Local Fields' Localization and Chaos and Nonlinear-Optical Enhancement in Clusters and Composites

Transcription

1 Local Fields' Localization and Chaos and Nonlinear-Optical Enhancement in Clusters and Composites Mark I. Stockman Department of Physics and Astronomy, Georgia State University, Atlanta, GA This Chapter is devoted to linear and nonlinear optical properties of disordered clusters and nanocomposites. Linear and nonlinear optical polarizabilities of large disordered clusters, fractal clusters in particular, and susceptibilities of nanocomposites are found analytically and calculated numerically. A spectral theory with dipole interaction is used to obtain quantitative numerical results. Major properties of systems under consideration are giant fluctuations, inhomogeneous localization and chaos of local fields that cause strong enhancement (by many orders of magnitude) of nonlinear optical responses. The enhancement and fluctuations properties of the local fields are intimately interrelated to the inhomogeneous localization of the systems eigenmodes ( plasmons ). Due to these fluctuations, mean-field theory completely fails to describe nonlinear optical responses.. INTRODUCTION.... EQUATIONS GOVERNING OPTICAL (DIPOLAR) RESPONSES, SPECTRAL REPRESENTATION AND SCALING...4. LINEAR OPTICAL RESPONSES INHOMOGENEOUS LOCALIZATION OF EIGENMODES GIANT FLUCTUATIONS OF LOCAL FIELDS AND ENHANCEMENT OF NON- RADIATIVE PHOTOPROCESSES CHAOS OF EIGENMODES ENHANCEMENT OF RADIATIVE PHOTOPROCESSES AND NONLINEAR POLARIZABILITIES IN CLUSTERS ENHANCED NONLINEAR SUSCEPTIBILITIES OF COMPOSITES MACROSCOPIC AND MESOSCOPIC FIELDS AND INTEGRAL FORMULAS FOR OPTICAL RESPONSES OF COMPOSITES HYPERSUSCEPTIBILITY OF A COMPOSITE FOR THE CASE OF NONLINEARITY IN INCLUSIONS HYPERSUSCEPTIBILITY OF A COMPOSITE FOR THE CASE OF NONLINEARITY IN THE HOST CONCLUDING REMARKS...5 REFERENCES...7 CAPTIONS TO FIGURES...4

2 . Introduction Clusters and nanocomposites belong to so-called nanostructured materials. Such materials typically are nanoparticles either bound to each other by covalent or van der Waals bonds, or dispersed in a host medium. Description of electromagnetic properties of such system is a longstanding problem going back to such names as Maxwell Garnett, Lorentz and Bruggeman. Properties of such materials may be dramatically different from those of bulk materials with identical chemical composition. A characteristic property of such systems is confinement of electrons, phonons, electric fields, etc., in small spatial regions. Such a confinement, in particular, modifies spectral properties (shifts quantum levels and changes transition probabilities), and also changes the interaction between the constituent particles. As we will be discussing in this Chapter, local (near-zone) electromagnetic fields are strongly fluctuating in space. Their magnitude is greatly (by orders of magnitude) enhanced with respect to the external (exciting) fields. A phenomenon closely related to the enhancement and fluctuations of local fields is localization of elementary excitations (eigenmodes) in the composites 4-8. The relevant excitations are polar waves that are traditionally called plasmons (this term originates from theory of metallic nanoparticles containing electron plasma, but is now often used in application to other nanocomposites). Plasmon-resonant properties leading to enhancement of local fields are especially pronounced in some metallic (especially silver, gold, or platinum) colloidal clusters, metal nanocomposites and rough surfaces. A typical example of such responses is surfaceenhanced Raman scattering (see, e.g. a review of Ref. 9 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. Imagine two fields with the same average intensity I E. For the sake of argument, let us say, the first field has the same constant intensity I in N >> points, and the second is strongly localized at one point where its intensity then should be I = NI. Consider a n th order nonlinearity where the nonlinear n n response is proportional to E I. The ratio of the nonlinear response for the first (constant) field is proportional to ( n n NI ) N ( NI N n ) n = I. In contrast, the response to the second (strongly n localized) field is = I. In such a way, the enhancement coefficient (the ratio of N n the nonlinear response in the second case to that in the first case) is N. Hence the localization has a potential to bring about strongly enhanced nonlinear responses where the enhancement increases with the order of nonlinearity and the degree of localization (spatial fluctuations). To maximize this effect, our goal is to find systems with the maximum spatial fluctuations of the local densities. We certainly expect that the density fluctuations will cause correspondingly large fluctuations of the local fields. There exists a class of systems that stands out in this respect. These are self-similar (fractal) systems, which (on average) repeat themselves at different scales. In other words, looking at such a system and not seeing its boundaries (neither at the maximum or a minimum scales), one cannot say what fraction of the system is observed, and what is the actual size of the

3 objects seen. For such systems, the number N of constituent particles (monomers) contained within a radius R scales as N R R D, () where R is a typical distance between monomers, and D is the fractal (Hausdorff) dimension of the cluster. The density of monomers as given by Eq. () is asymptotically zero for large clusters, ρ R R R () However, this does not mean that the interaction between monomers can be neglected. The underlying reason for that is a strong correlation between monomers in a cluster, with the pairpair correlation function scaling similar to Eq. (). Thus we have a unique system whose macroscopic density is asymptotically zero, but the interaction inside the system is strong. This idea has been proposed by us in an earlier papers. 4, An example of a fractal cluster, obtained by cluster-cluster aggregation (CCA) is shown in Fig.. In this figure, one can trace rarefied nature of fractal systems (represented by voids of density) and strong fluctuations of the density of constituent particles (monomers). To avoid possible misunderstanding, we point out that other, non-fractal systems also possess significantly enhanced optical responses, especially those that are tailored to have optimally-chosen dielectric properties changing in space. The physical origin of the enhancement in this case is the same as above. As an example of such systems we will consider a random Maxwell Garnett composite, where dielectric or metallic spheres are embedded in a host medium at random positions. Such a composite has earlier been considered in a mean-field approximation. We will consider below a model of such composites where the inclusion spheres are positioned on a cubic lattice in a host medium and call it a random lattice gas (RLG). We will use RLG as a model of random but not fractal composites. There has been an increase of interest in the optical properties (both linear and, especially, nonlinear) of composites during the last decade. - This revival of interest is due to improved theoretical understanding of the origin of the optical enhancement in composites and to demonstrated possibility to engineer composites whose desired nonlinear properties are better than those of their constituents. Theoretical advances in this field are based to a significant degree on the spectral methods. A general spectral method -5 and the dipolar spectral method 5,6 have proved to be very useful in both analytical theory and numerical computations. In particular, the spectral method has efficiently been used in the theory of electrorheological fluids, i.e., liquid composites whose hydrodynamic properties depend on the applied electric field. 6,7 In Sec., we present the coupled dipole equations governing optical responses and summarize the dipolar spectral theory. In that section we also summarize some predictions of D.

4 scaling. In Sec., on the basis of the spectral theory, we consider linear optical polarizabilities of clusters and susceptibilities of composites. Section 4 is devoted to the problem of localizationdelocalization of the elementary excitations (eigenmodes) of disordered clusters and composites. We summarize computations that have led us to introduce a concept of inhomogeneous localization of eigenmodes. Section 5 deals with a phenomenon of giant fluctuations of local electric fields in clusters and composites, which contribute to the strong enhancement of their nonlinear optical responses. Chaos of eigenmodes considered is Sec. 6 is a phenomenon similar to quantum chaos. Individual eigenmodes (surface plasmons) and even their averaged correlation functions exhibit very strong fluctuations on all scales. In Sec. 7, we consider enhancement of Raman scattering and nonlinear parametric mixing in clusters. Nonlinear polarizabilities of Maxwell Garnett composites in the spectral theory in comparison with a mean field theory are discussed in Sec. 8. Concluding remarks are presented in Sec. 9.. Equations Governing Optical (Dipolar) Responses, Spectral Representation and Scaling We concentrate on the dipole-dipole interaction, which is a universal interaction between polarizable particles at large distances. We consider a cluster (or a composite) whose particles (called below monomers) are positioned at points r i. Let us assume that the system (a cluster or composite) is subjected to the electric field E of the incident optical wave. This field induces the dipole d iα at an i th monomer (here α = xyz,, denotes the Cartesian components of the vector, and similar notations will be used for other vectors). The dipole moments satisfy well-known system of equations α d N ( ) = Ei δ j= iα α αβ ( rij ) ( rij ) d α β jβ rij rij ( ) Here E iα is the wave-field amplitude at the i th monomer, rij = ri r j is the relative vector between the i th and j th monomers, and α is the dipole polarizability of the monomer. We assume that the size of the system is much less than the wavelength of the exciting wave, and ( ) therefore the exciting field E α 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 justified if intermediate-to-large scales predominantly contribute to the properties under consideration. The dipolar spectral theory of the optical response of fractal clusters has been developed in Refs. 5 and 6. We note that a similar spectral approach has been independently introduced R.Fuchs and collaborators. 8,9 The material properties of the system enter Eq. () only via the combination Zr ij, where we have introduced the notation Z α. This along with the (approximate) self-similarity of the system is a prerequisite for scaling in terms of the spectral variable Z. A principal requirement for the scaling of a certain physical quantity F is that the. () 4

5 system eigenmodes contributing to F should have their localization radii L intermediate between the maximum scale (size of the cluster R c ) and the minimum scale, R. Then this quantity will not depend on any external length, leading to scaling. Because the quantity F is not sensitive to the maximum scale R c, it should have the functional dependence F = F( ZR ). On the other hand, the eigenmodes contributing to F are insensitive to a much smaller minimum scale. Therefore the dependence on R can be only power (scaling), and consequently where γ is some scaling index. ( ZR) FZ ( ) In accord with the above arguments, it is convenient to express all results not in terms of frequency, but in terms of Z, separating the imaginary and real parts, Z = X iδ. The choice of signs in this expression makes the dissipation parameter δ positive, while the spectral parameter X is positive when the frequency is blue-shifted from the plasmon resonance, and negative otherwise. For the sake of reference, we give here the expressions for X and δ for a metallic nanosphere in the Drude model, ( ε + ε ) h p γ, m ( ε + ε ) h p ( ); = γ X h h = ω ω δ s, ε ω R m R ε ω where ε is the intersubband dielectric constant of the metal, ε h is the dielectric constant of the ambient medium (host), ω p is the metal plasma frequency, ω s = ω p ε + ε h is the surface plasmon frequency, and R m is the nanosphere s radius. The spectral dependence of X and δ for silver is illustrated in Fig.. The most important feature in the figure is that in the yellow-red region of visible light, the real part of the polarizability greatly exceeds its imaginary part. Their ratio has the meaning of the quality factor of the surface-plasmon oscillations Q defined as X Q δ (6) This factor shows how many times on the order of magnitude the amplitude of the local field in a vicinity of a resonant monomer exceeds that of the exciting field. This important fact is considered in detail below [see Sec. 5, the discussion of Eqs. (9)-()]. The fact that for many metals Q may be large (as large as ) plays an important role in the theory since the enhancement of the optical responses is a resonant phenomenon, and strong dissipation would completely suppress it. (4) (5) 5

6 We note that in some earlier work the quality factor has been defined differently, R Q. δ Convenience of this definition is that in the Drude model [see Eq. (5)] this factor, R 6ε hω m p Q =, is a constant. In reality, Q is not a constant, because the dissipation R ( ) γ ε + ε h parameter δ depends on frequency (cf. Fig. and its discussion). We use in this Chapter the definition of Eq. (6) because it directly determines the enhancement factor of the local fields M n [cf. Eq. () and its discussion], that is the most important characteristic for the present theory. To introduce the spectral representation, 5,6 we will rewrite Eq. () as one equation in the ( ) N-dimensional space. To do so, we introduce N-dimensional vectors d) E ) projections give the physical vectors ( α ), ( α ) ( ) ( ) i d = diα i E = Eiα,.,,, whose Equations () then acquire the form ( ( Z + W) d) = E )) where W is the dipole-dipole interaction operator with the matrix elements ( iαw jβ) = ( rij ) ( rij ) α β δ αβ, r ij r ij i j, i = j., (7) (8) (9) We introduce the eigenmodes (plasmons) n ) ( n=,..., N) as the eigenvectors of the W- operator, Wn) = w n) n, () where w n are the corresponding eigenvalues. Practically, the eigenvalue problem () can be solved numerically for any given cluster. Having done so, one can calculate the Green s function G i α, jβ = N ( iα n)( jβ n) n= Z + which carries the maximum information on the spectrum and linear response of the system. We note that due to the time-reversal symmetry of the system (absence of a magnetic field), all w n, () 6

7 eigenvectors can be chosen and will be assumed real. Therefore, all amplitudes are symmetric, in i α n = n iα. particular, ( ) ( ). Linear Optical Responses The polarizability of a cluster or finite volume of a composite α and its density of eigenmodes ρ are expressed in terms of G as α αβ (i) (i) α, α. =. i j = G ρ = G, iα, jβ, iα, iα i, j (i) where α. is a polarizability of an i th monomer in the cluster (or a composite), and summation over repeated vector indices is implied. The dielectric constant of a cluster (composite) is given by N ε c = ε h + 4πα, V () where ε h is the dielectric constant of a host, V is the volume occupied by the cluster (composite) and i, jβ N i, j () α = G β is a polarizability of a monomer in the cluster (composite). Below we will consider results of numerical computations using Eqs. () and () and will compare them to some analytical predictions. First, let us consider scaling predictions. For this purpose we have to invoke a large magnitude of the quality factor of the optical resonance (6), Q >>. In this case, a dependence of type (4) becomes F( X) ( R X) and ρ( X ) γ. We have introduced 5,6 such a dependence for α αβ ( X ) and argued that the two quantities have the same scaling, d o Imα (X) ρ(x) R R X, (4) where d o is an index that we called the optical spectral dimension. We have also argued that the physical range of d o is d o. The strong localization has been essential for the derivation of Eq. (). It implies that all eigenmodes (at least all contributing eigenmodes) of a cluster are strongly localized The strong localization, as discussed by Alexander, means that for any given frequency parameter X there exists only one characteristic length L X of these eigenmodes playing the role of simultaneously their wavelength and their localization length. Using scale invariance arguments, we have shown 5,6 that L X should scale as 7

8 L X d o D R R X. (5) We have subjected the scaling predictions of Eqs. (4) and (5) to an extensive comparison with the results of large-scale computations. 7 Similar results have been obtained 7 for other types of clusters. These results are quite unexpected. One of those, a polarizability and eigenmode density for cluster-cluster aggregates (CCA), is shown in Fig.. The conclusion that one can draw from the figure is that neither the polarizability, nor density of eigenmodes scale. Interestingly enough, they still appear to be quite close to each other, supporting the conclusion of Refs.5 and 6 that all eigenmodes of a fractal cluster contribute (almost) equally to its optical absorption. This conclusion can be understood physically from an idea that a fractal is disordered and does not possess any geometric (point-group) symmetry on all intermediate scales (between R and R c ). Consequently, such strong disorder does not impose any selection rules that would otherwise govern the contribution of a specific eigenmode to optical absorption. Another relation to check is that of Eq. (5). First, one has to formulate how to calculate the localization radius. We use the definition of Ref.7, L ρ L where ρ ( w ) ( n iα ) r ( n iα ) n n n X =, where Ln = ri ρ n iβ iβ [ ] + n = X n δ, L n is the localization radius of a given eigenmode, and LX is the localization radius at a given frequency. The computed dependence of L X is shown in Fig. 4. As one can clearly see, there is no scaling in these data too. This finding is in contradiction with the conclusion of Ref. (precision of our calculations is much higher than that of Ref.). Evident failure of the scaling implies that at least one of the assumptions lying at its foundation is incorrect. Because we used the same model (dipole-dipole) for both the scaling theory and the numerical computations, the non-applicability of the model to system is out of question. In our consideration we consistently used high values of Q, so that the condition Q >> is also satisfied. The only cause of the failure of scaling appears to be the strong localization assumption. Now let us discuss the linear polarizability of a Maxwell Garnett composite, as calculated in Ref.. We show in Fig. 5 the results of a computation of the linear dielectric constant for a composite consisting of silver nanospheres in a dielectric host with a dielectric constant of ε =.. This figure displays both the results of the spectral theory computations accordingly to h Eq. () and those of a mean field approximation known as Maxwell Garnett formula (or an equivalent Lorentz-Lorenz formula), see, e.g., Ref.. As we see, the mean field theory gives a satisfactory description in the wings of the spectral contour, but fails in the region of the resonant absorption of the inclusions, where it considerably overestimates ε h. i, (6) 8

9 Undoubtedly, there should be reasons for the failure of both scaling theory and mean field theory. As we understand now, two interrelated phenomena can be blamed for this failure. These are inhomogeneous localization of eigenmodes and giant fluctuations of the local fields in space, see below in Secs. 4 and 5. In the case of the inhomogeneous localization, there are eigenmodes of all localization radii from the minimum distance between monomers (inclusions) R to the total size of the system R c. Both of these two extremes render the scaling theory inapplicable. It is important to emphasize that the eigenmodes with so vastly different localization radii co-exist at the same frequency (Sec. 4). Obviously, the strong fluctuations contradict to the basic assumption of the mean field theory. While we consider the inhomogeneous localization and giant fluctuations for clusters below in Secs. 4 and 5, here in Fig. 6, we demonstrate fluctuations of the local fields for the Maxwell Garnett composite. As one can clearly see, the spatial fluctuations (i.e., a change from an inclusion to an inclusion particle) of the local fields is significant in the resonant region (see the left panel in Fig. 6), where the intensity of local fields changes in space by orders of magnitude. These fluctuations cause the mean field theory to fail. As the right panel in Fig. 6 shows, in the spectral wings (in the off-resonant region), these fluctuations are much smaller. Consequently, the mean field approximation describes the polarizability reasonably well. 4. Inhomogeneous Localization of Eigenmodes To introduce the inhomogeneous localization, 8, (see also Refs. 4 and 5), we consider all eigenmodes of a single cluster. In Fig. 7 we show a special plot where each eigenmode is represented by a point in the coordinates its localization length L n versus the spectral parameter X. As one can see, at any frequency (value of X ) within a wide range of X, the eigenmodes have a very broad spectrum of their localization radii L X, from the minimum scale R of the distance between the monomers to the maximum scale Rc of the cluster total radius. As we have already mentioned above in Sec., either of these extremes ( Ln Rc and L n R ) violates a necessary scaling condition R << L n << Rc, causing failure of scaling for at least some quantities, such as linear responses (see Sec. ). However some other quantities, as we will see below in Secs. 5-7, do still scale, because they are insensitive to those extremes). It is also useful to take a closer look at some members of the ensemble of eigenmodes of a cluster. We take two pairs of eigenmodes that have almost equal frequencies (within a pair). The spatial distribution of the intensity of these eigenmodes is presented in Fig. 8. As we can see, at a comparatively large eigenvalues ( R w n =. 9 ), the eigenmode is indeed a very well localized, demonstrating a very sharp peak over just a few monomers. The localization radius for this eigenmode is indeed very small, L n / R c =. 4. Counterintuitively, an eigenmode with a very close frequency ( R. ) is almost completely delocalized (for this eigenmode, w n = L n R c =.75 ). However, the examination of its spatial profile shows that in reality this eigenmode consists of two sharp peaks separated by a distance on the order of the total size of the cluster. As we go closer to the plasmon resonance (smaller w n ), the minimum width of the peaks increases, as one would expect in view of Eq. (5). At the same time, the internal structure of the 9

10 eigenmode remains highly irregular with large spatial fluctuations. Again, the eigenmodes at the same frequency possess very different localization radii. The typical minimum size of an eigenmode (its core ) is of interest by itself. It is understandable that a small core implies large spatial fluctuations of the eigenmode intensities. These bring about strong enhancement of non-linear photoprocesses, as we have already noted in Sec. and discus below in Secs. 5 and 7. Below in this Section, in conjunction with Fig., we argue that the characteristic size of this core l X scales as a function of the spectral variable X. The behavior described above is very unusual. For the known localization patterns, eigenmodes are strongly localized for short wavelength (high frequencies) and delocalized in the long-wavelength wing. The physical reason for this is that a long wave (wavelength λ >> the typical size of the scattering inhomogeneities) sees almost homogeneous medium and propagates almost freely. In contrast, a short wave is strongly scattered from inhomogeneities with sizes on the order of λ (strongly here means that the scattering length itself is on the order of λ ). Thus, there exists the mobility edge, i.e., a frequency above which waves are localized and below 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 comparable to λ. This may suggest that all the eigenmodes are strongly localized, as assumed in Refs. 5 and 6. However, the result presented above and all numerical modeling of Refs. 7, 8, and -5 have shown that the strong localization is not the case. In actuality, the inhomogeneous localization 8, takes place, where the eigenmodes in a wide range of the localization radii coexist at any given frequency. Apart from the individual eigenmodes shown in Fig. 8, it is of interest to study a distribution of local intensities induced by an external exciting field, similar to what we have shown for a non-fractal Maxwell Garnett composite in Fig. 6. For fractal CCA clusters, an example of such a distribution is shown in Fig. 9 for two frequencies (parameters X ) that are very close to each other, and two perpendicular linear polarizations of the exciting light. These distributions are indeed extremely singular and fluctuating in space, even between the nearestneighbor monomers. This property of the local fields is the reason underlying the giant fluctuations of the local fields (see Sec. 5). The overall width of the distribution is of the order of the total cluster size. This is explained by the fact that the external radiation at a given frequency excites a group of individual eigenmodes, within which there always are delocalized modes. Because the interaction is very long-ranged and the clusters are self-similar, there is no intrinsic length scale characteristic of the problem. Consequently, the spatial extent of the intensity distribution is limited only by the clusters' size. Change of polarization of the exciting radiation at a given frequency brings about a dramatic redistribution of local intensities and change in the maximum intensities (cf. the left and right distributions in Fig. 9). The physical reason for this is that the resonant configurations of the monomers in most cases are highly anisotropic. This explains the high selectivity of the cluster photomodification in the radiation polarization observed experimentally. 6-8 The change of frequency of the exciting radiation by less than one percent also brings about pronounced changes in the intensity spatial distributions (cf. upper and lower panels in Fig. 9). Generally, the observed intensity distributions are in a good qualitative agreement with the direct experimental observation

11 by Moskovits and collaborators of the near-field optical fields in large silver clusters. 9 Recently a similar behavior qualitatively consistent with the inhomogeneous localization has been observed in more details in silver colloids having fractal (supposedly, self-affine) geometry in near-field scanning optical microscope experiments. 4-4 (A quantitative comparison is not possible because the distributions for individual clusters are inherently chaotic, strongly fluctuating from one cluster to another.) However, the conclusion of Ref. 9 that the observed phenomena support the strong localization hypothesis contradicts our conclusions. We have commented that the observations of Ref. 9 do not support its stronglocalization conclusions. In fact, these observations do support the inhomogeneous localization picture described in this Chapter (see Sec. 4), incompatible with the strong localization. The patterns of the local fields discussed above show that the inhomogeneous localization scenario of polar excitations (plasmons) in large self-similar clusters is principally distinct from both the strong and weak localization scenarios of non-polar excitations. The above-discussed individual eigenvectors (eigenstates) are chaotic. Consequently, they are difficult to compare quantitatively to each other. Therefore we consider below statistical characteristics (measures) of the eigenvectors. To examine the statistical properties of an eigenmode distribution, we introduce the distribution function P ( L, X ), which is the probability density that an eigenmode at a given X has the localization radius L, ( L L ) δ ( X ) P( L, X ) = δ n w n n, (7) We show this distribution calculated for CCA clusters in Fig. and for Maxwell Garnett composites modeled as a random lattice gas (RLG) in Fig.. The most conspicuous feature of the distribution of Fig. is its very large width. This width extends from almost the total size of the system Rc to some minimum cut-off size l X that is a function of frequency ω (X ). The cut-off is clearly seen in Fig. where it is also indicated in the lower panel by a thick dotted line. This cut-off is seen in the intensity spatial distributions of individual eigenmodes as a core, i.e., the characteristic minimum width of an eigenmode [cf. Fig. 8 and Fig. 9 and their discussion in Sec. 4]. The width of the distribution P ( L, X ) is so large that its characterization by a single dispersion relation L X [see Eq. (6)] is absolutely insufficient. For most of the spectral region, the cut-off length l X by magnitude is intermediate between the maximum and minimum scales, R c and R. This, along with the self-similarity of the clusters, suggests that l X scales with X, λ i.e., l X X. Indeed, Fig. supports the possibility of such a scaling with the corresponding index λ. 5. This illustrates general property of the inhomogeneous localization of eigenmodes for fractal (self-consistent) clusters and composites. A different situation exists for non-fractal composites, as one can see with an example of the Maxwell Garnett composite (simulated as RLG) shown in Fig.. The distribution for

12 X. is similar to that of CCA (Fig. ) characteristic of inhomogeneous localization. The major distinction from Fig. is that the distribution in Fig. shows the complete delocalization of the eigenmodes for X. that appears in a narrow range. Such a delocalization is expected for the low- X part of the spectrum, i.e., at frequencies close to the plasmon resonance of the individual inclusions (monomers). In contrast, there is no such delocalization for fractal (CCA) clusters, as seen in Fig.. 5. Giant Fluctuations of Local Fields and Enhancement of Non-Radiative Photoprocesses The picture of the intensities in any individual eigenmode (see Fig. 8) shows very large random changes of the intensity from one monomer to another, i.e., fluctuations in space. When a cluster is subjected to an external exciting radiation, its response is due to the excitations of eigenmodes. Therefore, we may expect that the eigenmode fluctuations will cause strong fluctuations of the local fields at individual monomers. We have already discussed spatial distribution of the local fields above in Secs. and 4. In this Section, following Ref. 4, we discuss distributions of the externally-induced local fields over their intensity. This distribution determines enhancement of optically-nonlinear incoherent processes (such as nonlinear photochemical reactions, optical modification, local melting, etc.). The local field at an i th monomer is expressed in terms of the Green s function () as E iα = Z jβ G E. iα, jβ (8) In terms of this field, the local field-intensity enhancement coefficient G i for an ith monomer and the corresponding distribution function PG ( )are defined as () β G i = E E i (), P( G) = N i δ ( G G ) We introduce also an n th moment of this distribution: n n M n = G = P( G) G dg. i. (9) () By its physical meaning, M n is the enhancement coefficient of an n th -order non-radiative (i.e., without emissions of photons) nonlinear photoprocess. If, for instance, a molecule is attached to a monomer of a cluster, then M n shows how many times the rate of its n -photon optical excitation exceeds such a quantity for an isolated molecule. A similar estimate is valid for the enhancement 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 and the dissipation parameter δ are shown in Fig.. The data in this figure are scaled by the

13 δ factor R Q ( n ), with the resonant quality factor given by Eq. (6). The most remarkable feature of Fig. is an almost perfect collapse of the data into a universal curve in an intermediate region of X for the case of very low dissipation ( R δ <. ). Moreover, this curve is actually close to a straight line in the intermediate region indicating a scaling behavior of M n ( X). We ( n ) conjecture this scaling as the dependence M Q M. n For the first moment M we have previously obtained 5,6 the exact relation ( X ) M = +δ Im α δ. Because the absorption Imα does not scale in X (see Sec.), the enhancement coefficient M n should not scale either. However, the dependence Im α ( X ) in the intermediate region of X is flat (see Fig. ). Therefore, for Q >> the apparent scaling in X takes place with a trivial index of n, n n X n M n Q X Im α = Imα X, n δ () in agreement with Fig.. A major result of Ref. 4, given by Eq.() is that the excitation rate of a non-radiative nth -order nonlinear photoprocess in the vicinity of a disordered cluster is resonantly enhanced by n a factor of M n Q. This quantity can be understood qualitatively in the following way. For each of the n photons absorbed by a resonant monomer, the excitation probability (rate) is increased by a factor of Q (proportional to the local field intensity), therefore the total rate is increased by a factor of Q n. However, the fraction of monomers that are resonant is small, Q n. Consequently, the resulting enhancement factor is M n Q, in agreement with Eq. (). For instance, for silver in the red spectral region Q (the optical constants for silver are adopted from Ref. 44), so each succeeding order of the nonlinearity gives enhancement by a factor of Q. We emphasize that the origin of this enhancement is the high-quality optical resonance in the monomers modified (shifted significantly to the red) by the structure of cluster. Among interesting effects related to the enhanced non-radiative excitation, we mention one, the selective photomodification of silver clusters. 6-8 We have considered above the moments (averaged powers) of the local fields. Now we consider another characteristic of the fluctuations, the distribution function PG ( ) (9) of the local-field intensity. Because the change of the minimum scale R implies also the change of the local fields, it is possible that the distribution function scales in some intermediate range, PG ( ) G Under the scale-invariance assumptions, the index ε does not depend on the minimum scale R. Consequently, ε does not depend on frequency (the spectral parameter X ) either. ε, ()

14 A simple model that allows one to calculate the scaling index ε is the binary approximation. 5,6 In this approximation, the eigenmode is localized at only a pair of the monomers. In this case we have found 4 the enhancement factor G i for a pair separated by a distance r, located at an angle Θ to the direction of the exciting field, G r ( Θ) i, = δ sin Θ cos ( X r ) + δ ( X r ) Using this, we calculate the distribution function as + δ Θ + + δ ( G G ( r, ) C( r) d r P( G) = δ Θ. i Here Cr ( ) r D is the density correlation function of the cluster. Taking into account that large values of G are of interest, we obtain from Eq. (4): PG ( ) G (5) Thus, in the binary approximation we obtain a universal scaling index of. Surprisingly, this index does not depend on the cluster s dimension D. Its value is determined merely by the vector nature of the fields. It is interesting to compare both the scaling prediction and the calculated value of the index ε = with the numerical results. These are shown in Fig. for CCA clusters and for random walk (RW) clusters. The main feature is an unusually wide distribution. The local intensities are on the order of the exciting intensity ( G = ), as well as three orders of magnitude 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. We note that the local regions of high intensity ( hot spots ) have also been observed directly with the scanning photon-tunneling (near-field optical) microscope 9 (see also a comment on Ref. 9). The value of the index ε is indeed almost independent from the frequency (parameter X ), as expected from the scaling theory. Interestingly enough, these values ( ε =. 45 for CCA and ε =. 44 for RW) are quite close to the prediction of the binary theory ( ε = ). This agreement is unexpected because there are no grounds to believe that the binary theory is applicable in a wide range of frequencies. 6. Chaos of Eigenmodes The eigenmode equation () has the same form as the quantum-mechanical Schrödinger equation. In quantum mechanics it is not uncommon that highly-excited states or states of complex systems possess chaotic behavior (see, e.g., Refs. 45, 46). One may expect a similar situation for eigenmodes of large disordered clusters and composites. The extreme sensitivity of the individual eigenmodes to a very small change of their frequency that is discussed in Sec. 4 and illustrated by Fig. 8 is a direct indication of such chaos. Even more than individual eigenmodes, statistical properties of chaotic eigenstates are of great interest. The giant fluctuations of the. () (4) 4

15 intensities of local fields discussed above in Sec. 5 provide one of the statistical descriptions. In this section we will consider spatial correlations of the chaotic eigenmodes. 4,5 A principal property that distinguishes this problem from quantum-mechanical chaos is a long-range nature of the dipole-dipole interaction. A similar tight-binding problem of quantum mechanics (Anderson model) is usually formulated with only next-neighbor hopping. In studied quantum-mechanical problems, chaotic quantum states do not possess long-range spatial 45, 47 correlations. The long ranged interaction on one hand tends to induce the long-range spatial correlations. On the other hand, it may tend to establish a mean field, suppress fluctuations, and eliminate the chaos. As we demonstrate below in this section, either of those trends may dominate, depending on the system geometry and spectral region. We expect that chaos is the most pronounced in clusters and composites with fractal geometry. The rationale for it is the following. A mean field is established and spatial chaos is eliminated when the correlation range of eigenmodes exceeds a characteristic size of the density variations in the system. However, fractal (self-similar) geometry implies that the system repeat itself on all spatial scales and, consequently, there exists no such characteristic spatial scale. This is a prerequisite for the coexistence of chaos and long-range correlations. To characterize the spatial correlations of eigenmode amplitudes, we introduce the amplitude correlation function (also called dynamic form factor), S αβ (, X ) = ( iα n)( jβ n) δ ( r rij ) δ ( X w n ) r, n, i, j (6) where δ () is the Dirac s δ -function (not to be confused with the dissipation parameter δ. It is useful to note that in the limit δ << X, the correlator S αβ ( r, X ) is related to the Green s function (), S ( r, X ) = Im G iα, jβ δ ( r rij ) αβ, π i, j This function expresses the correlation factor of amplitudes ( iα n) and ( n) (7) jβ of two eigenmodes with polarizations α and β at two points separated by a spatial interval of r at an eigenvalue ( frequency ) of X, averaged over ensemble of the systems. Similarly, we introduce a correlation function of the intensities of eigenmodes, 5

16 (, X ) = ( iα n) ( jβ n) δ ( r rij ) δ ( X w n ) C r, n, i, j (8) where the summation in the repeated indices α and β is implied as is done throughout the paper. In a similar way, this function yields a correlation of two eigenmode intensities, ( iα n) and ( jβ n). We have calculated 4,5 the above-defined correlation functions for an ensemble of CCA clusters (composites) with N = 5 monomers (inclusions). The corresponding result for S ( r, X ) = S ββ ( r, X ) is shown in Fig. 4. We see a developed pattern of irregular, chaotic correlations in almost the whole spatial-spectral region. The landscape observed in Fig. 4 well deserves the name of a devil s hill, where narrow regions of positive and negative correlations are interwoven, resulting in a turbulence-like pattern. This chaotic behavior is indeed a reflection of the chaos of individual eigenmodes. However, this chaos is in some sense stronger, because it is in a quantity averaged over an ensemble of statistically independent composites (consequently, it is fully reproducible). The deterministically chaotic pattern of Fig. 4 is likely to depend on a specific topology of the composite (CCA clusters). The straight lines shown in Fig. 4 are given by the binary-ternary approximation, see the discussion of Fig. 6 below. To distinguish between fluctuations of phase and amplitude in the formation of the devil's hill in Fig. 4, we will compare it to the second-order correlation function C ( r, X ) shown in Fig. 5. We see that that this function differs dramatically from the dynamic form factor. The relief in Fig. 5 is very smooth, in contrast to Fig. 4. This implies that the devil's hill is formed due to spatial fluctuations of the phases of individual eigenmodes, while their amplitudes are smooth functions in space-frequency domain. Another important feature present in Fig. 5 is the long range of the correlation. The correlation decay in r is indeed weaker than any power. As we discussed above in this section, self-similarity (fractality) of the CCA composite is of principal importance for the coexistence of chaos and long-range correlations of the eigenmodes. To emphasize this point, we show in Fig. 6 the results for the dynamic form factor S ( r, X ) for RLG composites. We see that for extremely large eigenvalues, X, the form factor S ( r, X ) has a quasi-random structure that is reproducible under the ensemble averaging. This structure is similar to what is observed for CCA composites (cp. Fig. 4). In the middle of the spectral region, X., the form factor is dominated by a few branches of excitation. The strongest one is marked on the lower panel by a solid white line X = r that is the corresponding branch of binary approximation 5,6. In this approximation, each eigenmode consists of two excited regions ( hot spots ) and the form factor is given by S (, ) δ ( ) ρ( ) + ( )[ δ ( + ) δ ( ) + δ ( ) δ ( + )], r X = r X f r X r X r X r X r (9) 6

17 where (r) f is a smooth distribution of the inter-monomer distances, f r) = N δ ( r ) r ij (. The weaker branch, marked by the dotted line X = ( + 57) r, is due to the ternary excitations, i.e., eigenvectors consisting of three hot spots. At X., we see a sharp transition to delocalization, where the positive-correlation region becomes uniform, spreading over most of the system. This transition occurs when the correlation length becomes comparable to the size of the system R c, i.e. at X R c N. Consequently, this transition is mesoscopic, i.e., it phases out as size of the system becomes very large. This is in contrast to an Anderson transition where local density of scatters is the determining parameter. This difference from Anderson localization is due to the long-range interaction in our case. As a result, most of the eigenvectors are not propagating waves, but change from binary/ternary excitations to delocalized surface plasmons as X decreases. To briefly conclude this section, chaos of eigenmodes is present for both conventional and fractal geometries of the composites. However, fractal composites demonstrate this chaos in the whole spectral region, while conventional composites do only in the extremes of the spectrum. For conventional geometries (RLG composites in particular), the eigenmodes are dominated by binary/ternary excitations with a mesoscopic transition to uniform surface plasmons near plasmon resonance of the inclusion particles (monomers), i.e. at X. In the next section we will study the role of the fluctuations and chaos of eigenmodes on the nonlinear radiative photoprocesses. 7. Enhancement of Radiative Photoprocesses and Nonlinear Polarizabilities in Clusters Among enhanced radiative photoprocesses, one of the best studied experimentally is surface-enhanced Raman scattering (SERS). 9 The intensity of Raman scattering for molecules adsorbed at rough surfaces or colloidal-metal particles is known to be greatly enhanced, by a factor of up to 6. 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, as we have shown 48 the exact expression of the SERS enhancement factor G RS can be obtained as X δ G RS + = Imα Q X Imα. δ () This result predicts that on the order of magnitude the enhancement is the same (by the factor of Q ) as for non-radiative photoprocesses of the second order. Qualitatively, we may interpret this result in the following way. For a large frequency shift, the outgoing photon is out of resonance and does not considerably interact with the system. Correspondingly, the entire enhancement is due to the local field of the absorbed photons, the same as for non-radiative processes [cp. the discussion after Eq. ()]. A much more dramatic enhancement is found 48 for the second case, namely that of Raman frequency shifts smaller than monomer s linewidth. This situation is characteristic of the most 7

18 typical and interesting cases of SERS. We have shown 48 that in this case there exists an approximate expression for the enhancement coefficient G RS of SERS, G RS Q R X Imα = X δ R X Imα. In this case we have G RS Q, where the physical interpretation is the following. Enhancement by Q is due to the first power of the local-field intensity. Another power Q appears because the outgoing photon is not emitted freely, but in the resonant environment. Finally, one power of Q vanishes because only a small fraction Q of the monomers is resonant to the exciting radiation. We compare the functional dependence predicted by Eq. () with the numerical simulation in Fig. 7. As we see, this equation describes the numerically found dependencies quite well, especially the dependence on X. Equation () shows that the Raman scattering is enhanced for an adsorbed molecule by a factor of G RS Q. For noble metals in the red region of visible light, we have Q, so the predicted enhancement is very large, G RS 4 6. This explains the range of enhancements found experimentally. 9 The comparison 48 of theoretical spectral dependence with the experiment is shown in Fig. 8. As we can see, the theory explains the experimental observations qualitatively. Note, however, that recently an extremely high enhancement of the Raman scattering has been observed 49 RS with G. This enhancement allowed observation of Raman scattering from a single molecule. Such an enhancement is not understood in the framework of the electromagnetic theory of Ref. 48. It is possible that the so-called chemical enhancement mechanism in addition to the electromagnetic one contributes to yield enhancements so high. Now, we will very briefly discuss coherent, or parametric, nonlinear photoprocesses. These processes are due to nonlinear wave mixing. One of the most interesting is frequencydegenerate four-wave mixing (a third-order process), responsible for such an interesting effect as phase conjugation. We have found the corresponding enhancement coefficient in Ref. in the binary approximation and Ref. 5 numerically. Further extensive computations of linear and nonlinear responses of clusters have later been done in Refs. 5 and 5. Physically, the enhancement of an nth -order parametric photoprocess can be estimated in as we have suggested in Ref. 4. The enhancement of the amplitude of each of the n + participating photons (including the emitted photon) is Q. This yields the enhancement of the + amplitude of the process as Q n. For any coherent process, the amplitude, not probability, is the quantity to average. The averaging is done by multiplying by the fraction of resonant X monomers, which found to be δ Imα Imα [see the derivation Eq. (5) in Ref. 4]. This leads Q to mean amplitude as Q n X Imα. Finally, the amplitude enhancement should be squared, yielding the intensity-enhancement coefficient () 8

19 For n =, this reduces to G ( X ) Q. ( n ) n Imα ( ) ( Im ) 6 ( ) G X δ X α () in agreement with the corresponding result of Ref. 5. The comparison of Eq. () to the numerical calculations shown in Fig. 9 supports this result. A remarkable feature of Eq. () is that the predicted enhancement is quite large. For the realistic dielectric parameters of silver in the visible range, Q, and correspondingly G ( ) 4 7. The experimental investigation 5 has indeed found a very strong enhancement for the phase conjugation, with G ( ) 6, confirming the theoretical predictions. () 8. Enhanced Nonlinear Susceptibilities of Composites 8. Macroscopic and Mesoscopic Fields and Integral Formulas for Optical Responses of Composites The enhancement discussed above in Sec. 7 is calculated for nonlinear-optical polarizabilities of molecules adsorbed at the surface of monomers of a cluster. In this section, based on Ref., we follow Refs. 5, 6, and 54 and consider susceptibilities of composites that possess the conventional non-fractal geometry. For such composites one can introduce mesoscopic electric field e (r) and induction d (r) varying in space due to the inhomogeneity of the composite s material. The conventional averaging of these fields yields the corresponding macroscopic quantities = E e( r) d r, D = d( r) d r, V V V V (4) where V is a sample volume of the composite. We will concentrate on Maxwell Garnett (MG) composites. Such a composite is a suspension of nanospheres in a uniform host. These nanospheres (monomers) in the theory of composites are conventionally called inclusions. Here we will not consider another type, Bruggeman composites that are a mixture of two continuous materials, whose geometry may be similar or identical. We consider the third-order nonlinear susceptibilities of the MG composites. Note that we have already discussed linear dielectric constants of the MG composites above in Sec.. We assume that the constituent materials of the composite are isotropic. Then the thirdorder optical nonlinearity is defined in a general case by the following material relation between the induction and field: 9

20 where ε (r), (r) * d( r) = ε( r) e( r) + 4π A( r) e( r) e( r) + B( r) e( r) e ( r), D = εce + 4π Ac E E + A, and B(r) B i in the inclusion material and the values ε h, * BcE E, are functions of the coordinates that acquire the values ε i, A h, and (5) A i, and B h in the host. These parameters for the host are assumed for simplicity to be coordinate-independent, and for the inclusions they may depend on the coordinate, while similar macroscopic (averaged) quantities for the composite ε c, A c, and B c are always coordinate-independent. As before, we consider optical (oscillating in time) electric fields. By e (r), d (r), E and D we understand the time-independent amplitudes of those fields. In what follows, we assume the linear polarization of the exciting radiation. Then the third-order nonlinear susceptibility (hypersusceptibility) of an isotropic composite is completely characterized by one constant, () χ c = Ac + Bc. (6) We impose a standard boundary condition on the mesoscopic electrostatic potential at the surface of the composite, where the mesoscopic field becomes the macroscopic field, ϕ ( r ) = Er, r S, (7) where ϕ (r) is the mesoscopic potential. From this, transforming from a bulk to surface integral and back in Eq. (4), one derives an exact equation for the induction, D = VE V d ( r) e( r) d r. (8) Using Eq. (8) and applying a generalized Gauss theorem along with the boundary conditions (7), in the first order in the field E, one obtains an expression for the linear dielectric constant of the composite () ( e ( r ) ε c = ( r) ) d r VE ε. V (9)

21 Here and below, the order of optical nonlinearity (in the external field) is denoted in parenthesis as a superscript. Alternatively, this dielectric constant ε c can be found directly from the macroscopic first-order field (induction) as () ε c = ε( r) e z ( r) d r. VE V To interpret this expression, we define the dipole moment induced on an a th inclusion 55 as (4) h a = () d d r s π e ( r), ( r), 4 s( r) ε V h ε i a where V a is the volume of this a th inclusion. We introduce by definition the polarizability tensor of an a th inclusion in the composite (a) a βλ by the relation 56 ε (4) d =. ( a) aβ αβλ Eγ It is easy to see that Eq. (4) is equivalent to Eq. () used to compute ε c above in Sec.. (4) To compute the nonlinear susceptibility (6) of the composite we collect terms of the third order in the optical nonlinearity. Then we use the generalized Gauss theorem and the boundary () conditions (7) to eliminate the unknown third order field e ( r). As a result, one obtains an exact expression 5, 54 for the third-order susceptibility of the composite () () () () χ c = χ ( r) e ( r) ( e ( r) ) d r, V E E V (4) where the hypersusceptibility of the (isotropic) constituent materials of the composite is defined as () χ ( r) = A ( r) + B( r). (44) () The most intriguing property of Eq. (4) is that e ( r) does not enter at all, thus greatly () simplifying the task of computing the hypersusceptibility of the composite χ c. The price to pay for this advantage is that one cannot compute each of the constants A c and B c characterizing the most general response, but only their combination (44). From Eq. (4), it follows that the hypersusceptibility of the composite is a sum of two contributions. The first contribution is due to nonlinearity in the inclusions, and the second is due to the nonlinearity in the host. These contributions can be considered independently.

22 8. Hypersusceptibility of a Composite for the Case of Nonlinearity in Inclusions For a Maxwell Garnett composite, the inclusions are spheres. We assume the characteristic distance between the inclusions to greatly exceed the radius R of a sphere. This ensures applicability of the dipole interaction approximation. Also, it allows one to find the linear (first-order) field in an a th inclusion particle, () ( a) ε h eaβ = qα βλ Eγ, q. R ε i ε h (45) This field, of course, is uniform in this approximation. Substituting Eq. (45) into Eq. (4), we obtain a closed expression for the hypersusceptibility of the composite, ( ) ( ) ( ) () ( a ) a c χ i f q q α β z α βz χ =, where f is a fill factor (a fraction of the total volume occupied by the inclusions) and ( a ) α = G βz aβ, bz b (47) is the polarizability of an a th inclusion in the composite. Formulas of Eqs. (46) and (47) give in a closed form a general expression for the third-order (frequency-degenerate) hypersusceptibility in the case of nonlinear inclusions. Apart from the general expression obtained above, we would like to obtain a compact () formula for χ c in a mean field approximation, similar to what has been done in Ref. 4. To do so, we note that a mean field approximation implies that variations of the local fields are neglected (cf. the discussion of Fig. 6). Assuming that and taking into account the spherical symmetry of an inclusion we can model the composite in the following way. A dielectric sphere of radius R of an inclusion particle with a dielectric constant of ε i is surrounded by a spherical shell of the host material of radius R with a dielectric constant ε h. The space outside of this shell is the homogeneous composite with the dielectric constant ε c. At all interfaces, the conventional boundary conditions of continuity of potential and the normal component of the induction are imposed. The solution of this model for the field inside an inclusion (for r R ) is (46) e ε h ε c ε h = El, E l E, ε + ε ε ( ) + i h h (48)

23 where we have introduced the effective local (Lorentz) field E. The solution outside of the inclusion r R is ( re ) r r E ε ε () l l i h e ( r) = El + α, α R 5, r ε i + ε h where we have introduced the polarizability of an isolated inclusion α. Substituting Eqs. (48) and (49) into Eq. (4), we obtain the familiar Maxwell Garnett formula that we write in the explicit form + fβ α ε i ε h R ε c =, β =, f, fβ R ε + ε R (5) where we introduce the fill factor f (i.e., a fraction of the volume of a composite occupied by inclusions). Substituting the internal field given by Eq. (48) into Eq. (4), we obtain the required expression for the hypersusceptibility in the case of nonlinearity in the inclusions, () () ε c + ε h ε c + ε h χ = c χ i f. ε i + ε h ε i + ε h (5) This expression precisely coincides with the corresponding result of Ref.. We model the Maxwell Garnett (MG) composite as a random lattice gas (RLG) of inclusions embedded in a uniform host. For the sake of numerical computations we realistically assume that the inclusions are silver nanospheres whose dielectric constant ε i is close to the bulk values, 44 and the embedding medium (the host) has the dielectric constant of ε =.. () () χc We have calculated numerically the enhancement coefficient g = for the thirdorder susceptibility of a MG composite using both the spectral theory [Eq. (46)] and the mean- () χi field theory [Eq. (5)]. The results are shown in Fig.. As one can immediately conclude, the mean-field theory completely fails to reproduce the results of the spectral theory that is exact within the framework of the dipole approximation. The underlying cause of this failure is in strong fluctuation and chaos of the eigenmodes and local fields discussed throughout this Chapter (see Secs. 4-6). The degree of the failure of the mean-field theory for nonlinear processes is much more dramatic than for linear susceptibility (cp. Fig. 5 with Fig. ). It is understandable, because nonlinear processes are much more sensitive to fluctuations of the local fields. This is due to the fact that a nonlinear response is proportional to a high power of the acting field. The difference between the results of the present theory and the mean-field theory is so large that it is impossible to show both simultaneously on the same linear scale. Therefore, we i h h (49)

24 show in Fig. these results on the logarithmic scale. As we can see, the spectral profiles of predicted by these two theories are very different. The mean field theory overestimates the enhancement in its peak (close to the surface plasmon resonance) by more then two orders of magnitude and underestimates the enhancement off the plasmon resonance frequency by about the same order of magnitude. The spectral profile in the mean-field theory is much narrower than follows from the spectral theory. The cause of these differences is in the neglect of the spatial fluctuations of the local field in the mean field theory. An important feature seen in Fig. and Fig. is a strong enhancement of the thirdorder hypersusceptibility by over three orders of magnitude. Still, this enhancement is much smaller than the one predicted and observed for fractal clusters (see above in Sec. 7). The reason is that the fluctuations of the local fields are much stronger in fractals, leading to the much stronger enhancement of the nonlinear optical responses. () χ c 8. Hypersusceptibility of a Composite for the Case of Nonlinearity in the Host In this case to compute the hypersusceptibility from Eq. (4), one needs to know the mesoscopic field in the host (outside of the inclusion particles). This is obtained from Eqs. () and (8) as e () β W βγ ( r) = Eβ W ( r) b ( r r ) d [ δ r r r ] r. βγ β βγ γ (5) The hypersusceptibility of the composite in this case has been computed by substituting (5) into Eq. (4) and performing the numerical integration by the Monte Carlo method. Similarly, the hypersusceptibility in the mean field approximation has been obtained by integrating an expression that follows from Eq. (4) in our coated-sphere model of the composite unit cell, b bγ, which yields g () χ χ () c () h ε c + ε h p ε h = R () () ( ) () e ( r) e ( r) χ c = χ h d r 4 R, π R E E f ( +. p p ( f )[8 f ( + f + f ) β β + f ( + f f ) β ) β β + 8 f ( β + β ) + 5], (5) (54) 4

25 This result reduces to the corresponding formula of Ref., if one retains only the lowest powers of f. This difference is due to the nature of approximations made in Ref.. We show in Fig. the real and imaginary parts of the third-order enhancement coefficient () () χc g = for the case of nonlinear host under consideration. As we can see, there is a () χ h disagreement between the present spectral theory and the mean field approximation by orders of magnitude. The enhancement in this case is significantly larger than for nonlinear inclusions, and the spectral dependence is different (cf. Sec. 8. and Fig. ). Because the difference between the spectral theory and the mean field theory is too large to compare them on the same plot, we show in Fig. the magnitude of the enhancement () () χ c coefficient g = for both the theories. We can see that the magnitude of the enhancement () χ i is about an order of magnitude higher than for the case of nonlinearity in the inclusions (cf. Fig. ). There is enhancement also in the wings of the spectral profile, while for the case of internal () nonlinearity the composite has lower χ c in the spectral wing than that of its nonlinear constituent. The difference is due to a different spectral dependence of the fields inside and outside of an inclusion. Specifically, a large value of the dielectric constant close to the plasmon resonance leads to small values of the fields inside the inclusions (dielectric screening), but not outside. Similar to Sec. 8., the mean field enhancement has a much narrower and higher spectral profile peaking about the plasmon resonance frequency of the composite. In contrast, the actual enhancement, as given by the present spectral theory, has a much wider and lower spectral profile. This is due to the fact that it is formed by many chaotic eigenmodes widely distributed in frequency. This feature is interdependent with the giant spatial fluctuations of the local fields (see above Secs. 4-6). Finally, we point out a principal distinction between the optical enhancement in fractal systems (clusters) and non-fractal systems (MG composites). As we have already mentioned, for the MG composites, the enhancement is peaked about the plasmon resonance frequency, though the enhancement profile is very wide (see, e.g., Fig. and Fig. ). In contrast, the enhancement for fractal clusters increases from the plasmon resonance frequency toward the extreme spectral wings (cf. Fig., Fig. 8, and Fig. 9). The root of this difference is in much large ( giant ) fluctuations of the local fields in fractals, which increase in a scaling manner toward spectral wings. 9. Concluding Remarks We have considered a variety of the photoprocesses mediated by disordered clusters and composites. We employed models with both fractal (self-similar) and conventional (non-fractal) geometries to simulate different existing systems. A remarkable feature of the disordered clusters and composites is giant fluctuations of the local fields that bring about their enhanced nonlinear-optical responses. Namely, the enhancement 5

26 is due not only to the high averaged value of the local fields, but principally due to their fluctuations in space, from one monomer (inclusion particle) to another. Nonlinearity, causing a higher power of the local fields to be an acting parameter, enhances the effects of these fluctuations. Therefore, the enhancements are found to increase dramatically with the order of nonlinearity. The chaos of the local fields bears many similarities to quantum chaos, but differs due to the long range of the dipole interaction. Not only individual eigenmodes are chaotic, but also their spatial correlation factors. These chaos and fluctuations are responsible for the dramatic failure of the mean-field theory to describe the nonlinear susceptibility of composites. Both fractal clusters and non-fractal composites exhibit strong enhancement of the radiative and non-radiative nonlinear optical responses. However there are significant qualitative differences and strong quantitative differences between these two classes of systems. Specifically, the enhancement in fractals is much greater. It also has a different spectral dependence: for fractals the enhancement increases monotonically from the plasmon resonance frequency toward the red (infrared) edge of the spectral contour. In contrast, for non-fractal composites, the enhancement is a broad peak around the plasmon resonance. The underlying reason of these differences is giant (spatial) fluctuations of local fields dominating the optical properties of fractals. At the same time, similar fluctuations for conventional (non-fractal) geometries are much smaller. Finally, many of the effects predicted have been verified experimentally, and we have mentioned some of them. However, many very interesting effects are not discussed due to the space limitations. Among them, we recognize the enhanced laser production of plasma by colloidal-metal clusters. 57 6

27 References. J.C.Maxwell Garnett, Philos. Trans. R. Soc. London, 85 (96).. H.A.Lorentz, Theory of Electrons (Dover, New York), 95.. D.A.G.Bruggeman, Ann. Phys. (Leipzig), 4, 66 (95). 4. V.M.Shalaev and M.I.Stockman, Optical Properties of Fractal Clusters (Susceptibility, Surface Enhanced Raman Scattering by Impurities), ZhETF 9, 59-5 (987) [Translation: Sov. Phys. JETP 65, (987)]. 5. V.A.Markel, L.S.Muratov and M.I.Stockman, Theory and Numerical Simulation Of The Optical Properties of Fractal Clusters, ZhETF 98, (99) [translation: Sov. Phys. JETP 7, (99)]. 6. V.A.Markel, L.S.Muratov, M.I.Stockman, and T.F.George, Theory and Numerical Simulation of Optical Properties of Fractal Clusters, Phys. Rev. B 4, (99). 7. M.I.Stockman, L.N.Pandey, L.S.Muratov and T.F.George, Optical Absorption and Localization of Eigenmodes in Disordered Clusters, Phys. Rev. B 5, (995). 8. M.I.Stockman, L.N.Pandey and T.F.George, Inhomogeneous Localization of Polar Eigenmodes in Fractals, Phys. Rev. B 5, 8-86 (996). 9. M.Moskovits, Surface Enhanced Spectroscopy, Rev. Mod. Phys. 57, (985).. A.V.Butenko, V.M.Shalaev and M.I.Stockman, Giant Impurity Nonlinearities in Optics of Fractal Clusters, ZhETF 94, 7-4 (988) [translation: Sov. Phys. JETP, 67, 6-69(988)]. 7

28 . T.A.Witten and L.M.Sander, Diffusion-Limited Aggregation, A Kinetic Critical Phenomenon, Phys. Rev. Lett. 47, 4-4 (98).. 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, (995).. J.E.Sipe and R.W.Boyd, Nonlinear Susceptibility of Composite Optical Materials in the Maxwell Garnett Model, Phys. Rev. B 46, (99). 4. R. W. Boyd and J. E. Sipe, Nonlinear Susceptibility of Layered Composite Materials, J. Opt. Soc. Am., B, 97-, R.W.Boyd, Influence of Local Field Effects on the Nonlinear Optical Properties of Composite Materials, Proc. Of the SPIE, 54, 6-4 (996). 6. R.J.Gehr, G.L.Fisher, R.W.Boyd, Nonlinear Optical Response of Layered Composite Materials, Phys. Rev. A 5, (996). 7. R.J.Gehr and R.W.Boyd, Optical Properties of Nanostructured Optical Materials, Chemistry of Materials, 8, 87-89, R.W.Boyd, R.J.Gehr, G.L.Fisher, and J.E.Sipe, Nonlinear Optical Properties of Nanocomposite Materials, Pure and Applied Optics, 5, 55-5 (996). 9. R.J.Gehr, G.L.Fisher, and R.W.Boyd, Nonlinear Optical response of Porous-Glass-Based Materials, J. Opt. Soc. Am., B 4, -4 (997). 8

29 . D.B.Smith, G.L.Fisher, R.W.Boyd, and D.A.Gregory, Cancellation of Photoinduced Absorption in Metal Nanoparticle Composites through a Counterintuitive Consequence of Local Field Effects, J. Opt. Soc. Am. B 4, 65-6 (997).. G.L.Fisher and R.W.Boyd, Third-Order Nonlinear-Optical Properties of Selected Composites, ACS Symposium Series 679, 8-4 (997).. G. W. Milton, Bounds on The Complex Permittivity of a Two-Component Composite Material, J. Appl. Phys. 5, (98).. G. W. Milton, Bounds on the Transport and Optical Properties of a Two-Component Composite Material, J. Appl. Phys. 5, (98). 4. G. W. Milton, Bounds on The Electromagnetic, Elastic, and other Properties of Two- Component Composites, Phys. Rev. Lett (98). 5. D.J.Bergman and D.Stroud, In: Solid State Physics, edited by H.Ehrenreich and D.Turnbull, v.46, p.47 (Academic Press, Boston, 99). 6. H.Ma, W.Wen, W.Y.Tam, and P.Sheng, Frequency Dependent Electrorheological Properties: Origins and Bounds, Phys. Rev. Lett. 77, (996). 7. W.Y.Tam, G.H.Yi, W.Wen, H.Ma, M.M.T.Loy, and P.Sheng, New Electrorheological Fluid: Theory and Experiment, Phys. Rev. Lett. 78, (997). 8. K.Ghosh and R.Fuchs, Spectral Theory for Two-Component Porous Media, Phys. Rev. B 8, 5-56 (988). 9

30 9. R.Fuchs and F.Claro, Spectral Representation for the Polarizability of a Collection of Dielectric Spheres, Phys. Rev. B 9, (989).. S.Alexander, The Vibration of Fractals and Scattering from Aerogels, Phys. Rev. B 4, (989).. V.M. Shalaev, R.Botet and A.V.Butenko, Localization of Collective Dipole Excitation on Fractals, Phys. Rev. B 48, (99).. M.I.Stockman, K.B.Kurlayev, and T.F.George, Linear and Nonlinear Optical Susceptibilities of Maxwell Garnett Composites: Dipolar Spectral Theory, Phys. Rev. B (999) (In press, to appear tentatively in December 5, 999 issue).. M.I.Stockman, L.N.Pandey, L.S.Muratov and T.F.George, Comment on Photon Scanning Tunneling Microscopy Images of Optical Excitations of Fractal Metal Colloid Clusters, Phys. Rev. Lett. 75, (995). 4. M.I.Stockman, Chaos and Spatial Correlations for Dipolar Eigenmodes, Phys. Rev. Lett. 79, (997). 5. M.I.Stockman, Inhomogeneous Eigenmode Localization, Chaos and Correlations for Large Disordered Clusters, Phys. Rev. E 56, (997). 6. A.V.Karpov, A.K.Popov, S.G.Rautian, V.P.Safonov, V.V.Slabko, V.M.Shalaev and M.I.Stockman. Observation of a Wavelength- and Polarization-Selective Photomodification of Silver Clusters, Pis'ma ZhETF 48, 58-5 (988) [Translation: JETP Lett. 48, (988)].

31 7. 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 85, 6-65 (99). 8. V.P.Safonov, V.M.Shalaev, V.A.Markel, Yu.E.Danilova, N.N.Lepeshkin, W.Kim, S.G.Rautian, and R.L.Armstrong, Spectral Dependence of Selective Photomodification in Fractal Aggregates of Colloidal Particles, Phys. Rev. Lett. 8, -5 (998). 9. D.P.Tsai, J.Kovacs, Z.Wang, M.Moskovits, V.M.Shalaev, J.S.Suh and R.Botet, Photon Scanning Tunneling Microscopy Images of Optical Excitations of Fractal Metal Colloid Clusters, Phys. Rev. Lett. 7, 44945, P.Zhang, T.L.Haslett, C.Douketis, and M.Moskovits, Mode Localization in Self-Affine Fractal Interfaces Observed by Near-Field Microscopy, Phys. Rev. 57, (998). 4. V.A.Markel, V.M.Shalaev, P.Zhang, W.Huynh, L.Tay, T.L.Haslett, and M.Moskovits, Near-Field Optical Spectroscopy of Individual Surface-Plasmon Modes in Colloid Clusters, Phys. Rev. B 59, 9-99 (999). 4. S.I.Bozhevolnyi, V.A.Markel, V.Coello, W.Kim, and V.M.Shalaev, Direct Observation of Localized Dipolar Excitations on Rough Nanostructured Surfaces, Phys. Rev. B 58, (998). 4. M.I.Stockman, L.N.Pandey, L.S.Muratov and T.F.George, Giant Fluctuations of Local Optical Fields in Fractal Clusters, Phys. Rev. Lett. 7, (994).

32 44. P.B. Johnson and R.W.Christy, Optical Constants of Noble Metals, Phys. Rev. B 6, (97). 45. M.V.Berry, Regular and Irregular Semiclassical Wavefunctions, J. Phys. A, 8-9 (977). 46. M.C.Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York), V.I.Fal co and K.B.Efetov, Long Range Correlations in the Wave Functions of Chaotic Systems, Phys. Rev. Lett. 77, 9-95 (996). 48. M.I.Stockman, V.M.Shalaev, M.Moskovits, R.Botet, and T.F.George, Enhanced Raman Scattering by Fractal Clusters: Scale Invariant Theory, Phys. Rev. B 46, 8-8 (99). 49. K.Kneipp, Y.Wang, H.Kneipp, L.T.Perelman, I.Itzkan, R.D.Dasari, and M.Feld, Single Molecule Detection Using Surface-Enhanced Raman Scattering (SERS), Phys. Rev. Lett. 78, (997). 5. V.M.Shalaev and M.I.Stockman, Resonant Excitation and Nonlinear Optics of Fractals, Physica A 85, 8-86 (99). 5. V.A.Markel, V.M.Shalaev, E.B.Stechel, W.Kim, and R.L.Armstrong, Small-Particle Composites. I. Linear Optical Properties, Phys. Rev. B 5, (996). 5. V.M.Shalaev, E.Y.Polyakov, and V.A.Markel, Small-Particle Composites. II. Nonlinear Optical Properties, Phys. Rev. B 5, (996).

33 5. A.V.Butenko, P.A.Chubakov, Yu.E.Danilova, S.V.Karpov, A.K.Popov, S.G.Rautian, V.P.Safonov, V.V.Slabko, V.M.Shalaev and M.I.Stockman, Nonlinear Optics of Metal Fractal Clusters, Z. Phys. D 7, 8-89 (99). 54. D.Stroud and P.M.Hui, Nonlinear Susceptibilities of Granular Matter, Phys. Rev. B 7, (988). 55. In this section and below we will number the inclusions with the indices a,b, to prevent possible confusion with the notation i for the inclusions. 56. The inclusion s number (a) in the superscript should not be confused with the order in the field that can be only () or (). 57. M.M. Murnane, H.C.Kapteyn, S.P.Gordon, J.Bokor, E.N.Glytsis and R.Falcone, Efficient Coupling of High-Intensity Subpicosecond Laser Pulses into Solids, Appl. Phys. Lett. 6, 68-7 (99).

34 Captions to Figures Fig.. Cluster-cluster aggregate (CCA) of N = monomers. Fig.. Dependence of the spectral parameter X and the dissipation parameter δ (in the units of R ) on light frequency expressed as photon energy. Fig.. Numerically obtained absorption ( Im α ) (upper panel) and density of eigenmodes (lower panel) for cluster-cluster aggregates. Fig. 4. Averaged localization (coherence) radius L X as a function of the spectral parameter X. Fig. 5. Comparison of the dielectric susceptibility of a composite in present spectral theory (ST) with the result of the Maxwell Garnett formula (MG). The relative dielectric constant of a composite ε is shown as a function of the photon frequency in energy units (ev). Fig. 6. Spatial distribution of the intensity of induced dipoles at different inclusions for a Maxwell Garnett composite. This distribution is plotted in the following way: A composite is generated and local (i) dipole polarizabilities α βγ are found for each inclusion from Eq. (). Then the inclusions are projected onto the xy -plane. If there are several inclusions projected onto the same site, one of (i) them is randomly left. The square of the local polarizability α βz for each of the inclusions left is plotted as the vertical coordinate. The left panel shows the distribution for the resonant region (in terms of the spectral parameter, R X =., while the right one is for the off-resonant region (spectral wing, R X =. ). Fig. 7. All eigenmodes with negative eigenvalues for a CCA cluster of N = 4 monomers. Fig. 8. Spatial distribution of the local-field intensities for an individual CCA cluster ( N = 5 ) shown over the two-dimensional projection of the cluster for the eigenvalues w n (in the units of R ) as indicated. The coordinates are shown in units of the lattice spacing R. The value of the gyration radius of the individual eigenmodes is given relative to the cluster radius R c. Fig. 9. Spatial distribution of the local-field intensities for external excitation of an individual CCA cluster ( N = 5 ) for the values of the spectral parameter X and the polarization of the exciting radiation shown in the figure. The value of the dissipation parameter is R δ =. Fig.. Localization-length distribution P ( L, X ) of eigenmodes for CCA clusters ( N = 5 ). The position of the lower X cutoff is qualitatively illustrated by the dashed bold line. Fig.. Same as in Fig. but for a Maxwell Garnett composite, simulated as a random lattice gas (RLG). Fig.. Normalized enhancement factors values of δ and n shown. n ( n ) G δ Q as functions of X for CCA clusters for the 4

35 Fig.. Distribution function of the local field intensity P (G) calculated for R δ =., 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. 4. Dynamic form factor S ( r, X ) for CCA composite (the number of inclusions N = 5, averaged over an ensemble of composites. The upper panel shows a -dimensional representation of the function, while the lower panel is the corresponding contour map. The vertical scale is pseudo-logarithmic to show simultaneously positive and negative values of S ( r, X ). To obtain it, 4 a small region of the plot for S ( r, X ) is removed. The function plotted is 4 log[ S ( r, X ) ]sgn[ S( r, X )]. The horizontal scales are logarithmic. Fig. 5. Intensity correlation function C ( r, X ) calculated for CCA composite ( N = 5 ) averaged over ensemble of systems. The upper panel is the function plotted in the triple-logarithmic scale and the lower panel is the corresponding contour map. Fig. 6. The same as in Fig. 4, but for RLG composite. Fig. 7. Scaled enhancement coefficient of SERS from silver colloid clusters in comparison with the prediction of Eq. (). The theoretical dependence is calculated for CCA clusters. Fig. 8. Theoretical and experimental spectral (in terms of wavelength) dependencies of the enhancement RS coefficient G for silver colloid clusters. Fig. 9. Scaled enhancement coefficient for third-order degenerate parametric process, calculated for CCA. () () χ c Fig.. Real (left panel) and imaginary (right panel) parts of the enhancement coefficient g = for () χ i the case of nonlinearity in inclusions. The curves are shown for a MG composite with a fill factor of f =. as functions of the light frequency (expressed as the photon energy). The results of the present spectral theory are indicated by ST and those of the mean-field theory by MF. () () χ c Fig.. Magnitude g = of the enhancement coefficient for a MG composite ( f =. ) () χ i computed in the present spectral theory (ST) and in the mean field approximation (MF) for the case of nonlinearity in inclusions. Note the logarithmic scale. Fig.. Same as in Fig., but for the case of nonlinearity in the host. Fig.. Same as in Fig., but for the case of nonlinearity in the host. 5

36