ENHANCED NONLINEAR-OPTICAL RESPONSES OF DISORDEREDCLUSTERS AND COMPOSITES

Mark I. Stockman*, Lakshmi N. Pandey#, and Thomas F. George§

*Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303-3083#Departments of Physics and Chemistry, Washington State University, Pullman, WA 99164-4630§Office of the Chancellor / Departments of Chemistry and Physics & Astronomy, University of Wisconsin-Stevens Point, Stevens Point, WI 54481-3897

1. IntroductionClusters and nanocomposites belong to so-called nanostructured materials. Properties of suchmaterials may be dramatically different from those of bulk materials with identical chemicalcomposition. Confinement of atoms, electrons, phonons, electric fields, etc., in a small spatialregion modifies spectral properties (shifts quantum levels, changes transition probabilities), andalso changes the interaction between the constituent particles. In this paper we concentrate onan important source of the modification of properties, namely on local fields.

When an aggregate of particles (we will call such particles monomers) is subjected to ahomogeneous (or smooth) external field, the local fields at different monomers differ from theaverage (macroscopic) field. If the system is random, then the local fields also strongly fluctuatein space from one monomer to another. We will consider optical fields created by externalelectromagnetic wave, mainly their electric component.

Some systems have pronounced resonant properties and react to an external field with high localfields. As an example, we can name metallic (especially silver, gold, or platinum) colloidalclusters or rough surfaces. The local fields affect monomers and molecules surrounding themand create strongly-enhanced optical responses. A typical example of such responses is surface-enhanced Raman scattering (see, e.g. a review of Ref. 1 and reference therein).

The most pronounced effect of the fluctuating local fields is on nonlinear optical susceptibilities.The reason for that can be understood qualitatively. Imaging two fields with the same average

intensity E2 , one of them is constant in space and the second changing (fluctuating). Of

course, the value of an n-th power of the intensity, E2n , will always be greater (and possibly,

much greater) in the second case, and the higher the n, the greater the difference.

To maximize this effect, our goal is to find systems with the maximum spatial fluctuations of thelocal densities. We certainly expect that the density fluctuations will cause correspondingly largefluctuations of the local fields.

There exists a class of systems that stands out in this respect. These are self-similar (fractal)systems, which (approximately) repeat themselves at different scales. In other words, looking atsuch a system and not seeing its boundaries (neither at the maximum or a minimum scales), onecannot say what fraction of the system is observed, and what is the actual size of the objectsseen. For such systems, the number N of constituent particles (monomers) contained within aradius R scales as

In: Nonlinear Optical Materials, IMA Volumes in Mathematics and Its Applications,v.101, p.225, J.V.Moloney, ed., Springer-Verlag, New York, 1998.

2

NR

R

D

≅

0

,

(1)

where R0 is a typical distance between monomers, and D is the fractal (Hausdorff) dimensionof the cluster. The density of monomers as given by Eq. (1) is asymptotically zero for largeclusters,

ρ ≅

→−

−

RR

R

D

03

0

3

0 .

(2)

However, this does not mean that the interaction between monomers can be neglected. Theunderlying reason for that is a strong correlation between monomers in a cluster, with the pair-pair correlation function scaling similar to Eq. (2). Thus we have a unique system whosemacroscopic density is asymptotically zero, but the interaction inside the system is strong. Thisidea has been proposed by us in an earlier papers.2,3

An example of a fractal cluster, obtained by cluster-cluster aggregation4 (CCA) is shown in Fig.1. This figure illustrates many properties of fractal clusters, including the low overall density andstrong correlation in the positions of monomers.To avoid possible misunderstanding, we point out that other, non-fractal systems may alsopossess enhanced optical responses, especially those that are tailored to have optimally-chosendielectric properties changing in space5,6. The physical origin of the enhancement in this case isthe same: the local fields change in space, and consequently the higher moments of theirintensity are increased.

2. Equations Governing Optical ResponsesWe concentrate on the universal interaction between the monomers, namely the dipole-dipoleinteraction. We assume that the system under consideration is subjected to the electric field E ofthe incident optical wave. This field induces the dipole diα at an i-th monomer of the cluster

(here α = x y z, , denotes the Cartesian components of the vector, and similar notations will beused for other vectors). The dipole moments satisfy well-known system of equations

( ) ( )α δα α αβ

α β β0

1 02

133−

== − −

∑d E

r r

r

d

ri i

ij ij

ijj

Nj

ij

( ) .

(3)

Here Eiα( )0 is the wave-field amplitude at the i-th monomer, r r rij i j= − is the relative vector

between the i-th and j-th monomers, and α0 is the dipole polarizability of the monomer. Weassume that the size of the system is much less than the wavelength of the exciting wave, andtherefore the exciting field Eα

( )0 is the same for all the monomers of a cluster. We note that the

dipole interaction is not valid in the close vicinity of a monomer. Our choice of interaction is

3

justified if intermediate-to-large scales predominantly contribute to the properties underconsideration.

The normal procedure applied to Eq. (3) would be the mean-field approximation, where weneglect the difference between the different induced dipoles and replace the summation in Eq.(3) by the integration with the density (2). However for a nontrivial fractal (D < 3), thisprocedure leads to an “ultraviolet” divergence, d Ri

D∝ → ∞−0

3 for R0 0→ . This implies thatthe mean-field approach fails in application to nontrivial fractal systems.

3. Scaling and Spectral RepresentationThe scaling theory of the optical response of fractal clusters have been developed in Refs. 7 and8. The material properties of the system enter Eq. (3) only via the combination Zrij

3 , where we

have introduced the notation Z ≡ −α01 . This along with the (approximate) self-similarity of the

system is a prerequisite for scaling in terms of the spectral variable Z . The important additionalrequirement for the scaling of a certain physical quantity F is that the system eigenmodescontributing to F should be in their localization radii L intermediate between the maximumscale (size of the cluster Rc ) and the minimum scale, R0 . Then, on one hand, the quantity F has

the functional dependence ( )F F ZR= 03 . On the other hand, the eigenmodes contributing to F

are insensitive to a much smaller minimum scale. Therefore the dependence on R0 can be onlypower (scaling), and consequently

( )F Z ZR( ) ∝ 03 γ

,

(4)

where γ is a scaling index.

Due to the above arguments, it is convenient to express all results not in terms of frequency, butin terms of Z , separating the imaginary and real parts, Z X i= − − δ . The choice of signs in thisexpression makes the dissipation parameter δ positive, while the spectral parameter X ispositive when the frequency is blue-shifted from a resonance, and negative otherwise. We notethat a similar spectral approach has been independently introduced and used by R.Fuchs andcollaborators.9,10 For the sake of reference, we give here the expressions for X and δ for ametallic nanosphere in the Drude model,

( ) ( ) ( )X

R Rm

a

a ps

m

a

a p

=+

−+1 2

3

1 2

3 230

3 2

30

3 2ε ε

ε ωω ω δ

ε εε ω

γ; = ,

(5)

where ε0 is the intersubband dielectric constant of the metal, εa is the ambient dielectric

constant, ω p is the metal plasma frequency, ωω

ε εs

p

a

=+0 2

is the surface plasmon

frequency, and Rm is the nanosphere’s radius.

4

The spectral dependence of X and δ for silver is illustrated in Fig. 2. The most importantfeature in the figure is that in the yellow-red region of visible light, the real part of thepolarizability greatly exceeds its imaginary part. Their ratio has the meaning of the quality factorof the surface-plasmon resonance Q,

Q X= δ .

(6)

This factor shows how many times on the order of magnitude the amplitude of the local field ina vicinity of a resonant monomer exceeds that of the exciting field. The fact that for manymetals Q may be large (as large as ≅ 102 ) plays an important role in the theory since theenhancement of the optical responses is a resonant phenomenon, and strong dissipation wouldcompletely suppress it.

To introduce the spectral representation,7,8 we will rewrite Eq.(3) as one equation in the 3N-

dimensional space. To do so, we introduce 3N-dimensional vectors ) )d E, ,( ) 0� , whose

projections give the physical vectors

( ) ( )i d d i E Ei iα αα α= =, ( ) ( ) ,0 0� .

(7)

Equations (3) then acquire the form

( ) ) )Z W d E+ = ( )0 ,

(8)

where W is the dipole-dipole interaction operator with the matrix elements

( )( ) ( )

i W j

r r

r ri j

i j

ij ij

ij ijα β δαβ

α β

= −

≠

=

31

0

2 3 ,

, .

,

(9)

We introduce the eigenmodes (surface plasmons) | )n ( ,...., )n N= 1 3 as the eigenvectors of theW-operator,

) )W n w nn= ,

(10)

where wn are the corresponding eigenvalues. Practically, the eigenvalue problem (10) can besolved numerically for any given cluster. Having done so, one can calculate the Green’s function

5

( )( )G i j

nn

N i n j n

Z wα β

α β, =

+=∑

1

3

,

(11)

which carries the maximum information on the spectrum and linear response of the system.

4. Linear Optical ResponsesThe polarizability of a cluster α and density of eigenmodes ρ are expressed in terms of G as

α ραβ α β α α= G = Gi ji j

i ii j

,,

,,

,∑ ∑ ,

(12)

where summation over repeated vector indices is implied. Let us invoke a large magnitude ofthe quality factor of the optical resonance (6), Q >> 1. In this case, a dependence of type (4)

becomes ( )F X R X( ) ∝ 03 γ

. We have introduced7,8 such a dependence for ααβ ( )X and

ρ( ) X and argued that the two quantities have the same scaling,

Im Imα ρ(X) (X) R R Xdo≅ ≅

−

03

03 1

,

(13)

where do is an index that we called the optical spectral dimension. We have also argued thatthe physical range of do is 1 0≥ ≥do .

Essential for the derivation of Eq.(12) has been the strong localization hypothesis stating that alleigenmodes (at least all contributing eigenmodes) of a cluster are strongly localized.11 The lattermeans that for any given frequency parameter X there exists only one characteristic length LX

of these eigenmodes playing the role of simultaneously their wavelength and their localizationlength. Using scale invariance arguments, we have shown7,8 that LX scales as

L R R XX

d

Do

≅−

−03

1

3 .

(14)

We have subjected the scaling predictions of Eqs.(13) and (14) to an extensive comparison withthe results of large-scale computations.12 The results are quite unexpected. One of those, apolarizability and eigenmode density for cluster-cluster aggregates (CCA), is shown in Fig. 3.The conclusion that one can draw from the figure is that neither the polarizability, nor density ofeigenmodes scale. Interestingly enough, they still appear to be quit close to each other,supporting the conclusion of Refs.7 and 8 that all eigenmodes of a fractal cluster contribute(almost) equally to its optical absorption. Similar results have been obtained12 for other types ofclusters.

6

Another relation to check is that of Eq.(14). First, one has to formulate how to calculate thelocalization radius (an exact definition is not important, if the strong localization really holds).We use the definition of Ref.12,

( ) ( )L

n i n i

X

n ii

iin

n

=

−

∑ ∑∑

∑

ρ α α

ρ

β βr r2 2 2

2

,

(15)

where ( )[ ]ρ δn nX w= − +−2 2

1

. The computed dependence is shown in Fig. 4. As one can

clearly see, there is no scaling in these data also. This finding is in contradiction to the oppositestatement of Ref.13. Considering this contradiction, one should take into account that theprecision of our calculations is much higher than in Ref.13.

Evident failure of the scaling implies that at least one of the assumptions lying at its foundationis incorrect. Because we used the same model (dipole-dipole) for both the scaling theory and thenumerical computations, the non-applicability of the model to system is out of question. In ourconsideration we consistently used high values of Q , so that the condition Q >> 1 is alsosatisfied. The only cause of the failure of scaling appears to be the strong localizationassumption. This implies that in any frequency (X ) range, there are eigenmodes that arecomparable in size either to the minimum scale R0 or to the maximum scale Rc , or both.

To examine this problem, we have undertaken a large-scale computational investigation of thelocalization properties of eigenmodes for several types of clusters. The results are presented inthe next section. They reveal a new pattern of localization.

5. Inhomogeneous Localization Of Eigenmodes

To go to the root of the problem, we have considered14,15 all eigenmodes of a single cluster. Wehave designed a special plot: each eigenmode is represented by a point in the coordinateslocalization length LX versus the frequency parameter X . All clusters demonstrate essentiallythe same localization pattern that we call the inhomogeneous localization. An example of such abehavior is shown in Fig. 5. As one can see, at any frequency (value of X ) within a wide rangeof X , the eigenmodes have a very broad spectrum of their localization radii LX , from theminimum scale ≅ R0 of the distance between the monomers to the maximum scale ≅ Rc of thecluster total radius.

It is also useful to take a closer look at some members of the ensemble of eigenmodes presentedin Fig. 5. We take two pairs of eigenmodes that have almost equal frequencies (inside a pair).The spatial distribution of the intensity of these eigenmodes is presented in Fig. 6. As we cansee, at a comparatively large R X0

3 128= − . , the eigenmode is indeed a very well localized,demonstrating a very sharp peak over a few monomers. The localization radius for thiseigenmode is indeed very small, L RX c/ .= 0 034. Surprisingly, an eigenmode with a very

close frequency (R X03 120= − . ) is almost completely delocalized (for this eigenmode,

L RX c = 0 89. ). However, the examination of its spatial profile shows that in reality this

7

eigenmode consists of two sharp peaks separated by a distance on the order of the total size ofthe cluster. As we go closer to the plasmon resonance (smaller X ), the minimum width of the

peaks increases, as one would expect in view of (14). At the same time, the internal structure ofthe eigenmode remains highly irregular with large spatial fluctuations. Again, the eigenmodes atthe same frequency possess very different localization radii.

The behavior described above is very unusual. For the known localization patterns, eigenmodesare strongly localized for short wavelength (high frequencies) and weakly localized (delocalized)in the long-wavelength wing. The physical reason for this is that a long wave (wavelengthλ >> the typical size of the scattering inhomogeneities) sees almost homogeneous medium andpropagates almost freely. In contrast, a short wave is strongly scattered from inhomogeneitieswith sizes on the order of λ (strongly here means that the scattering length itself is on the orderof λ ). Thus, there exists the mobility edge, a frequency above which waves are localized andbelow which they propagate.

This logic leading to the existence of the mobility edge is obviously inapplicable to fractals.They are self-similar systems and, therefore, do not possess any characteristic scattering length.For any eigenmode wavelength λ , there always exist inhomogeneities of the sizes comparableto λ . This might mean that all the eigenmodes are strongly localized, as assumed in Refs.7 and8. However, the result presented above and all numerical modeling of Refs.12, 14 and 15 haveshown that the strong localization is not the case. In actuality, the inhomogeneous localization15

takes place, where the eigenmodes in a wide range of the localization radii coexist at any givenfrequency.

To examine statistical properties of the eigenmode distribution, we introduce the distributionfunction G rX ( ) , which is the density of the probability that an eigenmode at a given X islocalized within the radius of r . The intensity of an nth eigenmode at an ith monomer is given

as the sum over polarizations, ( )n iαα

2∑ . So the probability density that the eigenmode is

present simultaneously at i - th and j - th monomers separated by the distance r is

( )( ) ( )g r r n i n jn i ji j

( ), ; ,

= − −∑ δ α βα β

r r2 2

,

(16)

where ( )δ � is Dirac’s δ -function.

The required distribution function G rX ( ) can be obtained by summing g rn ( ) over alleigenmodes with eigenvalues in the vicinity of a given X . Assuming that the eigenmodes arehomogeneously broadened with the width of δ , the corresponding weight factors are naturallychosen in the Lorentz form,

8

( )ν

δn

n

XX w

( ) =− +

12 2

.

(17)

We note that the total density of eigenmodes is then given by ν ν( ) ( )X Xnn

= ∑ . Taking this

into account, we obtain the following ansatz for the distribution function:

( ) ( ) ( ) ( )G r g r X XX n nn

= ∑ ν ν ,

(18)

where ( )g rn and ( )ν n X are given by Eqs.(16) and (17), and � denotes the average over

an ensemble of the clusters.

The self-similarity of the clusters suggests that there may exist a region of distances in which thefunction ( )G rX does not possess a characteristic length and, therefore, should scale,

( )G r rXDe∝ −1 ,

(19)

where the index De plays the role of the Hausdorff dimension of the eigenmodes. Analternative hypothesis would be a strong localization in the whole spectral region. In this case,the correlation function (18) may be given by, for instance, a so-called superlocalization ansatz,

( ) ( )[ ]G r r LX X

d∝ −exp φ ,

(20)

where dφ is the superlocalization exponent.

An example of the numerically obtained correlation function( )G rX is shown in Fig. 7. As wesee from the figure, the distribution is very wide. Its shape and width are incompatible with thestrong localization ansatz (20). Far from the plasmon resonance (R X0

3 2 5= − . ), the correlationfunction does scale in some intermediate region of distances. Close to the plasmon resonance, asX decreases, the region of scaling collapses. It is likely that scaling should take place for

R R X Lc X>> >>03 , but upon approaching the plasmon resonance, LX increases [cp.

Eq.(14)], closing down the scaling region.

The scaling of the correlation function of the eigenmode intensities suggests that theeigenmodes are themselves fractal objects. However, the scaling ( )G r rX ∝ −2 3. implies that theapparent Hausdorff dimension (18) is negative, De ≈ −13. . This shows that such a fractalcannot be a material one. The negative index shows that most of the eigenmode energy iscontained at small r , implying an essential singularity at the small scale.

9

6. Giant Fluctuations of Local Fields and Enhancement of Non-RadiativePhotoprocesses

The picture of the intensities in any individual eigenmode (see Fig. 6) shows very large randomchanges of the intensity from one monomer to another, i.e., fluctuations in space. When acluster is subjected to an external exciting radiation, its response is due to the excitations ofeigenmodes. Therefore, we may expect that these fluctuations will cause strong fluctuations ofthe local fields at individual monomers. We have investigated this phenomenon in Ref.16.

The local field at an ith monomer is expressed in terms of the Green’s function of Eq.(12),

E Z Ei i jj

α α β ββ

= ∑ G ,( )0 .

(21)

Then the local field-intensity enhancement coefficient Gi for an ith monomer and thecorresponding distribution function P G( ) are defined as

( )G P GN

G Gii

i= = −∑E

E

2

02

1( )

, ( ) δ .

(22)

We introduce also the nth moment of this distribution:

M G P G G dGnn n= = ( )∫ .

(23)

By its physical meaning, M n is the enhancement coefficient of an nth -order non-radiative (i.e.,without emissions of photons) nonlinear photoprocess. If, for instance, a molecule is attached toa monomer of a cluster, then M n shows how many times the rate of its n -photon opticalexcitation exceeds such a quantity for an isolated molecule. A similar estimate is valid for theenhancement of a composite consisting of the nonlinear matrix and resonant inclusion clusters.

The spectral dependencies of M n for different combination of the degree of nonlinearity n andthe dissipation parameter δ are shown in Fig. 8. The data in this figure are scaled by the factorR Q n

03 2 1δ − −( ) , with the resonant quality factor given by Eq.(6). The most remarkable feature of

Fig. 8 is an almost perfect collapse of the data into a universal curve in an intermediate region ofX for the case of very low dissipation (R0

3 0 01δ < . ). Moreover, this curve is actually close toa straight line in the intermediate region indicating a scaling behavior of M Xn ( ) . We

conjecture this scaling as the dependence M Q Mnn≅ −2 1

1( ) .

For the first moment M 1 we have previously obtained7,8 the exact relation

( )M X12 2= + δ α δIm . Because the absorption Imα does not scale in X (see Sec.4), the

enhancement coefficient M n should not scale either. However, the dependence Im ( )α X inthe intermediate region of X is flat (see Fig. 3). Therefore, for Q >> 1 the apparent scaling inX takes place with a trivial index of 2n ,

10

M Q XX

Xnn

n

nn≅ = ∝−

−2 1

2

2 12Im Imα

δα ,

(24)

in agreement with Fig. 8.

The major result of Ref.16, given by Eq.(24) is that the excitation rate of a non-radiative nth -order nonlinear photoprocess in the vicinity of a disordered cluster is resonantly enhanced by afactor of M Qn

n≅ −2 1. This quantity can be understood qualitatively in the following way. Foreach of the n photons absorbed by a resonant monomer, the excitation probability (rate) isincreased by a factor of ≅ Q2 (proportional to the local field intensity), therefore the total rate

is increased by a factor of ≅ Q n2 . However, the fraction of monomers that are resonant is

small, ≅ −Q 1 . Consequently, the resulting enhancement factor is M Qnn≅ −2 1, in agreement

with Eq.(24).

For instance, for silver in the red spectral region Q ≅ 30 (see, e.g., Ref.17), so each

succeeding order of the nonlinearity gives enhancement by a factor of Q2 1000≅ . Weemphasize that the origin of this enhancement is the high-quality optical resonance in themonomers modified by the cluster. Among interesting effects related to the enhanced non-radiative excitation, we mention one, the selective photomodification of silver clusters.18,19

We have considered above the moments (averaged powers) of the local fields. Now we consideranother characteristic of the fluctuations, the distribution function P G( ) (22) of the local-fieldintensity. Because the change of the minimum scale R0 implies the change of the local fields,the distribution function is likely to scale in some intermediate range,

P G G( ) ≅ ε ,

(25)

Due to scale invariance, the index ε does not depend on the minimum scale R0 . Consequently,ε does not depend on frequency (the spectral parameterX ) either.

A simple model that allows one to calculate the scaling index ε is the binary approximation.7,8

In this approximation, the eigenmode is localized at only a pair of the monomers. In this case wehave found16 the enhancement factor Gi for a pair separated by a distance r , located at anangle Θ to the direction of the exciting field,

( )( ) ( )

G rX r X r

i ,sin cos

ΘΘ Θ

=− +

++ +− −

δδ

δδ

2 2

3 2 2

2 2

3 2 22.

(26)

Using this, we calculate the distribution function as

11

( )( )P G G G r C r d ri( ) , ( )= −∫ δ Θ 3 .

(27)

Here C r r D( ) ∝ −1 is the density correlation function of the cluster. Taking into account thatlarge values of G are of interest, we obtain from Eq.(27) that

P G G( ) ∝ − 32 .

(28)

Thus, in the binary approximation we obtain a universal scaling index of − 3 2. Surprisingly,this index does not depend on the cluster’s dimension D . Its value is determined merely by thevector nature of the fields.

It is interesting to compare both the scaling prediction and the calculated value of the indexε = − 3 2 with the numerical results. These are shown in Fig. 9 for CCA clusters and forrandom walk (RW) clusters. The main feature is an unusually wide distribution. The localintensities present are on the order of the exciting intensity (G = 1), as well as three orders ofmagnitude smaller or greater. This feature is referred to as giant fluctuations of the local field.The regions of high intensity are responsible for enhanced nonlinear responses. They have alsobeen observed directly with the scanning photon-tunneling microscope.20 The value of the indexε is indeed almost independent from the frequency (parameter X ), as expected from thescaling theory. Interestingly enough, these values are quite close to the prediction of the binarytheory (ε = − 3 2). This agreement is unexpected because there are no grounds to believe thatthe binary theory is applicable in a wide range of frequencies.

7. Enhancement of Radiative Photoprocesses

Among enhanced radiative photoprocesses, one of the best studied experimentally is surface-enhanced Raman scattering (SERS).1 The intensity of Raman scattering for molecules adsorbedat rough surfaces or colloidal-metal particles is known to be greatly enhanced, by a factor of upto 106 . There are two limiting cases of SERS. In the first case, the Raman shift is very large,much greater than the homogeneous width of monomer’s absorption spectrum. In this case, aswe have shown17 the exact expression of the SERS enhancement factor GRS can be obtained as

GX

Q XRS =+

≈2 2δδ

α αIm Im .

(29)

This result predicts that the on the order of magnitude the enhancement is the same (by thefactor of ≅ Q ) as for non-radiative photoprocesses of the second order. Qualitatively, we mayinterpret this result in the following way. For a large frequency shift, the outgoing photon is outof resonance and does not considerably interact with the cluster. Correspondingly, all theenhancement is due to the local field of the absorbed photons, the same as for non-radiativeprocesses [cp. the discussion after Eq.(24)].

Much more dramatic enhancement is found17 for the second case, namely that of Ramanfrequency shifts smaller than monomer’s linewidth. This situation is characteristic of the most

12

typical and interesting cases of SERS. We have shown17 that in this case there exists anapproximate expression for the enhancement coefficient GRS of SERS,

G Q R XX

R XRS ≈ =

1

2

1

23

03

3

03Im Imα

δα .

(30)

In this case we have G QRS ≅ 3 , where the physical interpretation is the following.

Enhancement by ≅ Q2 is due to the first power of the local-field intensity. Another power

≅ Q2 appears because the outgoing photon is not emitted freely, but in the resonantenvironment. Finally, one power of Q disappears because only a small fraction ≅ 1 Q of themonomers is resonant to the exciting radiation.

We compare the functional dependence predicted by Eq.(30) to the numerical simulation in Fig.10. As we see, this equation describes the numerically-found dependencies quite well, especiallyfor the dependence on X .

Equation (30) shows that the Raman scattering is enhanced for an adsorbed molecule by afactor of G QRS ≅ 3 . For noble metals in the red region of visible light, we have

Q ≅ −30 100, so the predicted enhancement is very large, G RS ≅ −10 104 6. This explainsthe range of enhancements found experimentally.1 The comparison17 of theoretical spectraldependence with the experiment is shown in Fig. 11. As we can see, the theory explains theexperimental observations quite well.

Finally, we very briefly discuss coherent, or parametric, nonlinear photoprocesses. Theseprocesses are due to nonlinear wave mixing. One of the most interesting is frequency-degenerate four-wave mixing (a third-order process), responsible for such an interesting effectas phase conjugation. We have found the corresponding enhancement coefficient in Ref.3 in thebinary approximation and Ref.21 numerically.

Physically, the enhancement of an nth -order parametric photoprocess can be estimated in thefollowing simple way. The enhancement of the amplitude of each of the n + 1 participatingphotons (including the emitted photons) is ≅ Q . This yields the enhancement of the amplitude

of the process as ≅ +Qn 1 . For any coherent process, the amplitude, not probability, is thequantity to average. The averaging is done by multiplying by the fraction of resonant monomers,that is ≅ −Q 1 , leading to a mean amplitude as ≅ Qn . Finally, the amplitude should be squared,

yielding the enhancement coefficient G Qn n( ) ≈ 2 . More precise theory would give

( )G Q Xn n( ) Im≈ 2 2α .

(31)

For n = 3, this reduces to

13

( ) ( )G X X( ) Im3 6 2≈ δ α ,

(32)

in agreement with the corresponding result of Ref.21. The comparison of Eq.(32) to thenumerical calculations shown in Fig. 12 supports this result.

A remarkable feature of Eq.(32) is that the predicted enhancement is quite large. For therealistic dielectric parameters of silver in the visible range, Q ≅ −10 30, and correspondingly

G ( )3 4 710 10≅ − . The experimental investigation22 has indeed found a very strongenhancement for the phase conjugation, with G ( )3 610≅ , confirming the theoreticalpredictions.

8. Concluding Remarks

We have considered a variety of the photoprocesses mediated by disordered clusters. Most ofthe theory and numerical computations have been based on the fractal model of clusters. This isindeed a convenient model for a very strong disorder. In this case, a very big cluster has a verylow density (asymptotically zero as the size of the cluster increases). However, the interactionsinside the cluster, among its monomers, are not weak due to a very strong correlation in thepositions of the monomers. We believe that many of the results obtained are qualitatively, oreven quantitatively, valid for a general disordered clusters, especially those results that do notcontain fractal dimensions explicitly.

The remarkable feature of the systems considered is giant fluctuations of the local fields. Incontrast, these fields averaged over all monomers yield a value that is not enhanced.. This showsa unique property of such systems underlying their enhanced responses. Namely, theenhancement is due not to the high averaged value of the local fields, but to their fluctuations inspace, from one monomer to another. Nonlinearity, causing a higher power of the local fields tobe an acting parameter, enhances the effects of these fluctuations. Therefore, the enhancementsare found to increase dramatically with the order of nonlinearity.

Finally, many of the effects predicted have been verified experimentally, and we have mentionedsome of them. However, many very interesting effects are not discussed due to the spacelimitations. Among them, we recognize the enhanced laser production of plasma by colloidal-metal clusters.23

Acknowledgments

The authors greatly appreciate assistance by Dr. L. S. Muratov in the computer generation ofclusters and are grateful to him for useful discussions. One of us (MIS) acknowledges withgratitude the hospitality of the Institute for Applied Mathematics, University of Minnesota,Minneapolis, Minnesota during workshop Mathematical Methods in Material Sciences. MIS isalso grateful to the Departments of Physics and Chemistry of Washington State University,Pullman, Washington for support during his long-term Visiting Professorship. This researchwas supported the Pittsburgh Supercomputing Center under Grant No. PHY890020P.

14

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Surface Enhanced Raman Scattering by Impurities), ZhETF 92, 509 (1987) [Translation:Sov. Phys. JETP 65, 287 (1987)].

3. A. V. Butenko, V. M. Shalaev and M. I. Stockman, Giant Impurity Nonlinearities in Opticsof Fractal Clusters, ZhETF 94, 107 (1988) [translation: Sov. Phys. JETP, 67, 60 (1988)].

4. T. A. Witten and L. M. Sander, Diffusion-Limited Aggregation, A Kinetic CriticalPhenomenon, Phys. Rev. Lett. 47, 1400 (1981).

5. J. E. Sipe and R. W. Boyd, Nonlinear Susceptibility of Composite Optical Materials in theMaxwell Garnett Model, Phys. Rev. B 46, 1614 (1992).

6. G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe and L. A.Wellerbrophy, Enhanced Nonlinear-Optical Response of Composite Materials, Phys. Rev.Lett. 74, 1871 (1995).

7. V. A. Markel, L. S. Muratov and M. I. Stockman, Theory And Numerical Simulation Of TheOptical Properties of Fractal Clusters, ZhETF 98, 819 (1990) [translation: Sov. Phys.JETP 71, 455 (1990)].

8. V. A. Markel, L. S. Muratov, M. I. Stockman, and T. F. George, Theory and NumericalSimulation of Optical Properties of Fractal Clusters, Phys. Rev. B 43, 8183 (1991).

9. K. Ghosh and R. Fuchs, Spectral Theory for Two-Component Porous Media, Phys. Rev. B38, 5222 (1988).

10. R. Fuchs and F. Claro, Spectral Representation for the Polarizability of a Collection ofDielectric Spheres, Phys. Rev. B 39, 3875 (1989).

11. S. Alexander, The Vibration of Fractals And Scattering From Aerogels, Phys. Rev. B 40,7953 (1989).

12. M. I. Stockman, L. N. Pandey, L. S. Muratov and T. F. George, Optical Absorption andLocalization of Eigenmodes in Disordered Clusters, Phys. Rev. B 51, 185 (1995).

13. V. M. Shalaev, R. Botet and A. V. Butenko, Localization of Collective Dipole Excitationon Fractals, Phys. Rev. B 48, 6662 (1993).

14. M. I. Stockman, L. N. Pandey, L. S. Muratov and T. F. George, Comment on “PhotonScanning Tunneling Microscopy Images of Optical Excitations of Fractal Metal ColloidClusters”, Phys. Rev. Lett. 75, 2450 (1995).

15. M. I. Stockman, L. N. Pandey and T. F. George, Inhomogeneous Localization of PolarEigenmodes in Fractals, Phys. Rev. B 53, 2183 (1996).

16. M. I. Stockman, L. N. Pandey, L. S. Muratov and T. F. George, Giant Fluctuations ofLocal Optical Fields in Fractal Clusters, Phys. Rev. Lett. 72, 2486 (1994).

17. M.I.Stockman, V.M.Shalaev, M.Moskovits, R.Botet, and T.F.George, Enhanced RamanScattering by Fractal Clusters: Scale Invariant Theory, Phys. Rev. B 46, 2821 (1992).

18. A. V. Karpov, A. K. Popov, S. G. Rautian, V. P. Safonov, V. V. Slabko, V. M. Shalaev andM. I. Stockman. Observation of a Wavelength- and Polarization-SelectivePhotomodification of Silver Clusters, Pis'ma ZhETF 48, 528 (1988) [Translation: JETPLett. 48, 571 (1988)].

19. Yu. E. Danilova, A. I. Plekhanov and V. P. Safonov, Experimental Study of Polarization-Selective Holes Burned in Absorption Spectra of Metal Fractal Clusters, Physica A 185, 61(1992).

15

20. D. P. Tsai, J. Kovacs, Z. Wang, M. Moskovits, V. M. Shalaev, J. S. Suh and R. Botet,

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16

CAPTIONS TO FIGURES

Fig. 1

Cluster-cluster aggregate (CCA) of N = 1000 monomers.

Fig. 2

Wavelength dependence of the spectral parameter X and the dissipation parameter δ .

Fig. 3

Numerically-obtained polarizability (a) and density of eigenmodes (b) for cluster-cluster aggregates.

Fig. 4

Localization (coherence) radius LX as a function of the spectral parameter X .

Fig. 5

All eigenmodes with negative eigenvalues for a CCA cluster of N = 1024 monomers.

Fig. 6

Spatial intensity for four eigenmodes corresponding to [for (a) to (c)]: R X03 = -1.28, -1.20, -0.097 and -

0.099. The intensity profile is shown over a two-dimensional projection of the cluster.

Fig. 7

Pair correlation function ( )G rX for CCA clusters, found by Monte Carlo simulation for an ensemble of

1000 clusters of N = 300 monomers each. For comparison, the density autocorrelation function ( )C r isalso shown.

Fig. 8

Normalized enhancement factors G Qn nδ − −2 1( ) as functions of X for CCA clusters for the values of δand n shown.

Fig. 9

Distribution function of the local field intensity P G( ) calculated for R03 0 001δ = . , for the values of X

shown. The data on the upper panel are for CCA clusters and on the lower panel for random walk (RW)clusters.

Fig. 10

Scaled enhancement coefficient of SERS from silver colloid clusters in comparison with the prediction ofEq.(30). The theoretical dependence is calculated for CCA clusters.

17

Fig. 11

Theoretical and experimental spectral (in terms of wavelength) dependencies of the enhancementcoefficient GRS for silver colloid clusters.

Fig. 12

Scaled enhancement coefficient for third-order degenerate parametric process, calculated for CCA.

Fig. 1

19

Fig. 2

20

Fig. 3

(a)

(b)

21

Fig. 4

22

Fig. 5

0 .01 0 .1 1 10 100

E igenm odes, s ing le c lu ste r

A ve rage , 1000 c luste rs , = 300

A ve rage , s ing le c luste r

0 .01

0 .1

1 N =1024CCA, X<0

23

Fig. 6

(b)(a)

(c) (d)

24

Fig. 7

R30 X

C (r)

-0 .5

-2

25

Fig. 8

R03X

n, R 0 3

2 , 0 .0001

3, 0 .0001

1, 0 .001

2, 0 .001

1, 0 .01

2, 0 .01

26

Fig. 9

R03X

0

0.5

1

3

CCA

RW

R30X

0

0.5

1

3

(a)

(b)

27

Fig. 10

G R S(R 03 )-3

0.5 (R 30 |X |)4 Im

|X |3.3

28

Fig. 11

Theo ry

E xpe rim en t

29

Fig. 12