Quenching of second-harmonic generation by epsilon-near-zero media

Chenglin Wang; Ran Shi; Lei Gao; Alexander S. Shalin; Jie Luo

doi:10.1364/PRJ.491949

Abstract

Epsilon-near-zero (ENZ) media were demonstrated to exhibit unprecedented strong nonlinear optical properties including giant second-harmonic generation (SHG) due to their field-enhancement effect. Here, on the contrary, we report the quenching of SHG by the ENZ media. We find that when a tiny nonlinear particle is placed very close to a subwavelength ENZ particle, the SHG from the nonlinear particle can be greatly suppressed. The SHG quenching effect originates from the extraordinary prohibition of electric fields occurring near the ENZ particle due to evanescent scattering waves, which is found to be universal in both isotropic and anisotropic ENZ particles, irrespective of their shapes. Based on this principle, we propose a kind of dynamically controllable optical metasurface exhibiting switchable SHG quenching effect. Our work enriches the understanding of optical nonlinearity with ENZ media and could find applications in optical switches and modulators.

1. INTRODUCTION

Nonlinear optical harmonic generation is of great significance in a broad range of technologies and has been attracting much attention in photonics, chemistry, and biosensing. The understanding and manipulation of nonlinear optical properties of nanostructures is a major issue for realizing nonlinear nanophotonic devices. In the past decades, a lot of efforts have been made to enhance the nonlinear frequency conversion efficiency, including the new materials and systems [1 –7]. Recently, with the advent of plasmonics, metamaterials, and metasurfaces that go beyond natural materials in many aspects, significant attention has been devoted to the understanding and the observation of nonlinear optical processes in nanophotonics [1 –7]. For boosting the second-harmonic generation (SHG) [8 –13], as one of the most important nonlinear optical effects, many approaches have been proposed. For example, through exciting resonance modes like surface plasmon polariton resonances [8], Fano resonances [9], anapole modes [10,11], and bound states in the continuum [12], the fundamental fields inside the nonlinear materials can be greatly enhanced, so as to boost the SHG. Notably, epsilon-near-zero (ENZ) media [7,14 –17] with a vanishing permittivity are found to exhibit pronounced nonlinear optical properties [7,18 –38]. The ENZ media can provide large field enhancement due to the continuity of the normal component of the electric displacement field across the interface [18,19], and at the same time provide unique opportunities for realizing phase-matching conditions [23,38]. Consequently, the ENZ media can give rise to unprecedented strong nonlinear optical responses including the significantly enhanced SHG [18 –22,27 –29].

In this work, on the contrary, we demonstrate that the ENZ media can be exploited to “turn off” the SHG of a nonlinear particle. We show that when a tiny nonlinear particle is placed very close to a subwavelength ENZ particle, the SHG conversion efficiency can be reduced by more than 2 orders of magnitude as compared to that from the nonlinear particle alone (Fig. 1). The SHG quenching effect attributes to the extraordinary local evanescent fields occurring near the ENZ particle due to evanescent scattering waves. Remarkably, the prohibition of electric fields appears besides the ENZ particle, and therefore, the SHG from the tiny nonlinear particle placed there would be suppressed. We find that this extraordinary local evanescent field exists for both isotropic and anisotropic ENZ particles, irrespective of their shapes. Furthermore, based on the principle of SHG quenching effect, we propose a kind of dynamically controllable optical metasurface integrated with anisotropic ENZ media, which consist of alternative layers of semiconductor material cadmium oxide (CdO) and phase-change material germanium telluride (GeTe). We find that through changing the phase states of GeTe, the SHG from the metasurface can be switched on or off. Our work demonstrates a feasible approach for controlling nonlinear responses with ENZ media, which may find applications in optical switches and modulators.

(a) Schematic illustration of SHG from a tiny nonlinear particle. (b) The SHG is quenched when a linear ENZ particle is placed very close to the tiny nonlinear particle.

Figure 1.(a) Schematic illustration of SHG from a tiny nonlinear particle. (b) The SHG is quenched when a linear ENZ particle is placed very close to the tiny nonlinear particle.

2. SHG QUENCHING AND THE UNDERLYING PHYSICS

First, to explore the physical mechanism of the unique SHG quenching effect, we begin with a simple configuration comprising a linear ENZ spherical particle (relative permittivity $ε_{e} \approx 0$ , relative permeability $μ_{e} = 1$ ) and a tiny nonlinear spherical particle (relative permittivity $ε_{n} = 2$ , relative permeability $μ_{n} = 1$ ) in free space, as illustrated in Fig. 2(a). Both particles are in the deep-subwavelength scale with radii of $r_{e} = λ_{0} / 100$ (ENZ particle) and $r_{n} = 0.1 r_{e}$ (nonlinear particle), where $λ_{0}$ is free-space wavelength. The gap (i.e., edge-to-edge distance) between the two particles is $d$ . The two particles are illuminated by a plane wave propagating along the $z$ direction with electric field $E_{0}^{ω}$ polarized in the $x$ direction.

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Figure 2.(a) Schematic layout of the configuration for exploring the SHG quenching effect. It is composed of a linear ENZ spherical particle accompanied with a tiny nonlinear spherical particle in the deep-subwavelength scale. The green curve denotes the trajectory of the nonlinear particle moving around the ENZ particle. Positions 1 and 4 are close to the poles, positions 2 and 3 are on the equatorial plane, and position 5 is far from the ENZ particle. The two particles are illuminated by a plane wave propagating along the $z$ direction with the electric field polarized in the $x$ direction. (b) The blue solid line denotes the normalized scattering cross section of SHG ${SCS}^{2 ω}$ as a function of the position of the nonlinear particle along the trajectory in (a). The red dots show the ${SCS}^{2 ω}$ from the nonlinear particle alone, which is normalized to 1.

The SHG of the tiny nonlinear spherical particle can be accounted for by effective surface contribution and characterized by a surface nonlinear current source $J^{2 ω} = - i 2 ω ε_{0} {\overset{\leftrightarrow}{χ}}_{a}^{(2)} : E^{ω} E^{ω}$ , where ${\overset{\leftrightarrow}{χ}}_{a}^{(2)}$ is the nonlinear susceptibility tensor, and $E^{ω}$ is the fundamental frequency electric field inside the nonlinear particle. $ε_{0}$ is the permittivity of free space. For isotropic and centrosymmetric materials, the second-order surface susceptibility tensor ${\overset{\leftrightarrow}{χ}}_{a}^{(2)}$ can be reduced to only three independent nonzero components ${\overset{\leftrightarrow}{χ}}_{⊥ ⊥ ⊥}^{(2)}$ , ${\overset{\leftrightarrow}{χ}}_{⊥ ∥ ∥}^{(2)}$ , and ${\overset{\leftrightarrow}{χ}}_{∥ ∥ ⊥}^{(2)}$ [39,40]. Here the symbols $⊥$ and $∥$ represent the directions perpendicular and parallel to the particle’s surface, respectively.

Figure 2(b) shows the scattering cross section of SHG ( ${SCS}^{2 ω}$ ) from the tiny nonlinear particle when it is successively moved from position 1 to position 5 along the trajectory $1 \to 2 \to 3 \to 4 \to 5$ [green curves in Fig. 2(a)]. Along the trajectory $1 \to 4$ , the edge-to-edge distance $d$ is kept unchanged at $r_{e} / 20$ , and position 5 is at a distance of $6 r_{e}$ from the ENZ sphere’s center. Here we assume that the nonlinear susceptibility of the tiny particle is several orders of magnitude larger than that of the ENZ particle, and therefore, the nonlinear response of the ENZ particle is negligible. Such an assumption is reasonable when using appropriate optical materials [30], as we will show in the following practical implementations. The calculation is performed using the finite-element software COMSOL Multiphysics. The solid blue lines and red dots denote the normalized ${SCS}^{2 ω}$ of the cases with and without the ENZ particle, respectively. The latter case is normalized to 1. As expected, in the absence of the ENZ particle, the ${SCS}^{2 ω}$ will not change as varying the position under the illumination of a plane wave. Interestingly, we observe that the presence of the ENZ particle has a great influence on the SHG conversion efficiency of the nonlinear particle. The ${SCS}^{2 ω}$ reaches the maximal value when the nonlinear particle is on the equatorial plane of the ENZ particle (positions 2 and 3), while it is largely reduced when the nonlinear particle is close to the poles of the ENZ particle (positions 1 and 4). Compared with the case without the ENZ particle, we find that the ${SCS}^{2 ω}$ from the nonlinear particle at position 1 or 4 is reduced by more than 1 order of magnitude, indicating that the SHG is quenched. Such an SHG quenching effect would fade away if the nonlinear particle is at a distance comparable with the radius of the ENZ particle (position 5), indicating that the quenching of SHG is a short-range effect. The influence of the geometrical parameters on the SHG quenching effect is elaborated in Appendix A.1.

The underlying physics of such a unique position-dependent characteristic lies on the extraordinary evanescent fields near the ENZ particle. When encountering the ENZ particle, the incident light will be scattered into various directions. If the size of the ENZ particle is comparable to or smaller than the wavelength, strong scattered evanescent waves that are evanescent in the forward and backward directions but capable of transferring energy flux along the perpendicular directions would emerge [41 –43]. Such evanescent fields vary dramatically with positions, thus leading to the position-dependent SHG efficiency, as observed in Fig. 2. Particularly, near-zero electric field occurs on two sides of the ENZ particle perpendicular to the direction of incidence and along the direction of polarization, which is also denoted as the “side scattering shadows” [41]. When the tiny nonlinear particle is placed there, the incident light cannot “see” it, and therefore the SHG is suppressed.

The generation of the dramatically varying evanescent fields stems from the continuity boundary condition at the ENZ particle–air interface. Since the ENZ particle is at the deep-subwavelength scale, the fundamental electric field inside it can be considered uniform and can be approximately expressed as $E_{ENZ}^{ω} = \frac{3}{2 + ε_{e}} E_{0}^{ω}$ [43]. Thus, the continuity of the electric displacement field at the poles of the ENZ particle yields $E_{air, p}^{ω} = \frac{3 ε_{e}}{2 + ε_{e}} E_{0}^{ω},$ (1)where $E_{air, p}^{ω}$ is the electric field at the poles on the air side. On the other hand, the electric field on the equator of ENZ particle on the air side can be obtained as $E_{air, e}^{ω} = \frac{3}{2 + ε_{e}} E_{0}^{ω}$ (2)due to the continuity of the electric field. When $ε_{e} \to 0$ , we have $E_{air, e}^{ω} ≫ E_{air, p}^{ω} \to 0 .$ (3)

Equation (3) indicates the dramatically varying fundamental electric field around the ENZ particle and near-zero electric field at the poles. When a tiny nonlinear particle is placed close to the ENZ particle, it will experience significantly different fundamental electric fields at different positions, thus leading to the position-dependent SHG efficiency. Remarkably, the SHG is quenched when the nonlinear particle is located at the poles as the fundamental electric field there is shielded by the ENZ particle. We note that the extraordinary evanescent fields only exist in the regions very close to the ENZ particle, and therefore the quenching of SHG is a short-range effect.

More strict proof can be obtained based on the Mie scattering theory [44] and the electrostatic theory [45]. Under the limit of $k_{0} r_{e} ≪ 1$ (here $k_{0} = 2 π / λ_{0}$ ), the fundamental electric field outside the ENZ sphere ( $r > r_{e}$ ) can be derived as (see Appendix A.3 and A.4) $E_{air}^{ω} = E_{0}^{ω} (1 + 2 \frac{ε_{e} - 1}{ε_{e} + 2} \frac{r_{e}^{3}}{r^{3}}) \sin θ \hat{r} + E_{0}^{ω} (- 1 + \frac{ε_{e} - 1}{ε_{e} + 2} \frac{r_{e}^{3}}{r^{3}}) \cos θ \hat{θ},$ (4)where $r$ is the distance from the center of the ENZ sphere, and $θ$ is the polar angle (the angle measured from the $z$ axis). When $ε_{e} \to 0$ , we obtain the amplitude of the fundamental electric field on the ENZ sphere’s surface (i.e., $r = r_{e}$ ) on the air side as $E_{air}^{ω} = \frac{3}{2} E_{0}^{ω} \cos θ .$ (5)

Equation (5) implies that the fundamental electric field is enhanced (i.e., $E_{air}^{ω} = \frac{3}{2} E_{0}^{ω}$ ) on the equator (i.e., $θ = 0$ ), while is zero (i.e., $E_{air}^{ω} = 0$ ) at the poles (i.e., $θ = π / 2$ ), coincident with the above analysis.

For numerical verification, in Fig. 3(a) we show the simulated fundamental electric-field amplitude on the ENZ sphere’s surface on the air side in the absence of the nonlinear sphere. The numbers 1–5 represent five different positions corresponding to the positions of the tiny nonlinear sphere in Fig. 2. Apparently, the electric field on the equator (positions 2 and 3) is enhanced compared with that of the incidence, while it is largely reduced to near-zero at the poles (positions 1 and 4). At position 5 which is far away, the influence of the ENZ particle fades away, and the electric field there tends to that of incidence. We note that these extraordinary local electric fields are observed for relatively ideal ENZ particle with $ε_{e} = 10^{- 4}$ . In order to explore the influence of the $ε_{e}$ on the local electric fields, in Fig. 3(b) we plot the normalized electric-field amplitude $| E_{i}^{ω} | / E_{0}^{ω}$ at position $i$ ( $i = 1, 2, 3, 4, 5$ ) as a function of $ε_{e}$ . We observe that the electric field $| E_{1, 4}^{ω} |$ (or $| E_{2, 3}^{ω} |$ ) decreases (or increases) quickly as the $ε_{e}$ decreases from unity and becomes stable as long as $ε_{e} < 10^{- 2}$ . Since the evanescent fields only exist very near the ENZ particle, the electric field $| E_{5}^{ω} |$ remains almost unchanged.

Figure 3.(a) Distribution of normalized fundamental electric-field amplitude on the ENZ particle’s surface on the air side in the absence of the nonlinear particle. The numbers 1–5 denote five different positions of the nonlinear particle in Fig. 2. (b) Normalized fundamental electric-field amplitude $| E_{i}^{ω} | / E_{0}^{ω}$ at position $i$ ( $i = 1, 2, 3, 4, 5$ ) as a function of the $ε_{e}$ . (c) The ratio $| E_{2}^{ω} | / | E_{1}^{ω} |$ with respect to the ratio $r_{e} / d$ . Here $r_{e}$ is kept unchanged at $λ_{0} / 100$ . The red dashed line shows the ideal value of $ε_{e}^{- 1}$ . The inset illustrates the configuration. (d) The ratio $| {SCS}_{2}^{2 ω} | / | {SCS}_{1}^{2 ω} |$ with respect to the ratio $r_{n} / d$ . Here $d$ is kept unchanged at $λ_{0} / 2000$ . The ${SCS}_{1}^{2 ω}$ (or ${SCS}_{2}^{2 ω}$ ) denotes the SHG scattering cross section when the nonlinear particle is placed at position 1 (or 2), as illustrated by the inset.

We note that the electric field $| E_{1, 4}^{ω} |$ is not that close to zero because the positions 1 and 4 have a distance of $d = r_{e} / 20$ from the poles of the ENZ sphere. Actually, with reducing $d$ , the $| E_{1, 4}^{ω} | / E_{0}^{ω}$ will tend to be $1.5 ε_{e}$ , as implied in Eq. (1). On the other hand, the $| E_{2, 3}^{ω} | / E_{0}^{ω}$ tends to be 1.5 according to Eq. (2). For verification, in Fig. 3(c) we plot the ratio $| E_{2}^{ω} | / | E_{1}^{ω} |$ with the varying ratio $r_{e} / d$ ( $r_{e}$ is kept unchanged at $λ_{0} / 100$ ). Here we set $ε_{e} = 10^{- 4}$ . The results clearly show that the ratio $| E_{2}^{ω} | / | E_{1}^{ω} |$ approaches the ideal value (i.e., $ε_{e}^{- 1}$ , red dashed lines) when $d$ is considerably small, consistent with the above theory.

However, a practical nonlinear particle cannot be infinitely small. The presence of the nonlinear particle could affect the evanescent fields of the ENZ particle. Therefore, the nonlinear particle should be much smaller than the ENZ particle, such that the original evanescent fields are not largely disturbed, and the nonlinear particle can experience enhanced or weakened local fields instead of the averaged fields. In order to show the influence of the particle’s size, the ratio of SHG scattering cross section $| {SCS}_{2}^{2 ω} | / | {SCS}_{1}^{2 ω} |$ as a function of the ratio $r_{n} / d$ is plotted in Fig. 3(d). Here $d$ is kept unchanged at $λ_{0} / 2000$ . The ${SCS}_{1}^{2 ω}$ (or ${SCS}_{2}^{2 ω}$ ) denotes the SHG scattering cross section when the nonlinear particle is placed at position 1 (or 2), as illustrated by the inset in Fig. 3(d). We see that the ratio $| {SCS}_{2}^{2 ω} | / | {SCS}_{1}^{2 ω} |$ increases with the reducing $r_{n}$ , indicating better SHG quenching performance with smaller $r_{n}$ .

The above results suggest that in order to achieve excellent performance of SHG quenching effect, the conditions of $ε_{e} \leq 10^{- 2}$ and $d \sim r_{n} ≪ r_{e}$ shall be satisfied. In addition, it is noteworthy that the SHG quenching effect is robust against the material loss of the ENZ particle. We find that even when the $ε_{e}$ possesses an imaginary part 3 orders of magnitude larger than its real part, the good performance of SHG quenching effect can still be obtained (see Appendix A.2).

3. SHG QUENCHING IN GENERAL SITUATIONS

The extraordinary evanescent waves and the consequent SHG quenching effect are general and could be observed in arbitrary-shaped ENZ particles and anisotropic ENZ particles. As an example, in Fig. 4(a) we present the simulated fundamental electric-field distribution on the surface of a finite-sized isotropic ENZ film ( $ε_{e} = 10^{- 4}$ , with dimensions $0.02 λ_{0} \times 0.02 λ_{0} \times 0.005 λ_{0}$ ) on the air side under illumination of a plane wave. Prohibition of electric field above (or below) the cuboid’s upper (or lower) surface is clearly observed. Under this circumstance, a tiny nonlinear particle placed above the ENZ film cannot “see” the incident light, and therefore the SHG will be suppressed. For a demonstration, in Fig. 4(b) we compare the normalized SHG scattering cross section ${SCS}^{2 ω}$ from a nonlinear particle alone (red lines) and that from the nonlinear particle placed above the ENZ film at an edge-to-edge distance of $λ_{0} / 2000$ (blue lines) as a function of $ε_{e}$ . Here the nonlinear particle is the same as that in Fig. 2. It is seen that the ${SCS}^{2 ω}$ can be reduced by more than 2 orders of magnitude by the ENZ film with $ε_{e} \leq 10^{- 2}$ compared with the ${SCS}^{2 ω}$ from the nonlinear particle alone, indicating the suppression of SHG by the ENZ cuboid.

Figure 4.(a) and (c) Distribution of normalized fundamental electric-field amplitude on the surface of an (a) isotropic, (c) anisotropic ENZ film on the air side in the absence of the nonlinear particle. The incident light is polarized in the $z$ direction and propagates along the $x$ direction. (b) and (d) Normalized ${SCS}^{2 ω}$ from a nonlinear particle alone (red) and the nonlinear particle placed above the (b) isotropic ENZ film with different $ε_{e}$ , (d) anisotropic ENZ film with different $ε_{e, z}$ at an edge-to-edge distance of $λ_{0} / 2000$ (blue). The nonlinear particle is the same as that in Fig. 2. The insets illustrate the configurations.

Moreover, we consider an anisotropic ENZ film with $ε_{e, x} = 1$ , $ε_{e, y} = 1$ , and $ε_{e, z} = 10^{- 4}$ , which are the $x$ , $y$ , and $z$ components of the relative permittivity tensor, respectively. We see from Fig. 4(c) that the prohibition of electric field above (or below) the cuboid’s upper (or lower) surface still exists irrespective of the existence of anisotropy. This is because the $z$ component of the electric field on the air side is proportional to $ε_{e, z}$ and is therefore near zero, as a result of the continuity of the electric displacement field. The SHG quenching effect is verified in Fig. 4(d). We see that the anisotropic ENZ film with $ε_{e, z} \leq 10^{- 2}$ can significantly reduce the SHG scattering cross section ${SCS}^{2 ω}$ from the nonlinear particle to near 2 orders of magnitude (blue lines) compared with the ${SCS}^{2 ω}$ from the nonlinear particle alone (red lines). These results clearly manifest the universality of the SHG quenching effect.

4. DYNAMIC CONTROL OF SHG QUENCHING WITH OPTICAL METASURFACES

The universality of the SHG quenching effect suggests that it could be extended to other systems besides the above particle-scattering system to explore more efficient ways to control the SHG conversion efficiency. In fact, metasurfaces [46 –49], as two-dimensional artificial electromagnetic materials structured at the subwavelength scale, provide a versatile platform for manipulating light–matter interactions and realizing significant nonlinear responses [4 –6,12,13]. In the following, we would like to show a kind of optical metasurface exhibiting switchable SHG quenching effect.

Figures 5(a) and 5(b) show the schematic graphs of the optical metasurface, consisting of a square array of meta-atoms with a lattice constant of 200 nm on a 2 μm thick silica ( ${SiO}_{2}$ ) substrate. Each meta-atom comprises a large cuboid ( $100 nm \times 100 nm \times 50 nm$ in dimension) accompanied with two small ${SiO}_{2}$ cuboids ( $20 nm \times 20 nm \times 25 nm$ in dimension) on the upper and lower sides, as illustrated by the insets. The top surface of the ${SiO}_{2}$ cuboid is covered by a layer of graphene. In practical realization, such patterned graphene could be fabricated through photolithography [50], electron beam lithography [51], plasma etching [52], laser direct patterning [53], etc. The graphene is known as a unique nonlinear optical material with outstanding optical properties [13,54 –56]. Generally, the SHG is forbidden of free-standing graphene due to its centrosymmetry. Interestingly, when the graphene is placed on a substrate with broken inversion symmetry (e.g., the ${SiO}_{2}$ utilized here), SHG would be allowed [13]. Considering the fact that graphene belongs to the $D_{6 h}$ symmetry group, its surface second-order nonlinear optical conductivity tensor ${\overset{\leftrightarrow}{σ}}^{(2)}$ only has three independent nonzero components, i.e., $σ_{⊥ ⊥ ⊥}^{(2)}$ , $σ_{∥ ∥ ⊥}^{(2)} = σ_{∥ ⊥ ∥}^{(2)}$ , and $σ_{⊥ ∥ ∥}^{(2)}$ [13,57,58]. Here, we set $χ_{⊥ ⊥ ⊥}^{(2)} = \frac{i}{2 ω ε_{0} h_{eff}} σ_{⊥ ⊥ ⊥}^{(2)}$ , $χ_{⊥ ∥ ∥}^{(2)} = \frac{i}{2 ω ε_{0} h_{eff}} σ_{⊥ ∥ ∥}^{(2)}$ , and $χ_{∥ ∥ ⊥}^{(2)} = \frac{i}{2 ω ε_{0} h_{eff}} σ_{∥ ∥ ⊥}^{(2)}$ , where $σ_{⊥ ⊥ ⊥}^{(2)} = - 9.71 i \times 10^{- 16} {AmV}^{- 2}$ , $σ_{⊥ ∥ ∥}^{(2)} = - 2.09 i \times 10^{- 16} {AmV}^{- 2}$ , $σ_{∥ ∥ ⊥}^{(2)} = - 2.56 i \times 10^{- 16} {AmV}^{- 2}$ , and $h_{eff} = 0.33 nm$ is the effective thickness of graphene [13].

Figure 5.(a) and (b) Schematic graphs of an optical metasurface which can be dynamically switched to exhibit (a) high, (b) low SHG conversion efficiency through controlling the phase states of its constituent GeTe. The metasurface is composed of a square array of meta-atoms on a ${SiO}_{2}$ substrate. Each meta-atom consists of a central CdO–GeTe multilayered structure and two small graphene-covered ${SiO}_{2}$ cuboids, as illustrated by the insets. The length unit is nanometer. (c) The real part of $ε_{y, eff}$ , i.e., $Re (ε_{y, eff})$ , of the central CdO–GeTe multilayer with GeTe in the crystalline (blue) or amorphous (red) phase as a function of wavelength. The inset shows the imaginary part of $ε_{y, eff}$ , i.e., $Im (ε_{y, eff})$ . (d) The SHG energy from the metasurface with its constituent GeTe in the crystalline (green) or amorphous (yellow) phase. The SHG energy in the amorphous phase is enlarged by 10 times for better visualization. In each phase, the left and right bars correspond to the case with the central CdO–GeTe multilayer considered as an effective medium and the case with the original multilayer structure, respectively. The incident light is polarized in the $y$ direction and propagates along the $z$ direction. The operating wavelength is 6.93 μm.

The central large cuboid is composed of eight alternative layers of semiconductor CdO and phase-change material GeTe. The thickness of each CdO (or GeTe) layer is $t_{CdO} = 13.1 nm$ (or $t_{GeTe} = 11.9 nm$ ). The relative permittivity of CdO can be described using the Drude model as $ε_{CdO} = ε_{\infty} - \frac{ω_{p}^{2}}{ω (ω + i γ)}$ [59], where $ε_{\infty} = 5.5$ , $ω_{p} = 1.23 \times 10^{15} rad / s$ , and $γ = 1.77 \times 10^{13} rad / s$ . The GeTe possesses the ability of rapidly switching between the crystalline and amorphous phases in a reversible way, and the phase transition can be achieved optically or through Joule heating and is proven to be reliable, fast, and repeatable [60,61]. Remarkably, the optical properties of GeTe in different phases are significantly different, and thus the optical properties of the CdO–GeTe multilayer can be efficiently manipulated through changing the phase states of GeTe. Due to the deep-subwavelength size, the CdO–GeTe multilayer can be approximately homogenized as an effective anisotropic medium, whose $y$ component of effective relative permittivity can be expressed as $ε_{y, eff} = (ε_{CdO} t_{CdO} + ε_{GeTe} t_{GeTe}) / (t_{CdO} + t_{GeTe})$ [62]. In Fig. 5(c), we plot the real part of $ε_{y, eff}$ , i.e., $Re (ε_{y, eff})$ , as a function of wavelength when the GeTe is in the crystalline (blue lines) or the amorphous (red lines) phase. We see that $Re (ε_{y, eff}) = 0$ at the wavelength of 6.93 μm in the amorphous phase, in which case the imaginary part of $ε_{y, eff}$ , i.e., $Im (ε_{y, eff})$ , is not very large (see the inset). Under this circumstance, the central CdO–GeTe multilayer in the amorphous phase can be viewed as an effective anisotropic ENZ medium with $ε_{y, eff} \approx 0$ . However, when the GeTe is switched to the crystalline phase, we have $Re (ε_{y, eff}) = 0.81 ≫ 0$ and $Im (ε_{y, eff}) = 5.29$ , indicating the breakdown of the ENZ condition.

This unique property provides us with a route to dynamically control the occurrence and quenching of SHG of the optical metasurface through switching the phases of GeTe, as schematically shown in Figs. 5(a) and 5(b). When the GeTe is in the amorphous phase, the CdO–GeTe multilayer effectively serves as an anisotropic ENZ medium, which can generate near-zero local electric fields near the surfaces normal to the $y$ direction (see field distributions in Appendix A.5). The graphene placed there will be hard to “see” the incident light, and therefore the SHG from the graphene is suppressed [Fig. 5(b)]. Interestingly, when the GeTe is switched to the crystalline phase, the ENZ condition breaks down, and the SHG from the graphene emerges [Fig. 5(a)].

For numerical verification, we set the relevant parameters of graphene as $σ_{⊥ ⊥ ⊥}^{(2)} = - 9.71 i \times 10^{- 16} {AmV}^{- 2}$ , $σ_{∥ ∥ ⊥}^{(2)} = σ_{∥ ⊥ ∥}^{(2)} = - 2.56 i \times 10^{- 16} {AmV}^{- 2}$ , and $σ_{⊥ ∥ ∥}^{(2)} = - 2.09 i \times 10^{- 16} {AmV}^{- 2}$ at the operating wavelength of 6.93 μm [13]. At this wavelength, the second-order nonlinear susceptibility of graphene ( $\sim 10^{- 10} m / V$ ) is several orders of magnitude larger than those of CdO ( $\sim 10^{- 18} m / V$ ) [22], GeTe ( $\sim 10^{- 18} m / V$ ) [63], and ${SiO}_{2}$ [13], and therefore, here we consider the nonlinear responses of graphene only. In practical implementations, the graphene can be replaced by other optical materials with large nonlinear susceptibility such as gallium arsenide [64]. Figure 5(d) presents the SHG energy from the metasurface with GeTe in the crystalline (green bars) or amorphous (yellow bars) phase under illumination of normally incident light (polarized in the $y$ direction). In each phase, the left and right bars correspond to the case with the central CdO–GeTe multilayer considered as an effective medium and the case with the original multilayered structure, respectively, showing good agreement. We note that the SHG energy in the amorphous phase is enlarged by 10 times for better visualization. We see that the SHG energy in the amorphous phase is much lower than that in the crystalline phase, demonstrating the dynamically controllable SHG quenching effect in the optical metasurface.

We note that the pump laser used to enhance the SHG of graphene generally will not inadvertently change the phase of GeTe for two reasons. First, the intensity of the laser pulse that can effectively enhance the SHG of graphene is generally lower than that required to change the phase of GeTe because the transition temperature of GeTe is high [65,66]. Second, the activation energy of GeTe is $\sim 3.14 eV$ [67]. This indicates that the wavelength of the laser pulse used to change the phase of GeTe is required to be short, e.g., $\sim 248 nm$ [65,67], much shorter than the operating wavelength for SHG quenching (i.e., 6.93 μm).

In practical implementation, the constituent materials of the metasurface can be flexibly chosen. For example, graphene could be replaced by other two-dimensional materials with pronounced nonlinear optical properties, such as transition metal dichalcogenides (TMDs) including ${MoS}_{2}$ , ${WS}_{2}$ , and ${MoSe}_{2}$ [68,69]. The CdO–GeTe composite could be replaced by other materials such as indium tin oxide, aluminum-doped zinc oxide, and metal–dielectric composites operating at the ENZ frequency [70,71], but the drawback is that the ENZ wavelength cannot be dynamically controlled.

5. DISCUSSION AND CONCLUSION

The SHG conversion efficiency of a nonlinear material strongly depends on its molecular-scale constituents and their macroscopic ordering [8]. Based on this property, a conventional approach to manipulate the SHG signal is to engineer the local and global symmetries of meta-atoms in metamaterials and metasurfaces [72 –74]. In addition, the enhancement of SHG due to surface plasmon polariton resonance in metallic nanostructures is very sensitive to each meta-atom’s geometry, dielectric environment, and polarization of incidence, thus providing another approach to manipulate the SHG signal [75,76]. Here, unlike the previous endeavors, our approach paves a route to control the SHG conversion efficiency through exploiting the near-field effect of a linear material [77], that is, the extraordinary local fundamental electric fields of an ENZ particle, thus further realizing the unique SHG quenching effect. Our approach could be extended to other nonlinear optical processes, including sum- and difference-frequency generation.

It is noteworthy that TMDs, such as ${MoS}_{2}$ , ${WS}_{2}$ , and ${MoSe}_{2}$ , are capable of controlling the SHG conversion efficiency [68,69]. Due to their unique electronic band structures and strong light–matter interactions at the atomic scale, the SHG process of TMDs is sensitive to the number of layers, crystallinity, and interlayer coupling in TMD heterostructures [68,69,78 –81]. Interestingly, it was demonstrated that the SHG is quenched in even layered ${MoS}_{2}$ as it belongs to the centrosymmetric $D_{3 d}$ space group [78,79]. Fundamentally different from the SHG quenching with TMDs, the SHG quenching by the ENZ media attributes to the macroscopic electromagnetic responses.

Our findings could have an impact on the experimental investigations of optical nonlinearity with ENZ media. A frequently used configuration in experiments is to exploit plasmonic or dielectric nanoantennas placed on a thin ENZ film [25,27 –30,37]. Usually, the boost of nonlinear responses induced by the large field enhancement in the ENZ film is reported. Nevertheless, our results in Fig. 4 indicate that the thin ENZ film could significantly suppress the nonlinear responses from the nanoantennas. This breaks the traditional conception that the ENZ media are always responsible for SHG enhancement and enriches the understanding of the experimental observation of optical nonlinearity.

In conclusion, we have demonstrated the SHG quenching effect when a tiny nonlinear particle is placed very close to a subwavelength ENZ particle. The SHG quenching effect is found to originate from the extraordinary near-zero local electric fields occurring near the ENZ particle due to evanescent scattering waves and is proved to be universal with both isotropic and anisotropic ENZ particles, irrespective of their shapes. Based on the principle of SHG quenching effect, we have further demonstrated a kind of dynamically controllable optical metasurface integrated with phase-change material GeTe. Through changing the phase states of GeTe, the SHG from the metasurface can be switched on or off. Our work creates pathways for applications such as nonlinear optical switches and modulators based on the integration of nonlinear materials and ENZ media.

Acknowledgment

Acknowledgment. The authors sincerely thank Professor Yun Lai at Nanjing University for the insights he offered in discussions.

APPENDIX A

1. Influence of Geometrical Parameters

r_{e}

Figure 6.(a) Schematic layout of the configuration, which is the same as that in Fig. 2 in the main text. Here, the tiny nonlinear particle is placed at position 1 where the SHG quenching effect is expected. (b)–(d) Normalized SHG scattering cross section ${SCS}^{2 ω}$ with respect to the (b) $r_{e} / λ_{0}$ in the case of $r_{n} / λ_{0} = 0.001$ and $d / λ_{0} = 0.0005$ , (c) $r_{n} / λ_{0}$ in the case of $r_{e} / λ_{0} = 0.01$ and $d / λ_{0} = 0.0005$ , and (d) $d / λ_{0}$ in the case of $r_{e} / λ_{0} = 0.01$ and $r_{n} / λ_{0} = 0.001$ . In (b)–(d), the ${SCS}^{2 ω}$ is normalized to the ${SCS}^{2 ω}$ in Fig. 2, in which case the geometrical parameters are set as $r_{e} / λ_{0} = 0.01$ , $r_{n} / λ_{0} = 0.001$ , and $d / λ_{0} = 0.0005$ (marked by the vertical dashed lines).

Figure 7.(a) Normalized scattering cross section of SHG ${SCS}^{2 ω}$ from the tiny nonlinear particle when it is successively moved from position 1 to position 5 along the trajectory $1 \to 2 \to 3 \to 4 \to 5$ . The configuration is the same as that in Fig. 2. (b)–(d) Simulated fundamental electric-field amplitude on the ENZ sphere’s surface on the air side for the cases with material losses.

Figure 8.Distributions of normalized fundamental electric-field amplitude in one unit cell of the optical metasurface on the plane that contains the graphene. The metasurface is the same as that in Fig. 5. (a) Effective-medium model in the crystalline phase; (b) effective-medium model in the amorphous phase; (c) multilayer-structure model in the crystalline phase; and (d) multilayer-structure model in the amorphous phase.

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