties of WMGs for different types of m=4 are very different. As shown in Fig. 2(b), only the transmission mode is activated for both {\theta }_{in}=\pm 30°. Therefore, the retroreflection/transmission mode of WMGs can be flexibly modulated by varying the parity of the unit cells in each supercell.

For a comprehensive study of WMGs, we simulate the diffraction effect of water waves passing through these two types of WMGs over a range of incident angles [-45°,45°], as shown in Figs. 2(c)−(e). In general, anomalous reflection and refraction in WMGs occur not only at {\theta }_{in}=\pm 30° but also at a wide range of incident angles. Specifically, for the case of m=3, the diffraction wave of +1diffraction order is strongly reflected, with more than 90% for {\theta }_{in}\in [-45°,0°). However, the transmission coefficient is greater than 90% for {\theta }_{in}\in (0°,45°] at a diffraction wave of −1 diffraction order. Consequently, for the case of m=3, water waves can be modulated to produce anomalous reflection or refraction by simply switching the incident angle. For the case where m=4, as shown in Fig. 2(e), the transmission coefficient of the +1 diffraction order is more than 90% for {\theta }_{in}\in [-45°,0°), which is quite similar to the results when m = 3. For {\theta }_{in}\in (0°,45°], the transmission coefficient of the -1 diffraction order is more than 90%, which is drastically different from the results of m = 3. Therefore, for the WMGs of m=4, water waves are manipulated to produce anomalous transmission for all incident angles between {\theta }_{in}\in [-45°,45°]. We also derive the analytical results via coupled mode theory (see details in Supplementary Note 4), which agree well with the simulation results. Furthermore, we also calculate the results of WMGs with the types of m=5 and m=6, and the results again prove similar diffraction results depending on the parity of the unit cells (see detail in Supplementary Note 5). These results unambiguously demonstrate the universality of the parity effect in WMGs and serve as the guideline for further experiments.

3 Results and discussion

We then design water wave experiments on the basis of the theoretical calculations. We quantitatively obtain schlieren images by fast Fourier demodulation of a checkered backdrop [44], and the experimental setup is shown in Fig. 3(a). The aforementioned two types of WMGs were printed from acrylic, as shown in Fig. 3(b). To reduce the surface tension of water, we used deionized water with added surfactant (Kodak Photo-Flo), with a volume ratio of deionized water to surfactant was 50:1. Figures 3(c) and (d) shows the wave diffraction effect of odd-parity WMGs (m = 3). The incident wave on the WMGs is retroreflected when the angle of incidence is {\theta }_{in}=-30°, as shown in Fig. 3(c). Conversely, water waves can be manipulated to exhibit anomalous refraction (T_{-1}) when the incident angle {\theta }_{in}=30° [Fig. 3(d)]. For the case of m=4, only anomalous refraction is observed for {\theta }_{in}=\pm 30°, as shown in Figs. 3(e) and (f). These experimental observations agree with the theoretical prediction discussed in Section 2. We find that the diffraction efficiency of water waves is much lower than the theoretical prediction. For example, the measured maximum efficiency is 7.2% in the case of Fig. 3(e) and 56.3% in the case of Fig. 3(f), which are significantly lower than the theoretical results. This discrepancy results mainly from the inevitable scattering losses in WMG units and the dissipation loss of water waves (see Supplementary Note 6). Overall, the designed WMGs can not only be used to manipulate water waves effectively but also be easily tailored to switch the retroreflection/refraction of higher-order diffractions.

Furthermore, we analyzed and experimentally tested the broad incident angle response of WMGs. The iso-frequency contours are employed to intuitively uncover the underlying physics. In Fig. 4(a), if there is no phase shift along the interface, the iso-frequency contours in free space are merely indicated by two identical black circles, where the left one is for the incident wave, while the right one is for the refracted wave. Once the phase shifts are introduced and water waves propagate to WMG at an incident angle of 30°, then at the refracted side, the black circle will be shifted up with a displacement of \xi, as marked by the light blue circle. Then the direction of the transmitted wave is marked by a deep blue arrow, and the reflected wave is marked by a red arrow. Therefore, we derive the relationship between the diffraction angle and incident angle with such an analytical approach, as shown in Fig. 4(b). The typical water-wave field profiles are shown in Fig. 4(c). The red and blue points represent experimental data when m=3 and m=4. When the angle of incident waves is 13°, the angle of transmission waves is -48° [Fig. 4(c1)]. When the angle of incident waves is 36°, the angle of transmission waves is -19° [Fig. 4(c2)]. These observed results agree well with theoretical calculations. Notably, for m=3 type WMG, when the angle of incidence is -30°, the diffraction wave is exactly retroreflected as discussed previously. To unambiguously demonstrate the retroreflection effect, we shift the incidence angle to -15° and -45° and observe the wave field profiles, as shown in Fig. 4(d). It is clear that the reflected wavefront is no longer regular due to the interference between the incident and reflected waves. We also simulate other diffraction orders by tailoring the structures of WMGs in Supplementary Note 7.

Owing to the versatility of the gradient water depth of WMGs [45], we demonstrate a water wave focusing phenomenon via WMGs. The phase distribution of the focal WMG satisfies \varphi \left(y\right)={-k}_0(\sqrt{y^2+f^2}-f) [46], which is widely used to manipulate incident planar wavefronts into spherical waves, where f is the focusing length of the WMGs. Figure 5(a) shows the designed phase distribution along the y-axis (black curve) and modulated wavefront (gray curve). This schematic shows that an incident wave, after passing through the engineered WMGs, can accumulate the required phase shift and is directed toward the focal point. The designed WMG consists of 7 supercells separated by a perfect rigid body, and the two sides of the middle unit cell are symmetrical. In this work, we use the following parameters to fabricate devices: a_0=1.8 cm, d=1.5 cm, w=2 mm, and water depth h_0=1.5 cm. To reduce impedance mismatch, a gradient water depth is also utilized in each unit cell. We choose the maximum refractive index of seven unit-cells (C1 to C7) is {\beta }_{m,1}=1, {\beta }_{m,2}=2.9, {\beta }_{m,3}=4.4, {\beta }_{m,4}=4.9, {\beta }_{m,5}=4.4, {\beta }_{m,6}=2.9, and {\beta }_{m,7}=1, respectively. Figure 5(b) depicts the simulated phase change of each unit cell to a different frequency (analytical results are shown in Supplementary Note 8), which reveals the potential of broadband focusing. We also calculate the phase delay curves for different water wave frequencies (see Supplementary Note 8), which show quasi-quadratic curve features. Consequently, the broadband response in focusing WMG can be obtained.

Figures 5(c)−(f) show the typical wave focusing profiles of WMGs when the excited frequency is 5.2, 7.2, 7.7, and 8.2 Hz, respectively. A quasi-plane wave is normally incident on the WMGs from below, with its energy modulated and focused at a predefined focal point. In particular, clear wavefront convergence is evident at 7.2 Hz and 7.7 Hz [Figs. 5(d) and (e)], with the strongest focusing occurring at 7.2 Hz, where the phase modulation is most accurately realized. Despite being further from the optimal frequency, the wavefronts at 5.2 Hz and 8.2 Hz still exhibit well-defined focusing. Overall, the focusing phenomenon is experimentally sustained across a broader range of 5.2−8.7 Hz (see Fig. S6 in Supplementary Note 9). Taking 7.2 Hz as the center frequency, the relative bandwidth is calculated as \frac{\mathrm{\Delta }f}{f_c} = 50.4%, indicating broadband focusing of water waves enabled by the WMGs.

4 Conclusion

In conclusion, WMGs based on gradient water depth exhibit remarkable capabilities in manipulating wave propagation. In particular, they enable the excitation of high-order diffraction modes, representing a notable advancement in water-wave control. In general, the scattering efficiency can be decreased with the increase of diffraction orders. In our WMG designs, we mainly focus on the +1 and −1 order diffractions, which show high diffraction efficiency theoretically when neglecting the propagation loss of water waves. Through meticulous design and experimental validation, we demonstrate that the integer parity of WMGs controls the wave diffraction beyond the conventional GSL rule. In the experiment, the water waves experienced significant losses from the scattering losses of WMG units and dissipation in the confined channels, resulting in transmission efficiency much lower than the theoretical calculations. It would be promising to design more effective modulation units, such as using the logarithmic triangle and cubic root curve, in which the adiabatic evolution and impedance mismatch are improved at the interface and inside the channels. Additionally, the surface treatments and materials choices can be exploited to optimize the transmission efficiency. We further develop the focusing WMGs that efficiently concentrate broadband wave energy at a focal point. Such devices provide practical strategies for harnessing and redirecting ocean wave energy. For instance, anomalous diffraction WMGs offer a means to steer wave direction, relevant to coastal protection and wave energy harvesting. Meanwhile, focusing structures present a promising route toward enhanced energy concentration, contributing to the sustainable and efficient utilization of marine energy resources.

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