Harmonic Generation in Metallic, GaAs-Filled Nanocavities in the Enhanced Transmission Regime at Visible and UV Wavelengths

We have conducted a theoretical study of harmonic generation from a silver grating having slits filled with GaAs. By working in the enhanced transmission regime, and by exploiting phase-locking between the pump and its harmonics, we guarantee strong field localization and enhanced harmonic generation under conditions of high absorption at visible and UV wavelengths. Silver is treated using the hydrodynamic model, which includes Coulomb and Lorentz forces, convection, electron gas pressure, plus bulk X(3) contributions. For GaAs we use nonlinear Lorentz oscillators, with characteristic X(2) and X(3) and nonlinear sources that arise from symmetry breaking and Lorentz forces. We find that: (i) electron pressure in the metal contributes to linear and nonlinear processes by shifting/reshaping the band structure; (ii) TEand TM-polarized harmonics can be generated efficiently; (iii) the X(2) tensor of GaAs couples TE- and TM-polarized harmonics that create phase-locked pump photons having polarization orthogonal compared to incident pump photons; (iv) Fabry-Perot resonances yield more efficient harmonic generation compared to plasmonic transmission peaks, where most of the light propagates along external metal surfaces with little penetration inside its volume. We predict conversion efficiencies that range from 10-6 for second harmonic generation to 10-3 for the third harmonic signal, when pump power is 2GW/cm2.


Introduction
Since the first observation of enhanced optical transmission (EOT) [1] efforts have multiplied to prove that the conditions for EOT coincide with strong field localization on the metal surface and in proximity of the apertures [2][3][4]. Second harmonic generation (SHG) has been observed for a single aperture surrounded by grooves [5], and for arrays of sub-wavelength holes of different shapes [6] arranged in either periodic or irregular patterns [5][6][7][8][9]. Third harmonic generation (THG) has been demonstrated for a gold film patterned with nano-holes [10]. Metals like silver are centrosymmetric and lack a second order nonlinear term. However, they possess a relatively fast third order nonlinear response that may be among the largest of any known material ( (3)~1 0 -6 -10 -8 esu) that arises from smearing of the Fermi surface, and slower thermal contributions [11,12], and is certain to dominate third order processes. SHG arises from a combination of symmetry breaking at the surface and from volume contributions, in part due to the magnetic Lorentz force, and calculations are performed by introducing effective surface and volume sources each having suitable weight [13][14][15][16].
Another relevant feature in harmonic generation from sub-wavelength patterned metal is the nature and topology of the apertures. Field localization and harmonic generation are significantly different whether (slits, annular structures) or not (holes) resonant modes are excited inside the aperture. The ability of slits and annular structures to support TEM-like resonant modes [17][18][19] indeed allows more opportunities to efficiently generate harmonic fields when the apertures are filled with nonlinear materials like LiNbO 3 or GaAs [20][21][22]. This notwithstanding, a number of important aspects are ignored in theoretical descriptions: (1) detailed dynamical contributions of the metal to SHG and THG from electron gas pressure, convection, inner core electrons and a  (3) response; (2) harmonic generation due to symmetry breaking and magnetic forces at work in GaAs; (3) phase-locking between the pump and the harmonics that allows generation in wavelength ranges below the absorption edge; (4) spectral shifts due to electron gas pressure and third order processes in metal and semiconductor sections of the grating. These elements complicate the theoretical picture, and have not been investigated in this context. At the same time, these aspects have the potential to make these structures functional in regimes where absorption is substantial and always decisive for many potential applications.
Our study of harmonic generation from metallic structures unfolds without imposing any separation between surface and volume sources by adopting the hydrodynamic model [23][24][25][26][27] to describe free (conduction) electrons in the metal, by making no a priori assumptions about charge or current distributions, and by including Coulomb (electric), Lorentz (magnetic), convective, electron gas pressure and linear and nonlinear contributions to the dielectric constant of the metal arising from inner core electrons as outlined in reference [28]. When harmonic generation is tackled in plasmonic contexts there is an understandable but critical tendency to simplify the approach by focusing on the nonlinear proprieties of the material that fills the cavity, and by ignoring any role the metal may play other than being a mere vessel [20][21][22]. Indeed surface sources and magnetic forces that drive bound electrons in the active material (e.g. GaAs) are routinely overlooked, along with electron gas pressure and harmonic generation that arises from the surrounding metal walls. Yet, these same elements can play a catalytic role by activating new interaction channels and should be investigated.
Another ingredient that emerges as pivotal at wavelengths below the absorption edge is a phase-locking mechanism that dominates harmonic generation in the phase-mismatched regime, and renders materials transparent at the harmonic wavelengths. In bulk materials this is exemplified by the existence of a double-peaked generated SH signal. The evidence for this phenomenon, which is produced by the homogeneous and inhomogeneous solutions of the wave equation [29][30][31][32], can be found in experimental works carried out in bulk media, where large phase and group velocity mismatches between the fundamental field (FF) and the SH waves allows the observation of two distinct SH pulses traveling at different phase and group velocities [33][34][35]. The homogeneous solution propagates with the phase and group velocity dictated by material dispersion, and walks off from the pump. The inhomogeneous solution is trapped by the pump and is impressed with the pump's phase and group velocity. This peculiar behavior corresponds to a phase-locking mechanism that occurs in negative index [35,36] and absorbing materials [37,38] as the FF co-propagates with harmonics tuned below the absorption edge [34][35][36][37][38]. The effect persists in a GaAs cavity, with improvements predicted and observed for SH (612nm) and TH (408nm) efficiencies [38]. There is evidence that phase-locking and transparency also occur in ranges where the dielectric constant of materials like GaP displays metallic behavior (223nm) [39]. In what follows we tune the pump at 1064nm, where GaAs is transparent, so that SH and TH fall in ranges where absorption is dominant: the components that survive experience dramatic enhancement of conversion efficiencies.

Linear response of a silver metal grating filled with GaAs
We first examined the properties of a single slit of size a filled with GaAs carved on an otherwise smooth silver [40] layer having thickness w ( Fig. 1(a)). At the FF ε GaAs (1064nm) ~12.10. In the absorbing region ε GaAs (532nm) ~17.08+i2.86 and ε GaAs (354nm)~8.81+ i14.36. We optimized the linear transmission using incident TM-polarized light (H-field points into the page −x-axis− in Fig. 1(a)). The slit supports TEM-like modes that exhibit field intensities more than 100 times larger than input intensities [41,42]. We varied silver film thickness and aperture size and obtained a transmission map that reveals the resonant nature of the structure ( Fig. 1(b)).
Simulations were then carried out on an array of slits 60nm wide on a 100nm-thick film. The periodicity p of the array was varied from 200nm to 3200nm. In Fig. 2(a) we show the transmission for TM-(red line-square markers) and TE-polarizations (blue line-circle markers), normalized with respect to the energy that impinges on the geometrical area of the slits. In Fig.   2(b) we report the total transmittance as a function of wavelength for p=540nm, near the transmission maximum. An EOT of ~280% turns into a total transmission of 30% with a ~100nm-wide Fabry-Perot (FP) resonance centered at 1064nm. The second peak near 350nm is due mostly to the intrinsic transparency of silver. Plasmonic features (gap and resonance near 550nm) are scarcely visible and have little impact on harmonic generation compared to readily available and much more prominent cavity modes.    If the periodicity is chosen to be a multiple of the surface plasmon wavelength of the dielectric/metal unperturbed interface then a plasmonic band gap appears [43][44][45][46]. Slits have no cut-off for TM-polarized light, and a TEM-like resonant mode is always available, e.g. Fig. 1(b).
The interference of cavity and surface modes modulates the linear transmission profile - Fig.   2(a)-and opens a gap when the bare surface plasmon wavelength matches array periodicity.
Transmission values for an incident, TE-polarized pump field - Fig. 2(a)-are well below 1% for large periodicities, and approach 1% when slit-to-slit distance is small. The reason for the large difference between the two polarizations is due to the fact that resonant FP modes are not accessible to TE-polarized light, which at 1064nm are well below cut-off.

Linear and nonlinear models for Silver and GaAs
We now illustrate linear and nonlinear dynamics of a sub-wavelength patterned silver film filled with GaAs in the pulsed regime, and identify the origins of the generated signals. The model is outlined in details elsewhere [28], and here we summarize the most salient points. The hydrodynamic model describes the metal, and includes electric and magnetic forces, convection and electron gas pressure. We study a wavelength range where interband transitions are important (visible, UV) and use a Drude-Lorentz model to account for core electron contributions to the linear dielectric constant and to harmonic generation. GaAs is modeled as nonlinear Lorentz oscillators with absorption resonances at ~400nm and ~268nm [40], and all electrons are under the influence of electric and magnetic forces: we account for surface and volume contributions in all sections of the grating. Both silver and GaAs are assigned  (2) (zero for the centrosymmetric metal) and  (3) tensors typical of their crystallographic groups.
Conduction electrons in the metal are described as follows [28]:      P P , respectively. As we will see later, in nanocavity environments the linear pressure term can shift the band structures by tens of nanometers. If we assume that the fields and their respective polarizations and magnetizations may be decomposed as a superposition of harmonics, Eq.1 then represents a set of three coupled equations [28]. We distinguish between TE-and TM-polarized fields and define: The polarization directions are noted in Fig. 1(a). For TE-polarization the H-fields point along the y-and z-directions; for TM-polarization the E-field has components along y and z. The oscillator model is exemplified by the following scaled equations that describe generic, rapidly varying field envelope functions valid in the pump depletion regime [28]: The scaled coefficients are ; N is an integer that denotes the given harmonic order; the subscript b stands for bound. For further details about the model and the integration scheme employed we direct the reader to reference [28].
Bulk second and third order nonlinearities may be introduced directly into each of Eqs.3, or by defining a nonlinear polarization in the usual way, i.e.
(2) 2 , as we do. The second order polarization vector of GaAs may be written as follows [47]: 14 , Substituting the E-field vector defined in Eq.2 into Eq.4 leads to the following equations: TMy  TMz  TMy  TMz  TMy  TMz  TMy  TMz   i t  TMy TMz  TMy  TMz  TMy  TMz TEx  TMz  TEx  TMz  TEx  TMz  TEx  TMz   i t  TEx TMz  TEx  TMz  TEx  TMz  NL y TEx  TMy  TEx  TMy  TEx  TMy  TEx  TMy   i t  TEx TMy  TEx  TMy  TEx  TMy  Once TE-polarized fields are generated, all interaction channels become active and the generation of all harmonic fields is assured. However, the generation of down-converted, orthogonally polarized pump photons depends on the structure of the  (2) tensor. For example, a  (2) tensor whose only non-zero components are d 11 , d 22 , and d 33 cannot couple TM-to TEpolarized pump photons. We will return to this issue below.
The description of  (3) contributions may begin with the general expansion of the third order polarization as follows [47]: (3) , , , , 1,3 1,3 1,3 For a material like GaAs having cubic symmetry of the type 43m , Eq.6 reduces to: 3 3 x x x x x x y y x x z z y y y y x x y y y y z z For the metal the situation is similar, except that for isotropic crystal symmetry the relations between the tensor components allow one to write: It is evident that substitution of Eqs.2 for the electric field into Eqs.7-8 leads to  (3) contributions to all harmonic components, with self-and cross-phase modulation of the fields along with terms that couple orthogonal polarization states. The introduction of phase modulation effects on the pump fields is important, especially in metals [11,12], because given the right combination of intense fields and large  (3) band shifts may be substantial.

Nonlinear results for Silver gratings filled with GaAs
Whether it is due to vertical or horizontal resonances or a combination of both, EOT at near-IR, visible and UV wavelengths is characterized by field localization, absorption, and penetration inside the metal because in these ranges metals display dielectric constants of order unity. The interaction of light with free and bound electrons in metals becomes more efficient especially if the light is concentrated in small volumes. We consider the same system described in Figs . An incident TMpolarized field generates at least four harmonic fields: TM-polarized SH and TH (Fig. 3(a-b)), and TE-polarized SH and TH (Fig. 4(a-b)). If these results are examined together with Fig. 2 Although the choice χ (2) =10pm/V in GaAs leads to predicted TE-and TM-polarized SH efficiencies having similar values, efficiencies for TM-polarized SHG [20] have not been reported. The efficiency that we predict for the SH TM-polarized signal (which is independent of GaAs and arises mostly from the metal) is nearly 100 times larger than SHG from smooth metal layers, and 10 times larger than SHG from metal-dielectric stacks [28]. The lack of plasmonic resonances at the SH and TH wavelengths (see Fig. 2(b)) suggests that this behavior depends almost exclusively on (FP) cavity-Q [41], field overlap, and phase-locking.

Spectral features and field profiles
We now examine the individual features of the harmonic signals emitted by the grating, including spectra and field profiles. We choose pump pulses having peak intensity 2GW/cm 2 transmitted spectra for all generated fields. We note that the spectra display a small shift between transmission and reflection maxima but have similar amplitudes. In Fig.6 we show the spectra for TE-polarized pump photons. The conversion efficiencies that we predict for this novel process are already comparable to TE-polarized, TH conversion efficiencies. Albeit relatively small, the efficiency of this down-conversion process may be enhanced in bulk GaAs or by pumping the grating with TM-polarized SH and TH seed fields, as Eq.5a suggests, via the term: In Fig.7 we show snapshots of the corresponding field profiles inside the nanocavity when the peak of the pulse reaches the grating. All fields are well-localized inside the cavity, including the TH tuned at 354nm. Similar resonant behavior was obtained for planar cavities with harmonic fields tuned below the absorption edge [38,48]. One of our objectives is to also show that phaselocking [29][30][31][32][33][34][35] is playing a non trivial role. A smooth, 100nm-thick GaAs layer is only 20% transparent at 532nm, and completely opaque at 354nm. In a multi-pass geometry or a resonant cavity environment [48] the homogenous portion of the SH signal is removed more efficiently compared to bulk, so that all generated components that survive in the nonlinear medium propagate under phase-locking conditions. In our case - Fig. 2(b)-linear transmission is less than 0.05% at 532nm. At 354nm the grating is a bit Fig.7. Pump and harmonic field intensities inside and near the nano-cavities. The magnetic field intensities are depicted for TM-polarization; the transverse electric field is shown for TE-polarization. The pump magnetic field intensity (a) is amplified 250 times; the transverse pump electric field (not shown) is amplified by approximately two orders of magnitude. This combination gives way to TH conversion efficiencies that are unusually large (~10 -5 ).
transmissive thanks to the natural transparency of silver, but GaAs remains completely opaque.
More convincing numerical evidence of phase-locking may be achieved by increasing substrate thickness up to 170nm, and by reducing the width of the slits down to 20nm, so that we are still operating under resonant conditions (see Fig. 1(b)). As a result conversion efficiencies and localization properties vary little compared to Figs.6-7, except for an increased number of longitudinal peaks, as the fields resonate inside the cavity even though all the TE-generated fields are far below cut-off. This is a sure sign that phase-locking is the mechanism that drives the harmonic fields to resonate, despite the fact that the cavity should resonate only at the pump frequency [38,48].

Nonlinear results for Silver grating filled with a diagonal nonlinear material
We now show that it is possible to improve SH and TH conversion efficiencies if we assume the medium has a χ (2) tensor whose only non-zero components are d 11 =d 22 =d 33 , and if a χ (3) is triggered inside the metal. The second order polarization of Eq.4 takes the following form: (2) 2 ,   (2) tensor boosts SHG by at least three orders of magnitude compared to GaAs thanks to the full exploitation of field localization properties. Allowing a non-zero χ (3) in the metal improves THG well in excess of one order of magnitude compared to GaAs alone. Fig.9: Typical pump (a), SH (b) and TH (c) magnetic field intensities when the peak of a 50fs pulse reaches the grating. One should compare the field with the corresponding harmonics in Fig.7. While the entire metal surface has a non-zero χ (3) , only nonlinear sources on the metal walls inside the cavity matter to the process. what was reported for Cu ( (3) 6 10 Cu   esu) [48][49][50][51] in a multilayer environment [11], but is in line with nonlinearities reported for silver particles. For illustration purposes we fill the cavity with a material that has the same index of refraction as GaAs, the χ (2) of Eq.9, and we compare efficiencies with all else being equal. In Figs.8 we depict the conversion efficiencies for TMpolarized harmonics (TE-polarized fields remain null) as a function of pulse duration. The figures show that efficiencies increase as a function of pulse width and saturate for pulses longer than ~60fs. This behavior is typical of cavity phenomena [48]. The most important features in Figs.8 are perhaps the total efficiencies, i.e. ~10 -6 for SHG and ~10 -3 for THG. By choosing a suitable material and by allowing a non-zero χ (3) in the metal, results in exceptional conversion efficiencies in wavelength ranges that are usually deemed inaccessible, independent of phase matching conditions and absorption at the harmonic wavelengths.
Finally in Fig.9 we show typical field profiles that correspond to Fig.8. The figure should be compared directly with the corresponding intensities of Fig.7. While in Fig.7 most of the TMpolarized SH-signal came mostly from the metal, now the SH field originates mostly from the bulk nonlinearities of GaAs. Nevertheless, in Fig. 9(b) one can still see remnants of the localization displayed in Fig. 7(b), since the metal remains an active participant. In contrast, THG originates at the walls of the slit: the field spills into the cavity, becomes phase-locked and resonates with the pump, leading to large conversion efficiencies in the UV range.

Electron gas pressure
The discussion above suggests that in sub-wavelength regions and where metallic edges or corners are present the linear dielectric response is modulated by the pressure term. We analyze the linear regime of Eq.1, which may be written as follows: Fourier transformation of Eq.10 leads to:  (12) The poles are removed by the complex nature of  and  and the polarization is subject to resonant behavior. In Fig.10 we show the shifts that occur by varying the pressure coupling coefficient for the array of Fig.1. These shifts are substantial by any measure, even though cavity-Q is in the hundreds. The process can impact the entire band structure of the grating, with shifts and possible reshaping of linear and nonlinear spectra, suggesting that in these environments electron gas pressure should probably always be assessed.

Conclusions
We have performed a study of second and third harmonic generation in semiconductor-and dielectric-filled silver gratings, in the EOT regime. We find that using a 2GW/cm 2 pump it is possible to achieve conversion efficiencies of 10 -6 and 10 -3 for SHG and THG, respectively, in regions dominated by FP cavity resonances at the pump wavelength, and by absorption at the harmonic wavelengths. A phase-locking mechanism binds the harmonics to the pump field and creates conditions that allow SH and TH fields to resonate despite the large nominal absorption.
Plasmonic phenomena are unimportant to the process, and do not seem to offer an alternative to the sort of efficiencies that we predict by exploiting field enhancements in more traditional cavity environments. We have analyzed the interaction in two dimensions, by including surface and volume nonlinear phenomena in both metal and semiconductor sections of the grating, by considering different types of  (2) tensors, and by allowing the metal to display a  (3) response.
We have shown that it is possible to trigger a novel down-conversion process that can re- 10 Cu   , then a few tens of MW/cm 2 may suffice to begin to deplete the pump, within the limits imposed by inevitable band shifts arising from selfand cross-phase modulation. Both qualitative and quantitative aspects of the nonlinear response are promising, considering that typical material thickness is far smaller than the coherence length of the nonlinear crystal, and confined to deeply sub-wavelength slits.