Andreas Liapis, Luke Bissell and Xinye Lou. December 20, Abstract

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1 Scattering of Surface Plasmon Polaritons from Nano-Scale Features Andreas Liapis, Luke Bissell and Xinye Lou December 20, 2007 Abstract In this report, a theoretical treatment of the elastic scattering of surface plasmon polaritons from spherical nanoparticles is presented, and a simple formula for the scattered field is obtained. Scattering from a number of such features was observed experimentally, and was found to be in qualitative agreement with the theoretical expectations. The experimental setup used to observe these is also briefly described. Introduction The field of plasmonics has recently seen a great rise in popularity, as it may offer a way to overcome the current limit in the miniaturization of optical circuitry. Surface plasmon polaritons (SPP s) are inherently two-dimensional, and therefore ideal building blocks for all-optical integrated circuits [1]. Such circuits would be made out of protrusions or defects in the metal surface, designed in such a way so as to guide, reflect or even localize the SPP s. In order to design these elements efficiently, the process of surface plasmon scattering needs to be understood. In this report, the elastic scattering of surface plasmon polaritons from spherical nanoparticles deposited on a metal surface is considered, and a simple formula for the scattered field is obtained. A setup for observing such scattering is then described, followed by select examples of the experimental results obtained. Theory A number of theoretical models of varying complexity have been published on the subject of SPP scattering. In this report, perhaps the simplest and most intuitive model will be adopted, the scalar pointdipole approximation [2]. Under this approximation, nanoscale defects in the metal surface (scatterers) are treated as isotropically radiating point dipoles characterized by an effective polarizability, α eff, which depends on the geometry of the defect. This approximation is valid whenever the defects are small enough that the electric field does not vary appreciably across their volume, so that the higher orders in the multipole expansion vanish. For the case of a metal nanoparticle of radius R p and dielectric function ɛ p resting at a height z p above the metal surface (Figure 1), the effective polarizability is given by [2]: ε 0 ε m ε p z R p z p Figure 1: Schematic representation of the physical geometry. (After [2]) x 1

2 Figure 2: Calculated interference patters for scattering from one (a), two (b), and multiple (c) scatterers. Calculations were performed to 6 th Born order. α eff 8πk 3 0R 3 p(ɛ p 1) (ɛ p + 2 2(ɛ p 1)(R p /(2z p )) 3. (1) The field scattered by such a dipole will then be given by E scat (r) = k 0 2 α eff G(r, r p )E exc (r p ), (2) ɛ 0 where G(r, r p ) is the Green s function relating the point r to the location of the scattering particle, r p. Recall that the Green s function is a solution of the wave equation driven by unit impulse and, as such, it will include contributions towards both in plane and out of plane scattering. For the purposes of this report, however, it will be assumed that the scattering into propagating modes is much weaker than the scattering into plasmon modes. Moreover, since the distance scales considered here are significantly smaller than the plasmon propagation length, any damping associated with the SPP modes will be ignored. For a two-dimensional plane geometry with point symmetry about ρ p, the solutions to the wave equation are circularly symmetric, and may be represented by 0 th order Hankel functions as G(ρ, ρ p ) = i ( ) 4 H(1) 0 ksp P ρ ρ p. (3) The total field can therefore be expressed as a superposition of the initial plasmon wave, E 0 (ρ), and the sum of the contributions by each of the scatterers: E tot (ρ) = E 0 (ρ) + + i N α j eff 4 E exc(ρ p )H (1) ( 0 kspp ρ ρ j ). (4) j=1 When only one scatterer is present in the system, the excitation field may be identified with the incident plasmon. Multiple scatterers will, in general, affect each other, and so this assignment is only valid in the 1 st order Born approximation, E exc(ρ (1) j ) = E 0 (ρ j ). Higher orders of approximation should include contributions from all scatterers present, with the n th Born order given by E (n) (ρ j ) = E 0 (ρ j ) + + i N α l E exc (n 1) (ρ 4 l )H (1) ( 0 ksp P ρ j ρ l ). (5) l=1,l j Experimental Setup Surface plasmons were excited on a 58nm thick silver film using supercritical reflection in the standard Kretschmann configuration with a frequency doubled Nd:YAG laser emitting at 532nm (Figure 3). The angle of incidence was controlled by a tiltable mirror while the intensity of the reflected beam was being monitored; the fulfillment of the resonance condition 2

3 Figure 4: Amplitude (a) and phase (b) of a tuning fork driven at a frequency ω for two different probesample distances d. (c,d) show the amplitude and phase at frequency ω 0 as a function of the probesample distance [4]. Figure 3: Schematic diagram of the experimental setup. for the excitation of SPP s is accompanied by a significant reduction in the reflected intensity. The nearfield distribution was mapped out by photon scanning tunneling microscopy (PSTM), whereupon the SPP field is probed directly by raster scanning a sharpened glass fiber over the sample surface [3]. The tip of the fiber may be assigned an effective polarizablity according to which it will scatter the evanescent field into propagating modes. These are then collected by the fiber and relayed to a photomultiplyer tube (PMT), whose output is therefore directly proportional to the local intensity. Typically, the probe-sample distance is required to be smaller than the size of lateral field confinement [1, 4, 5]. Therefore, with a sub-wavelength sized probe, the probe is required to be scanned in such close proximity to the surface that it is necessary to prevent catastrophic probe-sample collisions. As in other types of scanning probe techniques, an active feedback loop was employed to maintain the probe-sample distance constant during the scanning process. A quartz tuning fork was used as a shear force sen- sor, whose vibrations in the direction parallel to the sample surface are influenced by the proximity of the sample (Figure 4). This non-optical sensing method was described in detail by Karrai et al [5]. Typically, the probe is oscillated at the resonance frequency of its mechanical support (tuning fork) and the amplitude, phase, and/or frequency of the oscillation are measured as a function of the probe-sample distance [4]. The feedback loop implemented here makes use of the height-dependent phase of the probe s oscillations, and is illustrated in figure 5: A signal generator generates a reference and a driving signal with the same frequency and phase. As the probe is brought closer to the sample surface, the shear force between the probe and the surface induces a phase change, which is measured and compared to an externally defined set point. The difference between the set point and the measured phase change gives the error signal, which is fed into a Proportional-Integral controller. A PZT scanner then adjusts the probe-sample distance accordingly. This setup provides simultaneous topographical and optical imaging and allows one to directly correlate the optical field distribution over the surface to its topography. 3

4 Figure 5: Functional block diagram of the feedback loop 1500 nm Experimental Results Gold nanoparticles were spin-coated on the Ag film. A Labview Vi was used to raster scan the probe tip across the surface and record both the film topography and the photocounts registered by the PMT. Figure 6, top, shows the topography of several nanoscale features, probably Au nanoclusters. Figure 6, bottom shows the interference fringes created between the incident SPP, and the SPPs scattered by these features. Parabolic fringes are evident, as one would expect from the interference of plane and cylindrical waves. The image is qualitatively similar with the results of the theoretical model presented in fiigure 2c. Figure 7 shows the reflection of a SPP from the edge of a scratch made in the Ag film with a screwdriver. In scratching the film, some of the material removed from the film is deposited along the sides of the scratch. Here, a section of the scratch is shown where the excess material forms a localized feature that lends itself to the scattering of SPP s. Interference fringes between the incident, reflected and scattered plasmons are seen. WSxM software [7] was used to analyze the fringe spacing in figure 7, which was found to be consistent with an incident plasmon wavelength of 273(29)nm. The theoretical value of the SPP wavelength is given by λ SPP (ɛ m + 1)/(ɛ m)λ = 509nm [4], where ɛ m = is the real part of the dielectric function of silver excited at 532 nm [6], and λ = 532nm is the excitation wavelength. The reason for the discrepancy between the theoretical and experimental value is uncertain, but is believed to be due to an Figure 6: Top: Au nanoclusters on an Ag film. Bottom: Interference between the incident and scattered fields. The SPP is incident from the bottom. incorrect calibration of the PZT scanner. Figure 7: Left: Topography of a localized feature in close proximity to a scratch made in the Ag film. Right: Interference pattern between the incident, reflected and scattered fields. The diagonal blue lines indicate the line cuts taken for the analysis of the fringe spacing. The SPP is incident from the top. Figure 8 shows another section of the same scratch, approximately 4µm away from the feature in figure 7. 4

5 Once again, straight fringes indicative of a reflected SPP are seen. The interference also shows a spherical wave, which may be due to scattering from another nanocluster below the area that was scanned. Since these fringes seem to extend beyond the edge of the metal film, it is believed that they may the result of propagating modes and not surface plasmons. Photon Counts *exp(-2*kz*z) data k zfit = nm -1 k zth = nm z (nm) 2 µm 2 µm Figure 8: Left: Topography of a scratch made in Ag film. Right: Interference pattern caused by the reflected and scattered SPP s. The spherical fringes observed may be an example of inelastic scattering. This last result puts the previous examples into question. In order, therefore, to investigate whether the observed fringe patterns were due to SPPs, a static lift experiment was performed over an area to the left of the edge in figure 7, whereupon the photocounts collected by the fiber were recorded as the tip was slowly lifted from the surface. The intensity of a field generated by a SPP should decay according to I 0 = E 0 2 exp( 2k z z), where k z, the imaginary part of the wavevector in the z direction, can be calculated according to k z = 1/(1 + ɛ m)ω/c. It was found that, for heights less than about 150 nm, the measured signal decreased slower than an exponential, while for heights greater than that the decay was exponential, with a decay constant that is within a factor of two from the theoretical prediction (Fig. 9). Two conclusions may be drawn from this result. Either the short probe-sample distance allows for frustrated total internal reflection so that a portion of the excitation field is detected along with the plasmon field, or there is local suppression of the SPP s due to the presence of the probe tip. A perhaps more Figure 9: Measured decay of SPP s and comparison with theory. Blue line: data. Red dashes: Best fit to experimental data using k z,fit = 0.002nm 1. Green line: Theoretical decay curve, k z,th = nm 1. fruitful examination of the nature of the observed interference patters would have been to compare the fields measured when the plasmon excitation beam is p-polarized with those obtained for s-polarization. Since the latter does not allow for the excitation of surface plasmons, the interference patterns are expected to vanish. Acknowledgments The authors would like to thank B.Deutsch, G. Piredda and L. Novotny for their constant advice and help, without which, this project would not have been possible. References [1] Zayats, A.V., Smolyaninov, I.I., and Maradudin, A.A., Physics Reports 408, 131 (2005). [2] Evlyukhin, A.B., Bozhevolnyi, S.I., Phys. Rev. B 71, (2005). [3] Reddick, R.C., Warmack, R.J. and Ferrell, T.L., Phys. Rev. B 39, 767 (1989). 5

6 [4] Novotny, L. and Hecht, B., Principles of Nano- Optics, Cambridge University Press, Cambridge, UK (2006). [5] Karrai, K. and Grober, R.D., Appl. Phys. Lett. 66, 1842 (1995). [6] Johnson, P.B. and Christy, R.W., Phys. Rev. B 6, 4370 (1972). [7] Horcas, I. et al., Rev. Sci. Instrum. 78, (2007). 6