1. Introduction

Beams possessing phase singularities have been studied extensively in far- and near-field optics in order to reveal their advantages beyond ordinary light [1–3]. The amplitude distribution of the beams represents a doughnut-shaped field. The beams often have optical vortices, which are spiral wavefront dislocations with orbital angular momentum. Remarkable applications of the doughnut beam include the trapping of microscopic particles and, furthermore, the transfer of the orbital angular momentum [2, 3]. This distribution is formed by a complex exponential term exp(inϕ) in cylindrical coordination, where n and ϕ are an integer called the topological charge of the phase singularities and the azimuthal angle, respectively. Laguerre-Gaussian beams [2] and Bessel beams [4, 5] are familiar cases possessing the topological charge. The electromagnetic field of the Bessel beam is the solution of a Bessel function with cylindrical coordinates (r,ϕ, z)defined as

where k_r, k_z , and k ₀ are wavenumbers in the radial and vertical components and a wavenumber of light in free space, respectively. The topological charge corresponds to the order of the Bessel beam. The Bessel beam is usually referred to as diffraction-free in far-field optics. The evanescent Bessel beam has been investigated for other interests in near-field optics [6, 7]. It is a typical case of the Bessel beam with a condition of $k_r^2$ >${\epsilon k}_0^2$, and the field is exponentially decayed in the z-direction. A zero-order evanescent Bessel beam produces a tiny optical spot beyond the diffraction limit when a radially polarized light is irradiated on a smooth surface in a total reflection condition. It is anticipated that the chirality of the incident light continues in the evanescent Bessel beam through the symmetric structure.

Interactions between light and nanoscale metal with chiral structures have been investigated intensively. A polarization of transmitted or reflected light is efficiently rotated by nanoscale planar gammadion gratings resulting from chiral optical response associated with surface plasmons [8, 9]. Optical transmission through a single planar gammadion hole is sensitive to whether the incident light is circularly polarized in a clockwise or a anticlockwise direction. Concentration of the transmitted field at the center of the hole is observed when the gammadion hole and the circular polarization have the same rotational direction. On the other hand, a polarization conversion appears at the rim of the hole in a case where the rotational directions are opposite [10, 11].

In this work, we investigate surface plasmons diffracted by a planar spiral groove on a metal film, which is also a well-known chiral structure called the Archimedes’ spiral, that show analytically and numerically interesting behaviors. The circularly polarized incident light is converted to a zero-order, a first-order, or a high-order evanescent Bessel beam as an interference of the surface plasmons. We define that the chirality extensively means the rotational direction along a propagation direction of the incident light (clockwise: +, or anticlockwise: -), an order of structural azimuthal variation, and the topological charge. The order of the evanescent Bessel beam n, i.e., the chirality of the surface plasmon, is given by the chiral sum of the incident light σ_i and the spiral σ_s . Finally, some remarks on possible applications of the chiral sensitivity and the beams are given.

2. Analysis of surface plasmonic evanescent Bessel beams

Surface plasmons have received much attention regarding energy transportation, high resolution, and optical throughput of a subwavelength aperture [12]. An incident light can be converted into surface plasmons by a periodical structure [13] and nonchiral concentric grooves [14] on metal films such as silver, gold, and aluminum. The surface plasmons that transport energy inward on an aperture placed in the center of the concentric grooves tremendously enhance the throughput of the aperture in comparison with the aperture on a smooth film. A recent noteworthy study of the dynamic response of the inward surface plasmons indicates that the phase of the surface plasmon depends on the location of the groove [15].

Figure 1 shows a schematic diagram of a second azimuthal variation (two-fold) anticlockwise spiral groove in free space with a chirality of σ_s =-2 impinged by a circularly polarized light with a chirality σ_i =+1. The groove consists of two spiral grooves with azimuthal angles 0 and π as origins of the spirals, and thus they are placed alternately without any overlaps. A position of one of the grooves in cylindrical coordination is formed as

where σ_sg and c are a constant repetition of the groove and a constant number, respectively. Since the number of the grooves is equal to the azimuthal variation |σ_s |, an actual repetition of the groove is g. The repetition is given by

to excite the surface plasmon, where θ is an incident angle of the incident light [12].

 

Fig. 1. Schematic diagram of second azimuthal variation anticlockwise spiral grooves on a metal film.

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A normalized electric field of a radial component of the circularly polarized incident light with a normal direction is a form of

When the groove is sufficiently narrow, it acts as a cutoff for the incident light of a parallel component to the walls of the groove; hence the surface plasmon is dominantly excited by the radial component of the incident light. An electric field of the surface plasmon at a concentric position is

Finally, a relative phase of the surface plasmon on any concentric position inside the spiral is simply obtained as

and the chirality of the surface plasmon -1 is obtained. It clearly means that the chirality of the surface plasmon σ_sp is given by the chiral sum of the incident light σ_i and the spiral σ_s . Here the electric field of the evanescent Bessel beam based on the surface plasmons is refined from Eq. (1) to

where E_z is the electric field in the z-direction. In a system of a pure surface plasmon where the attenuation of the surface plasmon is sufficiently low, i.e., Im(k_sp )≪1, the real part of the wavenumber in the z-direction is negligible, Re(k_z )≪1, and the third term induces the exponential decay along the z-direction. The first term means that the high-order Bessel pattern distributions revolve around the origin with the topological charge, i.e., planar vortices, present.

3. FDTD modeling

We numerically verify the rule of the chiral interactions between the incident light and the groove by a FDTD simulation. Structures in the computation are a half-infinite thick silver film with a permittivity of ε=-17.58+1.175i and anticlockwise spiral grooves 200 nm deep and 150 nm wide. The film is impinged by a circularly polarized light with a 658 nm optical wavelength along a normal direction to the film. The repetition of the grooves at 637 nm is calculated from

A 25 nm cubic cell, a computation space of 250 (x) by 250 (y) by 20 (z) cells, and a time step of dt=0.433 are specified. Figure 2 shows phases and normalized absolutes of electric fields normal to the film, _z E on screen A, which is located 62.5 nm above the film as shown in Fig. 1. The size of the figures is 4.0 µm by 4.0 µm. The phase is linearly changed as a contrast from -π (black) to π (white), and the strength of the field is shown as a linear contrast from 0 (black) to maximum (white). The two columns correspond to the cases of the clockwise circular polarization σ_i =+1 (left) and the anticlockwise circular polarization σ_i =-1 (right). The top and bottom rows display distributions induced by the anticlockwise spiral grooves possessing two-fold and four-fold variations, respectively. In the two-fold spiral, topological distribution with (a) one-fold and (c) three-fold variations are present in the circumference when rotations of the circular polarized incident light are clockwise and anticlockwise, respectively. Similarly, topological distributions with (e) three-fold and (g) five-fold variation are obtained on the four-fold spiral. The arrows indicate the rotational direction of the phases; hence every rotation of the fields is anticlockwise.

Figure 3 shows the envelopes of the simulated |E_z | on the x-axis (solid diamond: σ_s =-2, σ_i =+1; open diamond: σ_s =-2, σ_i =-1; solid triangle: σ_s =-4, σ_i =+1; open triangle: σ_s =-4, σ_i =-1; and the evanescent Bessel beams defined as absolute of Eq. (7) (solid curves) as a function of k_spr. The amplitudes are normalized by the first lobes of the evanescent Bessel beams where both chiralities are σ_s =-2 and σ_i =+1 for the simulations, and n=1 for the analytical curves. The simulations and the analysis are mostly matched for the points of the envelopes and the ratios between the first, third, and fifth orders. The small mismatches at the high-order lobes would be induced by the attenuation of the surface plasmons or localized fields on the grooves. We thus determine that the spiral grooves create the surface plasmonic evanescent Bessel beams on the film. The radii of the first lobes are 1.84/k_sp , 4.20/k_sp , and 6.42/k_sp in the cases of the first-, third-, and fifth-order evanescent Bessel beams, respectively. It is important in nano-optics that the radius can be shrunk as the wavenumber of the surface plasmons increases by maintaining the wavelength, the metal, the dielectrics covering the metal, and the metal thickness [12].

 

Fig. 2. Phases (a, c, e, g) and normalized absolute (b, d, f, h) of electric fields normal to the film Ez on screen A, which is 62.5 nm above the film. Scales: 4.0 µm by 4.0 µm. Linear contrast from black to white: -π to π for the phase; 0 to maximum for the strength of the field. Arrows in the figures: rotational direction of the phases. Left column: clockwise incident light. Right column: anticlockwise incident light. Top row: anticlockwise two-fold spiral grooves. Bottom row: anticlockwise four-fold spiral grooves. half infinite thick

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Fig. 3. Envelopes of |E_z| on the x-axis (solid diamond: σ_s =-2, σ_i =+1; open diamond: σ_s =-2, σ_i =-1; solid triangle: σ_s =-4, σ_i =+1; open triangle: σ_s =-4, σ_i =-1) and first-, third-, and fifth-order evanescent Bessel beams defined as absolute of Eq. (7) (solid curves) as a function of k_spr. The amplitudes are normalized by the first lobe of the solid diamonds and the curve of n=1.

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4. Discussion

We point out several possible applications of the tested behaviors. First is the detection of the chirality of the incident light by using the spiral structure. For example, in the case of the four-fold spiral groove, the dark circle at the center of Fig. 2(h) is much larger than that in Fig. 2(f). We can perceive the chirality of the incident light high-sensitivity by detecting a scattered field from a scatter or an aperture located at the center on the film, which is smaller than the large dark zone, but larger than another small one. Since the thickness of the metal film is several hundred nanometers, the system will be tremendously smaller than a general detection system consisting of a quarter wavelength plate and a polarizer. Second, several orders of electric multipoles can be obtained on the film with the topological distributions, since one phase oscillation of the field corresponds to one dipole: the dipole (a), the hexapole (c, e), and the decapole (g). Such fields possessing the planar vortices and chiralities would be used to sense, excite, and control some small chiral objects with high resolution. The vortices would transfer the orbital angular momentum to trapped particles [2, 3]. Another point of interest is that an electric monopole, i.e., the zero-order evanescent Bessel beam, is minimally excited by the circularly polarized incident light through a single spiral groove [(σ_s =-1, σ_i =+1) or (σ_s =+1, σ_i =-1)]. A tiny confined perpendicular field to the film at the center would couple to some surface plasmon waveguides with a longitudinal mode, which leads to extremely high resolution [16, 17].

5. Conclusion

We analytically and numerically investigated surface plasmons diffracted by Archimedes’ spiral grooves on a silver film impinged by a circularly polarized light. The surface plasmons have a selective chirality, which is given by chiral sum of the incident light and the spiral. The surface plasmons form several orders of evanescent Bessel beams with planar vortices. These insights could be used to detect the chirality of the incident light in optical communication. A subwavelength electric monopole or rotating multipole on the metal film induced by the evanescent Bessel beam could be treated as an optical source in nanophotonics, biophotonics, and information storage systems with high recording areal density.