When fractal aggregates of silver colloid nanoparticles are illuminated at visible frequencies, their dipolar surface-plasmon modes localize on sub-wavelength scales, producing frequency- and polarization-selective 'hot spots'. The 1994 photon scanning tunneling microscopy (PSTM) experiment of Tsai et al. (Phys. Rev. Lett. 72, 4149) imaged these localized modes and thereby verified the main concepts of the resonant optical theory of fractal objects, but left a linked set of quantitative questions open: whether the light field itself can be localized (trapped) by fractal aggregates, whether the near-zone (quasistatic, Hermitian) reduction of the dipole-dipole interaction remains valid for modes with interparticle phase factors krij≳1k r_{ij}\gtrsim1, how the quality factors QQ and absolute local-field enhancement factors of the localized modes can be measured directly, and how the three-dimensional cluster structure and finite tip-sample coupling affect the quantitative interpretation of PSTM images. These quantitative questions remain open for random three-dimensional fractal metal colloid clusters at optical frequencies.

Background

Small metal nanoparticles (diameters of roughly 10-100 nm, e.g. colloidal silver) support a localized surface plasmon: a collective oscillation of the conduction electrons that resonates at an optical frequency controlled by particle size, shape, and dielectric environment. When such particles aggregate slowly in solution they form random, scale-invariant 'fractal' cluster-cluster aggregates with no characteristic length scale. In the resonant optical theory of such fractals, each particle is treated as an induced dipole and the whole aggregate is described by coupled-dipole equations: each dipole responds to the external wave plus the fields of all the other dipoles. In the near-zone (quasistatic) limit krij<1k r_{ij}<1, where kk is the light wave number and rijr_{ij} the interparticle separation, the interaction is dominated by the 1/rij31/r_{ij}^{3} dipole term, phase factors are set to unity, and the (loss-free part of the) interaction operator becomes Hermitian, i.e. a real symmetric matrix whose eigenvectors are the dipolar eigenmodes. Because of the scale invariance and strong disorder of fractal aggregates, these eigenmodes localize on anomalously short length scales, with spatial positions that change strongly with excitation frequency and polarization; this is the theory developed by Shalaev, Botet, Markel, Stockman, and co-workers (see e.g. Shalaev et al., Physica A 1994; Shalaev, Phys. Rep. 1996; Shalaev, Handbook of Nanostructured Materials, 2000). The resulting 'hot spots' of large local field underlie surface-enhanced Raman scattering (SERS) and enhanced nonlinear optical responses of fractal colloid films. Photon scanning tunneling microscopy (PSTM) is a near-field optical technique in which the sample is excited by the evanescent field of total internal reflection and a sharpened optical-fiber tip, scanned at nanometric distance, converts the near field into a propagating detected signal, giving sub-wavelength spatial resolution; the detected intensity is commonly modeled as proportional to the squared local field amplitude. In the 1994 experiment of Tsai, Kovacs, Wang, Moskovits, Shalaev, Suh, and Botet (Phys. Rev. Lett. 72, 4149), PSTM images of laser-excited silver colloid fractal clusters showed highly localized optical modes with the frequency and polarization selectivity predicted by numerical simulations, verifying the main concepts of the resonant optical theory. Intrinsic mode properties enter that comparison only indirectly: the localization length was inferred from the measured PSTM image radii using a nominal mode quality factor QQ (ratio of resonance energy to linewidth, here taken as Q≃102Q\simeq10^{2} for silver), the three-dimensional clusters were collapsed to a surface-proximal projection for imaging, and the image model used simplifying assumptions about the tip response. A published Comment by Stockman, Pandey, Muratov, and George (Phys. Rev. Lett. 75, 2450 (1995)) with a Reply by Shalaev and Moskovits (Phys. Rev. Lett. 75, 2451 (1995)) shows that the theoretical basis of the Letter was contested from the start. What remains unresolved is a quantitative, experimentally anchored account of the localized optical excitations of fractal metal colloid clusters: the localization of the light field itself, the intrinsic resonant linewidths (quality factors) and absolute field enhancements of the localized modes, the regime of validity of the near-zone Hermitian reduction, and a controlled mapping between near-field images and intrinsic mode properties.

Problem Statement

For random fractal aggregates of metal (e.g. silver) colloid nanoparticles at optical frequencies - the system of Tsai et al., Phys. Rev. Lett. 72, 4149 (1994) - establish a quantitative, experimentally anchored account of the localized optical excitations and of the theory used to describe them. The common objective is to convert the qualitative 1994 verification of the resonant optical theory of fractals into measured intrinsic mode properties and a validated interpretation of near-field images. Concretely, determine: (1) whether the light field itself is localized (trapped) by fractal aggregates, as distinct from the localization of induced dipolar eigenmodes that PSTM already images, and if so, how this trapping is observed and quantified; (2) the validity range and physical consequences of the near-zone Hermitian reduction of the dipole-dipole interaction operator (retaining only the quasistatic term, setting phase factors to unity) that underlies the theory, in particular for modes or aggregates with krij≳1k r_{ij}\gtrsim1 - weakly localized modes, larger interparticle separations, or larger aggregates - including what changes when retardation, radiative terms, and damping make the interaction non-Hermitian; (3) the direct experimental determination of the quality factors QQ of the observed localized modes, and quantitative measurements of the absolute local-field enhancement factors, together with their relation to enhanced Raman and nonlinear scattering from such aggregates; and (4) a controlled quantitative mapping between photon scanning tunneling microscopy images and intrinsic mode properties, assessing how the three-dimensional structure of the clusters (which imaging collapses to a surface-proximal projection), finite tip-sample coupling, and departures from the assumption that the detected signal is proportional to the squared local field amplitude affect the inferred localization lengths and other mode parameters. Each of these four questions is posed independently and can be answered, refuted, or left open on its own: resolving one settles that component while the others remain open, and the complete account sought addresses all four.

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