1. Introduction

Peaks in scattering and absorption cross sections and local fields are manifestations of localized plasmonic resonances [1] that are sensitive to the dielectric properties of the metal and the surrounding medium, and that lend themselves to applications in sensing and detection. Nonlinear plasmonics is an emerging area of nanophotonics [2]: plasmonic nanoparticles are excellent candidates to engineer optical nonlinearities at the nanoscale, without resorting to phase-matching techniques. Metallic nanoparticles usually resonate at UV-visible wavelengths. There are two ways to engineer plasmonic resonances. The nanoantenna-theory approach modifies the particles’ shape. Elongated nanoparticles support standing waves formed by counter-propagating surface plasmon-polaritons, and behave as resonant antennas for optical fields [3,4]. Nanoantennas and nanocircuits [5] offer new opportunities for applications in energy harvesting [6,7], sensing [8] and nonlinear optics [2]. The other approach focuses on new plasmonic media. Metal-oxide semiconductors like indium-tin-oxide (ITO) and aluminum-zinc-oxide (AZO) have near-zero permittivity at telecom wavelengths with additional benefits of tunability and CMOS compatibility [9]. Enhanced nonlinearities in the epsilon-near-zero (ε≈0) regime have been demonstrated with second- and third-harmonic generation (SHG and THG) [10–12], z-scan and pump-probe experiments [13], so far using only thin-film, planar geometries. Here the two approaches are combined and applied to devise a nanostructure that supports two nested resonances, i.e., a plasmonic resonance within a plasmonic resonance. The system consists of a resonant ITO nanoparticle placed inside the gap of a resonant, gold dipole nanoantenna so that the ITO nanoparticle and the gold nanoantenna resonate near the ε≈0 frequency of ITO. The nanoantenna thus provides a preliminary field-enhancement inside its gap that permeates the region surrounding the ITO nanoparticle. This magnified field in turn drives the resonance sustained by the ITO nanoparticle inside the gap producing a further field-enhancement factor. We perform detailed calculations that take into account second- and third-order surface and volume nonlinear sources [14], and adopt a model based on nanocircuit theory [5,15] to design the nanoantenna and estimate local field enhancements. Unlike previous studies where metallic nanoparticles were introduced in the gap of metallic dipole nanoantennas [16,17], the pair of nested plasmonic resonances yields significant field localization and enhancement at both the surface and inside the bulk of the ITO nanoparticle, rather than in the metallic antenna or in the residual, empty space between the nanoparticle and the antenna, thus creating new opportunities for enhanced light-matter coupling.

Hybrid approaches that combine metals with different materials have been previously proposed to enhance the conversion efficiency of SHG in metal-dielectric, core-shell nanocavities [18] and THG in hybrid ITO-gold nanoantennas similar to the structure presented here [19–21]. However, in [19–21] ITO was regarded as a mere dispersion-less and loss-less dielectric with ε»2.89; for nanoantennas with gaps loaded with non-plasmonic materials, the origin of THG has been clarified in [22]. Crucially, however, other physical phenomena were ignored in [19–21] that we consider here: (i) the plasmonic behavior of ITO at infrared wavelengths; (ii) the nonlinear response due to free electrons; (iii) the presence and influence of SHG on third-order processes. For example, we predict that the quadratic, quadrupolar nonlinearity of the ITO free electrons induces a dipole-like, TH response that overwhelms the cubic nonlinearities of gold and ITO. Despite its quadrupolar nature, this TH source is amplified by the field enhancement that occurs in proximity and within the nanoparticle, and is then radiated very efficiently in the far field thanks to the coupling to dipole-like modes of the nanoantenna.

2. Field enhancement

The structure shown in the top-left inset of Fig. 1 consists of two gold strips of thickness w separated by a gap of height d. The total nanoantenna length is L_ant. The gap region is defined as $(x,y)\in \left[-w/2,w/2\right]\times \left[-d/2,d/2\right]$; an ITO nanoparticle (or nanowire in the 2-d geometry that we adopt) with side length a, is centered in this region. The structure is located in vacuum and is assumed to be invariant in the z direction. The permittivity of gold [23] is interpolated with a Drude-Lorentz dispersion model [24]. For the linear calculations, ITO is modeled as a free-electron gas in a positive ion background: its Drude-like relative permittivity is ${\epsilon }_{\text{ITO}}(\omega )={\epsilon }_{\infty }-{\omega }_{\text{p}}^2/({\omega }^2+i\omega \gamma )$, with ${\epsilon }_{\infty }=3.8$, plasma frequency ${\omega }_{\text{p}}^{}=2.975\times {10}^{15}\text{rad/s}$, and scattering rate $\gamma =2.04\times {10}^{14}\text{rad/s}$ [25]. The incident plane wave is assumed to have time-harmonic dependence $e^{-i\omega t}$, and to be polarized along the antenna long axis (y-axis). We designed the gold nanoantenna so that the main antenna mode would overlap the zero-crossing wavelength of the ITO-permittivity real-part, λ_ENZ ≈1.25 μm. Hence, we set w = d = 10 nm and L_ant = 560 nm. In Fig. 1 we plot the average field enhancement |Ey|/E₀| in the gap region, for the following scenarios: (i) vacuum gap, ${\epsilon }_{\text{g}}=1$; (ii) gap filled uniformly with a medium having ${\epsilon }_{\text{g}}=2.89$; (iii) gap filled uniformly with ITO, ${\epsilon }_{\text{g}}={\epsilon }_{\text{ITO}}$; (iv) gap partially loaded with an ITO nanowire with side-length 6 nm. The field enhancement in the gap of these structures is found by retrieving the Thévenin-equivalent nanocircuit of the antenna in the receiving mode (see [15] and Appendix A for details on the circuit model) and then compared with full-wave simulations. When the gap is filled with a dispersion-less, loss-less dielectric (i.e., ${\epsilon }_{\text{g}}\ge 1$), maximum field enhancement is achieved for ${\epsilon }_{\text{g}}=1$, i.e., by maximizing the magnitude of the gap reactance $Z_{\text{gap}}={(-i\omega {\epsilon }_0{\epsilon }_{\text{g}}w/d)}^{-1}$.

 

Fig. 1 Gap field enhancement vs. wavelength evaluated with the nanocircuit model (solid lines) in the gap of a nanoantenna under plane wave illumination (top-left inset), for four cases: (i) vacuum gap (black); (ii) gap filled with a dielectric of permittivity 2.89 (red); ITO-filled gap (green); gap with an ITO nanowire in vacuum (blue). Dashed curves refer to full-wave predictions. The blue, dashed-dotted curve is the full-wave field-enhancement averaged inside the ITO nanowire. $E_0=E_0\widehat{y}$ and $k_0=k_0\widehat{x}$ are input electric field and wavevector, respectively. The gold strips (yellow domains) are separated by a gap d. Top right: the distribution of the electric-field y-component for a plane-wave tuned at 1.45 μm.

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Another way to increase the field enhancement consists in filling the gap with a plasmonic medium, as in the case (iii) where ${\epsilon }_{\text{g}}={\epsilon }_{\text{ITO}}$. In this scenario a field enhancement of about 30 is predicted over a broad band above λ_ENZ: this effect is due to the complex-conjugate impedance matching between the inductive gap impedance, Z_gap, and the capacitive antenna impedance, Z_ant (see [15] and Appendix A for details on the circuit model). The nested resonances are excited in scenario (iv), in which the gap is loaded with an ITO nanowire: the field is not uniformly distributed in the gap region, especially for wavelengths near and above λ_ENZ, where ITO is metallic.

Although the average field-enhancement in the nanowire-filled gap (solid and dotted blue curves) is lower than the enhancement experienced by the vacuum- and ITO-filled gaps (black and green curves), the local field enhancement inside the ITO is much higher, as one may infer from the top-right inset of Fig. 1. This field distribution is due to the simultaneous excitation of the main nanoantenna resonant mode and the localized surface plasmon resonance of the ITO nanowire, which culminates in the plasmonic-resonance-within-a-plasmonic-resonance condition. The full-wave calculation predicts an average field-enhancement peak, inside the ITO nanowire, larger than 40 at 1.5 μm, as illustrated by the dashed-dotted blue curve in Fig. 1. We ascribe the origin of such large field enhancement to the excitation of the nested plasmonic resonances. The vacuum-gap nanoantenna provides a field-enhancement peak of about 20 in the gap region. If we now introduce an ITO nanoparticle inside the gap, by adopting the quasi-static approximation, we find a further field-enhancement factor of $\left|2/(1+{\epsilon }_{\text{ITO}})\right|$ inside the ITO particle. The peak of such extra field-enhancement is about 3 and occurs at the surface-plasmon resonance wavelength (≈1.4 μm), where${\epsilon }_{\text{ITO}}=-1$. Although this description summarizes the fundamental aspects of the resonance-within-a-resonance effect, it does not account for the shape of the nanoparticle and tends to underestimate the fringe fields due to coupling between the nanoparticle and the metallic edges that face the gap region (see inset of Fig. 1). This discrepancy is not negligible even for the 6-nm nanowire under investigation, considering that the gap height is only 10 nm. For these reasons, the actual field enhancement in the ITO nanoparticle calculated with full-wave simulations (dashed-dotted blue curve in Fig. 1) shows an enhancement factor closer to 40 rather than 60.

3. Second- and third-harmonic generation

We now study the nonlinear response of the gold-ITO nanoantenna, specifically SHG and THG. In these systems, second-order processes arising from the free-electron response cannot generally be neglected. The free-electron nonlinearity is due to a combination of convective, Lorentz and gas pressure forces; its nature is quadrupolar inside ITO and gold, and dipolar on the surfaces due to symmetry breaking. In contrast, the bound-electron response yields a dipole-allowed cubic nonlinearity. The calculations are performed using a spectral technique that integrates Maxwell’s equations in the time domain. This method simultaneously takes into account second- and third-order surface and volume nonlinear effects, nonlocal free-electron response in both ITO and gold, and models third-order cubic nonlinearities by introducing nonlinear Duffing oscillators. Details on the method can be found in [14,26] and are summarized in Appendix A.

In order to understand the type of nonlinear contributions arising from ITO and gold, we write the frequency-domain expressions of the nonlinear polarization density sources at the fundamental, second-harmonic (SH) and third-harmonic (TH) frequencies:

We consider a linear array of nanoantennas with periodicity 800 nm, illuminated by a y-polarized pump pulse at normal incidence, as illustrated in Fig. 2(a). In Fig. 2(b) we display the extinction spectrum (reflectance plus absorptance) of the array in the linear regime (very low pump intensity), with and without the ITO nanowire inside the gap. As in the isolated case, the purely metallic nanoantenna supports a broadband resonance. The plasmonic resonance associated with the ITO nanoparticle interferes with this broadband resonance and induces a double-peaked extinction spectrum.

 

Fig. 2 (a) A periodic array of dipole nanoantennas consisting of two 275nm-long gold arms and a square gap with side length of 10 nm: the gap is partially filled with an ITO square nanoparticle with side length of 6 nm. The pulsed pump signal (red arrow) excites the array at normal incidence and it is polarized along the antennas’ long axis. SH and TH scattered signals are represented with green and blue symbols, respectively. (b) The black curve is the linear extinction spectrum (reflection plus absorption) of the periodic array of dipole nanoantennas sketched in (a); the dashed red line is the same quantity evaluated for an empty gap.

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The integration of equations of motion for free and bound electrons together with Maxwell’s equations leads to the predictions of SH and TH absorption efficiencies (α_2ω and α_3ω, respectively) and far-field emission efficiencies (η_2ω and η_3ω, respectively) depicted in Fig. 3. The efficiencies are calculated as second- and third-harmonic power absorption and power emission divided by the pump power impinging on a unit cell of the array. The pump signal has a carrier wavelength swept from 0.8 to 2 μm, peak intensity of 4 MW/cm², and pulse duration of 50 fs. All things being equal, the same incident peak intensity causes the ITO-loaded structure to significantly magnify SH absorption and emission efficiencies by a remarkable six orders of magnitude compared to the bare gold array [see solid and dashed curves in Fig. 3(a)]. Removing the second-order nonlinear contributions from gold produces very similar results (not shown), suggesting that the dominant nonlinear source is indeed to be found at the surface and inside the ITO. The gold second-order nonlinearity dominates over the free-electron nonlinear response of ITO only for pump wavelengths smaller than 1 μm, a regime in which ITO behaves as a low-loss dielectric rather than a plasmonic material. Similar conclusions apply to THG and are summarized in Fig. 3(b): broadband gain, at least six orders of magnitude enhancement of TH absorption for pump fields tuned above 1500 nm, and more than three orders of magnitude enhancement of TH emission in the same wavelength range.

 

Fig. 3 (a) SH and (b) TH absorption (α) and emission (η) efficiencies for incident pulses with peak intensity 4 MW/cm² for an ITO-loaded array (solid curves) and the same array with empty gap (dashed curves). Absorption and emission efficiencies are defined as the energies absorbed and emitted (transmitted plus reflected) at the harmonic wavelengths normalized by the incident pump pulse energy. The efficiencies are plotted on a log-scale. Nonlinear SH and TH absorption and emission of the loaded structure display broadband enhancement near and above the ITO resonance region compared to the bare nanoantenna array.

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We note that while harmonic-generation efficiency is considerably enhanced by the presence of ITO, most of that same energy is re-absorbed by the structure and turned into heat. In fact, across most of the displayed spectrum nonlinear absorption rates ${\alpha }_{2\omega ,3\omega }$are predicted to be between two and four orders of magnitude larger than the far-field, nonlinear conversion efficiencies ${\eta }_{2\omega ,3\omega }$. Dissipation of harmonic light in metallic nanostructures is due to the high confinement of the induced nonlinear current sources and strong one-photon absorption in the metal bulk at the harmonic frequency [30]. In the hybrid, dipole nanoantenna under investigation, ITO in the gap is virtually lossless at the second- and third-harmonic frequencies therefore the only dissipation channel for harmonic light is the absorption in the gold arms surrounding the gap. Although geometrical variations can improve emission relative to absorption (for example, a gap entirely filled by ITO outperforms the nanoparticle-loaded gap, but displays higher nonlinear thresholds) the fact that most of the generated energy is transformed into heat challenges the notion that the structure is actually an efficient nonlinear emitter, if indeed it absorbs most of the generated energy at the harmonic frequencies. This is a common behavior in many metallic nanostructures, especially those that support hot spots for the pump field – e.g., metallic lamellar gratings and coupled nanoparticles similar to the dipole nanoantenna analyzed in the present paper, as recently demonstrated in [30].

Although the relative importance of second-order nonlinear processes on THG is generally overlooked, in Fig. 4 we show the results of TH conversion efficiencies of two separate calculations, one that explicitly includes the second-order sources such as $P_{3\omega }^{vol}+P_{\text{3}\omega }^{surf}$ in Eq. (1)(c) [solid curves, identical to those in Fig. 3(b)] while the other does not (dashed curves). The discrepancy between the two cases is obvious and ranges between two and four orders of magnitude near and beyond the ITO’s plasmonic resonance, making it clear that in that wavelength range the dominant TH sources are due to the second-order nonlinearity and are of the type:$P_{3\omega }^{vol}={\alpha }_{3\omega }\left[\left(E_{2\omega }\cdot \nabla \right)E_{\omega }+\left(E_{\omega }\cdot \nabla \right)E_{2\omega }\right]+{\beta }_{3\omega }\nabla \left(E_{\omega }\cdot E_{2\omega }\right)$ in the volume of the ITO nanoparticle, and $\widehat{t}\cdot P_{3\omega }^{surf}={\epsilon }_0{\chi }_{ttn}^{(2)}[E_{2\omega ,t}{E}_{\omega ,n}+E_{\omega ,t}{E}_{2\omega ,n}]$ and $\widehat{n}\cdot P_{3\omega }^{surf}={\epsilon }_0{\chi }_{nnn}^{(2)}[E_{2\omega ,n}{E}_{\omega ,n}+E_{\omega ,n}{E}_{2\omega ,n}]$ on the ITO nanoparticle surface. Given the quadrupolar nature of the SH field $E_{2\omega }$, a property due to the symmetry of the nanoantenna geometry, the dominant TH source $P_{3\omega }^{vol}+P_{\text{3}\omega }^{surf}$ has a non-zero electric dipole moment oriented along the nanoantenna direction and therefore generates a radiation pattern qualitatively identical to the pattern associated with the cubic nonlinear response ${\epsilon }_0{\text{χ}}^{(3)}(E_{\omega }\cdot E_{\omega })E_{\omega }$.

 

Fig. 4 TH absorption and conversion efficiencies for incident peak intensity of 4MW/cm² for an ITO-loaded nano-antenna array, with (solid curves) and without (dashed curves) the inclusion of the SH signal in the calculation.

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We note that for incident peak intensities that approach 0.5 GW/cm² the local field intensity inside the ITO nanowire reaches a magnitude close to 1 TW/cm², while overall pump depletion rates due to SH and TH absorption and generation are of order 0.1%. This is quite unprecedented in terms of both peak power and material thickness, and comes as a direct result of the nested resonances phenomenon. While the pump still remains mostly undepleted, different localization properties may lead to local SH and TH having amplitudes of the same order of magnitude as the pump. As a result of these highly unusual circumstances, it may not suffice to describe the dynamics in this regime by using equations of motion in the undepleted pump approximation. The breakdown of ITO for focused, nanosecond pulses has been reported to be of order 10 GW/cm² [31], while it is not known for incident femtosecond pulses. Finally, we note that harmonic generation and absorption appear to be sensitive to the magnitude of the linear, free-electron pressure of ITO, and thus its Fermi energy, because it can rearrange and even modulate the free-electron distribution well within the volume of the ITO. This generally does not occur for noble metals, where the charge remains well confined near the surface.

4. Conclusions

In summary, we have proposed a novel, nested plasmonic resonator based on a resonant nanoparticle placed inside the gap of a resonant dipole nanoantenna. This system provides unprecedented levels of field enhancement in the volume and on the surface of the nanoparticle, boosting by several orders of magnitude optical nonlinearities originating from free and bound electrons, as well as the nonlinearities of substances placed on the surface or inside the nanoparticle, e.g., Raman media. Interestingly, in this case we find that when the nanoantenna is pumped near the nested resonance, THG is due mostly to the second-order nonlinearities of free electrons within the plasmonic nanoparticle, rather than the cubic response of bound electrons.

Appendix A Circuit-theory model of the nanoantenna in the receiving mode

Here we describe the nanoantenna-theory model of the system under investigation. The equivalence is illustrated in Fig. 5. The antenna impedance Z_ant and the open-circuit voltage V_OC are retrieved by driving the antenna with a delta-gap voltage excitation and excluding the gap region from the calculation, as reported in [15]. Then, the electric-field enhancement in the gap region may be evaluated for any material and structural composition of the gap by applying the voltage-divider rule. With reference to Fig. 5, a voltage V₀ is applied between the two antenna edges facing the gap, corresponding to the application of an electric-field, V₀/d, on the boundaries of the gap region; then, the Helmholtz equation is solved by using a finite-element solver (COMSOL) over the whole structure except for the gap region. The antenna impedance per unit length is $Z_{\text{ant}}=V_0/I_{\max }$. The maximum current across the antenna, $I_{\max }=2H_z{|}_{x=w/2,y=0}$, is derived from the z-component of the magnetic-field (H_z) at the antenna center.

 

Fig. 5 Left: Gold-ITO nanoantenna under plane wave illumination. $E_0=E_0\widehat{y}$and $k_0=k_0\widehat{x}$are input electric field and wavevector, respectively. The gold strips (yellow domains) are separated by a gap d. An ITO nanoparticle of square cross section (green) is centered in the gap region. Right: the Thévenin equivalent circuit of the structure. Field enhancement in the gap region is given by |V_gap/(E₀d)|.

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The open circuit voltage,$V_{\text{OC}}=E_0{L}_{\text{eff}}$, is the product of the input plan-wave amplitude multiplied by the effective antenna length $L_{\text{eff}}={\displaystyle {\int }_{-L_{\text{ant}}/2}^{L_{\text{ant}}/2}2H_z(x=w/2,y)dy}/I_{\max }$.

For deeply subwavelength gaps, a quasi-static approximation may be adopted to evaluate the gap impedance Z_gap. For example, a gap filled uniformly with a material of relative permittivity ε_g exhibits an impedance per unit length$Z_{\text{gap}}={(-i\omega {\epsilon }_0{\epsilon }_{\text{g}}w/d)}^{-1}$, where ${\epsilon }_0$ is the free-space absolute permittivity. The sign of the real part of ${\epsilon }_{\text{g}}$determines the nature of the gap reactance, which is either capacitive if $\mathrm{Re}({\epsilon }_{\text{g}})>0$ or inductive if $\mathrm{Re}({\epsilon }_{\text{g}})<0$. When present, the imaginary part of the permittivity accounts for the resistive portion of the gap impedance, equal to $\mathrm{Re}(Z_{\text{gap}})$. For partially filled gaps, as in the case of the ITO nanowire in the middle of the gap, ${\epsilon }_{\text{g}}$ is estimated as an effective permittivity. The effective permittivity in the gap region, ${\epsilon }_{\text{g}}$, is evaluated in the quasi-static approximation. Poisson’s equation is solved in the gap region,$x,y\in \left[-w/2,w/2\right]\times \left[-d/2,d/2\right]$, by applying a static electric field $E_{\text{DC}}=E{}^{DC}\widehat{y}$. The effective permittivity is retrieved as ${\epsilon }_{\text{g}}={\epsilon }_{\text{eff}}=1+〈P_y^{DC}〉/({\epsilon }_0{E}^{DC})$, where $〈P_y^{DC}〉$ is the average polarization density in the gap region.

Once ${\epsilon }_{\text{g}}$ and $Z_{\text{gap}}$ are known, the field enhancement in the gap may be calculated as $FE=\left|V_{\text{gap}}/(E_{0}d)\right|$. By applying the voltage-divider rule, the following frequency-dependent expression may be written for the field enhancement,

(2)$$F{E}_{\text{C}}=\left|\frac{L_{\text{eff}}}{d}\frac{Z_{\text{gap}}}{Z_{\text{gap}}+Z_{\text{ant}}}\right|,$$

where we assume the dominant electric field component in the gap region is the one aligned with the antenna axis. As expected, field enhancement is inversely proportional to gap size. However, we note that Eq. (2) is valid strictly when quantum tunneling currents are negligible in the gap (d > 1 nm [32]). Full-wave calculations of the field-enhancements, averaged over the gap region, $F{E}_{\text{FW}}=\left|{\displaystyle {\int }_{-w/2}^{w/2}{\displaystyle {\int }_{-d/2}^{d/2}{E}_y}}dxdy\right|/(E_{0}wd)$, are in good agreement with FE_C, as illustrated in the main body of the paper.

Appendix B Calculation of second- and third-harmonic absorption and emission efficiencies in the time domain

In general, the vector field is decomposed into its harmonic components as follows: $F(r,t)={\displaystyle {\sum }_1^3}{F}_{n\omega }(r,t)e^{ni\left(k\cdot r-{\omega }_{c}t\right)}+\text{c}\text{.c}\text{.}$, where $F(r,t)$ stands for a field, a polarization or a current density; r is the position vector, ω_c is the fundamental carrier frequency and $F_{n\omega }(r,t)$ are complex envelopes of functions oscillating near the harmonic frequencies nω_c, with n=1,2,3 indicating the fundamental, second-harmonic (SH) and third-harmonic (TH) frequencies, respectively. The initial condition consists of a y-polarized, Gaussian pulse located in free space (k= ω_c /c) approaching the structure along the direction $\widehat{k}=k/k$. The vector Maxwell’s equations are integrated together with a system of equations that simultaneously describes linear and nonlinear optical properties of free and bound electrons inside both the metal and the ITO. Unlike other treatments, our approach includes up- and down-conversion processes deriving from all surface and volume sources, and accounts for pump depletion. The materials are described by an equation that determines the dynamics of nearly free-electrons as follows:

(3)$$\begin{array}{l}{\ddot{P}}_f^{}+{\gamma }_f{\dot{P}}_f=\frac{n_{0f}{e}^2}{m_0^{*}}E-\frac{1}{n_{0f}e}\left[\left(\nabla \cdot {\dot{P}}_f\right){\dot{P}}_f+\left({\dot{P}}_f\cdot \nabla \right){\dot{P}}_f\right]-\frac{e^{}}{m_0^{*}}E\left(\nabla \cdot P_f\right)\\ +\frac{{\mu }_0{e}^{}}{m_0^{*}}{\dot{P}}_f\times H+\frac{5}{3}\frac{E_F}{m_0^{*}}\nabla \left(\nabla \cdot P_f\right)-\frac{10}{9}\frac{E_F}{m_0^{*}}\frac{1}{n_{0f}^{}e}\left(\nabla \cdot P_f\right)\nabla \left(\nabla \cdot P_f\right)\end{array}$$

where $P_f$is the free-electron polarization density; $m_0^{*}$, e, $n_{0f}$are the free-electron’s effective mass, charge, and equilibrium density in either ITO or the metal, respectively; μ₀ is the free-space permittivity; ${\gamma }_f$ is the scattering rate and E_F is the Fermi energy. Additionally, an extended, nonlinear Duffing oscillator equation [26] describes bound electrons acted upon by an external electromagnetic field:

(4)$${\ddot{P}}_b+{\gamma }_b{\dot{P}}_b+{\omega }_0^2{P}_b-b_b\left(P_b\cdot P_b\right)P_b=\frac{n_{0b}{e}^2}{m_b^{*}}E+\frac{{\mu }_{0}e}{m_b^{*}}{\dot{P}}_b\times H,$$

where the subscript b stands for “bound.” While one free and one bound electron species with a resonance in the soft x-ray range may generally suffice to describe ITO, a metal like gold usually requires at least one free and two bound electron types to realistically reproduce the linear data from the IR well into the UV region of the spectrum. Equations (3) and (4) decomposes into three equations that separately describe the active harmonic components. The nonlinear third-order coefficient is $b_b={\omega }_0^2/\left(n_{0b}^2{e}^2\left|r_0^2\right|\right)$ [29] and $\left|r_0^{}\right|$is a rough measure of either lattice constant or atomic size.

The full nonlinear calculations are performed using a fast Fourier transform (FFT) beam propagation method outlined in [14,26].The FFT is carried out on a truncated spatial grid of 800 ´ 4000 points, with unit cell 1 nm ´ 1 nm, and temporal integration step 3.3′10⁻¹⁸ seconds. For gold, the free electron parameters in Eq. (3) are: $m_0^{*}=$ m_e $n_{0f}=$ 5.8´10²² cm⁻³, ${\gamma }_f=$ 2π 1.5′10¹³ rad/s and E_F ≈ 4 eV, which is estimated using the formula $E_F=\frac{{\hslash }^2}{2m_0^{*}}{\left(3{\pi }^2{n}_{0f}\right)}^{2/3}$. For ITO, the free electron parameters are: $m_0^{*}=0.5 m_e$, $n_{0f}=$10²¹ cm⁻³, ${\gamma }_f=$ 2π 3.3′10¹³ rad/s and a corresponding E_F ≈ 1 eV. These choices yield Fermi velocities that are similar for both materials, and of order 10⁶ m/s. The bound electron response in gold is modeled with two Lorentz-type oscillators [see Eq. (4)] having same electron mass and density as the free electrons, and the following coefficients: ${\gamma }_{b,1}=$ 2π 3.6´10¹⁴ rad/s, ${\omega }_{0,1}=$2π 8´10¹⁴ rad/s for the first oscillator, and ${\gamma }_{b,2}=$2π 3.6´10¹⁴ rad/s, ${\omega }_{0,2}=$2π 1.35´10¹⁵ rad/s for the second oscillator. The bound electron response in ITO is modeled with one Lorentz-type oscillator having the following coefficients: $m_{0b}^{*}=$ m_e, $n_{0b}=$ 10²¹ cm⁻³, ${\gamma }_b=$2π 3′10⁶ rad/s, ${\omega }_0=$2π 10¹⁶ rad/s. The coefficient $b_b={\omega }_0^2/\left(n_{0b}^2{e}^2\left|r_0^2\right|\right)$varies with oscillator parameters and approximate atomic/lattice size, density and resonance frequency. For gold, $\left|r_0^{}\right|$can range between 0.5-3Å, while for ITO $\left|r_0^{}\right|$may be as much as 1 nm (molecular size). For simplicity we will assume $b_b=$ 10³⁷ m⁴/A² for both ITO and gold, corresponding to the resonance frequency and density of ITO, and $\left|r_0^{}\right|$=5Å. This choice probably over estimates the effective bulk ${\chi }^{(3)}$in the metal, which ultimately contributes little to TH generation.

Acknowledgments

This research was performed while M. A. Vincenti and D. de Ceglia held a National Research Council Research Associateship award at the U.S. Army AMRDEC.

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