Theoretical and computational analysis of second- and third-harmonic generation in periodically patterned graphene and transition-metal dichalcogenide monolayers

Remarkable optical and electrical properties of two-dimensional (2D) materials, such as graphene and transition-metal dichalcogenide (TMDC) monolayers, offer vast technological potential for novel and improved optoelectronic nanodevices, many of which rely on nonlinear optical effects in these 2D materials. This paper introduces a highly effective numerical method for efficient and accurate description of linear and nonlinear optical effects in nanostructured 2D materials embedded in periodic photonic structures containing regular three-dimensional (3D) optical materials, such as diffraction gratings and periodic metamaterials. The proposed method builds upon the rigorous coupled-wave analysis and incorporates the nonlinear optical response of 2D materials by means of modified electromagnetic boundary conditions. This allows one to reduce the mathematical framework of the numerical method to an inhomogeneous scattering matrix formalism, which makes it more accurate and efficient than previously used approaches. An overview of linear and nonlinear optical properties of graphene and TMDC monolayers is given and the various features of the corresponding optical spectra are explored numerically and discussed. To illustrate the versatility of our numerical method, we use it to investigate the linear and nonlinear multiresonant optical response of 2D-3D heteromaterials for enhanced and tunable second- and third-harmonic generation. In particular, by employing a structured 2D material optically coupled to a patterned slab waveguide, we study the interplay between geometric resonances associated to guiding modes of periodically patterned slab waveguides and plasmon or exciton resonances of 2D materials.

Figure 1

Schematic of a generic multilayer structure periodic along the $x$ and $y$ directions under plane-wave incidence, and a closeup of the unit cell. The distribution of the 2D material (blue) is defined by the position-dependent sheet conductance ${\sigma }_s(x,y,z_s)$ at $z=z_s$. The bulk part of the periodic structure, which can consist of slanted (olive green), straight walled (red), or embedded parts (sand brown), is defined by the permittivity function ${\epsilon }_r(x,y,z)$.

Figure 2

(a) Dispersion of the complex sheet conductance ${\sigma }_s\left(\omega \right)$ of graphene in interband and intraband wavelength range. (b), (c) Frequency dependence of the real and imaginary parts of the sheet conductance of several TMDC monolayer materials, respectively.

Figure 3

Schematic of the atomic structures of graphene (symmetry group ${\mathcal{D}}_{6h}$, left) and TMDC monolayers (symmetry group ${\mathcal{D}}_{3h}$, right). The hexagonal lattice of graphene is centrosymmetric and thus SHG is forbidden. By contrast, the lattice of TMDC monolayers is noncentrosymmetric and as such SHG is allowed.

Figure 4

(a) Real and imaginary parts of the third-order nonlinear surface conductivity of graphene ${\sigma }_s^{\left(3\right)}$. (b) Spectral dependence of the effective bulk quadratic nonlinear susceptibility $|{\chi }_b^{\left(2\right)}|$ of three TMDC monolayer materials and the value of the dominant component of ${\chi }_{b,xyz}^{\left(2\right)}$ for GaAs, given for reference. For ${\mathrm{WS}}_2$, the experimental noisy data (black line) has been smoothed out (blue line).

Figure 5

Schematic of a multilayer structure located between cover and substrate and containing a 2DM sheet with surface conductance ${\sigma }_s$, the neighboring layers $A$ and $B$, and the evaluation layer $E$, in which the electromagnetic field is to be determined. (b) Linear (${\mathbf{c}}^{\pm },{\mathbf{e}}^{\pm },...$) and nonlinear (${\tilde{\mathbf{a}}}^{+},{\tilde{\mathbf{b}}}^{\pm },{\tilde{\mathbf{s}}}^{-}$) mode coefficients in the respective layers and the $\mathcal{S}$ matrices that connect them. (c) Combination of layers $B$ and $S$ yields the combined $\mathcal{S}$ matrix ${\mathsf{S}}^{ABS}$ and effective nonlinear coefficients ${\tilde{\mathbf{a}}}_{\mathrm{eff}}^{+}$ and ${\tilde{\mathbf{s}}}_{\mathrm{eff}}^{-}$. (d) Repeated combination of $\mathcal{S}$ matrices allows the calculation of the field coefficients $\mathbf{e}$ from the effective nonlinear mode coefficients ${\tilde{\mathbf{e}}}_{\mathrm{eff}}^{+}$.

Figure 6

(a) Cross section of the periodic structure through the plane $z=z_s$ containing the 2DM distribution ${\chi }_{2\mathrm{DM}}(x,y)$ with the boundary contour $\mathrm{\Gamma }$ showing a unit cell and the local coordinate system at a generic location $\tilde{\mathbf{r}}$. (b) Local coordinate system at $\tilde{\mathbf{r}}$. (c) Qualitative behavior of the surface quantities $E_{{\mathbf{t}}_n}^{\left(i\right)}$ and $j_{{\mathbf{t}}_n}^{(\omega ,\mathrm{lin})}={\sigma }_s^{\left(\omega \right)}{E}_{{\mathbf{t}}_n}^{(\omega ,s)}$.

Figure 7

Generic structures for numerical convergence analysis. (a) 1D periodic array of graphene ribbons with period $\mathrm{\Lambda }$ and width $w$. (b) 2D array of circular graphene disks with radius $r$ and periods ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2=\mathrm{\Lambda }$.

Figure 8

(a) Linear absorption spectra obtained by setting the added conductivity to zero ${\sigma }_{s,\mathrm{add}}=0$ (solid line), by using ${\sigma }_{s,\mathrm{add}}={\sigma }_{s,\mathrm{add}}\left({10}^{-5}\right)$ (dashed line), using a standard RCWA with $h_{\mathrm{eff}}=0.33\mathrm{nm}$ (dashed-dotted line) and results from [49] (green circles). (b) Top panel shows the dispersion map of the absorption vs $N$ and $\eta$, determined using the algorithm with added conductivity, whereas the bottom panel shows the dependence of absorption on $N$, calculated using a standard RCWA (dashed-dotted line), the algorithm with ${\sigma }_{s,\mathrm{add}}=0$ (solid line), and the algorithm with added conductivity ${\sigma }_{s,\mathrm{add}}={\sigma }_{s,\mathrm{add}}\left({10}^{-5}\right)$ (dashed line).

Figure 9

(a), (b) Electric field component $E_x$ around the graphene ribbon calculated for $N=100$ and the fields at the surface of the graphene ribbon corresponding to the improper (colored lines) and correct (black lines) field evaluation, respectively. In both cases, ${\sigma }_{s,\mathrm{add}}=0$. (c), (d) The same quantities as in (a) and (b), respectively, but determined for ${\sigma }_{s,\mathrm{add}}={\sigma }_{s,\mathrm{add}}\left({10}^{-5}\right)$. Only a part of the left half of the unit cell, defined by $x\in \left[-4\mu \mathrm{m},0\right]$, is shown, as the results are symmetric in $x$ and rather featureless far away from the graphene ribbon located at $\left|x\right|<2\mu \mathrm{m}$.

Figure 10

(a) Nonlinear radiation spectrum for $N=50$ (dashed lines) and $N=200$ (solid lines) calculated using the improper field evaluation (blue lines) and correct field evaluation (black lines) at the FF. (b) Intensity of TH radiation at ${\lambda }_{\mathrm{TH}}=22\mu \mathrm{m}$ vs $N$ determined for different values of added conductivity ${\sigma }_{s,\mathrm{add}}$. (c), (d) Intensity of TH radiation at ${\lambda }_{\mathrm{TH}}=11$ and $22\mu \mathrm{m}$, respectively, vs $N$ determined using improper field evaluation (blue lines) and correct field evaluation (black lines) at the FF. In both cases ${\sigma }_{s,\mathrm{add}}={\sigma }_{s,\mathrm{add}}\left({10}^{-5}\right)$.

Figure 11

(a) Third-harmonic near field around a graphene ribbon determined using the improperly evaluated interface field at the FF, $E_x^{\left(i\right)}$, to calculate the nonlinear polarization ${\mathbf{P}}^{\mathrm{nl}}$. (b) The same as in (a) but using the correctly evaluated field at the FF, ${\tilde{E}}_x^{\left(i\right)}$, to compute ${\mathbf{P}}^{\mathrm{nl}}$.

Figure 12

(a), (b) Linear absorption and intensity of TH radiation spectra calculated using a finite added conductivity ${\sigma }_{s,\mathrm{add}}={\sigma }_{s,\mathrm{add}}\left({10}^{-3}\right)$, and number of harmonics $N=10$, 15, 20, and 25.

Figure 13

(a), (b) Relative self-error for linear absorption and intensity of TH radiation, respectively, determined at the plasmon resonance wavelengths $\lambda =11.09\mu \mathrm{m}$ (dots), $5.081\mu \mathrm{m}$ (crosses), and $3.925\mu \mathrm{m}$ (circles).

Figure 14

Dominant electric field component $|E_x|$ at the FF (top panels) and TH (bottom panels) at the surface of a graphene disk for the resonance wavelengths, from left to right, ${\lambda }_{\text{FF}}=11.09$, $5.081$, and $3.925\mu \text{m}$. The number of harmonics used in the simulations was $N=40$ at the FF and $N=30$ at the TH.

Figure 15

(a) The FF absorption spectra for ribbons made of ${\mathrm{WS}}_2$, ${\mathrm{MoS}}_2$, ${\mathrm{WSe}}_2$, and ${\mathrm{MoSe}}_2$. (b) The SH radiation spectra for ribbons made of ${\mathrm{WS}}_2$, ${\mathrm{MoS}}_2$, and ${\mathrm{WSe}}_2$.

Figure 16

Waveguide structures comprising a periodically patterned slab waveguide with permittivities ${\epsilon }_r$ and ${\epsilon }_r^{\prime }<{\epsilon }_r$ covered by air and placed on a dielectric substrate with $\sqrt{{\epsilon }_s}=1.46$. On top of the slab waveguide different 2DMs are placed: (a) a uniform monolayer of ${\mathrm{WS}}_2$ or graphene, (b), (c) graphene ribbons with width $w$ distributed over the slab material with lower and higher permittivity, respectively. (d) Graphene ribbons with width $w$ distributed over both regions of the slab waveguide.

Figure 17

(a) Linear reflection, absorption, and transmission for waveguide height $h=0.18\mu \mathrm{m}$ demonstrate the effect of the intrinsic material absorption at $\lambda \approx 0.645\mu \mathrm{m}$ and the Fano resonance due to a waveguide resonance at $\lambda =0.6215\mu \mathrm{m}$. (b) Electric near field in and around the waveguide for $\lambda =0.6215\mu \mathrm{m}$ exhibits the spatial profile of the ${\mathrm{TM}}_0$ mode, with strong field enhancement at waveguide interfaces. (c) Nonlinear radiation intensity spectra near the fundamental frequency corresponding to the ${\mathrm{TM}}_0$ mode, determined for three values of the waveguide height $h$. (d) Electric near field at SH wavelength ${\lambda }_{\mathrm{SH}}=0.311\mu \mathrm{m}$ for $h=180\mathrm{nm}$.

Figure 18

(a) Map of linear reflection spectra vs waveguide height $h$ exhibits Fabry-Perot resonances, resonances due to exciton generation ($\downarrow$) and resonances due to excitation of waveguide modes ($\searrow$). Interaction between the ${\mathrm{TM}}_0$ waveguide mode and excitons of ${\mathrm{WS}}_2$ monolayer as well as Fano resonances can be observed. (b) Map of nonlinear radiation spectra determined for different waveguide height $h$.

Figure 19

(a) Absorption spectra (solid lines) of the homogeneous graphene sheet and the graphene ribbons on top of inner and outer parts of the slab waveguide [cf. Figs. 16–16] and reflection spectrum (dashed line) of the homogeneous graphene sheet. (b) The nonlinear radiation spectrum of the same structures as in (a). (c) Linear spectra (reflection, transmission, and absorption) and TH spectrum for the graphene ribbons placed on top of both parts of the waveguide, as per Fig. 16.

Figure 20

Dominant component of the electric field $E_x$ at the FF (top parts) and the TH (bottom parts) for selected fundamental wavelengths $\lambda$ and TH wavelengths ${\lambda }_{\mathrm{TH}}=\lambda /3$: (a) Field enhancement due to excitation of the ${\mathrm{TM}}_0$ waveguide mode for $\lambda =8.05\mu \mathrm{m}$. (b) Near-field profile of the ${\mathrm{TM}}_0$ mode at the TH wavelength ${\lambda }_{\mathrm{TH}}=4.4\mu \mathrm{m}$. (c), (d) Plasmonic field enhancement in the inner and outer ribbons for $\lambda =13.82$ and $15.17\mu \mathrm{m}$, respectively. (e), (f) Excitation of surface plasmons in graphene ribbons located on top of waveguide sections with ${\epsilon }_r^{\prime }$ (inner ribbons, $\lambda =7.1\mu \mathrm{m}$) and ${\epsilon }_r$ (outer ribbons, $\lambda =7.6\mu \mathrm{m}$), respectively. The labels of the panels correspond to the labels of the peaks in Fig. 19.

Figure 21

(a) Map of absorption spectra vs angles of incidence $\theta$ determined in the wavelength range of the second-order surface plasmon of graphene ribbons. (b) The map of nonlinear radiation spectra shows the inherited (red labels) and intrinsic, nonlinear (green labels) modes of the slab waveguide. The left and right branches of intrinsic modes of the same order exhibit anticrossing mode interaction, which leads to the formation of spectral band gaps.