Structural second-order nonlinearity in plasmonic metamaterials

Nonlinear processes are at the core of many optical technologies whose further development require optimized materials suitable for nanoscale integration. Here we demonstrate the emergence of a strong bulk second-order nonlinear response in a plasmonic nanorod composite comprised of centrosymmetric materials. We develop an effective-medium description of the underlying physics, compare its predictions to the experimental results, and analyze the limits of its applicability. We demonstrate strong tunable generation of the p -polarized second-harmonic light in response to either s - or p -polarized excitation. High second-harmonic enhancement is observed for fundamental frequencies in the epsilon-near-zero spectral range. The work demonstrates emergence of structurally tunable nonlinear optical response in plasmonic composites and presents a new nonlinear optical platform suitable for integrated nonlinear photonics.


INTRODUCTION
Second-harmonic generation (SHG), a phenomenon in which the incoming radiation of a frequency ω is converted into the signal at a double frequency 2ω, is a fundamental nonlinear optical process that enables high-resolution microscopy, laser technology, and surface studies [1][2][3][4][5]. Materials with strong second-order nonlinear response can further advance a broad class of photonic applications, including frequency conversion, optical information processing, sensing, security, and healthcare. Unfortunately, natural optical materials with strong second-order nonlinearity are few, and new solutions are needed to develop nonlinear optics in compact, wavelength-scale, and integrated systems.
Recent advances in nano-and microfabrication have brought into play a new class of composite media, often called metamaterials, whose optical properties are determined by shape and mutual arrangement of their components [6][7][8][9][10]. Metamaterials provide a flexible platform for engineering linear optical behavior that can range from isotropic [6,9] to anisotropic and hyperbolic [11,12] to chiral and bianisotropic [10]. As a rule, linear optical response of metamaterials can be related to averaged linear optical response of their components via the effective-medium theory (EMT) [9,10]. Similarly, the effective nonlinear susceptibility of the composite can be related to the nonlinear susceptibilities of constituent materials [13][14][15][16]. Recently, nonlinear metamaterials have been used for engineering third-order (Kerr-type) nonlinearity, achieving on-demand spectral response, including its sign and polarization control [17][18][19][20][21].
In this work, we show that re-shaping of electromagnetic fields in metamaterials with plasmonic components can be used to transform SHG from surface-to volume-dominated regime and engineer strong tunable bulk nonlinear response in plasmonic composites. We experimentally demonstrate tunable SHG from plasmonic nanorod metamaterials, develop a theoretical description of the observed phenomena, and prove that the nonlinear response can be engineered by changing structural parameters of the composite.

FABRICATION AND LINEAR OPTICAL RESPONSE
We consider SH response of the metamaterial comprising an array of gold nanorods in an alumina matrix [ Fig. 1(a)]. When the nanorod radius r and inter-rod separation a are much smaller than the operating wavelength λ, such metamaterial behaves as a uniaxial crystal with an optical axis parallel to the nanorods [11,22,23].
The linear optical response of the metamaterial is described by a diagonal permittivity tensorε with components ϵ xx ϵ yy ϵ ⊥ and ϵ zz ≠ ϵ ⊥ . If the material absorption is not too small and r ≪ a ≪ λ, the effective-medium parameters can be related to the relative permittivity of the host ϵ h and nanorod ϵ Au materials, and the nanorod concentration p πr 2 ∕a 2 via In the limit of small absorption, long nanorods, or large unit cells, the deviations from the local EMT predictions can be quantitatively explained by incorporating retardation effects into nonlocal (wavevector-dependent) EMT [24][25][26][27]. Importantly, components of the effective permittivity tensor ϵ ⊥ and ϵ zz can be of different signs (Fig. 1), tuning the nanorod composite between anisotropic dielectric, epsilon near zero (ENZ) media, and a unique "hyperbolic metamaterial," which enables propagation of waveguided modes with subwavelength light confinement that in turn enhance light-matter interactions in the metamaterial [10,12,23,28].
Plasmonic nanorod metamaterials were fabricated via Au electrodeposition into nanoporous anodized aluminum oxide (AAO) templates on a glass substrate [29]. An Al film of 500 nm thickness was deposited on a substrate by magnetron sputtering. The substrate comprises a glass cover slip with a 10-nm-thick adhesive layer of tantalum pentoxide and a 7-nm-thick Au film acting as a weakly conducting layer. Highly ordered, nanoporous AAO was synthesized by a two-step anodization in 0.3M oxalic and 0.3M selenic acid at 40 V and 48 V, respectively. Gold electrodeposition was performed with a three-electrode system using a non-cyanide solution. The length of nanorods was controlled by the electrodeposition time. Fabricated metamaterials were ion-milled to smooth the top surface and ensure that the nanorods are of the same length [30]. We estimate that the variation of the nanorod height is on a few nm level, so that optical properties of the composites are not affected by surface roughness. In particular, SHG, which is very sensitive to surface roughness, exhibited neither an appreciable diffuse component nor unpolarized signal, which typically appear for rough surfaces. The samples were annealed at 300°C to improve Au optical properties.
Two samples were used in this study: sample A composed of 18-nm-diameter, 220-nm-long nanorods arranged in 110-nm period array, and sample B comprising 67-nm-diameter, 150-nm-long nanorods in the array of 100-nm period. Reflectivity and the effective-medium parameters of both samples are presented in Fig. 1. Sample A exhibits the effective plasma frequency at around λ 0 ≃ 1500 nm, while sample B operates in hyperbolic regime throughout visible and infrared spectral ranges. The linear reflection spectra of the composites are typical of anisotropic metamaterials, showing resonances due to the Fabry-Perot modes of the metamaterial slab [23,28]. The measured spectra correspond well to the numerical models for both the full-wave solutions of the Maxwell's equations using finiteelement method (FEM) [31] and the transfer matrix formalism that approximates metamaterials as homogeneous layers with anisotropic permittivity given by Eq. (1).
Slight disagreement between the experiment and numerical calculations in the visible range is due to the interband transitions in gold that are not well described by the Drude model, ϵ Au ϵ b − ω 2 p ∕ωω− iτ with plasma frequency ω p e 2 n 0 ∕m e ϵ 0 1∕2 1.36 × 10 16 s −1 , inelastic scattering frequency τ 2.1 × 10 14 s −1 , parameter ϵ b 9.5, and ϵ 0 , e, m e , and n 0 being the permittivity of free space, electron charge, electron mass, and free-electron density in gold, respectively [32].

NONLINEAR OPTICAL RESPONSE
SHG spectroscopy was performed using light from the optical parametric amplifier (200 fs pulse trains at the repetition rate of 200 kHz and average power up to 50 mW in near-IR wavelength range 1100-1800 nm). The laser light polarization was controlled to achieve por s-polarized fundamental light incident on the sample at an angle of incidence of 45°with a spot approximately 30-50 μm in diameter. The reflected por spolarized SH light was spectrally selected using the set of short-pass optical filters and measured with the spectrometer and the cooled chargedcoupled device (CCD) camera. In order to compensate for pulse energy and pulse duration fluctuations, the measured signal was normalized to a reference SHG measured in reflection from β-barium borate (BBO) crystal. Also, the SHG from each sample

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was compared to the signal from a z-quartz plate. As a result, the SHG data for samples A and B can be directly compared to each other.
SHG spectra of the two samples fundamentally differ from each other. Sample A exhibits strong p-polarized SHG emission in response to a p-polarized pump in the ENZ frequency range (Fig. 2). At the same time, SHG signal generated by sample B (Fig. 3) exhibits pronounced maxima associated with excitation of the metamaterial slab modes [28] for both pand s-polarized excitation. Interestingly, the SHG intensity from sample B under s-polarized excitation is approximately four times stronger than under p-polarized pump, indicating the important role of local fields inside the metamaterial, as was observed previously for the nanoparticle composites [33].
The spectral and polarization dependences of the SHG are in a good agreement with the full-wave numerical simulations (Figs. 2 and 3) that implement the hydrodynamic model of the SHG in plasmonic media [28,34,35] (see Supplement 1) with the nonlinear polarization of gold given by where ω and 2ω represent the fundamental and SH frequencies, respectively, index α represents the Cartesian coordinates, and E, B, j are the electric field, magnetic induction, and current density, respectively (the quantitative difference between numerical and experimental results for sample A can be explained by deviation of optical absorption of solution-derived gold from the Drude model used in this work [36]). The detailed analysis shows that SHG efficiency and polarization dependencies are complex functions of the effective medium parameters, thickness of the metamaterial slab, and angle of illumination θ [28]. Nevertheless, in all cases, the nonlinear polarization, and, therefore, SHG, is dominated by the terms related to the components of the electromagnetic fields that have nonvanishing unit-cell averages [37] and to ∂∕∂z derivatives of these components. These terms depend on polarization of the incident beam and are given by This is illustrated in Figs. 2 Note that the simplified model predicts P 2ω;y 0, resulting in only p-polarized SHG signal, in line with the experiment as well as with the predictions of the full-wave calculations. Using Eqs. (3) and the constituent relationship j iωϵ 0 ϵ Au E, it becomes possible to represent the unit-cell-averaged nonlinear polarization in the metamaterial as a quadratic form of the (unit-cell-averaged) fields, introducing the effective bulk secondorder nonlinear susceptibilities χ 2,e and χ 2,m : where the Greek subscripts represent the Cartesian coordinates x, y, and z.
Components of the effective nonlinear susceptibility were calculated in the limit of the validity of local EMT [Eq. (1)], which yields homogeneous fields across the cross section of the nanorods [10,24, and Supplement 1], by substituting explicit relationships between field components inside the nanorod, their unit-cell averages, frequency, and components of the wavevector, resulting in  4) and (5) Fig. 3. Same as Fig. 2 but for sample B.

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Vol. 5, No. 12 / December 2018 / Optica χ 2,e x;xx N p Lϵ Au k x ϵ ⊥ ϵ zz , Here, N p 2ϵ 0 ϵ Au e∕mω 3 ∕ω 4 p ϵ Au 2ω − ϵ b is the normalization parameter, k x ω sin θ∕c is the transverse component of the wavevector, and L 2pϵ h ∕ϵ Au ω ϵ h represents the relationship between the E x and E y components of the electric field in the nanorod and its unit-cell-averaged values. The first three components describe SHG excitation due to p-polarized fundamental light, while the latter two represent the SH generated by the s-polarized light.

DISCUSSION
Equations (4) and (5) represent the main result of this work: the metamaterial as a whole exhibits dipolar-like nonlinear response even though its material constituents lack bulk dipolar χ 2 . The effective nonlinear susceptibilities of plasmonic nanorod composite are determined primarily by the structure of the local fields inside it [37]. Components of the effective nonlinear susceptibility depend on an angle of incidence so that the symmetry of the metamaterial is broken by the internal fields, except at normal incidence when the electric-dipole SHG is forbidden due to symmetry considerations. The explicit dependence of the effective nonlinear susceptibility on the wavenumber reflects the structural origin of the nonlinearity of a metamaterial.
The developed nonlinear EMT adequately predicts both spatial distribution of nonlinear polarization (Supplement 1) and spectral SH response [Figs. 2(d) and 3(d)] with exception of a small red shift of the SHG spectra for sample B, which is related to the red shift in a linear reflectivity observed in Fig. 1. Calculated values of an effective nonlinear response of the composite χ 2 ∼ 10 −10 -10 −6 [electrostatic units, ESU] [Figs. 2(c), 3(c), and 4] indicate relatively strong nonlinearity, which favorably compares to common nonlinear-optical crystals, including quartz, potassium dihydrogen phosphate (KDP) (χ 2 ∼ 10 −9 ESU), and LiNbO 3 (χ 2 ∼ 10 −7 ESU) [1,2]. Experimental data are in line with calculated values for the nonlinear susceptibility for both studied metamaterials. SHG intensity from the composites can be further optimized by manipulating geometry and reducing losses.
The main limitation on the effective-medium nonlinear description, presented in this work, comes from the granularity of metamaterial. In particular, the local EMT that underpins the final expressions in Eq. (1) assumes local behavior of the constituents, dipolar quasistatic field between the nanorods, and does not account for propagation of cylindrical plasmons along the nanorods. Nonlocal response of a free-electron plasma [23,38] may become relevant for composites with ultra-thin nanorods (r ≪ 1∕k F ∼ λ 0 ∕100, with k F being the Fermi wavenumber). Contributions of retardation effects and spatial dispersion may affect field distribution in the composites with a nanorod concentrations p ≳ 0.3. In the limit r ≪ λ, these effects can be taken into account by including retardation effects [24] in the nonlocal Maxwell-Garnett formalism. Excitation of cylindrical plasmons primarily affects composites with low loss operating across elliptic and ENZ regimes. This limitation can be addressed by including propagation of additional electromagnetic waves (with linear response described in Ref. [25]) into the developed formalism. Nonlocal EMT can be further developed to incorporate non-quasistatic effects. Granularity of the composite must also be considered when emission of SH light is calculated. We expect that including the high-index "longitudinal" modes through nonlocal EMT [25,39] may further improve predictive power of the formalism presented in this work.
In contrast to common nonlinear optical crystals with fixed optical properties, the structural origin of the second-order nonlinearity in metamaterials provides a platform for engineering not only spectral but also polarization properties of a nonlinear response. For example, the structural parameters of the nanorod metamaterials can be tuned to achieve dominant SHG contribution from either p → p (Fig. 2) or s → p (Fig. 3) polarization configurations [28,33].
In the former case, the metamaterial operates in the ENZ (ϵ zz ≃ 0) regime at a fundamental frequency λ 0 ≃ 1600 nm. The relatively weak effective nonlinear polarizability is compensated by the strong enhancement of z component of the electric field (a similar response has been predicted for bulk, nontunable, ENZ materials [20,40]). Interestingly, numerical calculations [ Fig. 5(a)] suggest that material absorption in gold (which Research Article effectively limits the value of jϵ zz j ) plays the role of the limiting factor in the SHG process in the ENZ regime. Reducing losses in the gold by a factor of 2 (achieved in calculations by reducing scattering frequency τ to the value that is in line with bulk Au τ 1.05 × 10 −14 s [32]) has a potential to increase SHG efficiency in the metamaterial by an order of magnitude. In another limit, the metamaterial operates in the hyperbolic regime so that the enhancement of the local field is attributed mainly to the Fabry-Perot modes of the metamaterial slab of a finite thickness [28]. This modest enhancement of a local field does not significantly depend on material absorption [ Fig. 5(a)] and, being accompanied by a relatively strong nonlinear polarizability, once again results in a strong SH response of the metamaterial.

CONCLUSION
We have demonstrated the emergence of structural nonlinearity in composite metamaterials. The approach, presented here on the example of SHG from plasmonic nanorod metamaterials, can be extended to analyze nonlinear response of a broad class of composites, such as plasmonic nanoparticle metasurfaces [33] and metamaterials based on noncentrosymmetric, strongly nonlinear materials, such as AlGaAs nanopillars [41]. Structural nonlinearity opens the door to utilize composite media to engineer spectral and polarization nonlinear response beyond what is available with naturally occurring materials.
All the data supporting this research are presented in the article and in the supplementary material.