If a phase gradient is introduced at the interface of two media through a metasurface, disrupting the original phase continuity at the interface, then the direction of the transmitted wave no longer satisfies Snell’s law, but will follow the GSL proposed by Yu et al in 2011, written in the following form [4]: k 1 ⁡ sin ⁡ θ t − k 1 ⁡ sin ⁡ θ i = ξ (1) Where, i , t subscripts denote the incident and transmitted; k 1 is the wave number of the incident wave and has k 1 = 2 π / λ 1 , λ 1 denotes the incident wave wavelength; ξ denotes the phase gradient, and ξ = d ϕ / d x .

If the incident angle θ i is larger than the critical angle θ s (the critical angle will make sin ⁡ θ t = 1 ), it will produce the same diffraction phenomenon as the optical/acoustic grid structure, producing diffracted waves of different orders, as shown in Figure 1A. Thus, Xie et al. introduced the higher-order diffraction term into the classical GSL transmission law by analogy and induction, giving birth to a modified GSL based on the higher-order diffraction term, expressed in the following form [6]: k 1 ⁡ sin ⁡ θ t − k 1 ⁡ sin ⁡ θ i = ξ + n G G (2) Where G denotes the reciprocal vector, which is numerically the same as the phase gradient ξ , but is formed for different reasons, the former due to the periodic nature of the structure and the latter due to the discrete distribution of the transmitted phase; n G is the higher order diffraction term, which takes the value of n G ∈ ( 0 , − 1 , − 2 , ... , − N G ) . In some recent articles, it is also expressed 1 + n G as n , taking the value n ∈ ( 1,0 , − 1 , − 2 , ... , − N ) , the phase gradient ξ and the reciprocal vector G are written in the form of numerical equality ξ = G , so that the above equation can be written as the following formula [30]: k 1 ⁡ sin ⁡ θ t − k 1 ⁡ sin ⁡ θ i = n ξ (3)

The physical meaning of the equation can be better illustrated by the wave vector diagram in Figure 1B. For each incident angle θ i , the transmission angle can be found at the corresponding diffraction level. However, as the number of diffraction levels n increases, the incident wave angle θ i cannot find the corresponding transmission angle in the wave number circles of the corresponding level even if it reaches the maximum value of 90°, as in the case of n = − 3 in the figure. Therefore, for a specific spacing ξ (phase gradient, which is taken as ξ = 2 π / L in this paper) and radius of the wave number circle k 1 , the diffraction level n is taken to the highest order diffraction level − N . More precisely, the maximum diffraction level − N is affected by the wavelength of the incident wave λ 1 and the length of a single period of the metasurface structure L , and substituting | sin ⁡ θ t − sin ⁡ θ i | <2 into Eq. 3 yields N = r o u n d u p [ − 2 k 1 / G ] + 1 . The equations related to wavelength and length are written as follows [7]: N = r o u n d u p [ − 2 L / λ 1 ] + 1 (4) Where r o u n d u p is the upward rounding function.

However, not all of the sound waves will be transmitted along their respective diffraction channels, and Fu treats a unit cell in the metasurface structure as a Fabry-Perot resonator, as shown in Figure 2. When the angle θ i of the incident wave is less than or equal to the critical angle θ s , the incident wave will be transmitted directly from the diffraction channel n = 1 ; when θ i is greater than the critical angle θ s , multiple diffraction channels will be opened and the incident wave will be reflected back and forth in the unit cell until the accumulated wave path satisfies the condition before transmission or reflection occurs. It can be found that when the number of reflections J is odd, the incident wave passing through the metasurface interface behaves as transmission and otherwise it appears as reflection, both of which propagate directly from the diffraction channel of the nearest and largest diffraction level, because this channel represents the smallest wave path, so the energy propagated by the largest diffraction level tends to be the highest. And the reflection and transmission satisfy the GSL transmission and reflection formulas, respectively. In summary, the law can be succinctly expressed as the following Eq. 7: k 1 ⁡ sin ⁡ θ t − k 1 ⁡ sin ⁡ θ i = n G           ( J   i s   o d d ) (5) k 1 ⁡ sin ⁡ θ r − k 1 ⁡ sin ⁡ θ i = n G           ( J   i s   e v e n ) (6) where, J = m + n represents the number of reflections, controlled by the number of unit cells m and the number of diffraction levels n . This equation shows that for a certain value of J , the transmission and reflection angles of incident light and sound waves in air can be accurately predicted when they pass through a periodically arranged metasurface structure, and are generally expressed mainly as transmitted or reflected waves at the corresponding maximum diffraction level. However, this equation only reveals the laws followed by the metasurface in regulating the acoustic waves, but there is no corresponding discussion for the complex elastic waves, which leads to a large number of existing theories for the study of elastic wave metasurfaces still stuck in the classical GSL, Eq. 1.

In this paper, we design a helical metasurface for modulating Lamb waves, which consists of a series of helical curved beam cells, as shown in Figure 3, where L indicates the length of a single cycle and l is the vertical span of the metasurface. The helical curved beam is formed by sweeping a rectangular section along the helical line, where the height of the rectangular section h is equal to the thickness of the sheet, which is 1 mm, and the length a is related to the number of unit cells. The parameter equation r is an adjustable parameter, and different values of r in the curve equation can be set to achieve different curved beam lengths. In order to maintain the continuity of the wave front when regulating the transmitted waves, the incident waves should have a phase difference of at least 2 π , or generally a positive integer multiple of 2 π , when they are transmitted from the first and last cells in the structure. In this paper, we construct 0 ∼ 2 π equal gradients of phase in the structure, so according to d φ i = k 2 d s i , we can invert the difference of the length between the curved beams (where φ is the phase, s is the wave path, subscript i indicates the curved beam number, and k 2 is the number of waves in the curved beam), and then according to the length integral equation s i = ∫ 0 2 π [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 + [ z ′ ( t ) ] 2 d t , can calculate and design the length of each helical line segment.

It is worth noting that the waves in the thin plate exist in the form of Lamb waves, while in the curved beam exist mainly in the form of bending waves, and there is a difference in the wave numbers between them. The wave numbers k 1 in Eqs. 1–6 are the wave numbers of Lamb waves in A₀ mode, which can be obtained from the dispersion curves, while k 2 above is the wave number in the helical curved beam, which is obtained from the bending wave number equation k 2 = ρ S ω 2 / E I 4 [17], where ρ is the density, S is the area of the beam section, ω is the angular frequency of the excitation, and E I is the structural stiffness. At the same time, the curvature of this metasurface structure has a small rate of change, so it can maintain a high transmittance.

According to the theory in the previous section, if set L = λ 1 , the highest diffraction order is −1 and the critical angle is 0°. When the incident angle is greater than 0° and the number of unit cells in the structure is an even number, that is, when the number of reflections J = m + n is an odd number, the high-order diffracted wave behaves as a transmitted wave at this time, which satisfies Eq. 6, and can be written as follows by substituting L = λ 1 and n = − 1 into it form: sin ⁡ θ t − sin ⁡ θ i = − 1       ( θ i > 0 ) (7)

When the incident angle is less than or equal to 0°, it conforms to the classical GSL. Substitute L = λ 1 and n = 1 into (1.6) to get: sin ⁡ θ t − sin ⁡ θ i = 1       ( θ i ≤ 0 ) (8)

Combining Eqs. 7, 8, it can be seen that when the two incident wave directions are symmetric about the vertical axis, the transmitted wave directions will also behave as axisymmetric, and the incident wave angle will always remain opposite in sign to the transmitted angle (except for 0°). This also indicates that it is theoretically possible to achieve nearly full-angle negative refraction except for the incident angle of 0°. Therefore, in this paper, the number of unit cells is set to m = 8 , and the detailed parameters of the structure are shown in Table 1. The incident waves are incident at + -30°, +-45°, and +-60°, respectively, with the same excitation mode and frequency, and the simulation results are obtained as shown in Figure 8. It can be found that the wave field in the figure is exactly the same as the theoretical prediction. When the incident wave is incident at positive and negative angles, the transmitted wave is also transmitted at corresponding angles of -+30°, -+17°, and -+7.7°, respectively, and the sign is opposite to the incident angle, realizing a perfect wide-angle negative refraction phenomenon.

As an important acoustical phenomenon, asymmetric transmission enables unidirectional wave transmissibility, waves can propagate from one side to the other side of the interface, while they cannot transmit from the reverse direction, such phenomena have great potential for application. In the field of elastic waves, the asymmetric transmission of waves can also be realized by metasurface structures. Therefore, in this paper, an asymmetric transmission device is designed based on the theory of parity-related diffraction, which only allows the single-pass of waves and perfectly realizes the wide-angle asymmetric transmission of Lamb waves.

Setting L = λ 1 , at this time, based on the parity-related diffraction theory Eqs. 7, 8, it is known that if the number of unit cells m is designed as odd and m + n as even, the higher-order diffracted wave will be reflected when the incident wave angle is greater than the critical angle 0°, and the reflection angle satisfies Eq. 6. Therefore, when the waves are incident in opposite directions from the lower left and upper right angles, respectively, the transmission is not the same, resulting in an asymmetric propagation path of Lamb waves.

Therefore, the number of unit cells is set to m = 7 , and the parameters are shown in Table 1, and the rest of the settings are the same as above, and then the simulation results are obtained when the waves are incident at −30°, −45°, −60° at the lower left corner and 30°, 45°, 60° at the upper right corner, respectively, as shown in Figure 9. By comparison, it is found that the wave fields in the figure possess good asymmetry, and the transmission directions are 30°, 17° and 7.7°, and the reflection directions are −30°, −17° and −7.7°, and the two groups of wave fields form a perfect asymmetric transmission phenomenon with each other.

When Lamb waves are used for structural damage monitoring, they often cause wave packet mixing because they have multiple modes, which further causes the damage signal to be difficult to identify and the problem of inaccurate damage imaging. How to separate the different modes of Lamb waves quickly and accurately is one of the important problems in the field of structural health monitoring that needs to be solved.

In this section, based on the difference of wave velocities between symmetric mode (S) and antisymmetric mode (A) of Lamb wave, a simple mode separator is constructed based on the parity-related diffraction theory, and the separation of A₀ mode and S₀ mode waves at an excitation frequency of 100 kHz is initially realized.

First, the aluminum plate dispersion curve was plotted, as shown in Figure 10. According to this figure, the wave velocities of A₀ mode and S₀ mode wave at the frequency thickness product of 100 kHz*1 mm are 951.9 m/s and 5373 m/s, respectively, in other words, the wavelength of S₀ mode wave at this frequency is 5.6 times that of A₀, λ 3 = 5.6 λ 1 and only A₀ and S₀ mode waves exist at this frequency thickness product.

Second, according to the parity-related diffraction theory and Eq. 4, it is known that for the S₀ mode, λ 0 = 5.6 λ 1 in this equation, the maximum diffraction level - N can only be 0 when L = 1.1 λ 1 , and Eqs. 5, 6 are then rewritten as sin ⁡ θ t − sin ⁡ θ i = 0         ( J   i s   o d d ) (9) sin ⁡ θ r − sin ⁡ θ i = 0         ( J   i s   e v e n ) (10)

The above equation shows that the transmission or reflection angle is equal to the incidence angle when the S₀ mode wave at this frequency thickness product is incident in any direction. However, for the A₀ mode wave, when it is incident vertically in the 0° direction, it satisfies the classical GSL Eq. 1, with a theoretical transmission angle of 65.4°. In order to illustrate the effect of mode separation when Lamb wave field is excited in three cases: 1. A₀ mode (control group) 2. A₀ and a few S₀ modes 3. S₀ and a few A₀ modes, three excitations are selected in this section based on the principle that symmetric (S) and antisymmetric (A) modes are dominated by in-plane and off-plane displacements, respectively: (a) antisymmetric excitation about the xy plane (b) symmetric excitation (c) antisymmetric excitation about the xz plane, and the schematic diagram is shown in Figure 11.

Then the first excitation method (a) is chosen, the excitation frequency is selected as 100 kHz, the amplitude size is 1 mm so that the structure length of the single-cycle metasurface L = 1.1 λ 1 , the number of unit cells m = 7 , J is odd, and the rest of the settings remain unchanged, the total displacement wavefield is obtained as shown in Figure 12A, it can be clearly seen that the wavefield contains only a single mode wave. Comparing with the z-direction displacement wavefield in figure (b), it can be seen that the total displacement wavefield basically exists only in the z-directional off-plane displacement, and it is judged as A₀ mode according to the wavelength.

Next, the second excitation method (b) is adopted, about z-directional symmetric excitation, which can excite part of the S₀ mode wave. The theoretical analysis in this subsection shows that when the wave is incident at 0°, the A₀ wave will transmit at 65.4°, while the S₀ wave will maintain 0° transmission, and the two transmission directions are different, forming the beam separation phenomenon. The simulation results are shown in Figure 12C, and it can be found that the Lamb wave field in the figure achieves a mode separation phenomenon consistent with the theory, with the shorter wavelength being the A₀ wave and the opposite being the S₀, and the deflection angle is also consistent with the theoretical prediction.

Finally, the third excitation method is used, which can excite the S₀ mode with mainly in-plane displacement (y-direction displacement). The rest of the settings remain unchanged, and only the number of unit cells m is changed to 8, J is an even number. According to (1.10), most of the energy of the S₀ wave will be reflected, while the A₀ wave is still deflected in the direction of 65.4° after passing through the metasurface. The simulation results are shown in Figure 12D, which clearly shows the accuracy of the theoretical prediction and also achieves a more perfect separation of the two modes in Lamb waves.

In conclusion, this section designs this metasurface structure as a simple mode separator based on the parity-related diffraction theory, combined with the difference in the wavelengths of Lamb wave A₀ and S₀ modes. Moreover, a reasonable size can be designed based on the above theory to realize the separation of multiple modal waves of Lamb waves in any direction according to the actual needs.

Similar to the case of S₀ wave propagation in the previous section, it is known from Eq. 4 that when L ≤ λ 1 / 2 , the higher order diffracted waves of A₀ mode can only be coupled to level 0, the maximum diffraction level - N = 0 . Therefore, when m is odd and even, respectively, the incident wave will also satisfy Eqs. 9, 10 with the same angular magnitude as the transmitted or reflected wave. Thus, a waveguide structure is designed in this section based on the full-angle total reflection phenomenon in it. This structure not only prevents the wave from leaking out, but also prohibits the wave from transmitting to the inside of the structure from the outside, forming a kind of “cage” to imprison the wave.

First, set L ≤ λ 1 / 2 , the number of unit cells m to 7, and the rest of the settings remain unchanged. The incident wave angles are still set to 30°, 45° and 60°, and the simulation results are shown in Figure 13. It can be seen that the waves in Figs. (a), (b) and (c) are transmitted at the same angle as the incident direction, which is the same as the results of the theoretical analysis.

Then set m to 8, and the rest of the settings remain unchanged. The simulation results are shown in Figure 14, and it can be found that the wave fields in Figs. (a), (b) and (c) are exactly as expected from the theory, and total reflection occurs in all of them, and total reflection occurs when the wave is incident at any angle in theory.

Based on the above discussion, a waveguide structure is designed by using this total reflection phenomenon. The structure schematic is shown in Figure 15A, which consists of two layers of the metasurface, and the middle range is the waveguide area. A line excitation is applied on the small semicircular edge, and the excitation waves in different directions will reflect back and forth when they encounter the metasurface interface, thus achieving directional propagation. Considering the large amount of computation of the metasurface model, the width of the middle waveguide region is set to be narrower and the length is shorter. The final energy density diagram is shown in Figure 15B. Although the Lamb wave is essentially leak-free, the conduction region is too small, resulting in a difficult to distinguish the wave path when it propagates inside the waveguide. Even so, the idea of using the metasurface as a waveguide structure is still feasible. If the metasurface structure can be simplified and the waveguide region can be designed from straight lines to curves or loops, then better conduction and isolation effects of Lamb waves can be further achieved.