JP_Third-Order Nonlinear Plasmonic Materials_Enhancement and Limitations

PHYSICAL REVIEW A 88, 053838 (2013)

Third-order nonlinear plasmonic materials: Enhancement and limitations

J. B. KhurginDepartment of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA

G. SunDepartment of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA

(Received 1 July 2013; published 25 November 2013)We develop a rigorous and physically transparent theory of enhancement of third-order nonlinear optical

processes achievable in plasmonic structures. The results show that the effective nonlinear index can be enhancedby many orders of magnitude, but, due to high metal losses the most relevant figure of merit, the amount ofphase shift per one absorption length, remains very low. This makes nonlinear plasmonic materials a poor matchfor applications requiring high efficiency, such as all-optical switching and wavelength conversion, but does notpreclude the applications where overall high efficiency is not required, such as sensing.

DOI: 10.1103/PhysRevA.88.053838 PACS number(s): 42.70.Nq, 42.65.Pc, 78.67.Pt, 42.65.Ky

I. INTRODUCTION

Various nonlinear optical phenomena have been attractingthe interest of the scientific community ever since scientistsgained access to intense optical fields with the invention of thelaser in 1960 [1]. Very shortly after this invention practicallyall the major nonlinear optical phenomena of second and thirdorder were successfully demonstrated [24]. Simultaneouslythe theory of nonlinear optics was developed by Bloembergenand many others [57]. Today a clear understanding of thenonlinear optical effects in various media exists and can befound in a number of excellent textbooks and monographs[8,9]. The fascinating promise of nonlinear optics has alwaysbeen based on the fact that nonlinear optical phenomena allowone in principle to manipulate photons with other photonswithout relying on electronics. Hence a large number ofall-optical devices that allow light manipulation based on eithersecond- or third-order nonlinear effects, such as frequencyconversion, switching, phase conjugation, and others, havebeen proposed and demonstrated in different materials andconfigurations [9]. However, while there have been somespectacular success stories that lead to practical products(such as frequency converters, optical parametric oscillators,frequency combs for measurements, and a few others), thus farthe majority of nonlinear optical phenomena have not becomecompetitive for practical applications, and not for the lack oftrying.

The reason for this seeming incongruity is quite simpleallnonlinear optical phenomena can be divided into two broadclasses: slow and ultrafast. The slow nonlinear phenomena aregenerally classified as such by the fact that optical fields donot interact directly, but through the various intermediaries,such as electrons excited when the photons get absorbed,or through the temperature rise caused by the release ofenergy of the absorbed photons. For as long as these in-termediaries exist, i.e., while electrons stay in the excitedstate or until the heat dissipates, their effect on the opticalfields accumulates, hence these phenomena, such as saturableabsorption, photorefractive effect, or thermal nonlinearity,can be quite strong, but this very fact makes them slow,as their temporal response is limited by the time constantassociated with relaxation, recombination, or heat diffusion

process. Furthermore, the slow nonlinearity always involvesthe so-called real process of photon absorption, and, onceabsorbed, these photons are never recovered, which means thatslow nonlinearities are always associated with a significantloss. While there exists a legitimate niche for these slownonlinearities (which may not be all that slow after all, assome saturable absorbers do show picosecond response), it isthe other hand, the so-called virtual or ultrafast nonlinearitiesthat have been the object of interest as they carry the promiseof transforming the fields of information processing andcommunications.

The term virtual, that is commonly associated with theultrafast nonlinearity, implies that the nonlinear phenomenondoes not involve excitation of the matter to the real excitedstates as there exist no transitions between the states thatare resonant with the photon energy. When the non-energy-conserving virtual excitation does take place its duration isdetermined by the uncertainty principle, and thus can beas short as a few femtoseconds or even a fraction of afemtosecond which explicates the term ultrafast. But it isprecisely the fact that the excitation lasts such a short timeinterval that makes the ultrafast nonlinearities relatively weak.On a microscopic level one can explain this by the factthat for small electric fields the atoms and molecules alwaysbehave as essentially harmonic oscillators and only when theapplied fields become appreciable relative to the intrinsic fieldEi holding the electrons confined within the atom or thebond between the atoms does anharmonicity arise causingthe nonlinear response. Overall, it can be loosely stated thatthe order of magnitude of the nonlinearity of the nth order canbe determined as

(n) (1)En1i , (1)where (1) = n2d 1 is the linear susceptibility and nd isthe index of refraction, typically anywhere between 1.5 and3.5 for most solids in the visible to near-IR range. Thetypical value of the intrinsic field is on the order of thebinding energy of a few eV divided by the bond lengthof 12 A, i.e., 10101011 V/m. Therefore, the second-ordersusceptibility (2) in solids cannot exceed 1000 pm/V andis usually far less than that due to crystal symmetry, while

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J. B. KHURGIN AND G. SUN PHYSICAL REVIEW A 88, 053838 (2013)

FIG. 1. (Color online) Comparison of cw (a) and mode-locked(b) laser outputs with equal average power.

the third-order susceptibility (3) is expected to be less than1019 m2/V2. Through the relation between the third-ordersusceptibility and nonlinear refractive index, n2 = (3)0/n2where the impedance of free space 0 = 377 , one can seethat the latter is limited to a magnitude less than 1013 cm2/W.This is indeed the case, with a typical nonlinear indexbeing strongly dependent on the width of the transparencyregion and ranging from n2 5 1016 cm2/W for fusedsilica that is transparent all the way to UV, to perhapsn2 1 1013 cm2/W for chalcogenide glasses transparentonly in the IR range [10].

Clearly, very strong optical power density is required inorder to produce appreciable ultrafast nonlinear optical phe-nomena. The average optical power available from a compactlaser rarely exceeds a few hundred milliwatts; furthermore,if one wants to envision all-optical integrated circuits, thepower dissipation requirements constrain the power to evenmuch lower levels than that, possibly less than a milliwatt.Hence early on it was understood that to make nonlinearoptical phenomena practical one must concentrate the powerin both space and time. Concentration in space usually impliescoupling the light into a tightly confining optical waveguide ora fiber. But the attainable concentration is limited to roughlya wavelength in the medium due to the diffraction limit.In addition, one may consider the resonant concentration ofoptical energy in microcavities [11,12], ring resonators [13],photonic band-gap structures [14], and slow-light devices[15,16], but all these resonant effects inevitably limit thebandwidth [17]. It is the concentration of optical power inthe time domain provided by pulsed sources, particularlyby the Q-switched [18] and mode-locked lasers [19,20],that has proven to be the winning technique in nonlinearoptics.

The reasons for this can be easily grasped from the sketchin Fig. 1. Consider the light of a cw source with the power Pcwthat propagates over a distance l in the medium with nonlinearsusceptibility (n). At the far end the nonlinear wave of powerP (n)cw l2| (n)|2Pncw will emerge [Fig. 1(a)]. If, on the otherhand, one could use a periodically pulsed source with dutycycle and the same average power P = Pcw [Fig. 1(b)],the peak power would obviously be Ppeak = 1Pcw andthe peak output nonlinear power would increase to P (n)peak l2| (n)|2nP ncw while the average nonlinear output would

amount to P (n) 1nP (n)cw ; i.e., the efficiency is increasedby 1n. For a typical mode-locked laser with a picosecond-pulse duration and 100-MHz repetition rate = 104, thisindicates that the efficiency of the second-order nonlinearconversion can be boosted by 104 and for the third-order effectit is even higher.

Use of ultrashort low-duty-cycle laser pulses has becomethe ubiquitous method of obtaining excellent practical resultsfor both the second-order (frequency conversion, parametricoscillation, and amplification) and the third-order (opticalfrequency comb and continuum generation) phenomena. Andyet if one is thinking of applications in information processing,the switches are expected to operate at the same symbolrate and duty cycle as the data stream. In other words, ifthe signal itself is, say, a typical no return to zero (NRZ)stream of 5-ps FWHM pulses in 10-ps bit intervals, using1-ps pulses at low duty cycle will not allow one to be ableto fully switch each individual symbol. Then one shouldlook at other methods of concentrating the energy and theattention is inevitably drawn back to the space domain and thequestion arises: Can one transfer the mode-locking techniquesfrom time to space, i.e., to create a low-duty-cycle high-peak-power distribution of optical energy in space, rather than intime, and to use it to effectively enhance nonlinear opticaleffects.

Extending the time-space analogy, let us look at what limitsthe degree of energy concentration in time and space. In thetime domain it is obviously the dispersion of group velocity,while in the space domain it is the diffraction. While thereis obvious equivalence between the mathematical descriptionof dispersion and diffraction, there is a stark differencethegroup velocity dispersion can be minimized by a numberof techniques because it can be either positive or negative,while the diffraction is always positive and there exists ahard diffraction limit to the optical confinement in an all-dielectric medium. Therefore, the closest space-domain analogto the mode-locking technique is a one-dimensional arrayof coupled resonators with either ring or photonic band-gapimplementation, which does provide some enhancement ofnonlinearity, but, as mentioned above, always at the expenseof bandwidth [21,22].

It is important to realize, though, that the diffraction limitis applicable only to the all-dielectric structures with positivereal parts of dielectric constants. In all-dielectric structures theenergy oscillates between electric and magnetic fields, and ifthe volume in which one tries to confine the optical energyis much less than a wavelength the magnetic field essentiallyvanishes (so-called quasistatic limit) and without this energyreservoir for storage every alternative quarter cycle theenergy simply radiates away. But if the structure contains amedium with negative real part of dielectric constant, i.e., freeelectrons, an alternative reservoir for energy opens upthekinetic motion of these free carriers in metal or semiconductor,and the diffraction limit ceases being applicable. The opticalenergy can be then contained in the tightly confined sub-wavelength modes surrounding or filling the gap between thetiny metallic particles. These modes, combining electric fieldwith charge oscillations, are called localized surface plasmons(LSPs), and in the last decade the entirely new closely relatedfields of plasmonics and metamaterials have arisen with

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the ultimate goal of taking advantage of the unprecedenteddegree of optical energy concentration on the subwavelengthscale [23].

Following these arguments, the researchers have observedenhancement of both linear (absorption, luminescence) [24,25]and nonlinear (Raman) phenomena [2628] in the vicin-ity of small metal nanoparticles and their combinations.Enhancement of many orders of magnitude has been ob-served experimentally for surface-enhanced Raman scattering(SERS) [2628], while the enhancement for luminescenceand absorption was more modest. To address this issue wehave developed a rigorous yet physically transparent theoryexplaining this enhancement provided by single [29] orcoupled [30,31] nanoparticles in which we have traced therelatively weak enhancement of luminescence to the largeabsorption in the metal, which cannot be reduced in trulysubwavelength mode in which the field is concentrated [32,33].In that work [32] we have shown that the decay rate of theelectric field in the subwavelength mode is always on the orderof the scattering time in metal, i.e., 1020 fs in noble metals.This is the natural consequence of the aforementioned fact thathalf of the time all the energy is contained in the kinetic motionof electrons in the metal where it dissipates with the scatteringrate. As a result, a significant fraction of the LSP energy simplydissipates inside the metal rather than radiating away. Thenet result is that only very inefficient emitters [34] and alsoabsorbers [35] can be enhanced by plasmonic effects such as,of course, the Raman process that is extremely inefficient [36],while the relatively efficient devices such as light-emittingdiodes (LEDs) [37], solar cells [38], and detectors [39] donot exhibit any significant plasmonic enhancement relative towhat can be obtained without the metal by purely dielectricmeans [40].

Therefore, it would only be natural to look into whatplasmonic enhancement can do for the inherently weaknonlinear processes, and, although the first works along thisdirection are over 30 years old [4146] the interest has peakedsignificantly in the last decade [47]. There are a numberof ways where nonlinear optical effects can be enhancedby the surface plasmons (SPs). One is the coupling of theexcitation field to form the much stronger localized field nearthe surface of the metal structure that leads to the enhancementof optical processes [48]. Such a strong near-field effect isresponsible for the experimental observations of significantRaman enhancement that has resulted in single-moleculedetection [2628,49] and SP-enhanced wave mixing suchas second-harmonic generation (SHG) on random [5052]and defined plasmonic structures [5359], as well as theenhancement of linear processes such as optical absorption andluminescence [24,25]. Another is the fact that SP resonance isultrasensitive to the dielectric properties of the metal and itssurrounding mediuma minor modification in the refractiveindex around the metal surface can lead to a large shift ofplasmonic resonance [60]. Such a phenomenon brings aboutthe prospect of controlling light with another light wherethe latter induces optical property change in the plasmonicstructure which in turn modifies the propagation of the originallight. Motivated by this promise, researchers around the worldhave been pursuing the goal of practical all-optical modulation

or switching based on Kerr nonlinearities in either unconfinedplasmonic materials [6164] or waveguides [6569], whichhas remained elusive up to this date.

At this point it is important to differentiate betweenthe sources of nonlinearity in these works, because bothmetals and dielectrics possess nonlinearity. The nonlinearsusceptibility of metal can be due to either free carriers orto band-to-band transitions. The nonlinearity of band-to-bandtransition (typically involving d bands in noble metals) is nodifferent from the interband nonlinearity of dielectrics andsemiconductors, except it always occurs in the region of largeabsorption due to free carriers, and in addition, the nonlinearityis strongest in the blue region of the spectrum, while weprefer to concentrate on the telecommunication region of13001500 nm. As far as nonlinearity of free electrons isconcerned, it is extremely weak because LSPs (at least whenthere are only a few of them per nanoparticle) are nearly perfectharmonic oscillators. That can be easily understood from thefollowing back-of-the envelope calculation. To maintain just asingle LSP with, say, h0 = 1.25 eV in a subwavelength modethat, as we mentioned above, decays with a time constant of1014 fs, it would mean power dissipation of 20 W in asmall volume on the order of, say, 2 1017 cm3, i.e., veryhigh power density of 1012 W/cm3 and temperature rise on theorder of about 10 K per picosecond. Clearly, one cannot expectto find more than a few SPs per mode before the catastrophicmeltdown. But then, as we have mentioned before, the energyof SP for half the time is contained in the form of kineticenergy of electrons, hence one can write

NeVm020x

20/

2 = h0, (2)where Ne 8 1022 cm3 is the electron density, V is themetal volume, and x0 is the classical amplitude of eachindividual electron. From Eq. (2) we immediately find x0 0.002 A and with such a tiny amplitude of motion the freeelectron cannot see any anharmonicity of the potential.

Therefore, we shall consider the structure in which themetal nanoparticles are embedded into the nonlinear materialwith large nonlinearity and low loss. We shall limit ourconsideration to the third-order nonlinearity because it leadsto optical switching and other interesting phenomena withoutphase matching, and furthermore, we shall limit ourselves tothe nonlinear modulation of the refractive index (real partof susceptibility) rather than absorption (imaginary part).One reason for it is that for the amplitude modulation it isdesirable to maintain the zero bit level as close to realzero as possible, which can only be done by the interference(as in, for instance, Mach Zehnder interferometer). Anotherreason is that by modulating the index one can take advantageof advanced phase-modulation formats, such as quadraturephase-shift keying (QPSK), quadrature amplitude modulation(QAM), etc. Index modulation is typically broadband and,in addition to simple modulation and switching, can beused for frequency conversion, while absorption modulationis an inherently resonant phenomenon. Finally, changes inabsorption are usually associated with real excitations; hencethey are not truly ultrafast.

In Sec. III E we shall briefly consider the implicationsof using metal nonlinearity as well as modulation of the

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FIG. 2. (Color online) (a) Gold spherical nanoparticle (a =20 nm) with the electric field distribution. (b) Gold ellipticalnanoparticle [long (short) axes: 25 nm (5 nm)] with resonance at1550 nm and associated electric field distribution. (c) Extinctionspectrum of the elliptical nanoparticle.

absorption coefficient and show that essentially the samefigures of merit will apply to those plasmonically enhancednonlinear schemes as to the one which we shall considerat lengththe structure shown in Figs. 2(a) and 2(b) inwhich metal nanoparticles are surrounded by a nonlineardielectric medium. The goal of our treatment is to evaluatethe enhancement of the third-order nonlinear polarizabilityof this metamaterial, or one can use the term artificialdielectric consisting of metal nanoparticles that enhancelocal field. In the course of this work we shall introducefigures of merit relevant to practical applications and seehow the plasmonically enhanced nonlinear materials stackup against the conventional ones. To make our treatmentboth general and physically transparent we shall fully relyon analytical derivations which, of course, would requirecertain simplifications that are justified for as long as oneis looking just for the order of magnitude of enhancement.For instance, we shall consider spherical and elliptical (orspheroidal) nanoparticles, single and coupled, but we shallindicate how the treatment can be expanded to other shapes ofnanoparticles, including nanoshells [70], that can be definedby just three parameters: resonant SP frequency 0, qualityfactor Q, and effective SP mode volume Veff . For thispurpose, in Fig. 2(a) we show spherical nanoparticles and inFig. 2(b) we show elliptical nanoparticles with resonance at thetelecommunication wavelength of 1550 nm with actual fielddistribution calculated numerically. Also shown in Fig. 2(c)

FIG. 3. (Color online) Fields and polarizations in the plasmoni-cally enhanced nonlinear metamaterial: (a) average E and local Eelectric fields and dipole p at the pump frequency; (b) local nonlinearfield E , dipole moment p

nl , and average nonlinear polarization P

nl .

is the extinction spectrum of the elliptical particle obtainednumerically where the resonance can be observed.

II. LINEAR OPTICAL PROPERTIES OF METALNANOPARTICLES EMBEDDED IN A DIELECTRIC

A. Polarizability and local field enhancementConsider a rather general scheme for plasmoncially

enhanced nonlinearity shown in Fig. 3(a) consisting ofnanospheres of radius a surrounded by the nonlinear dielectricwith relative permittivity d and nonlinear susceptibility tensor (3). The density of these spheres is Ns . In the most generalcase (3) implies four wave interactions, with some of thewaves being the pumps (of switching signals) and some beingthe nonlinear output signals. In many practical cases, suchas cross- and self-phase modulation, degeneracy reduces thenumber of interacting waves. In Fig. 3(a) we show just onepump (or switching) wave of frequency and one signal waveof frequency .

As the pump wave propagates through the material with theaverage electric field of E, the nanospheres become polarizedby this field, and acquire the dipole moment [71]

p =m dm + 2d 40da

3E = 30V m d

m + 2d dE, (3)

as shown in Fig. 3(a). Using the Drude model for thedielectric constant of metal m = 1 2p/(2 + j ) withplasma frequency p and scattering rate we can write

m dm + 2d =

1 2p2+i d

1 2p2+i + 2d

= (1 d ) 2 2p + i (1 d )

(1 + 2d ) 2 2p + i (1 + 2d )

20 + 2 d12d+1

20 2 i, (4)

where 0 = p/

1 + 2d is the LSP resonant frequency [34].Not far from the resonance we obtain

p =20

E20 2 i

QL()

E, (5)

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FIG. 4. (Color online) Dispersions of Q factors for gold and silvernanoparticles.

where we have introduced the Q factor of the mode as

Q = 0/, (6)resonant Lorentzian denominator,

L() = Q(1 2/20) i, (7)and polarizability of the nanoparticle as

30dV(

1 + d 12d + 1

)= 30dV 3d2d + 1 = 30dV.

(8)Here, for the spherical nanoparticle, the factor =3d/(2d + 1), and for particles of different shapes it will besomewhat different and polarization dependent, yet still withinthe same order of magnitude. Similarly, the value of resonantfrequency will change; however, since we are interested only inthe order of magnitude results in this work, all the conclusionsobtained here for spherical particles and their combinationswill hold for the particles of different shapes. It should benoted that the Q factor for a particular shape depends onlyon its resonant frequency 0 since the decay rate does notdepend on the shape (or exact dimensions) as long as particlesare much smaller than the wavelength (which is of courserequired to avoid scattering and diffraction effects).

The Q factor for the gold and silver, the two least-lossyplasmonic materials are shown in Fig. 4 as functions offrequency. Near 1550 nm the Q factor of bulk gold is about 12and for bulk silver it is closer to 30, according to Johnson andChristy [72], although for silver nanoparticles the interfacescattering usually decreases the Q factor by a factor of a few.Also, gold is easier to work with than silver, as it does notget oxidized, so in the subsequent discussion we use gold asthe material of choice, although in the end changing Q by afactor of a few will not affect any of our conclusions, since, westress once again, our results are all just an order of magnitudeestimates.

Equation (5) can be construed as the solution of theequation of motion of the harmonic oscillator, or the LSPmode characterized by the dipole moment p,

d2 pdt2

+ d pdt

= 20 p + 20 E, (9)

and consisting of coupled oscillations of the free electroncurrent inside the nanoparticle, and the electric field insideand outside the spherical nanoparticle [34],

E(r) ={ p40da3 r < a

140d r3 [3( p r)r p] r > a

, (10)

with the maximum field near the surface of the nanoparticleequal to

Emax ,(a) = 140d2 pa3

= 2 p30dV

, (11)

where V = 4a3/3 is the volume of the nanosphere.The total maximum field near the surface is then a sum of

a dipole field and the original average field,

Emax, = E + 230dV Q

L()E

= E[

1 + 2QL()

] 2Q

L()E. (12)

Hence near the resonance the local field is enhanced roughlyby a factor of 2Q relative to the average field.

The dipole LSP mode equation (10) is the lowest order(l = 1) among the many orthogonal modes defined by theangular momentum number l. The electric field of the lthmode can be written as

El (r) = Emax ,l Fl(r), (13)where the normalized mode shape function Fl(r) has amaximum value of unity and one can introduce the effectivevolume Veff,l in such a way,

Fl(r)[r (r)]

Fl(r)d3r = dVeff,l , (14)

that the total mode energy can be found asUl = 12dE2max ,lVeff,l . (15)

The effective volume of the lth-order mode [30],

Veff1 = a3

d= 3

4dV, (16)

is commensurate with the volume of the nanoparticle itself.

B. Effective index and absorptionIf the nanoparticles are much smaller than the wavelength

of light in the dielectric, one can apply a classical polarizabilitytheory in which each nanoparticle is treated as a polarizableatom. The effective dielectric constant of the compositemedium (or a metamaterial if one wants to use a more modern,de rigueur terminology) can be found as the sum of the originaldielectric constant and the susceptibility of the nanoparticleswith a density Ns ,

eff = d + Ns0

Q

L() = d +Ns30dV

0

Q

L()= d

[1 + 3f Q

L()], (17)

where we have introduced the effective filling factor,f = NsV Q1. (18)

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The latter condition is practically always satisfied in themedium with Q 10 and is required to avoid taking intoaccount local field (collective) effects that would changethe LSP resonant frequency according to the Lorentz-Lorenzformula. But, once again, even for a very dense medium thefrequency renormalization is not going to change the mainconclusions of this work.

In this approximation we can find the effective index ofrefraction as

neff = 1/2eff nd[1 + 3f

2Q2

(1 2/20

)|L()|2 +

3f2

iQ

|L()|2],

(19)where nd = d . Obviously, the effective absorption coeffi-cient,

a = 2nd

3fQ|L()|2 , (20)

also gets resonantly enhanced by the Q factor.Next we shall mention what happens to energy propagation

described by the Poynting vector given as

S =E2

20neff,r =

E2nd

20

[1 + 3f

2Q2

(1 2/20

)|L()|2

]. (21)

At the same time the average energy density inside is

U = 0dE2

2+ Ns0dE

2max ,Veff

2

= 0dE2

2

[1 + Ns 34d V

42Q2

|L()|2]

= 0dE2

2

[1 + 3f

2

d

Q2

|L()|2]. (22)

Now we can define the energy velocity as

vE, =S

U= c

f nd

1 + 3f2Q2(12/20)

|L()|2

1 + 3fd

Q2

|L()|2. (23)

Note that away from resonance the energy propagation velocityis identical to the group velocity, but at resonance we prefer touse energy propagation velocity since the group velocity losesphysical meaning (for instance, by becoming negative). Theslowing down factor, or relative group index, at resonance is

ng = cndvE

= 1 + 3f2Q2

d, (24)

which is the consequence of the fact that a large fractionof energy being stored in the nonpropagating LSP modes.This result is important because if we now compare Eq. (24)with Eq. (12) we can see that local intensity enhancementat resonance, 42Q2, can be written roughly as (ng 1)/f .This indicates the connection between the plasmonics andslow-light structures [73] in which local intensity also getsenhanced, but note that for Q 10 and f < 0.01 most ofthe enhancement for plasmonic structures comes not from theslowing down of light but rather redistribution of energy onthe subwavelength scale.

III. NONLINEAR PROPERTIES OF METAMATERIALWITH ISOLATED NANOPARTCLES

A. Nonlinear polarizabilityLet us now turn our attention to Fig. 3(b) where the local

nonlinear microscopic polarization at the frequency ,

Pnl(r,t) = Pmax ,nl G(r)eit , (25)

is established near the nanoparticle due to the presence of astrong local pump field. As mentioned above could be thesame as or different from the pump frequency that drives thenonlinear polarization. The maximum nonlinear polarization is|Pnl(rmax)| = Pmax ,nl , occurring usually at the same locationwhere the local pump field reaches maximum, and G(r) isthe normalized shape of nonlinear polarization. The nonlinearpolarization can now drive the LSP oscillations at the samefrequency according to the wave equation for the electricfield of the LSP mode,

2 E(r,t) r (r)c2

2

t2E(r,t) = 1

0c22

t2Pnl(r,t) (26)

where the relative dielectric constant can be written inside ofthe metal as

r (r) = r (r) + ir (r) r (r) [1 + i (r)/] . (27)We look for the solution of the form

E(r,t) =

l

E

max ,l Fl(r)eit , (28)

where Fl(r) is the normalized electric field of the lth LSPeigenmode with l = 1 being the dipole mode described byEq. (10), whose amplitude Emax ,1 we are trying to determine.Each eigenmode is a solution of the homogeneous waveequation

2 Fl(r) = 20,lr (r)c2

Fl(r), (29)and the modes are orthogonal and normalized to the effectivevolume Veff,l of the lth mode as defined in Eq. (14),

Fm(r)[r (r)]

Fl(r)d3r = dVeff,llm. (30)

Substituting Eq. (28) into Eq. (26) and using Eq. (29), weobtain

l

{[20,l + 2 + i (r)]Emax ,l(t)+ 2i Emax ,l(t)

t

}Fl(r)

= 2

0r (r)P

max ,nl G(r). (31)

If we now multiply Eq. (31) by (r)

F1(r), integrate overthe volume, and take advantage of the orthogonality conditionequation (30) to obtain the steady-state amplitude of the l = 1dipole mode driven by the nonlinear polarization at frequency,

E

max =Pmax ,nl

0d

Q

L() , (32)

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where the overlap coefficient, assuming that the dielectric isnondispersive and nonlossy, is

= d

F1(r) G(r)dV (r)

F 21 (r)dV= d

F1(r) G(r)dV

Veff,1. (33)

Now, according to Eq. (11) we can find the amplitude of thedipole mode,

p

nl =32V 0dE

max ,nl =

32V

Q

L()Pmax ,nl . (34)

And overall effective nonlinear polarization of the metama-terial is

P

nl =32f

Q

L() Pmax ,nl, (35)

shown in Fig. 3(b). As one can see, local nonlinear polarizationgets enhanced by being at resonance with the nanoparticledipole mode and enhancement is once again proportional tothe Q factor of the resonance.

Before continuing we shall recap the chain of events thatleads to the establishment of enhanced nonlinear polarizationas shown in Fig. 3:

(i) The average pumping field E polarizes nanoparticlesengendering linear dipole moment p in each of them;

(ii) Dipole oscillations are coupled with the linear localfield E(r) in the vicinity of each nanoparticle. This field isresonantly enhanced by a factor of the order of Q relative toE;(iii) A local nonlinear polarization Pnl (r) is established

in the vicinity of each nanoparticle. Since this polarizationis proportional to the third order of the electric field, it isenhanced roughly by a factor of Q3;

(iv) This polarization resonantly couples into the dipole LSPmode of the nanoparticle thus establishing the local nonlinearfield E(r) and the dipole moment pnl . Resonance causesenhancement by another Q factor;

(v) Finally the localized dipoles pnl combine to establishthe average nonlinear polarization P

nl .

Of course all the steps outlined above occur simultaneouslyand instantly, but in our view tracing the process step by stepprovides the clarity of a physical picture. Let us now turn ourattention to specific third-order processes.

B. Effective third-order nonlinearityConsider now the third-order nonlinearity in which inter-

action of electromagnetic waves at three different frequenciesis described by the general local third-order susceptibility,

P12+3nl (r) = 0 (3)(3, 2,1)E1 (r)E2 (r)E3 (r).(36)

In general, when all four frequencies, 1, 2, 3, and4 = 1 2 + 3 are different (but typically close to eachother) the nonlinear process described by Eq. (36) is four-wave mixing (FWM), when 3 = 1 4 = 21 2 Eq. (36)describes optical parametric generation (OPG), when 1 = 2and 3 = 4 it describes cross-phase modulation (XPM), andfor the case when all frequencies are equal Eq. (36) describesself-phase modulation (SPM). FWM and OPG are both of

great interest in wavelength conversion while both XPM andSPM are important for optical switching.

In a composite medium (metamaterial) the local fieldsEk (r) in Eq. (36) in the vicinity of the nanoparticle are alllocally enhanced relative to the mean fields Ek according toEq. (12), i.e.,

Ek (r) =2QL(k)

Ek F1(r). (37)

Hence the local third-order nonlinear polarization is

P12+3nl (r) = 0 (3)(3,2,1)(2Q)3

L(1)L(2)L(3) F1(r)F1(r)F1(r) E1 E2 E3

= P12+3max ,nl G(r), (38)where its amplitude is

P12+3max ,nl = 0| (3)(3,2,1)|

(2Q)3L(1)L(2)L(3)

E1 E2 E3 , (39)the shape function is

G(r) = (3)

| (3)| F1(r)F1(r)F1(r), (40)

and (3)

| (3)| is the normalized fourth-order nonlinear susceptibil-ity tensor. Substituting Eq. (40) into Eq. (35) we obtain

P4nl =32f 3

(2)3 Q4L(1)L(2)L(3)L(4)

0(3)(3,2,1)

E1 E2 E3 0 (3)eff (3,2,1) E1 E2 E3 , (41)

where the coupling coefficient for the third-order nonlinearity,according to Eq. (33), is

3 = dVeff,1

r>a

F1(r) (3)

| (3)| F1(r)F1(r)F1(r)d3r. (42)

Since typically all the frequencies are close to each other, wecan see that the effective nonlinear susceptibility gets enhancedas

(3)eff

32f 3 (2)3 Q

4

L2() |L()|2 (3), (43)

i.e., by a factor proportional to Q4. This is an outstandingresult, exciting enough to attract the attention of both the plas-monic and nonlinear optics communities to this topic, whichhas witnessed a surge of research efforts and publications asreviewed in Sec. I. Indeed, even with filling factor f 0.01one can expect 100-fold enhancement of susceptibility. Itmeans 100-fold enhancement of the nonlinear refractive indexand indicates that one can achieve the same efficiency ofnonlinear phase modulation at about 1/100 of the length,and, more dramatically, the same efficiency of the wavelengthconversion in only 1/10 000 of the length. These are preciselythe results that prompted many scientists to enter the racefor the largest enhancement. It would be nice if we couldend our discussion right here on this optimistic note, but

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J. B. KHURGIN AND G. SUN PHYSICAL REVIEW A 88, 053838 (2013)

one needs to maintain caution when it comes to reportingthese giant plasmonic enhancements. Our prior researchof plasmonic enhancement of various emission processesincluding photoluminescence [29], electroluminescence [34],and Raman scattering [36] has shown that large enhancementsare feasible only for the processes that have very low originalefficiencies (such as Raman scattering) but are far moremodest for those efficient processes such as fluorescence andelectroluminescence. It is therefore reasonable to expect thatthere must exist an upper limit of the nonlinear plasmonicenhancement.

C. Effective nonlinear index and maximum phase shiftTo understand the limitations of the enhancement we shall

first consider XPM (or SPM) for which nonlinear polarizationin Eq. (36) can be written as

P2nl (r) = 20ndn2I1 (r)E2 (r), (44)

where n2 = (3)(1,1,2)0/d is the nonlinear indexof the dielectric, and I1 (r) = |E1 (r)|2nd/20 is the localintensity (of course the energy in the near field does notpropagate and this definition of local intensity is just aconvenient way to relate the susceptibility with the nonlinearindex). The local change of refractive index is then

n(r) = n2I1 (r) = n2 I1 (2)2Q2

|L(1)|2F 21 (r), (45)

where the average intensity I1 = nd |E1 |2/20.Similarly, we now introduce the effective nonlinear index

as n2,eff = (3)eff (1,1,2)0/d and rewrite Eq. (41) asP2nl = 20ndn2,eff I1 E2 . (46)

According to Eq. (43) the effective nonlinear index getsenhanced by the same giant factor proportional to Q4,

n2,eff 32f 3 (2)3 Q

4

L2(2) |L(1)|2n2. (47)

Since the pump intensity decreases due to absorption, thenonlinear phase shift can be found as

(z) = 2

n2,eff

z0

I1 (z)dz =2a

n2,eff I0(1 eaz),(48)

where I0 is the input pump intensity and a is the absorptioncoefficient defined in Eq. (20). This means that the maximumphase shift obtained after about one absorption length is

max = |L(1)|2

3fQn2,eff

ndI0 3 (2)2 Q

3

L2(2)n2

ndI0. (49)

Achieving the -phase shift required to get pho-tonic switching would then require at resonance I nd [3n2(2)2Q3]1. If we assume 1.35 (estimatednumerically for the actual ellipsoid resonant at 1550 nm), Q 12, and large nonlinear index characteristic of chalcogenideglass n2 = 1013 cm2/W, we obtain the required switchingintensity of the order of I 6 109 W/cm2.

This result indicates that the giant nonlinear index enhance-ment equation (47) can only be used to reduce the length

FIG. 5. (Color online) Change of dielectric constant eff causedby the shift of the resonance frequency of SP 0.

of the device, while the switching intensity remains veryhighrequiring peak powers of about 60 W into a 1-m2waveguide. But the situation is actually even less optimistic;according to Eq. (12) the maximum local intensity near thenanoparticle surface is

Imax = (2Q)2 I 6 1012 W/cm2 (50)

(or maximum local field is of 5 107 V/cm), which issignificantly higher than the damage threshold of the material,and besides as simple calculations may show, it leads to thelocal temperature rise on the scale of 104 K/ps! In fact, ifone searches through all the nonlinear materials, it is difficultto find one that is capable of achieving ultrafast refractiveindex change larger than 0.1%. In addition to limitationdue to overheating and optical damage, at high power thenonlinearities of higher than the third order, i.e., (5) and (7),become important, and they often have their signs opposite to (3) [74] which leads to actual decrease in the nonlinear indexchange at high intensities.

Therefore, let us define the maximum local nonlinear indexchange attainable in a given material as nmax. Then fromEqs. (45) and (47) we obtain

neff, max 3f 3 Q2

L2(2)nmax. (51)

As we can see now the enhancement is only proportionalto Q2. This result makes perfect sense if we recognizethat local change of dielectric constant d, max = 2ndnmaxsimply causes the shift of the LSP resonant frequency 0 =p/

1 + 2d , as shown in Fig. 5, which in turn changes

the effective dielectric constant of the metamaterial eff, maxaccording to Eq. (17) as

eff, max effd

d, max = effL(2)

L(2)20

20d

d, max

6df1 + 2d

Q2

L2(2)d, max, (52)

where we have kept only the largest term proportional to Q2and disregarded smaller terms proportional to Q.

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What is most important though is that the maximumobtainable phase shift equation (49) becomes

max = 2a

neff, max = 3Q |L(1)|2

L2(2)nmax

nd. (53)

The simple meaning of Eq. (53) is that, even if we assumeenormous local nonlinear index change of 1% (local intensityof 1011 W/cm2), we cannot expect to get phase shift higherthan 0.1, almost two orders of magnitude less than what isrequired for -phase shift switching.

D. Frequency conversion using FWMIt is easy to see that small maximum phase shift for XPM or

SPM corresponds to even smaller efficiency of the frequencyconversion for FWM or OPG. Indeed the growth of the idlerE3 (z) in the presence of pump I1 (z) = I0eaz and signalE2 (z) = Ese

a2 z can be described by

d E3 (z)dz

= 2

n2,eff I1 (z)E2 (z) a

2E3 (z) (54)

with the solution

E3 (z) =2a

n2,eff I0Es[1 eaz]eaz/2 (55)

that reaches a maximum near z = 1a ln 3 equal toEi, max = 2

a

233/2

n2,eff I0Es =2

33/2maxEs. (56)

Therefore, maximum conversion efficiency from signal to idleris

Ii

Is= 1

3

(23max

)2, (57)

and under no conceivable conditions can it exceed 30 dB.

E. Absorption modulationAlthough we have mentioned that absorption modulators

are not nearly as versatile as the phase modulators, we cansimply find the maximum change of the imaginary part ofthe refractive index from Eq. (51) which will occur when 0(1 2Q2) and is equal to

(Imneff)max 2f 3Q2

L2(2)nmax, (58)

which upon comparison with Eq. (19) translates to themaximum absorption coefficient modulation,

a

a 4

33Q

nmax

nd. (59)

Thus the change in absorption per unit length does not appearto increase far beyond a few percent. Since the absorptionprocess is exponential, one can, of course, achieve deepermodulation than that of the total transmission by propagatingover many absorption lengths, but the insertion loss would beoverwhelming. Therefore, if we use the figure of merit, thechange of transmission per one absorption length, i.e., 1 exp(a/a) a/a , we can see that one cannot achievesignificant absorption modulation without incurring enormousloss.

Here we should also briefly mention that one could usemodulation of the refractive index of the metal itself, but it isdifficult to see how one can change the index of metal by morethan 1% unless one operates near the interband transitionswhere the Q factor is greatly reduced, which defeats the wholepurpose of plasmonic enhancement.

IV. METAMATERIALS WITH DIMERS OR NANOLENSES

A. Local field enhancementNow we have concluded that while nonlinear susceptibility

and nonlinear index of refraction do get enhanced significantlyin the simple nanostructures, the strong absorption makesmaximum attainable phase shift less than desired. From theprevious work of ourselves [75,76] as well as others [77,78],we have established that local fields can be enhanced evenfurther in more complicated nanoparticle structures. In thegap between two identical nanoparticles (dimer) [75] or inthe vicinity of a smaller nanoparticle coupled to a largernanoparticle of the same shape (nanolens) [76], the maximumfield enhancement was proportional to Q2 rather than Qfor a single nanoparticle; hence much larger cascadedenhancements of absorption, Raman scattering, and in somecases photoluminescence could be achieved in these hotspots. Therefore, it is tempting to evaluate the possibilityof using the hot spots to enhance nonlinearity. Since we haveshown that in either dimer or nanolens the field enhancement

FIG. 6. (Color online) (a) Gold spherical nanoparticle dimer(a1 = 20 nm, a2 = 10 nm, gap g = 4 nm, r12 = a1 + a2 + d) withthe electric field distribution. (b) Gold elliptical nanoparticle dimer[long (short) axes: 30 nm (6 nm) and 10 nm (2 nm); 5-nm gap]with resonance at 1550 nm and associated electric field distribution.(c) Extinction spectrum of the elliptical dimer.

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J. B. KHURGIN AND G. SUN PHYSICAL REVIEW A 88, 053838 (2013)

is similar, we shall limit our analysis to the case of nanolensonly, as it is easier to describe analytically.

Consider two spherical nanoparticles of radii a1 and a2separated by a vector r12 as shown in Fig. 6(a). The dipoleoscillation equation (9) is augmented by the dipole-dipoleinteraction between the two dipoles associated with the twocoupled nanoparticles,

d2 p1(2)dt2

+ d p1(2)dt

= 20 p1(2) + 201(2) E + 201(2)2 p2(1)

40dr312. (60)

Using the definition of polarizability, Eq. (8), we obtain forthe harmonic field E parallel to the direction of r12,

(20 2 j

)p1 220

a31r312

p2 = 201 E,(61)(

20 2 j)p2 220

a32r312

p1 = 202 E,

or simply

L()Q1p1 2(

a1

r12

)3p2 = 1 E

(62)2

(a2

r12

)3p1 + L()Q1p2 = 2 E,

which leads to the solution

p1 = QL()1 + 2Q

(a1r12

)32

L2() 42Q2( a1a2

r212

)3 E

= 40dL()a31 + 2Q

(a1r12

)3a32

L2() 42Q2( a1a2

r212

)3 E,

p2 = QL()2 + 2Q

(a2r12

)31

L2() 42Q2( a1a2

r212

)3 E

= 40dL()a32 + 2Q

(a2r12

)3a31

L2() 42Q2( a1a2

r212

)3 E. (63)For the maximum electric fields we obtain

Emax ,1 =1

40d2p1a31

= 2Q L() + 22Q

(a2r12

)3L2() 42Q2( a1a2

r212

)3 E,

Emax ,2 =1

40d2p2a32

= 2Q L() + 22Q

(a1r12

)3L2() 42Q2( a1a2

r212

)3 E.(64)

Obviously the maximum enhancement will take place if onecan get this condition,

42(

a1a2

r212

)3

Q2, 2

(a1

r12

)3 Q1. (65)

Essentially, we are going to the limit of a2 0 a1 r12, whichbrings us to

Emax ,1 2QL()

E, Emax ,2

[2QL()

]2E. (66)

As one can see in Fig. 6(a) the field is greatly enhancedin the vicinity of the smaller particle. In our prior work [76],using more precise calculations we have shown that the simpleanalytical results of Eq. (66) can be used as an upper bound onthe field enhancement in the nanolens, or as a matter of fact, inthe nanogap between two particles. In Fig. 6(b) we show thedimer that resonates at the wavelength of choice of 1550 nm, aswell as its extinction spectrum in Fig. 6(c). These results havebeen obtained using precise numerical calculations. Thereforewe can get the enhancement on the order of Q2 for the smallernanoparticle and the prospect seems to look bright.

B. Nonlinear polarizabilityThe high field in the vicinity of the smaller nanoparticle

will cause nonlinear polarization, Eq. (25),

Pnl,2(r,t) = Pmax ,2G2(r)eit , (67)

where G2(r) is the normalized distribution of nonlinearpolarization near the smaller particle. Then according toEqs. (31)(34) this polarization will induce the nonlineardipoles of two particles via the new driving term on theright-hand side of Eq. (62),

L()Q1pnl,1 2(

a1

r12

)3pnl,2 = 0,

(68)2

(a2

r12

)3pnl,1 + L()Q1pnl,2 =

32V2P

max ,2.

We then arrive at

pnl,1 = 2a32Q22(

a1r12

)3Pmax ,2

L2() 42Q2( a1a2r212

)3 ,(69)

pnl,2 = 2a32QL()Pmax ,2

L2() 42Q2( a1a2r212

)3 .As one can see from comparison to Eq. (34), the nonlinear

dipole of the larger nanoparticle 1 experiences additionalenhancement relative to the dipole of the smaller nanoparticle2. But note that now the volume of the smaller nanoparticle ispresent in the numerator of Eq. (69), hence the situation thatis optimum for the external field enhancement in the nanolensequation (65), i.e., the limit of a2 0 a1 r12, is far frombeing optimal for the enhancement of nonlinear polarization.

C. Effective third-order nonlinearity of the nanolens mediumLet us now estimate the effective nonlinear susceptibility of

the nanolens. According to Eqs. (39) and (64) the maximum

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THIRD-ORDER NONLINEAR PLASMONIC MATERIALS: . . . PHYSICAL REVIEW A 88, 053838 (2013)

nonlinear polarization near the smaller nanoparticle is

P12+3max ,2 = 0| (3)(3,2,1)|(2Q)3

[L() + 2Q( a1

r12

)3]L() + 2Q( a1r12

)32[L2() 42Q2( a1a2

r212

)3]L2() 42Q2( a1a2r212

)32 E1 E2 E3 , (70)and substituting it into Eq. (69) we obtain the nonlinear dipole of the larger particle 1 being equal to

pnl,1 = 1630 (3)Q532V1

(a2

r12

)3 [L() + 2Q( a1r12

)3]L() + 2Q( a1r12

)32[L2() 42Q2( a1a2

r212

)3]2L2() 42Q2( a1a2r212

)32 E1 E2 E3 , (71)and the effective susceptibility becomes

(3)eff = 24f (3)3Q5

[L()( a2

r12

)3 + 2Q( a1a2r212

)3]L() + 2Q( a1r12

)32[L2() 42Q2( a1a2

r212

)3]2L2() 42Q2( a1a2r212

)32 . (72)

So what is the maximum attainable nonlinearity enhancement?According to Eq. (66), the local field gets enhanced by a factorproportional to Q2 instead of Q for a single nanoparticle.For Raman scattering, which is also a third-order nonlinearprocess, the enhancement with the nanolens system can beQ8 instead of Q4 for a single nanoparticle, a tremendousimprovement. Can we expect similar improvement for theFWM and other third-order nonlinear processes? The answeris no, because according to Eq. (65) the largest enhancement oflocal fields is always attained when the volume of the smallerparticle becomes negligibly small. But the key characteristic ofEq. (72), already noted above, is the presence of the volume ofthe smaller nanoparticle in the numerator; hence the optimumcondition for the maximum effective (3) will not coincidewith the condition for maximum local field enhancement andoverall enhancement will be less than Q8.

To find this condition we consider the resonant caseL2( 0) = 1 and assume 2Q(a1/r12) 1; then weneed to optimize,

(3)eff 48f (3)32Q6

( a1r12

)6 42Q2( a1a2r212 )3[1 + 42Q2( a1a2

r212

)3]4 . (73)This enhancement reaches its maximum when

42Q2(

a1a2

r212

)3= 1

3, (74)

and equals

(3)eff 5f (3)32Q6. (75)

Well, as one can see, the enhancement of (3) and nonlinearindex n2 provided by the nanolens system is only proportionalto the Q6. This is rather easy to interpret. The local intensityin the nanolens gets enhanced by a factor proportional toQ4 and then, according to Eq. (52) the nonlinear refractiveindex change gets enhanced by the same additional factor Q2,whether it is a single particle, nanolens, dimer, or nanoantenna.The additional enhancement provided by the coupled particlescomposite equation (75), compared to the isolate nanoparticle

composite equation (43), is about

(3)eff,2

(3)eff,1

512

Q2

, (76)

i.e., a factor on the order of 100. Overall enhancement forthe previously considered case of 1.35, Q 12, and f =0.01 in chalcogenide glass can be as high as 1 105, but therelevant question is what it means in terms of maximum phaseshift that can be obtained.

D. Maximum attainable phase shift in nanolensThis maximum shift can be obtained in a way similar to

Eq. (49),max 1.73Q5 n2

ndI0. (77)

Therefore the pump optical intensity required to achieve -phase shift is I 8 107 W/cm2, i.e., less than 1 W ofpeak power into a 1-m2 waveguide. This appears to be areasonable power, but, of course the problem is that the localintensity is enhanced according to Eqs. (64) and (74) roughlyby

ImaxI

=E

max ,2E

2

94Q4 6 105, (78)

where we have used L2 () 1 near the resonance, indi-cating that the local intensity is on the scale of Imax 5 1013 W/cm2 which is way beyond the optical damage value.If we introduce once again the maximum local nonlinear indexas nmax = n2 Imax = 94Q4n2 I0, Eq. (77) can be rewritten as

max 0.23 Q3

nmax

nd. (79)

This result for the nanolens is even worse (by a factor ofabout 5) than the result in Eq. (53) for the isolated nanopar-ticles. Clearly, the dependence 4Q is common to any typeof nanostructure, monomer, dimer, trimer, or nanoantenna.The maximum achievable index of refraction nmax changesthe resonant frequency according to Eq. (52) which providesenhancement by the factor of Q2, but then the absorptioncoefficient also gets enhanced by the factor of Q so only a

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J. B. KHURGIN AND G. SUN PHYSICAL REVIEW A 88, 053838 (2013)

single factor ofQ survives in the end. The factor in front of 3Qis reduced in dimers and more complicated structures relativeto the monomers simply because a smaller fraction of the modeenergy is contained in the region where the index change ismaximal. Hence one should not expect any improvement inthe maximum obtainable nonlinear phase shift max beyonda single factor of Q in more complicated structures such astrimers, bowtie antennae, and so on, even if the effectivenonlinear index can be enhanced beyond the already hugeenhancement in Eq. (75). Giant enhancement of nonlinearitywill only mean that the nonlinear phase shift will saturateat much shorter distance but at essentially the same value ofEq. (53) or less, indicating that, to the best of our knowledgeon existing materials, it is impossible to achieve true all-opticalswitching using plasmonic enhancement.

V. DISCUSSION

Everything said above can be summarized in a simple figurewhich tells it all (Fig. 7). We consider a 1-m2 cross-sectionwaveguide made of chalcogenide glass with nonlinear indexn2 = 1013 cm2/W. We consider the third-order nonlinearprocess of XPM (although the case of SPM is no different).The peak pump power is P0 = 1.6 mW so that the intensityinside the waveguide free of metal nanoparticles is I0 = 1.6 105 W/cm2 and the nonlinear index change is n0 = 1.6 108. If the waveguide is filled with solitary Au nanoparticles,the local intensity is enhanced by a factor of (2Q)2, toIs = 1.8 108 W/cm2, and the local nonlinear index changeis ns = 1.8 105. When the waveguide is filled with Audimers optimized according to Eq. (74), the local intensity isenhanced by Eq. (78) to Id = 1.0 1011 W/cm2 which causes

FIG. 7. (Color online) Nonlinear phase shift in the 1-m2chalcogenide waveguide: (a) input power of 1.6 mW, no plasmonicenhancement; (b) input power of 1.6 mW, nonlinearity enhanced bythe elliptical Au nanoparticles with the filling factor of 0.001; (c) inputpower of 1.6 mW, nonlinearity enhanced by the dimers of elliptical Aunanoparticles with the filling factor of 0.001; (d) input power of 0.8W, no plasmonic enhancement; (e) input power of 0.8 W, nonlinearityenhanced by the elliptical Au nanoparticles with the filling factor of0.001; (f) input power of 8 W, no plasmonic enhancement.

local index shift nd = 0.01a rather nonrealistic value inview of potential optical damage and saturation, but we shallconsider it to be an upper limit.

The nonlinear phase shift versus distance curves are shownin Figs. 7(a)7(c) for the three aforementioned cases. In thewaveguide without nanoparticles [curve (a)] the nonlinearphase shift increases gradually and reaches 2 103 rad atlength z = 3 cm. For the waveguide impregnated with Aumonomers with the volume fraction f = 0.001 [curve (b)]the nonlinear shift is much larger but at z 5 m the phaseshift saturates at a rather small value of 2 105 rad. In thewaveguide impregnated with Au dimers with the same volumefraction f = 0.001 [curve (c)] the enhancement is stronger andit also saturates at 5 103 rad. In all three cases the required180 shift is attainable at this low power.

If we increase the input power by a factor of 500 toP0 = 800 mW the nonlinear shift in the waveguide withoutnanoparticles [curve (d)] would now reach a value of 1 rad atz = 3 cm. For the waveguide with Au monomers [curve (e)]the phase shift will now saturate at 0.02 rad and this is themaximum phase shift attainable because local intensity nowapproaches the damage threshold of 1011 W/cm2. There is nocurve for the waveguide with dimers because the local intensitythere would be nearly 1014 W/cm2 which is way beyondoptical damage. Once again, no full optical switching canbe achieved over a reasonably small distance.

Finally we can further increase input power by anotherfactor of 108 W. Now one cannot use waveguides withplasmonic enhancement because local intensities would bebeyond optical damage threshold for both monomers anddimers. So the only curve (f) is that for the waveguide withoutnanoparticles and as one can see the phase shift indeed achieves -phase shift at a length of about 9 mm.

VI. CONCLUSIONS

Thus we arrive at a rather dichotomous conclusion. On onehand, using waveguides impregnated with metallic monomers,dimers, and other constructions (one may call them plasmonicmetamaterials) allows one to achieve huge enhancement ofeffective nonlinear index, up to the order of 105 and moredue to the high degree of field concentration in the hot spotsOn the other hand, strong absorption in the metal causessaturation of the nonlinear phase shift for SPM and XPMor frequency conversion efficiency in the case of FWM andOPG at very short distances. Given the fact that maximumlocal index change is limited, generously, to about 1% dueto optical damage, the nonlinear phase shift saturates at avery small value of a few tens of milliradianswhich isinsufficient for any photonic switching operation. Similarly,conversion efficiency saturates at values less than 30 dB,making use of plasmonic nonlinear metamaterials for thispurpose highly inefficient. It is also clear that changes in Q bya factor of 23 that might be attainable in silver (although notdemonstrated to date due to oxidation and surface scattering)will not change the results in any substantial way, and onlyassure earlier saturation of the nonlinear conversion. The oneand only advantage of nonlinear plasmonic metamaterial is thatnonlinear effects may be observable at very small propagationdistances of a few micrometers with reasonable (but not low!)

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THIRD-ORDER NONLINEAR PLASMONIC MATERIALS: . . . PHYSICAL REVIEW A 88, 053838 (2013)

optical powersbut there is a giant chasm between beingobservable and being practical, and at this point, with theexisting metals and nonlinear materials, one cannot see howthe nonlinear plasmonic metamaterials can bridge this chasm.

In retrospect, our rather unenthusiastic conclusion aboutthe prospects of using plasmonic resonances to enhancenonlinearity does not appear to be surprising at all. Numerousresonant schemes for enhancement nonlinearity have beenproposed and investigated at length [79,80]. Some of theschemes rely upon intrinsic material resonances; others tryto take advantage of photonic resonant structures, such asmicroresonators and photonic crystals. The Q factor of theresonances ranges from a few hundreds to tens of thousands,and yet in the end, none of the resonant schemes has foundpractical applications to this day, due to the fact that resonanceis always associated with excessive absorption and dispersion.To this day optical fiber remains the nonlinear medium ofchoice in which low nonlinear coefficients are more thancompensated by the long propagation length and high degree ofconfinement. All kinds of all-optical switching and frequencyconversion techniques have been successfully demonstrated infiber [81]. The only other media in which all-optical switchinghas been consistently demonstrated is semiconductor opticalamplifier (SOA) in which the loss simply does not exist due tooptical gain. Neither fiber nor SOA relies upon any resonance

despite its apparent appealone always tries to avoid loss andexcessive dispersion.

So if the numerous relatively high-Q resonant schemesfor enhancing optical nonlinearity have failed to achievepracticality, it would have been naive to expect plasmonicresonance in metal nanoparticles with Q barely of the order of10 to succeed where so many have failed. Thus in retrospectthis work only confirms the obvious. And yet this obvious facthas not been universally accepted by the community, and wehope that our effort has been useful as it has revealed the natureand limitations of the plasmonic enhancement of (3) in greatdetail and without reliance on excessive numerical modeling.

We emphasize that it has not been our purpose to makebroad predictions of where the research in nonlinear plas-monics research may go in the future; our modest goal wasto provide a set of simple expressions and relevant numbersfor others so they can ascertain the prospects for usingnonlinear plasmonic metamaterials in their own applications.Still we may attempt to make a very general statement, thatplasmonically enhanced structures in nonlinear optics mightnot find too many applications requiring decent efficiency, suchas switching, wavelength conversion, etc., but may be of usein such applications where efficiency is not much of an issuesuch as sensing, as well as in fundamental studies of opticalproperties of different materials under extremely high fields.

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