The nonlinear properties of plasmonic materials, in particular gold [29], are well understood. At the plasmon resonance, multiphoton processes may be amplified by the enhanced electric field within or in proximity of metallic nanoparticles. For example, it has been observed that size-dependent third harmonic generation (THG) is feasible in gold nanoparticles [29, 46, 47]. It is interesting to note that THG is induced at the particle surface, with a nonlinear dipole proportional to the area of the sphere rather than to its volume [29], due to the small skin depth of metals and the corresponding small field penetration. Nanoantennas operating at infrared and visible frequencies are largely dependent on the plasmonic properties of metals [34, 48], whose behavior in the linear optical regime is well described by free electrons and additional interband transitions that produce frequency dispersion and support surface-plasmons responsible for the characteristic color of metals [48]. When the size of the nanostructures becomes comparable with the electron mean free path, such as at corners, tips and gaps, nonlocal and quantum effects come into play [49, 50] and, interestingly, metals exhibit nonnegligible nonlinear response [48, 51]. In previous literature [48, 51], the χ⁽³⁾ value of gold has been reported to be 1 nm²/V² However, in general, optical nonlinearities of plasmonic metals are largely unexplored and so far only few noble metal nanostructures have been studied, including SHG [11, 31], THG [29, 30] and FWM [12, 13].

Recently, it has been reported that nonlinear multi-wave mixing can be excited at the nanojunction between plasmonic dimers, as a result of the strong local field intensity, rendering a highly confined, coherent and frequency tunable local photon source [12, 13, 52]. In reference [12], Novotny and his group have demonstrated that, by reducing the separation between a pair of gold nanoparticles, the intensity of the degenerate FWM signal can be enhanced by four orders of magnitude.

Figure 2 demonstrates the experimental setup and measured emission spectrum of this wave-mixing nanodevice, where a high-peak intensity and short-pulse laser with tunable frequency ω₁ pumps an optical parametric oscillator (OPO), which provides pulses of the same duration at a tunable frequency ω₂. Consequently, as observed in Figure 2, one may obtain boosted nonlinear signal ω_FWM=2ω₁–ω₂, which originates from the third-order nonlinear polarization P⁽³⁾ (2ω₁–ω₂)=χ⁽³⁾E(ω₁)E(ω₁)E(ω₂), as well as very weak second-order responses at frequencies 2ω₁, 2ω₂, and ω₁+ω₂ (due to the unintentional asymmetry in nanoantenna geometry). For this geometry the FWM conversion efficiency is highly polarization dependent, with the signal being strongest, when the excitation field is polarized along the dimer axis. It has been reported that, at very close interparticle distances

Figure 2

(A) Experimental setup for multi-wave mixing by a single nanoantenna formed by a pair of gold nanoparticles (with a diameter of 60 nm). The laser pulses, at center frequencies ω₁ and ω₂, both locally irradiate the dimer nanoantenna, which is adhered on a glass fiber [see SEM image in the inset of Figure 2B], with its motion being accurately controlled by a PEZ-nanomotor. (B) Emission spectrum of the nanoantenna excited with pulsed lasers of wavelengths ω₁ =830 nm and ω₂= 1185 nm. The peak nonlinear signal 2ω₂ –ω₁ is measured, and the superimposed gray-dotted curve shows the spectrum of two nanoparticles with unequal size (60 nm and 100 nm in diameter). (Source: Ref. [12]).

[13], sensitive fluctuations of the photon count rate may be observed, since small changes in the contact region may give rise to huge variations in the spectral response.

When the interparticle distance decreases, the linear plasmon resonance shifts towards longer wavelengths, due to the corresponding increase in load capacitance; a dispersion that can be easily explained using the nanocircuit approach [16–20]. Therefore, by tuning the nanogap distance, the plasmonic resonance of the nanoantenna may be tuned to the FWM frequency ω_FWM =2ω₁–ω₂, leading to greatly enhanced nonlinear optical response. In this scenario, bursts of narrowband photons can be generated with a tunable center frequency, depending on the gap size and frequencies of excitation ω₁ and ω₂. Once gold nanoparticles touch, or quantum effects arise as the gap dimensions shrink [53] to less than a few nm, the resonance abruptly shifts to much lower frequencies. FWM signals can be further enhanced by coupling to a third nanoparticle placed in close proximity to the dimer, which is somehow analogous to a closed-loop nanocircuit [26]. This concept has been further extended to near-field optical imaging and spectroscopy [13, 52], by using nonlinear responses generated by a pair of gold nanoparticles as a localized photon source for fluorescence and extinction imaging. In this context, photons at frequency ω_FWM=2ω₁–ω₂ can be highly localized, with large temporal coherence and, more importantly, with frequency tunability. As a result, high-resolution fluorescence imaging can be obtained by tuning ω_FWM into the absorption spectrum of fluorescent sample molecules. Since the wavelength of the FWM signal can be adjusted over a wide spectral range by slightly detuning the pumping frequencies ω₁ and ω₂, the plasmonic dimer is ideally suitable for high-resolution fluorescence imaging and local extinction measurements.

It may be also possible to excite second-order nonlinearities by synthesizing asymmetric nanostructures, such as split-ring resonators [32] or T-shaped nanodimers [31], which inherently possess asymmetric field distributions at the particle perimeter. It has been experimentally demonstrated [31] that the asymmetrically distributed local electric fields in noncentrosymmetric gold nanodimers may produce a large SHG response.

In the previously discussed plasmonic nanostructures, the SHG enhancement process was limited to the plasmonic interface, as fields rapidly decay inside metals. Recently, Pu et al. [11] have demonstrated that symmetric and resonant composite plasmonic structures loaded with χ⁽²⁾ – NOMs may also exhibit greatly enhanced SHG responses. Figure 3A shows a nonlinear plasmonic core-shell optical nanocavity [11], where a noncentrosymmetric (BaTiO₃) medium, which utilizes the entire mode volume for even-order nonlinear optical processes, is incorporated within a plasmonic (gold) nanocavity. Figure 3B shows the normalized emission spectra of BaTiO₃-gold core-shell nanocavities (blue solid circles) and pure gold nanoparticles (green solid squares). It is evident that the coreshell nanostructure shows remarkably higher SHG responses, compared to the unloaded plasmonic particle. If we compare the composite plasmonic nanoparticle with a bare 100 nm BaTiO₃ nanocrystal, an enhancement of over 500 times in the

Figure 3

(A) Principle and geometry of plasmonic-enhanced SHG nanocavity. (B) Emission spectra of nominally 100 nm BaTiO₃-gold core-shell nanocavities (blue solid circles) and 200 nm gold nanoparticles (green solid squares) normalized to the SHG peak intensity of the core-shell nanostructures. The inset shows the same data magnified by 10 in magnitude, where the two-photon photoluminescence of 200 nm gold nanoparticles is more visible. (Source: Ref. [11]).

SHG radiation power was obtained. This device may enable efficient, subwavelength, coherent nanosources.

Another exciting optical application of asymmetric nanoantenna geometries consists in electromagnetically induced transparency (EIT) [54, 55] or Fano resonances [56], supporting a sharp transmission band within a broader absorption region. Such phenomena show great potential for slow light, novel sensors and low-loss metamaterials [54–56]. It is expected that controllable, strongly localized electric fields within the small-volume of plasmonic nanostructures may pave the way to viable nanoscale optical sensors with high sensitivity. It has been theoretically studied [54] and experimentally verified [55] that a plasmonic nanostructure comprising of asymmetrically stacked nanoantennas can represent the electromagnetic analog of quantum Fano-type interference effects. Reference [54], as an example, reports one possible plasmonic EIT design, for which a dipole nanoantenna with a broad linewidth (bright mode) is coupled to an underlying quadrupole nanoantenna, whose narrow linewidth (dark mode) is limited by non-radiative Drude damping. Due to substantial suppression of nonradiative losses, a very narrow transparency window with high modulation depth was experimentally observed. Being able to engineer and optimize these asymmetric resonant nanostructures may open new exciting possibilities for nonlinear optical applications, including enhanced χ⁽²⁾ effects and nonlinear EIT at the nanoscale. Recently, also the combination of third-order nonlinearities with dipole-dipole Fano-type resonances was demonstrated [57], showing great potential towards the design of efficient all-optical switches and nanomemories.

As outlined above, plasmonic-enhanced nonlinearities in nanoantennas and related nanostructures may be achieved by tuning their resonant frequency at the pumping wave frequency or at the frequency of the resulting nonlinear response of interest. Tunable, localized photon sources hold great potential for high-resolution imaging, spectroscopy and sensing, like single quantum dots, molecules, or ions. In particular, nonlinear nanoantennas may serve as promising frequency conversion devices at a single-photon level to realize single-photon transistors [58], where the energy of a single photon is employed to switch between different states. Even if it may be convenient to exploit the relatively strong intrinsic nonlinearity of metals for these nonlinear operations, another possibility consists in loading suitably tailored nonlinear nanoparticles at the gap of dimers and dipole antennas (Figure 1A). This approach may provide even stronger and more penetrable second- or third-order nonlinearities and may make easier the full characterization and modeling of their nonlinear response using the nanocircuit concepts. Dipole antennas loaded with nonlinear circuit elements, such as diodes and varactors (Figure 1B), are commonly used at RF and microwaves. Translating these concepts to optics, nano-dipole antennas loaded with nonlinear optical materials are considered one of the most viable nonlinear nanoantenna geometries, which are suitable for general-purpose nonlinear optical applications. This design (Figure 1A) allows exploiting the large and controllable field enhancement achieved at the nanogap, associated with the nanoantennas’ plasmonic features.

In the following sections, we concentrate on this last possibility, reviewing modeling and applications of nonlinearly loaded nanodipole antennas that may be rigorously modeled in array configurations and may realize a flexible design tool to strongly enhance second-order and/or third-order optical nonlinearities. In this approach, nonlinear nanoparticles are specifically loaded at the nanogap of a dimer or dipole, taking advantage of the largely uniform and strong local field enhancement at the plasmon resonance.

It is well-known that the shortening of guided wavelength for a mode propagating along a plasmonic nanorod with respect to the free-space wavelength [12–20] makes a nanodipole antenna resonate at a much shorter length than a conventional RF dipole of similar electrical length. This implies that it may be possible to realize dense arrays of resonant nanoantennas with periods much smaller than the wavelength of excitation and effectively realize artificial nonlinear metasurfaces, as shown in Figure 4. In their linear operation, the main extinction resonances of the metasurface arise from the collective Rayleigh scattering (with dipolar radiation signature) of the nanoantennas, for which the reflection peak of the

Figure 4

Design and operation principle of a single nonlinear nanoantenna and of a nanoscale SHG frequency conversion system, consisting of a plasmonic metasurface made of an array of silver nanoantennas loaded with χ⁽²⁾ nonlinear optical materials, which induces nonlinear frequency mixing at each nanoload.

metasurface depends on the plasmon oscillation length of the nanoantennas. Here, an individual nanoantenna is assembled as a pair of closely spaced silver nanowires with a nanoparticle load in the gap, whose relative permittivity in the linear region is ε_L. The relative permittivity of silver is described by the Drude-dispersion model obtained from experimental data [59]${\epsilon }_{Ag}={\epsilon }_{\infty }-{\omega }_p^2/\left[\omega \left(\omega +i\gamma \right)\right]$, with ω_p/2π=2175 THz, γ/2π=4.35 THz and ε_∞ =5. The physical dimensions of each nanoantenna are L=135 nm, D=10 nm and h=12.5 nm (see Figure 4). Here, commercially available Maxwell’s equation solver, CST Microwave Studio [60], was used to perform the full-wave, time-domain simulations, based on the finiteintegration technique. Figures 5A and B show our calculation of the normalized polarizability α (normalized to $6\pi {\epsilon }_0/k_0^3$), which relates the induced linear dipole moment to the local electric field at a given frequency, and the field enhancement at the nanogap for an individual nanoantenna for different values of ε_L load. The inset of Figure 5A shows the corresponding far-field radiation pattern, consisting of a typical dipolar radiation, as expected due to the small nanoantenna size. It can be seen, comparing the two panels, that large local field enhancement may be obtained at the nanoantenna resonance inside the gap (Figure 5B) [61]. Using the optical nanocircuit paradigm [16], we can tune and control the resonance frequency to a large degree by properly selecting the equivalent impedance of the nanoload, as shown by the different curves in Figure 5. This is possible because the local confinement of the electric field, as well as its specific polarization at the nanoload, allows quantitatively defining the optical impedance of the nanoantenna and the load, and model the problem as a circuit resonance. To many degrees, this becomes the optical analog of changing the inductive load of a nonlinear

Figure 5

Magnitude of (A) normalized polarizability for a single plasmonic nanoantenna and (B) local field enhancement at the nanogap, for different values of relative permittivity ε_L of the nanoload. The inset of Figure 5A shows the far-field dipolar radiation pattern at resonance. The inset of Figure 5B shows the near-field electric field distribution of an individual nanoantenna in the linear regime, showing strong field intensity in the gap at the plasmonic resonance.

antenna by turning its knob or varying the applied voltage on a varactor [16].

We extend in the following our recently proposed homogenization model of linear metasurfaces [62, 63] to this nonlinear regime. Through the description of the nanoantenna array as a homogenized nonlinear impedance surface, we can greatly simplify its design and optimization for various nonlinear optical applications. For simplicity, we start with second-order nonlinear metasurfaces loaded with χ⁽²⁾ NOM nanoparticles. However, the same approach can be used to homogenize third-order nonlinear metasurfaces.

The wave interaction with nanodipole antennas is obviously maximized when the polarization of the exciting electric field is aligned parallel to the dipole-arm axis $\widehat{z}$ (Figure 4). We consider an illumination at frequencies ω₁ and ω₂ composed of transverse-electric (TE) plane waves with electric field polarized along the $\widehat{z}-\text{axis}$, with arbitrary wave vectors. We assume that the nanoantenna length is subwavelength at these frequencies, as they operate near their resonance, such that they may be described in terms of the polarizability α (Figure 4A). The collective scattering at the two frequencies and at the mixing frequency ω₁ +ω₂ is composed of only the dominant diffraction order. This is equivalent to assume that the array may be described in the far-field as a homogenous averaged surface current sheet radiating plane waves at the relevant frequencies on both sides of the metasurface. The inherently subwavelength dimension of each nanoantenna, and the condition that the individual elements of the array are not too close to each other, makes possible to neglect quadrupoles and higher-order multipoles in considering the mutual coupling in the array.

The nonlinear averaged polarization vector at the location of one nanoantenna can be related to the macroscopic local field E_loc at the mixing frequency as

where α(ω₁ +ω₂) is the linear polarizability at the mixing frequency, β(ω₁+ω₂; ω₁, ω₂)≃χ⁽²⁾ (ε₀πD²h/4)f_rad (ω₁+ω₂)f_g(ω₁)f_g(ω₂) is the nonlinear hyperpolarizability and f_rad, f_g represent the enhancement factors at the relevant frequencies due to nanoantenna radiation and localization properties. These factors may be obtained from linear full-wave simulations, as presented in Figure 5. Rigorously considering the whole dynamic coupling among nanoantennas in the array, we may write (1) as:

where N_y and N_z are integer numbers associated with the nanoantenna coordinates y=N_ya_y and z=N_za_z, with a_y and a_z being the array periods along the two Cartesian axis, G(r_lmn) is the electric dyadic Green’s function [62, 63]

After some algebraic manipulation, we may write (2) as:

where $\Im$ is the local field correction factor:

and C is the interaction constant, which takes into account the dynamic coupling within the array and can be calculated as a fast-converging series using Poisson summations [64]:

The averaged surface current on the metasurface, induced by the collective response of the induced nonlinear dipole moments p⁽²⁾ is then given by

For ultrathin metasurfaces with subwavelength periodicity, the radiated plane waves at the mixing frequency ω₁ +ω₂ will be emitted on both sides of the metasurface with equal complex amplitudes, as a function of the amplitudes of the pumping waves E_p1 and E_p2 at frequencies ω₁ and ω₂, respectively.

In this section, we discuss and review three potential applications of nonlinear plasmonic metasurfaces formed by nanodipole antennas, which may be described using the analytical model discussed in the previous section. The goal is to review these applications in order to highlight the powerful tools offered by the nanocircuit model of nonlinearly loaded nanoantennas in various practical scenarios of interest and present novel numerical results and physical insights on their functionality.

4.1. Second harmonic generation and frequency doubling

Second harmonic generation (SHG) is the nonlinear process of converting a portion of the impinging energy at the fundamental frequency ω into the double frequency 2ω. It may be obtained as a degenerate three-wave mixing (2ω=ω+ω) process based on χ⁽²⁾ nonlinearity. In this section, we will discuss enhanced SHG over a single planar metasurface using the nanoantenna arrays discussed in the previous section with χ⁽²⁾ nanoloads (Figure 4). We will show that, despite the small volume of χ⁽²⁾ NOM nanoloads, a metasurface nanomixer or frequency doubler may be realized at optical frequencies. In this case, ω¹ and ω² in the notation of Section 2 are both replaced by the fundamental frequency ω and the mixing frequency ω₁ +ω₂ becomes 2ω Since this ultrathin metasurface has a negligible thickness, there is no issue in phase-matching along the direction normal to the metasurface, which is usually the case when using thick nonlinear samples. For oblique illumination with angle of incidence θ_in =sin^-1(k_z/k₀), phase matching on the tangential plane implies that the SHG waves will propagate at the same angle θ_SHG =sin^-1 (±2k_z/2k₀)=±θ_in, where the ± signs refer to the reflected and transmitted waves, respectively. Here, we consider a normally incident pumping wave with electric field polarized parallel to the nanodipole axis ${\text{E}}_{inc}=\widehat{z}{E}_0\exp \left[i\left(k_{0}x-\omega t\right)\right]$. After calculating the averaged surface current density (6), we can evaluate the SHG radiated fields on both sides of the surface as

$${\text{E}}^{SHG,\pm }=-\frac{\left(2\omega \right){\mu }_0}{\left(2k_0\right)}\frac{J_{av}^{NL}}{2}\exp \left[\pm i\left(2k_0\right)x\right]\widehat{z}.\left(\text{7}\right)$$( 7 )

Figure 6

Spectrum of SHG conversion efficiency for a nonlinear metasurface (red solid) compared to a NOM thin film (black dashed), whose magnitude has been magnified 10⁴ times for visibility.

The SHG reflection coefficient R^SHG is found as the ratio of the amplitude of the nonlinear reflected wave normalized to the incident electric field at the fundamental frequency:

$$R^{SHG}=\frac{-i{\left(2\omega \right)}^2{\mu }_0}{\left(2k_0\right)d_y{d}_z}\beta \left(2\omega :\omega ,\omega \right)L\left(2\omega \right)L\left(\omega \right)L\left(\omega \right){\left[\frac{2I_0}{c{\epsilon }_0}\right]}^{1/2},\left(\text{8}\right)$$( 8 )

where I₀ is the power density of the impinging wave at frequency ω. As expected, the SHG transmission coefficient T^SHG for forward propagating SHG waves has the same value as R^SHG (T^SHG=R^SHG), due to the very small thickness of the metasurface. The conversion efficiency η is given by $\eta ={\left|R^{SHG}\right|}^2+{\left|T^{SHG}\right|}^2=2{\left|R^{SHG}\right|}^2$. We consider a pumping wave with frequency around the nanoantenna resonance frequency ω/2π=285 THz and a reasonably high power density I₀=50 MW/cm². A nanoload with linear permittivity ε_L=12.5 and second-order susceptibility χ⁽²⁾=250 pm/V is also considered (i.e., typical values for III–V compounds, such as GaAs and CdGeAs₂[65]). Figure 6 shows the calculated SHG conversion efficiency over the spectrum for this nonlinear metasurface with periods d_y =250 nm and d_z=1.8 d_y. A sharp peak of SHG signal is observed with a conversion efficiency η^SHG =0.0034%. Albeit small, this value is much larger than the SHG conversion efficiency of a NOM uniform thin film with same thickness, which is calculated as η^SHG =tanh²[ζ]/sech²[0]=1.898×10^-7% [65], where ζ=x/l is the normalized distance parameter, x is the distance from the crystal interface and l=[2(ε_L(ω))²ε_L(2ω)ε₀c³/I₀]^1/2/(ωχ⁽²⁾). This dramatic enhancement (∼10⁴ order) is due to the plasmonic nanoantenna resonance at the fundamental frequency. As expected, higher SHG efficiency may be obtained by stacking more of such metasurfaces (although then phase matching should be carefully considered), or by introducing in the design optical nanocavities or nanoresonators to further boost the SHG signal. This nonlinear metasurface may benefit various applications based on SHG effects, paving the way to near-field imaging and spectroscopy, chemical sensing, localized photon sources and active photonic and plasmonic nanodevices. We are currently exploring some of these applications. In addition, this concept can be straightforwardly applied to other frequency mixing or wavelength conversion applications: by loading a χ⁽³⁾ NOM, we may be able to obtain enhanced third harmonic generation (THG) or the sum/difference frequency generation, provided that nanoantennas are properly designed to support local field enhancement at each harmonic, which may be realized by suitable nanoantenna designs with dual or multiple frequency bands of operation.

4.2. Phase conjugation and time reversal

One of the most popular applications of metamaterials and metasurfaces is sub-wavelength imaging, beyond the limits of conventional diffractive optics [22, 66–70]. Maslovski and Tretyakov [67] have proposed an interesting alternative approach to ‘perfect lensing’, based on phase-conjugation (or time reversal in signal theory), where a pair of ideal phaseconjugating surfaces may electromagnetically resemble a negative-index slab, without requiring the presence of lossy metamaterial inclusions which are designed to amplify the evanescent spectrum. Phase conjugation, a typical nonlinear process [68, 69], has been experimentally realized at microwaves by employing signal-mixers and nonlinearly loaded RF dipoles. As discussed above, the possibility of exploiting plasmonic effects in nanoantennas may largely enhance these properties at higher frequencies. Indeed, in the realm of optics, Pendry [70] has theoretically shown that this imaging setup may be ideally translated to the visible FWM processes [71, 72], where an ideal perfect lens may be achieved using a pair of parallel ultrathin layers, supporting an infinitely large nonlinear response [71, 72]. However, the very weak optical nonlinearities of naturally available NOM thin films may make difficult the practical implementation of this concept, and this ideal setup may require unrealistic pumping levels to achieve sufficient level of nonlinearity for subwavelength focusing.

We note that the plasmonic metasurfaces described in the previous sections, which combine largely enhanced nonlinear response and an ultrathin geometry, may represent the ideal candidate to realize phase-conjugation at optical frequencies based on degenerate FWM. We have suggested and explored this concept in Ref. [63]. Here, we review this application and put it into context within the general formalism introduced in section 2.

In order to achieve a large phase conjugation effect, we load nanoantennas with χ⁽³⁾ nonlinearities in order to induce a degenerate four-wave mixing process over an ultrathin surface. In this case, the array density of metasurface was optimized with dimensions d_y =100 nm and d_z=2.5 d_y. In addition, we tune the nanoantenna design so that the metasurface collectively resonates at the signal frequency ω/2π=320 THz, causing large reflections and very weak transmission of the impinging signal in its linear operation. For large and undepleted pump levels the drastically enhanced and strong fields at the nanogap can produce a conjugate signal with amplitude significantly larger than the signal wave. The large field enhancement at the nanodipole gap ensures drastic enhancement of the third-order nonlinear response at the nanoload. This is supported over an ultrathin profile, which is similar to the ideal mathematical surface suggested by Pendry for the same purpose. This ensures the efficient generation of phase-conjugate signals through FWM process, as well as the equivalence between reflected and transmitted conjugate signals, necessary for subwavelength imaging.

Figure 7A illustrates the operation principle of one phaseconjugating metasurface: the degenerate FWM process is operated by exciting the metasurface with a monochromatic signal wave at frequency ω and two strong, undepleted counterpropagating pump waves at the same frequency (so that ${\overrightarrow{k}}_{p1}+{\overrightarrow{k}}_{p2}=0$). A nonlinear dipole moment with conjugate phase (${\overrightarrow{k}}_r^{pc}=-{\overrightarrow{k}}_s$) is then generated at the nanogap nonlinear load after considering phase-matching [71]. We assume the signal field to be composed of an arbitrary superposition of transverse-electric (TE) impinging wave, linearly polarized with electric field along the nanoantenna axis $\widehat{Z}$ (Figure 7A): ${\text{E}}_s\text{=}\widehat{z}{\int }_{\infty }{E}_0\left({\text{k}}_{\parallel }\right)\exp \left(i{\text{k}}_{\parallel }\cdot \text{r+}{k}_{x}x\right)\exp \left(-i\omega t\right)d{\text{k}}_{\parallel }$, with $\text{r}=y\widehat{y}+z\widehat{z}$,k_|| being the projection of the wave vector k_s on the array, and E₀ being an arbitrary amplitude coefficient weights for each component of the impinging signal. This effectively represents an arbitrary source distribution in the transverse plane, to be imaged. After degenerate FWM, each nanoantenna load will support, in addition to its linear dipole moment, a phase-conjugated dipole moment at the mixing frequency ω=ω+ω-ω. Similar to the SHG process considered in the previous section, the total dipole moment at frequency ω for each nanoantenna is then given by

$${\text{p}}^{pc}\left(\omega \right)=\alpha \left(\omega \right){\text{E}}_{loc}\left(\omega \right)+\gamma \left(\omega :\omega ,\omega ,-\omega \right){\text{E}}_{loc}\left(\omega \right){\text{E}}_{loc}\left(\omega \right){\text{E}}_{loc}^{*}\left(-\omega \right)\left(\text{9}\right)$$( 9 )

where γ(ω:ω, ω, –ω) is the second hyperpolarizability, accounting for hyper-Rayleigh scattering $\gamma \left(\omega :\omega ,\omega ,-\omega \right)$$={\chi }^{\left(3\right)}{\epsilon }_0\pi a^{2}h{f}_g^4\left(\omega \right)$[8, 11]. After some algebraic manipulation, we may write the phase-conjugate expression as:

$${\text{p}}^{pc}\left(\omega \right)=\gamma \left(\omega :\omega ,\omega ,-\omega \right)\Im \left(\omega \right)\Im \left(\omega \right)\Im \left(\omega \right)\Im \left(-\omega \right)={\text{E}}_{p1}\left(\omega \right){\text{E}}_{p2}\left(\omega \right){\text{E}}_s^{*}\left(-\omega \right)\left(10\right)$$( 10 )

Similar to (6), the polarization fields supported by the subwavelength periodic array collectively radiate the fundamental diffraction order, or zero-order Floquet mode, sustained by the averaged current density ${\text{J}}_{av}^{pc}=-i\omega {\text{p}}^{pc}\left(\omega \right)/d_y{d}_z$. This averaged current distribution supports phase-conjugate plane waves on either side of the array, as illustrated in Figure 7A. Each phase conjugation plane wave has wave vectors ${\text{k}}_r^{p.c.}=-{\text{k}}_{\parallel }-{\text{k}}_z$ and ${\text{k}}_t^{p.c.}=-{\text{k}}_{\parallel }+{\text{k}}_z$ on the two sides of the array, respectively. We may define the reflection coefficient for each phase conjugated wave R^pc as the ratio between the amplitude of the backwardly propagating phase-conjugate wave to that of the incident signal wave:

$$R^{pc}=-\frac{i{\omega }^2{\mu }_0}{2\sqrt{k_0^2{\text{-k}}_{\parallel }^2}{d}_y{d}_z}\gamma \left(\omega :\omega ,\omega ,-\omega \right)L\left(\omega \right)L\left(\omega \right)L\left(-\omega \right)\left(\frac{4I_{p1}{I}_{p2}}{c^2{\epsilon }_0^2}\right),\left(\text{11}\right)$$( 11 )

where I_p1, I_p2 are the impinging flux intensity of the pumping waves. For large pump levels, R^pc in equation (11) may be much larger than unity, and dominate the linear response of metasurface at frequency ω. Following the theoretical proposal in Ref. [63], we then combine two of such highlyreflective metasurfaces separated by a distance d (see sketch in Figure 7B). In the limit of large R^pc=T^pc, this combination can effectively reproduce the same effect as a negative-refraction planar slab. The total transmission of phase-conjugate waves behind the second metasurface (Figure 7B) may be written as [70]:

$$\text{E}\left(x\right)=\widehat{z}{E}_0\frac{\left(R^{pc}\right)e^{i{k}_{\parallel }\cdot r_{\parallel }-i\sqrt{k_0^2-k_{\parallel }^2}\left(\left|x-x^{\prime }\right|-2d\right)-i\omega t}}{1-{\left(R^{pc}\right)}^2},\text{for}\text{real}\sqrt{k_0^2-k_{\parallel }^2}\left(12\right)$$( 12 )$$\text{E}\left(x\right)=\widehat{z}{E}_0\frac{{\left(R^{pc}\right)}^2{e}^{i{\text{k}}_{\parallel }\cdot r_{\parallel }+i\sqrt{k_0^2-{\text{k}}_{\parallel }^2}\left|x-x^{\prime }-i\omega t\right|}}{1-{\left(R^{pc}\right)}^2{e}^{i2\sqrt{k_0^2-{\text{k}}_{\parallel }^2}d}},\text{for}\text{imaginary}\sqrt{k_0^2{\text{-k}}_{\parallel }^2},\left(13\right)$$( 13 )

Figure 7

(Color online) Schematic diagram (top view) of: (A) phase-conjugation with a single nanoantenna metasurface through nonlinear four-wave mixing; (B) imaging device formed by two phase-conjugating metasurfaces. Reprinted with permission from [63]. Copyright (2011) American Chemical Society.

which, in the limit R^pc →∞∀k_||, corresponds to an ideal imaging system in the region d<x<2d, for an object placed at -d<x<0. We note that for a single metasurface T^img is dramatically non-uniform and amplified, which may cause large image distortions for the superposition integral E_s[63]. However, by considering a pair of phase-conjugating metasurfaces with sufficiently large level of R^pc, the transmission on the image plane T^img is quite uniform over a wide range of transverse wave vectors (see Ref. [63]). We now consider a practical scenario with two equal-intensity pump waves with power density I_p1=I_p2=50 MW/cm² and χ⁽³⁾=10^-16m²/V² (consistent with realistic values for some polydiacetylenes [73, 74], and realistic silver losses at ω/2π=320 THz.

Figure 8A shows the corresponding image plane field distribution at x=3d/2, for a sub-wavelength source placed at x=-d/2. We show the different cases of: (1) object plane (solid black line), (2) no device between object and image plane (gray dashed line), where diffraction dominates and (3) a pair of metasurfaces (red solid line), which is found to significantly restore the sub-wavelength resolution in the image plane. It is also seen in Figure 8A that the frequency response for different frequencies around the resonance ω₀/2π=0.320 PHz, showing a relatively good bandwidth of the subwavelength focusing effect, despite the fact that deterioration is expected due to lower levels of R^pc off-resonance. We note that sub-wavelength resolution over a moderate bandwidth of operation is obtained, even after considering realistic levels of absorption in silver. Furthermore, this system allows increasing the distance among metasurfaces and obtains similar sub-wavelength imaging properties, which is much more robust than the equivalent setup obtained with a metamaterial perfect lens [70]. Figure 8B shows the imaging properties of a pair of metasurfaces with distance ranging from d=0.5λ₀ to d=3λ₀. Here, a robust subwavelength imaging performance is observed, even for larger distances. We have also considered a larger, but still realistic, pump intensity I_p1=I_p2=1 GW/cm² for the case d=3λ₀ (dash-dot light blue line), verifying that resolution may be enhanced over larger distances by increasing the pumping levels and the corresponding value of R^pc.

4.3. Optical bistability effects

In this section, we propose to use plasmonic metasurfaces loaded with χ⁽³⁾ NOM, similar to the arrays we considered in the previous sections, to realize enhanced effects of optical bistability (AC Kerr effect) and optical switching [32]. We analyze the scattering and effective homogenized properties of individual and collections of optical nanoantennas loaded with nonlinear χ⁽³⁾ Kerr nanoparticles, with relative permittivity ${\epsilon }_{load}={\epsilon }_L+{\chi }^{\left(3\right)}{\left|E_g\right|}^2$, ε_L =2.2, χ⁽³⁾=4.4×10^-20 m²/V², typical values for fused silica [65]. We have recently shown how the concepts of antenna loading, tuning and matching may be fully introduced in plasmonic nanoantenna design [16–20], where the nanoparticles’ interactions with light can be treated as lumped nanocircuit elements; note that at the infrared/optical frequencies metals [Re(ε)<0] and dielectrics [Re(ε)>0] behave as inductive and capacitive, respectively [26]. Since the nanoantenna properties can be straightforwardly controlled and tuned by the nanoload properties placed at the gap, where the electric field is strongly enhanced, we first investigate the scattering and radiation properties of a single nanoantenna loaded with χ⁽³⁾ NOM, and then extend the concept to 2-D arrays. In this case, we consider the same silver nanoantenna with metasurface array density d_y =100 nm and d_z =1.5 d_y. Here, we use the semi-analytical method proposed

Figure 8

(Color online) Field distribution at the source (black solid), and image plane for the case of: no metasurfaces (gray dash), and double phase-conjugating metasurfaces (colored solid) for different (A) frequencies of operation and (B) separation distances between two metasurfaces. The dash-dot orange line in (B) is for a larger pump intensity of 1 GW/cm².

in Ref. [32] to calculate the nonlinear response of these optical nanoantennas, using optical nanocircuit concepts. Figure 9A shows the maximum scattering cross-section σ_max vs. wavelength for an individual nanoantenna for light intensity I=12 MW/cm². It is evident that a bistable loop is obtained in terms of scattering response, associated with the bistability of the load impedance. Farther away from the resonant wavelength, in the long-wavelength regime, the maximum available scattering cross-section smoothly increases with the wavelength of light, following its linear response (red line in Figure 9A). At a certain critical point, however, the increase in enhanced field at the nanogap, associated with the plasmonic resonance, produces a sudden jump to the second stability branch. After this discontinuous transition, if we start decreasing λ, the scattering cross-section will jump back to the initial stability branch, forming a bistable loop associated with the nonlinear response of the nanoload. It is noticed that we can apply the nanocircuit concepts to predict and tailor this bistability using simple loading design formulas for the nanodipoles, as we have discussed in Ref. [32]. It is also shown in our previous work [75, 76] that the hysteresis loops may be tailored with light intensity, and a larger bistable loop may be obtained by increasing the impinging flux intensity. The nonlinear properties of the nanoload also allow envisioning a selftunable resonant nanoantenna, whose resonance frequency may depend on the impinging light intensity. These hysteresis effects may have appealing applications in realizing alloptical nanomemories and nanoswitches at the nanoscale, since the nanoantenna may have two different stable states for the same applied optical intensity, which depend on the previous history of wavelength or intensity applied to the system. The inset of Figure 9A proposes one possible realization scheme: the nonlinear nanoantenna may be fed by a scanning near-field optical microscope (SNOM) probe, collecting the far-field scattering with photodetectors. This set-up is advantageous over currently available configurations for near-field imaging. The levels of optical intensities that may trigger this behavior are relatively low, due to the large field enhancement at the gap associated with surface plasmon polarization effects of the nanoantenna, squeezing and enhancing the stored optical energy within a small gap height. Figure 9B reports the transmission through a nanoantenna metasurface for impinging intensity I=470 MW/cm², showing that a similar bistable loop is experienced in terms of the overall metasurface transmissivity. Depending on the metasurface design, a higher level of flux intensity is required to realize an optical bistable loop, due to quenching effects associated with too close coupling between neighboring nanoantennas [32, 64]. It

Figure 9

(Color online) (A) Normalized maximum scattering cross-section vs. wavelength for linear (black line) and nonlinear operation (red line) for an individual nanoantenna at a luminous flux of I=12 MW/cm². (B) Transmission coefficient vs. wavelength for linear (black line) and nonlinear operation (red line) for an array of nanoantennas at a relative high luminous flux ofI=470 MW/cm². The unstable branches of the bistable curves are shown with dashed lines.

is seen that when the operation wavelength is getting close to the resonance of nanoantenna arrays in the linear regime (at a weak incident flux; red line), an analogous bistable behavior is triggered for the overall metasurface response, producing an intensity-dependent, self-tunable reflection and transmission properties. Such hysteresis effect may be interesting to realize compact and chip-scale, all-optical nanomemory or nanoswitch nanodevices [shown in the inset of Figure 9B], as well as intensity-tunable infrared/optical impedance surfaces.

It is found that a relatively low incident power may support bistability and switching effects, of great interest in a variety of optical nanodevices [32]. Based on similar concepts, an actively-tunable bistable Yagi-Uda nanoantenna array has been recently proposed [77]. Unlike its narrowband unloaded counterparts, this nonlinear Yagi-Uda nanoantenna array exhibits a broadband unidirectional emission and a bistable response in a preferential far-field direction.

Strong bistability and self-tunability effects associated with the nonlinear response of nanoantennas may drastically enhance the available degrees of freedom of linear optical nanoantennas and metasurfaces, enabling fascinating applications for novel optical nanodevices and self-tunable surfaces. We also envision the employment of stacked metasurfaces for the realization of metamaterials with enhanced nonlinear properties, as well as tunable and switchable effective material profiles, which could be varied from positive to near-zero or negative values [75, 76].

In this work, we have reviewed the use and a few applications of nanoantennas to boost nonlinear optical effects and operations. We have first discussed the large body of recent works utilizing the surface nonlinear susceptibility of metallic nanoparticles. We have then discussed how, by loading suitably tailored nanoantennas with nonlinear particles, we may be able to fully characterize, enhance and control their optical nonlinear response similar to radio-frequency nonlinearly loaded antennas. We have shown that planar arrays of these elements may be described as nonlinear metasurfaces, providing interesting potentials for boosting optical nonlinearities, with application in nonlinear wave mixing, phase conjugation and optical bistability. The proposed ultrathin nonlinear metasurfaces may provide practical venues to realize nanoscale frequency converters (nanomixers) and super-resolving imaging devices, which may allow relatively long-distance subwavelength imaging. All-optical nanomemories and nanoswitches may also be operated with relatively low flux intensity values. Their figures of merit are far superior to currently available nonlinear optical devices and the demonstrated concepts may be applied to coherent and frequency-tunable local photon sources, near-field imaging and spectroscopy, chemical sensing, subwavelength imaging system, logical unit cells for optical computing, on-chip communication and active plasmonics. We finally note that the aforementioned effects may be further amplified when considering the strong real part of the inherent χ⁽³⁾ nonlinearity in metallic nanoparticles [78].

This work has been partially supported by the ARO STTR project “Dynamically Tunable Metamaterials”, AFOSR with the YIP award No. FA9550-11-1-0009 and the ONR MURI grant No. N00014-10-1-0942.