## Abstract

While metals benefit from a strong nonlinearity at optical frequencies, its practical exploitation is limited by the weak penetration of the electric field within the metal and the screening by the surface charges. It is shown here that this limitation can be bypassed by depositing a thin dielectric layer on the metal surface or, alternatively, using a thin metal film. This strategy enables us to enhance four-wave mixing in metals by up to four orders of magnitude.

© 2011 Optical Society of America

Nonlinear optical frequency conversion is exploited in applications as diverse as laser fusion and laser pointers. Efficient conversion requires the nonlinear response from individual atoms or molecules to be summed up coherently, a process referred to as phase matching [1, 2]. This is typically accomplished in nonlinear crystals many wavelengths in size. However, for various applications where integration is required (logic, switching, sensing, ...) frequency conversion must be achieved in structures with reduced dimensions, such as interfaces, particles, and arrangements thereof. For these applications materials with strong optical nonlinearities are required.

Second-order nonlinear optical processes, such as second-harmonic generation (SHG) and sum-frequency generation (SFG), have been extensively studied on surfaces and interfaces of various kinds [3]. The main reason for the interest in second-order nonlinear processes is associated with the surface specificity of the nonlinear response, i.e. the bulk response is suppressed in materials with inversion symmetry. This property makes SHG and SFG sensitive probes for dynamic and spectroscopic studies of molecules adsorbed on surfaces. The second-order nonlinear response has been found to be strongly enhanced at metal interfaces [4, 5] and metal nanostructures [6, 7]. More recently several studies also found a strong third-order response from noble metal surfaces [8–10]. For example, it has been demonstrated that it is possible to observe third-order nonlinearities from *single* nanoparticles [11] and that the nonlinear process can be controlled and manipulated at the nanometer scale [12, 13] which can also be used for 4WM nonlinear microscopy [14, 15]. Furthermore the nonlinear response of metallic particles can be increased when embedded in dielectric matrices [10, 16] The nonlinear cross-sections observed in these structures can exceed the nonlinearities of the most commonly used nonlinear crystals such as *LiNbO*_{3} or *KTP* by orders of magnitude [10, 17, 18]. However, the screening of the electromagnetic field by the surface charges at the metal interface and the exponential decay prevents the driving field to enter the metal, and hence, inhibits the efficient generation of a nonlinear response because the volume contributing is limited by the metal’s skin depth.

To quantitatively understand the third-order nonlinear response of metals it is necessary to perform experiments on well characterized structures. The planar geometry is particularly simple because the momentum conservation imposed by translational invariance leads to a directional response. A first strategy to enhance the field inside the metal consists of depositing a thin dielectric layer on top of the metal, which leads to Fabry-Perot resonances and associated field enhancements at the metal surface. Replacing the passively acting dielectric by an active medium contributing with its own nonlinearity can further boost the effective nonlinearity. Finally, the field inside the metal can also be increased by using a thin metal film, which follows from a simple analysis of the Fresnel reflections/transmission coefficients. In this paper, we investigate these strategies and demonstrate a third-order nonlinear signal conversion that is enhanced by four orders of magnitude compared to a bare metal surface.

## 1. Four-wave mixing at a coated metal interface (theory)

In this section, we summarize the theory of four-wave mixing (4WM) at a planar metal surface coated with a dielectric layer of thickness *d*. The limit of a bare metal surface corresponds to *d* → 0. As illustrated in Fig. 1, two coherent incident laser beams with frequencies *ω*_{1} and *ω*_{2} are incident from angles *θ*_{1} and *θ*_{2}, respectively. The angles are measured from the surface normal in clockwise direction. The two beams induce a nonlinear polarization at frequencies:

*θ*

_{4wm1}and

*θ*

_{4wm2}, respectively. Equations (1) are statements of energy conservation and define the frequencies of the outgoing radiation. Similarly, the in-plane momentum conservation at a planar surface defines the outgoing propagation directions according to:

*real*solutions for

*θ*

_{4wm1}and

*θ*

_{4wm2}exist only for certain angular ranges of

*θ*

_{1}and

*θ*

_{2}. Solutions represented by

*imaginary*angles correspond to evanescent 4WM fields such as surface plasmon polaritons [19].

In the following, for the sake of brevity, we restrict ourselves to the 4WM process giving rise to *ω*_{4wm1}. The induced nonlinear polarization can be expressed as

*ω*

_{4wm}=

*ω*

_{4wm1}.

**E**

*are the electric field vectors associated with the incident beams of frequencies*

_{i}*ω*with

_{i}*i*=1,2, and

*χ*

^{(3)}is the third-order susceptibility, a tensor of rank four. We next assume that the nonlinear response is associated with the bulk and that the material is isotropic. In this case, the 81 components of

*χ*

^{(3)}can be reduced to only three non-zero and independent components, namely ${\chi}_{1212}^{\left(3\right)}$, ${\chi}_{1221}^{\left(3\right)}$, and ${\chi}_{1122}^{\left(3\right)}$ [1]. Here, the indices ‘1’ and ‘2’ stand for any Cartesian index (x, y, or z), with the condition that ‘1’ ≠ ‘2’. The only other non-zero components are ${\chi}_{1111}^{\left(3\right)}=\left[{\chi}_{1212}^{\left(3\right)}+{\chi}_{1221}^{\left(3\right)}+{\chi}_{1122}^{\left(3\right)}\right]$. For the case of four-wave mixing considered here, two of the incident fields are identical [c.f. Eq. (3)] and hence ${\chi}_{1212}^{\left(3\right)}={\chi}_{1122}^{\left(3\right)}$. The two remaining components, together with the input fields

**E**

*define the nonlinear polarization, which can be written as*

_{i}**k**

_{4wm}= 2

**k**

_{1}–

**k**

_{2}. Here,

**k**

_{1}and

**k**

_{2}are the wavevectors of the exciting fields in the non-linear medium.

Using the coordinate system defined in Fig. 1, each of the fields **E*** _{i}* can be represented in terms of the angle of incidence

*θ*, polarization angle

_{i}*ϕ*, and the wavevectors

_{i}**k**

*of the incident waves in the nonlinear medium as*

_{i}*t*and

_{s}*t*are the Fresnel transmission coefficients for

_{p}*s*(=

*TE*)- and

*p*(=

*TM*)-polarized incident light, respectively, and ${E}_{i}^{o}$ is the amplitude of the incident field. As discussed later in Eq. (7),

*t*and

_{s}*t*depend on the material and the thickness of the dielectric layer deposited on top of the metal surface.

_{p}The nonlinear polarization **P** defines a source current and gives rise to electromagnetic fields at the four-wave mixing frequency *ω*_{4wm}. Following the theory outlined by Bloembergen and Pershan [20] the reflected field **E**_{4wm} can be calculated [17] as

*ɛ*

_{1}=

*ɛ*and ${k}_{\downarrow}^{2}={\left({\omega}_{4\text{wm}}/c\right)}^{2}{\varepsilon}_{2}$, with

_{air}*ɛ*

_{2}=

*ɛ*(

_{metal}*ω*

_{4wm}) and the z-components of the wavevector defined by ${k}_{z\uparrow ,\downarrow}=\sqrt{{k}_{\uparrow ,\downarrow}^{2}-{k}_{4\text{wm},x}^{2}}$. Because of momentum conservation along the interface

*k*

_{4wm,x}=

*k*

_{↓,x}=

*k*

_{↑,x}. Depending on which of the two four-wave mixing processes is being considered we further have

*k*

_{4wm,x}= 2

*k*

_{1,x}–

*k*

_{2,x}or

*k*

_{4wm,x}= 2

*k*

_{2,x}–

*k*

_{1,x}, which is a restatement of Eq. (2). Furthermore, since

*k*

_{1,y}=

*k*

_{2,y}= 0 for the incident waves we also have

*k*

_{4wm,y}=

*k*

_{↓,y}=

*k*

_{↑,y}= 0. The optical properties of the dielectric layer are contained in the transmission coefficients

*t*and

_{s}*t*. Notice, that

_{p}*t*and

_{s}*t*in Eq. (6) are evaluated at the nonlinear frequency

_{p}*ω*

_{4}

*, whereas in Eq. (5) they are evaluated at the frequencies of the incident radiation.*

_{wm}## 2. Four-wave mixing at a coated metal interface (experiment)

In our experiments we use a Ti:Sapphire laser providing pulses of duration ∼ 200 fs and center wavelength *λ*_{2} = 800 nm, and an optical parametric oscillator (OPO) providing pulses of similar duration and wavelength *λ*_{1} = 707 nm. The beams are first expanded to 10 mm diameter and then focused by two lenses of focal length *f* = 50 mm on the surface. The angle between the two laser beams is held fixed at *θ*_{2} – *θ*_{1} = 60° and the laser pulses are made to overlap in time by use of a delay line. The spot diameters at the surface are ∼ 4.5 *μ*m and are spatially overlapping. We use a detection angle that is fixed with respect to the angles of the excitation beams, namely *θ*_{det} = *θ*_{1} + 26°. The radiation is collected and collimated by a *f* = 75mm lens, filtered by optical stop-band filters to reject light at the two excitation frequencies, and then sent into a fiber-coupled spectrometer. Alternatively, the collected light is detected with a single-photon counting APD for intensity measurements.

The spectrum of radiation detected at the angle *θ*_{det} consists of peaks that correspond to the 4WM frequencies described by Eq. (1). These peaks are located to the blue and the red side of the excitation wavelengths *λ*_{1} and *λ*_{2}. The 4WM peaks are only observed if the angles *θ*_{1}, *θ*_{2}, and *θ*_{det} fulfill the resonance condition defined by Eqs. (1) and (2). For *λ*_{4wm1} = 633 nm we used *θ*_{1} = 6° and *θ*_{2} = 66°, whereas for *λ*_{4wm2} = 920 nm we chose *θ*_{1} = −72.8° and *θ*_{2} = −12.8°. The 4WM peaks disappear when the pulses of the excitation beams are temporally or spatially detuned or if the sample rotation does not allow for momentum conservation of all contributing waves. It is important to notice that the spectra at planar metal samples are essentially background free. While optical four-wave mixing can also be measured on metal nanostructures such as particles [12] and roughened surfaces, these spectra usually feature a strong background due to one-photon and two-photon excited photoluminescence [21]. Besides of being essentially background-free, the spectrum of the planar metal surfaces is highly directional and the angular emission can be tuned by sample rotation, as discussed in Ref. [17].

The samples studied were either silver surfaces overcoated with thin dielectric layers or thin gold films deposited on glass substrates. Thin dielectric films were created by sputter-deposition of either TiO_{2} or SiO_{2} and thin gold films have been fabricated by thermal deposition. The thicknesses *d* of either metal film or dielectric layer are varied from sample to sample. For every sample, the 4WM intensity has been measured as a function of the excitation angles. Special care has been taken to nicely overlap the incident beams in space and time and letting the focus at the surface coincide with the axis of sample rotation.

A dielectric layer deposited on top of a metal surface alters the in- and out-coupling of waves in/from the metal. This effect can readily be understood within the framework of Fabry-Perot resonances because the partial reflection of the waves at each interface and the thickness-dependent accumulated phase shift may let the transmitted waves interfere constructively. This constructive interference then leads to increased 4WM generation. The electromagnetic field transmitted into the metal in the case of a dielectric layer with thickness *d* deposited on the metal’s surface is

*k*is the perpendicular component of the wavevector in the dielectric, and ${t}_{p,s}^{\left(1\right)}$, ${r}_{p,s}^{\left(1\right)}$ and ${t}_{p,s}^{\left(2\right)}$, ${r}_{p,s}^{\left(2\right)}$ are the Fresnel transmission and reflection coefficients for the air-dielectric and the dielectric-metal interface, respectively. The ‘two-interface’ transmission coefficients

_{d,z}*t*(

_{p,s}*λ*,

*θ*,

*d*) influence the 4WM efficiency in several ways, namely through the in-coupling of the excitation fields in Eq. (5) at the excitation wavelengths

*λ*

_{1}and

*λ*

_{2}, and through the out-coupling of the 4WM field in Eq. (6) at the four-wave mixing wavelength

*λ*

_{4wm}.

Substituting the expression for *t _{p,s}*(

*λ*,

*θ*,

*d*) in Eqs. (5) and (6) yields the oscillatory intensity plot depicted in Fig. 2. The figure shows the 4WM enhancement as a function of layer thickness

*d*and index of refraction

*n*relative to a bare silver surface. The calculation assumes 4WM generation at

*λ*

_{4wm}= 633 nm and TM polarized excitation fields. According to this calculation, a 50 nm film with

*n*=3 yields a 4WM enhancement of more than two orders of magnitude. As discussed later on, considerably higher enhancements are found for TE incidence.

Evaluating the thickness dependence for the case of SiO_{2} (*n* ≈ 1.5) leads to a behavior as plotted in Fig. 3. These curves depict the 4WM intensity as a function of thickness *d* of a SiO_{2} layer deposited on a silver surface. The top part of Fig. 3 shows the results for TM polarized incident fields and the bottom part for TE polarized incident fields. The curves have been normalized with the 4WM intensity calculated for a bare silver surface (*d* → 0). For TM polarized fields the maximum 4WM enhancement at *λ*_{4WM1} = 633 nm is predicted to be ≈ 6, whereas for TE polarized fields we obtain a maximum 4WM enhancement at *λ*_{4WM1} = 633 nm of ≈ 25.

The superimposed dots represent our experimental results. Each data point has been obtained from a separate sample, for which the thickness of the SiO_{2} layer has been adjusted during the deposition process. The experimental data points follow the theoretical curves reasonably well. The only adjustable parameter in our theory is the nonlinear susceptibility *χ*^{(3)} of silver, which can be estimated by a comparison of theory and experiment. We obtain a value that is a factor of ≈ 1.5 times larger than the value of
${\chi}_{\mathit{Ag}}^{\left(3\right)}=2.8\cdot {10}^{-19}{m}^{2}/{V}^{2}$ listed in Ref. [1]. The difference can be attributed to the different wavelengths used in our experiment and to the highly dispersive nature of the nonlinear susceptibilities of metals [22].

Figure 3 also shows the theoretical 4WM enhancement for *λ*_{4wm} = 920 nm. For a SiO_{2} thickness of *d* =120 nm and for TE polarized excitation we find a predicted 4WM enhancement of more than two orders of magnitude. We were not able to experimentally verify the curves for *λ*_{4wm} = 920 nm because of nearly grazing incidence of the *ω*_{1} beam. In particular, for TE incidence and for layer thicknesses smaller than 80 nm we did not observe any 4WM. Furthermore, for TM polarization and *λ*_{4wm} = 920 nm we measured a fluorescence background, which most likely originates from local imperfections in the SiO_{2} layer.

The third-order susceptibility of silver is nearly three orders of magnitude larger than for fused silica ( ${\chi}_{Si{O}_{2}}^{\left(3\right)}=2.5\cdot {10}^{-22}{m}^{2}/{V}^{2}$).

Therefore, the nonlinear response from silver is much stronger and any nonlinearity from SiO_{2} can be neglected in our analysis. However, this is not the case for dielectrics with higher *χ*^{(3)} coefficients, such as TiO_{2} (
${\chi}_{Ti{O}_{2}}^{\left(3\right)}=2.1\cdot {10}^{-20}{m}^{2}/{V}^{2}$ [1]). Figure 4 shows our experimental and theoretical 4WM results for such an active dielectric surface layer. The refractive index of TiO_{2} (*n* ≈ 2.3) is significantly larger than the refractive index of SiO_{2}, which gives rise to much stronger Fabry-Perot resonances (see Fig. 2). Our initial theoretical calculations followed the same steps as those outlined for the SiO_{2} layer above, neglecting any nonlinear contributions from the TiO_{2} layer. The maximum calculated 4WM enhancement factors turned out to be 450× for TE incidence and *λ*_{4wm} = 633 nm, 290× for TE incidence and *λ*_{4wm} = 920 nm, 50× for TM incidence and *λ*_{4wm} = 633 nm, and 20× for TM incidence and *λ*_{4wm} = 920 nm. While these values are larger than the values calculated and measured for the SiO_{2} layer, they are significantly lower than the experimental data shown in Fig. 4. We therefore conclude that the TiO_{2} layer itself contributes to the nonlinear response.

To account for the nonlinear contribution of the dielectric surface layer we extended the theory outlined in Section 1. The results are shown in Fig. 4 as solid and dashed curves. For silver we assumed a ratio of *χ*_{1221}/*χ*_{1122} = 1/ – 0.1 for the nonlinear susceptibility components, whereas for TiO_{2} we chose *χ*_{1221} = *χ*_{1122} and a value that is a factor 0.03exp(*i*0.8*π*) smaller than that of silver. This choice is justified because sputter deposition without post-annealing leads to an isotropic composition. The data shown in Fig. 4 indicates that a 70 nm TiO_{2} layer enhances the nonlinear response by more than four orders of magnitude.

To demonstrate that the nonlinear response is not only due to the TiO_{2} nonlinearity we deposited a 100-nm-thin TiO_{2} film on a glass substrate and performed similar 4WM measurements. The measured 4WM intensity turned out to be only 3 (TM, 4WM @ 633 nm) or 8 (TM, 4WM @ 920 nm) times the 4WM intensity from a bare silver surface. Thus, the giant enhancement observed for a TiO_{2} coated silver surface must be the result of a combined effect. Note that the differences between the calculations and measurements for TM incidence and *λ*_{4wm} = 920 nm and *d* = 200..300 nm have to be attributed to experimental imperfections and fluorescence background generated inside TiO_{2}.

## 3. Four-wave mixing at a thin metal film

So far we used a thin dielectric surface layer to increase the field at the metal surface and to improve the out-coupling of the nonlinear response. In this section we demonstrate that an enhanced nonlinear response can also be achieved by reducing the thickness of a metal film without having a dielectric layer on top. At first sight one would expect that reducing the volume of the nonlinear medium must lead to a reduced nonlinear response. However, this is compensated by the improved in- and out-coupling efficiency of a thin metal film.

To account for the nonlinear response of a metal film of finite thickness, the theory outlined in Section 1 needs to be modified. In essence, the transmission coefficients *t _{p,s}*(

*λ*,

*θ*,

*d*) need to be replaced since we’re no longer interested in the energy transmitted through a film but in the energy deposited in a film.

The solid curve in Fig. 3 shows the calculated 4WM intensity as a function of the gold film thickness *d*. For *d* >50 nm the film behaves like bulk metal and no enhancement is observed. On the other hand, below 50 nm, when the thickness becomes comparable to the skin depth, the influence of the lower metal-glass interface comes into play and the field in the metal increases. The enhanced fields in the metal film can be seen as an interference effect: the wave reflected from the top air-gold interface and the wave emanating from the bottom gold-substrate interface are nearly out of phase, thereby lowering their combined intensity and leaving more energy in the metal film. As a result we find that the 4WM intensity of a 20 nm Au film can be enhanced by a factor of 6 over a thick gold film.

We were not able to study gold films thinner than 20 nm because of inevitable gold island formation when thermally evaporating gold on glass. The islands lift the momentum conservation (Eq. (2)) and give rise to non-directional emission. Moreover, two-photon excited photoluminescence sets in [21], which adds a background to the 4WM signal. Furthermore, the damage threshold for thin metal films is significantly lower than for thick films, which requires the use of lower laser excitation intensities and leads to lower signal-to-noise.

## 4. Conclusions

Engineering the light in- and out-coupling by either using thin metallic layers or dielectric layers on top of metals can significantly increase the nonlinear response, thereby boosting the efficiency of frequency conversion by several orders of magnitudes. We find that the nonlinear enhancement is particularly strong for metal surfaces coated with thin dielectric layers having a high refractive index. The nonlinear enhancement can be increased further by suitably engineered dielectric-metallic multilayers or by lateral structuring. Our study revealed that dielectric films with non-negligible nonlinearities increases the nonlinear response further. A substantial improvement can be expected by replacing silver by gold and TiO_{2} by silicon or GaAs.

The enhancement of the nonlinear response can be readily understood in terms of Fresnel reflection / transmission coefficients accounting for all the interfaces in the system and by including the nonlinear contribution of the dielectric layer. The Fresnel coefficients enter at the frequencies of the excitation fields and at the frequency of the nonlinear signal. The concepts shown can readily be combined in multilayer structures consisting of alternating ultrathin metallic-dielectric layers and might open the possibility of on-chip frequency conversion in highly integrated devices of reduced dimensions or for the generation of higher harmonics.

## Acknowledgments

This research has been funded by La Fundacio CELLEX Barcelona and the National Science Foundation (grant ECCS-0918416). We thank J. Osmond and N. Sayols Baixeras for ellipsometric sample characterization.

## References and links

**1. **R. Boyd, *Nonlinear Optics* (Academic Press, San Diego, 2008), 3rd ed.

**2. **Y. R. Shen, *The Principles of Nonlinear Optics* (J. Wiley & Sons, New York, 1984).

**3. **T. Heinz, *Nonlinear Surface Electromagnetic Phenomena* (Elsevier, Amsterdam, 1991).

**4. **F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. **14**, 1029–1031 (1965). [CrossRef]

**5. **H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B **44**, 1220–1224 (1991). [CrossRef]

**6. **A. Leitner, “Second-harmonic generation in metal island films consisting of oriented silver particles of low symmetry,” Mol. Phys. **70**, 197 (1990). [CrossRef]

**7. **A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-Field Second Harmonic Generation Induced by Local Field Enhancement,” Phys. Rev. Lett. **90**, 013903 (2003). [CrossRef]

**8. **N. A. Papadogiannis, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. **166**, 133–139 (1999). [CrossRef]

**9. **B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and off-resonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. **83**, 4421–4424 (1999). [CrossRef]

**10. **H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO_{2} composite films on femtosecond and picosecond time scales,” Opt. Lett. **23**, 388–390 (1998). [CrossRef]

**11. **M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. **5**, 799–802 (2005). [CrossRef]

**12. **M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. **98**, 026104 (2007). [CrossRef]

**13. **N. K. Grady, M. W. Knight, R. Bardhan, and N. J. Halas, “Optically-driven collapse of a plasmonic nanogap self-monitored by optical frequency mixing,” Nano Lett. **10**, 1522–1528 (2010). [CrossRef]

**14. **H. Harutyunyan, S. Palomba, J. Renger, R. Quidant, and L. Novotny, “Nonlinear dark-field microscopy,” Nano Lett. **10**, 5076–5079 (2010). [CrossRef]

**15. **Y. Wang, C.-Y. Lin, A. Nikolaenko, V. Raghunathan, and E. O. Potma, “Four-wave mixing microscopy of nanostructures,” Adv. Opt. Photon. **3**, 1–52 (2011). [CrossRef]

**16. **C. Flytzanis, F. Hache, M. Klein, D. Ricard, and P. Roussignol, “1. Semiconductor and metal crystallites in dielectrics:,” in “*Nonlinear Optics in Composite Materials:*,” vol. 29 of *Progress in Optics*, E. Wolf, ed. (Elsevier, 1991), pp. 321–411.

**17. **J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. **104**, 046803 (2010). [CrossRef]

**18. **P. Genevet, J.-P. Tetienne, E. Gatzogiannis, R. Blanchard, M. Kats, M. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. **10**, 4880–4883 (2010). [CrossRef]

**19. **J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. **103**, 266802 (2009). [CrossRef]

**20. **N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. **128**, 606–622 (1962). [CrossRef]

**21. **M. R. Beversluis, A. Bouhelier, and L. Novotny, “Continuum generation from single gold nanostructures through near-field mediated intraband transitions,” Phys. Rev. B **68**, 115433 (2003). [CrossRef]

**22. **P. Ginzburg, A. Hayat, N. Berkovitch, and M. Orenstein, “Nonlocal ponderomotive nonlinearity in plasmonics,” Opt. Lett. **35**, 1551–1553 (2010). [CrossRef]