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Giant fluctuations of second harmonic generation on nanostructured surfaces
Chemical Physics 318 (2005) 156–162 www.elsevier.com/locate/chemphys
Giant fluctuations of second harmonic generation on nanostructured surfaces Mark I. Stockman Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303, USA Received 9 March 2005; accepted 8 June 2005 Available online 18 July 2005
Abstract On the basis of spectral-expansion GreenÕs function theory, we theoretically describe the fundamental harmonic (FH) and second harmonic (SH) local fields at nanorough surfaces of silver and gold. The numerical simulation of these fields in the model of random planar composites has shown that the SH local-field intensity possesses power-law (scaling) distribution for a wide range of system and excitation parameters, which is indicative of giant fluctuations of the SH local fields. Though these scaling phenomena are universal, the scaling index of the SH intensity distribution is not universal depending on the material, excitation frequency, and the fill factor of the nanosystem. The results obtained are in qualitative agreement with the existing experiments. 2005 Elsevier B.V. All rights reserved. PACS: 78.67.n; 68.37.Uv; 05.40.+j; 73.20.Mf Keywords: Local fields; Nanosystem; Fundamental harmonic; Second harmonic; Localization; Giant fluctuations; Distribution function; Scaling index
1. Introduction From its initial development, nanooptics has been one of the most promising and rapidly developing fields at the junction of optics and materials physics [1]. This paper is devoted to one fascinating property of the local optical fields in complex nanostructured metal systems, namely, their giant fluctuations in the real space of their intensity [2]. This paper represents a theory inspired by the recent experiments on statistical properties of the second harmonic generation (SHG) at nanorough metal surfaces [3–5]. The giant fluctuations of local optical fields in nanostructured systems have been predicted for fractal clusters of metal nanospheres [2]. They manifest themselves by a scaling (power-law) behavior of the distribution E-mail address: [email protected]. URLs: http://www.phy-astr.gsu.edu/stockman. 0301-0104/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.06.020
function of the local optical field intensities. Using dipole–dipole interaction model to describe the interactions between the nanospheres, the power (index) of this dependence a has been shown to be universal (not depending on the frequency or fractal dimension) and close to the analytically found value of a ¼ 32. The giant fluctuations and enhancement of local fields in such resonant random systems lead to the corresponding giant enhancement of two- and many-photon processes. In the decade following the publication of [2], there has recently been a great interest in optical properties of metallic nanostructures, in particular, nanorough metal surfaces and clusters. The enhanced optical responses have been especially well studied for surface-enhanced Raman scattering (SERS) [6]. The record enhancement of SERS, J 1012, have been found for Raman observations of single molecules [7,8]. This single-molecule SERS has been definitely linked to the clusterization of metal nanospheres and formation of fractal aggregates [9].
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The SHG in systems built of isotropic metals is principally different from SERS in two respects. First, SHG is a coherent process implying the interference of waves emitted by different sites; therefore, dephasing of the second harmonic (SH) polarization is of importance. This dephasing is the randomization of the SH-polarization phase in the plane of the nanostructure due to the spectral detuning of localized eigenmodes from the excitation frequency. Second, the SHG is a second-order (three-wave) process where for a center-symmetric medium, the SH polarization is concentrated at surfaces and interfaces, while the bulk contribution is suppressed. This is in contrast to odd-order processes such as the phase conjugation [10] or third-harmonic generation where the coherent generation of radiation and giant enhancement were observed. The recent experimental study of SHG by rough gold surfaces with the spatial resolution of 400 nm [3] revealed intriguing properties of SHG: (i) SHG is incoherent, i.e., hyper-Rayleigh scattering, implying strong dephasing. (ii) SHG is almost completely depolarized despite the small detection area. (iii) The SHG has topography of localized bright ‘‘hot spots’’. This topography and depolarization were independently supported, and the enhancement of SHG in the hot spots up to three orders of magnitude was found [4], where rough gold films were studied by confocal microscopy. The most importantly, this study of the distribution function of the SH intensity has revealed its scaling behavior where the index depended on the frequency, but was reasonably close to the value of b = 2.9. Note that if the SH intensity distribution would simply copy the distribution of the square of the fundamental field intensity, the relation between the corresponding indices would be b = a 1, which would give b 2.5, so it is not quite satisfied but not that far off. These findings have been confirmed in the later publication [5]. Theoretical study [11] of the depolarization, dephasing, and correlation of the SHG on rough surfaces has shown that the SH depolarization originates from the peculiarity of the SH being generated at the surfaces. Because noble metals are good conductors even at optical frequencies, the vector of the SH field is directed almost at the normal to the surface. Since the surface on the nanoscale is highly faceted and random, the SH is highly depolarized. The spatial correlation function of the SH field decays in space at the minimum scale. Thus, the depolarization of the SHG is a phenomenon that is originated at the nanoscale and transferred to the larger scales of the system. The origin of the dephasing is actually the delocalization of the linear local fields. As shown in [12], the delocalization of the optical fields on the nanoscale is a fundamental property: any eigenmode that is Anderson-localized is necessary dark and cannot be excited or observed from the far zone. This theorem shows that some published statements on the Anderson
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(strong) localization of local optical fields [13] and its being an underlying cause of the giant fluctuations [14] are completely incorrect. The actual delocalization of the nanoscale optical fields causes their concentration areas (‘‘hot spots’’) to overlap for different modes. Because these modes have randomly different frequencies, this leads to the random phase shifts and results in the dephasing [11]. Finally, the decay in space of the correlation function of the local SH fields is completely dominated by depolarization and occurs at the minimum (nano-) scale of the system. In this paper, we concentrate not on the spatial correlations of the vector SH fields but on their intensity fluctuations. We show below that these fluctuations are giant by nature, i.e., they have scaling distribution functions. However, different from the dipole-approximation case of [2], the corresponding index b is not universal but changes in a wide range depending on the material, frequency, and the density of the nanocomposite medium. In contrast, the fluctuations of the fundamental harmonic (FH) local field are much weaker, the scaling takes place in a much narrow region if at all pronounced. The reason for these profound differences is that the SH field is generated locally, in the near zone. Therefore, it is not a subject of the delocalization theorem of [12] and can be Anderson localized, which contributes to the giant fluctuations.
2. Theory We will follow mainly the spectral theory of [12] that we briefly summarize here for the sake of completeness and extend to the distribution functions, as needed. Because the phenomena to be discussed originate at the nanoscale, we consider a nanosystem whose entire extent is much smaller than the light wavelength and use the quasistatic approximation. The rough surface is described by the local dielectric function e(r, x) = e(x)H (r) + eh[1 H(r)] depending on coordinate r and excitation frequency x, where e(x) is the dielectric function of the metal, eh is the permittivity of the dielectric host, and H(r) is the characteristic function equal to 1 for r inside the metal component and 0 otherwise. The expression for the linear electric potential at the fundamental frequency x is Z uðrÞ ¼ u0 ðrÞ u0 ðr0 Þr02 Gr ðr; r0 ; xÞd3 r0 ; ð1Þ where u0(r) is the external excitation potential and Gr(r, r 0 ; x) is the retarded GreenÕs function [12,15], which is given by the following spectral expansion over the eigenmodes of the system: X Gr ðr; r0 ; xÞ ¼ /n ðrÞ/n ðr0 Þsn =½sðxÞ sn ; ð2Þ n
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where s(x) = eh/[eh e(x)] is the spectral parameter [16], and /n(r) and sn are the eigenfunctions and eigenvalues of the surface plasmon (SP) eigenproblem [12]. The GreenÕs function approach is highly stable numerically due to the fact that the spectral expansion (2) satisfies the exact analytical properties (analyticity, firstorder poles, sum rules, and causality) irrespectively of the accuracy with which the eigenproblem is solved. Consider the form of the SH nonlinear polarization ð2Þ PNL . In an isotropic medium, it contains three terms originating in the bulk of the metal [17], which have quadrupolar and magnetic-dipolar origin:
We define the dimensionless local intensities I(r) and ISH(r) of the FH and SH, respectively, as the corresponding enhancement factors 2
IðrÞ ¼ jPðrÞ=E0 j ;
ð9Þ
where E0 is the excitation field amplitude and PðrÞ ¼ EðrÞ½eðr; xÞ 1=ð4pÞ
ð10Þ
is the FH polarization. Finally, we introduce the corresponding distribution functions F(I) and FSH(ISH) for FH and SH local intensities as F ðIÞ ¼ hd½I IðrÞi;
ð2Þ
PNL ¼ aHEðr EÞ þ bHðE rÞE þ cHE ½r E;
2
I SH ðrÞ ¼ jPð2Þ ðrÞ=ðBE20 Þj ;
F SH ðI SH Þ ¼ hd½I SH I SH ðrÞi;
ð3Þ
ð11Þ
where H = H(r), E = $u, and a, b, and c are scalar functions of x (coefficients of the SH hyperpolarizability). Likewise, there are three surface contributions:
where d[ ] denotes the Dirac delta-function. The angular brackets denote both the ensemble and spatial averaging. This spatial averaging is extended over a narrow layer embedding the nanostructure. Note that we use the total polarizations at the FH and SH frequencies, not the electric fields, to find the corresponding intensities. Though the these distribution functions and those computed with electric fields have very similar behavior, we have taken into account that the polarizations determine the dipole moments radiating into the far zone and better correspond to the experiments [3–5].
ð2Þ
PNL ¼ AEðE rHÞ þ BE2 rH þ CE ½rH E.
ð4Þ
The last term in Eq. (3) vanishes in the quasistatic approximation. Due to displacement flux conservation, the a term in Eq. (3) has exactly the same structure as the A term in Eq. (4). We assume a good metal for which in the optical region je(x)j 1. Then the fields inside the metal are small as well as their tangential components at the surfaces. Resultantly, the a, b, A, and B terms in Eqs. (3) and (4) are all equivalent, and the C-term can be neglected. Therefore, assuming a good metal, without sacrificing the generality of our theory, we can set, e.g., ð2Þ
PNL ðrÞ ¼ BE2 ðrÞrHðrÞ.
ð5Þ
Here, nonlinear polarizability B determines only the magnitude of the SHG but vanishes from the enhancement of the polarization, spatial coherence, or SHG intensity with respect to the flat metal surface; therefore its magnitude and spectral dependence are not important for the present paper. The SH (second order) field potential induced due to polarization (5) can be calculated as a contraction Z 4p r ðr; r0 ; 2xÞr0 Pð2Þ ðr0 Þd3 r0 . uð2Þ ðrÞ ¼ ð6Þ G NL eh r ðr; r0 Þ is a retarded GreenÕs function, different Here, G from Gr ðr; r0 Þ (2) introduced in [15], which is given by the following spectral expansion over the eigenmodes of the system: X r ðr; r0 ; xÞ ¼ G /n ðrÞ/n ðr0 Þ=½sðxÞ sn . ð7Þ n
The SH electric field, E(2)(r) = $u(2)(r), in turn induces a contribution to the SH polarization due to the linear polarizability at the SH frequency of 2x. Resultantly, the total SH polarization is computed as ð2Þ
Pð2Þ ðrÞ ¼ PNL ðrÞ þ Eð2Þ ðrÞ½eðr; 2xÞ 1=ð4pÞ.
ð8Þ
3. Numerical computations and results: giant fluctuations and spatial correlations of SH local intensity We have solved the eigenproblem numerically as described in [11,12,18]. For statistical averaging, we employed ensembles of 64 systems of each kind. Every system was a random planar composite (RPC) in the space of 32 · 8 · 32 grid steps; to check the accuracy, we carried out some computations with 32 · 16 · 32 and 32 · 32 · 32 grid steps. Each grid cell is independently of the others filled with metal with probability f (the fill factor) or left unfilled. The metal layer is in the xz plane, its thickness in the normal (y) direction is two grid steps. We set the excitation field to be unity (E0 = 1) and z-polarized. The materials considered are silver and gold whose dielectric responses are taken the same as in bulk [19]. An example of RPC geometry for fill factor f = 0.5 is illustrated in Fig. 1(a) where H(r) is displayed in the plane of the RPC. To regularize the underlying partial differential equations, as in [12], we smooth H(r) by applying a Gaussian filter with the width of one grid step; this greatly improves the convergence with decreasing the grid step. This procedure smoothes out sharp edges and vertices and, actually, makes the RPC model more realistic, cf. Fig. 1(a). The local FH intensity I(r) in the plane of such an RPC made of gold at hx = 2.0 eV calculated from
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subject of the theorem [12] that states that eigenmodes (local fields) induced by the excitation from far zone should be delocalized (all Anderson-localized eigenmodes are dark). This localization of the SHG is certainly a contributing factor to its giant fluctuations in space that are considered below. Note that the SH hot spots are present at the positions of the FH hot spots but not at all of them. At those hot spots, the SH intensity does not correlate significantly with FH intensity. Such a behavior has been explained in [4] as being due to the fact that significant SHG occurs where there are strong hot spots of the eigenmodes at 2x frequency but not at all of them. The presence of the FH eigenmodes is also necessary at these locations to supply the excitation. Thus, the observed SHG hot spots appear due to the overlap of the plasmonic eigenmodes at the FH and SH frequencies. The distributions of the FH intensity for silver are shown in Fig. 2 for the excitation frequency hx = 2.0 eV and in Fig. 3 for hx = 1.5 eV. Similar results we have obtained for gold (data not shown). As we can see from Figs. 2 and 3, for low fill factors (f 6 0.5) the distribu-
Fig. 1. (a) Geometry of random planar composite (cross-section through the plane of symmetry xz): characteristic function H(r) displayed as density plot. The axis unit is the grid step; one unit may correspond to a length between 2 and 5 nm. Fill factor p = 0.5. (b) Spatial distribution of the magnitude of the local FH intensity I(r) in the plane of the RPC shown as a contour map; max I(r) = 35. (c) Same as (b) but for SH intensity (polarization enhancement) ISH(r); max ISH(r) = 250.
Eqs. (1) and (9) is shown in Fig. 1(b). We see that this intensity distribution is highly non-uniform, concentrated in some ‘‘hot spots’’ as we expect for the result of superposition of the chaotic eigenmodes excited by the excitation radiation [12,20–22]. At the same time, as we clearly see, there is a considerable delocalization of the local fields: the hot spots in Fig. 1(b) are spread out and overlap considerably, which is in line with the general theorem [12] on the delocalization of quasielectrostatic eigenmodes. In contrast, the SH intensity ISH(r) displayed in Fig. 1(c) shows well-localized, isolated hot spots. This localization is possible due to the fact that the SH local fields are not directly excited by the external fields, but instead are generated in the near zone by the SH nonlinear polarization of Eq. (5). Therefore, the SH is not a
Fig. 2. Probability distribution functions F(I) of the local FH intensity for RPCs of silver. The excitation frequency hx = 2.0 eV. The fill factors f and the found values of the scaling index a of this distribution function are shown at the top of the corresponding graphs. The scaling fits to the data for I 1 are shown by the dashed lines. Note the double logarithmic scale.
Fig. 3. Same as in Fig. 2 but for hx = 1.5 eV.
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tion function is scaling, F(I) / Ia, for I 1. As f increases, the index increases but the scaling area of the distribution contracts, and this index ceases to be meaningful. Note that the theoretical prediction of [2] that a = 3/2 is obtained in the dipole approximation for low dielectric losses and is therefore valid only for f small enough. As the results of Figs. 2 and 3 show, this prediction is, indeed, in qualitative agreement with the present computations for f = 0.1. In Figs. 4 and 5, we display the distribution functions FSH of the SH intensity ISH for hx = 2.0 and 1.5 eV, respectively, for silver as a material. There is clearly a scaling behavior of the distribution functions for ISH 1 and for low fill factors (f 6 0.5), FSH(ISH) / Ib. This behavior extends over four orders of magnitude in intensity manifesting the giant fluctuations of the SHG. However, in contrast to the incoherent excitation processes studied in [2], here the SH index b does not follow the simple relation b = a 1. In fact, jbj is significantly
Fig. 4. Probability distribution functions FSH(ISH) of the local SH intensity for RPCs of silver. The excitation frequency hx = 2.0 eV. The fill factors f and the found values of the scaling index b of these distribution functions are shown at the top of the corresponding graphs. The scaling fits to the data for ISH 1 are shown by the dashed lines. Note the double logarithmic scale.
Fig. 5. Same as in Fig. 4 but for hx = 1.5 eV.
less than ja 1j; also, the typical magnitude of the SH is less than one would expect from the square of the fundamental by two orders of magnitude. This means that the SHG does not copy the square of the local fields but instead is additionally suppressed and differently distributed. Physically, this suppression and non-following are due to the fact that SHG in an isotropic materials requires lower spatial symmetry. To excite SH, there should be a spatial coincidence of the FH and SH eigenmodes, as was suggested in [4] and already mentioned above in conjunction with Fig. 1. We note that the scaling index b of FSH(ISH) is not universal: as the fill factor f increases, absolute value of b also increases. As we see by the comparison of Figs. 4 and 5, this index also depends on frequency. Note that the universality of scaling indices in our initial computations [2] is due to the use of the dipole–dipole interaction approximation, which is only applicable for f 1. To explore the material dependence of the SHG scaling, we display the data for FSH(ISH) for gold as a material for hx = 2.0 eV in Fig. 6 and for hx = 1.5 eV in
Fig. 6. Probability distribution functions FSH(ISH) of the local SH intensity for RPCs of gold. The excitation frequency hx = 2.0 eV. The fill factors f and the found values of the scaling index b of these distribution functions are shown at the top of the corresponding graphs. The scaling fits to the data for ISH 1 are shown by the dashed lines. Note the double logarithmic scale.
Fig. 7. Same as in Fig. 6 but for hx = 1.5 eV.
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Fig. 7. The general conclusion that we draw from the comparison of these figures with the corresponding Figs. 4 and 5 for silver is that there is a pronounced scaling in both these cases, which is characteristic of the giant fluctuations of the SH local intensity. Thus, the existence of the giant fluctuations of the SHG (at not very high fill factors f) is a universal phenomenon. However, the scaling index b is material-dependent and not universal. It actually depends on all of the factors present in the problem: excitation frequency, fill factor, and material composition of the system. Finally, we discuss the spatial correlations of the FH vs. those of SH local fields. The corresponding correlation functions are introduced as 2
C FH ðrÞ ¼ hIðr1 ÞIðr2 Þd½ðr1 r2 Þ r2 i;
ð12Þ
C SH ðrÞ ¼ hI SH ðr1 ÞI SH ðr2 Þd½ðr1 r2 Þ2 r2 i;
where the averaging denoted by h i is over the statistical ensemble of the RPCs and also over the coordinates r1 and r2 within the layer of the metal RPC. The computed correlation functions for silver as a material are shown in Figs. 8 and 9. Similar results (data 1. 0.5
f=0.1 CSH CFH
1. 0.5
f=0.5 CSH CFH
r
r
2 4 6 8 10 12
2 4 6 8 10 12
Silver ω = 2.0 eV 1. 0.5
f=0.75 CSH CFH
1. 0.5
f=0.85 CSH CFH
r
r
2 4 6 8 10 12
2 4 6 8 10 12
Fig. 8. Spatial correlation functions CSH(r) of the local SH intensity (squares and solid line) and CFH(r) of the local FH intensity (small rhombuses and dashed line) for RPCs of silver. These functions are normalized to 1 for r ! 0. The excitation frequency hx = 2.0 eV. The fill factors f are shown at the top of the corresponding graphs. The distances are shown in the grid steps.
1. 0.5
1.
f=0.1 CSH CFH
0.5
f=0.5 CSH CFH
r
1.
f=0.75 CSH CFH
We have already discussed the results obtained as they have been presented above in the preceding Sections. Therefore here we only summarize the most important results of the paper. We have used GreenÕs function method based on the spectral expansion over quasistatic eigenmodes of a nanosystem to find the fields of the fundamental and second harmonics for random planar composites of silver and gold. The FH local intensity obeys a scaling distribution law (which is indicative of the giant fluctuations of the local fields) only at low enough fill factors where the scaling index is close to the theoretical value of 3/2. The SH local intensity has a pronounced scaling behavior in wide range of f, which is indicative of the developed giant fluctuations of the SH. This behavior of the SH is related to the fact that the SH near-zone field can be strongly localized, in contrast to FH. While the scaling behavior of the SH local intensity is universal, the corresponding scaling index is not: it depends on frequency, material, and fill factor of the nanosystem. The giant fluctuations of the SH are in agreement with the scaling behavior of the corresponding distribution functions observed experimentally [4,5].
0.5
Acknowledgments f=0.85 CSH CFH
r
r 2 4 6 8 10 12
4. Conclusion
2 4 6 8 10 12
Silver ω = 1.5 eV
0.5
not shown) have been obtained for gold. As we see from the graphs, the SH is much more localized than the FH: the corresponding correlation functions CSH(r) have a sharp decay within r = 1–2 grid steps. The FH intensity is much less localized; there is an appreciable delocalization of CFH(r) for f P 0.5 (note that f = 0.5 is the static percolation point of the RPC). These data are in agreement with fluctuation–distribution data discussed above: when the FH harmonic fields are localized, one can expect the dipole approximation to be qualitatively applicable, which would bring about the universal index of a = 3/2. As we have already discussed above, the delocalization of FH is due to the general delocalization theorem of [12] that states that any eigenmode of a nanosystem excited from the far zone is either delocalized or dark. This theorem does not apply to the SH that is generated by the corresponding polarization directly in the near zone.
r
2 4 6 8 10 12
1.
161
2 4 6 8 10 12
Fig. 9. Same as in Fig. 8 but for hx = 1.5 eV.
This work was supported by grants from the Chemical Sciences, Biosciences and Geosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, and a grant from the USIsrael Binational Science Foundation. I appreciate useful discussions with S. Bozhevolnyi and J. Zyss.
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References [1] D.S. Chemla, Phys. Today 46 (1993) 22. [2] M.I. Stockman, L.N. Pandey, L.S. Muratov, T.F. George, Phys. Rev. Lett. 72 (1994) 2486. [3] C. Anceau, S. Brasselet, J. Zyss, P. Gadenne, Opt. Lett. 28 (2003) 713. [4] S.I. Bozhevolnyi, J. Beermann, V. Coello, Phys. Rev. Lett. 90 (2003) 197403-1-4. [5] J. Beermann, S.I. Bozhevolnyi, Phys. Rev. B 69 (2004) 155429–9. [6] M. Moskovits, Rev. Mod. Phys. 57 (1985) 783. [7] K. Kneipp, Y. Wang, H. Kneipp, L.T. Perelman, I. Itzkan, R.R. Dasari, M.S. Feld, Phys. Rev. Lett. 78 (1997) 1667. [8] S. Nie, S.R. Emory, Science 275 (1997) 1102. [9] Z.J. Wang, S.L. Pan, T.D. Krauss, H. Du, L.J. Rothberg, Proc. Natl. Acad. Sci. USA 100 (2003) 8638. [10] S.G. Rautian, V.P. Safonov, P.A. Chubakov, V.M. Shalaev, M.I. Stockman, PisÕma ZhETF 47 (1988) 200 [Translation: JETP Lett. 47 (1988) 243]. [11] M.I. Stockman, D.J. Bergman, C. Anceau, S. Brasselet, J. Zyss, Phys. Rev. Lett. 92 (2004) 057402-1-4.
[12] M.I. Stockman, S.V. Faleev, D.J. Bergman, Phys. Rev. Lett. 87 (2001) 167401-1-4. [13] A.K. Sarychev, V.A. Shubin, V.M. Shalaev, Phys. Rev. B 60 (1999) 16389. [14] V.M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal–Dielectric Films, Springer, Berlin, 2000. [15] M.I. Stockman, S.V. Faleev, D.J. Bergman, Phys. Rev. Lett. 88 (2002) 067402-1-4. [16] D.J. Bergman, D. Stroud, Properties of macroscopically inhomogeneous media, in: H. Ehrenreich, D. Turnbull (Eds.), Solid State Physics, vol. 46, Academic Press, Boston, 1992, pp. 148–270. [17] C.K. Chen, T.F. Heinz, D. Ricard, Y.R. Shen, Phys. Rev. B 27 (1983) 1965. [18] M.I. Stockman, D.J. Bergman, T. Kobayashi, Phys. Rev. B 69 (2004) 054202. [19] P.B. Johnson, R.W. Christy, Phys. Rev. B 6 (1972) 4370. [20] M.I. Stockman, L.N. Pandey, T.F. George, Phys. Rev. B 53 (1996) 2183. [21] M.I. Stockman, Phys. Rev. Lett. 79 (1997) 4562. [22] M.I. Stockman, Phys. Rev. E 56 (1997) 6494.
Giant fluctuations of second harmonic generation on nanostructured surfaces Mark I. Stockman Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303, USA Received 9 March 2005; accepted 8 June 2005 Available online 18 July 2005
Abstract On the basis of spectral-expansion GreenÕs function theory, we theoretically describe the fundamental harmonic (FH) and second harmonic (SH) local fields at nanorough surfaces of silver and gold. The numerical simulation of these fields in the model of random planar composites has shown that the SH local-field intensity possesses power-law (scaling) distribution for a wide range of system and excitation parameters, which is indicative of giant fluctuations of the SH local fields. Though these scaling phenomena are universal, the scaling index of the SH intensity distribution is not universal depending on the material, excitation frequency, and the fill factor of the nanosystem. The results obtained are in qualitative agreement with the existing experiments. 2005 Elsevier B.V. All rights reserved. PACS: 78.67.n; 68.37.Uv; 05.40.+j; 73.20.Mf Keywords: Local fields; Nanosystem; Fundamental harmonic; Second harmonic; Localization; Giant fluctuations; Distribution function; Scaling index
1. Introduction From its initial development, nanooptics has been one of the most promising and rapidly developing fields at the junction of optics and materials physics [1]. This paper is devoted to one fascinating property of the local optical fields in complex nanostructured metal systems, namely, their giant fluctuations in the real space of their intensity [2]. This paper represents a theory inspired by the recent experiments on statistical properties of the second harmonic generation (SHG) at nanorough metal surfaces [3–5]. The giant fluctuations of local optical fields in nanostructured systems have been predicted for fractal clusters of metal nanospheres [2]. They manifest themselves by a scaling (power-law) behavior of the distribution E-mail address: [email protected]. URLs: http://www.phy-astr.gsu.edu/stockman. 0301-0104/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.06.020
function of the local optical field intensities. Using dipole–dipole interaction model to describe the interactions between the nanospheres, the power (index) of this dependence a has been shown to be universal (not depending on the frequency or fractal dimension) and close to the analytically found value of a ¼ 32. The giant fluctuations and enhancement of local fields in such resonant random systems lead to the corresponding giant enhancement of two- and many-photon processes. In the decade following the publication of [2], there has recently been a great interest in optical properties of metallic nanostructures, in particular, nanorough metal surfaces and clusters. The enhanced optical responses have been especially well studied for surface-enhanced Raman scattering (SERS) [6]. The record enhancement of SERS, J 1012, have been found for Raman observations of single molecules [7,8]. This single-molecule SERS has been definitely linked to the clusterization of metal nanospheres and formation of fractal aggregates [9].
M.I. Stockman / Chemical Physics 318 (2005) 156–162
The SHG in systems built of isotropic metals is principally different from SERS in two respects. First, SHG is a coherent process implying the interference of waves emitted by different sites; therefore, dephasing of the second harmonic (SH) polarization is of importance. This dephasing is the randomization of the SH-polarization phase in the plane of the nanostructure due to the spectral detuning of localized eigenmodes from the excitation frequency. Second, the SHG is a second-order (three-wave) process where for a center-symmetric medium, the SH polarization is concentrated at surfaces and interfaces, while the bulk contribution is suppressed. This is in contrast to odd-order processes such as the phase conjugation [10] or third-harmonic generation where the coherent generation of radiation and giant enhancement were observed. The recent experimental study of SHG by rough gold surfaces with the spatial resolution of 400 nm [3] revealed intriguing properties of SHG: (i) SHG is incoherent, i.e., hyper-Rayleigh scattering, implying strong dephasing. (ii) SHG is almost completely depolarized despite the small detection area. (iii) The SHG has topography of localized bright ‘‘hot spots’’. This topography and depolarization were independently supported, and the enhancement of SHG in the hot spots up to three orders of magnitude was found [4], where rough gold films were studied by confocal microscopy. The most importantly, this study of the distribution function of the SH intensity has revealed its scaling behavior where the index depended on the frequency, but was reasonably close to the value of b = 2.9. Note that if the SH intensity distribution would simply copy the distribution of the square of the fundamental field intensity, the relation between the corresponding indices would be b = a 1, which would give b 2.5, so it is not quite satisfied but not that far off. These findings have been confirmed in the later publication [5]. Theoretical study [11] of the depolarization, dephasing, and correlation of the SHG on rough surfaces has shown that the SH depolarization originates from the peculiarity of the SH being generated at the surfaces. Because noble metals are good conductors even at optical frequencies, the vector of the SH field is directed almost at the normal to the surface. Since the surface on the nanoscale is highly faceted and random, the SH is highly depolarized. The spatial correlation function of the SH field decays in space at the minimum scale. Thus, the depolarization of the SHG is a phenomenon that is originated at the nanoscale and transferred to the larger scales of the system. The origin of the dephasing is actually the delocalization of the linear local fields. As shown in [12], the delocalization of the optical fields on the nanoscale is a fundamental property: any eigenmode that is Anderson-localized is necessary dark and cannot be excited or observed from the far zone. This theorem shows that some published statements on the Anderson
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(strong) localization of local optical fields [13] and its being an underlying cause of the giant fluctuations [14] are completely incorrect. The actual delocalization of the nanoscale optical fields causes their concentration areas (‘‘hot spots’’) to overlap for different modes. Because these modes have randomly different frequencies, this leads to the random phase shifts and results in the dephasing [11]. Finally, the decay in space of the correlation function of the local SH fields is completely dominated by depolarization and occurs at the minimum (nano-) scale of the system. In this paper, we concentrate not on the spatial correlations of the vector SH fields but on their intensity fluctuations. We show below that these fluctuations are giant by nature, i.e., they have scaling distribution functions. However, different from the dipole-approximation case of [2], the corresponding index b is not universal but changes in a wide range depending on the material, frequency, and the density of the nanocomposite medium. In contrast, the fluctuations of the fundamental harmonic (FH) local field are much weaker, the scaling takes place in a much narrow region if at all pronounced. The reason for these profound differences is that the SH field is generated locally, in the near zone. Therefore, it is not a subject of the delocalization theorem of [12] and can be Anderson localized, which contributes to the giant fluctuations.
2. Theory We will follow mainly the spectral theory of [12] that we briefly summarize here for the sake of completeness and extend to the distribution functions, as needed. Because the phenomena to be discussed originate at the nanoscale, we consider a nanosystem whose entire extent is much smaller than the light wavelength and use the quasistatic approximation. The rough surface is described by the local dielectric function e(r, x) = e(x)H (r) + eh[1 H(r)] depending on coordinate r and excitation frequency x, where e(x) is the dielectric function of the metal, eh is the permittivity of the dielectric host, and H(r) is the characteristic function equal to 1 for r inside the metal component and 0 otherwise. The expression for the linear electric potential at the fundamental frequency x is Z uðrÞ ¼ u0 ðrÞ u0 ðr0 Þr02 Gr ðr; r0 ; xÞd3 r0 ; ð1Þ where u0(r) is the external excitation potential and Gr(r, r 0 ; x) is the retarded GreenÕs function [12,15], which is given by the following spectral expansion over the eigenmodes of the system: X Gr ðr; r0 ; xÞ ¼ /n ðrÞ/n ðr0 Þsn =½sðxÞ sn ; ð2Þ n
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where s(x) = eh/[eh e(x)] is the spectral parameter [16], and /n(r) and sn are the eigenfunctions and eigenvalues of the surface plasmon (SP) eigenproblem [12]. The GreenÕs function approach is highly stable numerically due to the fact that the spectral expansion (2) satisfies the exact analytical properties (analyticity, firstorder poles, sum rules, and causality) irrespectively of the accuracy with which the eigenproblem is solved. Consider the form of the SH nonlinear polarization ð2Þ PNL . In an isotropic medium, it contains three terms originating in the bulk of the metal [17], which have quadrupolar and magnetic-dipolar origin:
We define the dimensionless local intensities I(r) and ISH(r) of the FH and SH, respectively, as the corresponding enhancement factors 2
IðrÞ ¼ jPðrÞ=E0 j ;
ð9Þ
where E0 is the excitation field amplitude and PðrÞ ¼ EðrÞ½eðr; xÞ 1=ð4pÞ
ð10Þ
is the FH polarization. Finally, we introduce the corresponding distribution functions F(I) and FSH(ISH) for FH and SH local intensities as F ðIÞ ¼ hd½I IðrÞi;
ð2Þ
PNL ¼ aHEðr EÞ þ bHðE rÞE þ cHE ½r E;
2
I SH ðrÞ ¼ jPð2Þ ðrÞ=ðBE20 Þj ;
F SH ðI SH Þ ¼ hd½I SH I SH ðrÞi;
ð3Þ
ð11Þ
where H = H(r), E = $u, and a, b, and c are scalar functions of x (coefficients of the SH hyperpolarizability). Likewise, there are three surface contributions:
where d[ ] denotes the Dirac delta-function. The angular brackets denote both the ensemble and spatial averaging. This spatial averaging is extended over a narrow layer embedding the nanostructure. Note that we use the total polarizations at the FH and SH frequencies, not the electric fields, to find the corresponding intensities. Though the these distribution functions and those computed with electric fields have very similar behavior, we have taken into account that the polarizations determine the dipole moments radiating into the far zone and better correspond to the experiments [3–5].
ð2Þ
PNL ¼ AEðE rHÞ þ BE2 rH þ CE ½rH E.
ð4Þ
The last term in Eq. (3) vanishes in the quasistatic approximation. Due to displacement flux conservation, the a term in Eq. (3) has exactly the same structure as the A term in Eq. (4). We assume a good metal for which in the optical region je(x)j 1. Then the fields inside the metal are small as well as their tangential components at the surfaces. Resultantly, the a, b, A, and B terms in Eqs. (3) and (4) are all equivalent, and the C-term can be neglected. Therefore, assuming a good metal, without sacrificing the generality of our theory, we can set, e.g., ð2Þ
PNL ðrÞ ¼ BE2 ðrÞrHðrÞ.
ð5Þ
Here, nonlinear polarizability B determines only the magnitude of the SHG but vanishes from the enhancement of the polarization, spatial coherence, or SHG intensity with respect to the flat metal surface; therefore its magnitude and spectral dependence are not important for the present paper. The SH (second order) field potential induced due to polarization (5) can be calculated as a contraction Z 4p r ðr; r0 ; 2xÞr0 Pð2Þ ðr0 Þd3 r0 . uð2Þ ðrÞ ¼ ð6Þ G NL eh r ðr; r0 Þ is a retarded GreenÕs function, different Here, G from Gr ðr; r0 Þ (2) introduced in [15], which is given by the following spectral expansion over the eigenmodes of the system: X r ðr; r0 ; xÞ ¼ G /n ðrÞ/n ðr0 Þ=½sðxÞ sn . ð7Þ n
The SH electric field, E(2)(r) = $u(2)(r), in turn induces a contribution to the SH polarization due to the linear polarizability at the SH frequency of 2x. Resultantly, the total SH polarization is computed as ð2Þ
Pð2Þ ðrÞ ¼ PNL ðrÞ þ Eð2Þ ðrÞ½eðr; 2xÞ 1=ð4pÞ.
ð8Þ
3. Numerical computations and results: giant fluctuations and spatial correlations of SH local intensity We have solved the eigenproblem numerically as described in [11,12,18]. For statistical averaging, we employed ensembles of 64 systems of each kind. Every system was a random planar composite (RPC) in the space of 32 · 8 · 32 grid steps; to check the accuracy, we carried out some computations with 32 · 16 · 32 and 32 · 32 · 32 grid steps. Each grid cell is independently of the others filled with metal with probability f (the fill factor) or left unfilled. The metal layer is in the xz plane, its thickness in the normal (y) direction is two grid steps. We set the excitation field to be unity (E0 = 1) and z-polarized. The materials considered are silver and gold whose dielectric responses are taken the same as in bulk [19]. An example of RPC geometry for fill factor f = 0.5 is illustrated in Fig. 1(a) where H(r) is displayed in the plane of the RPC. To regularize the underlying partial differential equations, as in [12], we smooth H(r) by applying a Gaussian filter with the width of one grid step; this greatly improves the convergence with decreasing the grid step. This procedure smoothes out sharp edges and vertices and, actually, makes the RPC model more realistic, cf. Fig. 1(a). The local FH intensity I(r) in the plane of such an RPC made of gold at hx = 2.0 eV calculated from
M.I. Stockman / Chemical Physics 318 (2005) 156–162
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subject of the theorem [12] that states that eigenmodes (local fields) induced by the excitation from far zone should be delocalized (all Anderson-localized eigenmodes are dark). This localization of the SHG is certainly a contributing factor to its giant fluctuations in space that are considered below. Note that the SH hot spots are present at the positions of the FH hot spots but not at all of them. At those hot spots, the SH intensity does not correlate significantly with FH intensity. Such a behavior has been explained in [4] as being due to the fact that significant SHG occurs where there are strong hot spots of the eigenmodes at 2x frequency but not at all of them. The presence of the FH eigenmodes is also necessary at these locations to supply the excitation. Thus, the observed SHG hot spots appear due to the overlap of the plasmonic eigenmodes at the FH and SH frequencies. The distributions of the FH intensity for silver are shown in Fig. 2 for the excitation frequency hx = 2.0 eV and in Fig. 3 for hx = 1.5 eV. Similar results we have obtained for gold (data not shown). As we can see from Figs. 2 and 3, for low fill factors (f 6 0.5) the distribu-
Fig. 1. (a) Geometry of random planar composite (cross-section through the plane of symmetry xz): characteristic function H(r) displayed as density plot. The axis unit is the grid step; one unit may correspond to a length between 2 and 5 nm. Fill factor p = 0.5. (b) Spatial distribution of the magnitude of the local FH intensity I(r) in the plane of the RPC shown as a contour map; max I(r) = 35. (c) Same as (b) but for SH intensity (polarization enhancement) ISH(r); max ISH(r) = 250.
Eqs. (1) and (9) is shown in Fig. 1(b). We see that this intensity distribution is highly non-uniform, concentrated in some ‘‘hot spots’’ as we expect for the result of superposition of the chaotic eigenmodes excited by the excitation radiation [12,20–22]. At the same time, as we clearly see, there is a considerable delocalization of the local fields: the hot spots in Fig. 1(b) are spread out and overlap considerably, which is in line with the general theorem [12] on the delocalization of quasielectrostatic eigenmodes. In contrast, the SH intensity ISH(r) displayed in Fig. 1(c) shows well-localized, isolated hot spots. This localization is possible due to the fact that the SH local fields are not directly excited by the external fields, but instead are generated in the near zone by the SH nonlinear polarization of Eq. (5). Therefore, the SH is not a
Fig. 2. Probability distribution functions F(I) of the local FH intensity for RPCs of silver. The excitation frequency hx = 2.0 eV. The fill factors f and the found values of the scaling index a of this distribution function are shown at the top of the corresponding graphs. The scaling fits to the data for I 1 are shown by the dashed lines. Note the double logarithmic scale.
Fig. 3. Same as in Fig. 2 but for hx = 1.5 eV.
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tion function is scaling, F(I) / Ia, for I 1. As f increases, the index increases but the scaling area of the distribution contracts, and this index ceases to be meaningful. Note that the theoretical prediction of [2] that a = 3/2 is obtained in the dipole approximation for low dielectric losses and is therefore valid only for f small enough. As the results of Figs. 2 and 3 show, this prediction is, indeed, in qualitative agreement with the present computations for f = 0.1. In Figs. 4 and 5, we display the distribution functions FSH of the SH intensity ISH for hx = 2.0 and 1.5 eV, respectively, for silver as a material. There is clearly a scaling behavior of the distribution functions for ISH 1 and for low fill factors (f 6 0.5), FSH(ISH) / Ib. This behavior extends over four orders of magnitude in intensity manifesting the giant fluctuations of the SHG. However, in contrast to the incoherent excitation processes studied in [2], here the SH index b does not follow the simple relation b = a 1. In fact, jbj is significantly
Fig. 4. Probability distribution functions FSH(ISH) of the local SH intensity for RPCs of silver. The excitation frequency hx = 2.0 eV. The fill factors f and the found values of the scaling index b of these distribution functions are shown at the top of the corresponding graphs. The scaling fits to the data for ISH 1 are shown by the dashed lines. Note the double logarithmic scale.
Fig. 5. Same as in Fig. 4 but for hx = 1.5 eV.
less than ja 1j; also, the typical magnitude of the SH is less than one would expect from the square of the fundamental by two orders of magnitude. This means that the SHG does not copy the square of the local fields but instead is additionally suppressed and differently distributed. Physically, this suppression and non-following are due to the fact that SHG in an isotropic materials requires lower spatial symmetry. To excite SH, there should be a spatial coincidence of the FH and SH eigenmodes, as was suggested in [4] and already mentioned above in conjunction with Fig. 1. We note that the scaling index b of FSH(ISH) is not universal: as the fill factor f increases, absolute value of b also increases. As we see by the comparison of Figs. 4 and 5, this index also depends on frequency. Note that the universality of scaling indices in our initial computations [2] is due to the use of the dipole–dipole interaction approximation, which is only applicable for f 1. To explore the material dependence of the SHG scaling, we display the data for FSH(ISH) for gold as a material for hx = 2.0 eV in Fig. 6 and for hx = 1.5 eV in
Fig. 6. Probability distribution functions FSH(ISH) of the local SH intensity for RPCs of gold. The excitation frequency hx = 2.0 eV. The fill factors f and the found values of the scaling index b of these distribution functions are shown at the top of the corresponding graphs. The scaling fits to the data for ISH 1 are shown by the dashed lines. Note the double logarithmic scale.
Fig. 7. Same as in Fig. 6 but for hx = 1.5 eV.
M.I. Stockman / Chemical Physics 318 (2005) 156–162
Fig. 7. The general conclusion that we draw from the comparison of these figures with the corresponding Figs. 4 and 5 for silver is that there is a pronounced scaling in both these cases, which is characteristic of the giant fluctuations of the SH local intensity. Thus, the existence of the giant fluctuations of the SHG (at not very high fill factors f) is a universal phenomenon. However, the scaling index b is material-dependent and not universal. It actually depends on all of the factors present in the problem: excitation frequency, fill factor, and material composition of the system. Finally, we discuss the spatial correlations of the FH vs. those of SH local fields. The corresponding correlation functions are introduced as 2
C FH ðrÞ ¼ hIðr1 ÞIðr2 Þd½ðr1 r2 Þ r2 i;
ð12Þ
C SH ðrÞ ¼ hI SH ðr1 ÞI SH ðr2 Þd½ðr1 r2 Þ2 r2 i;
where the averaging denoted by h i is over the statistical ensemble of the RPCs and also over the coordinates r1 and r2 within the layer of the metal RPC. The computed correlation functions for silver as a material are shown in Figs. 8 and 9. Similar results (data 1. 0.5
f=0.1 CSH CFH
1. 0.5
f=0.5 CSH CFH
r
r
2 4 6 8 10 12
2 4 6 8 10 12
Silver ω = 2.0 eV 1. 0.5
f=0.75 CSH CFH
1. 0.5
f=0.85 CSH CFH
r
r
2 4 6 8 10 12
2 4 6 8 10 12
Fig. 8. Spatial correlation functions CSH(r) of the local SH intensity (squares and solid line) and CFH(r) of the local FH intensity (small rhombuses and dashed line) for RPCs of silver. These functions are normalized to 1 for r ! 0. The excitation frequency hx = 2.0 eV. The fill factors f are shown at the top of the corresponding graphs. The distances are shown in the grid steps.
1. 0.5
1.
f=0.1 CSH CFH
0.5
f=0.5 CSH CFH
r
1.
f=0.75 CSH CFH
We have already discussed the results obtained as they have been presented above in the preceding Sections. Therefore here we only summarize the most important results of the paper. We have used GreenÕs function method based on the spectral expansion over quasistatic eigenmodes of a nanosystem to find the fields of the fundamental and second harmonics for random planar composites of silver and gold. The FH local intensity obeys a scaling distribution law (which is indicative of the giant fluctuations of the local fields) only at low enough fill factors where the scaling index is close to the theoretical value of 3/2. The SH local intensity has a pronounced scaling behavior in wide range of f, which is indicative of the developed giant fluctuations of the SH. This behavior of the SH is related to the fact that the SH near-zone field can be strongly localized, in contrast to FH. While the scaling behavior of the SH local intensity is universal, the corresponding scaling index is not: it depends on frequency, material, and fill factor of the nanosystem. The giant fluctuations of the SH are in agreement with the scaling behavior of the corresponding distribution functions observed experimentally [4,5].
0.5
Acknowledgments f=0.85 CSH CFH
r
r 2 4 6 8 10 12
4. Conclusion
2 4 6 8 10 12
Silver ω = 1.5 eV
0.5
not shown) have been obtained for gold. As we see from the graphs, the SH is much more localized than the FH: the corresponding correlation functions CSH(r) have a sharp decay within r = 1–2 grid steps. The FH intensity is much less localized; there is an appreciable delocalization of CFH(r) for f P 0.5 (note that f = 0.5 is the static percolation point of the RPC). These data are in agreement with fluctuation–distribution data discussed above: when the FH harmonic fields are localized, one can expect the dipole approximation to be qualitatively applicable, which would bring about the universal index of a = 3/2. As we have already discussed above, the delocalization of FH is due to the general delocalization theorem of [12] that states that any eigenmode of a nanosystem excited from the far zone is either delocalized or dark. This theorem does not apply to the SH that is generated by the corresponding polarization directly in the near zone.
r
2 4 6 8 10 12
1.
161
2 4 6 8 10 12
Fig. 9. Same as in Fig. 8 but for hx = 1.5 eV.
This work was supported by grants from the Chemical Sciences, Biosciences and Geosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, and a grant from the USIsrael Binational Science Foundation. I appreciate useful discussions with S. Bozhevolnyi and J. Zyss.
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