1. Introduction

Nanoscale apertures in a metal show astonishing optical properties leading to enhanced and selective light transmission [1, 2], and light confinement [3, 4]. Several structures are considered. Efficient light transmission through a single subwavelength aperture in a metallic film can be achieved by optical tunneling through the fundamental guided mode in the hole. The key role of the aperture geometry on the light transmission process has been demonstrated experimentally [5, 6, 7] and theoretically [8, 9, 10, 11]. The design of such subwavelength metal apertures is still a major challenge for the conception of optimal probes for scanning near-field optical microscopy [12]. Indeed light confinement in tiny apertures has allowed an impressive enhancement of single-molecule analysis by fluorescence detection in subwavelength holes [3, 13].

Enhanced light transmission is also achieved through metallic films with arrays of subwavelength holes [1]. In this last case, the optical tunneling through the holes is boosted by the excitation of surface plasmon polaritons (SPP) on the flat metallic surfaces [14, 15]. However, the dispersion properties of surface waves are also strongly dependent on the hole geometry [16, 17]. The role of the shape and size of each aperture on the transmission resonance has been evidenced experimentally by several groups [18, 19, 20, 21]. Another very recent example of the crucial role of the aperture geometry on the optical properties of such structures is the enhancement of the nonlinear optical response of metallic films with hole arrays [22]. This properties are attributed to slow modes due the cut-off behaviour in each hole. In hole arrays, apertures act as uncoupled metallic waveguides due to the opacity of noble metals in the visible and infrared wavelength range if the metal thickness is greater than the penetration depth of the electromagnetic field (about 25 nm). Consequently, the optical modes of the holes can be studied independently in a each aperture.

In previous theoretical studies of light transmission through single holes or through hole arrays in metallic films, several approaches have been used to modelize the electromagnetic field in the apertures. The simpler model makes use of perfect metallic walls [9, 23, 14], which can be modified with an effective hole radius [24]. Alternatively, time consuming numerical methods allow accurate calculations of optical modes and light transmission (Fourier decomposition [15, 20, 25], FDTD [26], or finite elements [27]). Optical modes in cylindrical holes can be calculated by well-known waveguide theory [28, 29, 30], whereas approximation methods have been proposed recently for rectangular [31] and circular [32] metallic waveguides. Numerical methods allow to study other hole geometries [33]. However, these approaches do not allow to clearly distinguish the respective role of the hole shape, hole size, and metal permittivity. Here, we develop analytical formulae for the effective index of the fundamental optical guided mode in rectangular nanoscale holes. It provides simple physical interpretations of the influence of the shape and size of the holes, and the role of the finite permitivitty of the metal. The penetration depth of the electromagnetic field in the metallic walls and the role of coupled surface plasmon polaritons are emphasized.

In this article, we focus on the fundamental modes of 1D and 2D subwavelength metallic waveguides. We develop explicit analytical expressions for the 1D and 2D effective indices, in excellent agreement with numerical computations on broad size and wavelength ranges. In the first part, we study the fundamental TM mode of 1D planar metallic waveguides. This mode is composed of two coupled surface plasmon polaritons. We show that this coupling is driven by the metal skin depth and the waveguide width, leading to a wavelength independent effective index in the near- and mid-infrared wavelength. These results have important consequences in nanoscale hole waveguides. In the second part, we focus on 2D subwavelength rectangular metallic waveguides. We provide simple analytical equations for the effective index of the fundamental TE mode. Our model takes into account losses, and is in very good agreement with 2D finite element calculations. The role of coupled surface plasmon polaritons and metal skin depth on the effective index is analyzed. We propose very simple expressions for the effective index n and the cut-off wavelength λ_c of the fundamental guided mode. Taking into account the finite permittivity of the metal induces a red-shift of the cut-off wavelength of the fundamental guided mode of rectangular waveguides. The nanoscopic origin of this phenomenon is described. Universal curves of the effective index and the cut-off frequency of any rectangular metallic waveguide are presented. These results provide a straightforward analysis of the optical properties of nanoscale apertures, and a tool for the design of efficient metallic structures for extreme light confinement.

2. Coupled surface plasmon polaritons 1D metallic waveguides

We first study a one dimensional metallic waveguide. A dielectric slab of permittivity ε_d and width w is surrounded by two semi-infinite metallic regions of permittivity ε_m (see Fig. 1(a)). In the following, the subscripts d and m denote the dielectric and metallic regions, respectively. κ is the wavenumber of the fundamental guided mode, and k ₀ = 2π/λ is the free space wavenumber.

 

Fig. 1. Schematic of 1D planar metallic waveguide (a) and 2D rectangular metallic waveguide (d). The fundamental TM mode of planar 1D waveguides is composed of two surface plasmons waves (red). They are weakly coupled in wide waveguides (w ≫ λ, (b)) and strongly coupled in narrow waveguides (w ≫ λ, (c))

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In case of perfect metallic walls, the fundamental waveguide mode is a TEM mode whose axial wavenumber is κ= √ε_d k ₀. κ is independent of the waveguide width w, and there is no cut-off frequency. In case of non-perfect metal, the exact 1D mode equation for even TM modes in the waveguide is given by:

where $k_{\mathrm{xd}}=\sqrt{{\epsilon }_d{k}_0^2-{\kappa }^2}$ and $k_{\mathrm{xm}}=\sqrt{{\epsilon }_m{k}_0^2-{\kappa }^2.}$

In the limit of wide waveguides (w ≫ λ), an approximate solution is obtained for the fundamental TM₀ mode. When |ℑ(k_xdw)| ≫ 1, Eq. (1) is reduced to $\frac{k_{\mathrm{xd}}}{{\epsilon }_d}+\frac{k_{\mathrm{xm}}}{{\epsilon }_m}=0,$, which is the exact relation dispersion of surface plasmon polaritons (SPP) on a single semi-infinite dielectric/metal interface. The effective index of this mode is $n_{\mathrm{eff}}=\frac{\kappa }{k_0}=\frac{1}{\sqrt{\frac{1}{{\epsilon }_d}+\frac{1}{{\epsilon }_m}}}.$ The fundamental TM mode is composed of two uncoupled surface plasmon polaritons propagating along both dielectric/metal interfaces (see fig. 1(b)).

For very narrow (sub-wavelength) waveguides, the coupling between both SPP can not be neglected anymore. When $\mid k_{\mathrm{xd}}w\mid \ll 1,\frac{1-e^{\mathrm{ik}}{\mathrm{xd}}^w}{1+e^{\mathrm{ik}}{\mathrm{xd}}^w}~-{\mathrm{ik}}_{\mathrm{xd}}\frac{w}{2}$ and Eq. (1) becomes

With $k_{\mathrm{xd}}^2-k_{\mathrm{xm}}^2=\left({\epsilon }_d-{\epsilon }_m\right)k_0^2$ we can deduce the mode equation as a function of k_xd:

and we obtain a simple, analytical expression for the effective index of the fundamental TM ₀ mode (coupled SPP approximation):

 

Fig. 2. Effective index of the fundamental TM mode of a planar metallic waveguide (ε_d = 1) as a function of the width w for ε_m = -50.

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The expression $\frac{\lambda }{\mathrm{\pi w}\sqrt{-{\epsilon }_m}}\sqrt{1+\frac{{\epsilon }_d}{{-\epsilon }_m}}$ in Eq. (4) is a correction term due to the coupling of sur-face plasmon polaritons. Fig. 2 shows the comparison of the calculation of n _1D with Eq. (4) and with the exact mode equation (1) for ε_m = -50. The effective index is plotted as a function of the normalized width w/λ. Importantly, for very narrow waveguides (w < 0.1λ) the increasing coupling between both surface plasmon polaritons induces a strong increase of the effective index. For greater values of w/λ, the results are always close to the perfect metal solution (n _1D ~ √ε_d). Hence, no deviation occurs between the approximated and the exact solutions for large waveguides, even if the approximation was supposed to be most valid for very narrow waveguides. As a result, a very good agreement is achieved on a broad size range (three orders of magnitude).

Equation (4) can be simplified in two ways. First, the term $\sqrt{1+\frac{{\epsilon }_d}{{-\epsilon }_m}}$ can be omitted as soon as |ε_m| ≫ ε_d. Second, using a Drude model for the metal permittivity allows to write a very compact expression for n _1D:

where ${\delta }_c=\frac{c}{{\omega }_p}$ is a good approximation for the metal skin depth within the Drude model ${\epsilon }_m=1-\frac{{\omega }_p^2}{{\omega }^2+\mathrm{i\omega \gamma }}$ where ω = k _0c is the frequency. Precisely, Eq. (5) is valid when γ ≪ ω ≪ ω_p). For high conductivity metals (Ag, Au, Al), this corresponds to the near- and mid-infrared (NIR-MIR) range. In the following we use ${\omega }_p=1.2\times {10}^{16}{s}^{-1}$ and $\gamma =1.2\times {10}^{14}{s}^{-1}$ as a good approximation for the permittivity of gold [34] (the dielectric region is air, ε_d = 1). We obtain δ_c ~ 25 nm. The difference between the skin depth and δ_c is less than 14% in the 0.5 μm to 20 μm wavelength range, as illustrated on fig. 3.

The comparison of expressions (4) and (5) with exact mode calculation and with single SPP approximation is given on fig. 4 for both real and imaginary parts of the effective index n _1D. The waveguide width is w = 100 nm. A very good agreement between the 1D mode equation (1)

 

Fig. 3. Skin depth of the metal $\delta =\frac{\lambda }{\left(2\pi \sqrt{-{\epsilon }_m\left(\lambda \right)}\right)}$ (dark) as a function of the wavelength. The skin depth is nearly constant and equal to δ = c/ω_p (red) between 0.5 and 20 μm.

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and the coupled SPP approximation (Eq. (4)) is obtained for wavelengths greater than 1μm, for both the real and imaginary parts of the effective index. In this example, the condition |k_xdw| ≪ 1 is no more fullfilled below 1 μm. As a result, a slight discrepancy is found between the 1D mode equation and the coupled SPP approximation. On fig. 4, the effective index of the planar metallic waveguide is also compared to the effective index of a single SPP, and to the constant effective index n_w given by Eq. (5).

 

Fig. 4. Real (left) and imaginary (right) parts of the effective index of the fundamental TM mode of a planar metallic waveguide as a function of the wavelength λ. Dark solid line: effective index of a single metal/dielectric surface plasmon polariton. The metal permittivity is given by a Drude model: ${\epsilon }_m=1-\frac{{\omega }_p^2}{{\omega }^2+\mathrm{i\omega \gamma }},{\omega }_p=1.2\times {10}^{16}{s}^{-1},\gamma =1.2\times {10}^{14}{s}^{-1}$. The waveguide width is w = 100 nm.

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Let us emphasize the main properties of the effective index n _1D . The real part of n _1D is much greater than the effective index of a single SPP, due to the strong coupling between both SPP. The coupling mechanism between both SPP can be expressed by the ratio between the metal skin depth and the waveguide width δ/w. As a result, the constant metal skin depth induces a constant effective index n_w given by Eq. (5). It provides an accurate evaluation of the real part of n _1D in the 1 - 10 μm wavelength range (see fig. 4).

Equation (5) gives a very simple relation for the increase of the effective index when decreasing the distance between metal walls. This phenomenon induces a gradient of the effective index in V-shaped metal grooves, and allows light to be confined at the bottom of the groove. This light confinement mechanism has led to the achievement of channel plasmon-polariton waveguides made of subwavelength metal grooves [36, 37, 38].

3. 2D rectangular metallic waveguides

We now consider a 2D rectangular metallic waveguide (fig. 1(d)) with w_x > w_y and λ ≫ w_x, w_y: the NIR-MIR wavelength range is above the cut-off wavelength. We focus our study on the TE ₁₀ mode. This mode has the smallest attenuation along the waveguide, and it has been shown that it should play a predominent role in the enhanced transmission through subwavelength holes in a metallic film [14]. Recently, Gordon et al. have shown that for good metals this 2D problem can be approximated by two 1D problems, leading to accurate evaluation of the cut-off wavelength of rectangular metallic waveguides [31]. In the following, we use a similar approach to provide analytical formulae for the effective index of waveguide modes. Both 1D problems are described by mode equations of 1D metallic waveguides, which are simplified by first order approximations. We emphasize that absorption losses in metal are included in the model. It can be noted that an alternative approach based on perturbation of boundary conditions and Green’s theorem was described by Jackson [39]. However, we found that our model provides slightly different and more accurate results. Moreover, simple equations allow us to discuss the role of coupled surface plasmon polaritons, and of metal skin depth.

3.1. Analytical expression of TE₁₀ modes

Incase of perfect metal, the wavenumber of the TE₁₀ mode is ${\kappa }_0=\sqrt{{\epsilon }_d{k}_0^2-\frac{{\pi }^2}{w_x^2},}$ and this mode can be decomposed by two plane waves whose wavevectors are in the (x,z) plane. In case of good (but non perfect) metal, the TE ₁₀ mode can be decomposed by two pairs of coupled surface plasmons propagating in (x,z) planes at y = 0 and y = w_y [31]. Hence, its axial wavenumber κ can be calculated by the equation of the symmetric transverse electric TE ₁ mode of a planar metallic waveguide, replacing the dielectric permittivity ε_d by an effective permittivity λ′_d = n ² _1D where n _1D is the effective index of coupled SPP, given by Eq. (4). The TE ₁ mode equation can be written:

Using a first order approximation of the tan function at $k_{\mathrm{xd}}=\frac{\pi }{w_x},$ and using $k_{\mathrm{xd}}^2-k_{\mathrm{xm}}^2=\left({\epsilon }_d^{\prime }-{\epsilon }_m\right)k_0^2,$ we find that the effective index of the 2D TE ₁₀ mode can be written:

where the effective width w′_x is given by:

The analytical formulae (Eq. (7) and (8)) obtained for the effective index of the TE ₁₀ of 2D rectangular waveguide modes are compared to finite element calculations made with a commercial software [27] on fig. 5. The results are also compared to the perfect metal solution. The role of the exact metal permittivity is clearly shown. A perfect agreement between our analytical formulae and finite element calculations shows the accuracy of our approximations. We point out that the influence of the metal permittivity can not be taken into account by simply introducing an effective width [24]. The coupling of surface plasmon polaritons plays a key role. Both effective dielectric permittivity ε′_d and effective width w′_x are necessary to achieve a quantitative agreement with finite element calculations.

 

Fig. 5. Real (left) and imaginary (right) parts of the effective index of the fundamental TE₁₀ mode of a rectangular metallic waveguide as a function of the wavelength λ. The widths of the waveguide is w_x = 300 nm and w_y = 200 nm. The results of the perfect metal (dark), finite element calculation (blue) and coupled SPP approximation (Eq. (7) and (8)) (red) are compared.

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3.2. Role of the metal skin depth

Since we study subwavelength waveguides (w_x < λ) and ε′_d ≪ |ε_m|, it can be shown easily that the effective width w′_x can be approximated by:

where δ is the metal skin depth defined previously. We have calculated the imaginary part of n _2D with Eq. (7) and (9). Once again, the results are in quantitative agreement with finite element calculations (the results are not shown on fig. 5 since both curves are superimposed).

In the following, we neglect the imaginary part of the metal permittivity. Hence, the role of the metal on the effective index can be taken into account only by its constant skin depth in the NIR-MIR frequency range, leading to a very simple expression for the effective index n _2D of the TE₁₀ mode :

Equation (10) provides a very simple interpretation of the effect of the metal skin depth on the fundamental guided mode TE₁₀. Importantly, the penetration depth of the electromagnetic field in the metal δ does not play the same role on the different metallic walls of the waveguide. On the x-direction, we have shown that an effective width can be defined, simply increasing the waveguide width w_x by 2δ. On the y-direction, the effect of the finite permittivity of the metal is due to coupled surface plasmon polaritons, and is only sensitive for widths not greater than a few δ.

3.3. Red-shift of the cut-off wavelength due to the finite permittivity of the metal

From Eq. (10) we deduce the expression of the cut-off wavelength λ_c of rectangular waveguides as a function of the widths w_x and w_y and of the skin depth δ :

Equation (11) gives a straightforward interpretation of the red shift of the cut-off wavelength due to the finite permittivity of the metal. This effect may be responsible for the red shift of the transmission peak observed through single apertures in metal [7, 9, 10, 11, 31]. In each metallic walls, the increase of the penetration of the electromagnetic field increases the cut-off wavelength. However, the importance of this effect is not the same on each direction, as stated before for the effective index.

 

Fig. 6. Cut-off wavelength of the fundamental guided mode TE₁₀ as a function of w_y, for a silver rectangular waveguide with w_x = 270 nm. The approximated model from Eq. (11) (solid curve) is compared to reference [31] (red points), and to the perfect metal case (horizontal dashed line).

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This red-shift is illustrated on fig. 6. The cut-off wavelength λ_c is plotted as a function of w_y for a silver rectangular waveguide with w_x = 270 nm. The results are in very good agreement with reference [31] (red points). They show the influence of coupled surface plasmon polaritons for small w_y. In this example δ = 23 nm. The cut-off wavelength of perfect metal waveguides would be constant and equal to 540 nm (dashed line).

3.4. Universal curves for the effective index

Normalizing dimensions and wavelengths to the metal skin depth δ in Eq. (10) allows to plot universal curves for the effective index of the fundamental guided mode TE ₁₀. The results are valid for any rectangular metallic waveguide (for most noble metals in the NIR-MIR frequency range δ ≃ 25 nm). This is illustrated on fig. 7 for three different wavelengths.

 

Fig. 7. Universal curves for the real part of the effective index of the fundamental guided mode TE₁₀ for three different wavelengths (λ/δ = 100, 200 and 500). Dimensions of the rectangular waveguide (w_x,w_y) are normalized to the metal skin depth δ.

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On fig. 7, we clearly distinguish three different propagating conditions. In the dark region there is no propagating modes. From Eq. (10), we also note that in this case, for λ ≪ w _x, the attenuation length along the propagation direction z becomes independent of the wavelength and equal to: Λ = (w_x + 2δ)/π. This property is particularly important for the design of ’zero-mode’ metallic waveguides for single-molecule fluorescence analysis in tiny volumes [13, 3].

The grey region corresponds to classical waveguiding conditions with effective indices close to 1. The white region corresponds to very small values of w_y, and clearly shows the increase of the effective index due to the coupling between surface plasmon polaritons. As demonstrated previously (see Eq. 10), this effect is only sensitive when w_y is of the same order than δ.

3.5. Universal curves for the cut-off wavelength

 

Fig. 8. Universal curves for the cut-off wavelength of the fundamental guided mode TE₁₀ of rectangular metallic waveguides. Dimensions and wavelengths are normalized to the metal skin depth δ.

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Using the same normalization procedure in Eq. (11), we have also plotted universal curves for the cut-off wavelength of the fundamental guided mode TE ₁₀ for any rectangular metallic waveguide (see fig. 8). These curves give the waveguide geometries (w_x,w_y) which provide a given cut-off wavelength λ_c. They show that the effect of coupled surface plasmon polaritons on the cut-off wavelength is sensitive up to about w_y/δ = 10. For large values of w_y, the cut-off wavelength limit is equal to 2(w_x + 2δ).

4. Conclusion

In this paper, we have studied the fundamental optical modes of 1D and 2D subwavelength metallic waveguides. We have proposed a simple model based on coupled surface plasmon polaritons and first-order approximations. Analytical expressions have been developped, and are in very good agreement with numerical calculations. This approach reveals the role of non-perfect metallic walls on optical modes. These effects can be taken into account by introducing the metal skin depth, leading to very simple expressions for the effective index n and the cut-off wavelength λ_c of the fundamental guided mode. Universal curves of n and λ_c for any rectangular metallic waveguide have been presented. They provide a simple way to evaluate the role of the metal and of the geometry of nanoscale apertures on their optical properties. This model should be an efficient tool to design new metallic structures for the enhancement of single-molecule analysis in subwavelength apertures [13, 3], and nonlinear optical effects in metallic films [22].