1. Introduction

Over the past few years, metasurfaces, the two-dimensional counterparts of three-dimensional metamaterials, have proven to be particularly effective at controlling electromagnetic waves. However, most studies on metasurfaces have been restricted to purely linear structures and only few studies, as for instance [1–4], have investigated the synthesis and/or scattering from nonlinear metasurfaces, but without providing extensive discussion on the topic. Since nonlinearity may potentially bring about a wealth of new applications to the realm of metasurface-based effects, such as for instance nonreciprocity, second-harmonic generation and wave-mixing [5], we propose here a rigorous discussion on the synthesis and scattering analysis from second-order nonlinear metasurfaces.

In the following, we will use the generalized sheet transition conditions (GSTCs) to obtain the metasurface susceptibilities (linear and nonlinear components) in terms of specified incident, reflected and transmitted waves. Based on that, the conditions for no reflection for second-order nonlinear metasurfaces will be derived, which will be next used to analyze the scattering from such structures. The scattered field will be computed using perturbation analysis and the results will be compared with FDTD simulations.

2. Synthesis of second-order nonlinear metasurfaces

The GSTCs are boundary conditions that apply to zero-thickness discontinuities such as metasurfaces [6–8]. These conditions relate the discontinuities of the electric and magnetic fields to the presence of excitable surface polarization densities. We hereafter assume the harmonic time dependence e^jωt. In the case of a metasurface lying in the xy-plane at z = 0 and assuming only transverse polarizations, the GSTCs read

Let us now investigate the case of a metasurface with nonzero second-order nonlinear electric and magnetic susceptibility tensors. The presence of non-negligible electric nonlinearities may be found, for instance, in optical nonlinear crystals [5] while magnetic nonlinearities may be found in ferrofluids [9]. We restrict our attention to the case of monoanisotropic metasurfaces [8], whose electric and magnetic polarization densities are

To synthesize a nonlinear metasurface, one needs to solve (3) so as to express the susceptibilities as functions of the electromagnetic fields on both sides of the metasurface. As it stands in (3), the system has two equations for four unknowns, and is hence under-determined. If we consider two arbitrary transformations instead of just one [8], then the system becomes a full-rank one, reading

At this stage, one may think that substituting the specified arbitrary incident, reflected and transmitted fields into (5) would lead to well-defined susceptibilities. However, the specification of even “simple” transformations as, for instance, E_i = E₀e^{j(ω_it−k_iz)}, E_r = Re^{j(ω_rt−k_rz)} and E_t = Te^{j(ω_tt−k_tz)}, leads to susceptibilities that are time-varying, irrespectively of the values of ω_i, ω_r and ω_t. The fact that the susceptibilities are, in this case, time-varying is inconsistent with the implicit assumption in Eqs. (3) that they are not. Note that if the susceptibilities were time-dependent, insertion of (2) into the time-domain version of (1) would lead, from the derivation chain rule, to additional terms of the form $\left(\partial {\chi }_{uu}^{(k)}/\partial t\right)E_{\text{av}}$, H_av, where u =e,m and k = 1, 2. At this point, one may therefore wonder whether the problem has been adequately paused? However, fortunately, we shall see in the next section that this is not the case: the problem is adequately posed but the specified fields must satisfy a specific constraint, while being otherwise still synthesizable. This constraint will be established by considering specified fields satisfying the postulate that the susceptibilities in (5) are not functions of time. For this purpose, we will look at the problem from a different perspective. Instead of trying to synthesize the metasurface, we shall heuristically analyze its scattering with (arbitrary) known susceptibilities in order to understand what kind of reflected and transmitted fields are produced by such nonlinear metasurfaces, and hence deduce a proper way to perform the synthesis.

3. Scattering from second-order nonlinear metasurfaces

The fields scattered from a nonlinear metasurface may be obtained by solving (3). However, Eqs. (3) form a set of nonlinear inhomogeneous first-order coupled differential equations, that is not trivial to solve analytically. The problem may be simplified by assuming that the metasurface is reflectionless, which reduces (3) to a single equation. The conditions for no reflection in a nonlinear metasurface may be obtained by specifying E_r = H_r = 0 in (5) and assuming normally incident and transmitted plane waves, i.e. E = ±η₀H, where + corresponds to waves propagating in the +z-direction and vice-versa for −. To obtain the susceptibilities in (5), we have to consider the transformation of two sets of independent waves. One may assume that the incident and transmitted waves for the two transformations are either both propagating in the +z-direction, as in Fig. 1(a), or in the −z-direction, as in Fig. 1(b). In these figures, Ψ₁ and Ψ₂ correspond to two sets of independent x-polarized normally propagating plane waves being fully transmitted by the metasurface. The subscripts i and t refer to the incident and transmitted waves, respectively. One may also consider the case where the waves Ψ₁ are propagating in the +z-direction and the waves Ψ₂ are propagating in the −z-direction (or vice-versa), but it may be shown that conditions for no reflection do not exist in this case.

 

Fig. 1 Two different approaches to synthesize a nonlinear reflectionless metasurface. (a) Metasurface reflectionlessly transmitting two waves (Ψ₁ and Ψ₂) in the +z-direction. (b) Metasurface reflectionlessly transmitting two waves (Ψ₁ and Ψ₂) in the −z-direction.

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It will next be shown that the conditions for no reflection are not the same in the two cases depicted in Fig. 1. This is because of the presence of the square of the electric and magnetic fields in (5), which introduces an asymmetry in the definition of the susceptibilities in terms of the direction of wave propagation. This asymmetry is due to the different relations between the electric and magnetic fields (E = ±η₀H) for forward or backward propagating waves. Solving (5) for the case depicted in Fig. 1(a) leads to the following conditions for no reflection:

As of now, we continue the evaluation of scattering from the nonlinear metasurface considering the case in Fig. 1(a). Substituting (6) along with the difference of the specified fields, ΔE = E_t−E_i, the average of the specified fields, $E_{\text{av}}=\frac{1}{2}(E_{\text{t}}+E_{\text{i}})$, and the squared average of the specified fields, $E_{\text{av}}^2=E_{\text{av}}{E}_{\text{av}}^{*}=\frac{1}{4}\left(E_{\text{i}}^2+E_{\text{i}}{E}_{\text{t}}^{*}+E_{\text{t}}{E}_{\text{i}}^{*}+E_{\text{t}}^2\right)$, and similarly for the magnetic fields, transforms (3b) into

Equation (9) does not admit an analytical solution. To obtain an approximate expression of the transmitted field, we consider that the second-order susceptibilities are much smaller than the first-order ones, which is typically a valid assumption in the absence of second order resonance, χ⁽²⁾ ≈ 10⁻¹² χ⁽¹⁾ [5]. From this consideration, perturbation analysis [10] may be used to approximate the value of the transmitted field. Perturbation analysis stipulates that the approximate solution may be expressed in terms of a power series of the following form

According to (13), (15) and (16), the scattered field from the nonlinear metasurfaces may generally be expressed as

We shall now validate our theory with full-wave analysis. The metasurface is synthesized so as to satisfy the conditions for no reflection given in (6), and we consider the following arbitrary parameters: E₀ = 1.5 V/m, ${\chi }_{\text{ee}}^{(1)}=0.1\hspace{0.17em}\text{m}$ and ${\chi }_{\text{ee}}^{(2)}=0.04\hspace{0.17em}{\text{m}}^2/\text{V}$. The simulations are performed with the 1D FDTD zero-thickness metasurface simulation code developed in [11], modified to account for the nonlinear susceptibilities (see Appendix A). In the forthcoming simulations, we use the usual normalized constants c₀ = ∊₀ = μ₀ = f₀ = 1.

In the first simulation, whose results are plotted in Fig. 2(a), the metasurface is illuminated by a plane wave propagating in the +z-direction. In the figure, the metasurface is placed at the center and the simulation area is split into a scattered-field region (SF) and a total-field region (TF). The source is placed at the SF/TF boundary on the left of the metasurface. As expected, the metasurface is reflectionless (E = 0 in the SF region). A time-domain Fourier transform of the steady-state transmitted field is performed and the normalized (to E₀) result is plotted in Fig. 2(b). It may be seen that the metasurface generates a transmitted field with several visible harmonics satisfying E(ω_n) > E(ω_n+1), as expected.

 

Fig. 2 FDTD simulations of a nonlinear metasurface illuminated by a plane wave from the left with the following parameters: E₀ = 1.5 V/m, ${\chi }_{\text{ee}}^{(1)}=0.1\hspace{0.17em}\text{m}$ and ${\chi }_{\text{ee}}^{(2)}=0.04\hspace{0.17em}{\text{m}}^2/\text{V}$. (a) Electric field waveform. (b) Spectrum.

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In order to investigate its nonreciprocity, the metasurface is now illuminated from the left by a plane wave propagating in the −z-direction. The corresponding simulated waveform is plotted in Fig. 3(a), where the positions of the SF/TF regions have been changed accordingly. We can see that when this metasurface is illuminated from the right, it is not reflectionless anymore as evidenced by the nonzero electric field in the SF region. This is in agreement with the fact that different conditions for no reflection apply for different directions of propagation, according to the discussion that led to Eqs. (6) and (7).

 

Fig. 3 FDTD simulations of a nonlinear metasurface illuminated by a plane wave from the right with the following parameters: E₀ = 1.5 V/m, ${\chi }_{\text{ee}}^{(1)}=0.1\hspace{0.17em}\text{m}$ and ${\chi }_{\text{ee}}^{(2)}=0.04\hspace{0.17em}{\text{m}}^2/\text{V}$. (a) Electric field waveform. (b) Spectrum.

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The time-domain Fourier transform of the transmitted field in Fig. 3(a) is plotted in Fig. 3(b). It reveals that the frequencies that are odd multiples of ω₀ are transmitted, while the even multiples of ω₀ are strongly suppressed. Since the metasurface is lossless (purely real susceptibilities), we can infer that the frequencies that are even multiples of ω₀ are thus mostly reflected. The generation of transmitted waves, with corresponding frequencies that are even multiples of the excitation frequency, is thus a nonreciprocal process since $S_{21}^{{\omega }_0\to 2n{\omega }_0}\ne S_{12}^{{\omega }_0\to 2n{\omega }_0}$, where n = 1, 2, 3, ...

Next, we shall compare the theoretical results with FDTD simulations. For this purpose, two metasurfaces satisfying (6) with different susceptibilities are considered. The values of E₀ and ${\chi }_{\text{ee}}^{(1)}$ are specified while those of ${\chi }_{\text{ee}}^{(2)}$ are swept. The amplitudes of the first three harmonics (at ω₀, 2ω₀ and 3ω₀) of the transmitted field are obtained from (13), (15) and (16), and compared to the corresponding amplitudes found by FDTD simulation. Note that the amplitude of the harmonic at ω₀ is computed from both (13) and (16) since both of these equations include terms contributing to this harmonic. As explained in the Appendix A, the FDTD field update equations are only valid for specific values of E₀, ${\chi }_{\text{ee}}^{(1)}$ and ${\chi }_{\text{ee}}^{(2)}$, other specifications leading to a nonphysical behavior. In addition to choosing those parameters so as to follow this constraint, we have ensured ${\chi }_{\text{ee}}^{(1)}>{\chi }_{\text{ee}}^{(2)}$ in order to be consistent with the assumptions of the perturbation analysis method [see Eq. (11)].

The first comparison, presented in Fig. 4(a), considers E₀ = 1.5 V/m, ${\chi }_{\text{ee}}^{(1)}=0.1\hspace{0.17em}\text{m}$ and ${\chi }_{\text{ee}}^{(2)}$ swept in [0, 0.04] m²/V, while the second comparison, presented in Fig. 4(b), considers E₀ = 10 V/m, ${\chi }_{\text{ee}}^{(1)}=0.03\hspace{0.17em}\text{m}$ and ${\chi }_{\text{ee}}^{(2)}$ swept in [0, 0.01] m²/V. Both comparisons show good agreement between theory and simulation. In both cases, the discrepancies between the results of the two methods increase with increasing ${\chi }_{\text{ee}}^{(2)}$, as expected from the perturbation assumption ${\chi }_{\text{ee}}^{(1)}>{\chi }_{\text{ee}}^{(2)}$. Thus, as ${\chi }_{\text{ee}}^{(2)}$ increases towards ${\chi }_{\text{ee}}^{(1)}$, the error in the approximation of the transmitted field (Eq. (10)) progressively increases. Another source of error is the fact that Eq. (10) is truncated at the third term and that higher-order terms are neglected, which induces additional errors.

 

Fig. 4 Comparisons of the amplitudes of the first three normalized harmonics obtained by theory [Eqs. (13), (15) and (16)] (blue solid lines) and FDTD simulation (red dashed lines) for the following parameters: (a) E₀ = 1.5 V/m, ${\chi }_{\text{ee}}^{(1)}=0.1\hspace{0.17em}\text{m}$. (b) E₀ = 10 V/m, ${\chi }_{\text{ee}}^{(1)}=0.03\hspace{0.17em}\text{m}$.

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We have verified that the FDTD simulations in Figs. 2 and 3, as well as in Figs. 4(a) and 4(b) satisfy the losslessness and passivity power conservation condition ${\sum }_{n=1}^{\infty }{\left|E({\omega }_n)\right|}^2={\left|E_0\right|}^2$. In contrast, this condition does not hold in the case of the theoretical results, due to the truncation of (10), as is clearly apparent in Fig. 4(b), where ${\sum }_{n=1}^3{\left|E({\omega }_n)\right|}^2>{\left|E_0\right|}^2$ for values of ${\chi }_{\text{ee}}^{(2)}$ that are close to 0.01 m²/V.

4. Conclusion

We have investigated a particular case of second-order nonlinear isotropic metasurfaces that possess both electric and magnetic nonlinear susceptibilities. We have found the synthesis expressions relating the susceptibilities to the fields on both sides of the metasurface, which lead to the derivation of the reflectionless metasurface conditions. These conditions reveal the inherent nonreciprocal nature of nonlinear metasurfaces. Then, the scattered field from such metasurfaces was analyzed based on perturbation theory as well as full-wave simulations, and good agreement was found between the two approaches.