October 1985 / Vol. 10, No. 10 / OPTICS LETTERS 511

Surface-mediated enhancement of optical phase conjugation inmetal colloids

D. Ricard, Ph. Roussignol, and Chr. Flytzanis

Laboratoire d' Optique Quantique, Ecole Polytechnique, 91128-Palaiseau, Cedex, France

Received May 24, 1985; accepted July 30, 1985

We show that the optical phase-conjugated reflectivity from silver and gold colloids is enhanced by several ordersof magnitude. The reflectivity on resonance is comparable with that of CS2 for metal-particle volume concentra-tion of a few parts in 106. We trace this enhancement to the nonlinearities of the electrons in the metal particlesand extract the value of their optical Kerr-effect coefficient.

The amplitude of the electric field induced at the sur-face of a spherical metallic particle of diameter d by anoscillating electromagnetic field of wavelength X (>>d)exhibitsl 2 a dramatic enhancement at the surfaceplasma frequency w,, which is fixed by the condition'

Em'(ws) ± 2EdG(08) = 0, (1)

where Em = Em' + icm", complex, is the dielectric con-stant of the metallic particle and Ed, real, is that of thesurrounding dielectric. This surface-mediated en-hancement of the electromagnetic field is the maincause3 of the spectacular increase of the effective crosssections of optical effects emanating from molecules inthe dielectric at or close to a rough metallic surface(Raman scattering, second-harmonic generation). Itseffect, however, is drastically reduced in proportion tothe distance of the molecules from the center of themetal particle.

On the other hand, one expects this enhancement topreserve its full impact inside the entire internal areaof the metallic particle, provided that that area is muchless than the skin depth in the metal. In this quasi-static limit, a simple electrostatic approach 4 shows thatthis is indeed the case, and the optical properties of themetallic sphere will also be enhanced. If many such finemetallic spheres are dispersed in a dielectric, the opticalproperties of the spheres, linear and nonlinear, will besubstantially modified. Indeed, colloids of metal par-ticles and other similar metal-dielectric compositesfrequently exhibit2 optical properties, such as the op-tical anomaly in the visible range, strikingly differentfrom those of the constituent materials.

In this Letter we show that the same effect globallyendows such materials with large cubic optical non-linearities, much larger than those of the pure bulkmaterials. In particular, the value of their optical Kerrcoefficient measured at the surface-mediated reso-nance' in the optical phase-conjugation configurationis found to be enhanced by several orders of magnitudeover the one measured off resonance. For volumeconcentrations of the metallic particles not exceeding5 X 10-6 the measured values of x(3) are comparable withthat of CS2 and are entirely due to the electrons of themetallic particles. We also find that a simple extension

of the Maxwell-Garnett theory,5 used until now to de-scribe the linear optical properties at such low concen-trations, allows us to account for this enhancement ofthe optical nonlinearities and to extract the value of theintrinsic third-order susceptibility for the optical Kerreffect of the pure metal. To our knowledge this is thefirst time that this value has been measured, and thismethod opens new possibilities for the still unexplorednonlinear-optical properties of metals.

Let us first derive the expected enhancement factorfor a colloid of metal particles of low volume concen-tration that corresponds to our experimental situation.Each metallic particle of size much less than the wave-length of the probing optical beam is assumed to beentirely surrounded by the dielectric, and its overallfractional occupancy is p (<<1). Because of the aver-aging that takes place in this limit the composite can betaken as translationally and rotationally invariant. Inparticular, each metallic particle can be replaced by aspherical average one. The effective dielectric constant? of such a composite medium in the long-wavelengthlimit, neglecting interactions among particles, is simplyderived by Lorentz local-field arguments6' 7 and satis-fies

E-Ed - Em-EdZ + 2Ed Em + 2 Ed

(2)

The dielectric constant E m of a small particle, becauseof quantum size effects,8 may not necessarily be thesame as in a pure metal. For p << 1 one obtains

E = Ed + 3P(d m - EdEm + 2Ed

(3)

which exhibits a resonance at a frequency such that Em'

+ 2Ed = 0, the same as in Eq. (1). The width of this

resonance is determined by Em", the imaginary part ofem, and from Eq. (3) one can easily extract the extinc-tion coefficient

a=pEd' Emat = 9P C Ed 36/2 + )2 + e tr2 (4)c (Em' + 2Q) +Emd 4

The beautiful colors of the colloids of gold and silverreflect this dielectric anomaly, which occurs at thefrequencies given by the resonance condition [Eq. (1)],

0146-9592/85/100511-03$2.00/0 © 1985, Optical Society of America

512 OPTICS LETTERS / Vol. 10, No. 10 / October 1985

a translation to the visible of the anomaly that normallyoccurs at zero frequency in the pure metal. It is es-sential to stress that this originates only from the elec-trons of the metallic particles.

We now apply a strong optical field E0 of frequencyw and fix our attention on the component of the inducedcubic nonlinear polarization PfLs(co) that is relevant tothe optical Kerr effect that we measure. This compo-nent is characterized by a change of the effective di-electric constant M. In line with what was stated above,it is plausible to accept that this change be results froma change 3e ' of the dielectric constant of the metallicparticles; the relation between the two is simply ob-tained from Eq. (3):

6/ = f2Pbem, (5)

where, differentiating Eq. (3),

( I 3f5d 12(6f2 =(E 2d*(6)

If only the electrons contribute to bem, then it is simplyrelated to the third-order susceptibility Xi,(3)(W 2 -co,

co) of the metal by the relation

&,m = 127rXm(3) El2,

where E1 is the local field prevailing over the metallicparticles and is easily derived4 by electrostatictheory:

El 3 Ed E. = fE.. (7)Em+ 2 Ed

Since f2 = f 2 , Eq. (5) can now be written as

= 12rpfl4Xm (3)EEo2 . (8)

The relevant polarization is

Pfts = 3pf 14 X. (3)E. 3, (9)

which could also be obtained directly by expressing PALsas the density of dipoles induced on the metallic parti-cles with the appropriate local-field correction.9

The essential result is that the factor fl occurs in thefourth power, and consequently, when the frequency cosatisfies condition (1), the resonance enhancement isgreatly amplified. Since outside the sphere the localfield to the fourth power decreases rapidly4 with thedistance from the center, only a small volume of thedielectric (water in our case) would experience the en-hancement. Since the X(3) of water is small, our ex-perimental observations lead to a nonlinear responsedominated at resonance by the metallic spheres. In themeasurements of the phase-conjugated beam that wereport below, the relevant quantity actually is IPQLSI 2,

so the enhancement occurs as Ifi l.8 We notice that atresonance JflRj > 1, whereas off resonance 1flNR1 < I.Comparing a resonant and a nonresonant situation, wemay define a relative enhancement factor F =1fi'M 8/1 flNR1 8. Not withstanding the absorption loss atresonance, this leads to an increase of the phase-con-jugated reflectivity at resonance by several orders ofmagnitude above its value off resonance. The ab-sorption losses reduce this enhancement only by the

factor10 4e-LI(acL + 2)2, where a is given by Eq. (4)and L is the interaction length inside the composite.Our experimental results confirm these expectations.

In our experiments, we used the classical configura-tion for optical phase conjugations: A weak probebeam and two equally intense counterpropagatingpump beams were incident upon the sample, giving riseto a phase-conjugated wave whose intensity was mea-sured; the probe was incident at a small angle (-70) withrespect to the forward pump and was polarized parallelor perpendicular to the pump. All these beams, of thesame frequency, were delivered by the same source, amode-locked Nd:YAG laser, of pulse duration 28 X10-12 sec. The three pulses could be delayed with re-spect to one another.

The samples were gold and silver colloids prepared2

by reduction of a gold and a silver salt, respectively. Toavoid aggregation, freshly prepared hydrosols were usedand placed in a 2-mm cell. The volume fraction p was5 X 10-6 and 1.5 X 10-6 for gold and silver colloids, re-spectively. Transmission electron micrographs(TEM's) show that our freshly prepared hydrosols arefree of aggregates. Our gold hydrosols are fairly stable,showing a slow evolution on a month time scale. Oursilver hydrosols are less stable and evolve on a one-daytime scale. TEM micrographs also show that our me-tallic particles are spherical in shape to a high degree ofaccuracy, with an average diameter of 10 nm for goldand 7 nm for silver and standard size dispersions.

The spectrum of the gold colloids can be well ac-counted for in terms of expression (4) and the mea-sured'1 dielectric constant of pure gold. For the silvercolloids a correction had to be introduced, as the spec-trum calculated from Eq. (4) was sharper than themeasured one. This implies that em" for a small silverparticle is larger (by a factor of 5.5) than in bulk silver12

because of the size-limited scattering time experienced

3

ZI

-40 -20 0 20 40 60 80Time delay (10-12 sec)

Fig. 1. Normalized conjugated signal as a function of theforward or backward pump-pulse delay. Notice that it peaksat zero delay.

October 1985 / Vol. 10, No. 10 / OPTICS LETTERS 513

by the electrons. The effective mean free path is thusfound to be 2 nm. The (small) discrepancy betweenthis value and the average radius may be due to grainboundaries.

The phase-conjugated reflectivity for gold colloidswas measured off resonance at X = 1.064 Atm, thewavelength of the Nd:YAG laser, and at resonance atX = 0.532 Aum, its harmonic. For silver colloids thewavelengths 0.532 and 0.396 ,um were used off and atresonance, respectively, the second one being obtainedby stimulated Raman scattering of the third harmonicin ethanol. The phase-conjugated reflectivity mea-surements were done relative to that of a CS2 sample ofthe same thickness.

For both colloids, the reflectivity near resonance isless by a factor of 5 than for CS2 for parallel probe andpump polarizations. By varying the delay of eitherpump, we found that the main response was essentiallyinstantaneous and was superimposed upon a muchweaker slow component (see Fig. 1); the latter pre-sumably is of thermal origin. For crossed probe andpump polarizations the signal drops by a factor of 64 forthe gold colloid. This is consistent with a resonantelectronic process, given the good sphericity of ourmetallic particles. In the case of silver, we have alsoperformed measurements at X = 0.355 Am, the thirdharmonic of the laser. At that wavelength, the reflec-tivity of the colloid is about 15 times weaker than thatof benzene (CS2 is opaque at X = 0.355MAm), which is ingood agreement with our theoretical model. Off reso-nance the phase-conjugated reflectivity is less by afactor of 3000 than that of CS2 and is totally due to thewater and the glass windows of the cell. Taking intoaccount the reduction factor resulting from absorptionlosses at resonance, this implies a value for F well above3000 and 5000 for gold and silver colloids, respectively.Such factors should be considered only as lower bounds,in agreement with the theoretical predictions: 1.3 X 1010for gold and 3.6 X 106 for silver, as obtained from Eq. (7)using the appropriate value of Em.

Taking the value 13 X(3 )(G, -W, C) = 2 X 10-12 esu inCS2 at X = 0.532,um and assuming the same dispersionfor the optical Kerr X(3) of CS2 as for its static Kerrconstant, 14 we obtain

X(3)((J -a, W) = 1.5 X 10-8 esufor gold spheres at 0.53 Am,

X(3 )((s -W, W) = 2.4 X 10-9 esufor silver spheres at 0.40 Mm.

This is the first reported experimental determinationof optical Kerr coefficients for a metal. Although thephysical effects are not directly comparable, these val-ues are higher by 2 orders of magnitude than those ex-

tracted by third-harmonic reflectivity measurements. 15

Since there is now no theoretical model for the opticalnonlinearities in metals, no conclusions can be drawnregarding this difference. Furthermore, the suscepti-bility of a small metallic particle may be different fromthat of the bulk.

In conclusion, we have shown that the optical non-linearities of metal colloids are substantially enhancedby a resonance that is due to the effective medium, thesurface plasma resonance, as expected by direct ex-tension of the Maxwell-Garnett theory. This allowedus to extract values of the nonlinear optical coefficients.Since these systems and other composites in general canbe prepared with different concentrations and sizes ofparticles, we anticipate that this technique will be usefulfor studying the optical nonlinearities in metals byvarying different parameters over a wide range andsimilarly useful16 for studying semiconductor-dopedglasses.

We are indebted to K. C. Rustagi for numerous dis-cussions and helpful comments, to F. Hache for assis-tance during part of this work, and to P. Ballongue forthe TEM's. The Laboratoire d'Optique Quantique isaffiliated with the Centre National de la RechercheScientifique.

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3. See, for instance, Surface Enhanced Raman Scattering,R. K. Chang and T. E. Furtak, eds. (Plenum, New York,1982).

4. See, for instance, C. J. Bottcher, Theory of Electric Po-larization (Elsevier, Amsterdam, 1973), p. 78.

5. J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London 203,385 (1904); 205, 237 (1906).

6. R. Landauer, AIP Conf. Proc. 40, 2 (1978).7. D. K. Hale, J. Mater. Sci. 11, 2105 (1976).8. L. Genzel, T. P. Martin, and 0. Kreibig, Z. Phys. B21,3391

(1975).9. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P.

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