1. Introduction

Nonlinear optics has a wide range of applications in many areas mainly in communications and optical computing. For example, optical switches and modulators using nonlinear properties have been extensively used in modern telecommunication industry [1]. As such, miniaturization of optical components remains one of the biggest challenges in this domain. Metallic Photonic Crystal (PhC) structures are the principal candidates [2, 3] due to their Extraordinary Optical Transmission (EOT). This EOT is usually attributed to surface plasmon resonances [4] or both to the contribution of surface plasmon polaritons components and an evanescent wave component [5]. Other studies show that transmission up to 95% can be reached using specific geometries like Annular Apertures Arrays (AAA) [6, 7]. These authors demonstrated that the EOT is due to the excitation of an identified guided mode which induces a high confinement of the electric field inside the cavities between the metallic parts of the waveguide. This TE₁₁-like mode [8] is characterized by a reduced group velocity and by its large cutoff wavelength. Early studies focused on this enhanced transmission successfully demonstrated an increase of the Second Harmonic (SH) strength from hole [9], bowtie [10] or ”G” shaped [11] aperture arrays in addition to many other structures [12–14]. However, the conversion efficiency is still small compared to nano-structured nonlinear materials like (GaAs, LiNbO₃, KTP ...). In this paper, we suggest combining this light confinement and high nonlinear coefficient χ² to significantly enhance the nonlinear conversion from metallo-dielectric nano-structures. Due to slow light mode, we considerably improve the optical nonlinear efficiency of these nano-patterned structures. The dielectric material we are interested in is Lithium Niobate (LN) because it is one of the best known crystals that are used for various linear and non-linear optical applications [15]. Its excellent optical transparency in the visible and in the infrared ranges, its high electro-optic coefficient in addition to its nonlinear optical susceptibility make it an ideal material for optical waveguides, modulators, surface acoustic wave devices, OPO etc ...

Second Harmonic Generation (SHG) was the first nonlinear optical effect to be experimentally observed (crystalline quartz) [16]. It results from a light propagating at frequency ω through a crystal that exhibits a second order nonlinearity. This will induce a conversion of light into a double frequency 2ω or half the wavelength. In this paper, we suggest a novel device that enhances this nonlinear conversion by combining the EOT to a large nonlinear dielectric. Our concept consists on filling the cavities of a silver AAA by LN by lithium niobate. The suggested configuration is schematically illustrated in Fig. 1. The choice of the geometrical parameters and the metal in use is described in our previous work [17]. The simulated linear transmission spectrum exhibits double resonances due to the excitation of the TE₁₁-like mode at, first, its cutoff wavelength and, second, at the first FP harmonic. Briefly, their position varied as a function of the geometrical parameters. The radii have been chosen to locate the first peak at the fundamental wavelength. The silver thickness h is optimized for enhancing the SH response by locating the Fabry-Perot-like resonance at the second harmonic wavelength.

 

Fig. 1 Schematic showing embedded coaxial apertures into silver and filled by lithium niobate. (a) Geometry of one aperture of inner radius R_i = 65 nm and outer radius R_o = 135 nm. (b) Grating of squared array of apertures with periodicity p = 300 nm and thickness h = 120 nm = deposited on the same material.

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In order to test the feasibility of our new device configuration before fabrication, we developed a numerical algorithm. In our previous work [17], a full control on the transmission process which is mostly dictated by the geometrical parameters is explained. The nonlinear enhancement factor was estimated by dividing the strong electric field confined in sub-wavelength regions to the electric field generated in the same volume of an un-patterned LN at the pump wavelength. We considered that by improving this factor, we boost the nonlinear response of the structure. In this paper we show a realistic and detailed analysis taking into account the simulation of the the nonlinear optical response of the SHG in complex structures. The theory behind the nonlinear Finite Difference Time Domain (FDTD) [18, 19] algorithm used to simulate our device, as well as a description of its implementation is described. The purpose of developing this code is to obtain an understanding of the underlying physical processes of the metallo-dielectric nano-structures. Since the SHG is strongly dependent on the structural symmetry of the samples and thus on the polarization, nonlinear simulations are therefore crucial in order to study the different dependencies on the nonlinear responses. This code only takes into account the volume SHG so that only the LN is considered to generate SH signal. The surface SHG of silver is then neglected since the χ² of the metal is smaller than that of LN. While the largest second order nonlinear coefficient is d₃₃ = 33 × 10⁻¹²m/V (χ⁽²⁾ = 2d) for LN [20], the surface susceptibility is estimated to χ⁽²⁾ = 1.13 × 10⁻²⁰m/V for silver [21].

2. NL-FDTD algorithm

For materials that show both linear and nonlinear polarization properties and specifically the second order susceptibility function χ²(t), the relationship between the electric displacement field D⃗, the vacuum permittivity ε₀ and the electric field E⃗ is given by the electromagnetic constitutive relation for a dispersive medium:

Equation (1) includes the nonlinear polarization term for the material through:

Where P⃗ is the induced polarization, χ⁽ⁿ⁾ is the nth order nonlinear susceptibility and E⃗ is the electric field vector of the incident light. χ⁽¹⁾E⃗ describes the linear response of the material; χ⁽²⁾E⃗E⃗ defines the SHG, sum and the difference frequency generation; χ⁽³⁾E⃗E⃗E⃗ is both four-wave mixing and the third harmonic generation. Our work aims enhancing the second harmonic generation thus we limit the study to the second order nonlinear effect. The nonlinear FDTD method discussed here is based on the extension of the original Yee’s algorithm [22] that numerically solves Maxwell’s equations:

This is achieved by first calculating the linear electric and magnetic fields using the discretized equations. For example, the updated equation of the x-component of the electric field is expressed by:

Where n, Δt, Δy and Δz are the time index, time step, grid spacing in y and in z directions, respectively. Based on the x equations, we can calculate the two other components of E and H (E_y, E_z, H_y and H_z). The nonlinear polarization is calculated prior to the nonlinear electric field as follows:

The frequency domain product of the electric field and the second order frequency domain susceptibility function χ⁽²⁾ can be expressed in time domain by the linear convolution:

In this case, χ⁽²⁾ is the material time domain susceptibility. To get around the convolution product difficulty, we use a continuous source at a fixed wavelength.

In this case, Eq. (2) leads to a direct product in the time domain. Thus, the nonlinear polarization approximates to P⃗_NL(t) = 2ε₀d̃E⃗(t)E⃗(t) where d̃(χ⁽²⁾ = 2d̃) is the simplified nonlinear tensor of the material in use. Finally, the nonlinear electric field components are obtained through this equation:

In this paper, all the presented spectra are calculated wavelength by wavelength i.e. in a monochromatic regime through the 3D-NL-FDTD homemade code. To reduce the calculation time, we use a parallelized (OpenMp library) algorithm that runs on multiple cores simultaneously [23].

3. Numerical simulations and results

The parameters used in the FDTD calculations are δ_z = δ_y = 2.5 nm, a non-uniform meshing is applied in x direction to faithfully describe the structure. This latter varies continuously between 2.5 nm and 15 nm over 160 nm (20 spatial grids). The temporal step is set to δt = δ_z/2c where c is the speed of light in vacuum. We use the Perfectly Matched Layer (PML) as an artificial absorbing boundary for both linear and nonlinear signals. A Drude model is used to simulate the dispersion of the dielectric permittivity of silver. As shown on Fig. 1, the period is fixed to 300 nm and the dielectric constant of the LN is set to ε = n² = 2.138² at λ = 1550 nm and ε = n² = 2.179² at λ = 775 nm [24]. The thickness of the metallic film and the aperture radii are kept the same as in [17] (h = 120 nm R_i = 65 nm and R_o = 135 nm). Since the thickness of our structure is smaller than the coherence length of lithium niobate, no phase matching is needed to generate an optimum depletion of the nonlinear signal.

In addition, we assume that this signal is only generated by the embedded LN neglecting the contribution of the LN substrate. In other words, we consider that the thickness d of the substrate is an even multiple of the coherence length L_c (d = 2m × L_c, m ∈ IN). This length represents the distance over which the nonlinear polarization and the generated SH signal interfere constructively. By choosing d = 2m × L_c, the SHG signal generated from the substrate is avoided and only the SHG generated by the LN located inside the 120 nm-thick cavities contributes in the enhancement factor.

Our algorithm permits the simultaneous calculation of both linear and non linear responses of the structure; we will first explore the linear response at frequency ω. In Fig. 2(a), the zero-order transmission spectrum of the embedded structure shows two resonances. The main peak located at the fundamental wavelength (1550nm) corresponds to the excitation and the propagation of the TE₁₁-like mode inside the apertures at its cutoff wavelength. This induces a light confinement inside and nearby the cavities. Consequently, by filling out the cavity with a nonlinear material i.e. lithium niobate, we improve the nonlinear conversion, in the present case the SHG. The second peak, which corresponds to the first harmonic Fabry-Perot resonance of the same guided mode, is placed at the SH wavelength (775nm). This is obtained by adjusting the metal thickness [17] so that the generated SH signal can be propagated inside the cavities and transmitted in the optical far field.

 

Fig. 2 (a) Zero order transmission spectrum of the Metallo-Dielectric Photonic Crystal. (b) Nonlinear enhancement factor (blue solid curve) versus the harmonic wavelength compared to the linear transmission (green solid line) of the embedded structure. For both spectra, the geometrical parameters are Ri = 65 nm, Ro = 135 nm, h = 120 nm and p = 300 nm.

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To simulate the nonlinear response of the structure, we use the LN d̃ tensor that links the P⃗_NL to the electric field at the pump wavelength. This relationship is given by [20]:

Where d₂₂ = 3 pm/V, d₃₁ = 5 pm/V and d₃₃ = 33 pm/V. Note that in Eq. (9) the crystal axis of the LN is along the z direction, consequently, a basis change is necessary to take into account the wafer cut.

By looking at the second order nonlinear tensor of lithium niobate, we can observe that the highest nonlinear coefficient is d₃₃ which means that the z-component of the pump electric field plays the main role in the observed SH signal. As the field inside the cavities is dominated by the component parallel to the incident polarization, we chose to direct this electric field along the crystal axis. Therfore, we have chosen an X-cut LN wafer in our simulations.

In Fig. 2(b), we show the nonlinear enhancement factor as a function of the fundamental wavelength for an x-polarized incident plane wave illuminating the structure at normal incidence. The transmission enhancement factor is defined by the ratio of the SH signal generated from the embedded structure to the nonlinear signal generated by a LN layer of the same thickness. For a fundamental wavelength near the transmission peak (TE₁₁ wavelength), the patterned LN sample generates an enhanced SH signal, that is 27 times higher than that generated from the unpatterned lithium niobate layer. This value is higher than 17, the one predicted in ref [17] assuming an equitable contribution of the three components of the fundamental electric field.

In addition, we notice that the maximum of the nonlinear transmission is blue-shifted by 25 nm (λ_SH = 750 nm) from its expected position (775 nm) corresponding to the half of the main linear peak transmission wavelength (λ_fund = 1550 nm). Unlike previously published results by Fan et al. [25] who attributed this shift to the point dipole theory (emission of the nonlinear signal from the sub-wavelength apertures), the coupling between the ”horizontal” surface plasmon resonance and the ”vertical” TE₁₁ guided mode is the main reason of this observed blue-shift. That was numerically demonstrated by increasing the structure’s period allowing a redshift of the plasmon resonance far from the transmission peak of the guided mode. Thereby, the nonlinear signal transmission reaches its maximum at exactly half the fundamental wavelength.

In Fig. 3(a), the transmission and reflection spectra of the nonlinear SH signal are shown. The SH reflection coefficient defined by $I_{SH}^R\left(2\omega \right)/I_{\mathit{fund}}{\left(\omega \right)}^2$ shows a major peak at 810 nm which is due to the perturbation of the guided mode by the surface plasmon resonance excited at the metal-LN interface. This was verified by monitoring the surface plasmon resonance position as a function of the period. For large period values, the Surface Plasmon Resonance (SPR) occurs far from the guided mode position and no interferences are observed. Thus, the SPR follows a quite linear behavior given by its conventional dispersion relation. Contrarily, by decreasing the period, the SPR and the guided mode remain closer and a degeneracy breaking occurs inducing a redshift of the SPR position. That is why the SPR is located at λ_SPR ∼ 800 nm instead of 710 nm found from the dispersion relationship.

 

Fig. 3 (a): Normalized transmission and reflection spectra of the generated nonlinear signals inside the annular apertures. (b) and (c): Spatial distributions of the nonlinear electric field nearby one cavity at ${\lambda }_{T_{NL}}^{\mathit{max}}=750\hspace{0.17em}nm$ and at λ_SPRNL = 810 nm respectively.

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The normalized transmitted signal, presented on the same Fig. 3(a), shows a main peak at ${\lambda }_T^{\mathit{max}}=750\hspace{0.17em}nm$ which is attributed to the excitation of TE₁₁ guided mode inside the apertures. Let us emphasize that the SPR generated at the metal-LN interface also disturbs the transmitted signal from the metal-vacuum side. Consequently, a dip appears in the transmission spectrum around the SPR wavelength. Figures 3(b) and 3(c) show the nonlinear electric intensity distributions inside the cavity at these two wavelengths ( ${\lambda }_T^{\mathit{max}}$ and λ_SPR). For ${\lambda }_{\mathit{fund}}={\lambda }_T^{\mathit{max}}=1500\hspace{0.17em}nm$ the electric intensity is essentially confined at air interface whereas the light at λ_fund = λ_SPR = 1620 nm is confined at the lithium niobate side. In this latter case, the reflected optical signal remains smaller than the transmitted one as it is confirmed in Fig. 3(a).

Figure 4 shows the pump and the SH electric field intensities in two perpendicular planes. For a unit cell of an embedded aperture, the intensity is displayed in the zOy plane, 5 nm far from the surface, with the z-polarized incident wave at λ = 1500 nm. In Fig. 4(a), a localized electric field is observed in the cavity corresponding to the excitation of the TE₁₁ guided mode. The SH field in Fig. 4(b) is also confined nearby the cavities and follows the fundamental distribution of the electric field shown in Fig. 4(a). Figure 4(c) shows the intensity enhancement in the structure (zOx plane) at the pump resonance frequency of ${\lambda }_T^{\mathit{max}}=1500\hspace{0.17em}nm$. As expected, the light is uniformly distributed inside the cavity toward the propagation direction [8]. Note that, at the first harmonic Fabry-Perot resonance (λ_fund = 775 nm), the distribution of the linear electric intensity shows a node inside the apertures (Fig. 4(d)) corresponding to interferences between forward and backward propagating waves between the two metallic interfaces. Meanwhile the distribution of the SH signal presented in Fig. 4(e) is attributed to the excitation of the fundamental TE₁₁ mode inside the cavity and is consistent with the uniform distribution of the fundamental field of Fig. 4(c).

 

Fig. 4 The fifth root of the electric intensity distribution (|E|^0.4). (a),(c) are the linear distributions at ω, (d) is the linear distribution at 2ω. (b),(e) show the nonlinear distribution at 2ω.

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Figure 5 illustrates the time average Poynting vector at the pump and harmonic wavelengths respectively for the embedded structure. The incident light is z-polarized towards the lithium niobate substrate. In Figs. 5(a) and 5(b), the Poynting vector distribution is consistent to that of the electric intensity of Fig. 4(a): its maximum is located in the vicinity of the inner part of the aperture. Let us underline that the spatial average over the whole infinite structure of the temporal Poynting vector falls to zero in the zOy plane. This means that there is no energy flow in this transversal plane. However, in the zOx plane (the polarization plane), most energy flow is in the apertures and appears as a uniform vertical current (Fig. 5(c)). The Poynting vector of the nonlinear signal at λ = 750 nm is depicted in Fig. 5(d). Indeed, a nonlinear source is created inside the cavity and propagates in both directions of the Ox-axis. However it is clear that it is more transmitted on the air side than reflected in the substrate side. Note that the Poynting vectors in the vertical direction are not evenly spaced, this is due to the nonuniform meshing used in our calculations. In our older paper [17], we neglected the nonlinear part of the polarization in our simulations and we estimated that the phase distribution of the nonlinear electric field inside the cavity is the same as the fundamental one; having a uniform distribution along the cavity. Here we took into account the media asymmetry and thus the phase difference between the interfaces; this effect drives the nonlinear source to be positioned near the vacuum interface instead of the middle of the cavity. This phenomenon leads to have a transmitted signal greater to that reflected which can explain the difference in the obtained enhancement factor.

 

Fig. 5 Linear (a,c) and nonlinear (b,d) distributions of the Poynting vector in the (zy) and (xz) planes respectively. The incident field is Z–polarized.

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Due to the anisotropic character of the susceptibility tensor of LN, the polarization of the incident beam plays a key role in the SHG. Consequently, we have also performed a polarization analysis.

4. Polarization

In this section, we vary the angle θ between the incident electric field and the Oz–direction in order to vary the polarization of the impinging light (see Fig. 6(a)). This polarization angle is then changed from 0 to 360° by a step of 15 degrees. For every polarization angle, the SHG signal defined by |E(2ω)|/|E(ω)|² is calculated using the 3D-NL-FDTD for two different wafer cuts (Figs. 6(b) and 6(c)). The strong dependence of the incident polarization on the SH is only related to the asymmetrical character of the nonlinear material second order susceptibility tensor and to the crystal axis orientation. Thus every component of the electric field and the nonlinear polarization contribute to the total SH signal, its impact depends on the wafer cut chosen of the material in use. It is important to notice that we considered that the (xOy) plane of the crystal is the same as the FDTD algorithm. On the other hand, we assumed that the SHG is generated inside the cavity independently of each other so that the generated SHG are not affected or depolarized at the output of the cavity.

 

Fig. 6 (a) Schematic of the geometry employed for polarization. In (b) and (c) we plot the normalized SHG signal generated from an x-cut and z-cut nano-patterned PhC respectively versus the polarization angle.

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For an x-cut structure, we observe that the maximum of SH signal is obtained for θ = 0°. In fact the dominant polarization component (parallel to the crystal axis) follows a cosine square relationship that is in accordance with the polarization dependence of the SHG. The fundamental electric field is multiplied by the d₃₃ element of the simplified nonlinear tensor d̃, consequently the resulting nonlinear polarization component is predominant with regard to the two other components. Furthermore, for a z–cut nano-patterned LN wafer, we do not observe a similar polarization dependence. In fact, there is an additional effect of anisotropy and the SH strength is an order of magnitude lower than the SH generated from the nano-patterned x–cut LN. The amplitudes of the three P_NL components are quite similar and the resulting behavior of the nonlinear polarization can be mathematically described by a sum of cosine square and sine×cosine terms. Indeed, this can be attributed to the fact that the d₃₃ component does not intervene significantly in the case of z–cut LN.

5. Conclusion

In this paper, we investigate a new PhC device configuration that combines the EOT with a nonlinear material (lithium niobate) in the aim of enhancing the SHG signal. We develop a nonlinear FDTD code that takes into consideration the nonlinear polarization in Maxwell’s equations. On the other hand, we discuss the nonlinear behavior of such embedded structures in transmission as well as in reflection. We study the effects on the nonlinear response of the spatial distribution, the polarization effect and the nonlinear source position. It is crucial to take into account the excited guided mode at the fundamental wavelength as well as the created nonlinear polarization in order to obtain an efficient nonlinear device. Results show that the strength of the signal generated from the embedded structure is enhanced by a factor of 27 as compared to that generated from an unstructured x-cut lithium niobate. While the TE₁₁ guided mode improves the nonlinear conversion, the first Fabry-Perot harmonic helps to transmit the SH signal in the far field. Moreover, we show that the SH signal strength reaches its maximum value for an z-polarized incident light. We would like to also point out that in our structure, no phase matching is needed to generate the second harmonic signal. A forthcoming paper will detail the fabrication and the experimental results of this structure.