Optical whirlpool near absorbing metallic nanoparticle

The power-flow lines of light interacting with a metallic nanoparticle, in the proximity of its plasmon resonance, form whirlpool-like nanoscale optical vortices. Two different types of vortex have been detected. The outward vortex first penetrates the particle near its centerline then, on exiting the particle, the flow-lines turn away from the centerline and enter a spiral trajectory. Outward vortexes are seen for the wavelengths shorter then the plasmon resonance. For the wavelengths longer that the plasmon resonance the vortex is inward: the power-flow lines pass around the sides of the particle before turning towards the centerline and entering the particle to begin their spiral trajectory.

The structures of optical fi elds around metallic nanoparticles are of special interest due to their role in nanophotonic and plasmonic devices and metawaveguides [1][2][3]. Here for the fi st time, we report that light interacting with an absorbing metallic nanoparticle follows curly trajectories with curvatures on the subwavelength scale, creating whirlpool-like nanoscale optical vortices. These "energy sink" vortices with spiral energy flow line trajectories are seen in the proximity of the nanoparticle's plasmon resonance.
O ptical vortices have been identifi ed as features in scalar wavefront dislocations of monochromatic light fi elds and modal lines corresponding to nonmonochromatic light as well as in singularities in the maps representing vectorial properties of light [4]. It is now recogniz ed that singularities are often features of fi elds near sub-wavelength structures. A vortex structure in the streamlines of the Poynting vector has been detected for S ommerfeld's edge diff raction with discussion of the eel-like motion of light at the edge dating back to Newtonian times [5]. Recently vortices were found in light diff racted by narrow slits in silver and silicon [6,7]. However, to the best of our knowledge, vortex fi eld structures have never been detected in the vicinity of metal nanoparticles.
We studied the interaction of light with homogeneous isotropic spherical nanoparticles using Mie theory [8]an exact analytical wave theory giving time-harmonic electromagnetic fi elds E and H at freq uency ω that satisfy the wave eq uations where k 2 = ω 2 εµ. S olutions to these eq uations are presented in the form of a series of spherical B essel Functions inside the particle and spherical Hankel functions outside it. The nanoparticle is assumed to have a dielectric coeffi cient ε and permittivity µ. Mie theory gives exact solutions of the vector wave eq uation for the internal and scattered fi elds of the particle and has generated a massive body of literature in which fi eld patterns for angle-dependant scattering, modes of excitation, and in-tegral characteristics such as absorption and scattering cross-section have been calculated [9,10]. It has been shown that light can bend near a nanoparticle [11], however it has never been determined that the interaction of light with a nanoparticle can create a nanoscale vortex fi eld structure. Here we refer to vortices in the "trajectory" of light near the nanoparticle as defi ned by the lines of powerflow, i.e. lines to which the Poynting vector P = [E × H] is tangential. In the vortex regime of propagation the lines of powerflow are wound around the nanoparticle to create a nanoscale "whirlpool", comparable in siz e to the particle itself, whereby light seems to pass through the particle several times over.
We found that the vortex regime occurs in metallic (e.g. silver) nanoparticles in the vicinity of the plasmon absorbtion resonance. We analyz ed the fi eld structure around a nanoparticle excited by a plane electromagnetic wave. To illustrate the vortex structures graphically, we plotted solutions in the plane of polariz ation of the incident light using powerflow lines and a color scale for the absolute value of the Poynting vector (reg = high, blue = low). In the fi eld maps presented below the incident light is polariz ed in the plane of the page and propagates from left to right.
To relate the parameter fi eld for our calculations to observable values we shall defi ne the dimensionless scattering σ s and absorbtion σ a cross-sections of the nanoparticle. In the Rayleigh approximation, cross-sections for a particle much smaller than the wavelength are introduced via the particle's polariz ability α and its geometrical cross-section S: σ s = (k 4 /6π)|α| 2 /S and σ a = kIm(α)/S − σ s , where k = 2π/λ is the wave vector, and polariz ability is a function of the particle's shape and siz e [12].
We found that the existence of the vortex structure and the topology of the fi eld maps depend on the values of the real and imaginary parts of the particle's complex dielectric coeffi cient ε = ε + iε (see Fig. 2). Here and below we assume non-magnetic nanoparticles with µ = 1. Figures 2(a) and (b) show the modifi cation of the fi eld structure around a hypothetical nanoparticle for diff erent values of ε . In the case depicted in Fig. 2(a) the scattering and absorption cross-sections are much smaller than the geometrical cross-section and the particle is almost invisible to the external fi eld (σ a = 0.47, σ s = 0.03). M ost of the powerflow lines pass by the nanoparticle and only handful of them terminate on the particle, indicating small losses. In the case depicted in Fig. 2 Figure 1 shows the parameter fi eld where vortex structures can be observed. Two diff erent types of vortex have been seen. In the fi rst type, which we call an outward vortex, a bunch of powerflow lines fi rst penetrate the particle near its centerline then, on exiting the particle, they separate, turn away from the centerline and enter a spiral trajectory. O utward vortices are seen to the "left" of the the plasmon resonance i.e. for ε > −2.2 (in a spherical nanoparticle with a radius of 20 nm the plasmon resonance occurs at ε ∼ −2.2). In the second type of vortex, which we call an inward vortex, the powerflow lines pass around the sides of the particle before turning towards the centerline and entering the particle to begin their spiral trajectory. O utward vortexes are seen to the "right" of the the plasmon resonance i.e. for ε < −2.2.
We also found by numerical simulation that vortex fi elds can exist near non-spherical nano-objects. Nonspherical nanoparticles are of considerable interest for applications because flattened or elongated shapes tend to reduce the plasmon resonance freq uency, moving it from the blue-UV part of the spectrum to the more accessible visible-IR range. M ie theory is unsuitable for objects without spherical symmetry but computational methods provide an alternative to the analytical approaches and allow consideration of vortex fi elds around complex nanostructures. To analyz e the vortex fi elds near spheroidal nanoparticles we used 64-bit software, developed by C omsol Inc., which implements a true 3D fi nite element method [14] and applies Perfectly M atched Layer (PM L) [15] boundary conditions on all sides of the computational domain. We investigated a homogeneous oblate spheroidal nanoparticle with an aspect ratio of 2. Figure 3 shows the modifi cation of the fi eld structure around a spheroidal nanoparticle for diff erent values of ε . Here again, one can see the weak interaction regime in Fig. 3(a) (σ a = 0.42, σ s = 0.02), the high-loss regime in Fig. 3(b) (σ a = 3.7, σ s = 0.3), the creation of outward vortexes in Fig. 3(c) (σ a = 8.7, σ s = 2.9), and the creation of inward vortexes in Fig. 3 There are a number of intriguing q uestions that may be asked in relation to the nanoscale structuring of the energy flow near and inside the nanoparticle. For instance, a vortex structure with light passing through a nanoparticle several times backwards and forwards, resembles a standing wave in a dissipative Fabry-Perot resonator. O ne may therefore wonder if such a "nano-resonator" could provide conditions for a hysteresis and bistability in the nanoparticle's optical response if its dielectric properties depend on the intensity of light. The existence of vortex structures in nanoparticles could provide a graphical interpretation of the fact that the absorbtion crosssection of a particle can be much bigger that its geometrical cross-section. When a vortex is created, powerflow lines pass through the nanoparticle several times, "multiplying" the light-matter interaction and generating the high energy losses associated with the large optical crosssection.
The authors would like to thank M .V.B erry for important comments and useful references and K .F. M acDonald for discussions and assistance with manuscript preparation and also to acknowledge the support of the E ngineering and Physical S ciences Research C ouncil (UK ).