## Abstract

In this paper we explore the existence of electromagnetic surface bound modes on a perfect metal wire milled with arrays of subwavelength grooves. The surface modes are axially symmetric transverse magnetic (TM) waves and have the same polarization state with the dominant propagating surface plasmon polaritons on the real metal wires. The dispersion of the fundamental surface mode has close resemblance with the dispersion of the surface plasmon polaritons. Moreover, we note that for TM polarization this metallic structure can be equivalent to a dielectric coated metal wire with defined geometrical parameters and effective refractive index of the dielectric coating. This metallic structure is expected to have some potential applications in the optical research in microwave or THz region.

©2006 Optical Society of America

## Introduction

Recently, surface plasmon polaritons (SPPs) have attracted great attention for their relevant roles in subwavelength optics [1–6] and their applications in waveguides [7–14]. As we know, SPPs are surface-bound waves that only propagate along the surface of a conductor with finite conductivity [15] and the surface of a perfect electric conductor (PEC) which has infinite conductivity usually does not support surface plasmons. However, it has been shown that both one dimensional and two dimensional structured PEC surfaces can support surface-bound waves [16–20]. Further more, it was recently demonstrated that these surface waves have strong similarities with SPPs on the real metal surfaces [21–23].

In this letter, we explore the existence of surface bound states on a perfect metal wire with infinite periodic arrangement of subwavelength grooves. These grooves are circumferentially milled in the surface of the metal wire. This metallic structure can support axially symmetric transverse magnetic (TM) surface modes which have the same polarization state with the dominant propagating surface polaritons (DPSP) traveling along the real metal wires [24–26]. The dispersion of these surface modes has close resemblance with the dispersion of DPSP on the real metal wires. Interestingly, for TM polarization, this wave-guiding system can be regarded as a dielectric coated metal wire with defined geometrical parameters and effective refractive index of the dielectric coating. It is possible to use the structure to create a three dimensional (3D) effective dielectric medium with a high positive refractive index [18]. We also propose some potential applications for this metallic system in the optical research.

## 2. Theoretical analysis

First, consider arrays of grooves circumferentially milled in the surface of a perfect metal wire (see Fig. 1), the periodic structure is assumed to be infinite along the metal wire. *d* is the width of grooves and *p* the period of the array, *2a* and *2b* are the outer and inner diameter respectively, the depth of the grooves is defined as *h*=*a*-*b*. We assume vacuum for the groove regions and ambient environment around the metal wire. Because this system is a cylindrical structure, it is convenient to analyze the electromagnetic (EM) problem in the cylindrical coordinates.

We consider the eigenproblem for TM polarization, the magnetic filed component *H*_{z}
=0 indicates the TM polarization state. The procedure for solving the dispersion relation of surface modes supported by the system is the following. First the EM fields inside the grooves (in region 1) can be expressed as a superposition of circular parallel-plate waveguide modes supported by the grooves. As we assume that the wavelength of light is much larger than the width of the grooves, in the modal expansion of the fields inside grooves we only consider the fundamental waveguide mode. It is a propagating TEM mode with the electric field pointing in the z direction and the EM fields inside grooves hold the relations *H*_{z}
=0 and *E*_{r}
=*E*_{φ}
=0. Substitute the relations into Maxwell’s equations [27], we find that ∂*H*/∂*φ*=0 and ∂*E*/∂*φ*=0 which indicate that the EM fields inside grooves are independent of *φ* and have homogeneous angular distribution, thus have axially symmetric property. The magnetic field component *H*_{φ}
in the *mth* groove has the following form

$$\mathrm{for}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\begin{array}{c}\mid z-mp\mid \le \frac{d}{2}\\ b\le r\le a\end{array}$$

Where *ω* is the frequency, *C* is the speed of light. *A*^{m}
and *B*^{m}
are amplitudes of forward (in the r direction towards the *z* axis) and backward (in the r direction and deviates from the *z* axis) propagating TEM cylindrical waves inside the *mth* groove.
${H}_{\nu}^{\left(1\right)}\left(\frac{\omega}{c}r\right)$
and
${H}_{\nu}^{\left(2\right)}\left(\frac{\omega}{c}r\right)$
represent *νth* order Hankel functions for the first and second kind respectively. Here, we emphasize that in any waveguide system with cylindrical symmetry, the angular momentum is a conserved quantity and denoted by *ν* (*ν*=0, 1, 2…). In our case, the EM fields inside grooves are of homogeneous angular distribution, so *ν*=0.

Then, the EM fields outside the metal wire (in region 2, in the vacuum ambient environment) can be expressed as sum of diffractive waves. Because we are interested in surface modes which are bound state, all diffracted waves outside the metal wire are evanescent in the transverse direction in vacuum. The EM fields of each diffraction order has a form *K*_{ν}
(·) (modified Bessel functions) which decays in the region 2 from the metal-vacuum interface. As the angular momentum *ν* keep conservation in this system, the value of *ν* outside the metal wire should keep the same as that inside grooves. The magnetic field component *H*_{φ}
outside the metal wire has the form

$$\mathrm{for}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}r\ge a$$

Where *C*_{n}
is the coefficient associated with the diffraction order n.
${\beta}_{n}=\beta +\frac{2\pi n}{p}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}(n=-\infty ,...0,...\infty )$
, which is the momentum in the z direction of the nth diffraction orders, *β*=*k*_{z}
is the propagation constant along the z axis.
${\tau}_{n}=\sqrt{{k}_{0}^{2}-{\beta}_{n}^{2}}$
is the transverse momentum, here *k*
_{0} is the wave vector in vacuum. For evanescent diffractive waves, *k*
_{0}<*β*_{n}
so the transverse momentum *τ*_{n}
is purely imaginary. By substituting the relation: *H*_{z}
=0, ∂*H*/∂*φ*=0 and ∂*E*/∂*φ*=0 into Maxwell’s equations, one can determine all EM field components

Here, we denote by *ε* and *μ* the relative permittivity and relative permeability respectively, for vacuum *ε*=*μ*=1. From the above Eq. (6) and Eq. (7), we can determine the electric fields in terms of the magnetic field component *H*_{φ}
. By matching the boundary conditions (at *r*=*b*, *E*_{z}
must be zero; at *r*=*a*, continuity of *E*_{z}
at every point of the unit cell, and continuity of *H*_{φ}
only at the grooves location), we find the eigenmodes satisfy the dispersion relation

Where J and N are Bessel function and Newmann function, respectively. *f*=*d*/*p*,
${g}_{n}=\mathrm{sin}c\left(\frac{{\beta}_{n\xb7d}}{2}\right)$
.

From the above discussion we know that the surface waves supported by this structure have axially symmetric property and only one magnetic field component *H*_{φ}
, so they are axially symmetric transverse magnetic (TM) modes (we also call them azimuthally symmetric zeroth order TM modes [25, 26]).

It is worth commenting on the similarities between the surface modes in this case and dominant propagating SPPs (DPSP) along the real metal wires. In fact, the DPSP is the fundamental mode. Higher order SPPs modes can also be excited on the real metal wires, however, most of the field energy of these higher order modes is inside the metal, their attenuation are much larger than the DPSP [24]. The DPSP is the azimuthally symmetric zeroth order TM mode [24, 25] which has the same polarization state with the surface modes supported by the structure in our case. In Fig. 2, we plot the dispersion relation of the surface modes by numerically solving Eq. (8) for the case d/p=0.1, b/P=2.6, a/b=1.3. The dispersion has close resemblance with the dispersion of the DPSP supported by the real metal wires [28–30]. In the low frequency range, both of the dispersion curves approach the air line; while at large propagating constant *k*_{z}
, in the DPSP band the eigen frequencies *ω* approach the surface plasmon frequency *ω*_{sp}
[31]; whereas in our case *ω* approach the frequency *ω*_{CM}
which is the frequency location of a cavity waveguide mode inside the circular grooves. The frequency locations of the different cavity waveguide modes approximately correspond to the condition

In the limit *h*=*a*-*b*<<*b*, and *d*<<*p*, by using the asymptotic approximation of Bessel function and Newmann function, from the Eq. (10) the resonant frequencies of the cavity waveguide modes approximately satisfy the condition
$\mathrm{cot}\left(\frac{\omega}{c}h\right)=0$
, it coincides with the cavity waveguide modes inside the rectangular metallic grooves [21, 32].

## 3. Equivalent dielectric coated metal wire

It is interesting to note that for TM polarization the wave-guiding properties of the system can be equivalent to a dielectric coated metal wire with defined geometrical parameters and an effective refractive index of the dielectric coating (see Fig. 3). In the dielectric coated metal wire the TM modes are also axially symmetric waves and satisfy the dispersion relation [33].

The thickness of the dielectric coating is(*a*′-*b*′), the radius of the coated metal wire is *a*′. *n*
^{2} and ${n}_{0}^{2}$ are refractive index of the dielectric coating and vacuum respectively. *T*
^{2}+*β*
^{2}=(*ωn*)^{2}, *β*
^{2}-*τ*
^{2}=*ωn*
_{0})^{2},in which *β*=*k*_{z}
denotes the propagation constant of the surface modes.

In the dielectric coated wire, the cutoff of surface modes is due to the condition *τ*=0, which indicates that the surface waves are no longer bound states and become radiative modes. The cutoff frequencies of these modes correspond to the condition

In our case (the structured metallic system), by tuning the geometrical parameters of the structured metal wire, fundamental and higher order surface modes (TM_{0n}, n=0, 1, 2…,) can be excited, and the modal cutoff is due to the condition *τ*_{n}
=0 (n=0), which implies that when the zeroth-order diffracted wave outside the metal wire is not evanescent, there will exist a channel for light getting rid of the confinement around the metal wire. The cutoff frequencies satisfy the condition

We define the effective refractive index
$n=\frac{p}{d}$
for the dielectric coating of the equivalent coated wire. In order to make the modal cutoff frequencies of the equivalence keep the same with those of the structured metal wire, the geometrical parameters of the equivalent should have the relations
$a\prime =\frac{a}{\sqrt{{n}^{2}-1}}$
and
$b\prime =\frac{b}{\sqrt{{n}^{2}-1}}$
[see Fig. 3(b)]. Figure 4 shows the dispersion curves of the first two modes (TM_{00} and TM_{01}) in the first Brillouin zone of the structured metal wire and its equivalent dielectric coated wire. As the effective
$n=\frac{p}{d}$
increases, the dispersion curves of the equivalent system asymptotically approach those of the structured metal wire.

Figure 5 shows the modal patterns of the fundamental surface modes (TM_{00} mode) of the structured metal wire and the equivalent dielectric coated wire with an effective refractive index n=2.5. The normalized frequency of the surface mode is *ω*=0.05 in units of
$\frac{2\pi C}{p}$
, which is in the overlap region of the dispersion curves in Fig. 4(a). From Fig. 5(a) we can see that waves travel along the metal wire which is milled with arrays of subwavelength grooves, these waves are bounded around the metal wire and decay outside the grooves. Through evanescent coupling from one groove to the next, energy of light can create groove-to-groove propagation. By comparing with the equivalent dielectric coated wire, it is found that the two systems have almost the same external fields and modal wavelength (the spatial periodicity) of the surface modes. If the modal wavelength is much larger than the periodicity *p* of the array of grooves, the surface waves will be insensitive to the periodic geometry on the metal wire and the array of subwavelength grooves can be regarded as a quasi uniform effective medium along the propagation direction. This effective dielectric medium model is very valid in the long wavelength limit to characterize the wave-guiding properties along the periodic structures [18, 21].

## 4. Applications

The analysis and results presented in this paper are based on the perfect electric conductor (PEC) model. For frequencies range from microwave to THz regime, the PEC model is a feasible approximation for analyzing the propagation of surface modes supported by the structured metal surfaces, since the absorption loss and plasmonic effects in this frequency range can be largely neglected and it is valid to disregards the Zenneck wave solution for the structured metal surfaces [34].

The wave-guiding properties of the surface modes supported by the metallic system may have some potential applications. For example, quite recently researchers found that a simple metal wire which supports THz SPPs can be used as effective THz waveguides [12, 13]. However, the polarization mismatch between the light source and SPPs on the metal surface lead to a very low coupling efficiency [25, 26, 35], because the electric fields of light sources are generally linearly polarized but the DPSP on the metal surface is axially symmetric TM mode, the electric field is approximately radially polarized [25]. The polarization mismatch may be improved by the array of subwavelength grooves as a coupler integrated in the simple metal wire, since the SPP-like waves can be excited on the structured metal wire by the external light source. It is necessary to note that the notions of the improved input coupler and effective SPPs on the structured metal surfaces for THz radiation were hinted at by the group of Ajay Nahata in their recent works [36, 37, 38]. Notably, in the works of Ajay Nahata et al. they proposed a very promising method to facilitate the fabrication of the metal wires with subwavelength features: By fabricating arrays of linear grooves into the surface of a metal sheet and rolling the sheet about the axis perpendicular to the groove length, one can have a solid cylindrical metal conductor with circular grooves [38]. The micro fabrication techniques on surfaces of the planar structure are much easier than the fabrication on the cylindrical surfaces [39]. Here, we would like to point out that the theoretical analysis in our paper is based on the structures with infinite periodic elements (or the structures have great many periodic elements and can be approximately treated as infinite). So the theoretical treatment in our paper may not be suitable for the cases in the papers of Ajay Nahata et al., in which the structures have several finite periodic elements.

Another application is as waveguide filters integrated on the simple metal wire waveguides, since there exist stop-band between guided bands of the surface modes (see Fig. 4), waves with frequencies in the stop-band region can not propagate along the metal wire. Moreover, this system can be regarded as a dielectric coated conducting wire and can create 3D effective dielectric medium with a very high positive refractive index [see Fig. 4(b)], this implies that this system can support extra-slow speed for the surface wave propagation, and may be used as optic delay-line systems.

Moreover, at microwave or THz frequencies, the dispersion relation and sub-wavelength confinement of the effective SPPs on the structured metal wires do not rely on the finite conductivity of the metal, but are basically due to the geometrical parameters of the surface structure. In this way, the propagation characteristics of these effective SPPs can be controlled by the periodic surface geometry. Highly localized SPPs on the structured metal wires can be sustained in the THz region to overcome the considerable radiation loss due to bends, non-uniformities or nearby objects in the weakly-guided THz bare metal wires. More interestingly, by tapering the corrugated metal wires, the energy concentration and wave-guiding properties of these effective SPPs can be effectively and dramatically changed. This conical geometry may open up a new way for channeling THz radiation to micron-scale volumes for near-field imaging, spectroscopy and sensing applications [40, 41].

## 5. Conclusion

In conclusion, we have shown that the axially symmetric SPP-like waves can be exited on a perfect metal wire milled with arrays of subwavelength grooves. Moreover, for TM polarization, this system can be equivalent to a dielectric coated metal wire with defined geometrical parameters and an effective refractive index of the dielectric coating. This metallic structure is expected to have some potential applications for the optical research at microwave or THz frequencies. The results presented here are valid and can be directly applied in the wavelength range from microwave to THz region.

## Acknowledgments

We wish to thank Dr. J. B. Pendry from Imperial College for his encouragement and the careful reading of our manuscript, and we acknowledge careful reading of this manuscript and helpful suggestions by Dr. Weili Zhang and Dr. Jiaguang Han from Oklahoma State University. This work was supported by the National Key Basic Research Special Foundation of China (Grant Nos. 2003CB314904 and 2006CB806000), the National Natural Science Foundation of China (Grant No. 60678012) and National Natural Science Foundation of China (Grant No. 60578037).

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**41. ****Note**: After submitting our manuscript to Optics Express on 09/18/2006, we found that S.A. Maier et al. proposed basically the same structure but focusing on different aspects other than those in our manuscript. We think that the excellent work proposed by S. A. Maier et al. will open up possibilities to important applications in the THz optical research, and it is very necessary to incorporate their work into our paper in order to enrich our contents and emphasize the potential applications of this metallic structure in the optical research.