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Excitation, focusing and directing of surface plasmon polaritons (SPPs) with curved chains of nanoparticles located on a metal surface is investigated both experimentally and theoretically. We demonstrate that, by using a relatively narrow laser beam (at normal incidence) interacting only with a portion of a curved chain of nanoparticles, one can excite an SPP beam whose divergence and propagation direction are dictated by the incident light spot size and its position along the chain. It is also found that the SPP focusing regime is strongly influenced by the chain inter-particle distance. Extensive numerical simulations of the configuration investigated experimentally are carried out for a wide set of system parameters by making use of the Green’s tensor formalism and dipole approximation. Comparison of numerical results with experimental data shows good agreement with respect to the observed features in SPP focusing and directing, providing the guidelines for a proper choice of the system parameters.
©2007 Optical Society of America
1. Introduction
Modern nanotechnology has opened new possibilities for light manipulation (e.g., localization and directional propagation) with nanostructures whose sizes are about the light wavelength or even smaller. In this respect, studies of interactions of surface plasmon polaritons (SPPs) with metal nanostructures is one of the most promising research directions. SPPs are surface electromagnetic waves propagating along metal-dielectric interfaces [1]. Owing to the exponential localization of SPPs near the surface and their high sensitivity to surface inhomogeneities, SPPs can be advantageously used for creating planar micro-optical components and devices [2]. One can envision miniaturized integrated optical circuits utilizing SPPs that would enable manipulation of radiation at the length scale much smaller than the the light wavelength [2]. Over recent years a number of theoretical and experimental studies of artificially fabricated micro-components for two-dimensional optics of SPPs have been reported [3]. Thus, it has been experimentally demonstrated that nanoparticle ensembles on metal surfaces can be used to create efficient micro-optical elements for SPPs [4, 5, 6, 7, 8, 9]. SPP mirrors, beam splitters, and interferometers were successfully realized on this basis [7, 8, 9]. Moreover, it has been shown that periodically arranged surface nanoparticles exhibit the properties of 2D photonic crystals with respect to SPPs [10, 11]. If such an SPP band gap structure has narrow channels free from the particles, then SPPs can be confined to and guided along these channels [10]. As an alternative to manipulating SPPs by nanostructures at flat surfaces, an experimental realization of low-loss and well-confined channel SPP propagation along subwavelength metal grooves at telecom wavelengths has recently been reported [12, 13]. The transport of electromagnetic energy via chains of nanoparticles based on localized surface plasmons has been theoretically and experimentally considered for the case when the chain was located near a dielectric substrate [14, 15, 16, 17] or on a dielectric microwaveguide [18].
One of the main aspects of SPP micro-optics is the possibility to concentrate SPP fields at different points of integrated optical circuits. Consequently, the SPP focusing by different surface structures has attracted a great deal of attention. Most of previously considered realizations of the SPP focusing were based on the SPP reflection by properly arranged surface defects [7, 19, 20, 21, 22]. Another approach to achieving the focusing effect involves usage of the same surface structure for both SPP excitation and focusing. Quite recently, the SPP focusing by circular and elliptical slits milled into silver and aluminium films has been experimentally demonstrated and theoretically modelled [23]. SPP excitation and focusing by a hole array machined into a thick silver film has been studied as well [24].
In this paper we investigate experimentally and theoretically the SPP excitation, focusing and directing with curved chains of nanoparticles located on a metal surface. In contrast to the previous investigations, we consider here a relatively narrow laser beam interacting only with a portion of a chain and thereby exciting an SPP beam, whose divergence and propagation direction are dictated by the incident light spot size and its position along the chain. We show also that the SPP focusing regime is strongly influenced by the chain inter-particle distance. We conduct extensive numerical simulations of the configuration investigated experimentally by making use of the Green’s tensor formalism and dipole approximation, and compare numerical results with experimental data.
2. Sample preparation and experiment
The SPP optical elements were produced by making use of the two-photon polymerization (2PP) technique [25]. First, dielectric structures were fabricated upon a glass substrate by the 2PP of the inorganic-organic hybrid polymer ORMOCER provided by Microresist Technology. As a liquid, this polymer can be polymerized by use of a radical photoinitiator Irgacure 369 from Ciba Specialty Chemicals Inc. For the photopolymerization a femtosecond oscillator, Spectra-Physics Model Tsunami, was used. This system delivers laser pulses at the wavelength of 780 nm with a pulse duration of 80 fs (FWHM) and repetition rate 80 MHz. In the present experiments, the average power of 40 mW was applied. Schematic of the setup used for sample fabrication is illustrated in Fig. 1. Femtosecond laser pulses were focused by an oil-immersion objective (Nikon, 100×, numerical aperture of 1.3) through a covered glass filled with an oil.
A liquid polymer droplet was sandwiched between the glass substrate and a cover glass with the thickness of 150 µm. Their separation was fixed by a plastic frame with the size of 18×18 mm2 and thickness of 100 µm. For the fabrication of surface structures, the laser beam was focused through the cover glass and the ORMOCER layer on the substrate glass surface. During the structuring, the laser beam was scanned along the sample surface by galvo-scanner system from Scanlab. After completion of the 2PP and development of surface polymer structures, samples were washed in isobutyl-methylketon (4-methyl-2-pentanone) to remove a liquid non-irradiated polymer. At the final fabrication stage, 50-nm-thin gold films were deposited on dried samples with surface dielectric structures by electron sputtering resulting in nanostructured gold surfaces (Fig. 2). A scanning electron microscope (SEM) CamScan Serie 2 (Cambridge) was used for the visual inspection of fabricated samples. A resulting individual structure fabricated represents a curved (circular) line of protrusions (particles) on the gold surface (Fig. 3).
Fig. 3. Scanning electron microscope (SEM) (a) the top image of the structure, (b) the image of the structure obtained with view angle 45°. The radius of curved chains of nanoparticles is equal to 10 µm. The particle in-plane size (diameter) and inter-particle distance are estimated to be about 350 nm and 850 nm, respectively, the particle height is 300 nm.
The SPP excitation was conducted at the wavelength of 800 nm using a laser beam being incident on the sample surface (at normal incidence) and focused on the structure with a microscope objective. The incident radiation being scattered by nanoparticles was partially coupled into SPPs propagating away from the nanoparticles. In our experimental setup, the laser spot size and its position on the structure could be easily adjusted by changing the objective’s position. The SPP propagation along the surface was experimentally studied by the leakage radiation microscopy (LRM) as explained in detail elsewhere [9, 26]. This imaging technique relies on coupling of SPPs to leaky light modes, which penetrate through the gold film and propagate further (away from the sample surface) in the substrate being eventually captured by a high-numerical aperture objective used with an immersion oil.
In order to observe weak leakage radiation generated by propagating SPP the excitation laser beam transmitted through sample was filtered out by application of spatial filtering before the leakage radiation projected on the CCD sensor of the camera [26]. Only the leakage radiation of the propagating SPP without transmitted laser light was registered. The size of the excitation beam sport on the sample surface was measured in separated experiment by using a setup configuration with a temporally removed spatial filter. Since the excitation beam spot is not visible on the leakage radiation microscopy images we artificially denote the excitation beam spot by drawing the circle with proper dimension.
The SPP focusing by the curved chains of nanoparticles [Fig. 3(b)] was readily observed with the LRM for moderate sizes of the incident laser beam. The SPP excitation was obtained by a focusing laser beam (at normal incidence) directed to the center of the chain. The exciting laser spot was, in this case (Fig. 4), equal to approximately 10 µm in diameter. Interference of divergent SPPs excited by different particles of the structure gives rise to the SPP focusing with a focal point located at the center of curvature of the nanoparticle chain. Note that the focal SPP waist is very well localized and has relatively small value. In addition to the SPP focusing one can see that there is a system of SPP rays (fanning effect) on the other side from the surface structure. The origin of this effect is again in the interference between the SPPs produced by different chain particles. As will be demonstrated below with numerical simulations, this effect depends significantly on the inter-particle distance.
If the incident light spot size decreases, the SPP focusing effect becomes less pronounced (Fig. 5) due to the increase in diffraction divergence of the excited SPP beam (whose width decreases). Consequently, the SPP focal waist broadens resulting in a nearly parallel and relatively narrow SPP beam. It is interesting to note that one can change the propagation direction of such an SPP beam (always directed perpendicular to a local tangent of the particle chain) simply by changing the incident light spot position along the chain (Fig. 5). This effect might be found useful for the SPP manipulation in complex micro-optical elements utilizing chains of scatterers. Here it should also be borne in mind that the efficiency of SPP excitation decreases for the excitation directions deviating from the polarization orientation of the incident light beam. In order to further elucidate the role of main system parameters in the SPP focusing by chains of particles we carry out numerical simulations of the configuration investigated experimentally.
Fig. 4. Typical experimental LRM image of the SPP focusing by the surface structure shown in Fig. 3. Here the diameter of the incident light spot size for the SPP excitation (dashed circle) is equal to approximately 10 µm. The arrow indicates the incident light polarization.
Fig. 5. Experimental LRM images of the SPPs excited by the surface structure shown in Fig. 3b. The diameter of the incident light spot for the SPP excitation (dashed circles) is equal to 3 µm. (a) The exciting laser spot is at the center of the structure, (b) and (c) the laser spot is shifted away from the center. The arrow indicates the incident light polarization in all cases.
Fig. 6. Schematic top view of the considered system: a curved chain of identical gold particles with constant inter-particle (center-to-center) spacing D and with curvature radius R is located above the gold-air interface in the air half-space. The dashed circle indicates the incident light spot exciting SPPs. Only the particles located inside this circle interact efficiently with the incident light. The angle α determines the center of the incident light spot along the chain. The angle β determines the length of a circle segment filled with chain particles.
3. Numerical results and discussion
Emulating the configuration of surface structure studied experimentally, the system under theoretical consideration represents a curved chain of identical spheroid gold particles having the dielectric constant εg and a reference system consisting of air (in the region z>0)with the dielectric constant εr=1 and gold (in the region z<0) with the dielectric constant εg. The radius of the chain curvature is denoted by R. All particles are placed on the gold surface in the air half-space (Fig. 6) and, in general, have different dimensions with respect to the surface plane (typically, the in-plane dimension is larger than the perpendicular one).
The electric and magnetic fields in the system obey Maxwell equations and can be calculated using the Green’s function technique. In the framework of this formalism the total electric field in the reference structure with the particles is determined from the following Lippmann-Schwinger integral equation (all fields are assumed to be monochromatic at frequency ω)[27]
where E 0(r) is the external (incident) electric field at the point r, k 0 is the wave number in the vacuum, N is the total number of the particles in the chain; Vi is the volume occupied by the particle with number i. The propagation of light in the reference system without the particles is described by the Green’s tensor Ĝ(r,r ′) [28, 29]. In the point dipole approximation, each nanoparticle is treated as a dipolar scatterer, and the dipole moments of the nanoparticles can be found by solving the following system of equations [29, 30, 31]:
where r i=(xi,yi,zi) is the radius vector of the center of a particle with number i,ε 0 is the vacuum permittivity; α̂ is the free space polarizability tensor of the spheroid particle. The free space polarizability tensor in the long-wavelength (electrostatic) approximation is given by [32, 33]
where x̂, ŷ, ẑ are the coordinate unit vectors (x̂ and ŷ are in the metal dielectric interface plane, whereas ẑ is directed toward the dielectric),
Here V is the particle volume, mx, my and mz are the depolarizing coefficients [32]. mx=my=mz=1/3 corresponds to the case of spherical particles. In order to take into account spatial distribution of the incident field, which interacts with the particles, we assume that the incoming electric field is determined by the following expression E 0=êexp(-[x 2+y 2]/W 2), where ê is the unit polarization vector, W is the waist of the incident light beam.
We note that the dipole approximation has some restrictions and can be strict applied to the SPP scattering only if certain conditions are satisfied [34, 35, 36, 37]. However recently it has been shown [38] that in the far-field zone only one fitting parameter (the volume or size of scatterer) is sufficient to attain a good agreement between the strict consideration SPP scatting by a finite-size particle and the point dipole approximation results for any nonresonant wavelength range. Therefore in our simulations we will consider the size of the particles as a fitting parameter.
Since all particles are located in the air above the metal surface, the Green tensor Ĝ(r,r ′) in Eq. (2) splits into two separate contributions Ĝ 0(r,r ′) and Ĝs(r,r ′). The first contribution is the Green’s tensor of free space determining the direct electric dipole field [39], and the second contribution is the scattering part of the Green’s tensor accounting for the secondary electric field related to the reflection from the air-metal interface including the excitation of SPPs [28, 29]. Once the dipole moments of the particles are found from Eq. (2), the total electric field of the dipoles in the air outside the particles can be determined by
Note that for positions r sufficiently close to the metal-dielectric interface and for relatively large distances from scatterers (larger than several SPP wavelengths) the SPP scattering fields are dominant in the total scattered field in the system. This assumption is confirmed theoretically [40] and experimentally. For example, for an SPP being excited at relatively smooth metal surface, the near field intensity maps are directly related to the total SPP field [20]. Moreover for the SPP beam splitter configuration studied in [9], the SPP-to-SPP scattered fields are more than 80% of total scattered field in near-field zone. Therefore the scattered fields calculated from (5) on a close plane above the surface nanoparticle structure in the points being relatively far from the structure basically correspond to the SPP fields. In means that in this case the reference system Green’s tensor Ĝ(r,r ′) can be approximated by the tensor Ĝ SPP(r,r ′) describing the SPP propagation [30] in the system.
Fig. 7. Magnitude of scattered electric field calculated above the gold surface with a curved chain (with R=10µm and β=60°) of spheroid gold nanoparticles illuminated by a light beam at the wavelength of 800 nm being incident perpendicular to the gold surface and polarized along x-direction (Fig. 6). The waist W of the incident beam and the inter-particle (center-to-center) spacing D in the chain are (a) W=10 µm, D=400 nm;(b) W=10µm, D=800 nm; (c) W=1.5 µm, D=400 nm; (d) W=1.5 µm, D=800 nm.
In the numerical simulations we consider that the light beam with the wavelength λ=800 nm and with different waists is incident on a chain of gold equivalent spheroids placed on the gold surface (gold dielectric constant εg=26.3+1.8i)[41]. As it was mentioned the size of the particle is not a crucial parameter and chosen to be equal to 240 nm for the in-plane diameter (the spheroidal particles are characterized by their semi-axes hz/hx=hz/hy=1/3 and hz=40 nm) judging from the overall similarity between the simulated field intensity distributions and the experimental LRMimages (see below). The particles are separated with the center-to-center distance of D.
It should be stressed that, in our model, we do not consider the near-field (quasi-static) contribution in the total scattered (by a particle) field when calculating the intensity just above the particles. Since the comparison of the theoretical results with experimental intensity distribution is meaningful only away from the scattering particles, this omission is of no importance, but it does help to avoid plotting very high field intensities just above the nanoparticle chains.
The magnitude distributions of the scattered electric field calculated 160 nm above the airgold interface for the light beam incident on curved chains of nanoparticles are shown in Fig. 7 for different system parameters. From Fig. 7 it is immediately seen that the field distribution depends strongly on both the inter-particle distance D and the light spot size (the beam waist) W. When the inter-particle distance is smaller than the light wavelength, the pattern of the field magnitude distribution is relatively smooth [Fig. 7(a)]. In this case, the illuminated part of the chain exhibits scattering properties that are similar to those of a continuous ridge. Note that straight ridges are frequently used for excitation of a divergent SPP beam on a metal surface in SPP experiments [7, 8, 9]. When increasing the inter-particle distance the individual particles of the chain become relatively independent sources of the scattered waves whose phases differ considerably, resulting in a complex interference pattern [Fig. 7(b)] - a system of divergent SPP rays. A similar trend is also seen for a relatively narrow incident light beam [cf. Figs. 7(c) and 7(d)]. Comparison of the numerical results presented in Figs. 7(b) and 7(d) with the experimental images shown in Figs. 4 and 5 demonstrates good agreement with respect to the observed features in SPP focusing and directing, which is consistent with the fact that the experimental images were obtained with surface structures of bumps separated by relatively large distances (Fig. 3). Comparing the experimental and simulated results (Fig. 8), one notices well qualitative agreement.
Fig. 8. Normalized field intensities (a) along the cross section through the SPP focus parallel to the y-axis Fig. 6 (transverse cross section) and (b) along the cross section through the SPP focus parallel to the x-axis Fig. 6 (longitudinal cross section). For the calculation the chain curvature radius R, the angle β, the inter-particle distance D, and the waist W of incident light have been chosen correspondingly an experimental realization (R=10 µm; β=60°; D=850 nm; W=8 µm).
From Fig.7 it is also seen that, similarly to the experimental observations (cf. Figs. 4 and 5), the SPP focusing is more pronounced in the case of a larger spot size of the incident beam [cf. Figs. 7(a) and 7(c)]. As was suggested above the deterioration of SPP focusing with the decrease of the spot size occurs due to the increase in diffraction divergence of the excited SPP beam whose width is dictated by the incident beam width. Consequently, one should compare the diffraction angle λ/πW with the focusing angle W/R in order to evaluate the balance between these two effects. For example, the diffraction and focusing angles are practically equal to each other in the configuration considered in Figs. 7(c) and 7(d). It is then understandable that the waist of a focused SPP beam remains practically the same in a wide range of the interparticle distances, whereas the field intensity at the focus being the result of interference of SPPs formed by individual particles depends strongly on the inter-particle distance whose increase decreases the number of illuminated particles (Fig. 9). Stronger diffraction divergence of the excited SPP beam (due to a smaller incident light spot size) leads also to a larger shift (toward the chain) in the position of the SPP beam waist (Fig. 10) that also becomes wider in the y-direction [cf. Figs. 7(a) and 7(c)]. In general, these trends are well known in classical optics of Gaussian beams, and the presented simulations (Fig. 10) are thought to be of interest primarily from the point of view of the design of curved chain structures for practical purposes.
For a relatively narrow incident light beam, so that the focusing angle is larger but still comparable with the diffraction angle, one can realize a well-collimated SPP beam propagating in the direction perpendicular to the local chain tangent (Fig. 5). The results of appropriate simulations are shown in Fig. 11 demonstrating the main features observed experimentally. It should be understood that, in such a configuration, the excited SPP beam becomes weaker when the incident laser beam is shifted away from the chain center without changing its polarization. Taking into account the angular dependence of SPP excitation by an individual dipolar scatterer [29], one realizes that the power of a well-collimated SPP beam (excited as discussed above) should scale as cos2 α (Fig. 12). As a result, the intensity of the excited SPP beam decreases very little for small deviations of the excitation directions from the polarization orientation of the incident light beam (i.e., for small angles α). However when the light spot approaches to the chain end, the intensity decrease becomes more pronounced than that of cos2 α, because the effective number of particles interacting with the incident light beam decreases as well. Note that, for a narrow light beam and large inter-particle distance, the maximum SPP intensity exhibits a somewhat irregular dependence on the angle α (solid curve in Fig. 11), because in this case there are only a few particles interacting efficiently with the incident light, whose number is determined by the angle α.
Fig. 9. Field intensities calculated along the cross section through the SPP focus parallel to the y-axis (transverse cross section) for the different inter-particle distances D. The chain curvature radius R and angle β are as in Fig.7 and W=10 µm.
Fig. 10. Field intensities calculated along the cross section through the SPP focus parallel to the x-axis (longitudinal cross section). The chain curvature radius R and angle β are as in Fig.7 and D=400 nm. The power of the incident light is constant for the different values of W.
Fig. 11. Intensity of scattered electric field calculated above the gold surface with a curved chain of spheroid gold nanoparticles having the curvature radius R=10 µm, the angle β=60° and the inter-particle (center-to-center) spacing D=400 nm. The incident laser beam with the waist W=3 µm is shifted along the chain on the angle (a) α=30°, (b) α=-30° with respect of the x-axis. The incident light polarization is along x-direction.
Fig. 12. Maximum of field intensity in the beam propagating from the particles to the center of chain curvature as a function of position of the exiting light spot on the chain, the position is determined by the angle α (Fig. 6). The basic parameters are as in Fig. 11. The power of the incident light is constant for the different values of W.
In our model, the angle β and the chain curvature radius R are the external parameters that should correspond to the experimental configuration. In the experiment, they depend on the sample fabrication procedure. It is therefore important to know how the SPP focusing and directing are affected with the chain length. Here it should be borne in mind that the focusing efficiency is determined by the relationship between the focusing and diffraction angles. When the chain length L is smaller than the light spot size determined by W, the focusing and diffraction angles are determined by the ratios between the chain length L and the chain curvature radius R and between the light wavelength and L, respectively. Under these conditions, the diffraction divergence of the excited SPP increases progressively when decreasing the chain length (the angle β), resulting thereby in the decrease of the focusing efficiency (Fig. 13). If the chain length is larger than the light spot size, the focusing efficiency does not depend on the angle β, however the contrast between the excited SPP beam and the background in the system increases significantly when increasing the angle β (Fig. 14). Finally, the decrease of the focusing angle when increasing the chain curvature radius R leads to smaller SPP intensities at the focus due to the SPP damping over longer propagation distances involved. It was found, similarly to the simulations of SPP focusing by circular slits [23], that, under optimum conditions, the chain curvature radius, SPP propagation length and incident beam width should be close to each other.
Fig. 13. Field intensities calculated along the cross section through the SPP focus parallel to the x-axis (longitudinal cross section). The chain curvature radius R=10 µm, D=400 nm and W=20 µm.
Fig. 14. Field intensities calculated along the cross section through the SPP focus parallel to the y-axis (transverse cross section). The chain curvature radius R=10 µm and D=400 nm. The power of the incident light is constant for the different values of W.
4. Conclusions
Summarizing, we have considered experimentally and theoretically excitation, focusing and directing of SPPs with curved chains of nanoparticles located on a metal surface. The SPP excitation has been accomplished with a relative narrow laser beam (at normal incidence) interacting only with a portion of a nanoparticle chain. The experimental results obtained by the leakage radiation microscopy have shown that the curved nanoparticle chains (particles were placed on a circle segment) can focus the excited SPPs with the focal point located at the center of the chain curvature. It was found that the focusing efficiency strongly depends on the light spot size illuminating the chain nanoparticles. For relatively large spots, experimental SPP images demonstrated the efficient SPP focusing. If the incident light spot size decreased, the SPP focusing effect became less pronounced due to the increase in the diffraction divergence of the excited SPP beam. It has been shown experimentally that one can change the SPP propagation direction simply by changing the incident light spot position along the chain. In order to investigate the influence of different system parameters on the SPP focusing and directing, the theoretical model based on the Green function technique and dipole approximation has been applied. Comparison of numerically calculated images and experimental data showed excellent agreement with respect to the observed features in SPP focusing and directing. It was found that the focusing regime of SPPs is strongly influenced by the chain inter-particle distance, so that the focusing and directing effects with optimal properties can be obtained only when the chain inter-particle distance is smaller than the SPP wavelength. Following the experimental conditions, we have studied the role of the size of light spot exciting SPPs. If the light spot size being determined by W is sufficiently small in comparison with the chain curvature radius R so that the diffraction angle of a SPP beam is approximately equal to the focusing angle W/R, the focusing effect decreases and the maximum of SPP intensity moves toward the nanoparticle chain. Strong SPP focusing effects have been obtained for relatively lager W/R. We also analyzed the SPP field distribution in the system with the chains of different length and different curvature radius. Note that due to the dispersion law of SPPs the size of a SPP beam in the focus can be smaller than the size for the light beam with the same frequency. We believe that the presented results can be useful for the further understanding of various SPP scattering phenomena and development of subwavelength SPP-based optical devices.
Acknowledgements
The authors would like to thank the Network of Excellence “Plasmo-Nano-Devices” and “Plasmocom” for financial support. Two of the authors (A.B.E. and S.I.B.) acknowledge financial support from the Danish Technical Research Council, Contract No. 26-04-0158. A.B.E. is grateful the Deutscher Akademischer Austauschdienst (DAAD) and the Russian Foundation for Basic Research, Grant No. 06-02-16443, for the support. A.L.S. is grateful the Alexander von Humboldt Foundation and the Russian Foundation for Basic Research, Grant No. 06-02-08147, for the support.
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