Plasmon polariton assisted second-harmonic generation in graphene

TL;DR

The paper addresses enhancing second-harmonic generation from a graphene monolayer by embedding it in an attenuated total reflection configuration to exploit surface plasmon–polaritons. It develops a theoretical framework combining Maxwell equations with a Boltzmann-derived 2D nonlinear current in graphene, revealing two resonant SHG peaks corresponding to coupled SPP modes. The work shows that SPP excitation can dramatically boost SHG efficiency and even enable bistability under certain parametric conditions, offering design guidelines for robust nonlinear graphene photonics. The findings have implications for ultrathin plasmonic platforms and mid-IR to visible SHG in 2D materials and related systems.

Abstract

In this paper we present a theoretical examination of second-harmonic generation (SHG) in a graphene monolayer integrated within an attenuated total internal reflection (ATR) configuration. By embedding graphene in this optical setup, we explore the enhancement in the nonlinear optical response, particularly focusing on the efficiency of SHG. Our analysis reveals that the excitation of surface plasmon-polaritons (SPPs) plays a central role in significantly boosting the efficiency of SHG. The unique electronic properties of graphene, combined with the resonant characteristics of SPPs, create a synergistic effect that amplifies the nonlinear optical signals. This enhancement is attributed to the strong field confinement and the resonant nature of SPPs, which effectively increase the interaction between the incident light and the graphene monolayer. Furthermore, we analyze the underlying mechanisms that govern this process, providing a comprehensive theoretical framework that elucidates the interplay between graphene's electronic structure and the optical fields. Our findings suggest that the ATR scheme not only facilitates the excitation of SPPs but also optimizes the conditions for SHG.

Figures (11)

• Figure 1: Schematics of the graphene-based ATR structure. Although dielectric prism is depicted semicircular, like in experiments, throughout the paper it is considered to have an infinite radius, i.e. occupies the whole half-space $z<-d$.
• Figure 2: (a)(b) Reflectance $R^{(1)}$ (a) and SHG efficiency $R^{(2)}$ (b) versus angle of incidence $\theta$ and frequency $\omega$ of ATR structure with parameters: $d=10\,\mu$m, $\varepsilon_{3}=16$, $\varepsilon_{2}=1$, $\varepsilon_{1}=3.9$, $E_{F}=0.5\,$eV, $\gamma=0.25\,$meV, $E_{i}=1\,$kV/cm. (c) Maximal values of SHG efficiency $R^{(2)}_{max}$ vs angle of incidence $\theta$ for high-frequency (green line) and low-frequency modes (black lines) for the same parameters as those in (b). (d) SHG efficiency $R^{(2)}+T^{(2)}$ vs angle of incidence $\theta$ and frequency $\omega$ for the same structure as in (b), but with $\varepsilon_{3}=1$. (e) Frequency vs in-plane component of wavevector $k_{x}$ of nonlinear eigenmodes [Eq. \ref{['eq:dr']} with parameters $\varepsilon_{2}=1$, $\varepsilon_{1}=3.9$, $\gamma=0$, $E_{F}=0.5\,$eV, $E_{1,x}^{(1)}=0$ (green lines), or $E_{1,x}^{(1)}=1\,$MV/cm (red lines)].
• Figure 3: Effective refractive index $n_{eff}$ (a), (c) and second-harmonic in-plane component of electric field $\left|E_{1,x}^{(2)}\right|$ [in MV/cm, (b) and (d)] vs frequency $\omega$ and in-plane component of electric field of first harmonics $\left|E_{1,x}^{(1)}\right|$ of SFSPP [(a), (b)] and DFSPP [(c), (d)]. Upper insets in all panels show the cross section of main panels at fixed frequency $\omega=6\,$meV. All the parameters are the same as those in Fig.\ref{['fig:reflectance']}(d).
• Figure 4: The dependence of the in-plane component of the first-harmonic electric field $\left|E_{1,x}^{(1)}\right|$ (a) and SHG efficiency $R^{(2)}$ (b) on the electric field of the incident wave $\left|E_{i}\right|$ and frequency $\omega$ for the fixed angle of incidence $\theta=40^{\circ}$. Other parameters are the same as those in Figure \ref{['fig:reflectance']}(a). Upper insets demonstrate the cross sections of dependence in (a) and (b) at fixed frequency $\omega=4.37\,$meV.
• Figure 5: (a),(b) Examples of the bistability of the first-harmonic electric field $E_{1,x}^{(1)}$ (red lines, left $y$ axis) and of SHG efficiency $R^{(2)}$ (blue lines, right $y$ axis) for the parameters of the ATR structure $\theta=80^{0}, \omega=1\,$meV (a); or $\theta=40^{0}, \omega=11\,$meV (b). (c)-(e) Domains of bistability--minimal and $\left|E_{i}^{(min)}\right|$ and maximal $\left|E_{i}^{(max)}\right|$ values of bistability zone vs frequency $\omega$ and angle of incidence $\theta$ [(c) and (d), respectively] or vs frequency $\omega$ for fixed value of angle of incidence $\theta=60^{0}$ [(e), where $\left|E_{i}^{(min)}\right|$ and $\left|E_{i}^{(max)}\right|$ are depicted by green and blue solid lines, correspondingly]. In all panels, other parameters are: $d=10\,\mu$m, $\varepsilon_{3}=16$, $\varepsilon_{2}=1$, $\varepsilon_{1}=3.9$, $E_{F}=0.5\,$eV, $\gamma=0.25\,$meV.