## Abstract

A waveguide mode of a subwavelength rectangular hole in a real metal is analyzed. Due to coupling between surface plasmons on the long edges of the hole, the cut-off wavelength increases as the hole-width is reduced. The cut-off wavelength is found to be much larger than Rayleigh’s criterion for perfect metals — 2.3 times as large for a 15 nm wide hole. The analytical results are verified by finite-difference calculations. The finite difference calculations also show the influence of including material loss.

©2005 Optical Society of America

## 1. Introduction

The cut-off condition, for which there can be no propagation of light through a hole in a perfect metal, occurs when the wavelength of light is more than twice the hole-length across [1]. Below cut-off, Bethe showed that the transmission of light through an aperture rapidly decreases as the fourth power of the ratio of the aperture length to the optical wavelength [2]. Subwavelength arrays of metal holes [3] and single metal holes flanked with gratings [4] allow for extraordinary transmission as compared with Bethe’s theory, which has been explained in terms of resonant coupling to surface plasmon (SP) modes on the surface of the metal. Recently there has been an interest in the influence of the hole-shape on the transmission properties [5–7]. It was shown that in randomly distributed rectangular holes the transmission depends upon the aspect ratio of the hole, which was called the “shape-effect” in that paper [6]. It was postulated that the “shape-effect” was the result of localized SP resonances. A recent work has investigated the transmission through a single rectangular aperture, while systematically varying the aspect ratio of the rectangle [7]. That work found the surprising result that the maximum transmission through the hole was red-shifted as the hole was made smaller.

In this paper, we explain the shift to longer wavelengths when decreasing the aperture width in real metals. By considering the influence of a coupled SP mode from the long edges of the aperture and penetration of the electric field into the metal, we show that the Rayleigh’s cut-off condition is increased significantly for real metals. To verify the applicability of our analytic theory, we calculate numerically the mode profiles and the waveguide dispersion for rectangular holes in a metal.

## 2. Cut-off increase for a subwavelength hole in a real metal

Figure 1 shows the geometry of the problem under consideration. We break the 2D problem down into two 1D effective-index problems. This allows us to identify two separate contributions to the extended cut-off wavelength: penetration of the field into the metal along the x-direction, and coupling between SP modes along the y-direction. The latter effect explains how the cut-off wavelength actually increases as the hole size is reduced.

Although the effective index approach works for a perfect electric conductor (PEC), it is only an approximation for real metals because the boundary conditions cannot be consistently matched along orthogonal direction when the electric field penetrates into the metal. Nevertheless, for good metals, the field components that do not match at the boundaries are small, and the approximation gives good quantitative results. The validity of this approach is verified by numerical calculation with the finite-difference method. These calculations also show that material loss has a small influence on these results.

#### 2.1 Cut-off increase from penetration of the electric field into the metal

The propagation constant of the TE_{01} mode of a rectangular hole in a PEC is given by:

where *a* is the length of the rectangle and *λ* is the wavelength of light in vacuum [1]. The cut-off wavelength occurs when the propagation constant of the TE_{01} mode is zero, or when *λ*=2*a*. For longer wavelengths, the mode decays exponentially with a decay length that can be found from Eq. (1). This decay results because light at these wavelengths cannot propagate within the hole, and so it is reflected. For comparison with recent experiments [7], if the length of the rectangle is 270 nm, and the wavelength of light is 750 nm, the intensity of light in the mode is attenuated by 99.2% upon transmission through a 300 nm distance. Light with wavelength shorter than 540 nm should receive no attenuation in the mode at all.

In real metals at optical frequencies larger than the plasma frequency, the Drude model may be used to estimate the metal’s relative permittivity. Typically, the metal’s relative permittivity has a negative real part and a small imaginary part (which is neglected for this analysis, but reconsidered in the numerical model presented below). The electric field can penetrate into the metal by the skin-depth, which makes the hole appear larger. The propagation constant, *β _{TE}*, of a TE mode between two parallel plates of a real metal can be found from the characteristic equation:

where *ε _{m}* is the relative permittivity of the metal, and

*ε*is the relative permittivity of the dielectric between the metal sides,

_{d}*l*is the length between the metal sides, and

*k*=2

_{o}*π/λ*is the free-space wave-vector. By setting the propagation to zero, we find the cut-off wavelength:

At 750 nm in silver, the real part of the dielectric constant is -27.5 [8], which will give an increase in the cut-off wavelength of 14% over the PEC condition for a 270 nm wide hole.

#### 2. 2 Cut-off increase from coupled surface-plasmons

An even larger increase in the cut-off condition is found by considering coupled SP waves on the long edges of the aperture. This leads to the interesting result that the cut-off wavelength actually increases as the hole-size becomes smaller.

If we again consider the case of two parallel plates, we can formulate the TM mode of this configuration as coming from the sum of SP modes on the top and bottom plates. In the region between the plates, the field has a hyperbolic cosine dependence from the sum of two exponential decaying SP modes on the top and bottom plates. Not only does the field penetrate into the metal, as was found in the previous section, but the mode-shape within the hole is altered. The characteristic equation for this configuration is modified from the dielectric case to be:

For a 105 nm aperture in silver at 750 nm, this TM mode has a propagation constant that is 1.2 times the free-space wave-vector. As a result, the effective index squared of this mode is 1.46. Clearly, the effective index increases as the width of the hole is reduced.

If we assume, as is the case in a PEC rectangular waveguide, that the TE_{01} mode may be approximated by the TM mode in the transverse direction and the TE mode along the lateral direction, we can estimate the cut-off wavelength by using the effective index squared from Eq. (4) as the relative permittivity of the dielectric in Eq. (3):

As an example, for a 270 nm by 105 nm rectangular waveguide, we find that the cut-off wavelength for silver is 761 nm. This is 41% larger than the cut-off of a PEC waveguide with the same geometry.

Figure 2 shows the dispersion of *ε _{d}*=(

*β*)

_{TM}/k_{o}^{2}(as calculated from Eq. (4)), and the effective index squared, (

*β*)

_{TE}/k_{o}^{2}(as calculated using Eq. (2)). The Drude response was used to calculate the relative permittivity for silver [8], with an effective mass of 0.96 the free electron mass, a scattering time of 31 fs, and a background dielectric constant of 4.15. In a perfect metal, the hole-width does not influence the propagation constant for the TE

_{01}mode, and the cut-off wavelength is 540 nm. The shift for a TE mode of the 1D problem was found to be only 14% in section 2.1, which results in a cut-off around 620 nm. Therefore, reducing the hole-size to increase the SP-mode coupling increases the cut-off wavelength significantly. We may consider the extreme example of a hole-width of 15 nm (which would be challenging to fabricate), for which the cut-off wavelength increases to 2.3 times the PEC value.

The red-shift in the cut-off wavelength is mediated by the SP-coupling between the long-edges of the hole. Therefore, it depends on the hole-width rather than the aspect ratio of the hole; as the width is reduced, the coupling is increased.

The PEC limit is recovered as the real part of the metal permittivity, *ε _{m}*, approaches negative infinity. In this limit, the left-hand side of Eq. (4) vanishes, which means that in order to have the right-hand side vanish as well, ${\beta}_{\mathit{TM}}={k}_{o}\sqrt{{\epsilon}_{\mathit{air}}}={k}_{o}$ ; so no red-shift is seen in the cut-off wavelength as the width of the hole is reduced. The physical interpretation here is that a PEC cannot support SP modes at the metal-air interface, and so no SP-SP coupling is allowed.

#### 2.3 Numerical simulations

An analytic solution does not exist for the rectangular waveguide, except for the special case of the PEC. For this reason, we resorted to numerical simulation using the finite difference method to validate the effective index model of the previous sections. A detailed discussion of the finite difference method can be found elsewhere [9]. Convergence was ensured by reducing the grid-size and extending the artificial PEC boundaries. The smallest grid-size attempted was 0.5 nm. The largest PEC artificial boundary was set to a width of 1 micron, where all the field components are negligible. To speed up the calculations, symmetries in the x and y directions were exploited.

Figure 2 shows the simulation results with the symbols +, ×, □, and ○ for the 105 nm, 145 nm, 185 nm and 225 nm wide holes. We performed numerical calculations for hole-widths down to 15 nm, where the cut-off wavelength is extended to 1260 nm. In all cases, the numerically calculated effective index squared was found to be within 0.025 of the value found by the analytical model and the cut-off wavelength agreed to within 6 nm.

Figure 3 shows the field distribution for the E and H components as calculated numerically. Within the hole, the H-field along the x-direction and the E-field along the y-direction closely resemble those of the TE_{01} modes of a PEC rectangular guide. Along the other directions, and inside the metal, there are differences from the PEC case. These figures show the existence of E and H components normal to those found for the TE_{01} mode in a PEC. There is a penetration of the various field components into the metal at all boundaries. The field distributions along the top and bottom boundaries are clearly indicative of SP modes on these surfaces.

#### 2.4 Influence of material loss

Figure 4 shows the separate contributions of the cut-off attenuation and material losses, as calculated numerically when the imaginary part of the dielectric constant was added. Material loss plays a negligible role for wavelengths a few nanometers above the cut-off wavelength. For example, for a 300 nm film, at optical wavelength of 764 nm, the cut-off attenuation is 12000 times the material absorption. It should be noted that these two effects are distinct; material loss absorbs the photons, whereas cut-off attenuation occurs when the incident photons are reflected without absorption because they cannot propagate inside the hole.

## 3. Conclusions

We presented an analytic theory which showed a dramatic increase in the cut-off wavelength of real metals as compared to a perfect conductor. The cut-off wavelength was enhanced by 41% for the hole-sizes that were used in recent experiments, and it was more than doubled for hole-widths at the limits of present fabrication capability. This shows that care should be taken when using the perfect-metal approximation to decide if these holes are actually below cut-off. Furthermore, the analytic theory explains how the cut-off wavelength increases as the holes are made smaller; this enhancement results from coupling between the surface plasmons on the top and bottom edges of the hole. The results of the analytic theory agreed well with finite-difference numerical calculations. The numerical calculations also showed that the influence of material loss did not influence the cut-off wavelength and were negligible for wavelengths a few nanometers above the cut-off wavelength.

## Acknowledgments

The authors acknowledge financial support for this work from an NSERC Special Research Opportunity grant, and an in-kind contribution of the MODE Solutions software package from Lumerical Solutions Inc.

## References and links

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