By analogy to the three dimensional optical bottle beam, we introduce the plasmonic bottle beam: a two dimensional surface wave which features a lattice of plasmonic bottles, i.e. alternating regions of bright focii surrounded by low intensities. The two-dimensional bottle beam is created by the interference of a non-diffracting beam, a cosine-Gaussian beam, and a plane wave, thus giving rise to a non-diffracting complex intensity distribution. By controlling the propagation constant of the cosine-Gauss beam, the size and number of plasmonic bottles can be engineered. The two dimensional lattice of hot spots formed by this new plasmonic wave could have applications in plasmonic trapping.
©2013 Optical Society of America
Since their recent discovery, optical bottle beams [1–4] have found applications ranging from dynamical trapping of atoms to sorting of particles in colloidal suspensions [5–9]. The interest of bottle beams for optical trapping arises from the oscillating intensity distribution, creating dark (or bright) focii surrounded in all directions by intense (or low) electromagnetic fields. This distribution of light, in the shape of a bottle, creates suitable intensity gradients to trap low (or high) index particles inside the dark (or bright) spots. The emerging field of plasmonic trapping employs nano-structured metallic resonators [10–13] or propagating surface plasmon polaritons (spp)  to control the position of particles in the proximity of metallic structures. Plasmonic devices with reconfigurable 2-D grids of near-fields could be integrated into microfluidic systems, which is of interest for applications for example in biology . In this letter, we experimentally demonstrate the two dimensional analogue of optical bottle beams. These surface waves, which present oscillating low and high intensity focii, are generated at the interface between a metal and a dielectric. In the dielectric half-space, the component of the surface plasmon electric field normal to the metal-dielectric interface obeys the Helmholtz equation:
The surface plasmon propagation constant satisfies the equationEq. (1) . The field profile of the CGB is given by:16]. Compared to a plasmonic Airy beam, another non-diffracting surface wave [17–20], CGB is a conical solution which can be viewed as the 2D projection of a free-space Bessel beam with finite spatial extent. As previously discussed in Refs [16,21–23], CGBs are formed by the interference of two two-dimensional plane waves with intersecting directions of propagation. The straight lines along which constructive interference occurs define the propagation direction of the CGB, see Fig. 1(a). Its propagation constant is given by where is the half angle between the directions of propagation of the two plane wave’s components. By multiplying the two plasmonic plane waves by a Gaussian envelope of finite size, we demonstrated that the CGB remains non-diffracting in the paraxial approximation, while carrying finite amount of energy. While this envelope localizes the solution in a similar way as for Bessel Gauss beams , it also introduces negligible diffraction effects which are unnoticed due to the relatively short SPP the propagation distance, i.e. the main narrow lobe of the CGB diffracts very slowly as a beam with the same transverse dimension as the Gaussian envelope. Note that this envelope is significantly larger than the narrow width of the main intensity lobe of the CGB, explaining its non-diffracting behavior. The CGB propagates in a straight line with a constant and controllable phase velocity, which can be modified by changing the half angle between the two plane waves.
2. Theory and description
In the present work, we use the interference between co-propagating non-diffracting surface waves of different propagation constants to create an array of plasmonic hot spots. One way to achieve such plasmonic intensity grid could consist in superposing two different CGBs (Fig. 1(b)). Instead here, we superimpose a plasmonic plane wave propagating collinearly with the CGB (Fig. 1(c)). This scheme not only generates the characteristic intensity modulation of plasmonic bottle beams but is also more suitable for experimental implementation of plasmonic trapping. In this specific case, the plane wave coupler can be arranged such that it will not spatially overlap with the CGB couplers.
This allows us to dynamically control the position of the spots of intensity along the propagation direction (x): by adjusting the relative phase between the plane wave and the CGB, we can adjust the x-position of the trapping sites (see Media 1). In practice, this can be achieved by splitting an incident beam in three parts to focus each of them independently on each grating. The dynamic control can be performed by modulating the phase of the beam that generates the plasmonic plane wave via reflection on a piezoelectric kinematic mirror before sending to on the device. The electric field distribution normal to the plane of our two-dimensional bottle beam is given by:Fig. 1(c). The number of bottles along the propagation direction, N, is proportional to the decay length of SPP multiplied by the spatial frequency of the interference pattern, the latter being the wavevector mismatch between the CGB and the plane wave.
The wavevectors of the CGB and the plane wave are respectively and . The wavevector mismatch is expressed as, see illustration in Fig. 1(c). This gives . As the angle between the two components of the CGB increases, the bottles become more numerous and their size decreases. Figure 2(a) summarizes the evolution of both the length and the width of the bottle as a function of and shows the longitudinal and transverse field distributions for (Fig. 2(b) and 2(c), respectively). Due to the non-diffracting character of this solution, the interference pattern sets up a lattice of bottles with constant size and spacing (Fig. 1(c) and Fig. 2(a)). By changing the relative phase between the CGB and the plane wave, we can move the peaks in intensity back and forth along the x-direction (see movie in additional materials). Note that by changing the relative phase at a controlled rate, it is possible to impart precise momentum to trapped particles.
3. Experimental results and discussions
We used a focused ion beam (Zeiss NVision 40) to mill the plasmonic bottle beam couplers into a 150nm-thick gold layer. Prior to FIB milling, the gold film was template-stripped  from a silicon wafer onto a glass substrate in order to decrease surface roughness, thus decreasing scattering losses and increasing the SPP propagation distance. The two-dimensional SPP field distribution is acquired by collecting the in-plane field components using a near-field scanning optical microscope (NSOM) working in aperture mode [aperture].The structures are excited by a Ti:Sapphire laser collimated on the sample at normal incidence as described in Fig. 3(b). The results presented in this letter have been obtained by tuning the emission and the detection to a specific wavelength of 735nm, i.e. at the wavelength which maximizes the coupling efficiency of the gratings. The NSOM images obtained for half angle of 10, 20 and 30 degrees are shown in the left panel of Fig. 3(c).
In conclusion, we have demonstrated a new two-dimensional beam, the plasmonic bottle beam. This wave is generated by superimposing a non-diffracting cosine-Gauss beam (CGB), characterized by a small on-axis propagation constant, and a quasi-plane wave. Resulting from Talbot effect (previously discussed for the 3D case in ), the plasmonic bottle beam features a 1D line of intensity peaks and valleys which repeats itself during propagation, remaining homogeneous due to its non-diffracting character. Interferometric experiments have been realized with other recently-demonstrated non-diffracting solutions such as Airy beams , but this type of periodic field distribution cannot be achieved due to their self-accelerating nature. The plasmonic bottle beam will find applications in plasmonic optical trapping and in other fields of research where an intensity lattice is required. Plasmonic bottles could be used to optically sort particles by trapping those with a specific size. Since the location of the plasmonic bottles is defined by the relative phase between the CGB and the plane waves, it is possible to trap particles at intensity spots while shifting the phase of the plane wave to create plasmonic tractor bottle beams, i.e. plasmonic fields that can transport trapped particles even against the plasmon stream [30, 31]. By taking advantage of the cyclic nature of phase, it may be possible to build an optical ratchet, in which particles are continuously pulled or pushed along the beam. By arranging two or more bottle-beams around a central “working area”, it might be possible to achieve full 2-D control on the position of particles trapped close to a metallic surface.
The authors acknowledge support from the National Science Foundation (NSF). The devices fabrication was performed at the Harvard Center for Nanoscale Systems (CNS) which is a member of the National Nanotechnology Infrastructure Network (NNIN). This research is supported in part by the Air Force Office of Scientific Research under grant number FA9550-12-1-0289. M. Kats is supported by the NSF through the Graduate Research Fellowship Program. B. Cluzel and F. de Fornel acknowledge the burgundy regional council for financial support through the PHOTCOM project.
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