1. Introduction

The exploitation of sub-wavelength apertures in opaque films for light confinement into the subwavelength domain has been limited due to the negligible transmission efficiency of very small apertures. It has been recently noted [1–5], however, that cylindrical surface corrugations around a single, central, sub-wavelength-sized hole in a metal film can greatly enhance the otherwise weak transmission through the hole. This enhanced transmission is widely attributed to the influence of surface plasmon polaritons (SPPs) excited by the surface corrugations [1,2,3,4,6]. This effect can be explained also in terms of the diffracted evanescent wave model, proposed by Lezec et al. [5], in which the surface waves excited by the surface corrugations interfere with the incident plane wave that directly impinges on the aperture. According to this latter model, the phase difference between the incident plane wave and the surface waves determines the transmission maxima and minima.

Zakharian et al. [7] have developed an intuitive description governing light transmission through elliptical apertures in planar metal films. They concluded that achieving large transmission efficiency requires an aperture structure that can excite strong oscillator(s) on the upper surface of the metal film, which would then induce strong oscillations on the lower film surface in the neighborhood of the aperture. They also noted that the ability of a hole to support a guided mode, that can be excited by the incident polarization, appears to be critical for achieving large transmission, especially for thicker films. The latter observation agrees with results of our previous paper [8], where we reported that that by filling a sub-wavelength, cylindrical hole with a high index dielectric medium, the transmission efficiency of the hole can be increased over two orders of magnitude. We observed that the high index dielectric filling: (1) enables formation of the hybrid HE11 waveguide mode exhibiting a complex propagation constant, the imaginary part of which is relatively small; (2) transforms the hole into a Fabry-Pérot resonator, additionally boosting the light transmission through the sub-wavelength hole. To our best knowledge, this paper investigates for the first time a novel structure that combines high index dielectric aperture filling and concentric periodic surface corrugations around the aperture.

This paper is organized as follows. First, we analytically study the propagation constant (β) of guided modes that are supported by an infinitely long cylindrical, dielectric-filled hole in a silver medium. Then we study transmission properties of a sub-wavelength aperture, 50nm in radius, in a thick metal film versus the thickness of the metal film and the refractive index of the aperture. Finally, the influence of surface grooves on transmission efficiency is investigated in detail.

2. Light transmission through a single cylindrical aperture in a planar silver film

Throughout this paper, we consider a cylindrical sub-wavelength aperture, 50nm in radius, in a silver film (n_Ag=0.055+4.44i [9]) that is illuminated by a normally incident, linearly polarized plane wave having free space wavelength (λ ₀) of 650nm. First, we shall consider a cylindrical aperture that has infinite length along its axis (propagation direction of the incident wave). Such an aperture, which can be considered as a cylindrical waveguide having a silver cladding, can be characterized by examining the complex propagation constant (β=β _re+β _re i) of the hybrid HE₁₁ waveguide mode [9]. Figure 1 shows β of the HE₁₁ mode for the aforementioned cylindrical waveguide as a function of the refractive index of the waveguide core (n_c). Interestingly, the imaginary part of β starts to strongly decrease, while the real part exhibits a simultaneous increase, when n_c>2.2. The imaginary part attains its minimum when n_c=2.77, after which it starts to slowly increase. These results are obtained via standard waveguide theories with no weak guiding assumption [10,11].

Figure 2 shows the transmission efficiency (η) of a normally incident plane wave through a 50nm radius cylindrical hole in a free standing silver film as a function of the Ag-film thickness (t) for three different refractive indexes (n_c) of the hole: n_c=1.0 (solid line), n_c=2.4 (dashed line), n_c=2.8 (dashed-dotted line). We consider only completely opaque silver films, i.e., t≥100nm. The transmission efficiency is computed throughout this paper as a ratio of the total power emanating from the geometrical exit (lower film surface) aperture of the hole and the incident power falling into the geometrical entrance (upper film surface) aperture of the hole. The results are obtained via the BOR-FDTD method [12], in which the Yee cell size is chosen to be 2.5nm both in the radial and in the longitudinal z direction. The computation domain is terminated by uniaxial perfectly matched layers (UPML) [12], which provide highly absorbing boundary conditions. We observe that by filling the hole by a high index medium, η can be enhanced over a factor of 500. It is also seen that η exhibits a clear Fabry-Pérot kind of behavior; transmission maxima and minima appear periodically with thickness period of π/β_re. A similar behavior has been observed also for single slits in metal films [13,14].

 

Fig. 1. Complex propagation constant (β=β _re+β _im i) of the HE₁₁ mode as a function of the refractive index of the waveguide core (n_c) for an infinitely long cylindrical waveguide having a silver cladding and a core radius of 50nm. k ₀=2π/λ ₀, λ ₀=650nm, and n_Ag=0.055+4.44i.

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Fig. 2. Transmission efficiency (η) of a normally incident linearly polarized plane wave through a sub-wavelength (r=50nm), cylindrical dielectric hole in a silver film as a function of the film thickness (t) for different refractive indexes of the hole: n_c=1.0 (red line), n_c=2.4 (blue), n_c=2.8 (green). The refractive indexes of the incident and the exit medium are denoted by n₀ and n₁, respectively. n₀=n₁=1.0, n_Ag=0.055+4.44i, λ ₀=650nm. The solid line is magnified by a factor of 500.

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Fig. 3. (a) Cross-section of the modeled structure: A sub-wavelength hole (r=50nm) in a silver film (t=515nm) that is filled by a high index dielectric medium (n_c=2.8) is surrounded by a single groove having depth h and width w. (b) Transmission efficiency (η) of a normally incident, linearly polarized plane wave through the sub-wavelength hole shown in (a) as a function of the groove depth (h) and the distance (s ₁) between the central hole and the groove. In all cases, w=300nm, λ ₀=650nm, and n_Ag=0.055+4.44i.

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3. Light transmission through a single aperture in a corrugated silver film

We are now in a position to explore transmission properties of a single, high index, subwavelength (n_c=2.8, r=50nm) hole in a thick silver film surrounded by concentric circular grooves. First, we study the influence of a single circular surface groove on transmission efficiency as a function of the groove depth (h) and distance from the central hole (s ₁), see Fig. 3(a). To investigate relatively deep grooves with no light leakage through them, we consider a 515nm thick silver film. This thickness also provides resonant transmission trough the aperture according the results of Fig. 2. The groove width (w) is a fixed parameter, having value of 300nm, which is a common value used in recent experiments [4, 5]. Figure 3(b) shows the transmission efficiency (η) for a central hole that is surrounded by a single circular groove as a function of s ₁ and h. Figure 3(b) is obtained by linearly interpolating the results of simulations in which s ₁ is varied from 50 to 1000nm in 50nm steps and h is varied from 20 to 400nm in 20nm steps. Clearly, transmission efficiency depends periodically both on s ₁ and h. The periodic dependency on s ₁ is due to the phase difference (at the entrance of the central hole) between the incident plane wave and the surface wave excited by the concentric circular groove. This result is in accordance with the predictions of the diffracted evanescent wave model [5]. The periodic dependency on h is due to interference occurring within the groove. The maximum transmission efficiency also increases as a function of s ₁. This is due to fact that the length of the groove’s circular contour, which excites surface waves, increases linearly with s 1. However, when the groove is far enough from the central hole, the transmission efficiency starts to decrease due to attenuation of the surface waves.

 

Fig. 4. Transmission efficiency (η) of a normally incident, linearly polarized plane wave through a sub-wavelength hole (r=50nm, n_c=2.8) in a free standing silver film (t=180nm) when the central hole is surrounded by (a) a single groove, (b) two grooves (s ₁=300nm), and (c) three grooves (s ₁=s ₂=300nm). (d) Transmission efficiency as a function of the number of grooves when the radial distance (s_n) between the adjacent grooves is 300nm. In all cases, w=300nm, h=120nm, λ ₀=650nm, and n_Ag=0.055+4.44i.

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Figures 4(a)–(c) show results for the cases where the central hole is surrounded first by a single groove and then by two and three grooves. The metal film is now 180nm thick and groove is 120nm deep and 300nm wide. When s ₁=550nm in Fig. 4(a), light transmission through the central hole is almost zero. This is due to the fact that approximately equal power, with opposite phases, couples from the incident plane wave and the surface wave into the central hole, resulting in nearly zero transmission. In Figure 4(b), the distance between the first groove and the central hole (s ₁) is set to 300nm, which is the location of the first maximum in Fig. 4(a). It is seen that the transmission efficiency exhibits increased maxima and that the locations of the maxima and minima depend on the distance between the first and the second groove (s ₂). This occurs due to the same physical phenomena observed in the single groove case. The first maximum appears in Fig. 4(b) at s ₂=300nm, which is used as a distance between the first and the second groove in Fig. 4(c). Figure 4(d) shows how the transmission efficiency depends on the number of grooves (n) when the radial distance between the adjacent grooves is 300nm. The transmission efficiency increases until n=13 when η is about five orders of magnitude higher than η of a single aperture without the dielectric filling.

4. Summary and conclusion

This paper investigated for the first time transmission of a normally incident, linearly polarized plane wave through a high index dielectric-filled hole surrounded by surface corrugations. The dependency of transmission efficiency on a single 300nm wide groove was studied versus the groove depth and its distance from the central hole. It was observed that the phase difference between the incident plane wave and the surface waves determines the occurrence of transmission maxima and minima. A similar behavior was also observed with several grooves. When the central hole is surrounded by 13 grooves, the transmission efficiency is increased by five orders of magnitude compared to a single aperture without the dielectric filling.