### Nonlinear Optics

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Surface plasmon-polariton induced nonlinear polarization in metals was first reported for second harmonic generation in , where by exciting the surface mode at a Ag-air interface enhanced the overall nonlinear polarization [ 78 ]. First, the SPP excitation at the pump wavelength results in an enhanced electric field near the interface, which can significantly increase the nonlinear polarization in the interfacial region [ 79 ].

Second, the surface waves can be launched at the second-harmonic wavelength under the right phase-matching conditions [ 79 ]. This harmonic surface waves can then constructively interfere with other interfacial second-harmonic radiation, leading to an enhanced overall SH emission into the far-field [ 80 ]. In addition, launching surface polaritons at the FWM frequency can also be expected to generate strong nonlinear optical interaction if the wave-vector matching condition is fulfilled at the interface. Different from the SHG process, which only occurs at the metal surface, the surface polaritons mediated FWM emission can extend over the penetration depth of surface modes into the materials, which enables a sensitive spectroscopy probe for the third-order bulk nonlinear susceptibility,.

In addition to the surface FWM waves, the intrinsic FWM originating from the metal bulk also contributes to the total emission, although the SPP-enhanced FWM can be dominant in proper experimental configurations [ 82 ]. Similar phenomena have also been observed for third-harmonic generation.

More specifically, it has been demonstrated in recent experiments [ 80 , 83 ] that launching SPP modes at the FWM frequency was achievable by carefully selecting the incidence angles of the pump beams, as per figure 2 a. In this experimental configuration, the incidence waves were not directly coupled to the surface waves and thus enabled free-space excitation of SPP modes at the metal surface. In particular, the FWM signal can be further increased by coupling surface FWM waves with a nanostructured metal surface which is capable of more efficiently transferring the surface waves into outgoing, free-space propagating waves, as indicated in figure 2 b [ 84 ].

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Finally, the same group also demonstrated that a metal surface with locally structured features can dramatically enhance FWM, which has consequently inspired the development of nonlinear dark-field microscopes as depicted in figure 2 c [ 85 ]. Figure 2. Nonlinear excitation of surface plasmon polaritons at metal surfaces. Reproduced with permission from [ 83 ] by permission of the Royal Society.

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Reprinted figure with permission from [ 84 ], Copyright by the American Physical Society. Reprinted with permission from [ 85 ]. Copyright American Chemical Society. Distinct from the SPPs propagating at metal surfaces, localized surface plasmons supported by metallic nanostructures are non-propagating localized modes consisting primarily of evanescent waves. By efficiently confining the incident light into the nanoscale volume near the nanostructure surface, the LSP excitation creates substantially enhanced near-field intensities, which lays the ground for strongly leveraging various nonlinear effects both in plasmonic metals themselves and active nonlinear materials placed in their close vicinity.

As perhaps the simplest model to understand the nonlinear response from plasmonic nanostructures, a metallic nanosphere has been extensively investigated to reveal the origins of the nonlinearly generated signal, as illustrated in figure 3 a. According to the symmetry selection rule, the second-order nonlinear response originates from both the surface and bulk contributions [ 86 ]. Within the electrostatic approximation, the bulk nonlinear polarization, which dominates the total nonlinear emission intensity such as THG of individual plasmonic nanoparticles [ 87 ], has generally a dipolar character.

The dipolar bulk second-order nonlinear response, however, is forbidden for most metals because of their structural inversion centrosymmetry. Thus, observable second-order nonlinear emission, like SHG in bulk metals, originates mainly from the nonlinear polarization at the metal surface where symmetry breaking occurs.

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Specifically, the total SH emission from a plasmonic nanoparticle can be viewed as the interference between all the nonlinear dipoles distributed on the particle surface. Figure 3. Second-harmonic generation in Ag nanospheres. For a perfectly spherical nanoparticle that is much smaller than the wavelength, the local nonlinear dipoles at the opposite sides of the sphere surface cancel out between each other, leading to negligible SH emission into the far-field.

Indeed, the hyper-Rayleigh scattering results pertaining to spherical metal nanoparticles show a weak dipolar SH response for very small particles, as shown in figure 3 b , but a quadrupolar response for relatively large particles see figures 3 c and d [ 88 ]. Close inspection reveals that the dipolar response for the small particles originates from geometry asymmetry associated with shape deviation from a perfect sphere [ 89 , 90 ], and that the retardation effect is responsible for the quadrupole SH emission observed in the larger particles [ 91 , 92 ].

For metal nanoparticles of other shapes and more complex geometries, the multipolar characters of their nonlinear response can be derived following the same approach as for the spherical nanoparticles and the signature of their mode characters can be more pronounced due to plasmonic amplification and modulation [ 93 , 94 ].

It is generally believed that the following two enhancement mechanisms are critical in typical nonlinear spectroscopy of plasmonic nanostructures. The first one is the plasmon induced excitation enhancement of the incident field at the fundamental frequency, which substantially strengthens the multi-waves coupling in the nonlinear conversion process.

The second mechanism is related to the plasmon-amplified nonlinear emission efficiency, which is realized by coupling the outgoing nonlinear signals to radiative LSP modes or manipulating the local interference between the nonlinear sources at the subwavelength scale.

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In the first mechanism, complex metallic nanostructures having near-field plasmon coupling are favorable candidates for boosting the nonlinear effects see figure 4. These coupled nanostructures having nanometer-sized gaps possess stronger field localization than individual nanoparticles and typically display a few orders of magnitude field enhancement within the gap region. Since the nonlinear polarization strength scales quadratically or cubically with the fundamental field intensity for second- and third-order nonlinear interactions, respectively, the plasmonic field intensity enhancement can result in at least tens of thousands times stronger nonlinear emission intensities [ 95 , 96 ].

Figure 4. Nonlinear optical effects in coupled plasmonic nanostructures. Reproduced with permission from [ 95 ]. Reprinted figure with permission from [ 96 ], Copyright by the American Physical Society. When the near-field coupling across the nanoscale inter-particle gap increases the nonlinear excitation rate, the diffractive long-range coupling occurring in metallic nanoparticle arrays can also modify the nonlinear response.

As indicated in figure 5 a , two samples with subtle differences in ordering are found to have SH responses differing by a factor of up to 50 [ 97 ]. In these arrays, the linear optical properties are predominantly determined by the long-range diffractive coupling between the elemental particles. Since the nonlinear properties are strongly dependent on the linear optical properties, the orientational distribution and detailed ordering of the nanoparticles modify the plasmon resonance characteristics and thus affect the nonlinear excitation efficiency.

Similar effects have also been demonstrated in periodically distributed nanodisk monomers and dimers [ 98 ].

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The coexistence of localized plasmon modes and far-field photonic modes creates a Fano-type electromagnetic coupling in such arrays, which is responsible for the enhanced second-harmonic generation, as per figure 5 b. Figure 5. Tuning the nonlinear response of plasmonic nanoparticle arrays with diffractive long-range mode coupling.

The green bars in the bottom give the SH intensity of the two arrays, respectively. Reprinted with permission from [ 97 ]. The asymmetrical excitation dependence of the enhanced SH intensity red dots manifests the coupling effect. Reprinted with permission from [ 98 ].

The upper panels show the SEM images of two arrays with G - and mirror- G shaped structures, and the polar plots at the bottom present the SHG intensity of the two arrays as a function of the quarter-wave plate orientation in the pump beam path. Reprinted with permission from [ 99 ].

Another important representative manifestation of the geometry-ordering influence on the nonlinear response of plasmonic nanostructures is the presence of circular dichroism CD in second-harmonic generation termed as SHG-CD from structures with geometry chirality [ 99 — ]. As depicted in figure 5 c , the chiral structure consisting of G -shaped gold elements exhibits a pronounced SHG-CD effect. The distinct SHG intensity patterns see the four micrographic images rendered with false color resolved by nonlinear microscopy with circularly polarized pump beams indicate that the CD effect originates from the enantiomerically sensitive plasmon modes.

When all the G -shaped nano-elements in the two arrays are replaced with the same elements the same geometry and orientation , the SHG-CD effect vanishes because of the identical plasmonic response of the reordered arrays. Inspired by the concept of spin—orbit coupling of light in linear optics, we have recently witnessed growing research interest in nonlinear geometrical phase control, which opens up new ways for manipulating the harmonic generations at the nanoscale [ ]. Upon excitation by a circularly polarized pump wave, the phase of the nonlinear polarization of a plasmonic nanostructure can be controlled by using the structure geometry and orientation-dependent spin—orbit coupling.

Figure 6. Illustration of the geometry-phase-controlled nonlinear harmonic responses of individual metal nanostructures in plasmonic metamaterials. Reprinted by permission from Springer Nature: Nature Materials [ ], A single metal nanostructure itself or a nonlinear medium in the close vicinity of the metal nanoparticle locally forms an effective nonlinear dipole moment expressed by:. Coordinate rotation can be employed to facilitate the analysis of orientation angle dependent nonlinear dipole moment of the nanostructure.

As depicted in figure 6 a , the local coordinate frame is fixed to the nanostructure. In the local frame, the n th-order harmonic nonlinear polarizability is simply , and thus the corresponding nonlinear dipole moment can be written as:. This nonlinear dipole moment can be decomposed into two in-plane rotating dipoles according to the circular polarization states as:.

After the coordinate transformation from the local frame to the laboratory frame, these two rotating dipole moments can be written in the following form:.

Obviously, a geometry phase equal to either or is introduced in the nonlinear polarizability of the n th-order harmonic generation from the nanostructure with the same or opposite circular polarization to that of the pump wave. However, a nanostructure with four-fold rotational symmetry symmetry group , like the nano-cross in figure 6 c , permits only third-harmonic signals with circular polarization state opposite to the incident polarization. Importantly, under circularly polarized illumination, the linear optical response of a single nanostructure with rotational symmetry is independent of its spatial orientation [ ].

Thus, by properly assembling of rotation-symmetrical nanostructures with spatially varied orientation in a 3D or 2D lattice, one can form a nonlinear metamaterial or metasurface which shows homogeneous linear optical properties but well-defined local nonlinear polarizability distribution [ , , ].

In particular, plasmonic nanostructure based nonlinear metasurfaces are the most suitable candidates for manipulating the local phase-controlled nonlinear response. This is because the resonant excitation of localized surface plasmons in the constituent nano-elements can substantially increase the pump efficiency that consequently leverages the generated harmonic signals from the nonlinear metasurface.

As indicated in figure 7 a , the measured diffractive TH intensity patterns for the orientation-varied nanorod symmetry array suggest that TH waves with the same circular polarization are mainly diffracted into the first-order grating angle while those with the opposite circular polarization are concentrated in the second-order diffraction angle. In comparison, the metasurface consisting of nano-crosses with symmetry diffracts TH waves with opposite circular polarization into the first-order grating angle.

Similarly, the efficient nonlinear responses for a -type metasurface, such as the triangle-hole-arrayed 2D metamaterial shown in figure 7 b [ ], are limited to the second-harmonic generation with opposite circular polarization to that of the pump wave.