1. Introduction

A high nonlinear efficiency in frequency conversion process is a key factor for a number of photonic applications. The refractive index dispersion of semiconductors makes however difficult achieving the phase-matching (PM) condition which enables an optimal second-harmonic (SH) generation. In addition, the miniaturization of optical devices at the nanometer scale leads to deal with a reduced amount of nonlinear matter, which in turn limits the conversion process. Various methods have been proposed to overcome these limitations, including tailoring of photon dispersion relations by periodic index modulations [1] and the use of resonance effects in microcavities [2].

In this context, the pioneering work described in Ref. [3] have theorized the strong second-harmonic conversion processes in finite size 1D photonic crystals (PCs) which can be achieved by phase-matching resonant modes of frequencies close to the photonic band gap edge. The signature of this strong nonlinear interaction has been experimentally verified by the observation of a SH efficiency growth at the fifth power of length in Bragg mirror devices and in periodic waveguides [4]. These results are at the state of the art in the field of nanophotonic components and seem to attain an optimal limit. However, up to date the extension of this resonant PM condition has not been explored under sub-wavelength light confinement. In this Letter, we challenge to enhance the SH conversion by taking advantage of the strong light localization achieved in finite size dielectric nonlinear nanorod chains. Du et al. have recently shown [5] that in specific conditions dielectric chains support guided modes characterized by a lateral extension squeezed below the half wavelength. Here, we show that this sub-wavelength transversal confinement together with the resonant PM condition add an important property to the SH generation enhancement.

2. Phase-matching condition for periodic nanorod chains

Consider a chain of dielectric nonlinear nanorods of permittivity ε_r and radius r, and presenting a lattice period a. The χ ⁽²⁾ nonlinear polarization parameter is assumed to be oriented parallel to the axis of the rods. For the sake of simplicity, the refractive index dispersion is neglected since adjusting the filling factor in nanorods can compensate it [6]. We also consider a two-dimensional electromagnetic scattering problem by assuming that the height of the rods is larger four or five times than the wavelength. The validity of this assumption is justified by the several works showing the possibility to fabricate nanorods of exceptional length [7] and with diverse materials [8]. In the framework of the depleted pump condition and for s-polarized light, the fundamental (FF) and SH electric fields (z-component) E_z(r) satisfy the following set of nonlinear equations:

where k_ω and k _2ω are the wave vectors in vacuum at ω et 2ω, respectively. Before carrying on with nonlinear computations, the PM condition between the FF mode presenting the sub-wavelength confinement and the SH mode is first established. In Fig. 1 are depicted the first and second dispersion bands obtained for an 1D infinitely long array of rods made of GaAs (ε =12.25)in the case of s-polarized light. Note that the second band is partially above the light line since at these frequencies the quasi-guided modes can couple with the radiating air modes. Du and co-authors have shown [5] that the transverse sub-wavelength confinement occurs for the guided mode of the first band when the propagation constant is close to the edge of the first Brillouin-zone: k_‖(ω) ≃ π/a. Since a wave vector k_‖(ω) ≃ π/a is being imposed at ω, the PM condition implies to research at 2ω a SH mode possessing a wave vector k_‖(2ω) = 2k_‖(ω) ≃ 2π/a. This corresponds, in the first Brillouin zone, to the second photonic band in the neighborhood of the Γ point, i.e. for k_‖(2ω) ≃ 0. However, around the Γ point the SH waves are not perfectly guided since the modes are above the light line. At the Γ point the quasi-guided mode presents an odd symmetry in the chain direction (see insets in Fig. 1) preventing the coupling with the radiative air modes. This deduction is confirmed by the computation of the transversal Q factor, which diverges when the Γ point is approached. This property confines the SH mode inside the nanorods which again favors the nonlinear interaction with the FF mode. Finally, this photonic band diagram analysis allows us to determine the reduced FF frequency u_ω = a/λ = 0.22 and the radius of rods is r = 0.26a necessary to phase-match the FF guided mode for k_‖(ω) ≃ π/a with the SH quasi-guided mode for k_‖(2ω) ≃ 0.

 

Fig. 1. First and second dispersion bands for an infinitely long chain of dielectric nanorods. In the insets the Bloch modes (E_z component) associated with k = π/a and k = 0 for the first and second band respectively.

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3. Effective phase-matching condition in finite size structure

These PM conditions must now be studied for a finite number of nanorods N. The continuous quasi-guided modes observed for the infinitely long chain become indeed a set of discrete modes in the case of a finite size chain. The transmission diagram computed with the Multiple Scattering Method [9] for N = 20 shows a set of resonant frequencies, which indicates that the finite size chain can be viewed as an open cavity, Fig. 2(a). The resonance u ¹ _ω at the edge of the first photonic band gap [Fig. 2(a)] can be linked with the FF guided-mode of propagation constant k_‖(ω) ≃ π/a since it presents a strong lateral confinement [Fig. 2(b)]. Although this resonant mode is very similar along the longitudinal direction to that observed for 1D finite PCs, the effective PM condition technique developed in Ref. [10] cannot however be directly applied since we deal here with a 2D scattering problem.

In order to find the effective PM condition among the resonant modes lying in the second photonic band, we develop a coupled mode model based on a decomposition of the electric fields on the resonant eigenmodes of the finite chain. The resonant chain modes are the eigenfunctions Φ solution of the Helmholtz equation ΔΦ+k ²Φ = 0 without any incident field where the wave number is k ² = ε_rω ² _p/c ². In that case the frequency ω_p = ω ₀ − iγ_ω is complex and in the framework of the Multiple Scattering Method Φ and ω_p refer to the eigenfunctions and the poles of the scattering matrix S [9]. Finally, in the case of non degenerate eigenstates and for frequencies ω and 2ω corresponding to the real part of the poles (i.e. to the resonant frequencies), the FF and the SH electric fields are respectively $E_{\omega }(\mathbf{r},t)=a_{\omega }\left(t\right){\Phi }_{\omega }\left(\mathbf{r}\right)e^{-i{\omega }_{0}t}$ and $E_{2\omega }(\mathbf{r},t)=a_{2\omega }\left(t\right){\Phi }_{2\omega }\left(\mathbf{r}\right)e^{-i2{\omega }_{0}t}$ . The functions α_ω and α _2ω are related to the number of photons in the eigenmodes at ω and 2ω respectively. For the computations, we normalize the eigenmodes by fixing the electromagnetic energy stored in the mode: ∫_Ω ε_r|Φ|² dV = 1. Plugging these expressions into the nonlinear wave equation and applying the Slowly Varying Envelope Approximation in Time (SVEAT) [11] allows one to find the following set of coupled nonlinear equations:

 

Fig. 2. (a) Transmission resonances around the FF and SH frequencies for a chain composed by 20 nanorod. The frequencies at the SH are scaled of a factor of 2. (b) FF and SH mode intensity distribution.

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where the coupling coefficients ${\Gamma }_1=i\omega {\chi }^{\left(2\right)}{\int }_{{\Omega }_{\mathrm{NL}}}{\Phi }_{\omega }^{*2}{\Phi }_{2\omega }\mathrm{dV}$ and ${\Gamma }_2=i\omega {\chi }^{\left(2\right)}{\int }_{{\Omega }_{\mathrm{NL}}}{\Phi }_{\omega }^2{\Phi }_{2\omega }^{*}\mathrm{dV}$ refer to overlap integrals, which are calculated onto the nonlinear region Ω_NL, i.e. inside the nanorods. The function ϛ(E ₀) takes into account the incident pump field. By slightly varying the geometrical parameters of the chain and by keeping the permittivity fixed, the overlap integrals are performed when the FF frequency resonance u ¹ _ω is matched with each one of the SH resonances in order to get u ⁱ _2ω = 2u_ω, where u ⁱ _2ω with i = 1,2,3 refers to different resonances near the second band gap [Fig. 2(a)]. These computations allow us to determine that the resonance u ² _2ω is associated with a resonant mode satisfying the PM condition.

4. Results

The SH enhancement is now demonstrated by the direct computation of the conversion efficiency for two optimized chains constituted by 20 and 30 rods. To have an order of comparison with most of the optical nonlinear materials we consider a nonlinear susceptibility χ ⁽²⁾ of order of 10 pm/V. We consider a pumping field radiating at the normalized wavelength λ_F/a=0.222.

 

Fig. 3. (a) SH generation efficiency in function of the pump peak intensity for a chain of N = 20 and N = 30 nanorods. Dotted curves are obtained in the framework of the undepleted pump approximation. Solide curves show the rigorous calculation obtained with COMSOL. Square points are obtained with the model of Eqs. (2a) and (2b). (b) SH generation efficiency in function of the number of rods N (theoretical curve in solid line and numerical results in triangle points). The efficiency of the nanorod chain grows as a power of p = 5.3 of N.

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Remark that the choose of the operating wavelengths, in the case of semiconductors, is limited by the energy gap of the considered material. In Fig. 3(a) the conversion efficiency η, ratio of the total Poynting flow generated at 2ω with the incident Poynting flow at ω, is plotted with respect to the input pump intensity. Here, the incident pump field is a gaussian beam propagating toward the positive x-direction and presenting a waist of w ₀ = λ ₀/4 centered on the first nanorod (for negative x). The solid curves of Fig. 3(a) are obtained in the framework of the depleted pump by solving rigorously the nonlinear system of Eqs. (1a) and (1b) with COMSOL. These results which are in good agreement with the solution derived from the coupled mode model [Eqs. (2a) and (2b)] demonstrate a conversion of 10% and 37% for a 15GW/cm² input intensity and for 20 and 30 nanorods respectively. These results show that the nanorod chain is a very efficient device especially when the effective incident power is considered. A 10% conversion efficiency is reached with a peak power laser less than 200Wfor a real 3D structure composed by 20 nanorods of 5 µm height. This pump intensity can in addition decrease since the SH generation efficiency drastically increases with the number N of nanorods. From the coupled mode model in undepleted pump approximation it can be shown that the conversion efficiency is a quadratic function of the photon lifetime [12] at ω: η ∝ (τ_ω)². This photon lifetime τ_ω is related to the imaginary part of the scattering matrix’s poles by τ_ω = 1/|2γ|. Similarly to 3D scatterer chain [13], the computation of the poles associated with the FF mode reveals that the lifetime τ_ω increases with the cube of the numbers of rods, namely τ_ω ∝ N ³. A conversion efficiency η ∝ N ⁶ increasing at the six power of chain length can therefore theoretically be reached with 2D nanorod chains. This behavior approached in Fig. 3(b) by our nonlinear computations is the signature of the SH generation based on the resonant PM depicted for 1D PCs [10]. This strong nonlinear conversion provided by the 2D light localization in the nanorod chain is confirmed by the computation of the effective second order susceptibility χ ⁽²⁾ _eff which is 30 times larger, for a 20 rods chain, than the actual nonlinear susceptibility χ ⁽²⁾ of the bulk material. This result is more impressive if one considersf that it has been obtained by using an extremely small quantity of matter, even if compared to 1D-PBG structures.

5. Conclusion

In conclusion, we have investigated the SH generation process in nanorod chains in conditions of PM condition. A SH generation efficiency of 10% for 200 W of FF peak power is predicted, showing a χ ⁽²⁾ _eff about 30 times bigger than the actual second-order susceptibility of the nonlinear material. We have shown that sub-wavelength localization enhances the local field density reducing substantially the dimensions of the nonlinear devices. These results are based on a technologically feasible geometry [7, 8]. Moreover, in Ref. [5] authors have shown that nanorod chain structures are adequately robust also under experimental conditions. We think that our results open an interesting route for the design of highly integrated efficient nonlinear devices.