nonlinear plasmonics - nanjing university review... · ated reflection, transmission or absorption...

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Nonlinear optical effects (Box 1) have an important role in modern photonic functionalities, including control over the frequency spectrum of laser light, generation of ultra-

short pulses, all-optical signal processing and ultrafast switching1. Optical nonlinearities are inherently weak, because they are gov-erned by photon–photon interactions enabled by materials. They are superlinearly dependent on the electromagnetic field and can be strengthened in material environments that provide mecha-nisms for field enhancement.

An increased effective nonlinear optical response can be achieved through plasmonic effects. Such effects arise from coher-ent oscillations of conduction electrons near the surface of noble-metal structures2,3 (Box 2). For extended metal surfaces, this gives rise to surface plasmon polaritons (SPPs), which are surface elec-tromagnetic waves propagating at the metal–dielectric interface. For metal nanoparticles, the response arises from localized sur-face plasmons (LSPs), whose resonances depend on the particle size and shape. New concepts, including SPP crystals2 and wave-guides4, nano-antennas5 and plasmonic metamaterials6, can be used to tailor the optical responses further through the resonances of the individual units and their mutual electromagnetic coupling.

Plasmonic excitations can boost nonlinear optical effects in sev-eral ways. First, the coupling of light to surface plasmons can result in strong local electromagnetic fields2,5, significantly enhancing optical processes. A prime example is surface-enhanced Raman scattering, where plasmonic excitations at rough or engineered metal surfaces can enhance the inherently weak Raman process by orders of magnitude, allowing even single-molecule detection7. In nonlinear optics, this naturally translates to higher effective non-linearities of the metal itself or the surrounding dielectric material.

Second, plasmonic excitations can be extremely sensitive to dielectric properties of the metal and the surrounding medium. This is the basis for label-free plasmonic sensors: minute modifi-cations of the refractive index near the metal surface result in sig-nificant modifications of the plasmonic resonance8. In nonlinear optics, this extraordinary sensitivity can be exploited to control light with light, using a control beam to induce a nonlinear change in the dielectric properties of one of the materials, thus modifying the plasmonic resonances and the propagation of a signal beam.

Finally, plasmonic excitations can respond on the timescale of a few femtoseconds, allowing ultrafast processing of optical signals3. Nonlinear effects in plasmonics could therefore lead to several interesting nanophotonic functionalities, and we are now seeing the emergence of metal nanostructures designed to favour specific nonlinear processes.

Nonlinear plasmonicsMartti Kauranen1 and Anatoly V. Zayats2

When light interacts with metal nanostructures, it can couple to free-electron excitations near the metal surface. The electro-magnetic resonances associated with these surface plasmons depend on the details of the nanostructure, opening up oppor-tunities for controlling light confinement on the nanoscale. The resulting strong electromagnetic fields allow weak nonlinear processes, which depend superlinearly on the local field, to be significantly enhanced. In addition to providing enhanced non-linear effects with ultrafast response times, plasmonic nanostructures allow nonlinear optical components to be scaled down in size. In this Review, we discuss the principles of nonlinear plasmonic effects and present an overview of their main applications, including frequency conversion, switching and modulation of optical signals, and soliton effects.

In this Review, we discuss the principles of nonlinear plasmon-ics and present an overview of its main areas of application. We will consider the role of LSPs and SPPs on smooth films, in wave-guides and on periodically structured surfaces as well as plasmonic metamaterials. We will focus on the most traditional nonlinear processes, such as harmonic generation, nonlinear propagation and optical switching, all of which are important for nanopho-tonic functionalities. We will therefore exclude processes such as multiphoton-induced luminescence, which is a tool widely used to study field localization in metal systems. Lasing and related phe-nomena, such as spasing, are also inherently nonlinear but are not discussed here (for a recent review, see ref. 9). The emerging area of strong-field science in plasmonic systems10 is also beyond the scope of the present article.

Basic concepts in nonlinear plasmonicsNonlinear optical effects arise when electronic motion in a strong electromagnetic field cannot be considered harmonic. Expansion of the anharmonicity as power series in the field strength1 (Box 1) mixes the incident fields and produces new fields that oscillate at the sums and differences of the incident frequencies and can prop-agate in various directions. For applications, the most important effects occur at second and third order. The second-order response typically gives rise to wave-mixing effects that lead to frequency conversion, the most common example being second-harmonic generation (SHG), where the incident frequency, ω, is converted to its second harmonic, 2ω. New frequencies arise also from third-order nonlinearities. More importantly, however, the third-order response contains terms at the incident frequencies. This is known as the optical Kerr effect and results in nonlinear modifications of the refractive index (Box 1, equation (2)), allowing all-optical switching and modulation of light. By combining Kerr nonlin-earities with optical cavities or other systems with feedback, it is possible to obtain bistability, where one input signal allows two possible outputs. For optical beams with finite transverse size, dif-fraction and nonlinear effects can balance each other, giving rise to optical solitons.

The plasmonic structures can enhance nonlinear effects in two main ways. First, such structures provide field enhancement near the metal–dielectric interface, associated with the excitation of either SPPs or LSPs2,3. This is characterized by the frequency-dependent local-field factor L(ω, r) = |Eloc(ω, r)/E0(ω)|, where r is the position vector, Eloc(ω, r) is the local field at frequency ω associ-ated with the plasmonic excitation, and E0(ω) is the incident field. The resulting strong fields enhance the nonlinear response locally

Department of Physics, Tampere University of Technology, PO Box 692, FI-33101 Tampere, Finland. Department of Physics, King’s College London, Strand, London WC2R 2LS, UK. e-mail: [email protected]; [email protected]

FOCUS | REVIEW ARTICLESPUBLISHED ONLINE: 31 OCTOBER 2012 | DOI: 10.1038/NPHOTON.2012.244

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and, when carefully designed, even when the response is inte-grated over the whole sample. This concept is equally applicable to increasing the nonlinearity of the metal itself (intrinsic response) or of a material adjacent to the metal (extrinsic response). Second, the plasmonic excitation parameters, including the SPP wavevec-tor and the LSP or metamaterial resonance frequency, are very sensitive to the refractive indices of the metal and the surrounding dielectric. A nonlinear change in the index of either material can therefore significantly modify plasmonic resonances and associ-ated reflection, transmission or absorption of light.

Nonlinear effects are constrained by material symmetry1. Whereas all materials support third-order effects, second-order effects are possible only in non-centrosymmetric materials within

the electric-dipole approximation of the light–matter interaction. Although magnetic and quadrupole effects allow second-order effects even in centrosymmetric materials (Box 1), such effects are difficult to achieve. Centrosymmetry is therefore expected to be detrimental to second-order response even in the presence of local-field enhancement.

Intrinsic (ohmic) losses are always present in plasmonic sys-tems. They limit both the SPP propagation distance and the achiev-able local-field factors. For propagating waves, the nonlinear effect on the field amplitude scales linearly with the interaction length, typically up to 1 cm for phase-matched wave mixing (Box 1) in bulk materials and waveguides. For typical loss-limited field-enhancement factors of 10 for all interacting fields, the interaction

The response of materials to an optical field E is described by the material polarization P:

(1)P = ε0 χ(1) E + χ(2) E2 + χ(3) E3 + ∙∙∙

Here, ε0 is the vacuum permittivity and χ(s) is the sth-order sus-ceptibility of the material1. In general, the field E consists of time-harmonic at several frequencies, ωn. For moderate light intensities, only the first term in equation (1) is important, and each compo-nent gives rise to polarization oscillating at the corresponding frequency. Such a linear response describes conventional optical effects, such as refraction, absorption and scattering.

For strong fields, the higher-order (s>1) terms in equa-tion (1) need to be considered. They contain sums and differ-ences of the incident light frequencies and give rise to radiation at these new frequencies. The way the various positive and nega-tive frequencies combine is often described in terms of photon diagrams (Fig. B1). The most important effects arise from sec-ond- and third-order interactions. In second order, we obtain SHG, sum- and difference-frequency generation, and an elec-tro-optic response. Even more possibilities exist in third order, with third-harmonic generation and FWM as specific examples. Importantly, the third-order response contains oscillations also

at the original frequencies. This is known as the Kerr effect, where the permittivity and, thus, refractive index depend on intensity

(2)ε = ε0 χ(1) + 3χ(3)|E| 2 or

n = n0 + n2I

Here, ε is the permittivity of the material, n0 is its linear refractive index, n2 is the nonlinear index and I is the field intensity. In self-modulation, a beam modifies the index it experiences, leading to self-focusing or defocusing of beams of finite transverse size. In cross-modulation, a control beam modifies the index at the signal beam frequency, allowing all-optical modulation and switching.

A more complete description takes into account the vectorial character of the electric field and polarization. The susceptibilities are therefore tensorial quantities, which depend on material sym-metry. Third-order processes can occur in all materials. Second-order processes, however, are forbidden in centrosymmetric materials within the dipolar approximation, and a response can occur only at surfaces where centrosymmetry is broken. Beyond this, second-order effects can occur even in the bulk of centrosym-metric materials when higher-multipole (magnetic-dipole and electric-quadrupole) interactions are considered11,12.

The nonlinear response leads to strong, coherent signals only when individual nonlinear sources add up in phase. Such phase matching is important for frequency conversion and samples much larger than the wavelength. Otherwise, the nonlinear effects grow only over the coherence length of the interaction, typically on the order of 10 μm. For subwavelength nonlinear sources, phase-matching considerations are not important and the nonlin-ear signals can be emitted in all directions. The same is true for random ensembles of subwavelength sources, which lead to inco-herent signals emitted in all directions (hyper-scattering).

The basic formalism needs to be corrected by the fact that the opti-cal field experienced by a given dipole in the material is not the same as the macroscopic field. The local field at frequency ωn and point r is123

(3)Eloc, i(ω,rn) = Lij(ω,rn) Ej(ωn) Σj

where i and j refer to Cartesian components of the field. For clarity, the frequencies are shown as arguments of the fields. The local-field factor, Lij(ωn, r), is tensorial, as the field direction may also change through local-field effects. Such effects are particularly important for nanostructured materials, which can strongly mod-ify the local-field distribution, for example when a surface plas-mon is excited.

Box 1 | Nonlinear optics

a b

c

ω

ω

ω

ω

ω

ω

ω2

ω1

ω1

ω3

2ω

Figure B1 | Photon diagrams for common nonlinear processes. The solid horizontal lines correspond to real quantum-mechanical states of the material system, whereas the dashed lines are so-called virtual states, where the system resides only instantaneously. The thick arrows correspond to the input fields that can drive the material system up or down in energy as indicated. The thin downward arrow corresponds to the generated field, which returns the material to the initial state. a, SHG; b, degenerate FWM; c, coherent anti-Stokes Raman scattering (CARS). Degenerate FWM gives rise to an intensity-dependent refractive index, and CARS forms a basis for molecular vibrational spectroscopy.

REVIEW ARTICLES | FOCUS NATURE PHOTONICS DOI: 10.1038/NPHOTON.2012.244



length can thus be reduced by a factor of at least 1,000 (second-order process with three fields) to achieve an equivalent nonlin-ear response. The resulting lengths of 10 μm or less are shorter

than SPP propagation distance at most spectral ranges and are comparable to typical coherence lengths of photonic non-phase-matched interactions. For higher-order processes, the interaction

The optical properties of metal films and nanostructures are gov-erned by the coupling of the incident electromagnetic field to the coherent motion of free-electron plasma near the metal surface. Such systems can be classified on the basis of which type of plas-monic excitations they support2,3.

A surface plasmon polariton (SPP) is a propagating surface wave at the continuous metal–dielectric interface (Fig. B2). The electro-magnetic field decays exponentially on both sides of the interface, providing subwavelength confinement near the metal surface. On two-dimensional planar interfaces, an SPP is a longitudinal wave with electric field components both perpendicular to the metal inter-face and parallel to the wavevector. Light therefore needs to have an electric field component in the plane of incidence for SPP excitation.

The SPP wavevector is larger than the wavevector of light propa-gating in the adjacent dielectric medium. Coupling of light from free space to SPPs, in analogy with any type of guided photonic modes, therefore requires special arrangements such as a prism or diffrac-tion grating coupler. Owing to its dispersion, an SPP is a slow wave accumulating energy from the incoming light and providing field enhancement near the metal interface. As a result of ohmic losses in metal, SPPs have a finite propagation distance that depends on the geometry of the supporting structure.

The SPP dispersion and, therefore, the field confinement and enhancement, can be modified by structuring the interface (either the metal or the dielectric medium). By doing so, it is possible to make plasmonic waveguides2,4 and plasmonic crystals2 (periodi-cally structured plasmonic surfaces or films). In the latter case, one-dimensional (slits, grooves or ribs) or two-dimensional (holes, protrusions or nanoparticle arrays) nanostructures on the surface

or in the metal film, with period comparable to the SPP wavelength, are used. The SPP dispersion in plasmonic crystals exhibits a struc-ture of allowed and forbidden bands, similar to photon dispersion in two-dimensional photonic crystals.

Localized surface plasmons (LSPs) are associated with the elec-tron plasma oscillations in confined (subwavelength) geometries, for example metal nanoparticles. LSP resonances depend on the particle size, shape and refractive index of their surroundings. LSPs can be resonantly excited with light of the appropriate frequency and polarization irrespective of the excitation light wavevector. Because its field is confined near the nanoparticle, an LSP has a small mode volume and therefore provides significant electromag-netic field enhancement, which is limited by ohmic and radiative losses3 as well as quantum121 and nonlocal122 effects in the case of ultrasmall sizes.

LSPs can also be supported by subwavelength features on a metal surface. LSP resonances therefore have a considerable role in the behaviour of SPPs on rough surfaces if their frequency is close to the SPP frequency. The spectrum of LSPs associated with an ensem-ble of metallic particles (or voids) is determined by the interaction between the individual LSP resonances. The resulting spectrum and the field enhancement depend strongly on the shape and size of the individual particles and the distance between them. In such ensem-bles of interacting small metallic particles, a very strong enhance-ment of the electromagnetic field can be observed.

Recently plasmonic antennas based on pairs of, or several, plasmonic particles have been developed to control near-field-to-far-field light transformations5. Furthermore, sharp metal tips can produce strong local electromagnetic fields through the ‘lightning-rod’ effect2, even when LSPs are not resonantly excited. In all cases, a source of radiation localized to a subwavelength LSP mode can emit a signal in essentially all directions, independently of the direction of propagation of the incident field.

Plasmonic metamaterials consist of ensembles of electromag-netically coupled, subwavelength metal nanostructures6. Their optical properties can often be described by effective medium parameters. Typical metamaterials can be achieved with SRRs of different geometries, fishnets or nanoparticles. The optical response of plasmonic metamaterials can be designed by tailoring the response of individual units as well as the electromagnetic coupling between them. Of particular interest are properties that naturally occurring materials do not have. For example, certain resonances of plasmonic metamaterials can be interpreted to have a magnetic character even when the constituent materials are non-magnetic. Some metamaterials can be designed to exhibit ‘hyperbolic’ dis-persion, when the diagonal components of the permittivity tensor have opposite signs. In addition, the permittivity can be tailored to be very small at certain wavelengths giving rise to ‘epsilon-near-zero’ metamaterials104,105.

Universal properties of all types of plasmonic nanostructure are field enhancement near the metal surface, compared with the free-space field of the incident light; strong sensitivity to refractive index changes near the metal surface; and the possibility of engineering the dispersion and resonances by controlling the nanostructure geometry and dielectric surroundings. The resonances can thus be tuned to the operation wavelength where the nonlinear response needs to be enhanced.

Box 2 | Plasmonics

θSP

a

c d

b

Figure B2 | Examples of plasmonic structures. a, schematic of SPP on a planar metal film excited through a prism b, Electric field distribution of the SPP mode in a dielectric-loaded plasmonic waveguide-splitter. c, Dipolar LSP mode intensity distribution for nanorod-shaped particle. Brighter colour corresponds to higher intensity. d, SEM image of a plasmonic metamaterial consisting of an array of SRRs (from ref. 103).

FOCUS | REVIEW ARTICLESNATURE PHOTONICS DOI: 10.1038/NPHOTON.2012.244



length can be reduced even more. For both wave mixing at the nanoscale and Kerr-type processes involving LSP resonances, no phase matching is needed (Box 1), and the effect of losses is in lim-iting achievable local-field factors. Nevertheless, compared with other structures, the ratio of the resonance quality to the mode volume in plasmonic nanostructures is unsurpassed3. In all cases, however, to utilize the field enhancement at all frequencies, the structures must be designed with great care for good overlap of plasmonic modes at all frequencies.

To achieve optical signal modulation in plasmonic structures, either the plasmonic resonance can be shifted using control light (induced refractive index change) or absorption at the plasmonic resonance can be increased or decreased. Plasmonic nanostruc-tures can be beneficial here owing to the high sensitivity of their resonances to refractive index changes and because they localize the interaction volume below the diffraction limit. The small mode volume of plasmonic excitations provides the advantage for a low power requirement despite being lossy. Intrinsic metal nonlineari-ties also provide very high switching rates, limited only by the elec-tron relaxation time in metal: the response times range from tens of femtoseconds to a few picoseconds, depending on the electron plasma relaxation processes involved.

The nonlinear response can be described theoretically in terms of real and virtual electronic transitions in a material1. In metals, intraband transitions of conduction electrons—the crucial differ-ence from non-plasmonic materials—are essential, but interband transitions from the valence band to the conduction band can also be important. The conduction electrons can be treated as an iso-tropic electron gas in the hydrodynamic model11, an extension of the Drude model, where the electron velocity, v, obeys

(1)= − enE − v × B − p∆enc

дvдt

vτmn + (v• )v +∆

Here m is the electron effective mass, n the density and e is the elec-tron charge. The electric and magnetic components of the electro-magnetic field are denoted E and B, p is the pressure of the electron gas and c denotes the speed of light. On the left-hand side of equa-tion (1), the second term represents convection and the third term represents damping. The terms on the right-hand side are the electric and magnetic parts of the Lorentz force and the pressure gradient, respectively. Damping was neglected in the original hydrodynamic model11 but has proved to be important in more recent treatments of the nonlinear response12, especially for Kerr-type nonlinearity13.

The various nonlinear effects can be described by expanding all quantities to different orders of the interaction. Interestingly, the convective and magnetic terms give rise to a second-order response of higher-multipolar character even in the bulk of the electron gas12. The surface nonlinearities can be treated in the selvedge model11, where the electron density changes in a thin transition layer between the two bulk media.

The Kerr-type nonlinearity of metals is usually described by changes in the distribution of conduction-band electrons by the excitation light13, which can arise from the heating of the conduc-tion-band electrons and from the interband excitation of electrons from the d band to the conduction band. The change of the elec-tron distribution modifies both the real and the imaginary parts of the permittivity. The intraband part of the nonlinear response is modelled with the Drude-like response with modified param-eters13. The interband contribution is described within the random phase approximation14. Additionally, conservative ponderomo-tive potentials may also lead to third-order nonlinearities of the electron gas15.

Wave mixing in nanoplasmonic structuresSurface-enhanced nonlinear effects. Surface-enhanced SHG was first studied using electrochemically roughened silver sur-faces16. The SHG signal was found to be diffuse, but increased by four orders of magnitude relative to a flat reference surface when integrated over all detection directions. The results suggested that the signals are incoherent and enhanced by LSP resonances of the nanoscale surface features. Further support for the role of LSP resonances was obtained from SHG experiments on metal-island films and lithographic nanostructures17.

More-recent work has addressed surface-enhanced SHG from a number of viewpoints. Near-field SHG measurements have been performed using near-field detection18 and excitation19, the latter also correlated with surface topography. Far-field SHG micros-copy has emphasized the spatial overlap of the local fundamen-tal and SHG modes20. These and additional studies show that the SHG signals depend on the polarization of the fundamental field even if the light is focused down to the achievable resolution of a few hundreds of nanometres. The SHG emission, however, is inco-herent and depolarized21, theoretically down to the scale of a few nanometres22. Furthermore, semicontinuous metal-island films near the percolation threshold support LSP resonances over broad spectral ranges23.

LSP resonances can also enhance nonlinear processes other than SHG. In particular, surface-enhanced third-harmonic gen-eration has also been observed and its different symmetry rules, relative to those of SHG, discussed24.

Enhancement of SHG by propagating SPPs was also demon-strated early on25, and SPPs have been used to enhance four-wave mixing26 (FWM). Very recently, the roles of photonic and plas-monic modes in SHG from metal films have been addressed by momentum-space spectroscopy27.

Nonlinear scattering from individual nano-objects. The first formal electromagnetic description of incoherent second-har-monic scattering (hyper-Rayleigh scattering (HRS)) from small spherical particles was based on Mie theory but included only the local surface response28. This formalism and its extension to the local bulk response29 provided selection rules for signals arising from the various multipolar Mie terms at the fundamental and second-harmonic wavelengths.

Experiments on HRS30 from 13-nm gold particles yielded a very high hyperpolarizability of 2,000–3000 × 10−30 e.s.u. per atom, greatly exceeding that of the best molecular materials. Interference between dipolar and quadrupolar HRS was subsequently observed using 32-nm silver particles31. More detailed studies using parti-cles ranging in size from 20 to 150 nm have shown that small, non-spherical particles have a dipolar response arising from their lack of centrosymmetry32. For large particles, field retardation across the particle enhances the quadrupole interaction, allowing HRS even from spherical particles. Most recently, this interplay between multipoles has also been extended to octupoles33. Since these ensemble experiments, HRS from individual gold nanopar-ticles has been observed34.

The various multipoles are important in HRS because of their different resonance characteristics. Quadrupole resonances, for example, are weaker but have narrower linewidths than the usual dipole resonances. Unfortunately, the quadrupole resonance often occurs at the spectral onset of the dipole resonance and hence can-not be easily resolved. In HRS, however, the selection rules can be used to isolate the quadrupolar signal. This opens a possibility for nonlinear sensing of the refractive index of the surroundings with properties superior to those of traditional approaches35.

Non-spherical plasmonic objects have also been investigated. SHG from the sharp tip of a gold nanocone was strongest for




incoming light polarized along the tip axis36, as expected, as a result of the lowered symmetry relative to the sphere37. A rela-tively advanced structure for SHG consisted of a nano-aperture in a silver film surrounded by a concentric grating38 (Fig. 1a). The grating directed the incident light towards the aperture, whose SHG response was enhanced by four orders of magnitude com-pared with the aperture in the absence of the grating. The role of the nanogap between two nanospheres was investigated by FWM39. The signal increased as the gap size decreased to the ång-ström regime, where quantum effects start limiting the local-field enhancement40. The geometry of nanoantennas (one example shown in Fig. 1b) has also been shown to affect radiative damping of their LSP resonances, playing thereby a crucial role in efficiency third-harmonic generation41.

Recent work has utilized three-dimensional nanostructures. Non-centrosymmetric gold nanocups were proposed for SHG enhancement (Fig. 1c) and shown to be as efficient as traditional nonlinear materials42. The nonlinearity of an organic polymer has been combined with a plasmonic gap to allow tuning of SHG by voltage43 (Fig. 1d). SHG from non-centrosymmetric barium titanate nanoparticles has been enhanced by a factor of more than 500 by coating the particle with a gold shell44 (Fig. 1e), with possible applications as nonlinear labels in biological imaging. Furthermore, a tapered, hollow, plasmonic waveguide channelling energy to the narrow end yielded high-harmonic generation from xenon up to the forty-third harmonic45 (Fig. 1f).

Structured plasmonic surfaces for enhanced nonlinear effects. The first extended plasmonic structure designed for SHG was a metal grating that enhanced the local field at the SHG wavelength and gave rise to SHG emission into the first diffraction order17. A comprehensive theory of nonlinear metal gratings was subse-quently formulated46 and experimentally tested47.

The first example of a surface designed to be non-centrosym-metric consisted of an array of L-shaped nanoparticles48, fabri-cated to determine the femtosecond plasmon dephasing time by SHG autocorrelation measurements. Similar arrays of L-shaped gold particles were then investigated for their SHG properties (Fig. 2a). The efficiency was found to depend strongly on particle ordering in the array49, with the strongest signals when the funda-mental wavelength was at plasmonic resonance of the structure. It soon became evident, however, that the expected symmetry rules of the linear and nonlinear responses of similar samples are not obeyed50.

The split-ring resonators51 (SRR; Fig. 2b) have similar symme-try to the L-shaped nanoparticles. The plasmonic resonances of SRRs can be interpreted to have electric or magnetic character, although the constituent materials themselves are non-magnetic6. The initial measurements on arrays of SRRs suggested that the magnetic resonances are favourable for SHG52. However, no signif-icant difference between SHG from SRRs and that from comple-mentary SRRs, which consist of holes in a metal film, was found53. Because the roles of the electric and magnetic resonances of SRRs and complementary SRRs are reversed, the importance of mag-netic resonances in nonlinear responses has not been unambigu-ously confirmed.

SHG has also been discussed for centrosymmetric samples, including arrays of nanoparticles54 and nano-apertures55,56. In all cases, the strongest SHG was found when at least the fundamen-tal or SHG beam propagated at an oblique angle of incidence. An exception is the work on rectangular apertures at normal inci-dence57. There the origin of SHG was assigned to the focused beams that give rise to longitudinal field components, which results in coupling with the surface nonlinearity. The SHG effi-ciency correlated well with the cut-off of a waveguide mode within individual apertures.

GlassSilicaGold

Silica

EUVoutput

Tapered silverwaveguide

NIR input

EnhancedNIR field

x

zEFISH signal, I2 (t)

V(t) Fundamentalwave, I

O2–

Ba2+

Ti4+

+

–

+

–

+

–

p+

–

ω E0

2

r1r2

ts

d

a d f

e

b

c

ε 3

ε 1

ω

(2), (3)χ χ

2ω

2ω 2ω

ω

ω

Figure 1 | Examples of individual metal nanostructures for enhancing nonlinear effects. a, Nano-aperture in a silver film, surrounded by a circular grating for enhanced SHG (adapted from ref. 38). b, Gold bowtie antenna for third-harmonic generation (adapted from ref. 41). c, Non-centrosymmetric gold nanocup for SHG (adapted from ref. 42). d, Plasmonic grating and nanoslit filled with nonlinear polymer for electric-field-induced SH (EFISH) generation (adapted from Ref. 43). e, Barium titanate nanoparticle of radius r1 and nonlinear coefficient tensor of d coated by gold shell of thickness ts and outer radius of r2, which enhances the nonlinear polarization p for SHG (adapted from Ref. 44). f, Tapered plasmonic silver waveguide that focuses near-infrared (NIR) light to the tip for enhanced high-harmonic generation into the extreme-ultraviolet (EUV) regime (adapted from ref. 45).




Detailed inspection of refs 50, 52 and 54–57, however, shows that not L shapes, SRRs nor more symmetric shapes satisfied the symmetry rules of SHG. The results were thus modified by the extreme sensitivity of SHG to symmetry breaking, arising from the deviation of the shape from design or surface defects. The role of defects can also be interpreted in terms of multipole interference, which affects the directional properties of SHG58. This problem was resolved only recently when samples of much higher quality yielded SHG from the desired dipolar response59.

An attempt was also made to utilize nanodimers for SHG. Its dependence on the gap size of T-shaped nanodimers, however, was highly non-trivial60. In contrast to FWM from nanodimers39, which is not constrained by symmetry, the results for SHG were explained by the requirement that the strong local fields need to be asymmetrically distributed around the dimer.

The role of deliberate symmetry lowering in SHG has also been investigated. Modifying the arrangement of chiral G-shaped gold particles in an array (Fig. 2c) was shown to switch on or off the chi-ral nonlinearity61. There the particle size exceeded one micrometre, such that the arrays were well in the diffractive regime. A benefit of the large scale structures is that the hot spots within individual particles could be resolved by SHG microscopy. SHG microscopy has also been used to address magnetic domain ordering, which also modifies symmetry rules, in metal nanostructures62.

Nanostructured plasmonic surfaces have also been used to enhance the response of known nonlinear materials. One interest-ing sample consisted of arrays of coaxial holes in a 70-nm-thick gold film on a gallium arsenide (GaAs) substrate63. The structures exhibited the SHG response near the cut-off of a coaxial wave-guide mode. GaAs has also been used with SRR arrays64: samples were designed so that certain SHG signals were forbidden for both GaAs and SRRs alone but the combined structure gave rise to a

strong signal. Furthermore, arrays of silver nanoparticles have been used with a self-assembled film of organic molecules65, yield-ing an SHG enhancement factor of 1,600.

Very recent work has demonstrated more advanced plasmonic surfaces with enhanced nonlinear properties. The apparent cen-trosymmetry of dielectric gratings was broken for SHG by metal deposition at an oblique angle66. FWM from a gold grating was enhanced by a factor of 2,000 relative to FWM from a flat film, by tailoring the individual grating grooves to support cavity modes, which were further coupled by surface waves67 (Fig. 2d). Similarly, an array of spherical nanocavities in a gold film is beneficial for CARS spectroscopy68 (Fig. 2e). Arrays of bowtie antennas were used for high-harmonic generation up to the seventeenth har-monic69 and for enhancing several other nonlinear processes when the bowtie and array resonances were coupled70. Finally, SHG from arrays of L-shaped particles was tailored by subtle details of particle ordering (Fig. 2f), resulting in a 50-fold difference in SHG signals expected to be equivalent71.

Controlling light with light in plasmonic nanostructuresLSP and SPP nonlinear effects. The use of plasmonic structures for all-optical modulation or switching, or to achieve optically tun-able photonic properties, relies on enhancing Kerr-type nonlin-earities (Box 1, equation (2)) and detecting the modified refractive index of either the metal itself or the adjacent dielectric. Examples of such plasmon-enhanced nonlinear materials are metal nano-particles72,73 and bulk materials doped with such particles74–76. The excitation of such composites at the wavelength of the nanoparti-cle LSP leads to an increase in the effective nonlinear susceptibility compared with both materials separately77,78 (Fig. 3a). Similarly, the coverage of metal nanoparticle arrays, for example nanosphere or nanorod assemblies, with nonlinear material enhances the

Auw

E1

1ω 2ω

E2

d

xz

a b c

d e f

4WMω

Λg

Figure 2 | Examples of structured metal surfaces for nonlinear plasmonics. a–c, Some of the basic cases investigated for SHG: L-shaped gold particles (a; adapted from ref. 59); gold SRRs (b; adapted from ref. 51); G-shaped chiral particles ordered in different ways (c; adapted from ref. 61). d–f, Recent, more advanced, structures for enhanced nonlinear effects: Gold grating for enhanced FWM. The blue part represents the incident fields E1 and E2 at the respective frequencies ω1 and ω2 as well as the generated FWM signal at frequency ω4WM. The yellow part is the gold grating with critical dimensions indicated (d; adapted from ref. 67); gold nanocavities for enhanced CARS (e; adapted from ref. 68); a change in the ordering of L-shaped particles can suppress or enhance SHG compared with the case shown in a (f; adapted from ref. 71).




nonlinear response of the dielectric while also lowering the light intensities required for nonlinear action79,80. Specifically designed plasmonic nano-antenna resonances (Fig. 3c) have been con-trolled using the indium tin oxide free-carrier nonlinearity in the picosecond regime81.

The field enhancement associated with LSP modes has also allowed controlled photon tunnelling through nanoscale pinholes in metal films covered with a nonlinear polymer82,83. In particular, measurements of photon tunnelling through individual, naturally occurring, nanometre-scale pinholes have provided evidence of ‘photon blockade’, similar to Coulomb blockade in single-electron tunnelling82. Observations of photon tunnelling gated by light at a different wavelength have also been reported with somewhat larger pinholes83.

Propagating SPPs also can be used as the signal carrier. This was first shown with smooth metal films in the Kretschmann geometry, using thermal liquid-crystal phase transformations induced by light. Both switching of the reflected light and bistabil-ity were observed with the increase of the intensity of the incident light84. The latter was explained by positive feedback due to the

intensity-dependent refractive index of the liquid crystal illumi-nated by the evanescent field of the SPP wave.

All-optical switching and modulation in plasmonic wave-guides. To modulate and switch SPP signals by optical means in a waveguiding geometry, changes can be induced in the real or imaginary part of the permittivity by control light, and these changes can occur in the metal or the adjacent dielectric (Fig. 3d). Light-induced absorption modulation has been explored with the intrinsic metal (Au and Ga) nonlinearity85–87 as well as the die-lectric component in a plasmonic waveguide88,89. An integrated geometry has also been proposed, with signal SPP absorption modulation and stimulated emission controlled by an SPP beam at a different frequency90.

The intrinsic nonlinearity of gold is related to the interband excitation of non-equilibrium electrons and results in small changes in the real part of the permittivity but in more consider-able changes in the imaginary part. The associated relaxation time is on the picosecond scale72,73,85, and terahertz-rate direct modula-tion of SPP signals in plasmonic gold waveguides is thus possible.

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Figure 3 | Plasmonic systems for enhancing nonlinear optical Kerr effect. a, Controlling nonlinearity with metal nanoparticles: real (g’2 ) and imaginary (g’’2) parts of the nonlinear optical (NLO) refractive index enhancement factor with 10 nm diameter silver par-ticles embedded in carbon disulfide (from ref. 78) b, 500 × 440 nm2 nonlinear waveguide-ring resonator: the signal (red) and control (green) light are shown. Incorporated in the sketch is the simulated SPP field distribution 10 nm above the metal interface (adapted from ref. 92). c, Nonlinear gold nanoantenna with 20 nm gap embedded in high-conductivity indium tin oxide (ITO). Control (pump) changes the scattering efficiency of the antenna for probe light by modifying electron concentration near the Au/ITO interface (adapted from ref. 81). d, Direct modulation of the SPP signal: the signal beam is coupled to and from the waveguide by gratings at an aluminium–silica interface and modulated by the control pulses. Both the change in the SPP coupling efficiency and the induced absorption contribute to the modulation (adapted from ref. 91).




Similarly, the relaxation time of the interband transitions in alu-minium is much shorter and can provide even faster modulation speeds in aluminium-based plasmonic waveguides91.

The variation of the real part of the metal or dielectric per-mittivity will influence the propagation of SPP modes via modi-fication of their phase velocity and, thus, the phase of the wave. To convert this to intensity modulation, various phase-sensitive configurations, including Mach–Zehnder interferometers and waveguide-ring resonators, are needed. Both these configurations can be realized using various types of plasmonic waveguides, such as metal–insulator–metal, V-groove or dielectric-loaded wave-guides4. A dielectric refractive index variation of 0.001 can provide a 50% change in the transmission of a nonlinear, polymer-loaded, plasmonic waveguide-ring resonator (Fig. 3b), which needs only a 2.5-μm ring radius for operation at telecommunications wave-lengths92. The figures of merit (the ratio of modulation perfor-mance to size) of plasmon-based resonant nonlinear components are intrinsically better than those of photonic ones owing to the stronger changes in the effective refractive index of the mode, which is facilitated by the mode field distribution and the field-enhancement effects93.

Nonlinear plasmonic crystals: switching and bistability. Periodically structured metal films and surfaces, the so-called plas-monic or surface plasmon polaritonic crystals (SPPCs), provide considerable versatility for tailoring the wavelength-dependent plasmonic resonances and the electromagnetic field enhancement

in devices a few tens of wavelengths in size2,94. SPPCs have opti-cal properties analogous to those of two-dimensional photonic crystals. The spectral positions of their resonances, their disper-sion, and the phase and group velocities of SPP Bloch modes can be engineered by controlling the size and shape of the unit cell and its period. In particular, relatively flat SPP modes with low group velocity are possible, resulting in the high field enhance-ment beneficial for high effective nonlinearity. Moreover, with flat SPP bands, it is much easier to observe the shift of the resonance when nonlinear effects are induced, and the modulation contrast is higher. All the optical properties of plasmonic crystals, such as reflection, absorption and transmission, are determined to a large extent by the SPP Bloch modes, which can be easily excited, with-out any special arrangements, as a result of the periodic structure. All-optical control of the Bloch SPP mode structure by modifi-cation of the refractive index of the metal or adjacent dielectric therefore provides a very useful way of changing the reflection or transmission of signal light through SPPCs. Coupling of both con-trol and signal light to SPP Bloch modes further helps to enhance the nonlinear interaction94 (Fig. 4a).

These effects have been demonstrated using the nonlinearity of gold gratings to control SPP excitation at speeds determined by the subpicosecond response time of the gold nonlinearity95. Hybridization of SPPCs with nonlinear polymers and use of the strong electromagnetic field enhancement in cylindrical channels allowed efficient modulation of light transmission through the crystals by controlling the Bloch mode resonances96.

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Figure 4 | Controlling light with light in plasmonic nanostructures. a, Typical geometry of pump–probe experiments: signal (red) and control (green) light are coupled to plasmonic excitations and the effective nonlinearity is enhanced as a result (from ref. 94). b, Differential transmission spectra of a hybrid plasmonic crystal (a periodically structured gold film covered with 3BCMU polymer) for control light of wavelengths 488 nm and 514 nm. Depending on the nature of the plasmonic resonances, either more or less nonlinear transmission is observed (from ref. 97). c, Nonlinearity in a SRR-based metamaterial: the spectra of the effective two-photon absorption coefficient in the metamaterial (β̃ , blue and red) and an unstructured gold reference film (β, green). The inset shows a map of the electric field magnitude simulated 10 nm below the gold surface at the resonant wavelength (adapted from Ref. 103).” d, The spectra of the transient nonlinear changes of optical density, recorded at different time delay τ after the excitation, of the gold nanorod metamaterial shown in the inset. The red and blue colour scales correspond to positive and negative changes in optical density, respectively (adapted from Ref. 105). e, SPP soliton. Magnetic field distribution of the SPP beam at the metal–dielectric interface for linear diffraction at low power (left) and for a soliton generated with the increase of the incident power (right) (adapted from ref. 114).




Investigations of the spectral response of the SPPC switching in pump–probe configuration has shown that differential trans-mission of up to 60% can be achieved in some of the resonances with both increased and reduced transmission possible (Fig. 4b), depending on the origin of the plasmonic resonances, which can be related to either pure SPP Bloch mode or can involve interplay between Bloch modes and LSPs of the holes forming the crystal97. The observed SPP bandgap shift suggests that the control-light-induced changes in the average effective refractive index of the polymer are 10−3–10−4 and that local changes can be higher94. The spatial distribution of the control field intensity on the metal sur-face is also important. Owing to the feedback loop between the nonlinear changes in the refractive index of the polymer as the control intensity is varied, it is possible to observe bistable behav-iour in the transmission of signal light through the structure97. The self-action of light of varying intensity has also been observed in plasmonic structures hybridized with liquid-crystal mol-ecules98, showing good intensity-limiting properties of the plas-monic device. The response times of the above nonlinear effects in hybrid dielectric–metal systems are determined by the relaxation time of both the metal and the dielectric and is usually limited by the latter.

Full numerical simulations of nonlinear SPPCs in various geometries have been performed99,100. In particular, they have been modelled for one-dimensional structures (slit arrays) in relatively thin metal films, with nonlinear Kerr materials in the slits and on top of the metallic structures. In both cases, nonlinear trans-mission and its bistability were predicted in full agreement with experiments. Earlier modelling of gratings in a thick, perfect metal with the slits filled with Kerr-type nonlinear dielectric also pre-dicted bistability owing to waveguide modes in the slits101.

Nonlinear plasmonic metamaterials. Plasmonic metamaterials provide additional possibilities for all-optical switching, because their optical properties can be tailored through the plasmonic resonances of the constituents, for example SRRs or nanorods, and the electromagnetic coupling between them6. A change in the refractive index of the embedding dielectric or substrate there-fore leads to the modification of both individual plasmonic reso-nances and their interaction, resulting in an enhanced nonlinear response.

These properties have been demonstrated in nanorod-based metamaterials where the spectral position of the plasmonic mode was controlled by modifying the refractive index of an adjacent polymer layer79. Similarly, the dielectric and metal nonlinearity of SRR-based metamaterials has been explored. Using carbon nano-tubes as the nonlinear material hybridized with an SRR-based metamaterial, efficient all-optical modulation has been demon-strated102. Very short relaxation times, of less than 100 fs, may be achievable for bare gold SRR metamaterials, using the intrinsic nonlinearity of gold under two-photon excitation103 (Fig. 4c).

Plasmonic metamaterials also provide a new approach to enhancing nonlinearity by utilizing ‘nonlocal’ effects arising in the epsilon-near-zero regime104,105. The nonlocal effects depend strongly on losses and can be considerably modified by control-ling losses in gold nanorods105. The nonlinearity of gold under interband excitation leads to pronounced changes in the imagi-nary part of the permittivity. This gives rise to very large changes in the metamaterial transmission owing to the modification of the nonlocal response, with transmission changes of up to 80% with subpicosecond response times (Fig. 4d). For such performance in 100 × 100 nm2 devices, 10-fJ control pulses are sufficient.

Efficient all-optical modulation can also be achieved by con-trolling the coupling strength between molecular excitons and plasmonic excitations in metamaterials. This is extremely

sensitive to any perturbation of the system, including changes in the metal permittivity106,107.

Nonlinear SPP propagationNonlinearity is often used to control spatial and temporal charac-teristics of light pulses1. For SPP signals, it was initially proposed that the nonlinear effects be applied to diffraction management using self-focusing or defocusing nonlinearities and spatial soliton effects108–110. Owing to the short SPP propagation length, these effects are important only for strongly diffracting SPP beams. The problem of diffraction management may, however, become impor-tant if SPP amplification is considered to increase the SPP propa-gation distance.

The gain media for amplification often possess intrinsic nonlinearities that lead to self-focusing or defocusing. This is largely undesirable, for instance in lasers, but is sometime use-ful, for example to achieve diffractionless propagation. In plas-monic structures, the field enhancement may lead to this effect at smaller light intensities than in conventional photonic structures. Although no experimental observations of these nonlinear effects have been made so far, extensive theoretical treatments have been developed. Nonlinear propagation has been studied in both SPP- and nanoparticle-based waveguides, and a wealth of interesting effects has been predicted.

Light-intensity-dependent, and, thus, optically controlled, dis-persion can be achieved in plasmonic waveguides by relying on dielectric materials with Kerr nonlinearity111 or by utilizing the ‘ponderomotive’ nonlinearity of the metal112. These effects can be tailored so as to produce effective intensity limiters. Nonlinear waveguides consisting of silver nanoparticles have also been pre-dicted to show, under different conditions, bistability and modula-tional instability due to the nonlinearity of silver itself 113.

In principle, plasmonic soliton-like excitation is possible by choosing the correct balance between the gain and the nonlin-earity. This has been demonstrated with single-interface SPPs114 (Fig. 4e) and asymmetric, dielectric-clad metal films115. In the lat-ter, stable spatial solitons have been realized with the gain near one interface compensating the loss near the other.

In the absence of gain, self-trapping has been observed, but losses result in slowly decaying spatial SPP solitons. The self-trapping dimensions of about 100 nm have been predicted for plasmon-solitons115 which is much smaller than for solitons in dielectric waveguides with Kerr-type nonlinear medium. Discrete solitons have also been predicted in the case of plasmonic wires in a nonlinear medium116 as well as in planar plasmonic–dielectric multilayers117, requiring a relatively small gain of about 100 cm−1 for their observation.

Discussion and outlookNonlinear plasmonics is still a young research field, but its basic concepts have already been demonstrated in a wide variety of ways. Most structures so far have been based on relatively sim-ple concepts, yet their experimental implementation often pushed the limits of the nanofabrication techniques of the day. Great improvements in the quality of state-of-the-art samples now allow the principles of nonlinear effects to be tested ever more pre-cisely. At the same time, more sophisticated sample designs and functional concepts are emerging, and the activity in the field is increasing markedly.

The role of plasmonics in nonlinear optics is threefold. First, by enhancing the effective nonlinearity, plasmonic nanostructures contribute to conventional photonics by allowing nonlinear effects to be utilized with reduced optical power. Second, they make it possible to scale down nonlinear components in size, which is important for the development of integrable photonic devices




and, ultimately, fully functional nanophotonic circuitry. Third, the response time of plasmonic excitations is ultrafast, allowing opti-cal signals to be manipulated on femtosecond timescales.

A large fraction of the work on nonlinear plasmonics has stud-ied SHG, because it is straightforward to implement experimen-tally. Results on more complicated processes, such as FWM and CARS, are beginning to appear. However, with few exceptions the majority of experiments have involved free-space beams at least for signal or control light, even if they interact with each other through plasmonic excitations. For certain applications, it is nec-essary to design systems where all beams are maintained within the integrated plasmonic circuitry. The theoretical treatments of nonlinear propagation and soliton effects in plasmonic structures call for experiments and, if practically realized, will contribute a significant tool for all-optical signal manipulation.

Inherent losses in plasmonic systems are a notable challenge, as they limit the propagation length of SPPs to the range from 10 μm to several millimetres, depending on wavelength and geometry. As already mentioned, however, plasmonic field enhancement can ideally even overcompensate for the reduced nonlinear inter-action length. Losses are also important in the context of LSPs, where the local-field enhancement must be carefully balanced with radiative and ohmic losses to optimize the nonlinear inter-action. Nevertheless, it will be challenging to design structures where the local-field factors at all interacting frequencies are fully utilized. The design will also depend on the particular application and whether far- or near-field (integrated) devices are considered. The requirements on plasmonic resonances may indeed be differ-ent for integrated nonlinear components, allowing dark plasmonic modes with low radiative loss to be used.

Nonlinear plasmonics faces significant theoretical challenges. Although theoretical treatments exist for bulk metals and planar surfaces11, they need to be applied to nanoparticle geometries where the fields, nonlinear responses and nonlinear sources vary locally. For second-order effects, it is not even clear what the role of surface and bulk terms in the local response is. Experiments on planar metal films suggest that surface terms dominate12, whereas initial theories emphasized bulk terms118. Other theories have sug-gested that the importance of surface and bulk terms depends on the experimental details119. All these approaches12,118–120 need to be made mutually compatible if we are to understand how the non-linear responses can be optimized. In addition, quantum40,121 and nonlocal122 effects should be considered for structures with critical dimensions on the scale of single nanometres, as they may limit the attainable field enhancement.

In principle, a nonlinear response is enhanced when any combination of the interacting frequencies is resonant with the material. Unlike in conventional materials, where the resonances of the atomic-scale constituents are important, in plasmonic sys-tems the resonances occur mainly in the local-field factors123. For harmonic generation, a resonance at the fundamental fre-quency and its quality are crucial41,51,123,124. Recent work, how-ever, has shown that resonances at multiple wavelengths can also be utilized125,126.

For all-optical signal processing, the ultrafast nonlinearity of free electrons in plasmonic nanostructures provides record switching speeds, at switching rates greater than a terahertz. The switching energy per bit can also be low (down to 10 fJ), approach-ing the requirements for viable all-optical signal processing. Reduction of this energy to a few femtojoules would make all-optical switches competitive with electronic switches. Advanced tailoring of plasmonic resonances in metamaterials and utilization of their properties not available in conventional materials will be important for the development of next-generation, low-power, all-optical switches and all-optically-tunable plasmonic devices. Such

concepts have numerous applications as integrated and stand-alone photonic components.

Significant challenges need to be overcome and new types of nanostructure design investigated before the functional concepts of nonlinear plasmonics will be fully ready to move towards appli-cations. Nevertheless, we expect the field to become increasingly important in the development of the future nanophotonics, just as traditional nonlinear optics was in shaping the landscape of mod-ern photonics.

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