Efficient second harmonic generation using nonlinear substrates patterned by nano-antenna arrays

Doron Bar-Lev; Jacob Scheuer

doi:10.1364/OE.21.029165

1. Introduction

Recent advances in nanoscience have emerged the need for manipulating and controlling optical radiation at sub-wavelength scales [1]. Nano-antennas are one of the leading tools for fulfilling this need as they efficiently convert free-space propagating optical radiation to localized modes and vice versa [2,3]. This energy localization often results in large field enhancement mainly at the proximity of sharp metallic edges and structural gaps [4-7]. The enhancement, which is attributed to the electrostatic lightning-rod effect and to localized surface plasmon resonances [8,9], has been demonstrated in several types of gap nano-antennas, particularly Bowtie nano-antennas [4,5,10,11], exhibiting power enhancement factors exceeding 40dB [10]. In addition, it was shown that the enhancement is affected by the coupling between neighboring elements in nano-antenna arrays [11]. The energy localization and field enhancement can be utilized for enhancing the efficiency of various optical phenomena. In particular, they can enhance nonlinear nanometer scale optical phenomena which are important for diverse disciplines ranging from biology and physics to information science [12]. More specifically, nano-antennas were shown to enhance the efficiency of various nonlinear optical processes such as four wave mixing [13,14], two photon excited luminescence [4,15,16] and white light supercontinuum generation [17]. A special interest is of high harmonic generation [8,10-12,18-25], particularly second harmonic generation (SHG), as these processes may facilitate the generation of efficient light sources at the nano-scale. SHG in nano-antenna based structures stems from the symmetry braking at the surfaces and edges of the structure and from a nonlinear behavior of a surrounding host medium [9-12]. Thus far, the attempts to enhance the efficiency of SHG were focused mainly on increasing the field enhancement and on optimizing the resonant nature of the nano-antennas for the fundamental harmonic (FH) and the second harmonic (SH) [12,24]. In addition, several groups introduced a nonlinear media at the gap area [19-23], which as suggested in the seminal work of Pendry et al. [25] may enhance the effect compared to nonlinear response from a bulk nonlinear media with comparable dimensions. Nevertheless, it was recently shown that high field enhancement does not necessarily lead to an optimized nonlinear activity [26]. Moreover, in the presence of a nonlinear medium, the coupling between its nonlinear polarization, the nano-antenna structure, the incident FH and the generated fields may greatly affect the efficiency of the nonlinear processes. In addition, the buildup of the SH signal from nano-antenna arrays is different than that of bulk SHG and thus different optimal pump characteristics are required.

In this paper we explore several mechanisms that can potentially enhance the efficiency of SHG from nano-antenna arrays embedded in a nonlinear medium by several orders of magnitude. Using an FDTD based model combined with analytical and numerical calculations we study the impact of these mechanisms in an array of Au gap Bowtie nano-antennas embedded in congruent Lithium Niobate (LiNbO₃). These mechanisms are then generalized and lead to a set of general design rules for efficient second and high harmonic generation from nano-antenna arrays. First, we propose a recessed nano-antenna structure and show that is more suited for SHG from field enhancement, field profile and linear and nonlinear polarization considerations. Next we investigate the impact of the parameters of a FH Gaussian beam on the efficiency of the generated SH. This analysis leads to a derivation of a nano-antenna analog to the Boyd- Kleinman criterion [27] for efficient SHG. Finally we show that the dielectric properties of the nonlinear substrate lead to a preferable FH propagation and SH detection directions. Consequently an optimal method for introducing the FH and detecting the SH is suggested.

The reminder of the paper is organized as followed: In section 2 we describe the recessed nano-antenna structure and analyze its advantages. In section 3 we derive the attained SH due to a FH Gaussian beam incident on a nano-antenna array. In Section 4 we study the effects of the dielectric properties of the nonlinear substrate and in section 5 we discuss the results and conclude.

2. The effects of recessing the nano-antennas in a LiNbO₃ substrate

The SH field scales as the square of the FH field. Therefore it is desirable to utilize nano-antennas with strong field enhancement and to introduce the nonlinear material in the areas of maximal field enhancement. We chose to focus on Gap Bowtie nano-antennas as they provide strong field enhancement which is highly localized in the gap. Introducing a nonlinear material only in the gap of the nano-antenna constitutes a complex fabrication problem. Consequently we suggest a different approach in which the nano-antenna is embedded in the nonlinear substrate. This structure can be realized using conventional fabrication techniques such as focused ion beam milling or reactive ion etching. Note that to some extent this structure is complementary to an array of holes which is embedded in a nonlinear substrate as implemented in GaAs in ref [19,20], however we decided to apply a LiNbO₃ substrate due to polarization considerations as discussed in the following paragraph. To analyze the effect of embedding the nano-antennas in the substrate we first compare between the power enhancement factors of two Au gap Bowtie nano-antenna arrays. The first is placed on top of a Y-cut congruent LiNbO₃ (CLN), whereas the second is recessed in it. The analysis is performed using an FDTD model [28] which utilizes the measured dispersion of the employed metals [29] and CLN [30]. The properties of the nano-antenna arrays depend on the spectral characteristics of the combination of the nano-antenna elements and the corresponding loads [31]. In other words, the dielectric surrounding modifies the spectral response of the nano-antenna array. To set a common ground for comparison, the dimensions of the nano-antennas in both arrays are designed to correspond to a maximal field enhancement in the optical C-band (~1560nm), as shown in Fig. 1(a). This spectral region constitutes the FH region which consequently sets the SH at the vicinity of 780nm. The periodicity of the arrays is set to 650nm in both transverse dimensions in order to eliminate high order diffraction lobes at the FH while maintaining relatively wide array spacing and avoiding coupling looses. In addition, the Bowtie dimensions are set to suit realistic fabrication limitations [32]. Particularly, the gap width and the minimal radius of curvature are set to be 25nm to correspond to typical E-beam lithography limitations. The applied nano-antenna dimensions, arrays spacing and the x (ordinary) and z (extraordinary) axes of the crystal are described in Figs. 1(b) and 1(c). As expected, the recessed nano-antennas have smaller dimensions due to the red shift in the nano-antenna resonance originating from the presence of the dielectric in the gap [33].It is clearly seen that the recessed structure exhibits a larger power enhancement factor at the center of the gap (~550) compared to the on surface structure (~350). However, the SH power in the recessed structure is expected to be substantially larger than the square of the power enhancement ratio. This is due to the different profile and polarization of the localized fundamental field and generated second order polarization.

Fig. 1 (a) The wavelength dependence of the power enhancement factors of nano-antenna arrays on top and recessed in a y-cut congruent LiNbO₃. The enhancement factor is calculated at the middle of the gap (0, ± 20nm,0) for an S polarized (E-plane) normal incident plane wave. (b) and (c) describes the dimensions of the array and the orientation of the ordinary (x) and extraordinary (z) axes of the crystal relative to the nano-antenns, for the on surface and recessed arrays respectively.

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Generally speaking, the nonlinear medium contribution to the SH stems from the second order nonlinear polarization in the medium ${\underline{\tilde{P}}}^{(2)}$ which can be described as [34]:

{\underline{\tilde{P}}}^{(2)} = ε_{0} χ^{(2)} \underline{\tilde{E}} \cdot \underline{\tilde{E}}

where ε₀ is the vacuum permittivity, χ⁽²⁾ is the second order nonlinear susceptibility tensor and

\underline{\tilde{E}}

is the incident field. This nonlinear polarization acts as a source in the wave equation:

\nabla \times \nabla \times \underline{\tilde{E}} + \frac{1}{ε_{0} c^{2}} \frac{d^{2} {\underline{\tilde{D}}}^{(1)}}{d t^{2}} = - \frac{1}{ε_{0} c^{2}} \frac{d^{2} {\underline{\tilde{P}}}^{(2)}}{d t^{2}}

where

{\underline{\tilde{D}}}^{(1)}

is the linear electric displacement and c is the vacuum speed of light. Due to the definition of the polarization vector as a dipole moment density [34] and the localization of the FH field to areas much smaller than the first and second harmonic wavelengths, the second order polarization due to nano-antennas can be approximated as a point dipole source oscillating at the second harmonic frequency [19,20]. Note, that as our SH analyses is conducted at a single wavelength (780nm), the effect of the second order time derivative corresponds to a change in the dipole amplitude according to

ω_{S H}^{2}

. The SH dipole in our case is generated in the LiNbO₃. As shown in the YZ cross sections of the power enhancement factor depicted in Fig. 2, the localized field inside the LiNbO₃ is more intense for the recessed structure (Fig. 2(b)) and includes the peak of the enhanced FH. In contrast, for the on surface structure (Fig. 2(a)) this peak is obtained in the air gap and therefore it does not contribute to the SHG. Consequently the equivalent SH point dipole, which is an integration of the second order polarization over the LiNbO₃ volume where the field is localized, is further enhanced in the recessed structure, and is approximately 5.5 times larger than that of the on surface structure. This ratio corresponds to an expected SHG power enhancement of at least 14dB.

Fig. 2 Power enhancement factor (logarithmic scale) of the FH field in the LiNbO₃ substrate for Au gap Bowtie nano-antennas. (a) on top of the substrate; (b) recessed in the substrate.

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Another important aspect to be considered is the polarization of the localized FH field and the generated second order polarization in respect to the nano-antenna and the substrate nonlinear tensor. Nano-antennas are typically planar devices which most dominant localized field component is polarized in the antennas plane. In addition, polarization sensitive nano-antennas are expected to create a polarized localized field which corresponds to their preferred polarization. The tensorial form of the second order polarization can be described according to the following relation [34]:

(\begin{matrix} P_{x_S H} \\ P_{y_S H} \\ P_{z_S H} \end{matrix}) \equiv 2 ε_{0} (\begin{matrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{matrix}) (\begin{matrix} E_{x_F H}^{2} \\ E_{y_F H}^{2} \\ E_{z_F H}^{2} \\ 2 E_{y} E_{z} \\ 2 E_{x} E_{z} \\ 2 E_{x} E_{y} \end{matrix})

where d_ij (i = 1..3,j = 1..6) are the second order nonlinear coefficients and E_i (i = x,y,z) are the fundamental field components. To attain maximal second order polarization, the polarization of the FH localized field should correspond to the orientation of the maximal coefficient in the nonlinear susceptibility tensor and to match the polarization of the nano-antenna mode at the fundamental frequency. Nevertheless, in order to attain efficient re-emission of the SH, the generated second order polarization should also correspond to the polarization of the nano-antenna mode at the SH frequency. To verify this effect, we applied a point dipole source (oscillating at the SH frequency) at the center of the gap of each nano-antenna in the recessed array and calculated the power transmitted from the nano-antenna array to the air for different polarizations of the dipole source. As expected a polarization loss factor proportional to cos²(ψ) was obtained, where ψ is the angle between the SH dipole polarization and the nano-antenna localized field polarization axis at the SH frequency. Therefore, optimal nonlinear materials for efficient SHG using planar nano-antennas are those which generate an interaction between FH fields and SH polarization on the same plane (all components except d₁₄,d₂₅,d₃₆, see Eq. (3)). Moreover, for linearly polarized planar nano-antennas where both the fundamental and second harmonic localized fields are polarized in the same direction (as in the gap Bowties case), the optimal nonlinear materials for SHG are those which maximal nonlinear coefficient is one of the

d_{i i}

(i = 1..3) coefficients. In this case the material should be oriented such that its strongest nonlinear axis is parallel to the linear polarization axis of the nano-antenna. From these considerations it follows that 7 of the 21 crystal classes that lack inversion symmetry, classes

\begin{matrix} 222, & 622, & 422, & \bar{4} 2 m, & \bar{4} 3 m, & 23, & 432 \end{matrix}

, are less suitable for SHG from planar nano-antennas, while an additional class,

\bar{4}

, is less suitable for planar nano-antennas having linearly polarized response. Particularly, GaAs, is not optimal as its only non vanishing second order elements are d₁₄,d₂₅ and d₃₆. In contrast, for LiNbO₃ the strongest second order nonlinear coefficient is d₃₃ [35], which satisfies all of the criteria above. Nevertheless, as aforementioned in order to obtain efficient SH the crystal lattice should be oriented as described in Figs. 1(b) and 1(c) (note that an x-cut LiNbO₃ is suitable as well). These polarization considerations further favor the recessed structure as well. The polarization state of the gap Bowtie nano-antennas, stems from the boundary condition at the metal-LiNbO₃-Air interfaces. Hence the localized field in the gap is almost completely z-polarized, whereas beneath the nano-antenna the FH is polarized both in the z and y directions. As a result, only for the recessed case the nano-antenna polarization, the FH localized field polarization and the generated second order polarization can be matched.

To complete this section, we compared the SHG in the recessed nano-antenna structure to that of an on-surface bowtie-antenna where a nonlinear optical material (LiNbO₃ in this case) fills the gap area. Although such a structure is difficult to realize in practice, it constitutes a natural gauge for the efficiency of the SHG from the recessed structure. We found that SHG from such structure is lower by approximately 30% than that of the recessed nano-antenna structure, thus reinforcing our conclusion to focus on the later structure. Finally, it should be noted that we neglect the effect of the surface nonlinearity as it was shown experimentally that typically for metallic nano-structures on top of a nonlinear media, the bulk nonlinearity tends to be a more dominant contributor [18,19]. Nevertheless, most of the following discussion and conclusions can be applied for SHG due surface nonlinearity by applying the appropriate susceptibility tensor.

3. The impact of the beam parameters of a FH Gaussian beam

The buildup of the SH field in bulk propagation depends on the interference between the radiation emitted from atomic dipole sources that constitute the nonlinear medium. As such, obtaining maximal SHG requires the optimization of the bulk dimensions and phase matching conditions to the FH beam width and divergence as shown in the work of Boyd and Kleinman [27]. As mentioned before, the SH from a single nano-antenna embedded in a nonlinear substrate is a result of the localized second order nonlinear polarization which re-excites the nano-antenna. Hence, when considering SHG in nano-antenna arrays, the re-emitted SH field depends on the interference between the array elements which are excited by the localized SH dipole sources. Therefore, the FH beam characteristics may affect the generated SH in this case as well. To analyze the dependence of the SHG on the FH beam parameters, we consider an array of identical $M \times N$ nano-antenna elements on the XZ plane with uniform spacing in the x and z directions, d_x and d_z respectively (see Fig. 3).

Fig. 3 A FH beam incident on a nano-antenna array comprising uniformly spaced and identical elements. The axes are aligned according to the LiNbO₃ crystal system defined above.

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We assume that the array spacing is designed to minimize the coupling losses between the nano-antennas. The incident FH beam is assumed to be Gaussian and to carry CW power P_FH. Assuming that the beam propagates in air along the y axis, it can be described by the complex beam parameter: $q (y) = y + i π W_{0}^{2} / λ$ , where W₀ is the beam waist and y is the distance of the array from the beam waist. Hence the FH far-field can be described by [36]:

E (r, y) = E_{0} \frac{i y_{r}}{q (y)} e^{- i k \frac{r^{2}}{2 q (y)}}

Where y_r = πW₀²/λ is the Rayleigh length of the beam, r is the two dimensional radial coordinate and E₀ is the peak value of the FH field which is related to the beam power according to:

E_{0}^{2} = \frac{4 P_{F H}}{c ε_{0} π W_{0}^{2}}

The FH field is enhanced at each nano-antenna gap according to the field enhancement factor G resulting in the generation of second order polarization. According to the point dipole model (and assuming a CW FH) this second order polarization corresponds to a dipole source which amplitude is described by:

{\tilde{S} (z) |}_{G a p} = - \frac{1}{ε_{0} c^{2}} {\frac{d^{2} {\underline{\tilde{P}}}^{(2)}}{d t^{2}} |}_{G a p} \to {{(\frac{ω_{S H}}{c})}^{2} χ^{(2)} \underline{\tilde{E}} \cdot \underline{\tilde{E}} |}_{G a p} = {{(\frac{ω_{S H}}{c})}^{2} χ^{(2)} G^{2} E_{0}^{2} {(\frac{i y_{r}}{q (y)})}^{2} e^{- i k \frac{r^{2}}{q (y)}} |}_{G a p}

In deriving (6) we assume that the Gaussian profile is maintained even in the presence of the nano-antenna array. Consequently, assuming (without loss of generality) that M and N are odd numbers and that the peak of the Gaussian beam is aligned to the central nano-antenna, the far-field SH field contribution of each element is given by:

\begin{matrix} E_{S H} [m, n] = η (θ, ϕ) d_{x} d_{z} {(\frac{ω_{S H}}{c})}^{2} χ^{(2)} G^{2} E_{0}^{2} {(\frac{i y_{r}}{q (y)})}^{2} e^{- \frac{i k {(m d_{x} - \frac{M + 1}{2} d_{x})}^{2}}{q (y)}} e^{- \frac{i k {(n d_{z} - \frac{N + 1}{2} d_{z})}^{2}}{q (y)}} & m, n = 1.. M, N \end{matrix}

Note that we introduced the factor η(θ, ϕ), where the angles θ and ϕ are indicated in Fig. 3. This factor corresponds to the far-field SH radiation pattern of a single element and it consists of both the coupling between the generated localized field in the SH and the nano-antenna, and the radiation pattern of the specific nano-antenna element. Thus it is expected to be affected by linear and non-linear polarization considerations (as described in section 2) and by the radiative properties of the nano-antenna at the SH wavelength. In addition, we introduced the array unit cell area d_xd_z. This factor corresponds to the effective FH absorption cross section of a single element (under the assumption of a negligible coupling between the elements) and it relates to the portion of the FH power which is incident on a single unit-cell. In a more general case, this coefficient corresponds to the effective cross-section of the individual nano-antenna. Note that both η and d_xd_z are identical for all the elements in the array. It is known from antenna theory that the far-field radiation pattern of an antenna array can be described by the pattern multiplication [31], thus the far-field SH from the entire nano-antenna array can be described as:

E_{S H_f a r - f i e l d} = E_{S H_S i n g l e_E l e m e n t} \cdot A F

where E_{SH_Single_Element} is the element factor which consist of the radiation pattern of a single antenna at the reference point (in our case a single SH dipole fed nano-antenna) and AF is the array factor that describes the impact of the geometrical arrangement, the amplitudes and the phases of the elements in the array. Note that Eq. (8) is valid for any array consisting of identical elements, even if the magnitudes and the phases of the field of each element or even the spacing between them are not identical. Typically the spatial response of an antenna array arises from the AF. E_{SH_Single_Element} usually exhibits a relatively broad spatial response which is independent of the array size (assuming small coupling losses), whereas the AF generally becomes more spatially selective for large arrays. Therefore, in the reminder of this section we focus on the impact of the FH beam parameters on the AF. To derive the AF, we first note that E_{SH_far-field} stems from the SH far-field contributions of all the elements, hence for an M×N array it can be written as [31]:

\begin{array}{l} E_{S H_f a r - f i e l d} (ψ_{x}, ψ_{z}) = \sum_{m = 1}^{M} \sum_{n = 1}^{N} E_{S H} [m, n] e^{i (m - 1) ψ_{x}} e^{i (n - 1) ψ_{z}} \\ \begin{matrix} ψ_{x} = k d_{x} \sin θ \cos ϕ & , & ψ_{z} = k d_{z} \sin θ \sin ϕ \end{matrix} \end{array}

Note that we are interested primarily in the part of the SH which is radiating back to the air. Again, we assume that the elements comprising the array are identical and that the difference in their amplitudes and phases is caused by the FH Gaussian beam profile. Thus, by introducing Eq. (7) and Eq. (5) into Eq. (9), applying separation of variables and changing the origin to the center of the array, E_{SH_far-field} can be written as:

\begin{array}{l} E_{S H_f a r - f i e l d} (ψ_{x}, ψ_{z}) = C \cdot P_{F H} \cdot \frac{2 d_{x} d_{z}}{π W_{0}^{2}} \cdot {(\frac{i y_{r}}{q (y)})}^{2} \cdot \sum_{m = - \frac{M - 1}{2}}^{\frac{M - 1}{2}} e^{- \frac{i k d_{x}^{2} m^{2}}{q (y)}} e^{i m ψ_{x}} \sum_{n = - \frac{N - 1}{2}}^{\frac{N - 1}{2}} e^{- \frac{i k d_{z}^{2} n^{2}}{q (y)}} e^{i n ψ_{z}} \\ C = \frac{2 η {(\frac{ω_{S H}}{c})}^{2} χ^{(2)} G^{2}}{c ε_{0}} \end{array}

where C is a function of θ and ϕ and is independent of the FH beam parameters (up to the selection of the relevant element of the second order susceptibility tensor). We can identify that CP_FH, which describes the field due to a single element, corresponds to the element factor whereas the remainder of the right side of Eq. (10) (which depends on the geometrical properties of both the array and the FH Gaussian beam) constitutes the AF. The series in Eq. (10) has the form of a discrete time Fourier transform (DTFT), consequently, by applying the modulation theorem and utilizing the transforms of a sampled Gaussian and a Rect function [37], the AF can be rewritten as the following periodic convolutions:

\begin{array}{l} A F (ψ_{x}, ψ_{z}) = A F_{x} (ψ_{x}) \cdot A F_{z} (ψ_{z}) \\ A F_{x, z} (ψ_{x, z}) = \frac{1}{2 π} (\frac{i y_{r}}{q (y)}) \cdot \sqrt{1 - i \frac{y}{y_{r}}} \sum_{l = - \infty}^{\infty} \int_{- π}^{π} e^{- \frac{w_{0}^{2}}{8 d_{x, z}^{2}} (1 - i \frac{y}{y_{r}}) {(ψ_{x, z} - υ - 2 π l)}^{2}} \frac{\sin (\frac{υ N, M}{2})}{\sin (\frac{υ}{2})} d υ \end{array}

where AF_x,z(ψ_x,z) are the 1D array factors in the x and z directions respectively. If the standard deviation of the Gaussian in Eq. (11) is smaller than π/3, only the l = ψ_x,z/2π term will be significant in the interval -π≤υ≤π. This condition is equivalent to calculating the field at the center of one of the diffraction lobes, as it is analog to the diffraction condition. This condition is satisfied for Gaussian beams with waist that satisfy W₀≥6d_x,z/π = 1.91d_x,z, which is obtained in most practical cases as a typical nano-antenna array is composed of more than 2×2 elements. Hence, at the center of every diffraction lobe the AF can be approximated by:

\begin{array}{l} {A F (ψ_{x, z}) |}_{d i f f r a c t i o n_l o b e} = {A F_{x} (ψ_{x}) |}_{d i f f r a c t i o n_l o b e} \cdot {A F_{z} (ψ_{z}) |}_{d i f f r a c t i o n_l o b e} \\ {A F_{x, z} (ψ_{x, z}) |}_{d i f f r a c t i o n_l o b e} \approx \frac{1}{2 π} \sqrt{\frac{1}{1 - i \frac{y}{y_{r}}}} \int_{- π}^{π} e^{- \frac{w_{0}^{2}}{8 d_{x, z}^{2}} (1 - i \frac{y}{y_{r}}) υ^{2}} \frac{\sin (\frac{υ N, M}{2})}{\sin (\frac{υ}{2})} d υ \end{array}

Note that we introduced iy_r/q(y) = 1/(1-iy/y_r) in order to simplify AF_x,z. As can be seen, these points correspond to the maximal overlap between the functions in the integrant of the periodic convolutions, thus representing the maximal values of the AF. Typically, and particularly for recessed gap Bowties, the element factor E_{SH_Single_Element}, has a maximum at θ = 0, which corresponds to the 0th lobe. Therefore the expression in Eq. (12) also corresponds to the peak of the SH field. Let us now examine the impact of the beam parameters y and W₀ on the generated SH AF. Figure 4(a) depicts the amplitude and phase of the normalized AF at the center of any diffraction lobe, due to a square array comprising M = N = 151 elements with equal spacing d_x = d_z = d. The beam waist is equal to the width of the array, i.e. 2W₀/d = N-1 while the distance of the array from the beam waist is varied. The AF reaches a maximal value at y = 0 which corresponds to the array being located at the waist of the incident FH beam. This is because in this configuration all of the array elements are excited in phase thus resulting in a complete constructive interference at the diffraction lobes. In addition, the AF is strongly reduced when the distance of the array from the beam waist increases. This attenuation is symmetric along y, as expected from Eq. (12) and it is characterized by a full width half maximum (FWHM) of 2y/y_r = 3.32. Thus, unsurprisingly, attaining maximal efficiency necessitate precise positioning of the array at the pump beam waist. In order to explore the sensitivity of the AF to the exact waist position, we calculate the dependence of the FWHM of the AF on the incident beam waist as depicted in Fig. 4(b). To some extent, this calculation corresponds to the effective depth of focus of the FH beam for maximal SHG. It is clearly seen that there is an optimal waist size for which the generated SH is less sensitive to the exact position of the waist. It is obtained for a beam waist which satisfies 2W₀/d = 0.6·(N-1) and for this beam the FWHM is maximal and is equal to 2y/y_r = 4.52. Note that these relations are obtained for different array sizes as well. In addition, the FWHM reaches a lower bound which corresponds to 2y/y_r = 2 for wide FH beams. Thus, wide beams are relatively more sensitive to the exact beam location. For this bound, the effective depth of focus corresponds to the FH beam depth of focus, thus for wide beams the array should be in the confocal range of the beam for efficient SHG. Nevertheless, note that for wide FH beams the SHG efficiency drops, as the array is smaller than the beam waist and most of the FH beam power is not coupled to the nano-antennas.

Fig. 4 Normalized SH AF of NxN nano-antenna array (N = 151); (a) The dependence of the AF on y/y_r, for a FH beam that applies 2W₀/d = N-1; (b) The FWHM of AF due to different FH waist

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Following the last discussion, assuming that the FH beam waist location coincides with the array plane, the beam waist is still a parameter to be explored. To analyze its affect, we calculated the normalized AF as a function of the beam waist for the array discussed above. The results are depicted in Fig. 5. As can be seen, as long as the beam diameter is smaller than that of the array (most of the FH power is incident on the array) there is practically no change in AF. This means that as long as all of the FH beam power is utilized by the nano-antenna array, the SH peak is independent of the array size. The point where the AF is half of its maximal value corresponds to 2W₀/d = 1.92(N-1), whereas up to a point where 2W₀/d = 0.6(N-1), there is less than 1% change in the AF. These results lead to an important conclusion: a wide FH beam incident upon a large array and a highly focused FH beam incident upon a small number of elements are expected to exhibit similar SHG efficiencies (assuming that the total power of the FH beam and the array spacing remain unchanged). However, there are additional considerations that are discussed further in this section, which favors larger arrays. This analysis can be applied for higher harmonic generation as well. However, for any j>2 harmonic, using a wide beam that covers a larger number of elements is inefficient and impairs the AF. This results from the fact that the contribution of the j^th harmonic far-field of a single element (see Eq. (7)) is proportional to the peak of the FH field, E₀, to the power of j, thus inverse proportional to the FH waist to the power of j (see Eq. (5)). However, the product of the series in Eq. (10) is always proportional to the square of the waist. A different way to understand this generalization is from the element point a view. The contribution of each element to the j^th harmonic power decreases according to the number of elements to the power of j, whereas the contribution of the array to this harmonic is due to coherent constructive interference which is always proportional to the square of the number of elements. This is also the reason for the independence of the SH on the number of elements in the array (for beams smaller than the array).

Fig. 5 The dependence of the SH AF on the waist of the FH beam, calculated on an NxN nano-antenna array, N = 151.

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Nevertheless, there are other practical aspects to be considered. At first, as the FH waist decreases the beam intensity increases. Nonlinear media are typically affected by high intensity beams where three effects need to be considered: optical breakdown, photo-induced coloration and the photo-refraction effect [38]. The first effect damages the crystal mechanically while the latter two modify the optical transmission and the refractive index of the crystal and may change the response of individual nano-antenna elements. However, as the latter two effects cause a typically relatively small change, it is more likely that optical breakdown is the practical limitation. Though not as dramatic as the optical breakdown, changing the array size also modifies the SH beam diffraction. Typically the lobe of a single nano-antenna is substantially wider than that of the whole array. Thus, the width of the re-emitted SH beam is dominated by the AF. This width is inverse proportional to M (or N depending on the axis of interest). Thus smaller arrays exhibit stronger diffraction and larger divergence of the SH beam. Finally, let us compare our results to the results of Boyd and Kleinmann [27] for efficient SHG from a Gaussian beam FH which propagates in a bulk nonlinear crystal. For bulk SHG it was shown that the optimal SHG is obtained when the FH beam waist is located at the center of the nonlinear medium, the ratio L/2y_r equals 2.84 (where L is the optical path in the crystal) and that there is a wavevector mismatch of Δk = 3.2/L. This wave mismatch is required to compensate the different diffractions of the FH and the generated SH. In addition, it was shown that optimizing with respect to L/2y_r is not a critical adjustment and that for any waist location it is possible to find an appropriate phase matching condition which yields substantially the same results. We have shown that SH from a nano-antenna array requires the beam waist to be located on the array plane. Moreover, this was shown to be a critical parameter where a disposition of several Rayleigh ranges leads to a strong attenuation in the SH peak. Nevertheless we showed that there is a certain ratio between the beam waist and the array size which lead to a less strict waist position requirement (2W₀ = 0.6∙d∙(N-1)). In addition, in the nano-antenna case there is no need for phase mismatch as there is a strong localization of the FH field and the generated SH polarization to areas much smaller than the wavelengths. Note that the analog in our case to the L/2y_r parameter used in Ref. [27] here, is the ratio between the beam waist and the array size 2W₀/[d(N-1)]. As can be seen there are several different conclusions, most of which stem from the field localization which yield a spatial discretization in the SH sources and the transverse SH buildup in the nano-antenna case in respect to continuous longitudinal buildup in the bulk.

4. Preferred incident FH propagation and SH detection directions

A nonlinear substrate is typically characterized by a relatively high and real dielectric coefficient. More specifically, the dielectric coefficient of LiNbO₃ at 1.56µm varies between 4.57 and 4.89 depending on the field polarization relative to the crystal axes [30]. Following Sommerfeld work on antennas over dielectric substrates [39], it was shown that antennas on top of dielectric substrates transmit most of their power to the dielectric side [40-42]. Consequently, due to reciprocity, antennas on top of dielectric substrates are more susceptible to radiation which propagates from the dielectric side. Therefore, when considering SHG from nano-antennas which are recessed in nonlinear substrates, two related aspects need to be considered. This is because of the presence of the dielectric substrate which introduces a direction dependent FH field enhancement and SH re-emission. Recessed gap Bowtie nano-antenna arrays are most sensitive to radiation which propagates normal to the array. As such, to determine the propagation direction for which maximal FH field enhancement is obtained, it is sufficient to compare the two normal propagation directions, from the substrate and from the air. Figure 6 compares the cross sections of the FH power enhancement factor for both cases at the YZ (Figs. 6(a) and 6(c)) and XY (Figs. 6(b) and 6(d)) planes.The power enhancement was calculated for the recessed gap Bowtie nano-antenna array when the FH is incident from the air (Figs. 6(a) and 6(b)) and from the LiNbO₃ substrate (Figs. 6(c) and 6(d)). The FH was s-polarized (E_z, H_x) for both cases. Note that the incident fields were normalized according to the dielectric constant of the substrate to provide an equal power per unit cell. As can be seen, there is a strong localization to the gap area for both cases, as expected. However, the power enhancement factor for substrate propagation is larger by approximately $\sqrt{ε_{e o}} = \sqrt{4.57}$ , where ε_eo is the dielectric coefficient along the extraordinary axis that corresponds to the incident FH polarization. Therefore, the SH power is expected to be larger by the square of this ratio, i.e. by ε_eo. This ratio is identical to the radiated power ratio of perfect conductor antenna arrays over a semi-infinite dielectric [40]. This resemblance stems from the Lorentz reciprocity principle and the relatively low dielectric losses of Au nano-antennas at these wavelengths (shown experimentally in ref [43]).

Fig. 6 Power enhancement cross sections of the FH for normal incident wave at 1.56μm, propagating from the air (panels (a) and (b)) or from the substrates (panels (c) and (d)). The cross sections are along the E-plane (panels (a) and (c)) and the H-plane (panels (b) and (d)), which correspond to the YZ and XY planes respectively.

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Next, to analyze the spatial response of the re-emitted SH, we calculate the radiated power due to SH point dipole sources. Figure 7 depicts the SH irradiance at a 1m equidistance hemisphere in the far-field, where Fig. 7(a) is projected to the air and Fig. 7(b) into the substrate. The figures are calculated for a 10×10 Au gap Bowtie nano-antenna array embedded in LiNbO₃. Each nano-antenna is excited by a point dipole source oscillating at 780nm, located at the center of the gap (0,0,-20nm). All sources have equal phases and amplitudes. It is clearly seen that the SH beam propagating in the substrate exhibits grating lobes. This is due to an effective SH wavelength which is smaller than the array spacing. These lobes cannot be avoided in an effective way, as a further reduction in the array spacing would generate strong coupling losses for the FH field enhancement. Note that minor lobes around each grating lobe are also observed. These lobes result from the finite dimensions of the array, and if the number of elements would be increased to infinity they and the spot widths would reduce to infinitesimal size. It can also be seen that the 1st order lobes in the substrate are stronger then the 0th lobe. We attribute this to the fact that the 0th lobe is the only one that emits light towards the air clad. Consequently, the power emitted to 1st order lobes propagates only into the substrate while the power emitted to 0^st order lobe is divided between the upper (air) clad and the substrate, causing the 1st order lobes to have higher intensity than that of the 0th order lobe emitted to the substrate. The ratio between the peaks of the 1st and 0th lobes is, in our case, approximately 1.77 however this ratio may vary for different nano-antennas and spacing. Nevertheless, the irradiance peak of the SH 0th lobe is stronger for substrate re-emission compared to that of air re-emission, this time by a ratio of 3.42.This ratio stems from the more efficient radiation to the substrate from the nano-antennas and the smaller beam width (due to the smaller effective wavelength). Thus, it seems that the most efficient approach is to detect the 1st order SH substrate lobes. However, these lobes are propagating at an angle larger than the critical angle, thus rendering them much more complicated to detect. Therefore, extracting the substrates SH 0th order lobe seems preferable.

Fig. 7 SH irradiance at a 1m equidistance hemisphere due to a 10×10 Bowtie nano-antenna array recessed in y-cut LiNbO₃. The array is positioned at (0,0,0), where the coordinate system is aligned to the crystal axes and the E-plane is YZ plane whereas the H-plane is the XY plane; (a) The SH irradiance to the air; (b) The SH irradiance into the LiNbO₃; the dashed lines represent equi-polar angle circles.

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This discussion can be generalized for most SHG from nano-antennas embedded in (or on top) nonlinear substrates. Thus, the most efficient approach to detect and utilize SHG using nano-antennas embedded in a nonlinear substrate is to introduce the FH field from the substrate and to exploit the 0th lobe which is reflected back to the substrate. In the case of recessed gap Bowtie nano-antennas in LiNbO₃, applying this method may lead to approximately 12dB enhancement in the detected SH. Note that in practice the substrate is finite and the detector is usually set in the Air. Nonetheless, as we consider SHG from relatively narrowband signals, an anti-reflection coating for both the first and second harmonic can be applied.

5. Conclusion

We studied and analyzed several effects that exhibit substantial impact on the efficiency of SHG by nano-antennas embedded in nonlinear substrates. First, we showed that a recessed nano-antenna structure provides enhanced SHG characteristics compared to a similar on-surface structure. This structure was shown to 1) possess a stronger field enhancement; 2) have a localized field profile which is more efficient for SHG; 3) be preferable from both linear and nonlinear polarizations considerations over a corresponding on surface structure. As a part of this analysis we characterized the second order susceptibly tensor required for optimal SHG from planar nano-antennas. The strongest coefficients of this nonlinear susceptibility should coincide with the polarizations of the FH and SH nano-antenna modes and combine fields on a single plane which is parallel to that of the nano-antennas. Consequently, we showed that 7 of the 21 crystal systems lacking inversion symmetry are expected to exhibit nonlinear polarization losses when employed as nonlinear substrates for planar nano-antennas. Particularly we showed that in this sense LiNbO₃ is a more suitable substrate than GaAs (which is the most commonly used nonlinear substrate in experimental studies involving nano-antennas). The efficiency of SHG is also affected by the FH beam properties. For a Gaussian shaped FH beam, optimal SH is obtained, unsurprisingly, when the waist of the FH beam is located on the array plane. In addition, we showed that the generated SH is highly sensitive to the exact location of the waist, and that a deviation in the order of a few Rayleigh lengths results in a steep degradation of the SH peak. Furthermore, we showed that there is an optimal waist size, 2W₀ = 0.6·d·(N-1) for which the SH is less affected by the exact location of the waist. One of the interesting conclusions arising from this analysis is that SHG from a nano-antenna array does not depend on the FH beam waist as long as most of the beam power is coupled to the nano-antennas. Hence the benefits of using a nano-antenna array for SHG are essentially smaller diffraction and the ability to apply intense beams without damaging the nonlinear material. In addition, we found that for high (higher than two) harmonic generation it is advantageous to reduce the array size and to utilize highly focused FH beams (as long as the optical breakdown bound is not met). This analysis essentially constitutes the nano-antenna array counterpart to the Boyd-Kleinman analysis of the generation of SH in bulk material by Gaussian shaped FH beams. Finally, we showed that because of the dielectric properties of the nonlinear substrate it is advantageous to launch the FH beam from the substrate and to detect the re-emitted SH at the substrate grating lobes (assuming an anti-reflection coating for both the FH and SH is used). Particularly for recessed gap Bowtie nano-antenna arrays in LiNbO₃, we showed an order of magnitude enhancement in the peak of the far-field SH compared to launching the FH beam from the air and measuring the re-emitted SH to the air. It should be emphasized that most of our analysis can be applied to any nano-antenna array structure as long as the relevant second order susceptibility, substrate dielectric properties, and array characteristics are considered. Thus, by applying our design rules, the efficiency of second and higher harmonic generation from nonlinear materials patterned by nano-antennas can be enhanced by several orders of magnitude, and may lead to more effective nano-antenna based nonlinear optical devices as ultra-thin wavelength converters, sensors and modulators.

Acknowledgments

This work has been partially supported by the Israeli DoD and the Ministry of Trade and Industry.

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Efficient second harmonic generation using nonlinear substrates patterned by nano-antenna arrays

Abstract

1. Introduction

2. The effects of recessing the nano-antennas in a LiNbO₃ substrate

3. The impact of the beam parameters of a FH Gaussian beam

4. Preferred incident FH propagation and SH detection directions

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (12)

Optics Express

Abstract

1. Introduction

2. The effects of recessing the nano-antennas in a LiNbO3 substrate

3. The impact of the beam parameters of a FH Gaussian beam

4. Preferred incident FH propagation and SH detection directions

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (12)

Optics Express

2. The effects of recessing the nano-antennas in a LiNbO₃ substrate