1. Introduction

In recent years, the optical properties of transition-metal dichalcogenide (TMDC) monolayers have been the subject of studies because of the recent advances in the applications and physical understanding of two-dimensional materials learned from graphene [1–4]. As a centrosymmetric material, graphene exhibits large third-order nonlinear effects such as third-harmonic generation [5], strong optical Kerr nonlinearity [6], however the even-order nonlinear processes of graphene are inhibited. As a complementary the TMDC monolayers have noncentrosymmetric atomic lattice and hence allow even-order nonlinear optical processes [7–10]. These properties allow one to employ TMDC monolayers in active photonic devices with improved functionality, the strength of interaction between TMDC monolayers and electromagnetic (EM) waves plays a central role in the application. In general, such an interaction is very weak due to the single atom thickness of TMDC monolayers. The weak interactions block efficient generation of nonlinear effects and potential applications of TMDC monolayers in optoelectronic devices. Thus, how to improve the interaction between TMDC monolayers and EM waves and realize the efficient generation of nonlinear effects has become an important research topic.

Various methods to improve the interaction between TMDC monolayers and EM waves have been proposed. For example, the third-harmonic generation may be dramatically enhanced in graphene by inserting it in a properly designed one-dimensional photonic crystals (PCs) [11–13], using ultrastrong localized surface plasmon resonances [14–16] or combing with waveguide [17]. In our work, we attempt to explore whether or not this interaction can be realized by utilizing light bound states in the continuum (BIC) which are embedded trapped modes corresponding to discrete eigenvalues coexisting with extended modes of a continuous spectrum [18–20]. It is found that many structures support BIC such as the dielectric grating [18], waveguide structures [19, 21–23], surface of the object [20, 24], photonic crystal slabs [25, 26], they may be implemented more easily than the electron system in artificially designing the potential. The effect of the bound states in the continuum on the second harmonic (SH) generation from TMDC monolayer has not been investigated before, in our work we attempt to explore whether or not the large enhancement can be realized utilizing the field confinement.

To study this problem, a two layer structure where a TMDC monolayer is put on a grating is proposed, BICs which are realized in this structure can be view as a mode with an infinite quality factor Q which has no radiation leakage, however, the Fano resonances close to these BICs support radiation modes with ultrahigh Q factors [27, 28]. We find that the SH generation from the TMDC monolayer can be enhanced dramatically by using the Fano resonance near BICs. In order to complete this study, the scattering matrix method to study the layered structured has been developed to include the nonlinear effect. The physical origin for this phenomenon has also been analyzed. The rest of the paper is organized as follows: we will describe the theory and method in Sec. II, present the results and discussions in Sec. III, and give a summary in Sec. IV.

2. Theory and method

We consider a two layer structure where one TMDC monolayer is deposited on the one dimensional (1D) grating as shown in Fig. 1. Now suppose the grating is periodic along the x direction and the lattice constant is $a$. The width, thickness and relative permittivity of the grating are represented by $w$, $d$ and $\epsilon$, respectively, the grating is immersed in the air. The relative permittivity and effective thickness of the TMDC monolayer are denoted by ${\epsilon }_t$ and $d_t$, meanwhile, the EM properties of two dimensional material can alternatively be described by the conductivity $\sigma$, which is related to ${\epsilon }_t$ and $d_t$ through the following relation: $\sigma =-i{\epsilon }_0\omega d_t{\chi }_t$, where ${\epsilon }_0$ represents the permittivity of the vacuum and ${\chi }_t={\epsilon }_t-1$ is the linear polarizability, the angular frequency of EM wave is denoted by $\omega$. All these material are nonmagnetic. The EM can be well confined in the grating due to the formation of a bound states in a continuum (BICs) of radiation modes, near BICs the EM can be dramatically enhanced at the TMDC monolayer location, utilizing this property, we study the SH generation from TMDC monolayer near BICs.

 

Fig. 1 Diagram of the TMDC monolayer-grating structure and the SH generation process (oblique view shown in (a) and side view presented in (b)). The lattice constant, width, thickness and relative permittivity of the grating are marked by $a$, $w$, $d$ and $\epsilon$, respectively, the grating is immersed in the air. When a plane wave with angular frequency $\omega$ is incident on the structure at the incident angle ${\theta }_i$, the transmitted and reflected SH fields are generated due to the second-order nonlinear effect of TMDC monolayer.

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The relative permittivity of TMDC monolayers which is equal to a superposition of N Lorentzian functions is given by ${\epsilon }_{\text{TMDC}}=1+{\displaystyle {\sum }_{k=1}^N{f}_k/\left({\omega }_k^2-{\omega }^2-i\omega {\gamma }_k\right)}$ [29, 30], where $f_k$,${\omega }_k$ and ${\gamma }_k$ represent the oscillator strength, resonance frequency and spectral width of the kth oscillator, respectively, the values of these model parameters are provided in [30]. In contrast to graphene, TMDC monolayers are noncentrosymmetric and therefore the second order nonlinear effect is permitted. Based on the symmetry properties of their space group $D_{3h}$, it can be shown that the structure of their quadratically susceptibility tensor ${\chi }^{\left(2\right)}$ yields only one independent and nonvanishing component [7,8]: ${\chi }^{\left(2\right)}={\chi }_{yyy}^{\left(2\right)}=-{\chi }_{yxx}^{\left(2\right)}=-{\chi }_{xyx}^{\left(2\right)}=-{\chi }_{xxy}^{\left(2\right)}$, where we set the zigzag and armchair directions of the monolayer as the x and y directions. Numerical values for $d_t$ of the TMDC monolayers are also taken from [30]. In a similar manner to the linear property, the second-order optical property can be alternatively described by the corresponding nonlinear surface current ${\overrightarrow{J}}^{\left(2\right)}\left(2\omega \right)=-i{d}_{t}2\omega {\overrightarrow{P}}^{\left(2\right)}\left(2\omega \right)$, where $2\omega$ and ${\overrightarrow{P}}^{\left(2\right)}\left(2\omega \right)$ are the angular frequency of the SH EM wave and second-order polarization density. After giving the optical parameters of the two layer structure, in the following we describe detailedly the calculation of the EM fields at fundamental and SH frequencies in this structure via the surface current description of the TMDC monolayer.

When a plane wave ${\overrightarrow{E}}_0^{+}\left(\overrightarrow{r},\omega \right)$ at fundamental frequency is incident upon the two layer structure at the incident angle ${\theta }_i$ as shown in Fig. 1(a), the transmitted and reflected SH fields (${\overrightarrow{E}}_2^{+}\left(\overrightarrow{r},2\omega \right)$ and ${\overrightarrow{E}}_0^{-}\left(\overrightarrow{r},2\omega \right)$) are generated due to the second-order nonlinear effect of TMDC monolayer. Here we assume that each layer is surrounded by an infinitely thin film of air in both hand sides and the number of the subscript of the electric field represents the corresponding air region as shown in Fig. 1(b). In our work, the EM waves ${\overrightarrow{E}}_k\left(x,y,\omega \right)$ ($k=0,1,2$) at infinitely thin films of air are expanded by plane waves due to the periodicity of this structure

The EM field at fundamental frequency can be calculated by the scattering matrix theory, of which the details of the calculation process are described in [31, 32]. In the first place, we calculate every scattering matrix of these two layers at fundamental frequency [33]. elaborates the method to get the scattering matrix of the graphene via its conductivity, it is very easy to extend this method to include the TMDC monolayer, the scattering matrix of the photonic crystal slab can be calculated using plane wave expansion method in [34]. In the next place, the total scattering matrix of the two layer structure can be obtained by combing every scattering matrix of each layer. At last, the transmitted and reflected fields are calculated out via total scattering matrix, then the fields at every infinite thin film are obtained through every scattering matrix of each layer.

In the following, we describe the calculation of the SH electric field in great detail. The previous scattering matrix method is used to calculate the linear optical property, in our work, this method is extended enormously to include the nonlinear effect. Based on the calculated fundamental electric field at the TMDC monolayer location and the quadratically susceptibility tensor ${\chi }^{\left(2\right)}$, the nonlinear surface current ${\overrightarrow{J}}^{\left(2\right)}\left(2\omega \right)$ is obtained and can be also expanded in plane waves

Based on the scattering matrixes $Q_1^{\eta }\left(2\omega \right)$ $\left(\eta =I,II,III,IV\right)$, $C_1^{\kappa }\left(2\omega \right)$ $\left(\kappa =I,II\right)$ of the TMDC monolayer and $Q_2^{\eta }\left(2\omega \right)$ $\left(\eta =I,II,III,IV\right)$, $C_2^{\kappa }\left(2\omega \right)$ $\left(\kappa =I,II\right)$ of the grating, the total scattering matrixes $Q_{1,2}^{\eta }\left(2\omega \right)$ $\left(\eta =I,II,III,IV\right)$, $C_{1,2}^{\kappa }\left(2\omega \right)$ $\left(\kappa =I,II\right)$ of the two layer structure can be obtained

Utilizing the total scattering matrix shown in Eq. (7), we can obtain the generated SH electric fields. Then the electric field intensities of the transmitted ($I_t\left(2\omega \right)$) and reflected ($I_r\left(2\omega \right)$) SH fields can be expressed as

In order to compare the SH generation from the two layer structure with those from the freestanding TMDC monolayer, the enhancements of the transmitted and reflected SH field (ET and ER) are introduced as

3. Results and discussions

In this section, we present numerical results for the SH generation from the two layer structure. The relative permittivity of the grating is taken as $\epsilon =15.21$, which corresponds to ${\text{Si}}_{\text{0}\text{.7}}{\text{Ge}}_{\text{0}\text{.3}}$ [38, 39]. As a typical material of TMDC monolayers, we take ${\text{WS}}_{\text{2}}$ monolayer as the object to give the numerical results, similar phenomena can recur for the other TMDC monolayers. The effective thickness of ${\text{WS}}_{\text{2}}$ monolayer is $d_t=0.618\text{nm}$ and its relative permittivity can also be obtained [30]. The second-order nonlinear susceptibility of the monolayer ${\text{WS}}_{\text{2}}$ is chosen to be ${\chi }^{\left(2\right)}=100\text{pm}/\text{V}$ which is compatible with the results in [7] and the second-order effect of the grating can be neglected.

We plot the transmission and absorption as a function of the wavelength $\lambda$ of the EM wave and the thickness $d$ of the grating under normal incident EM wave in Figs. 2(a) and 2(b). The lattice constant and width of the grating are selected as $a=460\text{nm}$ and $w=0.6a$, the EM wave is incident normally and polarized along the y direction. It is obviously seen that the transmission and absorption spectrums exhibit many sharp resonance features with high Q factor, there are many embedded states of which the linewidth and absorption become zero, these are called BICs [40] indicated by the arrows and square in Figs. 2(a) and 2(b). These BICs are symmetry-protected bound states at zero in-plane wave vector, at these BICs the reason why leakage radiation to the surface normal direction and absorption are forbidden is that the symmetry of the EM field in the grating is incompatible with the external radiation. In order to give a further analysis, the square absorption region shown in Fig. 2(b) are magnified in Fig. 2(c) and three absorption spectrums indicated by the dashed lines in Fig. 2(c) for three different thicknesses $d=312.8\text{nm}$, $d=321.71434\text{nm}$, and $d=331.2\text{nm}$ are plotted in Figs. 2(d), 2(e) and 2(f), respectively.

 

Fig. 2 Transmission (a) and absorption (b) for the two layer structure as a function of the wavelength $\lambda$ of the EM wave and thickness $d$ of the grating. (c) Magnified absorption map for the square region in (b). (d)-(f) Corresponding absorption spectrums for different thicknesses of the grating $d=312.8\text{nm}$ (d), $d=321.71434\text{nm}$ (e) and $d=331.2\text{nm}$ (f). The parameters are assumed as follows: $a=460\text{nm}$ and $w=0.6a$.

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Exact BICs with no radiation leakage do not interact with the continuum states, hence the absorption at BICs are zero. A BIC is formed in Fig. 2(c) that can be seen clearly by the asymptotic behavior of the absorption spectrums shown in Figs. 2(d)-2(f), at $d=321.71434\text{nm}$ the absorption shown in Fig. 2(e) become nearly zero and the BIC appear, near this BIC sharp absorption resonances at $d=312.8\text{nm}$ and $d=331.2\text{nm}$ are shown in Figs. 2(d) and 2(f). We actually measure the BICs from the experiment in an asymptotic method that the transmission, reflection or reflection of the leaky modes asymptotically approach a singularity in the corresponding spectrums. As shown in Fig. 2(c) when the configuration approach the BIC, the absorption increases gradually and then decreases to nearly zero at the BIC, meanwhile, the linewidth of the absorption narrows gradually and becomes nearly zero at the BIC.

We are interested in the slow light effect near BICs which are very benefit for the design of optical devices. A critical point is the maximal electric field enhancement at the surface of the grating which is related to absorption in our work, this field enhancement is defined as (${\displaystyle \iint \left|\overrightarrow{E}/{\overrightarrow{E}}_i\right|dS}/{\displaystyle \iint dS}$), where the relative field $\overrightarrow{E}$ are integrated over the period surface of the grating and ${\overrightarrow{E}}_i$ represents the incident field. Figures 3(a) and 3(b) illustrate the maximal absorption and maximal field enhancement as a function of the thickness $d$ of the grating, respectively. One maximal point is gotten by selecting the maximum from the absorption or field enhancement spectrum at different wavelength for a known thickness of the grating. It can be found that the maximal absorption and maximal field strength exhibit similar phenomena, in order to disclose this, we plot the position, the wavelength $\lambda$ of the EM wave and thickness $d$ of the grating, of the maximal absorption and maximal field enhancement in Fig. 3(c) and find that these two positions are coincide with each other, that is to say the increase or decrease of the absorption is originate from the electric field enhancement at the ${\text{WS}}_{\text{2}}$ monolayer location. The valley points in Figs. 3(a) and 3(b) correspond to the BIC which exhibits lowest absorption and field enhancement, as the configuration approach to the BIC the maximal absorption and maximal field enhancement at the resonance increases gradually and then decrease to minimal value at the BIC, two peaks at $d=317.75972\text{nm}$ and $d=325.87182\text{nm}$ are formed near the BIC for each of these two figures, the corresponding peaks are 0.437 and 0.574 in Fig. 3(a) and are 18.966 and 20.615 in Fig. 3(b). We plot the absorption spectrum at the second peak in Fig. 3(d) and find that a very sharp resonance appears, the sharpness can be described by the relative line width $\Delta \lambda /{\lambda }_m$, where ${\lambda }_m$ is the center wavelength of the peak and $\Delta \lambda$ represents full width at half maximum of the peak, in this figure the relative linewidth is 0.0167% which is much less than that in other works [41]. The topic we are interested in is the effect of the critical field enhancement on the SH generation from the two dimensional material ${\text{WS}}_{\text{2}}$ monolayer.

 

Fig. 3 (a) Maximal absorption near BIC as a function of the thickness of the grating. (b) Maximal field enhancement near BIC as a function of the thickness $d$ of the grating. (c) The position spectrum of the maximal absorption (black line with black symbol) and maximal field enhancement (red line with red symbol) indicated by the wavelength $\lambda$ of the EM wave and the thickness $d$ of the grating. (d) Absorption spectrum at the maximum point in (a) $d=325.87182\text{nm}$. The other parameters are identical to those in Fig. 2.

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In order to study the effect of the resonant states near the BIC on the SH generation from the ${\text{WS}}_{\text{2}}$ monolayer, we plot in Fig. 4 the SH generation enhancement ET of the transmitted field and ER of the reflected field from the two layer structure compared with that from the freestanding ${\text{WS}}_{\text{2}}$ monolayer for different thicknesses of the grating. It is clearly seen that the resonant enhancements ET and ER appear and are redshifted that is in agreement with the resonant features in Figs. 2(c)-2(f) and 3(d). The enhancements ET and ER simultaneously reach 6.000 and 3.3326 at $\lambda =1015.076\text{nm}$ and $d=321.71434\text{nm}$ which is corresponding to the BIC, due to no radiation leakage and interaction with the continuum states at the BIC, the enhancements ET and ER are so small that these can be neglected. At $\lambda =1017.821\text{nm}$ and $d=325.87182\text{nm}$ which is corresponding to the maximum of maximal field enhancement in Fig. 3 ET and ER can reach as much as 19880.9 and 14912.7 shown in Fig. 4(c). For comparison we also provide the ET and ER spectrum for ordinary Fano resonances, specially, ET and ER reach 2637.22 and 887.492 at $\lambda =1008.362\text{nm}$ and $d=312.8\text{nm}$ in Fig. 4(a), 7458.30 and 9188.38 at $\lambda =1021.120\text{nm}$ and $d=331.2\text{nm}$ in Fig. 4(d). The relative linewidths of the ET spectrums are 0.04166%, 0.01101% and 0.02939% for Figs. 4(a), 4(c) and 4(d), respectively, hence the resonant peak is narrowed when the configuration approaches to the BIC.

 

Fig. 4 SH generation enhancement ET of the transmitted field (a)-(d) and ER of the reflected field (e)-(h) as a function of the wavelength $\lambda$ of the incident field for different thicknesses of the grating. The thicknesses of the grating are taken as follows: $d=312.8\text{nm}$ for (a) and (e), $d=321.71434\text{nm}$ for (b) and (f), $d=325.87182\text{nm}$ for (c) and (g), and $d=331.2\text{nm}$ for (d) and (h). The other parameters are identical to those in Fig. 2.

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In addition to the resonance, the SH generation enhancement ET and ER can also be tuned by the azimuthal angle $\varphi$ of the ${\text{WS}}_{\text{2}}$ monolayer. This is because its anisotropic second-order nonlinear susceptibility depends on $\varphi$, if the ${\text{WS}}_{\text{2}}$ monolayer is rotated $\varphi$ clockwise, the corresponding second-order nonlinear polarization can be expressed as

 

Fig. 5 SH generation enhancement ET of the transmitted field (solid line) and ER of the reflected field (dashed line) as a function of the azimuthal angle of the ${\text{WS}}_{\text{2}}$ monolayer. The wavelength of the EM wave and thicknesses of the grating are taken as $\lambda =1017.821\text{nm}$ and $d=325.87182\text{nm}$ corresponding to the maximum point in Fig. 4(c).

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So far the BICs are observed via tuning the thickness of the grating, in fact the BICs can be observed by changing the incident angle ${\theta }_i$ of the EM wave. Figure 6(a) shows the absorption as a function of the wavelength $\lambda$ and incident angle ${\theta }_i$ of the EM wave. The incident wave is TE (transverse electric) polarized along the y direction. Furthermore, the corresponding maximal field enhancement and maximal absorption as a function of the incident angle ${\theta }_i$ are shown in Figs. 6(b) and 6(c), respectively, it can be observed clearly that there are three BICs where the values of absorption are nearly zero and there are also peaks of the maximal field enhancement and maximal absorption around these three BICs that is similar to the previous study. In Fig. 6(d), we plot the absorption spectrum at ${\theta }_i=16.1723°$ which corresponds to the maximum point in Fig. 6(b) and find a very sharp resonant peak with relative linewidth 0.0184%.

 

Fig. 6 (a) Absorption for two layer structure as a function of the wavelength $\lambda$ and incident angle ${\theta }_i$ of the EM wave. Maximal field enhancement (b) and maximal absorption (c) near BIC as a function of the incident angle ${\theta }_i$ of the EM wave. (d) Absorption spectrum at the maximum point in (b) ${\theta }_i=16.1723°$. The parameters are assumed as follows: $a=460\text{nm}$, $w=0.6a$ and $d=1.0a$.

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Figures 7(a) and 7(b) display the SH generation enhancement ET of the transmitted field and ER of the reflected field from the two layer structure compared with the freestanding ${\text{WS}}_{\text{2}}$ monolayer at ${\theta }_i=16.1723°$ which corresponds to the peak of maximal field enhancement in Fig. 6(b). Like the case in Fig. 4, sharp resonant peaks appear for the transmitted and reflected fields, ET and ER reach 353.5 and 1893.1 at $\lambda =1030.075\text{nm}$, respectively, and the relative linewidth of the ET peak is 0.01651%.

 

Fig. 7 SH generation enhancement ET of the transmitted field (a) and ER of the reflected field (b) as a function of the wavelength $\lambda$ of the incident field at ${\theta }_i=16.1723°$. The other parameters are identical to those in Fig. 6.

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4. Conclusion

We have designed TMDC monolayer-grating structure to realize the large SH generation enhancement based on the BICs and proposed the extended scattering matrix theory including the nonlinear effect of TMDC monolayer. Two methods to observe the optical BICs in this structure are considered, one is tuning the thickness of the grating, the SH generation can be enhanced by more than four orders of magnitude, the other is changing the incident angle of the EM wave, the enhancement can reach three orders of magnitude. The enhancements in the two cases also exhibit outstanding monochromaticity due to the small linewidth about 0.01% and can be tuned in a large range by adjusting the azimuthal angle of the TMDC monolayer. We believe these phenomena are very beneficial for the design of optical devices.

Appendix Scattering matrix of TMDC monolayer at second harmonic frequency

1 Transmission and reflection of TMDC monolayer including second-order nonlinear effect

In order to obtain the transmission and reflection of the TMDC monolayer including second-order nonlinear effect [42], we assume an incident plane wave ${\overrightarrow{E}}^i$ at second harmonic angular frequency ${\omega }_{\text{SH}}=2\omega$ is incident from the x-z plane shown in Fig. 8, the transmitted and reflected plane waves (${\overrightarrow{E}}^t$ and ${\overrightarrow{E}}^r$) are generated. Due to the ultrathin thickness the TMDC monolayer is regarded as an interface, the electric and magnetic fields at the interface can be related to the following boundary conditions:

(12)$$\{\begin{array}{c}{E}_x^t-E_x^i-E_x^r=0,H_x^t-H_x^i-H_x^r=\sigma E_y+J_y^{\left(2\right)},\\ E_y^t-E_y^i-E_y^r=0,H_y^t-H_y^i-H_y^r=-\sigma E_x-J_x^{\left(2\right)},\end{array}$$

here ${\overrightarrow{H}}^i$, ${\overrightarrow{H}}^t$ and ${\overrightarrow{H}}^r$ represent the incident, transmitted and reflected magnetic field, $J_x^{\left(2\right)}$ and $J_y^{\left(2\right)}$ denote the nonlinear surface current originating from the second-order nonlinear effect of the TMDC monolayer.

 

Fig. 8 Transmission and reflection of a plane wave from a TMDC monolayer at the SH frequency. The incident, transmitted and reflected waves are denoted by ${\overrightarrow{E}}^i$, ${\overrightarrow{E}}^t$ and ${\overrightarrow{E}}^r$, the mediums up and down the interface are marked by the number 1 and 2.

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The magnetic field can be expressed by the electric field with the Maxwell equations, then after careful calculation the relation between the electric fields up and down the interface is obtained

(13)$$\{\begin{array}{c}{E}_x^t=T_x{E}_x^i+C_{tx},E_y^t=T_y{E}_y^i+C_{ty},\\ E_x^r=R_x{E}_x^i+C_{rx},E_y^r=R_y{E}_y^i+C_{ry}.\end{array}$$

The transmission and reflection coefficients $T_x$, $T_y$, $R_x$ and $R_y$ are given by

(14)$$\{\begin{array}{c}{T}_x=\frac{2q_{2z}/{\epsilon }_2}{q_{1z}/{\epsilon }_1+q_{2z}/{\epsilon }_2+q_{1z}{q}_{2z}\sigma /\left(\omega {\epsilon }_0{\epsilon }_1{\epsilon }_2\right)},T_y=\frac{2q_{1z}}{q_{1z}+q_{2z}+\omega {\mu }_0\sigma },\\ R_x=\frac{q_{2z}/{\epsilon }_2-q_{1z}/{\epsilon }_1-q_{1z}{q}_{2z}\sigma /\left(\omega {\epsilon }_0{\epsilon }_1{\epsilon }_2\right)}{q_{1z}/{\epsilon }_1+q_{2z}/{\epsilon }_2+q_{1z}{q}_{2z}\sigma /\left(\omega {\epsilon }_0{\epsilon }_1{\epsilon }_2\right)},R_y=\frac{q_{1z}-q_{2z}-\omega {\mu }_0\sigma }{q_{1z}+q_{2z}+\omega {\mu }_0\sigma },\end{array}$$

and the nonlinear terms $C_{tx}$, $C_{ty}$, $C_{rx}$ and $C_{ry}$ are written as

(15)$$\{\begin{array}{c}{C}_{tx}=\frac{-J_x^{\left(2\right)}{q}_{1z}{q}_{2z}/\left(\omega {\epsilon }_0{\epsilon }_1{\epsilon }_2\right)}{q_{1z}/{\epsilon }_1+q_{2z}/{\epsilon }_2+q_{1z}{q}_{2z}\sigma /\left(\omega {\epsilon }_0{\epsilon }_1{\epsilon }_2\right)},C_{ty}=\frac{-J_y^{\left(2\right)}\omega {\mu }_0}{q_{1z}+q_{2z}+\omega {\mu }_0\sigma },\\ C_{rx}=\frac{-J_x^{\left(2\right)}{q}_{1z}{q}_{2z}/\left(\omega {\epsilon }_0{\epsilon }_1{\epsilon }_2\right)}{q_{1z}/{\epsilon }_1+q_{2z}/{\epsilon }_2+q_{1z}{q}_{2z}\sigma /\left(\omega {\epsilon }_0{\epsilon }_1{\epsilon }_2\right)},C_{ry}=\frac{-J_y^{\left(2\right)}\omega {\mu }_0}{q_{1z}+q_{2z}+\omega {\mu }_0\sigma },\end{array}$$

here $q_{1z}$ ($q_{2z}$) is the z component of wave vector in the upper medium 1 (down medium 2), ${\epsilon }_0$ and ${\mu }_0$ are the permittivity and permeability of the vacuum respectively, ${\epsilon }_1$ and ${\epsilon }_2$ represent the relative permittivity of the upper medium 1 (down medium 2). Due to the consideration of the second-order nonlinear effect of TMDC monolayer, four terms ($C_{tx}$, $C_{ty}$, $C_{rx}$ and $C_{ry}$) emerge in the equation. This method can be extended to calculate the other nonlinear effect of other two dimensional materials.

This method in the x-z incident plane is required to extended to the arbitrary plane, we can construct a new axis $x\text{'}$, $y\text{'}$ and $z\text{'}$, the azimuthal angle between the $x\text{'}$-$z\text{'}$ plane and the x-z plane is represented by $\varphi$, then the electric fields in the new axis are related by

(16)$$\left(\begin{array}{c}{E}_{x\text{'}}^t\\ E_{y\text{'}}^t\end{array}\right)=\left(\begin{array}{cc}{\cos }^2\varphi T_x+{\sin }^2\varphi T_y& \sin \varphi \cos \varphi \left(T_x-T_y\right)\\ \sin \varphi \cos \varphi \left(T_x-T_y\right)& {\sin }^2\varphi T_x+{\cos }^2\varphi T_y\end{array}\right)\left(\begin{array}{c}{E}_{x\text{'}}^i\\ E_{y\text{'}}^i\end{array}\right)+\left(\begin{array}{c}\cos \varphi C_{tx}-\sin \varphi C_{ty}\\ \sin \varphi C_{tx}-\cos \varphi C_{ty}\end{array}\right),$$

where ${\left(E_{x\text{'}}^t,E_{y\text{'}}^t\right)}^{\text{T}}$ and ${\left(E_{x\text{'}}^i,E_{y\text{'}}^i\right)}^{\text{T}}$ represent the transmitted and incident electric fields in the new axis, a similar relation for the reflected electric fields can be obtained easily. Thus the transmission and reflection of the TMDC monolayer including second-order nonlinear effect in the arbitrary axis are calculated.

2 Scattering matrix of TMDC monolayer including second-order nonlinear effect

In the following we calculate the scattering matrix of the TMDC monolayer at SH frequency including second-order nonlinear effect based on the transmission and reflection derived in the former part. In Fig. 9 two electric fields ${\overrightarrow{E}}_1^{+}$ and ${\overrightarrow{E}}_2^{-}$ are incident on the TMDC monolayer and two electric fields ${\overrightarrow{E}}_2^{+}$ and ${\overrightarrow{E}}_1^{-}$ are generated.

 

Fig. 9 Scattering of two waves from a TMDC monolayer. These two waves indicated by ${\overrightarrow{E}}_1^{+}$ and ${\overrightarrow{E}}_2^{-}$ are incident from the up and down mediums, two waves ${\overrightarrow{E}}_2^{+}$ and ${\overrightarrow{E}}_1^{-}$ are generated.

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The electric fields in the up and down interface are expressed as $E_{1x\left(y\right)}={\displaystyle {\sum }_{mn}\left[E_{1mnx\left(y\right)}^{+}{e}^{i{\beta }_{mn}z}+E_{1mnx\left(y\right)}^{-}{e}^{-i{\beta }_{mn}z}\right]e^{i{k}_{mnx}x+i{k}_{mny}y}}$ and $E_{2x\left(y\right)}={\displaystyle {\sum }_{mn}\left[E_{2mnx\left(y\right)}^{+}{e}^{i{\beta }_{mn}z}+E_{2mnx\left(y\right)}^{-}{e}^{-i{\beta }_{mn}z}\right]e^{i{k}_{mnx}x+i{k}_{mny}y}}$, here $\left(k_{mnx},k_{mny}\right)$ is the Bragg wave vector, the superscript + (-) of the electric field denotes that the wave vector along z axis is positive (negative). These electric fields are related by the scattering matrixes

(17)$$\left(\begin{array}{c}{E}_2^{+}\\ E_1^{-}\end{array}\right)=\left(\begin{array}{cc}{T}^{\left(1,2\right)}& R^{\left(2,1\right)}\\ R^{\left(1,2\right)}& T^{\left(2,1\right)}\end{array}\right)\left(\begin{array}{c}{E}_1^{+}\\ E_2^{-}\end{array}\right)+\left(\begin{array}{c}{C}_1\\ C_2\end{array}\right),$$

where $E_1^{\pm }={\left(\cdots E_{1mnx}^{\pm },E_{1mny}^{\pm }\cdots \right)}^{\text{T}}$ and $E_2^{\pm }={\left(\cdots E_{2mnx}^{\pm },E_{2mny}^{\pm }\cdots \right)}^{\text{T}}$ are amplitude columns, $C_1$ and $C_2$ represent the columns resulting from the second-order nonlinear effect of TMDC monolayer. $T^{\left(1,2\right)}$ or $R^{\left(1,2\right)}$ ($T^{\left(2,1\right)}$ or $R^{\left(2,1\right)}$) represent the matrixes relating the transmitted or reflected electric field with the incident EM wave from medium 1 (2). Based on the transmission and reflection derived in former part, the elements of these scattering matrixes are given by

(18)$$T_{mn,pq}^{\left(i,j\right)}={\delta }_{mn,pq}\left(\begin{array}{cc}{\cos }^2{\varphi }_{mn}{T}_{mnx}^{\left(i,j\right)}+{\sin }^2{\varphi }_{mn}{T}_{mny}^{\left(i,j\right)}& \sin {\varphi }_{mn}\cos {\varphi }_{mn}\left(T_{mnx}^{\left(i,j\right)}-T_{mny}^{\left(i,j\right)}\right)\\ \sin {\varphi }_{mn}\cos {\varphi }_{mn}\left(T_{mnx}^{\left(i,j\right)}-T_{mny}^{\left(i,j\right)}\right)& {\sin }^2{\varphi }_{mn}{T}_{mnx}^{\left(i,j\right)}+{\cos }^2{\varphi }_{mn}{T}_{mny}^{\left(i,j\right)}\end{array}\right),$$

(19)$$R_{mn,pq}^{\left(i,j\right)}={\delta }_{mn,pq}\left(\begin{array}{cc}{\cos }^2{\varphi }_{mn}{R}_{mnx}^{\left(i,j\right)}+{\sin }^2{\varphi }_{mn}{R}_{mny}^{\left(i,j\right)}& \sin {\varphi }_{mn}\cos {\varphi }_{mn}\left(R_{mnx}^{\left(i,j\right)}-R_{mny}^{\left(i,j\right)}\right)\\ \sin {\varphi }_{mn}\cos {\varphi }_{mn}\left(R_{mnx}^{\left(i,j\right)}-R_{mny}^{\left(i,j\right)}\right)& {\sin }^2{\varphi }_{mn}{R}_{mnx}^{\left(i,j\right)}+{\cos }^2{\varphi }_{mn}{R}_{mny}^{\left(i,j\right)}\end{array}\right),$$

(20)$$C_{1mn}=\left(\begin{array}{c}\cos {\varphi }_{mn}{C}_{mntx}^{\left(1,2\right)}-\sin {\varphi }_{mn}{C}_{mnty}^{\left(1,2\right)}+\cos {\varphi }_{mn}{C}_{mnrx}^{\left(2,1\right)}-\sin {\varphi }_{mn}{C}_{mnry}^{\left(2,1\right)}\\ \sin {\varphi }_{mn}{C}_{mntx}^{\left(1,2\right)}+\cos {\varphi }_{mn}{C}_{mnty}^{\left(1,2\right)}+\sin {\varphi }_{mn}{C}_{mnrx}^{\left(2,1\right)}+\cos {\varphi }_{mn}{C}_{mnry}^{\left(2,1\right)}\end{array}\right),$$

(21)$$C_{2mn}=\left(\begin{array}{c}\cos {\varphi }_{mn}{C}_{mnrx}^{\left(1,2\right)}-\sin {\varphi }_{mn}{C}_{mnry}^{\left(1,2\right)}+\cos {\varphi }_{mn}{C}_{mntx}^{\left(2,1\right)}-\sin {\varphi }_{mn}{C}_{mnty}^{\left(2,1\right)}\\ \sin {\varphi }_{mn}{C}_{mnrx}^{\left(1,2\right)}+\cos {\varphi }_{mn}{C}_{mnry}^{\left(1,2\right)}+\sin {\varphi }_{mn}{C}_{mntx}^{\left(2,1\right)}+\cos {\varphi }_{mn}{C}_{mnty}^{\left(2,1\right)}\end{array}\right),$$

here the matrixes $T_{mn,pq}^{\left(i,j\right)}$, $R_{mn,pq}^{\left(i,j\right)}$ are the transmission and reflection matrixes derived in the former part when the wave vector is taken as $\left(k_{mnx},k_{mny},{\beta }_{mn}\right)$ and $\left(i,j\right)$ represents $\left(1,2\right)$ or $\left(2,1\right)$, $C_{1mn}$ and $C_{2mn}$ are the corresponding nonlinear terms.

Funding

National Natural Science Foundation of China (NSFC) (Grants No.11574400 and No. 11204379).

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