## Abstract

A numerical study on the complex propagation constants of the surface plasmon polariton (SPP) rectangular hollow waveguide by the method of lines (MoL) is performed. New cut-off conditions are proposed for the SPP waveguide. A SPP rectangular hollow waveguide constructed of gold is first considered. The dependences of complex propagation constants on the sizes of the waveguide and on the wavelength are investigated. Fundamental and unusual characteristics of the SPP waveguide are revealed. The validity and limitations of effective index method (EIM) are examined by comparing the numerical results obtained by the MoL with the approximate results obtained by EIM. The differences in the propagation characteristics among the various metals are then shown.

©2008 Optical Society of America

## 1. Introduction

Optical circuits on subwavelength or nanometric scales have attracted a great deal of attention recently. The size of the optical waveguide using surface plasmon polariton (SPP) can be decreased significantly compared to the conventional diffraction-limited optical device. Therefore, the SPP is expected to be employed as nanometric optical wire in integrated optical circuits and nanometric optical devices in the future. A number of interesting experimental or theoretical studies have examined practical and concrete nanometric optical waveguides using SPP [1-5] such as SPP gap waveguide [6]. One of the fundamental SPP waveguides is the rectangular hollow metallic optical waveguide. In particular, the propagation characteristics of SPP waveguides play an important role in the extraordinary large transmission through the hole arrays on the metallic film [7-9]. It is also related the SPP slot waveguides which have been proposed for the element of subwavelength optical devices [10-12].

Thus far, interesting phenomena of the SPP rectangular hollow waveguide have been reported in comparison with the waveguide made of perfect electric conductor (PEC) used in microwave technology. For example, the cut-off wavelength in the SPP rectangular hollow waveguide is increased significantly compared to the theoretical results of the PEC rectangular waveguide. This means that the size of the optical waveguide using SPP can be significantly decreased compared to that based on the theory of the PEC rectangular waveguide [13-16]. Since the permittivity of the metallic material that supports SPP has a complex value in the optical frequency region, the propagation constant of the SPP waveguide inevitably has a complex value below the propagation condition, and even below the cut-off condition. The electromagnetic fields penetrate into the metal walls of the SPP waveguide. It is difficult to obtain a rigorous and analytical solution of the guided mode for the SPP waveguide of rectangular cross section. The approximate propagation constants can be obtained by effective index method (EIM) [9, 14]. Recently, improved EIM has been proposed by Kumar and Srivastava [15] and a robust first-order analytical treatment has derived in [17].

The numerical techniques are effective, and the SPP rectangular hollow waveguide has been investigated in several numerical studies by the finite-difference method (FDM) [14], the finite-difference time-domain (FDTD) method [16], and the finite-element method (FEM) [17]. However, the basic characteristics of the complex propagation constants of the SPP rectangular hollow waveguide have not been sufficiently investigated in detail. In general, the accurate propagation constants in the SPP uniform waveguide can be obtained by solving the complex eigenvalue problem.

In the present paper, we present a numerical investigation of the SPP rectangular hollow waveguide by the method of lines (MoL) with high accuracy. We first consider the case in which the metal that constitutes the waveguide is gold (Au) of complex-valued relative permittivity. We calculate the dependences of the complex propagation constants on the waveguide sizes and on the optical wavelength. The fundamental, important, and interesting characteristics of the SPP rectangular hollow waveguide have been investigated in detail by comparing the numerical results obtained by the MoL with those obtained by EIM. The difference in the propagation characteristics among the various metals is then shown.

## 2. Geometry of the problem

The simple geometry of the problem considered in the present study is shown in Fig. 1. A rectangular hollow waveguide of width *a* and height *b* is created inside a metallic material. We assume that the length of the waveguide in the *z*-direction is infinite and calculate the propagation constants of this hollow waveguide in the z-direction. The permittivities in the hole (free space) and of the metal that surrounds the hole are given by *ε*
_{0} and *ε*
_{1}, respectively. We calculate the propagation constants of the mode along the *z*-direction. We employ the method of lines (MoL) to solve the complex and generalized eigenvalue problem in the two-dimensional cross section of the SPP waveguide shown in Fig. 1.

## 3. Method of lines

The method of lines (MoL) is a well-known numerical technique, and its application to various electromagnetic problems, including optical wave guiding, is well established. MoL has been examined in a number of studies for the purpose of calculating the propagation constant, including that of the SPP waveguide [18-21]. The formulations used in this study are based on the formulation in reference [20]. The basis for the numerical analysis of the waveguide is the following equations of the electromagnetic fields * E* and

*under the assumption of harmonic time dependence exp(jωt):*

**H**where *μ*
_{0} is the permeability in a vacuum, *k*
_{0}
^{2}=ω^{2}
*ε*
_{0}
*μ*
_{0} is the free space wave number, and *η*
_{0}=(*μ*
_{0}/*ε*
_{0})^{1/2}. The distribution of the relative permittivity in the *x*-*y* plane is assumed to be given by *ε*
_{r}(** x**)=

*ε*

_{r}(

*x*,

*y*) in Eq. (1). The two potentials ∏

_{e}and ∏

_{h}are defined as

where * i_{x}* is the unit vector in the

*x*direction in Fig. 1 and

*k*

_{z}is the complex propagation constant to be determined numerically. In this formulation, we assume that the wave is propagating in the z direction. Two potentials,

*ψ*

_{e}and

*ψ*

_{h}, satisfy the wave equations, and we discretize these equations by the MoL. We first enclose the appropriate region including the hole, as shown in Fig. 1, with electric and/or magnetic walls in order to establish fixed boundary conditions. Next, the guiding structure with a dielectric constant of

*ε*(

_{r}**)=**

*x**ε*(

_{r}*x*,

*y*) is modeled through a set of single layers. These layers show the relative permittivities, which depend on only one transverse coordinate

*ε*

_{r}(

*x*,

*const.*), and the

*x*axis and the function

*ε*(

_{r}*x*,

*const.*) are discretized using two shifted line systems parallel to the

*y*axis. The number of layers and their thicknesses can be chosen arbitrarily. We perform the following steps for each layer. Field and wave equations, which are partial differential equations, are set up. These equations are subject to the boundary conditions at the lateral walls. A partial discretization of the potentials and field components is performed. The partial replacement of derivatives with difference formulas yields systems of coupled ordinary differential equations. Applying suitable difference operators simultaneously satisfies the lateral boundary conditions. A transformation of the discretized potentials diagonalizes the systems of equations. The obtained uncoupled equations can easily be solved. Relationships between the tangential fields at the upper boundary of a layer and those at the lower boundary of a layer are achieved. Matching of the tangential fields, which must be continuous across all interfaces of the layered structure, yields a characteristic equation. The last field matching condition, applied near the center of the structure can be derived as follows [21]:

where the column vector * Ē* is composed of the transformed tangential components of the electric field strength

*at a suitably chosen interface, and Ȳ(*

**E***k*

_{z}) is a matrix equation containing

*k*. The complex propagation constant of the modes

_{z}*k*

_{z}can be obtained by searching for values that satisfy det[Ȳ(

*k*)]=0. This process constitutes the generalized and complex eigenvalue problem. Once the propagation constant of a mode has been determined, the field distributions of the guided modes are easily generated. In order to show the validity of the present results by MoL, we calculate values of α and β of the TE

_{z}_{01}mode at two wavelengths for the waveguide made of gold whose size is given by

*a*=200 nm and

*b*=300 nm. We obtained these values as β=0.029, α=2.672 at

*λ*=2.0 µm and β=0.081, α=5.662 at

*λ*=4.0 µm by MoL. The results agree well with those obtained by improved EIM and finite-element method (FEM) shown in the reference [15, 17].

## 4. Numerical results and discussion

We first assume that the wavelength is fixed at *λ*=652 nm, and the metal that constitutes the SPP waveguide is gold (Au) with a relative permittivity of *ε*
_{1}=-13.2-j1.08 [23]. In the present paper, as the first step, we consider only a narrow rectangular hollow waveguide and consider the first and second modes, the main electric fields of which are the *x*-components in Fig. 1, i.e., the TE_{01} and TE_{02} modes.

The dependences of the phase constant *β* and the attenuation constant *α* normalized by the free space wavenumber defined as j*k _{z}*/

*k*

_{0}=

*α*+j

*β*on the waveguide-height

*b*/

*λ*are shown in Figs. 2 and 3 with the waveguide-width

*a*/

*λ*as a parameter. Figure 2 shows the results for the TE

_{01}mode, and Fig. 3 shows the results for the TE

_{02}mode. The open symbols in Fig. 2 show the results calculated by the EIM [14]. The bold black curves show the results for the case in which the waveguide is made of a perfect electric conductor (PEC) in Figs. 2 and 3.

In order to intuitively understand the propagation characteristics of the SPP rectangular hollow waveguide shown in Figs. 2 and 3, we first consider the case of the PEC rectangular waveguide. The *x* component of the electric fields of the TE_{01} and TE_{02} guided modes in the PEC rectangular waveguide can be written as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\propto \mathrm{cos}\left({k}_{x}x\right)\mathrm{sin}\left({k}_{y}y\right)\mathrm{ex}p\left(-j{k}_{z}z\right).\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\left({\mathrm{TE}}_{02}\right)$$

The wavenumbers in the *x*, *y*, and *z* directions satisfy the dispersion relation inside the PEC waveguide as

For the TE_{01} and TE_{02} modes, we can set *k _{x}*=0 and

*k*=

_{y}*nπ*/

*b*(

*n*=1, 2 for TE

_{01}and TE

_{02}modes) from the boundary conditions at

*y*=±

*b*/2 in Fig. 1 and can obtain the following relations for normalized propagation constants

*α*and

*β*:

$$\alpha ={\left[\frac{{n}^{2}}{{\left(\frac{2b}{\lambda}\right)}^{2}-1}\right]}^{\frac{1}{2}},\beta =0\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\mathrm{for}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\frac{b}{\lambda}<\frac{1}{2}$$

The results of Eq. (7) are shown by the bold black curves for the cases of the TE_{01} and TE_{02} modes in Figs. 2 and 3, respectively. Note that the ordinates in Figs. 2 and 3 represents both *α* and *β* simultaneously.

As an example, in Fig. 2, for the case of the PEC waveguide, the bold black curve represents the phase constant *β* in the range of *b*/*λ*>0.5 and the attenuation constant *α* in the range of *b*/*λ*<0.5. In Fig. 3, the bold black curve represents only the attenuation constant *α* in the range of *b*/*λ*<1.0. The two black curves that represent *α* and *β* do not intersect in these figures, because the propagation region and the cut-off region can be clearly separated by the cut-off condition *b*/*λ*=0.5 for the TE_{01} mode and *b*/*λ*=1.0 for the TE_{02} mode shown in Figs. 2 and 3. Furthermore, these curves are independent of the waveguide width *a* and the normalized phase constant *β* is always smaller than unity, as shown in Figs. 2 and 3.

Next, we consider the case of the SPP rectangular hollow waveguide. The propagation constants of the SPP waveguide have always been complex values, because the metal has a complex-valued permittivity in the optical wavelength. Therefore, the curves that represent *α* and *β* cannot be clearly separated by a given cut-off condition, and these curves intersect in Figs. 2 and 3 inevitably in this case. However, we can consider an adequate cut-off condition for the SPP waveguide. For simplicity, we only consider the case of the TE_{01} mode because it is possible to apply the same concept as in the case of higher modes. We assume that the electric field of the TE_{01} mode in the SPP rectangular hollow waveguide can be expressed approximately as

as used in the EIM. The field distribution of Eq. (8) gives the dispersion relation in a free space inside the SPP waveguide as

We can see that the sign of *k*
_{x}
^{2} in Eq. (9) is opposite that in Eq. (6) and cannot set *k _{x}*=0, even for the TE

_{0n}mode, due to the boundary conditions on the boundary at

*x*=±

*a*/2 in Fig. 1. This characteristic derives the reduction, as compared with the PEC waveguide, in waveguide height

*b*that gives the cut-off condition. The boundary conditions at

*y*=±

*b*/2 are different from those of the PEC because electromagnetic fields penetrate the metal. From Eq. (9), we can write normalized propagation constants as

We can derive following relations from the above equation as follows:

under the condition 1+*Re*(*k*
_{x}
^{2}/*k*
_{0}
^{2})-*Re*(*k*
_{y}
^{2}/*k*
_{0}
^{2})≥0, and we call this region the propagation region. Under the condition 1+*Re*(*k*
_{x}
^{2}/*k*
_{0}
^{2})-*Re*(*k*
_{y}
^{2}/*k*
_{0}
^{2})<0, we can derive

and we call this region the cut-off region. Relations *α* and *β* shown in Eqs. (11)-(14) agree qualitatively with the results shown in Figs. 2 and 3. From these relations and the analogy of the PEC waveguide, we can define the cut-off condition as

for the SPP waveguide. When the dissipation of the metal is neglected, condition (15) agrees with the result calculated in reference [14]. At the cut-off condition (15), we can obtain the following relation:

from Eqs. (11)-(15). In Fig. 2, the values of *α* and *β* at the cut-off condition (15) represent a point of intersection between the two curves that represent *α* and *β*. The values *α* and *β* calculated by the MoL and by the EIM at cross points in Fig. 2 are shown in Table 1. When EIM with no dissipation of the metal is used, all of the values in Table 1 are zeros because, in this case, *Im*(*k*
_{x}
^{2}/*k*
_{0}
^{2})-*Im*(*k*
_{y}
^{2}/*k*
_{0}
^{2})=0. The relative errors defined by (value by MoL – value by EIM)/(value by MoL)×100 in Table 1 are obtained as 8.5%, 7.3%, and 6.4% for *a*=*λ*/24, *λ*/16, and *λ*/8, respectively.

The waveguide-heights *b*/*λ* that gives the cut-off condition (15) by the MoL and by EIM in Fig. 2 are shown in Table 2. The relative errors as shown in Table 2 are given as 12.6%, 9.1%, and 5.5% for *a*=*λ*/24, *λ*/16, and *λ*/8, respectively. These results show that the validity of the EIM degrades for small waveguide-width *a*. The difference between the results obtained by MoL and EIM in Tables 1 and 2 show the validity and limitation of the approximate field distribution Eq. (8) of the SPP waveguide. The results obtained by the EIM agree well with the numerical results obtained by MoL in the propagation region in Fig. 2. However, near the cut-off regions, the difference between the results of MoL and EIM becomes significant, as shown in Fig. 2.

For the case of the TE_{02} mode, the dispersion relation is given by Eq. (9), and the cut-off condition is given by Eq. (15). Therefore, the Eq. (16) is also valid. The values of *α*=*β* obtained by the MoL at the cut-off condition are 0.270, 0.220, and 0.155 for *a*=*λ*/24, *λ*/16, and *λ*/8, respectively, in Fig. 3. These values are similar to those listed in Table 1. Apparently *k _{y}* depends strongly on the modes, because

*k*defines the field distribution along the y-direction, i.e., the mode, as shown in Eq. (8). So,

_{y}*Im*(

*k*

_{y}

^{2}) may strongly depend on the modes. However, Eq. (16) which contains

*Im*(

*k*

_{y}

^{2}) does not strongly depend on the modes. The reasonable explanation of this result is as follows: The

*Im*(

*k*

_{y}

^{2}) is small compared with

*Im*(

*k*

_{x}

^{2}) for both the TE

_{01}and TE

_{02}modes, and the

*Im*(

*k*

_{x}

^{2}) does not depend strongly on the modes. Numerical results also show this result.

By comparing the results obtained by the SPP waveguide with those obtained by the PEC waveguide, the following fundamental characteristics are shown in Figs. 2 and 3.

- The sizes that give the cut-off condition are significantly decreased from those of the PEC waveguide.
- The normalized phase constant can become larger than unity, i.e., the phase velocity in the waveguide can be smaller than the light velocity in free space.
- The propagation constant depends on the waveguide-width, even for TE
_{01}and TE_{02}modes. - The EIM is appropriate for calculation of the propagation constants in the propagation region.

## 5. Dependence of propagation constant on the waveguide width

The dependences of the normalized phase constant *β* and the attenuation constants *α* on the waveguide width *a*/*λ* with waveguide-height *b*/*λ* as a parameter are shown in Fig. 4 for the TE_{01} mode and in Fig. 5 for the TE_{02} mode. The bold black lines also show the results for the PEC waveguide, which are independent of the waveguide width *a*/*λ*. In Fig. 4, the open symbols denote the results obtained by the EIM. In the case of the TE_{01} mode, the PEC waveguide is below the cut-off condition with *α*=1.732, *β*=0 for *b*/*λ*=1/4, is in the cut-off condition with *α*=*β*=0 for *b*/*λ*=1/2, and is below the propagation condition with *α*=0, *β*=0.866 for *b*/*λ*=1.0, as shown in Fig. 4. The SPP waveguide falls below the propagation condition in the region *a*/*λ*<0.06, even for the case of *b*/*λ*=1/4, and is below the propagation condition for *b*/*λ*=1/2 and *b*/*λ*=1.0, as shown in Fig. 4. For the case of the TE_{02} mode, the PEC waveguide is below the cut-off condition with *α*=1.732 and *β*=0 for *b*/*λ*=1/2 and in the cut-off condition with α=*β*=0 for *b*/*λ*=1.0. The SPP waveguide enters the propagation condition in the range of *a*/*λ*<0.0445 for *b*/*λ*=1/2 and is below the propagation condition for b/*λ*=1.0. The results shown in Figs. 4 and 5 indicate that it is possible to cause the SPP waveguide fall below the cut-off condition by increasing the waveguide width *a*.

When the dissipation of the metal is neglected, we can obtain the dependence of cut-off wavelength on the waveguide-width by MoL that can be compared with those shown in the references [14, 15, 17] as shown in Fig. 6. The results also shows the validity of the method used in this paper.

## 6. Dependence of propagation constant on the wavelength

The dependences of the normalized phase constant *β* and attenuation constant *α* of the SPP rectangular hollow waveguide made of gold on the wavelength are shown in Figs. 7 and 8, respectively, and agreement with earlier calculations [9, 15, 17]. The dependence of relative complex permittivity of the gold used in the calculation is shown in the inset in Fig. 7. The results obtained by the EIM are shown by open symbols and those obtained by the PEC waveguide are shown by the bold black curves in Figs. 7 and 8. We can find the propagation region of the wavelength for the given size of SPP waveguide in these figures. As an example, it is difficult to find the region of small attenuation for the small waveguide of 60 nm×120 nm because of the large dissipation in the small wavelength. For the small waveguide of 60 nm×120 nm, the difference in the results between EIM and MoL is significant.

## 7. Field distributions

The mode functions of the SPP rectangular hollow waveguide can be obtained by MoL. Typical distributions of electric field components *Re*[*E _{x}(x,y)*],

*Re*[

*E*], and

_{y}(x,y)*Re*[

*E*] of the TE

_{z}(x,y)_{01}and TE

_{02}modes of the SPP rectangular cross section are shown in Fig. 9 for the case in which the waveguide is given by

*λ*=652 nm and the cross section is given by

*a*×

*b*=200 nm×652 nm.

As shown in Fig. 9, the guided-mode is mainly composed of the *x* component. Since the PEC waveguide has only *E _{x}* components for TE

_{0n}modes, the

*y*and

*z*components of the electric fields are due to SPP propagation. The results shown in Fig. 9 (a) indicate the validity of the field distribution of Eq. (8).

## 8. Other metals

The difference in the propagation characteristics of the SPP waveguide among various metals should be investigated. Therefore, we consider waveguides constructed of silver (Ag), copper (Cu), aluminum (Al) and nickel (Ni) [22, 23] and calculate the dependences of the normalized phase and attenuation constants of a rectangular hollow waveguide *a*×*b*=60 nm×240 nm on the wavelength for only the TE_{01} mode. The results are shown in Figs. 10 and 11, respectively. The results for the PEC waveguide are also shown by the bold black curves in these figures. The cut-off wavelength of PEC waveguide is 480 nm in Figs. 10 and 11. The dependences of the relative permittivities of the metals are shown in the insets in Figs. 10 and 11. The dependence of the cut-off wavelength on the metals is shown. Except for silver, the attenuation constants are not small in the propagation region because of the dissipation of metal. Note that the attenuation constant of gold is much higher than that of silver in the propagation region shown in Fig. 10. We can see that the silver has the excellent characteristics of small α and large β in the short wavelength range (λ<600 nm) compared with other metals.

## 9. Conclusions

A numerical study on the complex propagation constants of a surface plasmon polariton (SPP) rectangular hollow waveguide made of the metals by the method of lines (MoL) has been performed. Fundamental, important, and interesting propagation characteristics, constants, and field distributions of the SPP rectangular hollow waveguide were investigated numerically in detail and were compared with the results obtained by perfect electric conductor waveguide. A new cut-off condition of the SPP waveguide has been proposed. The accurate numerical results obtained by MoL and approximate results obtained by the effective index method (EIM) were compared. The results obtained by MoL agree with the numerical results obtained by EIM, except in the case of the small waveguide. The differences in the propagation characteristics among the various metals are shown. The results of the present study are important for additional theoretical and experimental research into nanometric optical circuits as well as applications.

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