Dipole and quadrupole surface plasmon resonances contributions in formation of near-field images of a gold nanosphere

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1 Dipole and quadrupole surface plasmon resonances contributions in formation of near-field images of a gold nanosphere M. Shopa *, K. Kolwas **, A. Derkachova, G. Derkachov Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, Warszawa, Poland Abstract Multipolar plasmon optical excitations at spherical gold nanoparticles and their manifestations in the particle images formatted in the particle surface proximity are studied. The multipolar plasmon size characteristic: plasmon resonance frequencies and plasmon damping rates were obtained within rigorous size dependent modelling. The realistic, frequency dependent dielectric function of a metal was used. The distribution of light intensity and of electric field radial component at the flat square scanning plane scattered by a gold sphere of radius 95nm were acquired. The images resulted from the spatial distribution of the full mean Poynting vector including near-field radial components of the scattered electromagnetic field. Monochromatic images at frequencies close to and equal to the plasmon dipole and quadrupole resonance frequencies are discussed. The changes in images and radial components of the scattered electromagnetic field distribution at the scanning plane moved away from the particle surface from near-field to far-field region are discussed. Keywords: near-field imaging, multipolar surface plasmon resonances, Mie theory, optical properties of noble metal nanospheres, gold nanoparticles, plasmonic nanostructures, NSOM, PSTM, plasmonics. 1 Introduction The ability of metal/dielectric structures to sustain coherent electron oscillations known as surface plasmon polaritons (SPPs) has been recently intensively investigated [1], [2], [3], [4], [5], [6]. The most rapidly growing areas of physics and nanotechnology focus on plasmonic effects. Thanks to recent fabrication advances, nanoplasmonic applications have grown in number and diversity. In particular, the interaction of light with metal nanoparticles has been an issue of intense research (e.g. [7], [8], [9]). It has been shown that nanoscale plasmonic structures greatly enhance local electromagnetic fields in certain regions near their surfaces at specific wavelengths of light, controlled by the nanostructure geometry. The geometry- and size-dependent properties of nanoparticles [10], [11], [12], [13], [14], [15] have potential applications in nanophotonics, biophotonics and biomedicine [16], [17], [18], [19], near-field scanning optical microscopy [20], sensing [21], [22] and spectroscopic measurements. Much of the impetus for excitement connected with plasmon-related optical phenomena is driven by the discovery and development of surface enhanced Raman scattering (SERS) technique [23], [24], [25]. For example, gold nanospheres are used as enhancing surfaces and can provide enhancement up to intensity' magnitude [10], [11], [12], [13], [26]. These large enhancements * ** Mykola Shopa Mykola.Shopa@ifpan.edu.pl Krystyna Kolwas Krystyna.Kolwas@ifpan.edu.pl

2 are attributed to highly concentrated electromagnetic fields associated with strong localized surface plasmon (SP) resonances. Near-field distribution of the scattered light intensity in the proximity of a plasmonic nanostructure can give us valuable basic information about optical properties of that structure, valuable for diverse applications. The near-field's complicated pattern contains both: the components able to propagate and components confined to the surface and damped outside. Our study refers to the near-field imaging techniques such as near-field scanning optical microscopy (NSOM) [14], [27], [28] and photon scanning tunnelling microscopy (PSTM) [15] allowing to image the structure of a sample with nanometer resolution, well below the diffraction limit. However, interpretation of images still poses some problems in many applications. One of the reasons is that the electric field at the scanning tip can be highly non-homogeneous. An even more fundamental problem is the question what is the near-field homogeneous image itself; that is how the near-field distribution characterizing the object looks like and how it reflects the optical plasmonic properties of that structure. Our aim is to illustrate the diversity of the near-field images of a plasmonic spherical particle illuminated by a homogeneous light field at frequencies close to and equal to the frequencies of dipole and quadrupole plasmons and to understand, what the localization of the surface plasmon means in the case of spherical interface. We chose Au sphere of radius of 95nm as an example of a good scattering plasmonic nanostructure that is characterized by significant plasmonic radiative rate. Such particle offers both dipole and quadrupole plasmon resonance frequencies in the visible range. Let us notice, that in many experiments, particles used are very small in comparison with the light wavelength. Particles of the radii of up to about 10nm of well defined spherical shape and the controlled size can be fabricated more easily than the larger ones. However, the scattering abilities of much smaller than the light wavelength particle are negligible in comparison with the absorptive ones. Therefore, such small particles are not optimal for applications in which the enhancement of the plasmon near-field surrounding the particle is needed (for example SERS technique or expected transport of confined plasmon excitations from particle to particle at nanometer length scales in particle arrays). The near-field images of Au sphere of radius of 95nm at resonance dipole and quadrupole frequencies resulted from the spatial distribution of the light intensity that is proportional to the component of the full mean Poynting vector, that is normal to the detection scanning plane placed in the particle proximity. While for such large particles (compared to the light wavelength) the quasistatic approximation does not hold, we used the results of the rigorous size dependence modelling of the inherent particle plasmon size characteristics [29], [30], [31]. Using the Mie theory formalism we show, that the maxima in the total scattering or absorption spectra for such large particles (which are the manifestations of SP resonances), are blue shifted in respect to the plasmon resonance positions. The enhancement of the scattered or absorbed light intensity (arising from the interference of plasmonic electric and magnetic fields with other nonresonance contributions) is the manifestation of surface plasmons only, the notion related to TM electromagnetic field waves localized at the metal-dielectric interface (or to the surface densities free-electron waves coupled to these fields) [30], [31]. That is why a manifestation of plasmon resonance characterised by a large radiative plasmon damping rate (spectrally large maxima in light intensity) can be shifted. The change in the images with distance between nanosphere and detection surface is also discussed. While the near-field radial components of the electromagnetic field are expected to play considerable role in the image formation (as they are coupled with the SP waves at the particle surface) we completed our consideration by the spatial distribution of that component over the scanning plane.

3 2 Formalism and simulation background 2.1 Plasmon size characteristics Figure 1: Multipolar (l = 1,2,...) plasmon resonance frequencies for gold nanosphere as a function of particle radius. The dipole and quadrupole plasmon resonances for 95nm gold nanosphere can be excited with photon energy and respectively. Possibility of resonant excitation of the surface plasmon oscillations (collective oscillations of surface free-electron densities) and damping of these oscillations we treat as the inherent property of a conducting sphere, that is abstracted from the quantity, which can be observed [29], [30]. Excitation of SP oscillations is a resonant phenomenon that is possible when the frequency of the incoming light field approaches the characteristic eigenfrequencies, of a plasmonic sphere of a radius with for dipole and for the quadrupole plasmon mode. The dynamically induced surface charge density distributions can be of diverse complexity. With the increasing sphere surface, multipolar distributions of higher than dipole distribution can be induced [31]. If excited, plasmon oscillations are damped at the corresponding rates. SP resonance frequencies as a function of size for gold nanospheres used here are derived in [30]. The complete plasmon size characteristics result from self-consistent rigorous electromagnetic approach described in [29], [30] in more detail. The example of size dependence of multipolar plasmon oscillation frequencies completed by the corresponding damping rates are shown in Figure 1 for gold sphere.

4 Figure 2: Scattering, extinction and absorption cross-sections spectra for gold nanosphere of = 95nm As a scattering object we chose Au sphere of radius = 95nm. For such large particle the quasistatic approximation certainly does not work. According to the data in Figure 1, a particle of that rad '' size is characterized by significant plasmon radiative rates ( R) ( R) / 2, l = 1,2,3..., with equal 0,663eV and 0,066eV correspondingly (see Figure 1) in both dipole l=1 and quadrupole l=2 plasmon modes. A particle with such large radiative rates is an efficient radiative antenna able to scatter the light effectively in proportion to the radiative rate contribution in ' the total damping rate [31]. Both the dipole l= 1( R = 95nm) 1, 96eV and quadrupole ' l= 2 ( R = 95nm) 2, 57eV plasmon resonance frequencies fall in the visible range ( = 632,57nm and 482,43nm correspondingly) and are spectrally easy to resolve. Let us notice, that without knowing plasmon size characteristics, the resonance dipole and quadrupole plasmon frequency for the sphere of chosen radius would be unknown, while for such large particles (compared to the light wavelength) the quasistatic approximation does not hold and the rigorous size dependence must be used. Plasmonic features, that we could derive from plasmon size characteristics presented in Figure 1, are reflected in the spectra of the total absorption and scattering cross-sections calculated for a sphere of = 95nm with help of Mie theory [32], [33], [34] and presented in Figure 2. However, Mie theory does not deal with the notion of collective surface density oscillations (neither the notion of plasmon resonances). The enhancement of the light intensity (arising from the interference of plasmonic electric and magnetic fields with other nonresonant contributions) is the manifestation of the SP oscillations. Let us note, that the notion of surface plasmons is related to the electromagnetic fields localized at the metal-dielectric interface. These fields are coupled to the free-electron surface wave densities [30], [31]. Let us also notice, that Mie theory is not a handy tool in studying the size dependence of plasmon resonances that manifest in some maxima of the light spectra. In addition, maxima in the total scattering cross-sections can be "blue shifted" in respect to the multipolar plasmon position derived from eigenvalue problem and presented in Figure 1. The frequency shift is bigger if the l l

5 spectral width of the maximum is larger. It is due to the dispersion of the metal and complex interference of scattered fields of many multipolar orders of TM and TE modes [30] contributing to the maximum. The broadening of the maxima is approximately proportional to. Therefore, the assumption of the position of the maximum in the scattered light intensity as the surface plasmon resonance position can be inaccurate. 2.2 Input material parameters In order to take metal's plasmonic properties into account we consider the analytic form of the Drude dielectric function with the effective parameters: that describes the optical properties of many metals quite well within relatively wide frequency range. is the phenomenological parameter describing the contribution of the bound electrons to the polarizability. is the bulk plasmon frequency, is the phenomenological electron damping constant of the bulk material [35], [36]. The effective parameters of the function,, were chosen to satisfactorily reproduce the experimental values of Re [ n 2 ( )] ( Im [ n 2 ( )] ) [37] in the frequency ranges:. The optical properties of the dielectric medium outside the sphere were assumed to be Orthogonal polarization geometries Figure 3: Scheme of the orthogonal polarization geometries showing polarization directions of the incident and scattered light in respect to the observation plane Contribution of the dipole and quadrupole plasmon resonances to the scattered light intensity and its distribution (so the particle image), can be conveniently demonstrated by studying the two orthogonal polarization geometries and sketched in Figure 3a. The chosen orthogonal polarization geometries allow us to demonstrate the role of the dipole and the quadrupole plasmon mode contribution in distinction from plasmon mode contributions of higher multipolar order [37].

6 While and are the intensities of the purely scattered light (no influence of the interference with the EM field of the incoming light wave propagating in perpendicular direction), observation of and makes study of the pure scattering abilities of a plasmonic sphere possible. Figure 4: Spectra of the scattered light intensities and for gold nanosphere of radius 95nm, with dipole surface plasmon resonance at and quadrupole resonance at in the orthogonal polarization geometries (Figure 3) Such observation geometries enable the exposure of a single dipole plasmon contribution and a single quadrupole plasmon contribution and to separate these contributions spatially by observing and scattered intensities. Alternatively, when keeping the directions of illumination and observation unchanged, the rotation of linear polarizers by the angle 90 (Figure 3b) leads to the same observation opportunity ( and ), what is more convenient from the experimental point of view [37]. The effect of spatial separation of a single dipole and quadrupole plasmon contributions to the spectra of scattered light intensities and in the centre of the scanning plane calculated on the basis of Mie theory is demonstrated in Figure Numerical near-field images Plasmons excited at spherical gold nanoparticles contribute to the particle near-field distribution and to the particle images formatted in the particle proximity. Plasmons cause enhancement of the near-field intensity, but they introduce also important modification of the spatial field and intensity distribution. Contribution to the image of the electromagnetic fields coupled to plasmon oscillations are not straightforward, while the spatial distribution of the light field intensity is a result of constructive and destructive interference of all the fields scattered by the particle. The distribution of electric and magnetic fields of light scattered by a sphere can be found from Mie theory [35], [36].

7 A measure of the scattered light intensity is the magnitude of the Poynting vector, averaged over the time interval which is long compared with the light wave period : (1) The magnitude of the Poynting vector represents the averaged density of the scattered energy flow, the direction of corresponds the direction of energy propagation. The image plane (Figure 5) serves as a scattered light intensity detector that registers photons flux normal to the plane. The image in the XY plane is formed due to the distribution of the normal component of the Poynting vector over this plane: where is the unit vector normal to the XY plane. If we use the angles to parameterize the position (x, y) at the plane XY [38] according to the notation introduced in Figure 5, the normal component of the averaged Poynting vector can be represented as: (2) (3) Figure 5: Scheme of construction of the normal component vector over the XY scanning plane at distance of the averaged Poynting In near-field region the radial components of electric and magnetic fields and are comparable to the components,,, correspondingly. They play an significant role in image

8 formation, while: The amplitudes of the corresponding field components decay with distance and. In far field region (i.e. for such that ) the radial components of electric and magnetic fields and are negligible in comparison with the transverse components,,,. Therefore, while moving away from near-field to far field region the scattered wave gains transverse character. 3 Results and discussion We acquired a series of images for gold sphere of radius = 95nm in both parallel and perpendicular polarization geometries. The series illustrate changes in images and distribution with distance of the scattering plane from the sphere, as well as changes due to the applied frequency of the illuminating light wave. The angular dimensions of the scanning plane remain the same as the scanned angles. Linear dimensions of XY scanning plane are approximately 1000x1000nm at the close sphere proximity. The actual size of the scanning plane is given in each image. Polarization of incident light is assumed to be parallel or perpendicular to the scanning plane (Figure 3). While the brightness of images differs by orders of magnitude, each image possess its own scale, that allows associate colours and intensity which is proportional to the normal component of the full Poynting vector (Equation 4) to the scanning plane, as discussed above. The components of the scattered field electromagnetic field contributing to the Poynting vector are calculated on the basis of Mie theory. as: (4) Figure 6: Numerical images (upper row) and distribution of the radial component of the scattered electric field (lower row) at scanning plane placed at the close proximity of the particle surface:. Frequency of incident light is changing from below to above the dipole plasmon resonance frequency at (see Figure 1). a) polarization perpendicular to the scanning plane, b) polarization parallel to the scanning plane Figure 6 illustrates images at frequency equal to the dipole plasmon frequency and at frequencies below and above that frequency. The images in observation geometry (Figure 6a) are much larger than the object itself. They are strongly affected by

9 the dipole SP resonance contribution, as one could expect for perpendicular observation geometry (see also Figure 4). The important contribution of the near-field radial components of electromagnetic field is the reason for this effect, as illustrated in Figure 6a, lower row. The widespread distribution of the radial component of the electric field above the sphere (lower row in Figure 6a can be interpreted as a projection of the excited SP wave at the sphere surface. The spectral width of this effect is defined by the corresponding radiative rate included in the dipole plasmon rate (Figure 1) and is much larger than the corresponding rate for the quadrupole plasmon. These rates define half-widths at half maximum (HWHM) of the spectrum at any point of the plane including the plane center (see Figure 4). However, influence of distribution on the image formation is not straightforward. The corresponding intensity results from interference of all the components of the near-field (see Equation (4)). The images in orthogonal observation geometry (Figure 6b) are not affected by the surface-localized fields coupled to the dipole SP, according to the expectation (a dipole does not radiate in the axis direction). Therefore, in such observation geometry, the images better reflect the circular shape of the object and its size. Figure 7: Numerical images (upper row) and distribution of the radial component of the scattered electric field (lower row) at scanning plane placed at the close proximity of the particle surface:. Frequency of incident light is changing from below to above the quadrupole plasmon resonance frequency at (see Figure 1). a) polarization perpendicular to the scanning plane, b) polarization parallel to the scanning plane Figure 7b illustrates the images obtained in polarization geometry with intension to demonstrate the role of quadrupole SP contribution in image formation. The images in polarization geometry at the quadrupole plasmon frequency are much brighter than for neighbouring frequencies used for image simulations; as expected, the spectral width of the enhancement is much smaller than for the dipole case, and is given by the corresponding radiative rate. At the quadrupole SP resonance the radial electric field amplitude is strongly enhanced as well (Figure 7b, lower row). Figure 8 illustrates the change in images and in distributions of the amplitude of radial electric field component at the scanning plane moved away from the particle surface from near-field to farfield region. Figure 8a reflects the changes in spatial distributions in geometry at the quadrupole plasmon resonance frequency, while Figure 8b corresponds to

10 polarization geometry at the dipole plasmon resonance frequency. Square decay of the radial components of the near-field with distance causes not only the change in the image brightness, but also the image shape. The lower row in Figures 8a and 8b illustrate the changes in the distribution of of the scattered field with distanced correspondingly. Such distribution reflects the pattern of the plasma waves excited at the particle surface at the dipole and quadrupole frequencies. The normal to the interface component of the electric displacement field must be continuous at the interface. Resulting jump in must be caused by the induced free electron surface charge density. The asymmetry of the pattern at the scattering plane (Figure 8) is caused by the presence of the illuminating light field (coming from the left side of the picture) and its weak (due to the chosen observation geometry) but nonzero contribution to the patterns. Figure 8: Numerical images (upper row) and distribution of the radial component of the scattered electric field (lower row) at the scanning plane moving away from near-field region to the far field. Distance between the particle surface and the scanning plane is 0nm, 50nm, 100nm, 200nm, 400nm and 800nm, as seen from left to right. Frequency of incident light is equal to a) the quadrupole plasmon resonance frequency for polarization parallel to the scanning plane, b) the dipole plasmon resonance frequency for polarization perpendicular to the scanning plane 4 Conclusions Near-field images of a spherical plasmonic particle illuminated by linearly polarized light are deviated from circular shape. For given particle size, the shape of the image changes with the direction of illuminating field polarization in respect to the direction of observation and depends on

11 the frequency of the illuminating field. This last tendency is not only a result of a simple dependence of a metal index of refraction on frequency, but is mainly due to the plasmonic abilities of a particle and is size dependent. SP features are described by the SP resonance frequencies and damping rates with size dependence due to the radiative damping. These parameters define plasmonic features of a metallic nanosphere embedded in a dielectric environment of known index of refraction. As our analysis show, excitation of SP resonances possess substantial impact on the particle near-field image; not only its brightness, but also its shape. The larger the damping rate is, the broader is the spectral range around the SP resonance frequency for these effects. Near-field scattered by a sphere is strongly nonhomogeneous. That causes dramatic changes in the sphere images with distance to the detector. The amplitudes of the radial component of the electric and magnetic field, which decrease in proportion to the square of the distance from the scatterer, play a significant role in building the nearfield image of a plasmonic particle, and cannot be omitted in analysis of the near-field distribution in the vicinity of nanostructures of different shapes. In the near-field region the radial components modify the length and direction of the Poynting vector. Our preliminary study is aimed to demonstrate the diversity and complexity of near-field images of a plasmonic spherical particle illuminated by a light at frequency close to or equal to the frequency of dipole and quadrupole plasmon resonance. Our results, though being only a numerical model, can provide a starting point on what scattered field we can expect from a noble metal nanosphere. In spite only the example of gold sphere of radius = 95nm is studied, the conclusions are general enough to be used for any other plasmonic particle. Sphere is considerably a simple object, but has a unique advantage of precise solutions of interesting electrodynamic problems. Features of localized surface plasmon waves on a sphere are still not fully understood and revealed, By studying plasmonic sphere features we can broaden our knowledge of undergoing electromagnetic processes connected with SP and SP manifestation in measurable quantities before moving on to more complex objects, widely studied nowadays, for example [19, 20, 21]. Nevertheless spherical nanoparticles are widely used in all kinds of applications, exploiting plasmonic features of noble metal nanospheres. Acknowledgment. We would like to acknowledge support of this work by Polish Ministry of Education and Science under Grant No N N References 1. W.L. Barnes, A. Dereux, T.W. Ebbesen, Surface plasmon subwavelength optics, Nature 424, 824 (2003). 2. M. Quinten, A. Leitner, J.R. Krenn and F.R. Aussenegg, Electromagnetic energy transport via linear chains of silver nanoparticles, Opt. Lett. 23, 1331 (1998). 3. M.L. Brongersma, J.W. Hartman and H.A. Atwate, Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit, Phys. Rev. B 62, (2000). 4. S.A. Maier, P.G. Kik, H.A. Atwater, S. Meltzer, E. Harel, B.E. Koel and A.A.G. Requicha, Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides, Nat. Mater. 2, (2003)

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