Electrical control of second harmonic generation in a graphene-based plasmonic Fano structure

Fajun Xiao; Weiren Zhu; Wuyun Shang; Ting Mei; Malin Premaratne; Jianlin Zhao

doi:10.1364/OE.23.003236

1. Introduction

Localized surface plasmon resonances (LSPRs), the collective electron oscillations at surfaces of metallic nanostructures, have been a subject of intense studies owing to their ability to manipulate light at the nanoscale. The corresponding strong confinement of light leads to giant enhancement of electromagnetic field that makes incredibly weak physics processes observable. This can be exemplified by the enhanced nonlinear optical processes, including second harmonic generation (SHG) [1,2], multiphoton luminescence [3,4], and four wave mixing [5,6]. Among these phenomena, SHG is of particular interest for applications in high sensitive sensors [7], deep subwavelength imaging [8,9], and ultracompact frequency generators [10]. SHG in a metallic structure is proportional to the square of the local field intensity. Therefore, it can be dramatically augmented by having an enhanced local field at resonance. Recently, LSPR-enhanced SHGs have been theoretically and experimentally demonstrated in various metallic nanostructures, such as single nanoparticles [2,11], plasmonic antennas [12,13], nanoclusters [14–16], and structured metasurfaces [17–19]. In particular, significant attempts have been made to improve the electric field induced SHGs to operate as actively controllable frequency converters [10,20]. To control the SHG in a plasmonic structure, it is necessary to effectively modulate the SHG source. This can be realized by manipulating the radiative and nonradiative losses to alter the local field intensity at the fundamental frequency. It is also demonstrated that Fano resonance can be another promising solution towards the controllable SHG [21].

The plasmonic Fano resonance arises from the interaction between a dark (subradiant) mode and a bright (superradiant) mode. The dark mode, possessing a small dipole moment, is difficult to get activated by an interacting plane wave and has significantly low radiative loss in the far field. However, the dark mode can be excited by the highly scattered bright mode via near field coupling. The coupling of these two modes can be well adjusted by changing the geometry [22–25], the surrounding refractive index [26,27] and the excitation beam [28–30], resulting in tunable radiative and nonradiative losses. Thus, a Fano structure can exhibit adjustable local field intensity, making it a prime candidate for achieving controllable SHG. In this work, we propose a graphene-based Fano structure to deliver an actively controllable SHG. Even though graphene is single atom thick, its interaction with the local field can be dramatically strong [31,32]. More importantly, this interaction can be readily controlled by tuning the Fermi level of graphene through electrical doping [33–35]. We demonstrate that a broad tunable range (220 nm) in resonance wavelength and a 45 times increase in peak intensity can be achieved for the SHG emission from our proposed Fano structure by slightly increasing the graphene Fermi level from 65 to 95 meV. Compared with the traditional technique which relies on changing the geometry, the proposed graphene-based Fano structure provides a more flexible approach to engineer the SHG, which may potentially expand the application of SHG even further.

2. Linear response of graphene-based Fano structure

The Fano structure under consideration is a periodical silver dolmen stacked above a monolayer graphene, shown schematically in Fig. 1(a). The periods of this structure in both x and y directions are 2.3 μm and the thickness of dolmen is 50 nm. The geometry of a single dolmen is illustrated in the close-up view of Fig. 1(a), which is defined by the parameters: l₁ = 2 μm, l₂ = 1.4 μm, s = 1 μm, w = 0.2 μm, and g = 0.1 μm. The permittivity of silver is described by the Drude model ε(ω) = 1-ω_p²/(ω² + iγω) with ω_p = 1.37 × 10¹⁶s⁻¹ and γ = 8.2 × 10¹³s⁻¹. The graphene-based dolmen structure is supported by a silicon/silicon dioxide substrate. A back gate voltage is applied to electrically bias the graphene for tuning its Fermi level. Considering the contributions from both inter- and intra- band transitions, the real (ε_g^′) and imaginary parts (ε_g^″) of the graphene permittivity can be written as [32]

\begin{matrix} ε_{g}^{'} (ω) = 1 + \frac{e^{2}}{8 π ℏ ω ε_{0} d} \ln \frac{{(ℏ ω + 2 | E_{F} |)}^{2} + Γ^{2}}{{(ℏ ω - 2 | E_{F} |)}^{2} + Γ^{2}} - \frac{e^{2}}{π ε_{0} d} \frac{| E_{F} |}{{(ℏ ω)}^{2} + {(1 / τ)}^{2}} \\ \begin{array}{l} ε_{g}^{''} (ω) = \frac{e^{2}}{4 ℏ ω ε_{0} d} [1 + \frac{1}{π} (\tan^{- 1} \frac{ℏ ω - 2 | E_{F} |}{Γ} - \tan^{- 1} \frac{ℏ ω + 2 | E_{F} |}{Γ})] \\ + \frac{e^{2}}{π τ ℏ ω ε_{0} d} \frac{| E_{F} |}{{(ℏ ω)}^{2} + {(1 / τ)}^{2}} \end{array} \end{matrix} .

Here, e, ħ, ε₀, and E_F are the electron charge, the reduced Planck constant, the vacuum permittivity, and the graphene Fermi level, respectively. In the following simulation, we adopt the inter-band transition broadening Γ = 8.3 meV [36], the free carrier scattering rate 1/τ = 0 meV [32], the thickness of graphene d = 0.5 nm, the relative permittivities of silicon and silicon dioxide

ε_{Si} = 3.6

, and

ε_{{SiO}_{2}} = 3.9

.

Fig. 1 (a) Schematic view of the monolayer graphene-based dolmen structure. The geometry parameters l₁, l₂, s, w, and g are 2, 1.4, 1, 0.2, and 0.1 μm, respectively. The periods in both x and y direction are 2.3 μm. (b) Illustration of Fano resonance as an interference between a narrow state of dark mode and a broad state of bright mode. (c) Extinction spectra of the dolmen structure excited by plane waves with horizontal (red) and vertical (blue) polarization as indicated in (d). (d) Normalized surface charge distributions of the dipolar mode and Fano extinction dip as labeled in (c).

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The Fano resonance of a dolmen structure can be interpreted as the interference between a narrow state of dark mode (quadrupole) and a broad continuum state of bright mode (dipole), which is illustrated in Fig. 1(b). These two interacting modes can be also viewed as two coupled harmonic oscillators x_d(b) = C_d(b)(ω)eⁱ^ω^t that are governed by the relations [24]

\begin{matrix} \frac{d^{2} x_{d}}{d t^{2}} + γ_{d} \frac{d x_{d}}{d t} + ω_{d}^{2} x_{d} + g x_{b} = \frac{1}{2} (η_{d} \frac{d^{3} x_{d}}{d t^{3}} + η_{b} \frac{d^{3} x_{b}}{d t^{3}}) + η_{d} E \\ \frac{d^{2} x_{b}}{d t^{2}} + γ_{b} \frac{d x_{b}}{d t} + ω_{b}^{2} x_{b} + g x_{d} = \frac{1}{2} (η_{d} \frac{d^{3} x_{d}}{d t^{3}} + η_{b} \frac{d^{3} x_{b}}{d t^{3}}) + η_{b} E \end{matrix} .

Here, ω_d(b), γ_d(b), and η_d(b) are the resonance frequency, nonradiative damping and radiation coupling efficiency of the dark (bright) mode, respectively; g represents the coupling constant between the dark and bright modes; E denoted as E = E₀eⁱ^ω^t is the electrical field of the excitation beam. Using Eq. (2), C_d(b)(ω) can be analytically derived

\begin{matrix} C_{d} (ω) = \frac{- η_{b} E_{0} (g + \frac{i}{2} η_{b} ω^{3}) - η_{d} E_{0} (ω^{2} - i γ_{b} ω - ω_{b}^{2} - \frac{i}{2} η_{b} ω^{3})}{(ω^{2} - i γ_{d} ω - ω_{d}^{2} - \frac{i}{2} η_{d} ω^{3}) (ω^{2} - i γ_{b} ω - ω_{b}^{2} - \frac{i}{2} η_{b} ω^{3}) - (g + \frac{i}{2} η_{d} ω^{3}) (g + \frac{i}{2} η_{b} ω^{3})} \\ C_{b} (ω) = \frac{(g + \frac{i}{2} η_{d} ω^{3}) C_{d} (ω) - η_{b} E_{0}}{ω^{2} - i γ_{b} ω - ω_{b}^{2} - \frac{i}{2} η_{b} ω^{3}} \end{matrix} .

These values can then be used to calculate α_ext(ω) = I₀ + |C_d(ω) + C_b(ω)|² with I₀ accounting for the background, which is utilized hereafter to describe the extinction spectrum of the dolmen structure.

We carry out a full-wave numerical simulation [finite difference time domain (FDTD) method] to study the linear response of the proposed dolmen structure. Without loss of generality [14,19], we adopt an oblique incidence configuration [illustrated in Fig. 1(a)] for both linear and nonlinear excitations, where the incident angle is 45° and the incident plane is oriented along the symmetric axis of the dolmen. Figure 1(c) shows the extinction spectra of the dolmen structure with an ungated graphene, where the red dashed and blue solid curves correspond to the excitations with two orthogonal polarizations indicated in Fig. 1(d). For horizontal polarization, i.e. oriented along the symmetry axis of the dolmen, a broad resonance peak (red dashed curve) appears in the extinction spectrum, revealing that a dipolar mode is excited in this configuration. This dipolar mode can be confirmed by its charge distribution shown in the left panel of Fig. 1(d), where the surface currents at the two long antennas oscillate in phase. However, for the vertical polarization, an obvious asymmetric lineshape (blue solid curve) is found, indicating the occurrence of Fano resonance. The charge distribution at the Fano dip [labeled with the orange dot in Fig. 1(c)] is depicted in the right panel of Fig. 1(d), in which the quadrupolar charge distribution on the two long antennas tightly bonds with the dipolar charge distribution on the short antenna.

Figure 2(a) shows the real and imaginary parts of the graphene permittivities at different Fermi levels. The changes in the graphene permittivity are because the increase of the Fermi level leads to Pauli blocking of inter-band transitions, suppressing optical dissipation channel in graphene. When the dolmen structures are stacked above graphene, the variation of the graphene permittivity can bring a change to the resonance wavelength of the structure. According to the perturbation theory, the resonance wavelength shift of dark (bright) mode Δλ_R follows [37]

\frac{Δ λ_{R}}{λ_{R}} = - \frac{1}{2} \frac{〈 E_{d (b)} | Δ ε_{g} | E_{d (b)} 〉}{〈 E_{d (b)} | ε_{g} | E_{d (b)} 〉},

where E_d(b) is the local field of the dark (bright) mode, and Δε_g is the change in the graphene permittivity. From Eq. (4), it is expected that Δλ_R scales with Δε_g. Therefore, the overlapping of dark and bright modes can be controlled by Fermi-level-modulated graphene permittivity, giving rise to a tunable Fano resonance. As seen in Fig. 2(a), there is a great change in graphene permittivity for probe photon energy around 2|E_F|. E_F is tuned in a range of 65-95 meV to control the Fano resonance within wavelength of 6.5-9.5 μm. The achieved tunable Fano resonance is shown in Fig. 2(b). To acquire deep insight into this tunable Fano resonance, we use Eq. (3) to fit the extinction spectra of the dolmen structures. A good agreement is achieved between the simulation and the theory model. It can be examined in Fig. 2(c), where the red curves show the fitting to the extinction spectra at graphene Fermi levels of 70, 80 and 95 meV. The detailed dependence of resonance wavelength on the graphene Fermi level is displayed in Fig. 2(d) (red dots). An interesting feature is found in these results: the resonance wavelengths of both dark [upper panel of Fig. 2(d)] and bright modes [lower panel of Fig. 2(d)] exhibit initially a red shift, then a blue shift with the increasing of Fermi level. This behavior can be attributed to Fermi-level-modulated real parts of the graphene permittivity, depicted as blue triangles in Fig. 2(d). The similarity between the trend of λ_R and ε_g^′ reveals their linear scaling relation, and confirms that the tunability of Fano resonance is mainly due to the Fermi-level-controlled graphene permittivity.

Fig. 2 (a) Graphene permittivities at Fermi levels of 75, 80, 85, 90, and 95 meV, where the upper and lower panels are the real and imaginary parts of the permittivity, respectively. (b) Extinction spectra of the dolmen structure as a dependence of graphene Fermi level. (c) Representative extinction spectra of the dolmen structure with graphene Fermi levels E_F = 70, 80 and 95 meV, where the dots and lines correspond to the simulation and fitting results, respectively. (d) Resonance wavelengths of dark (upper panel) and bright modes (lower panel) versus graphene Fermi level. Here, the dots and triangles are the resonance wavelengths of the modes and the corresponding real parts of graphene permittivity, respectively.

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3. SHG in graphene-based Fano structure

As a second-order nonlinear process, SHG is forbidden for centrosymmetric materials. However, this nonlinear response is observable at a metallic surface where inversion symmetry is broken. The SHG sources of a metallic structure can be treated as a nonlinear polarization current oscillating at the second harmonic frequency. In the realm of dipole approximation, the nonlinear polarization can be described as

p^{(2)} (r, 2 ω) = χ_{s}^{(2)} : E (r, ω) E (r, ω),

where χ_s⁽²⁾ and E(r,ω) correspond to the surface second-order susceptibility tensor and the electrical field at fundamental frequency, respectively. According to symmetry argument, the surface second-order susceptibility can be divided into two families of non-vanishing quantities, which are χ_s,⊥⁽²⁾ = χ_s,_zzz⁽²⁾ and χ_s,||⁽²⁾ = χ_s,_xxz⁽²⁾ = χ_s,_xzx⁽²⁾ = χ_s,_yyz⁽²⁾ = χ_s,_yzy⁽²⁾. In our following calculations of SHG, the fundamental electrical field is evaluated at the interface between dolmens and the air by employing FDTD simulation. The nonlinear susceptibilities are taken from the experimental data in [38]. Then, the total SHG intensity can be obtained by integrating over the metal and air interface

I_{SHG} (ω) = \int_{s} χ_{s}^{(2)} : E (x, y, ω) E (x, y, ω) d x d y .

The blue solid curve in Fig. 3(a) shows the reflective SHG spectrum of the dolmen structure with an ungated graphene, for which the polarization of incident beam is along the short antenna of the dolmen. It shows that the lineshape of SHG is similar to that of the linear response, reflecting that the SHG characteristics stem from the enhanced resonance of the fundamental field. This can be further confirmed by the spectrum of the integral local field intensity |E_loc|², shown as the red dashed curve in Fig. 3(a). The agreement at the resonance peaks and dips can be found between the SHG and fundamental local field intensity. In Fig. 3(c), we depict the distributions of SHG intensity and 4th power of fundamental field at the interface between air and dolmen. It can be seen that these two quantities have almost the same distributions as well as the dependences on the wavelengths, indicating their direct scaling relation. Additionally, the polarization properties of SHG at three representative wavelengths labeled in Fig. 3(a) are examined with the SHG intensity as a function of polarization in Fig. 3(b). The predominant SHG signals can be found for polarization along the symmetric axis of the dolmen structure. However, the SHG emission is forbidden for polarization perpendicular to the symmetric axis. Such a selection rule reveals the mirror symmetry of the dolmen structure and was also demonstrated in other SHG studies of systems with mirror symmetry [1,39].

Fig. 3 (a) Spectra of SHG (blue solid curve) and local field intensity (red dashed curve) of the dolmen structure excited by a vertical polarization beam at graphene Fermi level E_F = 0 meV. (b) Polarization of the SHG emission represented as a polar diagram at the wavelength indicated in (a). (c) Normalized SHG intensity (upper panels) and the 4th power of fundamental field distributions (lower panels) at the dolmen and air interface with the excitation wavelength as labeled in (a).

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Figure 4(a) depicts the SHG emission of the dolmen structure as the function of graphene Fermi level. It exhibits significant changes in both the resonance peak and intensity. To quantitatively unravel these changes, we employ an empirical equation to describe the two resonances associated SHG [19]

I_{SHG} (ω) = {| A + \frac{B e^{i φ_{1}}}{ω - ω_{1} + i ξ_{1}} + \frac{C e^{i φ_{2}}}{ω - ω_{2} + i ξ_{2}} |}^{4} .

Here, A accounts for the non-resonant contribution, B and C are the strength of two resonances; ω₁₍₂₎, ξ₁₍₂₎ and φ₁₍₂₎ are the resonant frequency, the damping coefficient, and the phase of the corresponding resonance, respectively. The SHG model fittings are shown by the red curves in Fig. 4(b), where the theoretical model agrees well to the calculated SHG spectra at Fermi levels of 70, 80, 95 meV. The dependences of the resonance wavelengths of two modes on the graphene Fermi level are illustrated in Fig. 4(c), where the upper and lower panels correspond to the SHG resonance peaks for dark and bright modes, respectively. It is suggested the SHG peaks have an obvious shift (90 nm for dark mode, and 220 nm for bright mode) with the graphene Fermi level ranging from 65 to 95 meV. Also, these peak shifts obey almost the same trend as their corresponding linear dark and bright modes shown in Fig. 2(d). This is due to the fact that the resonance characteristics of SHG mainly arise from the resonance enhanced linear response. Therefore, the change of linear resonance with graphene Fermi level can be reflected as a similar variation in SHG. Another finding is that the maximum I_SHG displays a step-like increase with the graphene Fermi level [shown in the top panel of Fig. 4(d)]. It experiences a 45 times increase when the graphene Fermi level changes from 65 to 95 meV. We further extract integral |E_loc|⁴ and imaginary part of graphene permittivity at the fundamental wavelength of the maximum SHG, as shown in the middle and bottom panels of Fig. 4(d). Because of the linear scaling relation between I_SHG and |E_loc|⁴, a step-like increase is also found in the plot of integral |E_loc|⁴ versus E_F. However, ε_g″ shows an opposite trend against I_SHG that exhibits a step-like decrease with graphene Fermi level. We therefore interpret the increase of I_SHG as a direct consequence of the decrease of ε_g″, which reduces the absorption of the fundamental local field by graphene and increases the intensity of SHG source.

Fig. 4 (a) SHG spectra of the dolmen structure as a function of graphene Fermi level. (b) Representative SHG spectra at the graphene Fermi levels E_F = 70, 80 and 95 meV. The dots and lines are the simulation and fitting results, respectively. (c) Dependence of the resonance wavelength of SHG on graphene Fermi level. The upper and lower panels correspond to the SHG peaks of the dark and bright modes, respectively. (d) I_SHG (top panel), integral |E_loc|⁴ (middle panel), and imaginary part of graphene permittivity (bottom panel) versus the graphene Fermi level at the fundamental wavelength for maximum SHG.

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4. Conclusion

In this paper, we proposed a scheme to actively control the SHG using a graphene-based dolmen structure. A theoretical model was developed and numerical simulations were carried out to evaluate its performance. By slightly changing graphene Fermi level, we show a tuning range as large as 220 nm for SHG resonance wavelength. This effect can be understood by employing a general perturbation theory that the variations in real part of graphene permittivity lead to the resonance energy change for both linear and nonlinear responses. It is also worth noting that a 45 times increase in SHG peak intensity was achieved by appropriately rising the graphene Fermi level. This is because the imaginary part of graphene permittivity decreases with increasing Fermi level, which leads to a lower loss for SHG driving field, i.e. the fundamental local field, and boost the SHG signal. Our results provide a powerful way to tailor the SHG signal from the plasmonic structure and may push forward the applications of SHG.

Acknowledgments

F. Xiao, W. Shang, T. Mei and J. Zhao acknowledge financial support from the National Basic Research Program (973 Program) of China (grant no. 2012CB921900), the National Natural Science Foundation of China (grant no. 61308031, 61377035, 61176085 and 61377055), and the Natural Science Basic Research Plan in Shaanxi Province (grant no. 2014JQ1033). T. Mei acknowledges financial support from the Department of Education of Guangdong Province, China (grant no. gjhz1103). W. Zhu and M. Premaratne acknowledge financial support from the Australian Research Council through its Discovery Grant scheme under Grant DP140100883.

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Electrical control of second harmonic generation in a graphene-based plasmonic Fano structure

Abstract

1. Introduction

2. Linear response of graphene-based Fano structure

3. SHG in graphene-based Fano structure

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (7)

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