When polarized light passes through a sub-skin-depth metal, a nonclassical phenomenon is observed at the metal-dielectric interface, such as a metal exposed to air or a metal thin film coated on a dielectric. Surface plasmons are generated by the redistribution of free electrons in the metal by an external field at the metal-dielectric interface. As a result, the field distribution in the metal changes, leading to increased light transmittance and a beaming effect[1-9].

Active research has been conducted on metalenses for use as verstile multifunctional photonic components[10] and the design of metasurfaces for ultrathin, ultralight, and planar multifunction[11], including high-quality imaging by overcoming the diffraction limit and correcting aberrations[12]. Furthermore, on genetic algorithms to automate the metasurfaces design for absorption bandwidth broadening[13], and the application of metasurfaces to electromagnetics[14] has been reported. When light passes through a medium composed of a metal array of sub-skin-depth size, more complex effects appear. In a medium composed of metal plates, fields induced on the metal surface get coupled[15-17]. Additionally, in the metal array structure, multiple interactions occur between the light scattered in the process of passing through the medium and the induced fields around the metal. Interactions between gold nanoparticles[18] and quantification of the interactions between metal particles[19] have been published in previous studies.

This study investigates the optical properties in a medium composed of sub-skin-depth metal arrays through three-dimensional (3D) simulation. We analyzed the light intensity distribution in the metal plate medium with a thickness less than the skin depth. In addition, the characteristics of light passing through the metal cylinder array medium were comparatively analyzed with those of the metal plate. The validity of the simulation results was confirmed by comparing them with the results of previous studies that analyzed space and mesh size for the plate structure[16]. Furthermore, in this study, we analyzed the beam characteristics in a cylindrical metal array structure, which is a two-dimensional isotropic medium structure (with respect to two mutually perpendicular axes). We also discuss the results for the spherical metal array, which is a three-dimensional isotropic structure.

The metal plate and cylinder analyzed in this study were made of Au. The skin depth of Au is 25 nm, and the relative permittivity is ϵ=−11.74+1.2611i at an incident wavelength 632.8 nm. The Lorentz-Drude model was applied to obtain the metal properties. The incident light had a Gaussian distribution with a central wavelength of 632.8 nm and a half-width of 265 nm, and was polarized along the x-axis.

The simulation analysis area and mesh size are directly related to the computation time. The computation time decreases as the mesh size increases. However, if the mesh size is larger than the appropriate size, the nano effect may not be included in the calculation result. Therefore, determining the optimum mesh size at an appropriate level is very important in nano-optical research. In this study, considering the analysis time, the simulation space was set to 1.0 μm × 1.0 μm × 2.2 μm, and the FDTD simulation mesh spacing was set to 5 nm in order to include the nanostructure effect in the simulation results[16]. In addition, boundary conditions were set such that all electric and magnetic fields reaching the simulation interface were absorbed. Even when light propagates through empty space, it diverges and the intensity is reduced. In the simulation, absorption at domain boundaries reduces the intensity measured by the detector.

The graph in Fig. 1 shows the intensity measured at the output plane z = 2.19 μm for light emitted from a light source at the origin and propagating through the medium. In the inset of Fig. 1, the maximum intensities of incident and outgoing light are normalized to 1. The incident light has a Gaussian distribution with a single wavelength of 632.8 nm. The transmitted light has the same Gaussian distribution as the input light. The output (dashed line) has a very weak intensity compared to the input (solid line), and the line width is very large. Therefore, it is observed that the beam width widens as the light propagates.

Figure 1. (Color online) Light intensity distribution along the x-axis.

1. Metal plate array medium

The thickness of the nano-sized metal platesand the gap between them were 20 nm and 30 nm, respectively. The medium consists of 21 metal plates aligned in the z-direction, with air layers in between. The metal plates are made of Au, and their thickness is less than the skin depth. The incident light propagates from the left end of the medium (z=0) to the right. When light polarized in the x-direction is incident along the z-direction on the nano-sized metal plates, surface plasmons are generated on the metal surface. This redistributed field is transmitted to the opposite metal surface as the thickness is less than the skin depth of Au and affects the intensity distribution of the field in the medium by interacting with the field of the adjacent metal surface.

Figure 2 illustrates the simulated intensity distribution in the x-z plane. In the absence of a metal plate, the light beam gradually spreads, and the width of the beam increases as shown in Fig. 1. However, in a medium with a metal plate, the intensity is focused around z= 1.10–1.20 μm and then spreads again.

Figure 2. (Color online) X-direction electric field component, E_x intensity distribution for metal plate medium on the x-z plane.

The field intensity distribution in the x-y plane was investigated. Figure 3 shows the distribution of the Pointing S_z in the x-y plane. Figure 3(a) and 3(d) depict the incident plane and the exit plane, respectively, while Fig. 3(b) and 3(c) correspond to the plane inside the medium. In Fig. 3(c), the Pointing distribution shows a wide spread in the y-direction, while there is relatively little spread in the x-direction. By comparing the incident light distribution with Fig. 3(a), it is possible to evaluate the focus in the x-direction.

Figure 3. (Color online) S_z distribution on x-y plane (a) z = 0.01 μm (b) z = 0.40 μm (c) z = 1.15 μm (d) z = 2.19 μm.

Figure 4 shows the x-direction line profile of Fig. 3(a) to 3(d). The solid lines of the Gaussian-shaped distribution correspond to Fig. 3(a) and 3(d), while the gear-shaped lines correspond to Fig. 3(b) and 3(c). The gear-shaped line distribution is a result of the metal plate effect. It can be observed that, compared to the red gear-shaped distribution, the blue gear-shaped distribution has a narrower envelope width at both edges (ellipse in Fig. 4). Therefore, the beam width is reduced in the inner space of the medium in Fig. 3.

Figure 4. (Color online) S_z distribution along the x-axis.

Also, upon comparing the input and output distributions, i.e., the black dotted line and the green solid line, it can be observed that the widths of the two distributions are similar. Considering that light spreads as it passes through empty space, as shown in Fig. 1, it can be inferred that there is almost no beam spread even after the beam passes through the medium. This is because, as demonstrated in Fig. 2, the spread is minimal due to the focusing phenomenon, in which light is collected while passing through the metal plate medium, was minimal. By utilizing the focusing phenomenon, the diffraction of light, which is responsible for the image blur in imaging optical systems, can be controlled. The resolution of an image is limited by the diffraction of the optical system, which is commonly known as the diffraction limit. However, if this limit can be overcome, it has the potential to significantly improve image resolution and play a crucial role in microstructure analysis.

Let us consider a case where light is incident at +45 degrees with respect to the horizontal direction. Refracted light at the interface between the metal plate and the air must travel upwards above the horizontal line after being refracted according to Snell's law. However, as shown in Fig. 5, it proceeds below the horizontal line after refraction. That is, it was negatively refracted as it passed through the metal plate. The focusing phenomenon in Fig. 2 is explained by negative refraction, i.e., light traveling upward is refracted downward at the air-medium interface. On the other hand, light traveling downward is refracted upward and focused with in the medium.

Figure 5. (Color online) H_y intensity distribution for metal plate medium on the x-z plane for oblique incident beam.

The amplitude of field a_n of multiple waveguides satisfies the differential equation for interaction.

In this model, the field induced on each metal plate interacts only with the field of its adjacent plate, and the coupling coefficient is C. For plates on the edge, either an−1 or an+1 is zero. The coefficient D is related to the potential applied to the plate, but in this study, D was set to zero since no potential was applied. The transverse velocity of the field can be calculated using these coefficients.

κ is the product of wavenumber and period. In general, C is positive, but when it is negative, light is negatively refracted as shown in Fig. 5.

2. Cylinder-shaped metal array medium

The cylinder-shaped array elements (Au) analyzed in this study are lined up along the y-axis. The diameter and spacing of the cylinders are both 20 nm, which is smaller than the Au skin depth, approximately 25 nm. The medium features a cylinder array structure, with 16 cylinders along the x-axis and 47 cylinders along the z-axis. When x-polarized light is vertically incident on the air-medium interface, the light intensity is symmetrical in the direction perpendicular to the optical axis (z-axis), as shown in Fig. 8(a). To confirm the optical behavior of the cylinder medium, the light intensity distribution on the x-y plane was analyzed for the input and output surfaces, as illustrated in Fig. 8(b) and (c), respectively.

Figure 8. (Color online) (a) E_x intensity distribution for metal cylinder array medium on the x-z plane (b) interaction between induced fields on adjacent plate surfaces (c) multiple interactions between the fields induced on the cylinder surface in the periphery.

Figure 7 shows the distribution of the Pointing S_z on the x- and y-axis for the input and output surfaces shown in Fig. 6(b) and (c), respectively. By analyzing the light intensity along the x-axis (which is perpendicular to the cylinders) in Fig. 7(a), it can be observed that the width of the pointing distribution on the output surface is narrower than that on the input surface. This indicates that the light does not spread while propagating in the medium, rather it is focused. However, as shown in Fig. 7(b), the intensity distribution on the output surface is wider in the y-axis direction (parallel to the cylinders).

Figure 6. (Color online) (a) E_x intensity distribution for metal cylinder array medium on the x-z plane (b) S_z on the input plane (z=0.01 μm) (c) S_z on the output plane (z=2.19 μm).

Figure 7. (Color online) S_z line profile for the input and output light intensity in the x-y plane (a) x-direction (b) y-direction.

In Fig. 8(a), the electric field intensity distribution in the x-z plane is shown when light is incident in the +x direction at 45 degrees, and negative refraction is observed in the cylinder medium. Furthermore, in Fig. 8(a), the phenomenon of splitting of the incident light into two streams is observed. Both streams of light are refracted and propagate in the (-)x direction, resulting in negative refraction.

As compared to a medium with a plate structure, light propagates differently in a cylinder-shaped medium. Figure 8(b) and (c) show the cross-sectional structures of the metal plate medium and the metal cylinder medium, respectively. The metal plate medium in Fig. 8(b) has a structure in which metal plates and air layers of a certain thickness are alternately arranged periodically. Since it is a closed structure in the longitudinal direction, only interaction with adjacent plates is possible. On the other hand, the cylinder medium has a periodic structure in both the lateral and longitudinal directions as shown in Fig. 8(c), and there is no blockage. Therefore, multiple interactions are possible in the lateral, longitudinal, and diagonal directions in the cylinder structure. This difference in structure causes a difference in the light intensity distribution between the plate and cylinder structures.

The amplitude of field a_m,n after multiple interactions in the metal cylinder medium is

The second term in the equation above consists of nine items. The coefficient C_m,n represents the potential applied to the cylinder a_m,n. The remaining eight terms Cm±1,n±1 are the interaction coefficients with the surrounding cylinders. Here, the interaction coefficient Cm±1,n includes the interaction coefficient for the metal plate case in Eq. (1). If we assume that the interaction has no directional bias, Cα,β=Cβ,α is satisfied. The induced field in each cylinder is determined by multiple interactions. The field strength at any point in the medium is determined by two factors: the field induced in each cylinder, determined by the above equation, and the diffraction of light passing between cylinders. Therefore, there are 26 induced sources (lateral cylinders) and 25 sources (spaces between cylinders) present in the cylinder medium. In addition, the field strength at each point is determined by the superposition of the sources corresponding to at least three interacting layers.

Figure 9(a) shows the distribution of the x-y plane strength of the output, and Fig. 9(b) shows the corresponding intensity distribution along the x-axis. Beam splitting occurs in the cylinder structure, and the main (high and wide distribution) intensity distribution is narrower than that of the incident light.

Figure 9. (Color online) S_z distribution for cylinder array medium on output plane z=2.19 um (a) x-y plane (b) x-axis.

The birefringence observed in previous studies[20, 21, 22] is due to phase retardation caused by asymmetric structures. However, the beam splitting observed in this study is believed to be a multiple interference phenomenon resulting from the interaction between nano structures. This phenomenon does not occur in plate structures, where the space is closed and multiple interactions are subtracted. On the other hand, the light distribution for the spherical 3D array medium capable of multiple interactions was found to be very similar to that of the cylinder medium.

In this study, 3D simulations were used to investigate the light intensity distribution in sub-skin-depth metal arrays, optical properties of metal plate and cylinder array mediawere analyzed. Light incident perpendicular to the medium boundary was focused in all media, with light transmittance of 0.197 and 0.292 for plate and cylinder media, respectively. The light transmittance, ratio of output intensity to input, of the cylinder medium was 48.0% higher than that of the plate medium. The cylinder structure showed diffraction, similar to a diffraction grating, but without a decrease in transmittance due to the narrow tunnel. Further research is needed to understand the role of multiple interactions in increasing light transmittance. In all media analyzed, negative refraction occurred when light was incident obliquely on the interface. The cylinder medium also showed beam splitting due to multiple interactions of the metal structures. The optical characteristics on the x-z plane of the spherical metal array medium were found to be very similar to those of the cylinder medium.

Optical isotropy is a crucial property that widens the usability of a medium by giving the same optical result regardless of the incident angle of light. Negative refraction, which was earlier observed in plates, formed a one-dimensional interaction barrier. In contrast, the cylinder structure analyzed in this study provides a two-dimensional isotropic medium with respect to two mutually perpendicular axes, while the spherical metal array structure can ensure 3D optical isotropy. Complete 3D optical isotropy would require random spacing of the materials constituting the medium.