The distribution of phase-shift gradient is an important factor in the study of high-order-mode diffraction. Here, the phase shift is described as the difference in wave phases between the incident wave and the transmitted wave changed by the metasurface. By designing a phase-gradient metasurface, which is equivalent to a periodic lattice, the diffraction mechanism [17, 22] in the metasurface can be described as sin θ tra / r e f − sin θ inc k = ξ + n G G , (1) where θ _inc is the angle of the incident wave, θ _tra/ref is the angle of the transmitted or reflected wave, k = 2π/λ is the wavenumber, λ is the wavelength (assuming that the incident area and transmitted area have the same medium), ξ is the phase gradient along the metasurface, n _G is an integer which represents the effect of periodicity, and G is the lattice constant of reciprocal lattice. In Eq. 1, the phase gradient ξ can be expressed as ξ = dϕ/dx ₂, where ϕ is the phase shift and x ₂ is the coordinate axis along the metasurface. ξ can be assumed as positive because the positive and negative phase gradients are physically equivalent.

To simplify the design of the metasurface, a supercell with length S in the metasurface is introduced by including m functional units as shown in Figure 1A to realize the distribution of the phase gradient over a complete 0 ∼ 2π range. Therefore, the phase shift of the jth functional unit in the supercell is set at ϕ j = 2 π j / m j = 1 , 2 , … , m . For the functional unit with a fixed width W, the width of the supercell is determined by including m function units, that is, S = mW. Thus, ξ can be written as the gradient of phase shifts in adjacent functional units, that is, ξ = Δϕ/W. In the supercell with m functional units, it can also be written as ξ = (mΔϕ)/(mW) = 2π/S. In addition, G can be written as G = 2π/p for a one-dimensional case, where p = S is the lattice constant of primitive lattice. So, G = 2π/S, and the values of ξ and G are naturally equal. For a single supercell, Eq. 1 is rewritten as sin θ inc + n λ S = sin θ tra / r e f , (2) where n = n _G +1 is the diffraction order n = 0 , ± 1 , ± 2 , … . λ/S in Eq. 2 plays a role similar to the phase gradient, so it is not distinguished from the phase gradient in the following sections. The positive and negative angles are specified in Figure 1A, which is the same as traditional refraction and reflection theories.

For the case of n = 1, Eq. 2 can be reduced to the GSL. For the case of n ≠ 1, the wave diffraction can be described as the result of multiple reflections of the wave in the functional unit. Figure 1B illustrates the process of wave propagating in the functional unit for T times to reach its boundary. For example, if T = 1, it means that the incident wave passes through the functional unit directly as the transmitted wave. If T = 2, it means that the incident wave has one-time reflection in the functional unit and then leaves it at the incident side to be the reflected wave. Considering two adjacent functional units in the supercell, if both have (T−1) times inside reflections, the difference between their phase shifts becomes the cumulative phase of T-time wave propagation, that is, Δ ϕ = T ϕ j + 1 − ϕ j = 2 π T / m . Thus, the +1st-order diffraction (GSL) can also be considered to have the shortest path with T = 1. For the case of n > 1, because λ/S is positive, the GSL will also be satisfied, if Eq. 2 is satisfied for the nth-order diffraction. However, the diffraction orders with n > 1 can be neglected, and the +1st order diffraction can replace them with the shortest path; this is because the wave diffraction prefers to follow the tendency for the shortest path of wave propagation [40]. Therefore, the actual range of n is reduced to n = +1, 0, − 1, − 2, ….

For the case of n ≤ 0, the difference between phase shifts of two adjacent functional units can also be written as Δϕ′ = 2πn/m according to Eq. 2, which is the nth-order diffraction wave. Furthermore, the relationship of phase shifts between two adjacent functional units can be established by using the plane wave expression method as e ^iΔϕ = e ^iΔϕ ′, that is, e i 2 π T / m = e i 2 π n / m . For the diffraction order n ≤ 0, a positive integer Z is introduced to lead 2πT/m = 2πn/m + 2πZ, which is simplified to T = n + mZ. Because the wave propagates following the shortest path in the functional unit, the number Z is taken as 1, and then a relationship in a supercell with m functional units can be established as T = m + n . (3)

If T is an odd number, the incident wave transmits through the supercell, and if T is an even number, the total reflection occurs. Consequently, by combining Eqs 2 and 3, the diffraction relation is summarized as sin θ inc + n λ m W = sin θ tra , T = m + n i s o d d , sin θ inc + n λ m W = sin θ ref , T = m + n i s e v e n . (4)

Therefore, the relationship between transmission/reflection of the wave and diffraction order n of the supercell is established by the wave propagating path, which is related to the addition of m and n. It means that the performance of wave propagation can be altered by changing the number of functional units m in a supercell or the diffraction order n.

According to Eq. 4, the transmission or reflection of a wave can be controlled by the number of functional units m when the diffraction order n is determined. Actually, m and n are not independent in a strict sense. The applicable diffraction order of the existing metasurface is set by the distribution of the phase gradient in the supercell with fixed m. Recent studies [23–27] only used different structures of metasurfaces to realize different functional units m and then obtain the desired diffraction orders. In order to obtain a variable number m in the same metasurface, a basic functional unit based on the electromechanical coupling effect is designed by attaching PZT patches on both surfaces of the host plate as shown in Figure 2A. In the functional unit, four PZT patches (orange) with thin electrodes (blue) are symmetrically attached on both surfaces of the host plate (gray). To realize the manipulation of Lamb waves in the plate, each PZT patch is polarized along three directions [37], which are marked with red arrows in Figure 2A, and connected to an adjustable inductance L through the electrode on its surface. The effective capacitance of the PZT patch C _p and the shunting inductance L form a capacitance–inductance resonant system to alter the equivalent stiffness of the PZT patch [41, 42]. By adjusting the value of shunting inductance L, the equivalent stiffness of the PZT patch can be adjusted accordingly, and then the propagation of Lamb waves can be manipulated by the functional unit.

In fact, it is difficult to realize continuous adjustment of inductance using a real inductor. An adjustable synthetic inductor is achieved by using Antonio’s circuit [42] as shown in Figure 2B, which consists of two operational amplifiers, a capacitor (C ₁), three resistors (R ₁, R ₂, and R ₃), and an adjustable resistor (R ₄). The equivalent inductance of the synthetic circuit is induced by L = R ₁ R ₃ R ₄ C ₁/R ₂, which is easily and continuously tuned by changing R ₄. The host plate in the functional unit is designed as l ₁ × l ₂ = 12 mm × 8 mm with the height h = 2 mm. The size of the PZT patch is l _p × l ₂ × h _p = 5 mm × 8 mm × 1 mm, and the gap between two PZT patches is l ₃ = 2 mm. The material parameters of the host plate (aluminum) and PZT patch (PZT-5H) are listed in Table 1, where the crystal symmetry of PZT-5H is 4 mm.

Without the loss of generality, the propagation of the anti-symmetric mode Lamb wave A₀ in the aluminum plate is studied. To quantitatively evaluate the capability of a functional unit for manipulating A₀-mode Lamb wave propagation, wave propagation in a host aluminum plate with one functional unit is considered, as shown in Figure 3A. To illustrate the propagation of the A₀-mode Lamb wave in the plate, the incident A₀-mode Lamb wave is excited at the left side of the functional unit, periodic boundary conditions are applied to both upper and lower sides of the host plate, and the perfectly matched layers (PMLs) are attached to both left and right sides of the host plate. The wave propagation process is calculated by the finite element method (FEM) via commercial software COMSOL Multiphysics. To obtain reliable calculation results, the functional unit is meshed by quadratic serendipity elements with a size smaller than one-tenth of the wavelength, and the mesh is appropriately encrypted near PZT patches. For the stable A₀-mode Lamb wave excited at a frequency of 14 kHz with the wavelength λ = 0.037 m, the transmission ratio, which is described as the ratio of amplitudes of out-of-plane displacements (|w|) for the transmitted signal to the incident signal, and the phase shifts are calculated as the shunting inductance L changing from 0.095 H to 0.140 H as shown in Figure 3B. The results show that the phase shifts of the functional unit cover a full 0 ∼ 2π range, and the transmission ratio is relatively high (above 0.8). Consequently, the functional unit can be an elementary unit of the supercell to achieve an arbitrary phase shift in a 0 ∼ 2π range by only adjusting the shunting inductance. For a metasurface fabricated by a series of proposed functional units, the number of functional units m in a supercell can be changed by appropriately selecting the phase shift of each functional unit, and the supercell makes the diffraction order to be changeable in the same metasurface.

In addition, Figure 3C illustrates the sum of the square of the transmission ratio |t|² and square of the reflection ratio (the ratio of |w| for the reflected signal to incident signal) |r|² to show the energy loss of the A₀-mode Lamb wave at frequency 14 kHz. The wave phase changes due to the local resonance of the PZT patch with a shunting circuit, which will inevitably consume energy. Therefore, the total energy is slightly less than 1, especially near the formant frequency. However, the sum is close to 1 with altering inductance L in general. It means that there is almost no strong energy loss and mode conversion of the wave by passing though the functional unit. Furthermore, the phase shifts, transmission ratios, and reflection ratios of the functional unit at an arbitrary frequency in the range from 10 to 30 kHz can also be calculated by FEM via commercial software COMSOL Multiphysics, and the effectivity of the functional unit to manipulate the A₀-mode Lamb wave by adjusting the shunting inductance L is verified.

If two A₀-mode Lamb waves are incident with two opposite angles about the normal line of the metasurface, their transmitted angles are still opposite to each other, so they should satisfy the following condition: sin − θ inc + n 1 λ S = sin θ tra , sin + θ inc + n 2 λ S = sin − θ tra , (6) and n ₁ = −n ₂, for realizing the symmetric transmission. Obviously, the symmetric transmission cannot be achieved by the GSL. Considering Eq. 5, only diffraction of +1st, −1st, and 0th orders exists in the metasurface, so that the diffraction orders of symmetric transmission in Eq. 6 are n ₁ = −n ₂ = 1. The incident wave with the negative incident angle −θ _inc is noted as NI and with the positive incident angle +θ _inc is noted as PI. Eq. 6 indicates that for the angles of NI and PI with axial symmetry about the normal line of the metasurface, their transmitted waves manipulated by the metasurface can have the axial symmetry and travel at the same side of incident waves.

According to Eq. 4, T = m + n should be an odd number to guarantee that the wave will transmit through the metasurface. For the critical incident angle of the +1st-order diffraction (GSL) θ _c1 = arcsin(1 − λ/S), the incident angle of NI should satisfy −θ _inc < θ _c1, while the angle of PI should satisfy +θ _inc > θ _c1 with the existence condition of −1st-order diffraction, −1 < sin(+θ _inc) − λ/S < 1.

There are 64 functional units to be arranged as the tunable metasurface in an aluminum plate as shown in Figure 3A for manipulating the A₀-mode Lamb wave at frequency 14 kHz. An interval l ₄ = 0.001 m is set between all adjacent functional units to ensure that the phase shift modulation of the wave by each functional unit is independent and free from interference. Therefore, the length of the supercell can be obtained as S = m(l ₂+l ₄). When the difference in phase shifts between two adjacent functional units is π/2, a supercell needs four functional units, m = 4, to form the distribution of phase shifts in the range of 0 ∼ 2π as shown in Figure 4A, so the metasurface totally includes 16 supercells. In Figure 4A, the orange rectangles represent the functional units. The length of the supercell with four functional units as shown in Figure 4A is S _m=4 = 0.036 m, and it satisfies the condition expressed in Eq. 5. The critical angle of +1st-order diffraction is θ _c1 = −1.59°, which means that most of the incident angles can realize the symmetric transmission. For the incident angles of 30° and 15°, the transmitted angles θ _tra can be calculated by Eq. 6, and the values are 31.86° (θ _inc = 30°) and 50.26° (θ _inc = 15°), respectively. FEM is conducted to verify the analytical result, and the numerical results of out-of-plane displacement fields about incident and transmitted waves with NI and PI of 30° are shown in Figures 4B1, B2, respectively, while Figures 4C1, C2 show the results with NI and PI of 15°, respectively. The white arrows in Figures 4B, C point out the designed propagation directions of incident and transmitted waves. Compared with the displacement fields of wave motion, the propagation of refraction waves is in good agreement with designs. The results shown in Figures 4B, C illustrate that two incident angles with the same value but contrary signs can manipulate the refraction waves by the metasurface to transmit waves at the same sides along the designed transmitted angles with axial symmetry about the normal line of the metasurface.

In general, for the angles of NI and PI with axial symmetry about the normal line of the metasurface, their transmitted waves manipulated by the metasurface can be without axial symmetry and propagate along arbitrary directions and are called as asymmetric transmission. Here, we consider one special case of generalized asymmetric transmission, which is that the angles of NI and PI have axial symmetry about the normal line of the metasurface; however, NI can pass through the metasurface, while PI is reflected by the metasurface. The corresponding condition for this case is sin − θ inc + n 1 λ S = sin θ tra , sin + θ inc + n 2 λ S = sin − θ ref , (7) where n ₁ = 1 and n ₂ = −1 are the diffraction orders of NI and PI, respectively. In Eq. 7, the angles of NI and PI still have axial symmetry, and even the angles of transmitted/reflected waves maintain numerical axial symmetry. NI transmits through the metasurface following the GSL, and according to Eq. 3, T = m + n ₂ = m−1 should be an even number to make sure that the PI will be reflected by the metasurface. Figure 5A illustrates the distribution of phase shifts along the supercell with three functional units, m = 3. For manipulating the A₀-mode Lamb wave at frequency 14 kHz, according to Eq. 5, the ratio λ/S _m=3 = 1.370 makes sure that only −1st, +1st, and 0th orders of diffraction can exist in the metasurface. The critical angle of the +1st-order diffraction can be calculated as θ _c1 = −21.74°. Different from the symmetric transmission of NI and PI in the supercell with m = 4, the limitation to the incident angle should be considered to realize asymmetric transmission. If the incident angle is smaller than θ _c1 = −21.74°, according to Eq. 7, only n ₁ = n ₂ = 0 satisfies the diffraction equations and only reflected waves occur for both NI and PI.

Figures 5B, C show the out-of-plane displacement fields of A₀-mode Lamb waves calculated by FEM for the cases with the NI and PI angles of 30° and 45°, respectively. The transmitted or reflected angles θ _tra/ref of both cases can be calculated according to Eq. 7 to be 60.50° (θ _inc = 30°) and 41.55° (θ _inc = 45°), respectively. The white arrows in Figures 5B, C illustrate the designed propagation directions of waves, and the results are in good agreement with the designed angles. In Figures 5B1, B2, the angles of NI and PI are axial symmetry with 30°, but for the NI in Figure 5B1, the wave transmits through the metasurface with the designed angle 60.50°, while for the PI in Figure 5B2, the wave reflects backward by the metasurface with the designed angle −60.50°. The same tendency can be observed by the NI and PI cases of axial symmetry with angle 45° in Figures 5C1, C2. In addition, there are very low fluctuations to be observed in Figures  4B, C, 5B, C. There are two reasons that cause the chaotic disturbance. One is because the phase gradient of the realistic metasurface is discrete, so a few waves directly pass through the metasurface. The other one is because the 0th-order diffraction still exists in the symmetric and asymmetric transmissions based on +1st- and −1st- orders of diffraction but with negligible low amplitudes.

It should be noted that the transmission of NI and reflection of PI in asymmetric transmission are determined for the positive phase gradient. Their functions can be exchanged when the sign of the phase gradient is inverted, which is also applied in symmetric transmission. The symmetric and asymmetric transmissions are realized by the same metasurface based on the tunability of diffraction orders. It also means that the function is tunable to meet different demands and conquer the main drawback of the existing metasurfaces.

In Section 3, the relation between wavelength and length of the supercell is restricted by Eq. 5 to avoid the generation of higher order diffraction in the metasurface. The 0th-order diffraction has the longest path among all the diffraction orders with n ≤ 0 in the same supercell according to Eq. 3. Thus, if the 0th-order diffraction is used for designing the metasurface with a higher amplitude, all the other order diffraction should be suppressed. Based on Eq. 2, the supercell should be designed to satisfy the condition of λ/S > 2 to make sure that no diffraction wave exists except n = 0. As an example, for the A₀-mode Lamb wave at frequency 14 kHz, the supercell with two functional units, m = 2, can satisfy the condition λ/S _m=2 = 2.06 > 2. Accordingly, when n = 0, T = m + n = 2 is an even number and Eq. 4 becomes sin θ _inc = sin θ _ref, which means that the wave only reflects directly along the same angle as the incident one no matter which angle the incident wave is. Therefore, the omnidirectional reflection wave is easily realized by the 0th-order diffraction.

For the supercell with two functional units, m = 2, the distribution of phase shifts is illustrated in Figure 6A. There are three cases of A₀-mode Lamb waves at frequency 14 kHz with the angles of NI as 0°, 30°, and 60°, respectively, to be calculated by FEM, and their out-of-plane displacement fields are shown in Figure 6B. The white arrows in Figure 6B mark the designed directions of incident waves and reflected waves. The results in Figure 6B are in good agreement with the designed arrows and show nearly total reflections in these cases.

To further demonstrate that wave reflection is independent of the incident angle, a triangle-shaped metasurface is constructed by the supercell with m = 2. A point source (S ₁) is placed in a random position inside the triangle-shaped metasurface, and the energy field (represented by |w|²) is shown in Figure 6C1. The result shows that the wave energy is enclosed in the triangle, and no wave is transmitted anywhere outside the triangle. When the point source (S ₂) is moved outside the triangle-shaped metasurface, the wave energy can be isolated by the triangle as the energy field (|w|²) as shown in Figure 6C2. Therefore, the metasurface with the capability of omnidirectional reflection can make the wave energy to be concentrated inside the structure or isolated outside the structure, which exhibits exciting prospects for wave isolation.

The supercell with m = 2 shows the strong ability to reflect the wave energy with an arbitrary incident angle. To deeply utilize the property, a channel is constructed by two metasurfaces as the route for wave propagation. When the wave propagates in the channel, it will be reflected by the metasurface boundary regardless of the incident angle. The channel can provide only one way for the wave to travel, despite its shape, that is, straight or curved. Figure 7 shows two examples of channelized waveguides for the A₀-mode Lamb wave at frequency 14 kHz: one is a straight line and the other is in “L” shape. Figure 7A shows the energy field of out-of-plane displacement (|w|²) in a straight channel formed by metasurfaces based on the supercell with m = 2. The white dot with a red circle marks the location of the wave source. The width of the channel is selected smaller than the wavelength as 0 .8λ to avoid the generation of wave interference [27]. The result shows that the wave is trapped in the channel and propagates straight forward. If the channel is turned to form an “L”-shaped channel by metasurfaces, the energy field shows that the wave travels through the channel with high amplitude as shown in Figure 7B. Therefore, for two metasurfaces with omnidirectional reflection using the 0th-order diffraction, different shapes of waveguides can be designed to realize wave propagation along any designed route in the plate.