4000-enhancement of difference frequency generation in a mode-matching metamaterial

: In the wake of the control of light at the sub-wavelength scale by nanoresonators, metasurfaces allowing strong ﬁeld exaltations are an attractive platform to enhance nonlinear processes. Recently, high eﬃciency second harmonic and diﬀerence frequency generations were demonstrated in metasurfaces that generate a nonlinear polarization normal to the surface. Here, we introduce a mode matched resonator that is able to produce this particular nonlinear polarization in a layer of gallium arsenide associated with a gold metasurface. The nonlinear conversion mechanism is described as a two-step process in which eﬃciency is shown to yield a good colocalization and a strong enhancement of the pump ﬁelds, as well as a high extraction eﬃciency of the generated ﬁeld. This mode-matched metasurface is able to reach a diﬀerence frequency generation (DFG) eﬃciency of 10 − 2 W/W 2 . This opens a new paradigm where alternative nonlinear materials could be reintroduced in metasurfaces and yields even higher eﬃciency than high eﬀective χ ( 2 ) structures.


Introduction
Nanoantennas have the ability to concentrate light in subwavelength volume, which is particulary appealing for exalting light-matter interactions [1,2]. Taking advantage of this capacity, many works have been focused on developing nonlinear metasurfaces [3][4][5][6][7]. They exhibit effective nonlinearities several orders of magnitude higher than in bulk materials, and can be tailored to tune the wavelength, polarization or phase of the emitted wave.
Second order nonlinear conversion have been studied in all-dielectric metasurfaces based on nanocylinders [8][9][10], gratings [11] or waveguides [12], as well as in metallic metamaterials [13][14][15][16][17] and photonic crystals [18]. While they can be used to enhance the nonlinearities of metal [13,19], III-V semiconductors like gallium arsenide are particularly interesting since they have a high nonlinear susceptibility χ (2) 150 pm/V in the infrared [20]. It has been included in various antennas to enhance further the nonlinear SHG efficiency, for instance GaAs was included in a subwavelength metallic hole [21] or a subwavelength metallic slit [22,23], or associated with metallic split-ring resonators [24]. More recently, GaAs nanopatterned layer were shown to support Mie-type modes that enhanced further the SHG efficiency [25]. Difference frequency generation (DFG) was also enhanced by using split ring resonators [26]. DFG is a very promising way to produce sources of light spanning the infrared and terahertz regions. These processes provide high enhancement ratio of the nonlinear conversion, still, they have maximum nonlinear conversion effiencies around 10 −5 W/W 2 .
Another way to enhance the effective nonlinear susceptibility of materials is to engineer multiple quantum wells [27]. Naturally, with the uprising of metasurfaces, both concepts were associated and demonstrated to provide very high efficiency for the SHG [28,29]. High efficiency DFG has also been demonstrated for generating light in the infrared [30] and in the Terahertz [31]. These processes have demonstrated conversion effiencies up to 10 −2 W/W 2 for SHG and up to 10 −3 W/W 2 for DFG. One particularity of these multiple quantum wells coupled to a metasurface is that the nonlinear polarization is enhanced in the longitudinal direction (P z ), while in most metasurfaces, the nonlinear polarization is enhanced in the lateral directions.
Here, we introduce a multi-resonant nanostructure that includes a single epitaxial GaAs layer able to provide an even higher DFG efficiency (up to 10 −2 W/W 2 ) than multiple quantum wells structures. Since the generation of P z is very rarely studied, we introduce a theoretical framework to compute it in the case of modal methods. It takes into account the specific boundaries conditions encountered by the generated propagating waves in the nonlinear medium. Besides, the high efficiency is obtained thanks to a careful mode-matching optimization so that each pump and signal wavelengths is paired with a resonance. We also show how to describe the DFG as a two-steps process, each impacting its conversion efficiency. In the first step, the nonlinear polarization is obtained by a strong co-localization of the pump fields, then the emitted light depends on the extraction efficiency of the structure.

Waveguide resonator structure
The structure studied in the whole manuscript is a waveguide resonator, which is depicted in Fig. 1. The waveguide structure is composed of a layer of material with a χ (2) (thickness h) encapsulated between a continuous gold layer, which is thick enough to be optically opaque and is considered as a semi-infinite layer, and a gold grating (period d, thickness t and width of the gold ribbons w). The model of gold used here is a Drude model in agreement with experimental data in the infrared range: Au (λ) = 1 − 1/(λ p /λ(λ p /λ + iγ)) with λ p = 159 nm and γ = 0.0075 [32].  (2) ) and a gold grating (period d, thickness t and width w). Two pump waves are incoming on the structure with a normal incidence (ω 1 and ω 2 ).
Computing the field generated by the nonlinear polarization along the z axis is pretty unusual, and we introduce a theoretical framework for modal methods that takes into account the specific boundary conditions encountered by the generated field in this case. Noteworthily, the simulation method described in this section can be applied to any grating structures that includes nonlinear materials.

Nonlinear polarization computation
The nonlinear complex polarization generated by second order nonlinear effects is given by the equation: where E 1 (ω 1 ) and E 2 (ω 2 ) are the complex linear pump fields . These linear pump fields have to be computed inside the structure in each area that exhibits a nonlinear susceptibility. In the waveguide resonator structure, the pump fields are computed in the gallium arsenide layer thanks to a B-spline modal method (BMM) [33]. In this method, the electric and magnetic fields are interpolated on a basis of non uniform B-splines basis along the x axis in each layer. Eigenmodes of the electromagnetic fields are computed by solving Maxwell equations at each sampling coordinate x.
The xyz components of the electric field E, at the sampling coordinates x and fixed height z, can be represented as the superposition of two waves, one is up-going, while the other is down-going in the structure: The fields are written at time frequency ω thanks to their projections on the eigenmodes basis E ω C (see Fig. 2): where U ± are the vectors of the modes amplitudes, which are given relatively to a height z 0 . In order to compute them, the scattering matrices at each interface are determined thanks to the continuity of the tangential electric and magnetic fields [34]. Block sub-matrices are then used in the following of the manuscript. For instance, S α BA stands for the transmission scattering matrix from layer B to medium A, and S α BB for the reflection scattering matrix on interface α in layer B (see Fig. 2) [35].

Construction of the nonlinear field
Once the nonlinear polarization is computed, each area of the nonlinear layer generates a source term that contributes to the generated field. The total generated field is the sum of each of these elementary contributions, and in practice, the nonlinear layer is discretized along the z axis. The sublayer located at height z and of thickness dz generates a source term dΠ(z), and the scattering matrices and propagations undergone by the generated fields dU ± are shown in Fig. 2.
Due to our choice of the pump fields and the orientation of the crystal in layer C, the source term is oriented along the z axis. This orientation determines the boundary conditions encountered by the elementary generated fields. For other pump field choices, the source term can be oriented according to x or y, in which case, boundary conditions have been described by Heron et al. [36].
First, the Maxwell-Ampere and Maxwell-Faraday equations are integrated on a sub-layer (thickness dz) and a period d: where z ± = z ± dz 2 . The following equations describe the continuity of the field across the sub-layer: The upward field at the lower interface of the sublayer corresponds to the downward field that has propagated in the GaAs layer and been reflected by lower gold interface γ : The downward field at the upper interface of the sublayer corresponds to the upward field that has propagated in the GaAs layer and been reflected by upper gold interface β : with M i = Q i z S i CC Q z i , and Q z i is the propagation vector from the height z to the interface i. Equations (5)-(7) can be written as: and with dΠ(z) = µ 0 ω 2 (P z + 0 E z ).
Then the contribution of each sub-layer is computed as: where I stands for the unity matrix. Finally, the total generated fields are computed from E ω 3 C the linear field exalted at ω 3 and by integrating dU + and dU − on all the sub-layers: This total generated field, once computed in the nonlinear layer, can be propagated and computed in all the layers by straightforward scattering matrices computations.

Resonance behavior
We now consider the resonator with gallium arsenide as the nonlinear material. The gallium arsenide layer has a second-order nonlinear susceptibility χ (2) xyz , only these xyz coefficients are non zero due to its crystal symmetries (zinc blende type, 43m class). The nonlinear properties of gold are not considered in the following, as the nonlinear susceptibility of gold is far smaller than for gallium arsenide [37]. The structure is illuminated with two pump waves, the first one is transverse electric (TE) polarized at a ω 1 frequency while the second one is transverse magnetic (TM) polarized at a ω 2 frequency. They generate a nonlinear polarization that is particularly strong along the z axis that gives birth to different second order nonlinear effects, like SHG, SFG (sum-frequency generation) and DFG. Here we present a design that is exalting a generated signal by difference of frequencies at ω 3 = ω 1 − ω 2 . The structure can be engineered so that it supports Fabry-Perot and guided modes in the GaAs layer [38] that subsequently improve the efficiency of the DFG. The DFG can be described as a two-steps process that will drive the conversion efficiency. First, the nonlinear polarization is obtained by a good co-localization of the pump fields, then the emitted light depends on the extraction efficiency of the structure. The importance of these two steps are illustrated on a grating with dimensions d = 458 nm, w = 220 nm, t = 40 nm, h = 145 nm. Figure 3 shows the computed reflectivity spectra in both polarizations for this structure. It exhibits two resonances in TE polarization, which are due to a guided mode resonance [38] and a Fabry-Perot resonance, and three in TM polarization which are three harmonics of a Fabry-Perot resonance in the metal-insulator-metal (MIM) cavity [39].
These different types of resonances do not depend on the same geometrical parameters, as shown in Fig. 4.   Fig. 3. Reflectivity spectra of the waveguide resonator in TM polarization (red curve) and in TE polarization (green curve). Two scenarios of DFG are pointed out with the phase matching 1 λ 1 − 1 λ 2 = 1 λ 3 . The parameters of the grating are d = 458 nm, w = 220 nm, t = 40 nm, h = 145 nm.

Vertical Fabry-Perot resonance
In this resonance, the light undergoes successive reflections between the gold miroir and the gold grating, which acts as a partially reflective mirror (see Fig. 4(a)). This resonance appears in TE polarization, and is ruled by this equation [40]: where h is the height of the cavity (here the thickness of the GaAs layer), n the refractive index of the GaAs, and λ φ describes the reflecting condidions of the cavity. The dependance in h is shown in Fig. 4(a).

Guided modes resonance
In this resonance, the light propagates horizontally in the GaAs layer, by successive reflections.
Here the light comes from the top of the system, and is injected in the GaAs layer thanks to the diffraction created by the grating (see Fig. 4(b)). This brings to some rules on the geometrical parameters: First, in order to optimize the injection in the structure, we exclude the reflection diffraction [41]: with θ i the incident angle on the grating, θ the angle coming out of the grating (coming in the GaAs layer), n GaAs refractive index, m the order of diffraction and d the periodicity.
In normal incidence it remains to: d<λ<nd.
The dependance in d is shown in Fig. 4(b). Secondly, to prevent a cut-off phenomenon, we can write the phase condition: Which brings to a cut-off thickness of GaAs:

Horizontal Fabry-Perot resonance
This resonance results from the coupling between the surface plasmons propagating on both dielectric/metal interfaces on either side of the dielectric, which must be thin enough to allow the coupling. The coupling will change the index under the bar, which will create a horizontal Fabry-Perot cavity (Fig. 4(c)) [39]. The resonance wavelength is ruled by this equation: where δ si the skin depth of the metal (δ = 25 nm), m is the considered harmonic, and n eff is the effective index : n eff = 1 + 2δ h − d m , with d the dielectric permitivity (here GaAs) and m the metal permitivity (here gold). Figure 4(c) shows the dependance on w.
These different rules of conception allow us to drastically reduce the range of geometrical parameters on which we realize the optimisation with a brute force scanning. Actually, all resonances depends on the thickness h of the GaAs layer. Nevertheless, the guided modes resonances depends on the periodicity d, regardless the width of the ribbons w. On the contrary, the horizontal Fabry-Perot resonance is extremely dependent on w. These nearly independent behaviors provide us design rules for a triple-resonance matching for DFG generation. Two scenarios for generations of DFG that takes advantage of the concentration of the fields at the pump frequencies are introduced. In both cases, the DFG is given by: In the first case (pointed out in blue in the figures), there are resonances at each wavelengths λ 1 = 1.064 µm, λ 2 = 1.6 µm and λ 3 = 3.17 µm. In the second case (pointed out in orange in the figures), there is no resonance at the signal wavelength, and the wavelengths values are λ 1 = 1.2 µm, λ 2 = 1.55 µm and λ 3 = 5.3 µm. Figure 5 represents the field maps of pumps and the nonlinear polarization computed from Eq. (1) in both scenarios. The nonlinear polarization in the blue scenario ( Fig. 5(a)) is much higher than in the orange scenario (Fig. 5(b)), because the resonances at pump wavelengths leads to higher enhancement of the electric field, and there is a better overlap between the two pump fields. The enhancement of the field can be related to the quality factors [42,43], which are higher in the blue scenario (Q = 96 for TE and Q = 29 for TM in the blue scenario, Q = 14 for TE and Q = 54 for TM in the orange scenario). The effective nonlinear susceptibility can be computed from the following expression [30]:

Co-localization of the pump fields
The integral is defined on the cross-section of the GaAs layer. In the blue (resp. orange) scenario, it yields a value χ (2) eff = 13 nm/V (resp. χ (2) eff = 3 nm/V). Thus, only the blue case DFG scenario is studied hereafter: xyz E pump E idler ). (a) Blue scenario: the pump wave is TE-polarized at λ 1 = 1.064 µm and the idler wave is TM-polarized at λ 2 = 1.6 µm generate a nonlinear polarization at λ 3 = 3.17 µm Scenarios with other wavelengths could be studied that exploit the same resonances, by tuning the geometrical parameters of the structure. In these cases, a similar nonlinear efficiency is obtained.

Extraction of the nonlinear field
The nonlinear field generated in the layer has to be extracted from the structure through the grating. A structure that is efficiently trapping an incoming wave, is reciprocally able to efficiently extract a generated wave in the layer. This behavior is not fully intuitive, and it can be understood by writing the reflectivity and extraction ratio expressions from S-matrices. The reflectivity and extraction ratio amplitudes are written as: where and Q i j is the propagation vector from the interface i to the interface j.
The matrices S α AA and S α AB are the scattering matrices at the the gold grating interfaces, so are not depending on the GaAs layer thickness h. Thus, the derivative against h of the extraction amplitude is the same than the reflectivity multiplied by the matrix S α AB −1 . It means that they have the same extrema and that their variations are linked through the S α AB −1 matrix that depends only on the metallic grating geometry. Besides, when there is no resonance, the reflectivity amplitude is close to S α AA , and therefore the extraction ratio is close to zero. The schematics of S-matrices involved in the reflectivity and in the extraction are shown in Fig. 6. The reflectivity (orange curve) and extraction (purple curve) efficiencies are plotted at the signal wavelength λ 3 = 3.17 µm as a function of the gallium arsenide layer thickness. Both curves are exhibiting resonances with an alternance of peaks and dips, but they are in phase opposition. The extraction yield is maximum when the reflectivity reaches a minimum value. In the orange scenario, since there is no resonance at λ = 5.3 µm in the structure, there is a high reflectivity (mirror behavior) and thus a weak extraction. This low extraction ratio, combined to a poorer co-localization, leads to a 30 times smaller efficiency of conversion. The signals generated by other effects (SHG and SFG) present also a low extraction ratio, because the structure does not present resonances in the reflectivity spectrum at their respective signal wavelengths. Figure 7 represents the components of the nonlinear electric field generated by the P z nonlinear polarization. The generated electric field is mainly enhanced inside the grating layer with hot spots on the sidewalls of the metallic ribbons. The distribution of the electric field is quite different from that of the nonlinear polarization since it accounts for the effects of the resonance in the structure at λ 3 . In fact, this distribution is very close to the one obtained for an incoming wave at λ 3 (data not shown). This resonance behavior is paramount for the extraction and efficiency of the structure, and to evaluate this latter, the efficiency of the DFG process is defined as η sample = P DFG P 2 inc . Fig. 7. Normalized electric field maps E x and E z generated by the resonator at λ = 3.17 µm (blue scenario). The generated field is TM-polarized.

Efficiency
For the sake of comparison, the efficiency of the structure is compared to a GaAs layer deposited on a gold mirror without the gold metasurface with an efficiency η ref . This layer acts as a cavity, which resonance wavelength is tuned by the thickness of the layer. In order to have the same resonance wavelength as the sample, the height is set at h = 200 nm. Figure 8 shows the effiencies η sample and η ref as a function of the pump wavelength λ 2 and with the other pump wavelength set at λ 1 = 1.064 µm. As expected, the efficiencies are maximum near the resonance wavelength of the structure and are tremendously reduced away from this resonance. The efficiency of the DFG process can be increased by three orders of magnitude compared to the reference (η sample = 14 × 10 −3 W/W 2 and η ref = 35 × 10 −6 W/W 2 at λ 2 = 1.608 µm). This is one order of magnitude higher than metasurfaces associated to multiple quantum wells, albeit it was also demonstrated experimentally. To emphasize the key contributions of the colocalization and the extraction, the maximum efficiency was computed in the orange ratio and in this case η sample = 10 −5 W/W 2 at λ 2 = 1.608 µm).

Conclusion
In summary, we have demonstrated nonlinear metasurfaces that exhibit a very high efficiency of difference frequency generation in the infrared. By cautiously engineering the Fabry-Perot resonances and the guided mode resonances that takes place in the GaAs waveguide layer, it is possible to give birth to a 20-fold enhanced nonlinear polarization. In order for the signal wavelength to escape the layer where it was originally generated, the resonator has to act as a trap for the incoming linear light, this is achieved by matching an additional resonance. Our work demonstrates the possibility of achieving such high efficiency without using multiple quantum wells included in metasurfaces. This new paradigm can be applied to other nonlinear materials with greatly relaxed phase-matching conditions in order to conceive infrared or THz sources. Obviously, applying these design rules to quantum heterostructures could also yield even higher efficiencies, as well as increasing the quality factors of the resonances in MIM structures [42], even if it can be at the expense of a more complex fabrication.

Disclosures
The authors declare that there are no conflicts of interest related to this article.