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Detecting nonlocality by second-harmonic generation from a graphene-wrapped nanoparticle

Chenglin Wang, Dongliang Gao, and Lei Gao

Open Access Open Access

Abstract

With the rapid development of nanofabrication technology and nonlinear optics, the nonlinear detection by nanostructures is highly appreciated. In this paper, we study the second-harmonic generation by a spherical nonlocal plasmonic nanoparticle wrapped with graphene. We develop a simple method for calculating the electric field at second-harmonic frequency and analyze the influence of the nonlocal response of the metal on the second-harmonic. We find that this nanostructure can probe the material's properties by detecting the radiation intensity of the second-harmonic generation. In addition, the nonlocal response of the plasmonic core can promote the absorption efficiency of second-harmonic generation. Our study may offer a new way for studying the plasmonic quantum effects and nonlinear probing technology and improving the nonlinear conversion efficiency of photonic devices.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

With the rapid progress in nanofabrication technology and optical metamaterials, researchers are increasingly interested in the nonlinear effects of metallic nano-metamaterials [1–3]. Second-harmonic generation (SHG) is one of the basic processes in nonlinear optics, which is the nonlinear process of converting two photons at fundamental frequency into a new photon at twice the fundamental frequency [4]. SHG has various potential applications in optical detections, such as biosensing [5], imaging [6,7], optical communication [8] nonlinear switch [9], etc. Due to the potential applications value of SHG, researchers have proposed many methods to improve the conversion efficiency of SHG. For example, Lithium niobate metasurface has been proved to be able to control SHG performance and effectively enhance the conversion efficiency of SHG [10].

Under the dipole approximation, SHG radiation is generally forbidden for two-dimensional materials with centrosymmetry, such as graphene. However, when the influence of the spatial dispersion of graphene is considered, the second-order nonlinear response of graphene is non-zero [11]. In addition, for highly doped graphene, the inversion symmetry is effectively broken by optical dressing with an in-plane photon wave vector [12–16]. Therefore, the strength of graphene-induced SHG is comparable to that of non-centrosymmetric two-dimensional materials [11]. What is more, because of its high tunability and strong nonlinear response, graphene has become one of the most popular nonlinear materials in recent years. For example, the intensity of SHG in monolayer graphene is enhanced under the ultrastrong terahertz field pulse [17]. It has been demonstrated that the field-enhancement effect can significantly increase the second-order nonlinear responses of graphene metasurfaces [18]. Moreover, high nonlinear conversion efficiency has been achieved by using a metamaterial platform that combines graphene with a photonic grating structure [19].

Notably, due to the electron-electron interaction in the dielectric response of metals, the nonlocal effects should be taken into account when the size of the structure is reduced to nanometer scale. Fuchs et al. investigated the multipole response of small metallic sphere using nonlocal dielectric function [20,21]. In addition, the nonlocality of metal materials can enhances the nonlinear response of the nanostructures, such as optical bistability [22,23]. Furthermore, the nonlocal effects of free electrons will significantly change the nature of the matter interaction in the linear and nonlinear response of the material. In this case, the conventional local solution of Maxwell’s equations can no longer accurately describe the electromagnetic characteristics of nonlocal case. Hence, we need to consider the influence of nonlocal effects on SHG.

In this paper, we study the second-harmonic generation by a spherical nonlocal plasmonic nanoparticle wrapped into graphene. Firstly, within the quasistatic approximation, we developed a simple method for calculating the electric field at SH frequency. Then we analyzed the influence of the nonlocal response of the metal on the second-harmonic. The results show that SHG radiation can not only be suppressed well at a specific frequency, but also realize the super-radiation mode of SHG of this structure by adjusting the nonlocality strength of plasmonic core. This result provides a new path for detecting the properties of materials. We can probe the properties of the material by detecting the radiation intensity of the second-harmonic generation. Moreover, the nonlocal response of plasmonic core effectively promotes the absorption of SHG. This nonlinear optical phenomenon always exists whether below or above the plasma frequency. Notably, the absorption efficiency under nonlocal description can surpass the local case by two orders of magnitude. This nonlinear optical phenomenon provides an effective way for us to detect SHG over a broad frequency range.

2. Theoretical model and methods

Figure 1 illustrates the schematic diagrams to be considered, which consists of graphene- wrapped spherical nonlocal plasmonic nanoparticles with radius $a$ surrounded by the host medium with relative permittivity ${\varepsilon _h}$. We consider the incident electric field ${\boldsymbol E} = {E_0}{{\boldsymbol z}_0}{e^{ - i\omega t}}$ . There are two main steps to calculate the second-harmonic generation of the nanosystem. The first step (i) We introduce the semiclassical infinite barrier model [21,24] to obtain the fundamental frequency (FF) electrostatic potential and the FF electric displacement vector ${\boldsymbol D}$. Then the distribution of FF electrostatic potential and the FF electric displacement vector ${\boldsymbol D}$ in each region ($|r |> a$ and $|r |\le a$) is obtained using boundary conditions. The second step (ii) Calculate the second-harmonic generation (SHG) from the graphene layer's nonlinearity. The solution idea is similar to that of the fundamental frequency, but the boundary conditions are different in the case of SHG. The detailed derivation is as follows.

 figure: Fig. 1.

Fig. 1. Schematics of the graphene-wrapped nanoparticle illuminated by a plane wave. The radii of nanoparticle is $a$. ${\varepsilon _a}(k,\omega )$ and ${\varepsilon _h}$ are the relative permittivity’s of the nanoparticle and background medium, respectively.

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In general, the field intensity E and the displacement D being related through the nonlocal relation,

$$D({\boldsymbol r}) = \int {{\varepsilon _0}{\varepsilon _a}({\boldsymbol r} - {\boldsymbol r}^{\prime},\omega )} E({\boldsymbol r}^{\prime}){d^3}r^{\prime}.$$
The spatial dispersive permittivity of nonlocal plasmonic nanoparticle ${\varepsilon _a}({\boldsymbol k},\omega ) = {\varepsilon _\infty } - {{\omega _p^2} / {[{\omega ({\omega + i\gamma } )- {\beta^2}{k^2}} ]}}$ is taken from Ref. [25], where $\beta$ is proportional to the Fermi velocity ${\nu _F}$ as $\beta = \sqrt {{3 / 5}} {\nu _F}$. We have the relation between the electric displacement vector and electric displacement potential ${\boldsymbol D}({\boldsymbol r}) ={-} \nabla {\varphi _D}({\boldsymbol r})$ and assume charge sources located on the sphere surface,
$${\nabla ^2}{\varphi _D}({\boldsymbol r}) = C_l\delta (r - a){P_l}(\cos \theta ),$$
where ${P_l}(\cos \theta )$ is a Legendre polynomial of order l, and ${C_l}$ stands for an unknown l-dependent constant. Taking the Fourier transform of Eq. (2), we get,
$${\varphi _D}({\boldsymbol k}) ={-} \frac{{4\pi {a^2}}}{{{k^2}}}{C_l}{( - i)^l}{j_l}(ka){P_l}(\cos \theta ),$$
where ${j_l}(kr)$ is the spherical Bessel function of order l, ${\boldsymbol k} \equiv ({k,{\theta_k},{\phi_k}} )$ in spherical coordinate. By taking the inverse Fourier transform of the Eq. (3), we obtain,
$${\varphi _D}({\boldsymbol r}) ={-} \frac{2}{\pi }{a^2}{C_l}{P_l}(\cos \theta )\int_0^\infty {{j_l}(ka){j_l}(kr)dk} .$$

Using the formula $\int_0^\infty {{j_l}({a_1}x){j_l}({a_2}x)dx = } {{\pi a_1^l} / {({2({2l + 1} )a_2^{l + 1}} )}}$ with ${a_2} > {a_1}$ and the relation between the electric displacement potential ${\varphi _D}({\boldsymbol k})$ and the electrostatic potential $\varphi ({\boldsymbol k}){\varphi _D}({\boldsymbol k}) = {\varepsilon _0}{\varepsilon _a}({\boldsymbol k},\omega )\varphi ({\boldsymbol k})$, we get,

$$\varphi ({\boldsymbol k}) ={-} \frac{{4\pi {a^2}{C_l}{{( - i)}^l}{j_l}(ka){P_l}(\cos \theta )}}{{{k^2}{\varepsilon _0}{\varepsilon _a}({\boldsymbol k},\omega )}}.$$
Then by taking the inverse Fourier transform of the Eq. (5), the electric potentials $\varphi (r)$ and the radial component of the displacement potential ${D_r}(r)$ inside and outside the sphere can be written as,
$${\varphi _1}(r) = {V_l}\left( {{r^l} - \frac{{{\alpha_l}}}{{{r^{l + 1}}}}} \right){P_l}(\cos \theta ),$$
$${D_{r1}}(r) ={-} {\varepsilon _0}{\varepsilon _h}{V_l}\left( {l{r^{l - 1}} + ({l + 1} )\frac{{{\alpha_l}}}{{{r^{l + 2}}}}} \right){P_l}(\cos \theta ),$$
$${\varphi _2}(r) ={-} \frac{2}{\pi }{a^2}{C_l}{Y_{l0}}(\theta ,\phi )\int_0^\infty {\frac{{{j_l}(ka){j_l}(kr)}}{{{\varepsilon _0}{\varepsilon _a}(k,\omega )}}} dk,$$
$${D_{r2}}(r) = \frac{{{C_l}}}{{2l + 1}}{Y_{l0}}(\theta ,\phi )\frac{{l{r^{l - 1}}}}{{{a^{l - 1}}}},$$
where the subscripts 1 and 2 represent the outside and inside areas, respectively. ${V_l}\textrm{ = } - {E_0}$ is a constant. ${\alpha _l}$ is the l-polar polarizability. ${Y_{l0}}(\theta ,\phi )$ are the usual spherical harmonics. We apply the boundary conditions on $r = a$.${\varphi _1}(a)\textrm{ = }{\varphi _2}(a)$, ${D_{r1}}(a) - {D_{r2}}(a) = {{{\nabla _s} \cdot {{\boldsymbol j}^\omega }} / {({i\omega } )}}$[26], where ${{\boldsymbol j}^\omega } = {\sigma _g}E_\parallel ^\omega$ with $E_\parallel ^\omega = {\boldsymbol - }\nabla {\varphi _1}$, and ${\sigma _g}\textrm{ = }{{i{e^2}{E_F}} / {\pi {\hbar ^2}({\omega + i{\gamma_g}} )}}$ is the linear conductivity of graphene [25,27]. Then the distribution of FF electrostatic potential is given by,
$${\varphi _1}(r)\textrm{ = } - {E_0}r\cos \theta \textrm{ + }{E_0}\frac{{{E_L} - {\varepsilon _h} - 2{\Theta _1}}}{{{E_L} + 2{\varepsilon _h} - 2{\Theta _1}}}\frac{{{a^3}}}{{{r^2}}}\cos \theta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({|r |> a} ),$$
$${\varphi _2}(r) ={-} \frac{6}{\pi }{a^2}{E_d}{E_L}\cos \theta \int_0^\infty {\frac{{{j_1}(ka){j_1}(kr)}}{{{\varepsilon _a}(k,\omega )}}} dk{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({|r |\le a} ),$$
where ${E_L} = {\pi / {\left( {6a\int_0^\infty {{{({{j_1}(ka){j_1}(ka)} )} / {{\varepsilon_a}(k,\omega )}}} dk} \right)}}$, ${\Theta _1} = {{{\sigma _g}} / {i{\varepsilon _0}\omega a}}$, ${E_d} = {{3{\varepsilon _h}{E_0}} / {({{E_L} + 2{\varepsilon_h} - 2{\Theta _1}} )}}$ , and ${\varepsilon _0}$ is the vacuum permittivity.

Based on the calculation of the FF electrostatic potentials, the next step is to determine the nonlinear potential distribution at the second-harmonic (SH) frequency. It's similar to the linear case, the general expression of electric potentials ${\varphi ^{2\omega }}(r)$ and the radial component of the displacement potential $D_r^{2\omega }(r)$ in nonlinear case are written as:

$$\varphi _1^{2\omega }(r) ={-} {V_l}\frac{{\alpha _l^{2\omega }}}{{{r^{l + 1}}}}{P_l}(\cos \theta ),$$
$$D_{r1}^{2\omega }(r) ={-} {\varepsilon _0}{\varepsilon _h}{V_l}({l + 1} )\frac{{\alpha _l^{2\omega }}}{{{r^{l + 2}}}}{P_l}(\cos \theta ),$$
$$\varphi _2^{2\omega }(r) ={-} \frac{2}{\pi }{a^2}C_l^{2\omega }{P_l}(\cos \theta )\int_0^\infty {\frac{{{j_l}(ka){j_l}(kr)}}{{{\varepsilon _0}{\varepsilon _a}(k,\omega )}}} dk,$$
$$D_{r2}^{2\omega }(r) = \frac{{C_l^{2\omega }}}{{2l + 1}}{P_l}(\cos \theta )\frac{{l{r^{l - 1}}}}{{{a^{l - 1}}}},$$
where $\alpha _l^{2\omega }$ is the l-polar polarizability in nonlinear case, and stands for an unknown-dependent constant. At the boundary $r = a$, we also apply the boundary conditions, $\varphi _1^{2\omega }(a)\textrm{ = }\varphi _2^{2\omega }(a)$, $D_{r1}^{2\omega }(a) - D_{r2}^{2\omega }(a) = {{{\nabla _s} \cdot {\boldsymbol j}_{2\omega }^{NL}} / {({i2\omega } )}}$. Note that the nonlinear current source is ${\boldsymbol j}_{2{\omega _0}}^{NL} = {{\boldsymbol j}^{2\omega }} + \sigma _g^{2\omega }{\boldsymbol {\rm E}}_\parallel ^{2\omega }$, where ${{\boldsymbol j}^{2\omega }} = \sigma _g^{(2 )}{{\boldsymbol E}_\parallel }{{\boldsymbol E}_\parallel }$ is the current at the second-harmonic, ${{\boldsymbol E}_\parallel }$ is the FF field tangential to the graphene surface, and $\sigma _g^{(2 )}\textrm{ = } - 3k {e^3}v_F^2/8\pi {\hbar ^2}{\omega ^3}$ is the second-order nonlinear conductivity of graphene, $k$ is the wavevector of the mode at the fundamental frequency (FF) [28], ${v_{Fg}}$ is the Fermi velocity of graphene. In the quasistatic limit, the current can be simplified to ${{\boldsymbol j}^{2\omega }} = ({{{({i27{e^3}v_{Fg}^2E_0^2} )} / {({16\pi {\hbar^2}\omega_0^3{{({{E_L}\textrm{ + }2 - 2{\Theta _1}} )}^2}{R_0}} )}}} )\sin 2\theta {\boldsymbol \theta }$, where ${\boldsymbol {\rm E}}_\parallel ^{2\omega }$ is the second-harmonic field tangential to the graphene surface and $\sigma _g^{2\omega } \equiv {\sigma _g}({2\omega } )$ is the linear conductivity of graphene at at second-harmonic frequency [26,29,30]. Then the potential excited by quadrupole is expressed as follows,
$$\varphi _1^{2\omega }(r) = \frac{{1\textrm{ + }3\cos 2\theta }}{4}\frac{3}{8}\frac{{{e^3}v_{Fg}^2}}{{{\varepsilon _0}\pi {\hbar ^2}{\omega ^4}({2{E_N} + 3{\varepsilon_h} - 3{\Theta _2}} )}}E_d^2\frac{{{a^2}}}{{{r^3}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({|r |> a} ),$$
$$\varphi _2^{2\omega }(r) = \frac{{1\textrm{ + }3\cos 2\theta }}{4}\frac{{15}}{4}\frac{{{e^3}v_{Fg}^2{E_N}}}{{{\varepsilon _0}{\pi ^2}{\hbar ^2}{\omega ^4}({2{E_N} + 3{\varepsilon_h} - 3{\Theta _2}} )}}E_d^2\int_0^\infty {\frac{{{j_2}(ka){j_2}(kr)}}{{{\varepsilon _a}(k,\omega )}}} dk{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({|r |\le a} ).$$
And the corresponding quadrupole moment component,
$${Q_{SH}} = \frac{3}{8}\frac{{{e^3}v_{Fg}^2}}{{{\varepsilon _0}\pi {\hbar ^2}{\omega ^4}({2{E_N} + 3{\varepsilon_h} - 3{\Theta _2}} )}}E_d^2{a^2},$$
where ${E_N} = {\pi / {10a\int_0^\infty {{{({{j_2}(ka){j_2}(ka)} )} / {{\varepsilon _a}(k,\omega )}}} dk}}$, ${\Theta _2}\textrm{ = }{{\sigma _g^{2\omega }} / {({i{\varepsilon_0}a\omega } )}}$

To quantify the conversion efficiency of SHG, the radiation efficiency of SHG is given by [26],

$$Q_{Rad}^{2\omega } = \frac{{\frac{{\textrm{c}{{({2{k_0}} )}^6}}}{{\textrm{60}}}{{|{{Q_{SH}}} |}^2}}}{{{P_0}}}.$$
And the absorption efficiency of SHG is defined as [31,32],
$$Q_{Abs}^{2\omega } = \frac{{{\varepsilon _\textrm{0}}\omega {\mathop{\rm Im}\nolimits} \textrm{[}{\varepsilon _a}\textrm{]}\int_{vol} {{{|{{{\boldsymbol E}^{2\omega }}} |}^2}dr} }}{{{P_{FF}}}},$$
where ${{\boldsymbol E}^{2\omega }}$ is the electric field at SH frequency and ${P_0} = ({{c / {8\pi }}} ){|{{{\boldsymbol E}_0}} |^2}\pi {a^2}$ [21] is the incident power.

3. Results and discussion

On the basis of the above theoretical process, it is interesting to explore the influence of the nonlocality of plasmonic core on the second-harmonic generation. We first discuss the influence of nonlocality strength on second-harmonic radiation. To describe nonlocality strength of plasmonic core, we introduce nonlocality strength parameter $\delta = {\beta / c}$[33]. For different plasma materials, the nonlocality strength parameter can reach different values. For example, for alkali metals and semiconductors, the nonlocality strength parameter can reach 1/450 and 1/280. While in metamaterials, the nonlocality strength parameter can achieve larger values [33–36].

According to the nonlocality of metal, we studied the influence of nonlocality strength $\delta = {\beta / c}$ of plasma materials on second-harmonic generation of this nanoparticle. In order to compare the radiation efficiency of SHG under the localized and non-localized cases, we show the radiation efficiency of SHG in logarithmic ${\log _{10}}({Q_{Rad}^{2\omega }} )$ with different frequency under local description $({\delta = 0} )$ in Fig. 2(a). And Fig. 2(b) shows the radiation efficiency of SHG in logarithmic ${\log _{10}}({Q_{Rad}^{2\omega }} )$ with different frequency and nonlocality strength parameter $\delta$ under the nonlocal $({\delta = {\beta / c}} )$ descriptions. As can be observed from Fig. 2(a) and (b), the radiation efficiency of SHG changes significantly when the nonlocality strength of the plasmonic core is changed in nonlocal case. Interestingly, the radiation efficiency of SHG in nonlocal case $({\delta = 0.052} )$ could surpass the local prediction by four orders of magnitude. And compared with the local case, the radiation efficiency of SHG can also be well suppressed under specific nonlocal intensity $({\delta = 0.0125} )$. The above phenomena can be reasonably explained from Eq. (10). we convert the quadrupole moment component in local $({{Q_{LSH}}} )$ and nonlocal $({{Q_{NSH}}} )$ cases into the following form,

$${Q_{LSH}} = \frac{3}{8}\frac{{{e^3}v_{Fg}^2{a^2}{{({3{\varepsilon_h}} )}^2}}}{{{\varepsilon _0}\pi {\hbar ^2}{\omega ^4}}}\frac{1}{{({R{E_{L2}} + iI{M_{L2}}} )}}\frac{1}{{{{({R{E_{L1}} + iI{M_{L1}}} )}^2}}},$$
$${Q_{NSH}} = \frac{3}{8}\frac{{{e^3}v_{Fg}^2{a^2}{{({3{\varepsilon_h}} )}^2}}}{{{\varepsilon _0}\pi {\hbar ^2}{\omega ^4}}}\frac{1}{{({R{E_{N2}} + iI{M_{N2}}} )}}\frac{1}{{{{({R{E_{N1}} + iI{M_{N1}}} )}^2}}},$$
where $R{E_{L1}} = Re [{{E_L} + 2{\varepsilon_h} - 2{\Theta _1}} ],{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I{M_{L1}} = {\mathop{\rm Im}\nolimits} [{{E_L} + 2{\varepsilon_h} - 2{\Theta _1}} ]$ are the real and imaginary parts of ${E_L} + 2{\varepsilon _h} - 2{\Theta _1}$ in local case, respectively. $R{E_{L2}} = Re [{2{E_N} + 3{\varepsilon_h} - 3{\Theta _2}} ],{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I{M_{L2}} = {\mathop{\rm Im}\nolimits} [{2{E_N} + 3{\varepsilon_h} - 3{\Theta _2}} ]$ are the real and imaginary parts of $2{E_N} + 3{\varepsilon _h} - 3{\Theta _2}$ in local case, respectively. Similarly, the real and imaginary parts of ${E_L} + 2{\varepsilon _h} - 2{\Theta _1}$ and $2{E_N} + 3{\varepsilon _h} - 3{\Theta _2}$ in the nonlocal case are $R{E_{N1}} = Re [{{E_L} + 2{\varepsilon_h} - 2{\Theta _1}} ],{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I{M_{N1}} = {\mathop{\rm Im}\nolimits} [{{E_L} + 2{\varepsilon_h} - 2{\Theta _1}} ],$ $R{E_{N2}} = Re [{2{E_N} + 3{\varepsilon_h} - 3{\Theta _2}} ],$ and $I{M_{N2}} = {\mathop{\rm Im}\nolimits} [{2{E_N} + 3{\varepsilon_h} - 3{\Theta _2}} ]$ , respectively. The real parts of ${E_L} + 2{\varepsilon _h} - 2{\Theta _1}$ and $2{E_N} + 3{\varepsilon _h} - 3{\Theta _2}$ with different frequency under local and nonlocal description as shown in Fig. 2(c) and (d). According to Eq. (13) and (14), we can find that the radiation efficiency of SHG reaches the maximum when the real part of ${E_L} + 2{\varepsilon _h} - 2{\Theta _1}$ or $2{E_N} + 3{\varepsilon _h} - 3{\Theta _2}$ is equal to zero. Meanwhile, comparing Fig. 2(c) and Fig. 2(d), $R{E_{N1}}$ and $R{E_{N2}}$ have a larger adjustment range by changing the nonlocality strength of the plasmonic core. When $|{R{E_{N1}}} |$ or $|{R{E_{N2}}} |$ reaches the maximum, the radiation efficiency of SHG can reach the minimum, and when $|{R{E_{N1}}} |\to 0$ or $|{R{E_{N2}}} |\to 0$ , the radiation efficiency of SHG can reach maximum. From the above discussion, we can conclude that the radiation efficiency of SHG is more sensitive to the nonlocal effects due to the nonlocal nature of plasmonic core. This provides a new adjustment dimension for the adjustment of second-harmonic conversion efficiency. Meanwhile, we can also determine the nonlocality strength of plasmonic core by detecting the radiation intensity of SHG, so as to determine the property of the material.

 figure: Fig. 2.

Fig. 2. The influence of nonlocality strength on second-harmonic radiation. (a) and (b) The radiation efficiency of SHG in logarithmic ${\log _{10}}({Q_{Rad}^{2\omega }} )$ with different frequency and nonlocality strength parameter $\delta$ under the local $({\delta = 0} )$ and nonlocal $({\delta = {\beta / c}} )$ descriptions, respectively. (c) and (d) The real parts of ${E_L} + 2{\varepsilon _h} - 2{\Theta _1}$ (black lines) and $2{E_N} + 3{\varepsilon _h} - 3{\Theta _2}$ (red lines) with different frequencies under local ((c)) and nonlocal ((d)) descriptions, respectively. The corresponding parameters are $a = 10nm$, $\hbar {\omega _p} = 0.8eV$ [25], the incident intensity ${10^8}{W / {c{m^2}}}$ [29]. The loss rate of plasmonic core $\hbar {\gamma _D} = 1 \times {10^{ - 3}}eV$. The parameters of graphene: the loss rate $\hbar {\gamma _g} = 2 \times {10^{ - 4}}eV$, and Fermi energy ${E_F} = 1eV$.

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Figure 3(a) reveals the radiation efficiency of SHG in logarithmic ${\log _{10}}({Q_{Rad}^{2\omega }} )$ with different frequency under the local (black line), $\delta = 0.052$ (red line) and $\delta = 0.0125$ (blue line). It can be seen that the conversion between the suppressed mode and the super-radiation mode of the second-harmonic radiation can be achieved under the same incident frequency $({\hbar \omega = 0.6\textrm{eV}} )$ by adjusting the nonlocal intensity of the plasmonic core $\delta = 0.052$ to $\delta = 0.0125$. Fig. 3(b) and (c) show the corresponding near field images at second-harmonic frequency for point B and point C, respectively. It can be seen from the near field distribution that the second-harmonic field corresponding to point B is much larger than that corresponding to point C. And it is clear that the major contribution to the SH intensity is the quadrupole. It is further demonstrated that the nonlocal effect of metal can not only promote the nonlinear conversion of SHG, but also restrain the nonlinear process of SHG. In addition, it is worth noting that there are two peaks (peak A and peak B) in each case (black line (local), red line ($\delta = 0.052$) and blue line ($\delta = 0.0125$)). The reasons for these two peaks are explained below. The quadrupolar SH emission mechanisms (peak B in Fig. 3 (a)) $E_1^\omega + E_1^\omega \to E_2^{2\omega }$ (the two terms $E_1^\omega$ on the left of the arrow indicate fundamental modes, and the term $E_2^{2\omega }$ on the right indicates the SH emission mode.) is the dominant response of SHG because it can be excited at the fundamental frequency and does not require retardation effects [37,38]. Explicitly, the mechanism of second-harmonic quadrupolar emission mode is produced by the combination of an electric dipole $E_1^\omega$ and an electric dipole mode $E_1^\omega$[37,39]. Unlike the above second-harmonic emission mode, peak A in Fig. 3(a) is also the second-harmonic quadrupolar emission mode which is caused by the linear current source $\sigma _g^{2\omega }{\boldsymbol {\rm E}}_\parallel ^{2\omega }$ at second-harmonic frequency.

The nonlocal intensity of plasmonic core not only affects the radiation of the second-harmonic generation but also has a great influence on the absorption of SHG. Figure 4(a) shows the absorption efficiency of SHG in logarithmic ${\log _{10}}({Q_{Abs}^{2\omega }} )$ with different frequency under the local (black line) and nonlocal $({\delta = 0.276 \times {{10}^{ - 2}}} )$ descriptions. We find that the maximum absorption efficiency of SHG appears around $\hbar \omega = 0.77eV$ in local case, while for the nonlocal case, the absorption efficiency of SHG reaches maximum at $\hbar \omega = 0.78eV$. The blue shift of the absorption spectrum is clearly visible. This is caused by the quantum effect of small metal nanoparticles [40,41]. Surprisingly, we can observe that the absorption efficiency under nonlocal description could surpass the local case by two orders of magnitude. The reason for the phenomena can be reasonably explained from Eq. (12). Equation (12) shows that the absorption efficiency of SHG is proportional to the local field at the second-harmonic frequency and the imaginary part of the dielectric constant ${\mathop{\rm Im}\nolimits} \textrm{[}{\varepsilon _a}\textrm{]}$ of this nanostructure. Figure 4(b) and (c) show that the corresponding near field images at second-harmonic frequency for peak I and peak II, respectively. It can be seen that the second-harmonic field corresponding to peak I is ten times larger than that corresponding to peak II. Meanwhile, as shown in Fig. 4(d) and (e), the imaginary parts of ${E_L}$ and in nonlocal cases are generally larger than local descriptions about the frequency region we consider. Therefore, the nonlocal response of plasmonic core can effectively promote the absorption of SHG. Note that, as shown in the red lines Fig. 4(a) and (b), there are a number of peaks that appear above the plasma frequency due to the excitation of confined longitudinal polarization fields for small sizes [42,43].

 figure: Fig. 3.

Fig. 3. (a) The radiation efficiency of SHG in logarithmic ${\log _{10}}({Q_{Rad}^{2\omega }} )$ with different frequency under the local (black line), $\delta = 0.052$ (red line) and $\delta = 0.0125$ (blue line). (b) and (c) The corresponding near field images at second-harmonic frequency for point B and point C, respectively. Other parameters are the same as those in Fig. 2.

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 figure: Fig. 4.

Fig. 4. (a) The absorption efficiency of SHG in logarithmic ${\log _{10}}({Q_{Abs}^{2\omega }} )$ with different frequency under the local (black line) and nonlocal $({\delta = 0.276 \times {{10}^{ - 2}}} )$ descriptions. (b) and (c) The corresponding near field images at second-harmonic frequency for peak I and peak II, respectively. (d) and (e) The imaginary parts of ${E_L}$ and ${E_N}$ with different frequency under local (black line) and nonlocal (red line) descriptions, respectively. Other parameters are the same as those in Fig. 2.

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The radiation and absorption of SHG discussed above arise at frequencies $\omega < {\omega _p}$ . Finally, we discuss the effect of the nonlocal response of the plasmonic core on second-harmonic absorption above the plasma frequency ${\omega _p}$, as shown in Fig. 5. Compared to the plasmas arising at frequencies $\omega < {\omega _p}$, longitudinal bulk plasmas appearing at frequencies $\omega < {\omega _p}$ are very weak and difficult to detect. However, in this frequency region, the absorption efficiency of SHG is significantly improved due to the appearance of the longitudinal mode. In order to be able to clearly observe the absorption of SHG at frequencies $\omega > {\omega _p}$, here we only show the absorption spectrums of local (black line) and nonlocal (red line) case at frequencies $\omega > {\omega _p}$. As depicted in Fig. 5, for the local case, there is no longer the absorption peak of SHG in this frequency region. In the nonlocal description, while different, there are additional absorption peak of SHG in this frequency region. Notably the absorption efficiency could surpass the local case by two orders of magnitude. This provides an effective way to detect SHG over a large frequency range.

 figure: Fig. 5.

Fig. 5. The absorption efficiency ${\log _{10}}({Q_{Abs}^{2\omega }} )$ of SHG in local (black line $\delta = 0$) and nonlocal (red line $\delta = 0.276 \times {10^{ - 2}}$) descriptions above the plasmon frequency ${\omega _p}$. The corresponding parameters are the same as those in Fig. 2.

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4. Conclusions

In conclusion, we have studied the second-harmonic generation by a spherical nonlocal plasmonic nanoparticle wrapped into graphene. We demonstrated that the second-harmonic generation response in graphene-wrapped spherical nonlocal plasmonic nanoparticle is closely connected to the nonlocality strength of plasmonic core. Our results show that we can not only suppress the radiation of SHG at a specific frequency, but also realize the super-radiation mode of SHG of this structure by adjusting the nonlocality strength of plasmonic core. Inversely, we can also obtain the property of plasmonic core by detecting the radiation intensity of SHG. This offer a new way for nonlinear detection technology. Moreover, we have also demonstrated that nonlocal response of plasmonic core effectively promote the absorption of SHG. Notably, the absorption efficiency under nonlocal description can surpass the local case by two orders of magnitude. The nonlinear optical phenomenon always exists, that is, the absorption efficiency of SHG can be improved through the nonlocal response whether below or above the plasma frequency. This nonlinear optical phenomenon provides an effective way for us to detect SHG over a large frequency range. The above research results may provide a new path for the research of plasmonic quantum effects, nonlinear detection technology, and improving the nonlinear conversion efficiency of photonic devices.

Funding

National Natural Science Foundation of China (92050104, 12174281); Suzhou Prospective Application Research Project (SYG202039).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Schematics of the graphene-wrapped nanoparticle illuminated by a plane wave. The radii of nanoparticle is $a$. ${\varepsilon _a}(k,\omega )$ and ${\varepsilon _h}$ are the relative permittivity’s of the nanoparticle and background medium, respectively.

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Fig. 2.
Fig. 2. The influence of nonlocality strength on second-harmonic radiation. (a) and (b) The radiation efficiency of SHG in logarithmic ${\log _{10}}({Q_{Rad}^{2\omega }} )$ with different frequency and nonlocality strength parameter $\delta$ under the local $({\delta = 0} )$ and nonlocal $({\delta = {\beta / c}} )$ descriptions, respectively. (c) and (d) The real parts of ${E_L} + 2{\varepsilon _h} - 2{\Theta _1}$ (black lines) and $2{E_N} + 3{\varepsilon _h} - 3{\Theta _2}$ (red lines) with different frequencies under local ((c)) and nonlocal ((d)) descriptions, respectively. The corresponding parameters are $a = 10nm$, $\hbar {\omega _p} = 0.8eV$ [25], the incident intensity ${10^8}{W / {c{m^2}}}$ [29]. The loss rate of plasmonic core $\hbar {\gamma _D} = 1 \times {10^{ - 3}}eV$. The parameters of graphene: the loss rate $\hbar {\gamma _g} = 2 \times {10^{ - 4}}eV$, and Fermi energy ${E_F} = 1eV$.

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Fig. 3.
Fig. 3. (a) The radiation efficiency of SHG in logarithmic ${\log _{10}}({Q_{Rad}^{2\omega }} )$ with different frequency under the local (black line), $\delta = 0.052$ (red line) and $\delta = 0.0125$ (blue line). (b) and (c) The corresponding near field images at second-harmonic frequency for point B and point C, respectively. Other parameters are the same as those in Fig. 2.

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Fig. 4.
Fig. 4. (a) The absorption efficiency of SHG in logarithmic ${\log _{10}}({Q_{Abs}^{2\omega }} )$ with different frequency under the local (black line) and nonlocal $({\delta = 0.276 \times {{10}^{ - 2}}} )$ descriptions. (b) and (c) The corresponding near field images at second-harmonic frequency for peak I and peak II, respectively. (d) and (e) The imaginary parts of ${E_L}$ and ${E_N}$ with different frequency under local (black line) and nonlocal (red line) descriptions, respectively. Other parameters are the same as those in Fig. 2.

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Fig. 5.
Fig. 5. The absorption efficiency ${\log _{10}}({Q_{Abs}^{2\omega }} )$ of SHG in local (black line $\delta = 0$) and nonlocal (red line $\delta = 0.276 \times {10^{ - 2}}$) descriptions above the plasmon frequency ${\omega _p}$. The corresponding parameters are the same as those in Fig. 2.

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Equations (22)

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D ( r ) = ∫ ε 0 ε a ( r − r ′ , ω ) E ( r ′ ) d 3 r ′ .
∇ 2 φ D ( r ) = C l δ ( r − a ) P l ( cos ⁡ θ ) ,
φ D ( k ) = − 4 π a 2 k 2 C l ( − i ) l j l ( k a ) P l ( cos ⁡ θ ) ,
φ D ( r ) = − 2 π a 2 C l P l ( cos ⁡ θ ) ∫ 0 ∞ j l ( k a ) j l ( k r ) d k .
φ ( k ) = − 4 π a 2 C l ( − i ) l j l ( k a ) P l ( cos ⁡ θ ) k 2 ε 0 ε a ( k , ω ) .
φ 1 ( r ) = V l ( r l − α l r l + 1 ) P l ( cos ⁡ θ ) ,
D r 1 ( r ) = − ε 0 ε h V l ( l r l − 1 + ( l + 1 ) α l r l + 2 ) P l ( cos ⁡ θ ) ,
φ 2 ( r ) = − 2 π a 2 C l Y l 0 ( θ , ϕ ) ∫ 0 ∞ j l ( k a ) j l ( k r ) ε 0 ε a ( k , ω ) d k ,
D r 2 ( r ) = C l 2 l + 1 Y l 0 ( θ , ϕ ) l r l − 1 a l − 1 ,
φ 1 ( r )  =  − E 0 r cos ⁡ θ  +  E 0 E L − ε h − 2 Θ 1 E L + 2 ε h − 2 Θ 1 a 3 r 2 cos ⁡ θ ( | r | > a ) ,
φ 2 ( r ) = − 6 π a 2 E d E L cos ⁡ θ ∫ 0 ∞ j 1 ( k a ) j 1 ( k r ) ε a ( k , ω ) d k ( | r | ≤ a ) ,
φ 1 2 ω ( r ) = − V l α l 2 ω r l + 1 P l ( cos ⁡ θ ) ,
D r 1 2 ω ( r ) = − ε 0 ε h V l ( l + 1 ) α l 2 ω r l + 2 P l ( cos ⁡ θ ) ,
φ 2 2 ω ( r ) = − 2 π a 2 C l 2 ω P l ( cos ⁡ θ ) ∫ 0 ∞ j l ( k a ) j l ( k r ) ε 0 ε a ( k , ω ) d k ,
D r 2 2 ω ( r ) = C l 2 ω 2 l + 1 P l ( cos ⁡ θ ) l r l − 1 a l − 1 ,
φ 1 2 ω ( r ) = 1  +  3 cos ⁡ 2 θ 4 3 8 e 3 v F g 2 ε 0 π ℏ 2 ω 4 ( 2 E N + 3 ε h − 3 Θ 2 ) E d 2 a 2 r 3 ( | r | > a ) ,
φ 2 2 ω ( r ) = 1  +  3 cos ⁡ 2 θ 4 15 4 e 3 v F g 2 E N ε 0 π 2 ℏ 2 ω 4 ( 2 E N + 3 ε h − 3 Θ 2 ) E d 2 ∫ 0 ∞ j 2 ( k a ) j 2 ( k r ) ε a ( k , ω ) d k ( | r | ≤ a ) .
Q S H = 3 8 e 3 v F g 2 ε 0 π ℏ 2 ω 4 ( 2 E N + 3 ε h − 3 Θ 2 ) E d 2 a 2 ,
Q R a d 2 ω = c ( 2 k 0 ) 6 60 | Q S H | 2 P 0 .
Q A b s 2 ω = ε 0 ω Im [ ε a ] ∫ v o l | E 2 ω | 2 d r P F F ,
Q L S H = 3 8 e 3 v F g 2 a 2 ( 3 ε h ) 2 ε 0 π ℏ 2 ω 4 1 ( R E L 2 + i I M L 2 ) 1 ( R E L 1 + i I M L 1 ) 2 ,
Q N S H = 3 8 e 3 v F g 2 a 2 ( 3 ε h ) 2 ε 0 π ℏ 2 ω 4 1 ( R E N 2 + i I M N 2 ) 1 ( R E N 1 + i I M N 1 ) 2 ,
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