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1. Introduction

Due to the unique abilities to concentrate and enhance the light in the subwavelength scale [1–3], plasmon resonances can enhance nonlinear optical responses [4], such as second har-monic generation (SHG) [5, 6], multiphoton excited lumines-cence [7, 8], third harmonic generation [9, 10], and four-wave mixing [11, 12]. In particular, a number of plasmonic nano-structures, such as single nanoparticles [6, 13], nanoantennas [14, 15] and metasurfaces [16, 17], have been investigated for enhanced SHG to aid applications in bio-sensing, struc-tural properties and laser beam characterizations [18]. Owing to the inherently weak nature of optical nonlinear processes, one may tempt to enhance the signal level of weak SHG in

plasmonic nanostructures. This essentially requires increasing the near-field intensity at the fundamental frequency given that the SHG yield is proportional to the square of the fun-damental field intensity [19]. A particularly efficient way to boost the near-field intensity for enhancing SHG is to exploit the Fano resonances in plasmonic structures [20]. The Fano resonances, identifiable with their characteristic asymmetric lineshapes were first discovered in atomic systems, experi-encing interference between discrete and continuum states [21]. Analogously, the Fano resonances in plasmonic struc-tures can be described by the interference between a narrow dark mode and a broad bright mode via near-field coupling [22]. The plasmonic Fano resonances have been widely studied [20, 23–28] and the corresponding near-field interaction can produce an extremely strong local field which is beneficial to the SHG enhancement. Owing to its effectiveness for local field

Journal of Physics: Condensed Matter

Enhanced second harmonic generation from a plasmonic Fano structure subjected to an azimuthally polarized light beam

Wuyun Shang1 , Fajun Xiao1,3, Lei Han1, Malin Premaratne2, Ting Mei1 and Jianlin Zhao1,3

1 MOE Key Laboratory of Material Physics and Chemistry under Extraordinary Conditions, and Shaanxi Key Laboratory of Optical Information Technology, School of Science, Northwestern Polytechnical University, Xi’an 710129, People’s Republic of China2 Advanced Computing and Simulation Laboratory (AχL), Department of Electrical and Computer Systems Engineering, Monash University, Clayton, VIC 3800, Australia

E-mail: fjxiao@nwpu.edu.cn and jlzhao@nwpu.edu.cn

Received 4 November 2017, revised 24 December 2017Accepted for publication 3 January 2018Published 17 January 2018

AbstractWe show that an azimuthally polarized beam (APB) excitation of a plasmonic Fano structure made by coupling a split-ring resonator (SRR) to a nanoarc can enhance second harmonic generation (SHG). Strikingly, an almost 30 times enhancement in SHG peak intensity can be achieved when the excitation is switched from a linearly polarized beam (LPB) to an APB. We attribute this significant enhancement of SHG to the corresponding increase in the local field intensity at the fundamental frequency of SHG, resulting from the improved conversion efficiency between the APB excitation and the plasmonic modes of the Fano structure. We also show that unlike LPB, APB excitation creates a symmetric SHG radiation pattern. This effect can be understood by considering an interference model in which the APB can change the total SHG far-field radiation by modifying the amplitudes and phases of two waves originating from the individual SRR and nanoarc of the Fano structure.

Keywords: plasmonics, second harmonic generation, singular optics

(Some figures may appear in colour only in the online journal)

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enhancement, a large number of theoretical and experimental studies on this topic can be found [29–33]. Traditionally, in order to achieve a significantly enhanced near-field intensity, the Fano structure primarily requires either the geometry with sharp features such as small gaps and tips, or the dark modes with high Q factors [31, 33]. On the other hand, if the struc-ture is fixed, one can always manipulate the polarization states of the excitation for similar effects. It has been demonstrated that cylindrical vector beams (CVBs) [34], including radially and azimuthally polarized beams (APBs), can excite much brighter ‘hot spots’ in plasmonic lenses [35], multimers [36, 37] and split-ring resonators (SRRs) [38] than those produced by linearly polarized beams (LPBs). Strikingly, CVBs can be generated and reconfigured by a spatial light modulator [34, 39–41]. Thus, we can argue that the latter method may be a better solution for SHG in Fano structures by matching the polarization states of the excitation to the eigenmodes of the plasmonic structures.

In this paper, we study the SHG of a SRR/nanoarc Fano structure under illuminations of APB and LPB. It is found that an almost 30 times increase of SHG intensity is realized under illumination of APB with respect to LPB. This remark-able SHG enhancement is attributed to the increase of local field resulted from the spatial matching between the polariza-tion states of excitation and the plasmonic modes. Moreover, a more symmetric SHG far-field radiation pattern is observed under the excitation of APB, which can be well interpreted by an interference model. We envision that our results may pave the way for future applications in nonlinear optical devices.

2. Methods

The proposed Fano structure is built by the coupling gold SRR and nanoarc which are coaxially aligned with each other, as shown in figure 1(a). The dimensions of this optimized Fano structure are: the angular gap α = 45°, the inner radius r1 = 35 nm, the outer radius r2 = 60 nm for the SRR, the central angle β = 160°, the inner radius r3 = 90 nm, and the outer radius r4 = 190 nm for the nanoarc. The thickness T of the structure is kept constant (T = 40 nm) throughout this paper.

The APB is a type of CVBs whose polarization is aligned along the azimuthal direction in the transverse plane. In the framework of the vector diffraction theory, focused by a high numerical aperture (NA) objective, an APB will propagate with a pure transverse polarization through the entire focal region. The azimuthal electric field component at a given point in the focal region can be expressed as [42]

Eφ (ρ, z) = 2A∫ α0

0cos1/2 (θ) sin (θ) l0 (θ) J1 (kρsinθ)exp (ikzcosθ) dθ,

(1)where A is a constant, α0 = sin−1(NA/n) is the maximum focusing angle, k = 2π/λ is the wavenumber, J1 is the first order Bessel function of the first kind, l0 denotes the pre-scribed apodization pupil function and can be described by the Bessel–Gauss function [42]

l0 (θ) = exp

[−β2

0

(sinθ

sinα0

)2]

J1

(2β0

sinθsinα0

). (2)

Here, β0 = 1 is the ratio between the pupil radius and the beam waist.

As shown in figure 2, an objective with NA = 0.9 is used to tightly focus an LPB, with the polarization along y axis, or an APB illuminating on the structure at air ambiance. The intensity distribution of an APB (λ = 1 µm) at the focal plane derived from equation (1) is shown in figure 1(b). The com-putation of SHG from the Fano structure can be divided into three steps: calculation of the fundamental near-field, deriva-tion of the source of SHG (i.e. nonlinear polarization) from the fundamental electric field and evaluation of the second harmonic electric field in far-field domain [43].

The linear optical response of the designed Fano structure is numerically calculated by using a commercial software (Lumerical Solutions Inc., Canada) based on the finite-differ-ence time-domain method [44]. In our simulations, perfectly matched boundary conditions are employed to mimic embed-ding of the single nanostructure in an infinitely large free space. The structure is swept-meshed with a minimum size of 2 nm and is illuminated by a normally incident LPB or APB. The permittivity of gold is taken from experimental data [45], which is fitted to the Drude model in the visible and near-infrared regions. The omnidirectional scattering spectra for the wavelength ranging from 0.8 µm to 1.7 µm as well as the charge and near-field distributions at specific wavelengths are performed to demonstrate the Fano resonance of the designed structure.

In a nonlinear optical process of SHG, two photons at the fundamental frequency (ω) are converted into one photon at the second harmonic frequency (2ω). In classical theory, the second harmonic comes from nonlinear polarization (P(2ω)), and is forbidden in the bulk of centrosymmetric materials within the dipole approximation. However, the centrosym-metry is locally broken at the metal surface because of the finite dimension of the atomic lattice [18], allowing the second harmonic polarization at the surface [46, 47]

P(2ω) (rs) = χ(2)s : E(ω) (rs)E(ω) (rs) δ (rs − h (rs)) ,

where E(ω)(rs) is the fundamental electric field at the point rs, χ(2)s is the surface nonlinear susceptibility tensor taking effect

only at the surface boundary defined by h(rs). When the sur-face anisotropy is neglected, the non-vanishing components

of the nonlinear susceptibility tensor are χ(2)s,⊥⊥⊥, χ(2)

s,⊥‖‖ and χ(2)s,‖ ‖⊥ = χ

(2)s,‖⊥‖ [18]. Here, ⊥ denotes the normal component

and ‖ the tangential component at the surface. Since it has

been proved that χ(2)s,⊥⊥⊥ dominates the surface second har-

monic response of gold nanoparticles [48], only this comp-onent is considered in our analysis, giving

P(2ω)⊥ (rs) = χ

(2)s,⊥⊥⊥E(ω)

⊥ (rs)E(ω)⊥ (rs) δ (rs − h (rs)) , (3)

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where E(ω)⊥ (rs) is the normal component of fundamental elec-

tric field at the point rs.The second harmonic electric field detected at an external

point R(R, Θ, Φ) (as shown in figure 2) can be calculated by using the Green’s function approach. In the far-field domain (|R| � |rs|), the second harmonic electric field generated from the Fano structure can be expressed as [46, 49]

E(2ω) (R) = µ0ω2exp(iKR)πR

∫∫∫V dVexp

(−iKR·rsR

)

×

0 0 0cosΘcosΦ cosΘsinΦ −sinΘ−sinΦ cosΦ 0

P(2ω)⊥,x (rs)

P(2ω)⊥,y (rs)

P(2ω)⊥,z (rs)

iRiΘiΦ

,

(4)where iR, iΘ and iΦ are the standard unit vectors of the spherical coordinates, and K is the wavenumber at the second harmonic frequency. In our calculation, the SHG far-field radiation pat-tern for a specific fundamental wavelength is defined as the spatial radiation distribution of the |E(2ω)(R)|2 on a spherical surface with the radius |R| = 100 µm and the SHG intensity is obtained by integrating |E(2ω)(R)|2 over this spherical surface. The SHG intensity of the fundamental wavelength range from 0.8 µm to 1.7 µm, i.e. SHG spectra, and SHG far-field radia-tion patterns at specific fundamental wavelengths are used to evaluate the second harmonic response of the Fano structure under LPB and APB excitations.

3. Results and discussions

3.1. Linear optical response

Figure 3(a) shows the LPB excited scattering spectrum of the SRR/nanoarc structure (red solid curve), where a Fano line-shape can be found [50]. The Fano resonance of our designed structure stems from the interference between the subradiant magnetic dipole mode supported by individual SRR (blue dashed curve in figure  3(a)) and the superradiant electric dipole mode supported by individual nanoarc (orange dashed curve in figure 3(a)). The magnetic dipole mode and electric dipole mode can be viewed as a discrete state and a continuum state, respectively. These two modes destructively interfere at the far-field, leading to a spectral dip at the wavelength,

1.39 µm. The coupling between these two dipolar plasmon modes results in a hybridized lower-energy bonding mode and a higher-energy anti-bonding mode [51]. The bonding mode at the wavelength of 1.48 µm (see the lower-right inset of figure  3(a)) is the resonance of same azimuthally coupled oscillations, however, the anti-bonding mode at the wavelength of 1.15 µm (see the lower inset of figure 3(a)) is corre sponding to the oppositely aligned azimuthal oscilla-tions. The SRR/nanoarc structure is then illuminated by an APB, which is aligned coaxially with the SRR. As the red curve displayed in figure  3(b), the APB excited Fano line-shape exhibits a much stronger bonding mode resonance and a weaker anti-bonding mode resonance. This can be attributed to spatial matching between the polarization state of APB and the symmetry of the Fano structure [50]. As a driving force, APB can effectively excite and enhance the bonding aligned oscillations, but decrease the anti-bonding oscillations. This

Figure 1. (a) Schematic view of the SRR and nanoarc combined Fano structure with the geometry parameters: r1, r2, r3, r4, α, β and T. (b) Intensity and electric vector distributions of an APB.

Figure 2. Schematic diagram of second harmonic response from the Fano structure excited by tightly focused LPB and APB. rs and R are the position vectors of the structure surface and the second harmonic electric field detection point.

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improved conversion efficiency between the APB excitation and the bonding mode produces a higher local field enhance-ment than that with LPB excitation (shown in figure  3(c)), and thus such an effect can be exploited to augment nonlinear optical effects, in this case, specifically SHG.

3.2. Second harmonic response

The curves in figure 4(a) show the SHG spectra of the SRR/nanoarc structure excited by APB (red solid curve) and LPB (black dashed curve). It can be seen that the SHG intensity is significantly increased for the SRR/nanoarc structure illu-minated by APB as compared with that illuminated by LPB. Especially, SHG intensity has an almost 30 times increase at the fundamental wavelength, 1.46 µm, when the illumina-tion is switched from LPB to APB. To unveil the origin of the enhancement of SHG, we calculate the spectra of the integral

fundamental electric field intensity (∣∣∣E(ω)

⊥

∣∣∣2) over the struc-

ture’s surface under APB (blue solid curve in figure 4(b)) and LPB (green dashed curve in figure 4(b)) excitations. The line-shapes of SHG spectra are similar to that of integral funda-mental electric field intensity spectra, both for APB and LPB excitations, indicating that the SHG characteristics stem from the enhanced resonance of the fundamental electric field [52]. In addition, the fundamental electric field intensity and sur-

face second harmonic polarization (∣∣∣P(2ω)

⊥ (rs)∣∣∣) distributions

at peak positions of SHG spectra are shown in figures 4(c) and (d), respectively. For the excitation of APB, the fundamental electric field intensity is significantly enhanced as compared with the counterpart excited by LPB. This enhancement, which directly gives rise to the augment of SHG intensity, can be attributed to the improved conversion efficiency between the APB excitation and the bonding mode.

Far-field properties of SHG processes are critical to the design of practical applications in the nonlinear regime. Figures  5(a) and (b) display the normalized SHG far-field radiation patterns at SHG spectra peaks shown in figure 4(a), when the structure are illuminated by LPB and APB, respec-tively. For the excitation of APB, a more symmetric SHG radiation pattern is obtained as compared with its counterpart excited by LPB. For a clear comparison, the normalized SHG radiation patterns in the xy and xz planes are shown in fig-ures 5(c) and (d). To explain the difference of SHG far-field radiation patterns between figures 5(a) and (b), we introduce an interference model in which the total SHG radiation of the SRR/nanoarc structure is considered as the interference between SHG radiation from individual SRR and nanoarc [53]. In figures 6(a)–(f), we show the SHG far-field amplitudes nor malized by the maximum value of SHG amplitude of SRR and the phase differences (|ϕSRR − ϕnanoarc|) of SHG far-field between SRR and nanoarc, respectively. As can be seen from the phase differences in figure 6(e), two SHG radiations from SRR and nanoarc interfere destructively in xy plane for both

Figure 3. Scattering spectra of (a) SRR (blue dashed curve), nanoarc (orange dashed curve) and the combined structure (red solid curve) with the illumination of LPB and (b) SRR/nanoarc structure with the illumination of APB. The spectrum of SRR is magnified by 10 for comparison. The insets are the charge distributions at the labeled wavelengths and the polarizations of the excitation field. (c) Near-field intensity distributions on the top surface of the structure under LPB (up) and APB (down) illuminations at the labeled wavelengths.

Figure 4. (a) SHG spectra and (b) fundamental electric field intensity spectra of the SRR/nanoarc structure with the excitations of LPB (black and green dashed curves) and APB (red and blue solid curves). The SHG spectra and fundamental electric field intensity spectra for the excitation of LPB are multiplied by 10 and 2 for clear comparison. (c) Intensity of the normal component of fundamental electric field and (d) surface second harmonic polarization distributions at peak positions of the SHG spectra, under the excitations of LPB (left) and APB (right), respectively.

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Figure 5. Normalized SHG far-field radiation patterns at the SHG spectra peaks of SRR/nanoarc structure excited by (a) LPB and (b) APB. The corresponding SHG radiation patterns in (c) xy and (d) xz planes for the illuminations of APB (red solid curve) and LPB (black solid curve). The outer circle corresponds to the same intensity for each polar plot.

Figure 6. SHG far-field amplitude patterns of SRR (black solid curves), nanoarc (red solid curves) and SRR/nanoarc (blue solid curves) in the xy (top row) and xz (bottom row) planes for the illuminations of (a) and (c) LPB and (b) and (d) APB. Phase differences (|ϕSRR − ϕnanoarc|) of the SHG far-field between SRR and nanoarc in the (e) xy and (f) xz planes, where the black dashed curves and red solid curves are for excitations of LPB and APB, respectively.

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LPB and APB excitations. The SHG far-field patterns of SRR are almost the same for LPB and APB excitations as shown with the black solid curves in figures  6(a) and (b), respec-tively. However, the SHG far-field patterns of nanoarc excited by APB (red solid curve in figure 6(b)) is more symmetric than that excited by LPB (red solid curve in figure 6(a)). As the consequence of the destructive interference, the APB excited total SHG radiation (blue solid curve in figure 6(b)) becomes more symmetric than its LPB exited counterpart (blue solid curve in figure 6(a)). For the case in xz plane, SHG radiation amplitudes from SRR (black solid curves in figures 6(c) and (d)) and nanoarc (red solid curves in figures 6(c) and (d)) are mirror symmetry with respect to z axis for both the LPB and APB excitations. On the other hand, the APB excited phase difference of SHG far-field is more symmetric with respect to z axis (red solid curve in figure 6(f)) than that excited by LPB (black dashed curve in figure 6(f)). Therefore, in term of total interfered SHG far-field, one can see a more symmetric quadrupole-like radiation pattern excited by APB (blue solid curve in figure 6(d)) than its counterpart excited by LPB (blue solid curve in figure 6(c)).

4. Conclusions

In summary, we have proposed a method to enhance SHG from a gold SRR/nanoarc Fano structure by employing an APB as the excitation. Our results show that, compared with the LPB excitation, the APB can bring an almost 30 times increase in SHG peak intensity. This enhancement of SHG can be traced back to the boosting of the local field intensity at the funda-mental frequency due to the enhanced conversion efficiency between the APB excitation and the plasmonic modes of the designed structure. In the far-field domain, APB can excite a much more symmetric SHG radiation pattern than that excited by LPB. We explain these features by considering interfer-ences in the structure and its influence on the far-field prop-erties of SHG. It reveals that in contrast to the case of LPB, APB excitation creates symmetric SHG amplitude and phase patterns from individual SRR and nanoarc. Our results provide an effective and an efficient strategy to enhance the SHG in plasmonic Fano structures and may find applications in non-linear detectors, sensors, and other nonlinear optical devices.

Acknowledgments

This work was supported by the National key R&D Program of China (2017YFA0303800), National Natural Science Founda-tion of China (NSFC) (11634010, 61675170, 61675171, and 61377035), Australian Research Council (DP140100883), Natural Science Basic Research Plan in Shaanxi Province (2017JM6022), Fundamental Research Funds for the Central Universities (3102017zy017).

ORCID iDs

Wuyun Shang https://orcid.org/0000-0001-5087-5196

Ting Mei https://orcid.org/0000-0001-7756-040X

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