1. Introduction

Radially polarized beams, which possess electric field vectors arranged like the spokes of wheels pointing out from the center in the beam cross-section, have attracted much interest and are expected to be used in many applications in optics, such as improved optical data storage and optical microscopy, because of their peculiar focusing properties, for example, the generation of a strong longitudinal field on the beam axis [1], their ability to create a small focal spot size with a long depth of focus [2, 3], and zero Poynting vector on the beam axis even though the electrical field is strong [4]. Optical sources that generate radially polarized beams by using an external cavity [5], or more practical sources using optical fibers [6] and semiconductor lasers [7–11], have been developed in the last decade. Nevertheless, even though the peculiar focusing properties in free space have been shown and we can easily generate radially polarized beams due to the development of several optical sources, practical applications of radially polarized beams have not been shown yet. One of the reasons is surely that interactions between such beams and materials have not been investigated much, even though the focusing properties of these beams are very distinct from those of other beams.

We have reported on novel interactions between a half-wavelength sized gold cube and a focused radially polarized beam [12]. The beam energy can propagate through the cube, even though the cube is set on the beam axis in the focal plane, and the electric field enhancement due to the cube results in interaction between the focused electric field and the cube. As mentioned before, these effects are due to the focusing property that the beam creates zero Poynting vector on the beam axis even though the electrical field is strong. In other words, all of the energy can pass through the gold cube located on the beam axis, because the cube does not disturb the Poynting vector. At the same time, the longitudinal electric field interacts with the cube and induces a surface plasmon mode.

In this paper, by taking advantage of these focusing properties, we examine electric field enhancement in the tiny spaces between metal cubes placed on the beam axis along the propagating direction. In Section 2, we describe the calculation model based on the three-dimensional finite-difference time-domain (3D FDTD) method, using a surface impedance method to introduce metal effects. In Section 3, we discuss the electric field enhancement corresponding to the spacing between the two gold metal cubes placed on the beam axis by comparing a focused radially polarized beam and a focused linearly polarized beam. In Section 4, we show how increasing the number of metal cubes on the beam axis affects the electric field enhancement. Finally, we give some concluding remarks in Section 5.

2. Calculation model

In this study, we simulated the electromagnetic fields of focused radially polarized and linearly polarized beams in a three-dimensional finite difference time domain (FDTD) space. We excited a plane a few wavelengths away from the focal plane with initial electromagnetic fields that can be calculated by using vectorial diffraction theory [1, 13]. When both of the electric and magnetic fields were excited based on the vectorial diffraction theory over a large area with 10λ × 10λ, the electromagnetic fields would be propagated only in one direction and focused. Figure 1(a) shows the calculation model. The three-dimensional space was divided into λ/20 units, and the excitation plane was set at z/λ = −2.5 [12]. We excited the plane with focusing electromagnetic fields with NA (the numerical aperture) = 0.9, and β (the filling-factor of the beam in the lens pupil) = 1.5. Figure 1(b) shows the time-integrated electric field intensity profile and the intensity distribution of the Poynting vector in the propagating direction (z), with exciting a focusing radially polarized beam. Note the null intensity of the Poynting vector along the beam axis, where the maximum intensity of the electric field constructed with z-polarization was generated.

 

Fig. 1 Calculation model for three-dimensional FDTD analysis. (a) 3D model; (b) time-averaged distributions of the electric field intensity (Ei: i = total, x, z) and the Poynting vector (Sz) calculated by the FDTD method for a focused radially polarized beam.

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We introduced two gold metal cubes with a side length of λ/2 on the optical axis, separated by distance d (the bottom of the upper cube was located at z/λ = d/2, and the top of the lower cube was located at z/λ = -d/2). We used the surface impedance method to introduce these metal cubes. The metal cubes had a complex refractive index ε = −1.88 + i3.42 at λ = 500 nm [14]. Incidentally, it is valid to use the surface impedance method because these gold cubes are much larger than the gold skin depth [15, 16].

3. Electric field enhancement between two gold cubes

Figure 2 shows the maximum intensity, which is normalized with respect to the maximum intensity in free space, as a function of the separation between the gold cubes (d). When the incident beam is linearly polarized, the maximum intensity is almost constant, irrespective of the spacing. In the case of a radially polarized beam, on the other hand, the maximum intensity is almost constant at a separation distance of more than 0.5λ, whereas, at a separation distance of less than 0.5λ, the maximum intensity increases as the distance decreases. When the separation becomes 0.1λ, the electric-field enhancement is more than 20-times greater than the enhancement in free space.

 

Fig. 2 Maximum electric-field intensity as a function of the separation (d) between the gold cubes. The intensity is normalized with respect to the maximum intensity of the focused beam in free space.

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First, comparing a linearly polarized beam and a radially polarized beam, we consider the electric field intensity profiles of each beam when the cube separation is 0.1λ, as shown in Figs. 3(a) and 3(b). In case of the linearly polarized beam, most of the incident light is reflected by the cube placed at z/λ<0 and constructively interferes with the incident light, as shown Fig. 3(a). In the case of the radially polarized beam, on the other hand, the electric field intensity is concentrated between the two cubes, as shown in Fig. 3(b). We found that the electric field is transmitted to the space z/λ>0, when normalized with respect to the maximum intensity of the beam in free space. This is due to the null Poynting vector in the propagating direction on the beam axis, which causes the light to propagate without being obstructed by the gold cubes placed on the beam axis. Figures 3(c) and 3(d) show magnified images of the electric-field distributions with electric vectors at a certain time. The maximum electric-field intensity of the linearly polarized beam comes not from the surface plasmon mode of the gold cube but from constructive interference between the incident and reflected light, since a strong electric field is generated at a location away from the gold cube. Also, the reason why the strong intensity is not generated in the space between the cubes, in contrast to the case of the radially polarized beam, is that the electric field vectors that are diffracted at the edge of the cube become asymmetric towards x = 0 and destructively interfere, even though a surface plasmon mode is induced at the edge of the cube. On the other hand, the radially polarized beam generates the maximum intensity at precisely the center of the space between the two cubes, as shown Fig. 3(d). The electric field vectors reveal that the longitudinal polarization produced the strongest intensity.

 

Fig. 3 Calculated intensity profiles of total electrical field interacting with two half-wavelength sized gold cubes. The time-averaged intensity profiles of a linearly polarized beam (a) and a radially polarized beam (b) with a cube separation d/λ = 0.1. Inset images are normalized with respect to the maximum intensity in free space. The intensity profiles and electric field vector of a linearly polarized beam (c) and a radially polarized beam (d) with a cube separation d/λ = 0.1. The intensity profile and electric field vectors of a radially polarized beam with a cube separation d/λ = 0.5 (e). The intensity profile and Poynting vector of a radially polarized beam (f).

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Now, we discuss in detail the reason why the maximum electric field intensity is generated from the longitudinally polarized electric field. There are two ways to generate a longitudinally polarized electric field in the space between the cubes when a radially polarized beam is incident: i) One has a similar origin to the generation of longitudinal fields in free space, because the Poynting vector in the propagating direction is not obstructed by the gold cube placed at z/λ<0. ii) The other has its origin in diffraction of the components of the radial polarization, which has a maximum intensity around the beam axis. The components possess radial electric field vectors around the beam axis, and the electric field vectors point in the z-direction after being diffracted by the cube. Consequently, additional longitudinal fields can be generated. The longitudinal components generated in these ways satisfy the conditions for constructive interference. In addition, when the beam is diffracted at the edges, surface plasmon modes, which originate in edge-plasmon modes, are induced. Finally, all longitudinal components, irrespective of their origin, can induce surface plasmon modes in the x-y planes of the gold cubes: in the bottom surface of the upper cube and in the top surface of the lower cube. Therefore, the radially polarized beam shows strong enhancement of the electric fields in the space between the cubes due to these longitudinal components.

Second, we will now examine how the electric-field enhancement is affected by the distance between the cubes (d), in the case where a radially polarized beam is incident. Initially, the intensity is not changed in the region d/λ>0.5. This is because no interaction occurs between the cubes, due to the sufficiently large separation between them, as shown by the intensity profile at d/λ = 0.5, as shown in Fig. 3(e). In this situation, the cubes are influenced separately by the two effects mentioned above, that is, constructive interference and surface plasmon modes. In the region d/λ<0.5, however, the maximum intensity increases as the separation between the cubes becomes smaller, even if we assume gold cubes or perfectly conducting cubes. Such strong enhancement which is generated as the two cubes approach each other might be explainable by electromagnetic modes (resonance modes) generated in the tiny space between the cubes. To evaluate the assumptions by excluding the two effects given above, we simplified the calculation model so that an x-polarized plane-wave is incident on two perfect conductor cubes arranged in the x-direction. This model is just like the state shown in Fig. 3 rotated by 90°, because the z-polarized electric field components propagate in the x-direction between the two cubes which are arranged in the z-direction as shown in Fig. 3 (f). Figure 4 shows the calculated electric fields. By comparing the result at d/λ = 0.6 (a) and the result at d/λ = 0.1 (b), the incident plane wave in the latter case clearly creates another waveguide mode which has almost half the wavelength of the incident wave. This mode can create a constructive resonance mode in the model of Fig. 3, because there are incident electric field waves in Fig. 3 that have the same phases and propagate in the + x, -x, + y, and -y directions due to the nature of diffraction of the radially polarized beam. As a result of such modes, strong electric fields are generated in the region d/λ<0.5, even when we assume gold cubes or perfectly conducting cubes.

 

Fig. 4 Calculated electric fields excited by x-polarized plane wave. Two perfect conductors are placed a separation of 0.6λ (a) and 0.1λ (b).

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These three effects, namely, i) constructive interference of the longitudinal fields created by the focusing properties of the radially polarized beam and by the diffraction of the radial components, ii) coupling to surface plasmon modes in the gold surface by the longitudinal fields, and iii) generation of resonance modes in the space between the cubes, lead to electric-field enhancement 22-times larger than the maximum intensity in free space, when we arrange two gold cubes in the z-direction across the focus of the radially polarized beam. This enhancement is much greater than the enhancement observed when we arrange two gold cubes in a similar way in the x-direction across the focus of a linearly polarized beam.

4. Electric field enhancement between multiple gold cubes

As discussed in Section 3, the focused radially polarized beam can create strong electric fields between two gold cubes arranged in the beam propagation direction. One of the origins of this characteristic is that the beam can generate longitudinal fields as focused in free space, because the Poynting vector in the propagating direction is not shaded by the gold cube placed at z/λ<0. Then, to exploit this characteristic, we considered arranging multiple gold cubes along the propagating direction.

We place multiple gold cubes across the focus (z/λ = 0) on the beam axis at regular fixed intervals: d/λ = 0.1. Figure 5 shows the electric field intensity distributions when the numbers of cubes are two, three, four, five, and six. The intensity of the color bar is normalized with respect to the maximum intensity when the radially polarized beam is focused in free space. The maximum intensity is 22-times higher, 20-times higher, and so on. The results clearly show that enhanced electric fields can be generated in each space between the cubes, even though the number of cubes is increased, and that multiple enhanced electric fields can be created along the propagating direction. The reason for this is that the Poynting vector in the propagating direction is not blocked by any of the gold cubes placed on the beam axis, because of the null intensity of the Poynting vector on the beam axis. Interestingly, we noticed that the enhancement appears when more than five gold cubes are placed over the depth of focus of the beam, which is almost equal to the region where the longitudinal components are generated. This means that not only are longitudinal fields generated by the nature of focusing of a radially polarized beam, but also the fields created by the diffraction of the radially polarized components at the cubes, as discussed in Section 3, have an effect on the electric field enhancement.

 

Fig. 5 The time-averaged intensity profiles of a radially polarized beam with a cube separation d/λ = 0.1. The intensity is normalized with respect to the maximum intensity in free space. Two (a), three (b), four (c), five (d), and six (e) gold cubes were arranged.

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5. Conclusion

As part of an investigation into the interactions between focused radially polarized beams and materials, we examined the electric field enhancement in the tiny spaces between gold cubes arranged on the beam axis along the propagating direction. When we arrange two gold cubes with a separation larger than half the wavelength, electric field enhancement is generated by two separate effects, namely, coupling to the surface plasmon modes and constructive interference brought about by longitudinal electric fields, whose origins are the focusing properties of the radially polarized beam and the effects of diffraction of radial components at the cubes. When the cube separation is smaller than half the wavelength, in addition to the effect of those longitudinal fields on the field enhancement, we showed that resonance modes in such tiny spaces lead to even stronger electric field enhancement. Moreover, we showed that a focused radially polarized beam can generate a string of electrical field enhancement regions in the propagating direction when multiple metal cubes are located on the beam axis. All of those characteristics are peculiar to the focused radially polarized beam. Therefore, if we could bring about interactions with materials placed in such novel electric field enhancement regions, this could lead to the development of novel optical devices in future. In particular, we expect that the ability of such beams to generate multiple electric field enhancement regions along the propagating direction may lead to new functionality in the arena of near-field optics.