Local Fields' Localization and Chaos and Nonlinear-Optical

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 30303 E-mail: [email protected] http://www.phy-astr.gsu.edu/stockman 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. 1. INTRODUCTION .............................................................................................................2 2. EQUATIONS GOVERNING OPTICAL (DIPOLAR) RESPONSES, SPECTRAL REPRESENTATION AND SCALING.............................................................................4 3. LINEAR OPTICAL RESPONSES ...................................................................................7 4. INHOMOGENEOUS LOCALIZATION OF EIGENMODES .......................................9 5. GIANT FLUCTUATIONS OF LOCAL FIELDS AND ENHANCEMENT OF NONRADIATIVE PHOTOPROCESSES...............................................................................12 6. CHAOS OF EIGENMODES ..........................................................................................14 7. ENHANCEMENT OF RADIATIVE PHOTOPROCESSES AND NONLINEAR POLARIZABILITIES IN CLUSTERS ..........................................................................17 8. ENHANCED NONLINEAR SUSCEPTIBILITIES OF COMPOSITES ......................19 8.1

MACROSCOPIC AND MESOSCOPIC FIELDS AND INTEGRAL FORMULAS FOR OPTICAL RESPONSES OF COMPOSITES ........................................................................................................................................ 19

8.2

HYPERSUSCEPTIBILITY OF A COMPOSITE FOR THE CASE OF NONLINEARITY IN INCLUSIONS....................... 22

8.3

HYPERSUSCEPTIBILITY OF A COMPOSITE FOR THE CASE OF NONLINEARITY IN THE HOST.......................... 24

9. CONCLUDING REMARKS ..........................................................................................25 REFERENCES......................................................................................................................27 CAPTIONS TO FIGURES ...................................................................................................34

1. 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 Garnett1, Lorentz2 and Bruggeman3. 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 composites4-8. 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 1 ∝ E 2 . For the sake of argument, let us say, the first field has the same constant intensity I1 in N >> 1 points, and the second is strongly localized at one point where its intensity then should be I 2 = NI 1 . Consider a n th order nonlinearity where the nonlinear response is proportional to

E 2 n ∝ I n . The ratio of the nonlinear response for the first

(constant) field is proportional to

1 ( NI 1n ) = I 1n . In contrast, the response to the second (strongly N

1 ( NI 1 ) n = N n−1 I1n . In such a way, the enhancement coefficient (the ratio of N the nonlinear response in the second case to that in the first case) is N n−1 . 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). localized) field is

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

2

objects seen. For such systems, the number N of constituent particles (monomers) contained within a radius R scales as D

 R N ≅  ,  R0  (1) where R0 is a typical distance between monomers, and D is the fractal (Hausdorff) dimension of the cluster. The density of monomers as given by Eq. (1) is asymptotically zero for large clusters,  R ρ≅R    R0  −3 0

D− 3

→0 . (2)

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. (2). 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.4,10 An example of a fractal cluster, obtained by cluster-cluster aggregation11 (CCA) is shown in Fig. 1. 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 space12. 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 approximation13. 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.12-21 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 method22-25 and the dipolar spectral method5,6 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.26,27 In Sec. 2, we present the coupled dipole equations governing optical responses and summarize the dipolar spectral theory. In that section we also summarize some predictions of

3

scaling. In Sec. 3, 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. 2. 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 ri . 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 = x , y , z 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

−1 iα 0

α d

=E

( )( )

 rij rij  α − ∑  δαβ − 3 rij2 j =1   N

( 0) iα

β

  d jβ  r 3  ij

. (3)

Here E i(α0 ) 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 0 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 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 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.28,29 The material properties of the system enter Eq. (3) only via the combination Zrij3 , where we have introduced the notation Z ≡ α 0−1 . 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

system eigenmodes contributing to F should have their localization radii L intermediate between the maximum scale (size of the cluster Rc ) and the minimum scale, R0 . Then this quantity will not depend on any external length, leading to scaling. Because the quantity F is not sensitive to the maximum scale Rc , it should have the

functional dependence F = F (ZR03 ) . On the other hand, the eigenmodes contributing to F are insensitive to a much smaller minimum scale. Therefore the dependence on R0 can be only power (scaling), and consequently F (Z ) ∝ (ZR03 )

γ

, (4)

where γ is some scaling index. 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,

X =

1 3 Rm

(ε 0 + 2ε h )3 2 ( 3ε hω p

ω − ω s ); δ =

1 3 Rm

(ε 0 + 2ε h )3 2 γ 3ε hω p

2

, (5)

where ε 0 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 =

ωp ε 0 + 2ε h

is the surface

plasmon frequency, and Rm is the nanosphere’s radius. The spectral dependence of X and δ for silver is illustrated in Fig. 2. 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

Q≡

X

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. (19)-(21)]. The fact that for many metals Q may be large (as large as ≅ 10 2 ) 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.

5

R −3 We note that in some earlier work the quality factor has been defined differently, Q ′ ≡ 0

δ .

Convenience of this definition is that in the Drude model [see Eq. (5)] this factor, 6ε h ω p R3 Q ′ = m3 , is a constant. In reality, Q ′ is not a constant, because the dissipation R 0 γ (ε 0 + 2ε h )3 2 parameter δ depends on frequency (cf. Fig. 2 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. (21) and its discussion], that is the most important characteristic for the present theory. To introduce the spectral representation,5,6 we will rewrite Eq. (3) as one equation in the

)  , whose

3N-dimensional space. To do so, we introduce 3N-dimensional vectors d ), E ( 0 ) , projections give the physical vectors

(iα d ) = d iα , (iα E ( 0 ) ) = E i(α0 ) ,

. (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

( )( )

 rij rij α  δ − 3 αβ  (iα W jβ ) =  rij2   0,

β

  1  r 3 ,  ij

i≠ j i= j.

, (9)

We introduce the eigenmodes (plasmons) | n ) ( n = 1,....,3N ) as the eigenvectors of the Woperator,

W n) = w n n) , (10) where w n are the corresponding eigenvalues. Practically, the eigenvalue problem (10) can be solved numerically for any given cluster. Having done so, one can calculate the Green’s function 3N

G iα , jβ = ∑ n =1

(iα n )( jβ n ) Z + wn

, (11)

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

6

eigenvectors can be chosen and will be assumed real. Therefore, all amplitudes are symmetric, in particular, (iα n ) = (n iα ) . 3. 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 = ∑ α .(i) , α .(i) = ∑ G iα , jβ , i

j

ρ = ∑ G i α ,i α , i, j

(12) (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 1 + 4πα  , V   (13) where ε h is the dielectric constant of a host, V is the volume occupied by the cluster

1 ∑ G iβ , jβ is a polarizability of a monomer in the cluster 3N i, j (composite). Below we will consider results of numerical computations using Eqs. (12) and (13) and will compare them to some analytical predictions. (composite) and α =

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 >> 1 . In this case, a dependence

of type (4) becomes F ( X ) ∝ ( R03 X ) . We have introduced5,6 such a dependence for α αβ ( X ) and ρ ( X ) and argued that the two quantities have the same scaling, γ

Im α (X) ≅ ρ(X) ≅ R03 R03 X

d o −1

, (14)

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 1 ≥ d o ≥ 0 . The strong localization has been essential for the derivation of Eq. (12). It implies that all eigenmodes (at least all contributing eigenmodes) of a cluster are strongly localized The strong localization, as discussed by Alexander30, 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 shown5,6 that L X should scale as

7

L X ≅ R0 R X 3 0

d o −1 3− D

. (15)

We have subjected the scaling predictions of Eqs. (14) and (15) to an extensive comparison with the results of large-scale computations.7 Similar results have been obtained7 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. 3. 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 R0 and Rc ). 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. (15). First, one has to formulate how to calculate the localization radius. We use the definition of Ref.7,

∑ρ L = ∑ρ n

LX

n

n

n

, where Ln = ∑ r (n i α ) iβ

2 i

2

 2  −  ∑ ri (n i α )   iβ 

2

,

[

(16)

]

where ρ n = ( X − wn ) + δ 2 , Ln is the localization radius of a given eigenmode, and L X 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.31 (precision of our calculations is much higher than that of Ref.31). 2

−1

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 >> 1 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. 32. 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 h = 2.0 . This figure displays both the results of the spectral theory computations accordingly to Eq. (13) and those of a mean field approximation known as Maxwell Garnett formula (or an equivalent Lorentz-Lorenz formula), see, e.g., Ref. 13. 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 .

8

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) R0 to the total size of the system Rc . 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,8,33 (see also Refs. 34 and 35), 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 0 of the distance between the monomers to the maximum scale ≅ R c of the cluster total radius. As we have already mentioned above in Sec. 3, either of these extremes ( Ln ≅ Rc and Ln ≅ R0 ) violates a necessary scaling condition R0 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 33-35 have shown that the strong localization is not the case. In actuality, the inhomogeneous localization8,33 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.36-38 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

10

by Moskovits and collaborators of the near-field optical fields in large silver clusters.39 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.40-42 (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. 39 that the observed phenomena support the strong localization hypothesis contradicts our conclusions. We have commented33 that the observations of Ref. 39 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 ,

P ( L, X ) =

∑ δ (L − L )δ (X − w ) n

n

n

, (17)

We show this distribution calculated for CCA clusters in Fig. 10 and for Maxwell Garnett composites modeled as a random lattice gas (RLG) in Fig. 11. The most conspicuous feature of the distribution of Fig. 10 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. 10 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. (16)] 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, Rc and R0 . This, along with the self-similarity of the clusters, suggests that l X scales with X , i.e., l X ∝ X

λ

. Indeed, Fig. 10 supports the possibility of such a scaling with the corresponding

index λ ≈ −0.25 . 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. 11. The distribution for

11

X ≥ 0.1 is similar to that of CCA (Fig. 10) characteristic of inhomogeneous localization. The major distinction from Fig. 10 is that the distribution in Fig. 11 shows the complete delocalization of the eigenmodes for X ≤ 0.01 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. 10. 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. 3 and 4. In this Section, following Ref. 43, 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 ith monomer is expressed in terms of the Green’s function (12) as

E iα = Z ∑ G iα , jβ E β( 0 ) . jβ

(18) In terms of this field, the local field-intensity enhancement coefficient G i for an ith monomer and the corresponding distribution function P (G ) are defined as

Gi =

Ei E

2

(0) 2

, P (G ) =

1 N

∑ δ (G − G ) i

i

. (19)

We introduce also an n th moment of this distribution:

Mn = Gn

=

∫ P (G )G

n

dG

. (20)

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. 12. The data in this figure are scaled by the

12

factor R03δQ − 2 ( − 1) , with the resonant quality factor given by Eq. (6). The most remarkable feature of Fig. 12 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 ( R03δ < 0.01 ). Moreover, this curve is actually close to a straight line in the intermediate region indicating a scaling behavior of M n ( X ) . We n

n conjecture this scaling as the dependence M n ≅ Q 2 ( −1) M 1 .

For the first moment

M1

we have previously obtained5,6 the exact relation

M 1 = ( X 2 + δ 2 ) Im α δ . Because the absorption Im α does not scale in X (see Sec.3), 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. 3). Therefore, for Q >> 1 the apparent scaling in X takes place with a trivial index of 2n , M n ≅ Q 2 n −1 X Im α =

X 2n

δ

2 n −1

Im α ∝ X 2 n ,

(21) in agreement with Fig. 12. A major result of Ref. 43, given by Eq.(21) is that the excitation rate of a non-radiative n th -order nonlinear photoprocess in the vicinity of a disordered cluster is resonantly enhanced by n a factor of M n ≅ Q 2 − 1 . 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 2 (proportional to the local field intensity), therefore the total rate is n increased by a factor of ≅ Q 2 . However, the fraction of monomers that are resonant is small,

≅ Q −1 . Consequently, the resulting enhancement factor is M n ≅ Q 2 n −1 , in agreement with Eq. (21). For instance, for silver in the red spectral region Q ≅ 30 (the optical constants for silver are adopted from Ref. 44), so each succeeding order of the nonlinearity gives enhancement by a factor of Q 2 ≅ 1000 . 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.36-38 We have considered above the moments (averaged powers) of the local fields. Now we consider another characteristic of the fluctuations, the distribution function P ( G ) (19) of the local-field intensity. Because the change of the minimum scale R0 implies also the change of the local fields, it is possible that the distribution function scales in some intermediate range,

P(G ) ≅ G ε , (22) Under the scale-invariance assumptions, the index ε does not depend on the minimum scale R0 . Consequently, ε does not depend on frequency (the spectral parameter X ) either.

13

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 found43 the enhancement factor Gi for a pair separated by a distance r , located at an angle Θ to the direction of the exciting field,

G i (r , Θ ) =

δ 2 sin 2 Θ

(X − r )

−3 2

+δ2

+

δ 2 cos 2 Θ

( X + 2r )

−3 2

+δ2

. (23)

Using this, we calculate the distribution function as

P (G ) = ∫ δ (G − G i (r , Θ )) C ( r ) d 3 r . (24) D Here C ( r ) ∝ r −1 is the density correlation function of the cluster. Taking into account that large values of G are of interest, we obtain from Eq. (24):

P(G ) ∝ G

− 32

(25) 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 the vector nature of the fields. It is interesting to compare both the scaling prediction and the calculated value of the index ε = − 3 2 with the numerical results. These are shown in Fig. 13 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 = 1 ), 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) microscope39 (see also a comment33 on Ref. 39). 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 = 1.45 for CCA and ε = 1.44 for RW) are quite close to the prediction of the binary theory ( ε = − 3 2 ). 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 (10) 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

14

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.34,35 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 correlations.45, 47 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 (r, X ) =

∑ (iα n )( jβ n ) δ (r − r ) δ (X − w ) ij

n

,

n ,i , j

(26)



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