1 Introduction

Theoretical and experimental studies of nanometer scale optical integrated circuits employing surface plasmon polaritons (SPPs) have been carried out by many researchers [1–5]. Conventional optics cannot focus a laser beam into a region that is much smaller than the light wavelength due to the diffraction limit of light. Therefore, it is important to develop techniques such as superfocusing or nanofocusing using SPPs [6, 7]. A practical way to achieve this is to use tapered or conical metal structures, and many studies have used such probes [8–13]. However, the basic characteristics of the enhanced local fields that are nanofocused near the probe tip have been insufficiently investigated. So far, it has been considered that the enhanced local field at a tip decays monotonically with increasing distance from the probe tip, as determined by theory based on electrostatic approximations. Recently, the present authors have shown accurate calculations of enhanced local fields near the metal-coated conical dielectric probe tip by the volume integral equation method (VIE) method [14, 15]. It has been shown that the enhanced local fields interfere destructively near the tip of the probe and three-dimensional (3D) nanosized dark spots with a light shell can be generated in free space for specific structures on the probe tip [14, 15]. The generation of such spots is not only interesting from the view point of optical theory, but it can also be applied to the optical trapping of atoms whose refractive indices are smaller than that of that of their surroundings [16–21]. The sizes of 3D dark spots realized in the conventional optics must be limited by the light wavelength. However, those of 3D dark spots considered in this paper can become nanosize because they are due to the interference among near fields. In this paper, we consider nanofocusing of SPPs by a metal-coated dielectric conical probe with nano-rods. The basic characteristics of the 3D nanosize dark spots in free space generated by the enhanced local fields were investigated. Two kinds of dark spots were generated: one is disk shaped and is generated near the rod end, the other is generated around the nano-rod and its shape is toroidal.

2 Metal-coated conical dielectric probe with nano-rods

Figure 1a shows a schematic diagram of the probe considered in this study. The conical dielectric probe with a relative permittivity ε2 is placed on the x–y plane in the (x, y, z) coordinate system shown in Fig. 1a. The side of the conical dielectric probe is coated to a thickness d with metal with a relative permittivity ε1. The metal-coated conical probe has a base radius R and a height h. A thin metallic nano-rod with length l is connected to the tip of the conical probe. We assume that the nano-rod is made of the same metal as the coating metal. The surrounding free space has a permittivity of ε0. The conical probe with the nano-rod is assumed to be rotationally symmetric about its center axis, coincident with the z-axis (see Fig. 1a). A radially polarized Gaussian beam is normally incident on the bottom surface of the conical probe located at z = 0 from the negative z-direction below the probe. The beam axis is coincident with the z-axis. The radial and z components of the incident electric field at z = 0 can be written as [22, 23]:

Fig. 1
figure 1

a Geometry of the metal-coated conical dielectric probe with a nano-rod. The probe is placed on the x–y plane, and has a base-radius of R and a height h. It is coated with a metal whose permittivity is ε1. The metal coating has a thickness d. The rod is made of the same metal as the coating metal. The permittivities of the surrounding free space and the dielectric in the conical structure are denoted by ε0 and ε2, respectively. A radially polarized Gaussian beam is incident normally on the base of the probe from the negative z direction along the z-axis. b The structure is discretized by cubes of size δ. The arrangement of cubes making up the probe and nano-rod is shown. Only the cubes in the top 14 layers of the probe and 10 cubes of the nano-rod are shown. The probe contains a total 347 layers, which are parallel to the x–y plane. c Each cube shown in b is divided into eight smaller cubes of size δ/2 when applying the MoM to the VIE

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$$ \begin{gathered} E_{r} \left( {r, \, 0} \right) = 2A\left( {r/w} \right)\;\exp \left[ { - \left( {r/w} \right)^{2} } \right], \hfill \\ E_{z} \left( {r, \, 0} \right) = j4A\left[ {1 - \left( {r/w} \right)^{2} } \right]/\left( {k_{0} w} \right)\;\,\exp \left[ { - \left( {r/w} \right)^{2} } \right], \hfill \\ \end{gathered} $$
(1)

where r = (x 2 + y 2 ) 1/2, w is the spot size of the beam and k 0 is the wavenumber in free space. The beam amplitude is given by A = 1 so that the incident beam has the same energy as a linearly polarized beam of unit amplitude with the same spot size. The incident beam illuminates the metal-coated conical probe through the bottom surface of the probe. First, a radially polarized SPP is excited on the probe surface and the optical field of the SPP is strongly enhanced and focused near the tip by decreasing the cross section of the probe in the propagation direction. Then, the enhanced local optical field on the tip excites the SPP guided mode along the rod. The excited SPP mode in the rod is reflected by the rod end and generates a localized and enhanced field around the rod. We solve the scattering problem shown in Fig. 1a by the VIE method [14, 15]. In the VIE method, the structure is assumed to consist of tiny cubes. We first discretize the whole structure in Fig. 1a using tiny cubes; i.e., we consider that the conical structure and the rod are composed of cubes with dimensions δ × δ × δ as shown in Fig. 1b. We then discretize the VIE by the method of moments (MoM) using rooftop functions as the basis and testing functions in each cube. In this study, the value of δ is given by δ = 5 nm (k 0δ = 0.05). The nano-rod on the probe tip is approximated by a long right-angled parallelepipe with a cross section of δ × δ, i.e., a SPP rectangular waveguide. When we apply MoM to the VIE, we use cubes of size δ/2 (2.5 nm), whereas we discretize the structure using cubes of size δ. This means that each cube of size δ, that makes up the whole structure in Fig. 1b, is further subdivided into eight smaller cubes of size δ/2 in the numerical calculations by the MoM as shown in Fig. 1c. The structure used in the numerical calculations is approximately rotationally symmetric about the z-axis. Finally, we solve the resultant system of linear equations numerically using the general minimum residual method with fast Fourier transformation.

3 Optical intensity distributions near the nano-rod

In the numerical calculations, the wavelength is λ = 633 nm, and the complex permittivity of the coating metal and nano-rod is ε1/ε0 = −13.8 − j1.08 (Au). The permittivity of the dielectric is ε2/ε0 = 2.25 and the beam spot size is w = λ at z = 0. The thickness d of the metal coating cannot be specified precisely because the side of the metal-coated probe is bumpy since it is composed of cubes (see Fig. 1b, c). Thus, we specify the average thickness to be d = 27.4 nm (k 0 d = 0.27). We first consider the idealized mathematical model of a typical conical structure that has an apex angle of 37°, a base radius given by R = 755.6 nm (k 0 R = 7.5) and a height given by h = 1747.9 nm (k 0 h = 17.35). We then discretize this metal-coated conical probe using cubes arranged in 347 layers, i.e., parallel to the x–y plane. The cubes of the top five layers are illustrated in Fig. 2a by black squares. Though this structure is a purely conical shape, we consider that the structure shown in Fig. 2a consists of a conical probe and a nano-rod whose length is l = 5.0 nm (k 0 l = 0.05). We add cubes of size δ on the z-axis to this structure to form the nano-rod as shown in Fig. 1b. We show the optical intensity distributions |E(x, 0, z)|2 on the x–z plane near the nano-rod using a logarithmic scale for different rod lengths l, i.e., the number of cubes, from 1 to 17, corresponding to the nano-rods shown in Fig. 2a–p. The minimum rod length is l = 5.0 nm (k 0 l = 0.05) and the maximum rod length is l = 85.6 nm (k 0 l = 0.85) in Fig. 2a, p, respectively. The arrangement of cubes used in the discretization of the nano-rod and top four layers of the conical probe in the x–z plane is illustrated by black squares in each panel in Fig. 2. When the rod length is increased from l = 10.1 nm (k 0 l = 0.15) to l = 30.2 nm (k 0 l = 0.3), it can be seen that dark spots are generated in free space about 10 nm above the rod, as shown in Fig. 2b–f. We refer to these as dark spots of the first kind. When the rod length is l = 35.3 nm (k 0 l = 0.35), no dark spot is seen, as shown in Fig. 2g. With further increase in rod length, interestingly, dark spots are generated around the rod as shown in Fig. 2h–k. Since the shape of these dark spots is toroidal, i.e., very different from those of the first kind, we refer to them as dark spots of the second kind. For l = 60.4 nm, as shown in Fig. 2l, the dark spot vanishes again. The dark spots of the second kind reemerges as shown in Fig. 2m–o. It is interesting that the position of the dark spot in (m) is similar to that in (i). The same relation can be seen between (n) and (j), and (o) and (k). Consequently, it is found that two kinds of dark spots are generated by the structure shown in Fig. 1a.

Fig. 2
figure 2

Change in the optical intensity |E(x, 0, z)|2 distributions in the x–z plane near the nano-rod with increasing rod-length l. Notice that the scale is logarithmic, as shown below the figures. The area illustrated is 102.5 × 122 nm in size. The arrangement of cubes used in the discretization of the nano-rod and top four layers of the conical probe in the x–z plane are illustrated by black squares

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4 Three-dimensional dark spots near the nano-rod

In order to clarify the detailed structure of the dark spots of the first kind, shown in Fig. 2b–f, the optical intensities |E(0, 0, z)|2 along the z-axis are shown using a linear scale in (Fig. 3a) for the case of Fig. 2a–g. The abscissa z–h is the distance from the probe tip and the positions of the rod end are also shown in Fig. 3a by the colored arrowheads. Figure 3b shows the typical intensity distribution on the x–z plane for the case of l = 15.1 nm (k 0 l = 0.15) (see Fig. 2c). Since the results shown in Fig. 3 are rotationally symmetric around the z-axis, the 3D dark spots shown in Fig. 2b–f are perfectly surrounded by a light shell and have a disk shape. For the case of dark spots of the second kind, the optical intensity |E(x, y, given)|2 distributions along a line parallel to the x-axis are shown using a linear scale in Fig. 4a for various positions along the z-axis for the case of l = 50.4 nm (k 0 l = 0.5) (see Fig. 2j). The z-axis positions are illustrated in the inset with the top four cubes of the nano-rod (see Fig. 2j). The numbers on the curves correspond to those shown in the inset. Figure 4b shows the intensity distribution |E(x, y, given)|2 around the nano-rod for curve (5) in Fig. 4a. It can be seen that the dark spot is toroidal in shape and is perfectly surrounded by a light shell. For both kinds of dark spots, the optical intensities change promptly compared with those reported for conventional diffraction limited optics [16, 17].

Fig. 3
figure 3

a Distributions of optical intensity along the z-axis |E(0, 0, z)|2 with rod length l as a parameter. The abscissa z–h is the distance from the probe tip. The arrowheads on the upper axis show the position of the rod end for each rod length. The color of each arrowhead indicates the rod length. b Optical intensity |E(x, 0, z)|2 distributions in the x–z plane for the case of l = 15.1 nm (k 0 l = 0.15)

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Fig. 4
figure 4

a Distributions of optical intensity along an axis parallel to the x-axis |E(x, 0, given)|2 on the planes indicated by the inset for rod-length l = 50.4 nm. b Optical intensity |E(x, y, given)|2 distributions in the x–y plane for the case of z = 15.1 nm. This position corresponds to curve (5) in a. The numbers on the curves correspond to those shown in the inset. The boundary of the nano-rod exists at x = 2.5 nm (k 0 x = 0.025) shown by the dotted line in a

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5 Interference among enhanced local optical fields

It was previously shown that the dark spots of the first kind (Fig. 2b–f) occur due to destructive interference between two enhanced local fields at the probe tip and the rod end [14]. Similarly, the dark spots of the second kind are also generated by interference among enhanced local fields. The intensity distribution |E(x, δ, z)|2 in free space beside the rod for the case of l = 50.4 nm (k 0 l = 0.5) is shown in Fig. 5a. A strongly enhanced local field can be seen at three points, i.e., the probe tip, the midpoint of the rod and the rod end. The vector characteristics of the electric field in the x–z plane Re[E(x, 0, z)] near the rod is shown in Fig. 5b. The regions of the dark spots are shown by the dotted red ellipses in Fig. 5b. It can be seen that the dark spot is generated by vector cancellation due to phase differences among enhanced local fields at three points. These phase differences are caused by the SPP guided mode along the nano-rod. It is possible to calculate the propagation constant K of the guided mode along the nano-rod by approximating the SPP rectangular waveguide shown in Fig. 2b or c by an SPP cylindrical waveguide. For a metallic cylinder with a permittivity ε1/ε0 = −13.8 − j1.08 (Au) and a radius given by a = 2.82 nm (k 0 a = 0.028), which has the same area as the cross section δ × δ, the result is K = (11.7−j0.666) × k 0. The phase constant of 11.7 × k 0 agrees with that calculated from the intensity distribution along the rod shown in Fig. 5a, which represents a standing wave. Therefore, dark spots of the second kind are also generated by interference among the three enhanced local fields.

Fig. 5
figure 5

a Optical intensity |E(x, δ, z)|2 distribution on a linear flat scale in the x–z plane for the case of l = 50.4 nm. b Vector characteristics of the electric field Re[E(x, 0, z)] in the x–z plane for the case of l = 50.4 nm. The regions surrounded by the red ellipse indicate the regions of the dark spots. Notice that the dark spots are rotationally symmetric around the z-axis

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6 Conclusions

The basic characteristics of 3D dark spots generated by a metal-coated conical dielectric probe with a nano-rod were investigated numerically. It was found that two kinds of dark spots are generated; the one generated above the rod end is disk shape and the other generated around the nano-rod is toroidal shape. These dark spots are perfectly surrounded by a light shell. Since the dark spots considered in this study occur due to interference among enhanced near fields, they are nanosized and are generated in close vicinity to the object. Their intensity changes extremely rapid when compared to conventional diffraction limited optics. It is possible to apply these phenomena to optical atom trapping. However, for optical atom trapping, accurate calculations of the intensity distributions inside the dark spots are required. Unfortunately, it is difficult to perform such calculations at this point due to the limited ability of our system. It is expected that it will be possible to carry out more accurate calculations using a smaller mesh size in the near future due to recent developments in computer simulation techniques.