1. Introduction

Nonlinear wave mixing processes are employed for sensing [1–4], sum- and difference frequency generation [5–7], and the generation of entangled photon pairs [8–10]. In recent years, monolayers of transition metal dichalcogenides (TMDCs) such as MoS₂ have shown great promise as they can be combined with various photonic platforms including silicon compounds [11,12], but also show a strong second-order nonlinear response caused by their intrinsically broken inversion symmetry [13–15]. Hence, TMDCs allow to design of the linear and nonlinear response of photonic platforms to a certain extent independently. But, unfortunately, conversion efficiencies of TMDCs are limited by their atomic thickness. Thus, design strategies to enhance light-matter interaction in TMDCs are required. Here we focus on second harmonic generation (SHG) aiming at the up conversion of a fundamental harmonic (FH) into a second harmonic (SH) wave at double the frequency. But, our approach can likewise be applied to other nonlinear wave mixing processes based on light matter interaction in TMDCs.

Numerical simulations have shown that SHG in TMDCs is enhanced by orders of magnitude if the 2D material sheet is combined with resonant photonic structures [16,17] as optical cavities [18,19], waveguides [20,21], resonant plasmonic structures [22–24] and dielectric metasurfaces [25,26]. Experiments confirmed these predictions for one-dimensional (1D) and two-dimensional (2D) photonic crystals (PCs) [25,27]. All these structures have in common that they are resonant at the FH frequency. Further improvement is expected for double resonant structures simultaneously enhancing FH and SH fields. However, such an approach requires a kind of phase matching and is usually hindered by dissimilar refractive indices at FH and SH wavelengths. In a recent theoretical work, the authors proposed to circumvent that problem by using arrays of metallic nanostructures the dipolar and quadrupolar modes of which are resonant at FH and SH frequencies [28]. Numerical simulations suggest four orders of magnitude enhancement compared with the bare structure. In another experimental paper, the authors report 16-fold enhancement based on a TMDC embedded in a layered cavity structure and Bragg reflectors with stop bands at FH and SH frequencies [29].

Here we follow a different and scalable approach based on a very simple all-dielectric planar waveguide which is decorated with periodically arranged trenches and a TMDC on top. We show that deliberately breaking the symmetry in such 1D and 2D metasurfaces can be used to match resonances at FH and SH frequencies. In the case of the 2D structure limits of achievable enhancement are not set by the structure itself, but by fabrication tolerances and the linewidth of the exciting laser.

We numerically determine the resulting SH enhancement by a generalized source finite-difference time-domain (GS-FDTD) approach, which is well suited for low conversion efficiencies and allows us to apply the FDTD method to nonlinear 2D materials by introducing the concept of a nonlinear generalized source [30].

2. Results and discussion

Based on the GS-FDTD method, the nonlinear optical interaction is modeled by conducting the nonlinear simulation based on two distinct linear FDTD simulations. Since the 2D TMDC is modeled as a generalized surface current, its thin layer compared to the rest of the structure can be neglected during the simulations. In the initial linear simulation, the excitation via a continuous-wave (CW) source at the FH wavelength 880 nm is considered. It excites a second order polarization in the MoS₂ layer via the susceptibility tensor of the D_3h point-group $\chi _{x^{\prime} x^{\prime} x^{\prime} }^{(2 )} ={-} \chi _{x^{\prime} y^{\prime} y^{\prime} }^{(2 )} ={-} \chi _{y^{\prime} y^{\prime} x^{\prime} }^{(2 )} ={-} \chi _{y^{\prime} x^{\prime} y^{\prime} }^{(2 )} = 5{\; }{10^{ - 9}}\textrm{m}\; {\textrm{V}^{ - 1}}$ where $x^{\prime}$ corresponds to the arm-chair and $y^{\prime}$ to the zig-zag direction as reported in [2,31]. The time derivative of this second-order nonlinear polarization acts as a nonlinear generalized source current in another FDTD simulation modeling SHG in the structure. The employed generalized source current models the electromagnetic response of the 2D TMDC. Consequently, the need for a bulk layer representation of the 2D material is eliminated in simulation [30].

The dielectric materials (silica, silicon nitride) used in the simulations have negligible dispersion in the investigated frequency domain. Respective refractive indices were taken from [32,33].

2.1 MoS₂ on 1D PCs

First, we assume a simple grating structure with a MoS₂ monolayer on top (see a principal sketch of the structure in Fig. 1(a)). To induce resonances at FH and SH wavelength we merge, two nonequivalent unit cells (Fig. 1(b) and 1(c)) into a single bigger one (Fig. 1(d)).

 

Fig. 1. 1D PCs with a MoS₂ monolayer placed on top (a) principal sketch and (b-d) construction of the unit cell of a double resonant grating by combining a symmetric 1D PCs (b) with a period of Λ to couple the FH light into the guiding layer and one with a shorter period Λ’ (c) to couple the SH light. Thus, a unit cell of an asymmetric double periodic grating (d) with a period of Λ is formed. (e) Linear reflection spectra of symmetric and asymmetric 1D PCs shown in (b-d) for normal incident and polarization parallel to the grating bars.

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We start with a grating on top of a high index silicon nitride layer (see Fig. 1(b), period Λ = 540 nm, width w = 500 nm, depth h = 120 nm, thickness of silicon nitride layer on silica d = 180 nm). For an excitation under normal incidence at FH frequency its first diffracted order excites the fundamental guided mode of the high index layer. As a consequence, a leaky wave resonance (LWR) appears at FH frequency causing a sharp peak in the reflection spectrum at 880 nm (see dashed blue line in Fig. 1(e)). A further LWR, which is caused by the excitation of the same waveguide mode via the second order of the grating shows up at 480 nm. However, due to waveguide dispersion, this second LWR does not match the SH wavelength and no resonant enhancement of the SH wave is obtained.

In contrast, a grating with a much shorter period (see the blue area in Fig. 1(c), period Λ’ = 270 nm, width w’ = 130 nm) excites the waveguide mode via its first diffracted order at SH frequency as demonstrated by a peak in reflection at 440 nm (see dashed green line in Fig. 1(e)). But this grating now lacks the resonance at FH frequency.

To induce resonances at FH and SH frequencies we partially combine both structures (yellow and blue areas in Fig. 1(c)). Thus, we break the symmetry in the unit cell by slightly increasing the width of one of the bars (see Fig. 1(d), widths dw = 60 nm). The resulting asymmetric structure has still the right period Λ = 540 nm to excite the FH mode. But, it shows also signatures of the high frequency grating (displayed in Fig. 1(c)). In particular it still reproduces the LWR at 440 nm. Hence, this double periodic grating (see Fig. 1(d)) shows peaks in reflection at both FH and SH wavelengths (see red line in Fig. 1(e)).

In addition, a further resonance shows up in the reflection spectrum at 800 nm. These two LWRs around the FH frequency are caused by coupling forward and backward propagating guided modes by the second order of the grating (see intensity profiles of resulting super modes in Figs. 2(a) and 2(b)). Two similar LWR states, a symmetric and an antisymmetric one, also exist for the symmetric grating as displayed in Fig. 1(b), but only the symmetric one can be excited under normal incidence (see Fig. 1(e), blue dashed line). Breaking the symmetry of the unit cells makes both modes optically accessible (see Fig. 1(d)), although they still differ considerably with respect to their field shapes. Maxima are found either at the bars or the trenches. Consequently, the LWR at 880 nm, which is mainly localized inside the bars, is very sensitive to an increase in their width. In contrast, the LWR at SH frequency is located merely, below the grating trenches (see Fig. 2(c)) and is almost not affected by changes in the bars. Hence, the width of the bars is a convenient tuning parameter to shift the LWR at 880 nm with respect to that at 440 nm thus giving us a tuning parameter to obtain resonances exactly at FH and SH frequencies.

 

Fig. 2. Mode profiles of the asymmetric 1D PC at selected resonances (red line in Fig. 1(e)) via first-order diffraction (a) at FH wavelength, (b) at 800 nm, and (c) second-order diffraction at SH wavelength.

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The double resonant 1D PC structure provides not only a strong excitation of TMDCs placed on top by the FH field but also an efficient extraction of generated SH by resonant scattering towards free space. The asymmetric double resonant 1D PC structure allows for 20 times more efficient SHG compared to the symmetric 1D PC (see Fig. 1(b)), which is only resonant for the FH frequency (see Fig. 3) and 4600 times more enhancement compared with SH generation in the structure being resonant at SH wavelength only (see Fig. 1(c)).

 

Fig. 3. Normalized SH intensity from MoS₂ for varying wavelengths of the FH driving field. The electric field is polarized parallel to the bars and the TMDC sheet is assumed to be oriented such that its armchair axis is parallel to the bars.

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2.2 SHG in MoS₂ on a 2D PCs

The grating structure discussed above does not utilize the full flexibility of a 2D metasurface covered by a TMDC sheet. When considering a genuine 2D metasurface the strong polarization dependence of SHG in TMDCs must be considered. It is determined by the second-order susceptibility tensor with D_3h point-group symmetry as introduced above. The generated SH is parallel to the driving FH field only if both are directed in the armchair direction. If the FH field deviates from that direction by an angle $\varphi $ the SH polarization turns into the opposite direction without changing its magnitude. Therefore, the part of the SH field pointing into the excitation direction is modulated like $\propto \cos ({3\varphi } )$.

Provided that the orientation of the TMDC sheet is chosen correctly a SH waveguide mode propagating orthogonally to the original FH mode can be excited. Hence, a 2D PC allows disentangling the two resonances at FH and SH wavelength as we will demonstrate below.

Again, we study an asymmetric, but 2D PC. Here, asymmetry refers to two different periods Λ and Λ’ implemented into orthogonal directions (see the unit cell in Fig. 4(a), etching width: 490 nm, etching depth: h = 20 nm, silicon nitride layer of thickness d = 140 nm). The grating period Λ in the x-direction is adjusted to create a resonance at FH wavelength for z-polarized FH light exciting the structure under normal incidence (red line in Fig. 4(b)). In contrast grating periods Λ’ in the z-direction is chosen such that a grating resonance is created at SH wavelength for x-polarized light (blue line in Fig. 4(b),). The TMDC sheet on top is oriented such that its armchair direction has a 30-degree angle with FH z-polarized excitation. For that orientation of the TMDC, a z-polarized FH driving field will generate an SH with x-polarization, both fields being resonant for their specific propagation directions.

 

Fig. 4. (a) Schematic of a unit cell of asymmetric 2D PC with periods of Λ= 540 and Λ’ = 240 nm, etching widths of w = 490 nm, and w’ = 200 nm (red and blue arrows indicate FH and SH polarization, respectively). (b) Reflection spectra of asymmetric 2D PCs under normal incident.

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For comparison, we study a symmetric 2D PC with a square lattice the period of which is adjusted such that it resonates at FH wavelength (period 540 nm). It has a resonance at 880 nm, but no resonance at SH wavelength.

Again, the asymmetric double resonant structure performs 170 times better than the symmetric one with the square pattering (Fig. 5).

 

Fig. 5. Normalized SH intensity from MoS₂ at the SH wavelength.

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Please note that SH generation efficiency in this structure is critically determined by the etching depth and width. The shallower the etching, the higher the quality factor and the narrower the line width of respective resonances. In principle, even much higher q-factors can be obtained, but technical limitations of manufacturing, as well as the linewidth of the exciting laser, will set the limits for this kind of structure. But, for the assumed asymmetric 2D configuration SH generation is again about 60 times more efficient than that in the already optimized asymmetric 1D PC discussed first. This underlines the great potential offered by the second available transverse dimension.

3. Conclusion

To conclude, coupling a TMDC monolayer with a metasurface can greatly enhance SH generation, particularly if the linear structure shows resonant behavior at both FH and SH wavelengths. We have demonstrated two approaches to obtain that goal: First, we combined two 1D gratings thus breaking the mirror symmetry within the unit cell. In a second approach, we employed the second dimension of a metasurface and the polarization sensitivity of SHG in TMDCs. By using a rectangular lattice with dissimilar base lengths, we designed an optical structure that is resonant at FH and SH frequencies to enhance SH generation compared with structures that support only FH or SH resonances. The two optimization schemes demonstrated in this paper can likewise be applied to other excitation frequencies and even to other frequency conversion processes, as long as one aims at resonances at all interacting frequencies e.g. in spontaneous parametric down-conversion.