## Abstract

We investigate transmission of a normally incident, linearly polarized plane wave through a circular sub-wavelength hole in a metal film filled by a high index dielectric medium. We demonstrate for the first time that the transmission efficiency of such holes exhibits a Fabry-Pérot-like behaviour versus thickness of the metal film, similar to that exhibited by sub-wavelength slits in metal films illuminated by TM-polarized plane waves. We show that by reducing the imaginary part of the propagation constant of the hybrid HE_{11} mode and by fortifying the Fabry-Pérot resonance, the high index dielectric filling can greatly enhance light transmission through a circular sub-wavelength hole.

©2005 Optical Society of America

## 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*/λ_{0})^{4}, where *R* is the aperture radius and λ_{0} 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.

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 TM_{01} 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* = Δ*z* = *r*/20, where *r* is the radius of the cylindrical hole (see Fig. 1), and Δ*ρ* and Δ*z* 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 10Δ*z* to tens of micrometers.

## 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* ≤ λ_{0}, the silver waveguide supports only the HE_{11}, HE_{12} and HE_{13} modes with the effective cut-off radii of 0.21λ_{0}, 0.56λ_{0} and 0.8λ_{0}, 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 HE_{11} 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 HE_{11} 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 0.05λ_{0} 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* = λ_{0} 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 HE_{11} 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{${{n}_{\text{Ag}}}^{2}$} |.

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 HE_{11} 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 HE_{11} 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.

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 HE_{11} 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 HE_{11} mode.

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.

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 TM_{01} and TM_{02} 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* ≤ λ_{0}, the hole does not support any other TM_{0n} modes. It is seen that the effective cutoff radius is 033λ_{0} for the TM^{01} mode and 0.83λ_{0} for the TM_{02} 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 TM_{01} mode can never have a zero cutoff radius. Figure 8 shows a comparison between the propagation constants of the TM_{01} and HE_{11} modes when the refractive index of the hole is 2.0. Also in this case, the HE_{11} mode exhibits a smaller cutoff radius than the TM_{01} mode.

## 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 HE_{11} 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 HE_{11} 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 (λ_{0} = 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.

## References and links

**1. **H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

**2. **A. Taflove and S. C. Hagness, *Computational Electrodynamics: The Finite Difference Time Domain Method* (Second Edition, Artech House, Boston, 2000).

**3. **A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Transmission of light through small elliptical apertures,” Opt. Express **12**, 2631–2648 (2004),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2631. [CrossRef] [PubMed]

**4. **C. Hafner, *The Generalized Multipole Technique for Computational Electromagnetics* (Artech House, Boston, 1990).

**5. **R. Wannemacher, “Plasmon supported transmission of light through nanometric hole in metallic thin films,” Optics Comm. **195**, 107–118 (2001). [CrossRef]

**6. **J. Jin, *The Finite Element Method in Electromagnetics* (John Wiley & Sons, New York, 2002).

**7. **A. P. Hippins, J. R. Sambles, and C. R. Lawrence, “Gratingless enhanced microwave transmission through a subwavelength aperture in a thick metal plate,” Appl. Phys. Lett. **81**, 4661–4663 (2002). [CrossRef]

**8. **F. J. Garcia de Abajo, “Light transmission through a single cylindrical hole in a metallic film,” Opt. Express **10**, 1475–1484 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-25-1475. [PubMed]

**9. **T. Thio, K. M. Pellerin, and R. A. Linke, “Enhanced light transmission through a single subwavelength aperture,” Optics Lett. **26**, 1972–1974 (2001). [CrossRef]

**10. **A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express **12**, 3694–3700 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-16-3694. [CrossRef] [PubMed]

**11. **P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

**12. **M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple Treatment of Multi-Term Dispersion in FDTD,” IEEE Microwave Guided Wave Lett. **7**, 121–123 (1997). [CrossRef]

**13. **Y. Takakura, “Optical Resonance in a Narrow Slit in a Thick Metallic Screen,” Phys. Rev. Lett. **86**, 5601–5603 (2001). [CrossRef] [PubMed]

**14. **K. Okamoto, *Fundamentals of Optical Waveguides* (Academic Press, San Diego, 2000).

**15. **E. D. Palik, *Handbook of Optical Constants of Solids* (Academic Press, New York, 1985).