1. Introduction

Modern optical technologies such as near-field scanning optical microscopy, optical lithography, heat-assisted magnetic recording and optical data storage demand efficient near-field light sources that provide light confinement well beyond the fundamental diffraction limit. In theory, the fundamental diffraction limit can be exceeded – in a brute force fashion – by spatially limiting the extent of an incident wavefront by imposing a very small transmitting aperture in the path of an incident light beam. In practice, however, this approach suffers from the extremely small light transmission efficiency through the aperture. According to the Bethe’s law [1], the transmission coefficient of a sub-wavelength circular hole, which is defined as a ratio of the transmitted power through the hole to the incident power falling into the entrance pupil of the hole, in an infinitely thin perfectly conduction film is proportional to (R/λ₀)⁴, where R is the aperture radius and λ₀ is the free space wavelength. Due to the strong assumptions of Bethe’s law, it only superficially describes the transmission properties of a sub-wavelength hole. Analysis of practical physical realizations of small apertures requires rigorous electromagnetic modelling tools such as the finite difference time domain (FDTD) method [2,3], multiple multi-pole technique [4,5] or the finite element method [6,7], which can take into account the finite conductivity as well as the finite thickness of the metal film.

The transmission properties of sub-wavelength apertures have recently been studied by several authors [3,5,8]. For example, Wannemacher [5] has studied the fundamental characteristics of a single circular hole in thin metal films, as well as the role played by surface plasmon polaritons in the transmission process, while Zakharian et al. [3] have been developing an intuitive description of the behavior of electromagnetic fields in elliptical apertures. García de Abajo [8] has investigated light transmission through simple circular holes in perfectly conducting thin films and holes containing additional structure such as sphere or a high index dielectric filling, which seems to improve the transmission efficiency at specific wavelengths. Enhanced transmission through cylindrical holes in metal films surrounded by periodic surface corrugations has also gained lots of interest [9,10] in recent years. These transmission enhancements are attributed to the influence of surface plasmon polaritons, which are excited by the surface corrugations [9,10].

In this paper, we explore the characteristic properties of single sub-wavelength holes in metal films filled by a high index dielectric medium. We begin our analysis by analytically investigating the propagation constant (β) of guided modes that are supported by infinitely long cylindrical holes in a silver medium. Then, we study the dependency of β on the refractive index of the hole. Next, the body-of-revolution finite difference time domain (BOR-FDTD) method is employed to analyze the light transmission through sub-wavelength holes in a metallic (conductive) medium, when such holes are filled by a high index dielectric medium. The light transmission of these holes is compared to the transmission of similar nonfilled holes in the identical medium. We find that the high index dielectric filling improves the transmission efficiency by over two orders of magnitude and that the sub-wavelength cylindrical holes exhibit a Fabry-Pérot-like interference behaviour, similar to the transmission exhibited by two-dimensional metallic slits [13] under TM polarized illumination.

2. Numerical simulation model

The modeled geometry is schematically depicted in Fig. 1. A linearly polarized plane wave having free space wavelength of 488nm normally impinges on a cylindrical hole of radius r in laterally infinite silver film (n _Ag = 0.05 + 3.02j) of thickness t. The refractive index of silver was interpolated from the experimental data presented by Johnson et al. [11]. The symbols n _c, n ₀ and n ₁ denote the refractive indexes of the hole, incident medium and the exit medium, respectively. In this study, we consider only free standing structures, i.e., n ₀ = n ₁ = 1.0.

 

Fig. 1. Illustration of the modeled system. A cylindrical hole of radius r in a laterally infinite silver film of thickness t is illuminated by a normally incident, linearly polarized plane wave. The refractive indices of the hole, silver film, incident medium and the exit medium are denoted by n _c, n _Ag, n ₀, and n ₁, respectively.

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The light interaction with the metallic hole is modeled by the BOR-FDTD method [2], which assumes that the physical structure to be modeled is cylindrically symmetric. The BOR-FDTD method represents the electric and magnetic field components in cylindrical coordinates (ρ, ϕ, z), where ρ, ϕ, and z have their conventional meanings. Due to fact that, in the cylindrical coordinates, the electric and magnetic field components are periodic in ϕ, we can represent them as complex Fourier series:

The complex amplitudes of E _m (ρ,z) and H _m (ρ,z) are are solved in a two-dimensional mesh, which is a significant advantage compared to the three-dimensional FDTD method. As an additional advantage, the BOR-FDTD method resolves exactly the cylindrical shape of the structure. In the series expansions (1) and (2), the mode number m goes from zero to infinity. In practice, the incident field defines the number of modes that must be solved. For example, problems in which the incident field is a linearly polarized plane wave, or a Gaussian beam, or a hybrid HE_1n mode of an optical fiber, require only the solution of the mode m = 1. On the other hand, a radially polarized Bessel beam or the TM₀₁ mode of an optical fiber requires only the solution of the mode m = 0. The computation domain is terminated using uniaxial perfectly matched layer (UPML) absorbing boundary condition [2], and materials in which the real part of the refractive index is smaller than the imaginary part, as is the case with noble metals at optical frequencies, are modeled via the Lorentz dispersion model based on the auxiliary differential equation [12].

In this study, we model only the interaction of a normally incident, linearly polarized plane wave with a sub-wavelength hole in a metal film that has infinite extent in lateral directions. The laterally infinite metal film is accomplished by terminating the metal film directly by UPML. The incident plane wave is then excited by the TF/SF technique [2], which is modified to enable modelling of the problem as follows: Since the background structure in which the sub-wavelength hole resides is invariant in the x and y directions, propagation of normally incident plane waves through the background structure can be modelled using a one-dimensional FDTD simulation. This one-dimensional simulation, which is run simultaneously with the actual BOR-FDTD simulation, is then used as a look-up table that defines the incident field for the TF/SF technique. The use of look-up tables with the TF/SF technique is described in detail in Ref. [2].

To obtain accurate results with the BOR-FDTD method, the two-dimensional BOR-FDTD mesh has to be adequately sampled. In practice, a mesh with at least 30 mesh points per wavelength is required to keep numerical dispersion errors under control [2]. In this study, we use a uniform mesh with Δ;ρ = Δz = r/20, where r is the radius of the cylindrical hole (see Fig. 1), and Δρ and Δz are the mesh space increments in the ρ and z directions, respectively. With such a sampling, we obtain accurate results for film thicknesses starting from 10Δz to tens of micrometers.

 

Fig. 2. Complex propagation constant (β = β_re + jβ_im) of guided HE_1n modes (n = 1: solid line, n = 2: dashed line, n = 3: dotted-dashed line) as a function of the hole radius (r) for an infinitely long air hole in a silver medium. λ₀ = 488nm, k₀ = 2π/λ ₀ .

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3. Results

When a normally incident plane wave interacts with a cylindrical hole in a metal film, it can excite only hybrid HE_1n (n ≥ 1) waveguide modes, in which E_z ≠ 0 and H_z ≠ 0. This is due to the fact that only the HE_1n modes have the same angular dependency (in the azimuthal angle θ) as the linearly polarized plane wave propagating in the z direction. The complex propagation constants (β) of the HE_1n modes can be solved in the complex β domain using standard theories of dielectric optical fibers [14] without the weakly guiding assumption.

First, we shall consider the complex propagation constant of the HE_1n modes of an infinitely long air-filled hole in a silver medium as a function of the hole radius (r) at the wavelength of 488nm. The numerically obtained results are shown in Fig. 2. It is observed that when r ≤ λ₀, the silver waveguide supports only the HE₁₁, HE₁₂ and HE₁₃ modes with the effective cut-off radii of 0.21λ₀, 0.56λ₀ and 0.8λ₀, respectively. Second, we shall describe the dependency of the propagation wave vector β on the refractive index of the hole (n _c). In Fig.3, we show β of the HE₁₁ mode as a function the hole’s radius for three different refractive indices of the medium within the hole: n _c = 1.0, 2.0, and 3.0. We observe that by filling the hole by a high index dielectric medium, the cut-off radii of the HE₁₁ mode can be significantly reduced. When n _c = 3.0, the imaginary part of β has rather interesting behaviour: It has a local minimum around 0.05λ₀ after which β is an increasing function of hole radius. Now, one might think that there is no light transmission through a metal-clad circular waveguides for which, e.g, n _c = 3.0 and r = λ₀ due to the large value of the imaginary part of the waveguide’s β. However, this is not the case. Light propagation can occur via higher order HE_1n modes, which are not shown in Fig. 3. The just mentioned peculiar behaviour of the HE₁₁ mode, when n _c = 3.0, is probably due to fact that the absolute values of the real parts of the dielectric constants of silver and the hole are converging to each other, i.e., |Re{$n_{\mathrm{c}}^2$}|→|Re{${n_{\text{Ag}}}^2$} |.

 

Fig 3. Complex propagation constant (β = β_re + jβ_im) of guided HE₁₁ modes as a function of the hole radius (r) for an infinitely long dielectric-filled (n _c = 1.0: solid line, n _c = 2.0: dashed line, n _c = 3.0: dashed-dotted line) hole in a silver medium. λ₀ = 488nm, k₀ = 2π /λ₀.

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For the rest of this study, we shall restrict ourselves to two distinct hole radii: r = 25nm and r = 50nm. Figure 4 shows β for the HE₁₁ mode as a function of n _c for the selected hole radii. We stress that these radii do not support any other HE_1n modes. We find that the imaginary part of β starts to decrease, while the real part exhibits a simultaneous increase, when n _c becomes larger than 2.2 in the case of r = 25nm and larger than 1.6 with r = 50nm. The imaginary part of β gains its minimum value when n _c ≈ 2.5 (with r = 25 nm), or when n _c ≈ 2.0 (with r = 50nm). From the view point of light transmission through a sub-wavelength hole, we would expect the maximum transmission to occur when the imaginary part of the wave vector of the HE₁₁ mode is minimized. To verify this expectation, we next utilize the BOR-FDTD method to predict how the transmission efficiency of a sub-wavelength hole in a finitely thick silver medium depends on the refractive index of the hole.

 

Fig. 4. Complex propagation constant (β = β_re + jβ_im) of guided HE11 mode of a dielectric hole in an infinitely thick silver medium as a function of the refractive index of the hole (n _c) for two different hole radii: r = 50nm (solid line) and r = 25nm (dashed line).

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Figures 5 and 6 show the transmission efficiency (η) of a dielectric-filled hole in a metal film, versus thickness of the film, with different refractive indexes of the hole (n _c) when the hole radius r = 25nm and r = 50nm, respectively. The transmission efficiency is defined herein as a ratio of the transmitted power behind the aperture (hole) and the incident power at the entrance of the aperture. The incident and the transmitted power are computed from the z component of the incident and the transmitted time-averaged Poynting vectors, respectively, with the difference that the incident power is integrated over the aperture entrance while the transmitted power is integrated over the geometrical exit of the aperture. We observe that when r = 25nm, the high index dielectric filling enhances the transmission efficiency over two orders of magnitude. In the case of r = 50nm, the enhancement is in order of 30. It is also found, as expected, that the transmission efficiency is maximized when the imaginary part of β of the HE₁₁ mode exhibits its minimum value. Further, when the refractive indexes of the dielectric filling provide high transmission efficiency, the transmission efficiency exhibit Fabry-Pérot-like behaviour (versus film thickness), similar to the case of two-dimensional metallic slits [13] under TM polarized illumination. That is, the transmission maxima and minima appear periodically versus film thickness t, with a thickness period of π/β_re, where β_re is the real part of the propagation constant of the guided HE₁₁ mode.

 

Fig. 5. Transmission efficiency (η) of a dielectric hole in a silver film versus thickness t of the film. The silver film is illuminated by a normally incident, linearly polarized plane wave. The medium within the hole has refractive index n _c and the hole’s radius r = 25nm.

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Fig. 6. Same as Fig. 5 but with r = 50nm.

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We have also investigated cases in which the incident and the exit media are identical to that dielectric medium which fill the hole, i.e., n ₀ = n ₁ = n _c (see Fig. 1). We observed that the transmission efficiency reduces significantly and also the η versus thickness curve is less modulated. This implies that the reflecting entrance and exit interfaces of the dielectric-filled hole form a Fabry-Pérot resonator, which additionally boosts light transmission through the hole. The fact that light propagates several round-trips in the resonator magnifies the importance of the small imaginary part of the propagation constant.

 

Fig. 7. Complex propagation constant (β = β_re + jβ_im) of guided TM_0n modes (n = 1: solid line, n = 2: dashed line) as a function of the hole radius (r) for an infinitely long ai-filled hole in a silver medium. λ₀ = 488nm, k₀ = 2π/λ₀.

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Finally, we shall consider characteristic properties of the TM_0n modes. When a circular aperture is illuminated by a radially polarized beam, for example by a Bessel beam, that is focused on the central point of the aperture, the incident field can excite the TM_0n (n ≥ 1) mode of a cylindrical metallic waveguide. Figure 7 shows the complex propagation constant of the TM₀₁ and TM₀₂ modes as a function of hole radius for an air-filled hole in an infinitely thick silver medium at the wavelength of 488nm. We note that when r ≤ λ₀, the hole does not support any other TM_0n modes. It is seen that the effective cutoff radius is 033λ₀ for the TM⁰¹ mode and 0.83λ₀ for the TM₀₂ mode, which are significantly larger than the cutoff radii of the corresponding HE_1n modes. The physical explanation for this may be as follows: Under a radially polarized illumination, the surface charges generated by the incident field are in the same phase over the entire contour of the aperture edge, whereas under a linearly polarized illumination, the surface charges are in opposite phases at opposite sides of the aperture. Since the electric field can span directly only from a positive charge to a negative charge, it may explain why the HE_1n modes exhibit smaller cutoff radii than the TM_0n modes. This also indicates that the TM₀₁ mode can never have a zero cutoff radius. Figure 8 shows a comparison between the propagation constants of the TM₀₁ and HE₁₁ modes when the refractive index of the hole is 2.0. Also in this case, the HE₁₁ mode exhibits a smaller cutoff radius than the TM₀₁ mode.

 

Fig. 8. Comparison between the complex propagation constants (β = β_re + jβ_im) of guided HE₁₁ (solid line) and TM₀₁ (dashed line) modes as a function of the hole radius (r) for a dielectric-filled hole (n _c = 2) in an silver medium. The inset illustrates the electric field distributions of HE₁₁ and TM₀₁ modes.

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4. Summary and conclusion

In this paper, we have shown that the propagation characteristics of sub-wavelength holes in metal films illuminated by a normally incident, linearly polarized plane wave can be predicted by investigating the complex propagation constants of the hybrid HE₁₁ mode. We observed that light transmission through sub-wavelength holes can be remarkably enhanced by filling the hole with a high index dielectric medium. The high-index dielectric filling enables the formation of a guided HE₁₁ mode which has a relatively small β_im. Further, it creates a Fabry-Pérot-like resonator which provides additional improvement of light transmission through the hole. The fact that light propagates several round-trips in the resonator magnifies the importance of the small imaginary part of the propagation constant.

This study provides results only for one incident wavelength (λ₀ = 488nm), but the idea of high index dielectric filling is also valid at longer wavelengths. For example, crystalline silicon, which has the refractive index of 3.8+0.016j@650nm [15], could be used as a filler for cylindrical 40nm diameter holes in a Ag medium. A structure, which has even higher transmission efficiency, can be constructed by combining the idea of a high index dielectric filling with circular surface corrugations that surrounds the central hole. This will be considered in detail in our forthcoming paper.