## Abstract

A detailed study of guided modes in a nanoscale metallic rectangular waveguide is presented by using the effective dielectric constant approach. The guided modes, including both traditional waveguide mode and surface plasmon mode, are investigated for the silver rectangular waveguide. The mode evolution in narrow waveguide is also discussed with the emphasis on the dependence of mode dispersion with waveguide height. Finally, the red-shift of the cutoff wavelength of the fundamental mode is observed when the waveguide height decreases, contrary to the behavior of regular metallic waveguide with PEC boundary. The comprehensive analysis can provide some guideline in the design of subwavelength optical devices based on the dispersion characteristics of metallic rectangular bore.

©2007 Optical Society of America

## 1. Introduction

Metallic nanostructures have attracted the attention of many scientists and the study of surface plasmon polaritons (SPPs) has become one of the main research branches of nanophotonics[1]. Nanoscale apertures in metallic films show abnormal optical properties and lead to enhanced light transmission and light confinement [2, 3]. According to the waveguide theory in microwave frequencies, the guided modes cannot exist above the cut-off wavelength, which is the same magnitude as the waveguide dimension. However, in optical frequencies, the properties of the metals have to be described by complex permittivity with dispersion and loss. Furthermore, the real parts of the permittivity can be negative below the plasma frequency. This extraordinary mode of light transmission in subwavelength apertures has been investigated for metallic coaxial waveguide [4, 5], circular aperture[6, 7], and rectangular waveguide[8,9]. It was found that such nanoscale apertures can support SPP modes, which can propagate even when the dimension of the aperture is smaller than the guidance wavelength due to fact that such SPP modes have a longer cutoff wavelength.

In analyzing the light transmission through subwavelength hole in metallic films, several methods have been used to model the fields in the apertures. For subwavelength cylindrical and coaxial waveguides, the guided modes can be calculated by conventional waveguide theory. However, for a metallic rectangular waveguide where the metal is modeled as a plasma medium, there is no analytic solution, except when the metal is modeled as PEC. Numerically, the finite-difference method in time domain and frequency domain can be used to analyze the guided modes and transmission characteristics in such metallic rectangular waveguide [10]. Due to the large difference between the permittivities of the metal and the dielectric, the finite-difference method may cause calculation instability when applied to certain nanoscale metallic structures. Recently, the effective dielectric constant (or index) method (EDCM or EIM) [11], which has been successfully used in the analysis of optical waveguide for many years, has been used to analyze the fundamental mode in the metallic rectangular trench and hole [12–14].

In this paper, the EDCM is used to analyze both the traditional waveguide (TWG) modes and the SPP modes in the metallic rectangular waveguide where the permittivity of the metal is represented by the Drude model. The mode evolution in narrow waveguide is also discussed with the emphasis on the dependence of mode dispersion with waveguide height. Finally, the red-shift of the cutoff wavelength of the fundamental mode is observed when the waveguide aspect ratio decreases. These results can give some guideline in the design of subwavelength optical devices.

## 2. Analysis methodology

Figure 1(a) shows a schematic illustration of the metallic rectangular waveguide under consideration. The relative dielectric constant of the rectangular core is assumed to be *ε _{r}*

_{1}, and the relative dielectric constant of the metal at optical frequencies can be approximated by the Drude model

where *ε _{r}*

_{2}is the permittivity at infinite frequency,

*ω*and

_{p}*ω*are the plasma and collision frequencies respectively. The collision frequency, typically

_{τ}*ω*<0.1

_{τ}*ω*, is related to the metal loss and can be neglected for simplicity.

_{p}The concept of effective dielectric constant approach is also shown in Fig.1. The cross section of metallic waveguide can be treated as a combination of two coupled 1D slab waveguides in the *x* and *y* directions respectively [Figs. 1(b), 1(c)].

We first consider the *E ^{y}_{mn}* modes (TE-like), in which

*E*and

_{y}*H*are the dominant electromagnetic fields. The wave functions for the

_{x}*x*-direction slab can be written as [15]

Applying the boundary conditions at *y*=±*b*: *E _{z}*

_{1}=

*E*

_{z}_{2}and

*H*

_{x}_{1}=

*H*

_{x}_{2}, we get

Eliminating the constants in Eq. (3), we can get the guidance condition for the *x*-direction slab

where ${\rho}_{y}=\frac{{\epsilon}_{{r}^{1}}{\gamma}_{y}}{{\epsilon}_{{r}^{2}}\left(\omega \right){k}_{y}}$ and ${\gamma}_{y}^{2}={k}_{0}^{2}({\epsilon}_{r1}-{\epsilon}_{r2}(\omega ))-{k}_{y}^{2}$.

For real wave number *k _{y}*, Eq. (4) can be rewritten as

For imaginary wave number *k _{y}*, let

*k*=

_{y}*ik′*, and Eq. (4) can be rewritten as

_{y}The effective dielectric constant for the *x*-direction slab waveguide can be expressed as *ε _{er}*

_{1}=

*ε*

_{r}_{1}-(

*k*/

_{y}*k*

_{0})

^{2}. Thus, the metallic waveguide can be regarded as a slab waveguide [Fig.1 (c)] along

*y*-direction with the effective dielectric constant

*ε*

_{er}_{1}in the core and

*ε*

_{r}_{2}(

*ω*) in the cladding and the wave functions can be expressed as

Applying the boundary conditions at *x*=±*a*: *E _{y}*

_{1}=

*E*

_{y}_{2}and

*H*

_{z}_{1}=

*H*

_{z}_{2}, we get

Eliminating the constants in Eq. (8), we can get the guidance condition for the *y*-direction slab:

where *ρ _{x}*=

*γ*/

_{x}*k*and

_{x}*γ*

^{2}

*=*

_{x}*k*

^{2}

_{0}(

*ε*

_{er}_{1}-

*ε*

_{r}_{2}(

*ω*))-

*k*

^{2}

*.*

_{x}Since there is no solution when *k _{x}* is imaginary, Eq. (9) can be rewritten as

For the *E ^{y}_{mn}* modes, there are two types of guided waves in the metallic rectangular waveguide. The first type is the TWG modes which satisfy the guidance conditions Eq. (5) and Eq. (10), and the second type is the SPP modes which satisfy the guidance conditions Eq. (6) and Eq. (10). The TWG modes are denoted as

*E*, while the SPP modes are denoted as

^{y}_{mn}*E*or

^{y}_{me}*E*. The subscripts

^{y}_{mo}*m*and

*n*are integers and they correspond to the number of peaks of each field component in the

*x*and

*y*directions respectively, the subscripts

*e*and

*o*refer to the symmetry of the field distribution (even or odd) in

*x*or

*y*directions.

In the similar procedure, we can get the guidance condition for *E ^{x}_{mn}* mode as follows

where *γ*
^{2}
* _{x}*=

*k*

^{2}

_{0}(

*ε*

_{r}_{1}-

*ε*

_{r}_{2}(

*ω*))-

*k*

^{2}

*,*

_{x}*γ*

^{2}

*=*

_{y}*k*

^{2}

_{0}(

*ε*

_{r}_{1}-

*ε*

_{r}_{2}(

*ω*))-

*k*

^{2}

*-*

_{x}*k*

^{2}

*,*

_{y}*k*

*=*

_{x}*ik′*, and

_{x}*k*=

_{x}*ik′*(for SPP modes only).

_{x}To analyze the dispersion of the guided waves in metallic rectangular waveguide, we solve for *k _{x}*(or

*k′*) and

_{x}*k*(or

_{y}*k′*) numerically, and the propagation constant

_{y}*β*can be expressed as

## 3. Results and discussion

In the calculation, we choose *ε _{r}*

_{2}=3.7 and

*ω*=7.1085×10

_{p}^{15}rad/s to model the permittivity of silver. The parameters were calculated by Sönnichsen [16] from experimental data on the reflection and transmission of 25–50 nm thick silver films for wavelengths from 0.188µm to 1.9µm[17]. Figure 2 shows the dispersion curves for TWG modes and SPP modes in two different silver waveguides. The shaded region beginning from the plasma frequency (

*f*=1131THz) is the forbidden area where no guided mode exists. This region is delimited with the refractive index of the silver as solid line. It is shown that the SPP modes usually exist below a critical frequency (

_{p}*f*=1003THz) at which the relative permittivity is negative unity, and the lowest order TWG modes (

_{c}*E*

^{x}_{11},

*E*

^{y}_{11}) exist above the plasmon frequency, but some higher order TWG modes can exist both below and above the plasmon frequency. All the TWG modes locate in fast wave region (

*β*/

*k*

_{0}<1.0), while the SPP modes can migrate from slow wave region to fast wave region as the operating frequency decreases. Although each mode has its own dispersion characteristics, the dispersion curve of some modes can continue smoothly from one mode to another mode, for example: in both cases,

*E*

^{y}_{12}and

*E*

^{y}_{1o},

*E*

^{x}_{31}and

*E*

^{x}_{11}are continuous. Moreover, it is found that some of higher order TWG modes have discontinuous dispersion curves, for example

*E*

^{x}_{21}in 400nm×200nm silver waveguide, one part exists below the plasma frequency and connect with the mode

*E*

^{x}_{o}_{1}, and another part exists above the plasmon frequency and connects with

*E*

^{x}_{41}. Thus, some of TWG modes and SPP modes can coexist at the same frequency with certain waveguide geometries.

Figure 3 shows the dispersion curves for the SPP modes in silver waveguides with different waveguide heights. The waveguide width 2*a* is assumed to be 250nm. For comparison, the dispersion curves for the SPP modes *TM _{e}* and

*TM*in the corresponding slab waveguides with the same heights are also shown in Fig. 3. As the waveguide height decreases, the SPP modes

_{o}*E*

^{x}_{e}_{1}and

*E*

^{x}_{o}_{1}are being suppressed and their cutoff frequencies move up and eventually approach the critical frequency. Also, the slope of curve for the odd SPP mode

*E*

^{y}_{1o}can change from positive to negative when the waveguide height is very small. This means that there exists backward surface wave in the narrow metal waveguide.

Figure 4 shows the calculated cutoff wavelength dependence of primary SPPs modes on the waveguide aspect ratio. In a rectangular waveguide with the PEC boundary, the waveguide height does not influence the propagation constant for the fundamental *TE*
_{10} mode, and the cut-off wavelength of *TE*
_{10} is twice the waveguide width. However, the red-shift of cutoff wavelength of fundamental mode *E ^{y}*

_{1e}appears when waveguide aspect ratio decreases. This allows light with longer wavelength to propagate through the rectangular aperture. Moreover, it is found that when the aspect ratio is less than 0.45, it is more suitable for single mode operation.

## 4. Conclusion

In this paper, the EDCM is used to analyze the modes in the metallic rectangular waveguide. The results show that the metallic waveguide supports not only the TWG modes, but also the SPP modes. The SPP modes usually exist below the critical frequency. As the operation frequency decreases, the SPP modes migrate from slow wave region to fast wave region. When the waveguide height decreases, the SPP modes *E ^{x}_{e}*

_{1}and

*E*

^{x}_{o}_{1}are suppressed and the red-shift of cut off wavelength of the fundamental mode is observed. Furthermore, the slope of curve for the odd SPP mode

*E*

^{y}_{1}

_{o}can change from positive to negative when the waveguide height is very small. This means that there exists backward surface wave in the narrow metal waveguide. Moreover, it is found that when the aspect ratio is less than 0.45, it is more suitable for single mode operation. These results can provide some guideline in the design of nanoscale optical devices based on the dispersion characteristics of metallic rectangular hole.

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