## Abstract

An explicit formula for metal wire plasmon of terahertz wave is analytically derived. The derivation is based on the huge relative permittivities of nonmagnetic metals in the spectral region of terahertz wave, some important properties of modified Bessel functions, and a suitable Taylor expansion. The obtained formula is further checked by many numerical tests. We find that, for all 11 tested nonmagnetic metals, for the whole spectral region of terahertz wave, and for the wide radius range from 10 μm to infinity, the relative deviation for the effective index is always smaller than 5%. This good agreement clearly shows that the derived expression can be conveniently used for the analysis and design of metal wire plasmon of terahertz wave.

© 2009 OSA

## 1. Introduction

Terahertz (THz) wave, locating between the infrared and microwave bands in the electromagnetic spectrum, is one of the hot research topics. It is normally defined as the range from 0.1 to 10 THz (or correspondingly, from 30 μm to 3 mm in wavelength). In recent years, terahertz technology has shown potential applications in many fields, such as in sensing, imaging, and spectroscopy [1–3]. Among those research works, effective THz waveguides have attracted more and more interests [4–35]. In 2004, Wang and Mittleman [4] reported that a simple metal wire can effectively guide THz wave. Since then, many interesting theoretical and experimental works on metal wire THz waveguide have been carried out [5–21].

It was quickly shown that the THz waveguide effect of metal wire comes from the azimuthally polarized surface plasmon [5]. The effective index of this kind of metal wire plasmon is implicitly given in an eigen-value equation. To get the effective index, one has to use various numerical techniques to solve the related eigen-value equation [5–21]. Therefore, an explicit approximate formula for the effective index is welcome.

In this paper, we shall derive such an approximate explicit formula for metal wire plasmon of terahertz wave. We shall also further test the accuracy and the validity range of the formula. The paper is organized as follows. In Section 2, we shall provide a rough solution to the eigen-value equation and test its accuracy. In the derivation process, the huge relative permittivities of nonmagnetic metals in the spectral region of terahertz wave and some important properties of modified Bessel functions will be used. In Section 3, we shall further transform the rough solution to an approximate solution, and test the accuracy and the validity range of the latter. In the related derivation process, a suitable Taylor expansion and some important properties of the eigen-value equation will be used. And in Section 4, we shall conclude this paper. For simplicity, in this paper, we only discuss the case of nonmagnetic metals whose relative magnetic permeabilities are always 1.

## 2. A rough solution to the eigen-equation

For a flat metal-dielectric interface, there exists an electromagnetic bound state which is TM polarization, and this bound state is called surface plasmon (SP) [36]. While, in the case of metal wire, SP can also exist at a cylindrical metal-dielectric interface [37,38]. It has one magnetic field component H_{φ}, and two electric field components E_{r} and E_{z}. The only transverse magnetic field component H_{φ} indicates the TM polarization.

Consider a cylindrical metal wire surrounded by air. We are only interested in axially symmetrical eigenmodes, that is to say, the relations ∂**E**/∂φ = 0 and ∂**H**/∂φ = 0 hold in the cylindrical coordinates. For TM polarization of a nonmagnetic metal, by substituting the above-mentioned relations into Maxwell’s equations [39] and using the continuities of E_{z} and H_{φ} at the interface, one can get the following eigen-equation [5,37,38]:

_{a}= [(n

_{eff})

^{2}-1]

^{1/2}, κ

_{m}= [(n

_{eff})

^{2}-ε

_{m}]

^{1/2}. n

_{eff}is the effective index of the eigenmode, ε

_{m}denotes the relative permittivity of the metal. I

_{0}(.), K

_{0}(.), I

_{1}(.) and K

_{1}(.) are modified Bessel functions. k

_{0}= 2π/λ

_{0}, where λ

_{0}and k

_{0}denotes wavelength and wave number in free space, respectively. R is the radius of the metal wire.

We first consider the first term of Eq. (1). Through a lot of numerical calculations, we find that the effective index n_{eff} is always about 1 provided that the radius R is not extremely small. That is to say, n_{eff}≈1. On the other hand, the relative permittivity of a metal is huge in the spectral region of THz wave. By taking these two properties into account, one can obtain the following relation.

From Eq. (3) one can find that the first term of Eq. (1) is approximately a constant for a pre-given R. For convenience, we here define this constant as a:

We now further consider the second term of Eq. (1). Unlike κ_{m} in the first term, the parameter κ_{a} in the second term changes apparently with the change of radius R, so does the ratio K_{1}(k_{0}κ_{a}R)/K_{0}(k_{0}κ_{a}R). In order to find an approximate expressions for κ_{a}, we need first to find a suitable approximate for K_{1}(k_{0}κ_{a}R)/K_{0}(k_{0}κ_{a}R). For convenience, we define the ratio K_{1}(k_{0}κ_{a}R)/K_{0}(k_{0}κ_{a}R) as a function f(u):

In terms of the parameter a and the function f(u), Eq. (1) can be approximately expressed as

Equation (7) is the fundament of our further analytical derivation. In other words, our analytical work will be based on Eq. (7).

We denote by f_{r}(u) a rough expression for f(u). We choose f_{r}(u) as the following form

1) By use of the asymptotic properties of modified Bessel functions, one can prove that f (u)~1 + 1/(2u) for very large u. One can further find that both f(u) and f_{r}(u) approach 1 when u approaches infinity.

2) By use of the asymptotic properties of modified Bessel functions, one can prove that K_{1}(u)~1/u, K_{0}(u)~-ln(u/2) for very small u. However, the change of -ln(u/2) is much slower than that of 1/u. As a result, f(u) approaches infinity basically as the function 1/u does when u is very small. One can find that, except for a coefficient α, both f(u) and f_{r}(u) approach infinity with the same functional form when u approaches 0.

The constant α can be optimally chosen. Through many numerical tests, we find that u is basically larger than 0.001 when the radius R is larger than the wavelength in the spectral region of terahertz wave. To optimally control the deviation between f_{r}(u) and f(u) in the wide range from u = 0.001 to u = ∞, we let f_{r}(u) = f(u) at the point u = 0.01. Accordingly, the coefficient α is determined to be 0.2018. In the remainder of this paper, we shall use the relation

Both the functions f(u) and f_{r}(u) in the range of 0.001≤u≤10 are shown in Fig. 1 (a)
. The corresponding relative deviation [f(u)-f_{r}(u)]/f(u) is shown in Fig. 1 (b). One can see that the two functions agree well when u approaches 0.01 or u becomes very large. In the range of 0.01≤u≤10, the maximum relative deviation is about 25%. When u is smaller than 0.01, the deviation becomes apparently. In particular, the deviation becomes about 40% at the point u = 0.001. We consider the deviation of about 40% as an acceptable value because f_{r}(r) is only a rough expression for f(u). Obviously, f_{r}(u) becomes invalid for smaller u, because the relative deviation becomes larger and larger in this case.

We denote by κ_{ar} a rough expression for κ_{a}. Replacing the function f(u) and κ_{a} in Eq. (7) by f_{r}(u_{r}) and κ_{ar}, respectively, one can get

Substituting Eq. (8) and (9) into Eq. (10), one can obtain the following equation

whereEquation (12) has two roots in mathematics. However, only one of them has physical meaning. Because the field distribution of H_{φ} decays with the increase of radial coordinate r, κ_{ar} should have a positive real part [5]. Accordingly, one should choose the following root

_{effr}for n

_{eff}can be further obtained from the relation

To get an intuitive impression on the rough solution, we calculate the κ_{ar} values, as a function of R. The metal is chosen to be copper and the frequency is chosen to be 0.5 THz (i.e., λ_{0} = 0.6 mm). The corresponding ε_{m} value is ε_{m} = −6.3 × 10^{5} + j2.77 × 10^{6} according to a fitted Drude formula for copper [40]. The scope of metal wire is chosen to be 0.01mm≤R≤10^{4}mm. For comparison, the exact values κ_{a} are also calculated by numerically solving Eq. (1). The values of κ_{ar} and κ_{a} are shown in Fig. 2 (a)
. The relative deviations for the real part and the imaginary part of κ_{ar} are shown in Fig. 2 (b). The rough values n_{effr} and the exact values n_{eff} are shown in Fig. 3 (a)
. And the relative deviations for the real part and the imaginary part of n_{effr} are shown in Fig. 3 (b).

From the above two figures, we can see that, in the chosen range of 0.01mm≤R≤10^{4}mm, the maximum relative deviation of κ_{a} is about 20%, and the maximum relative deviation of n_{effr} is about 40%. These deviations are rather large. Therefore, the rough solution needs to be further improved.

## 3. Transform from the rough solution to an approximate solution

In this Section, we shall use a suitable Taylor expansion and the properties of modified Bessel functions to derive an approximate solution with high accuracy. For convenience, we denote by κ_{aa} the approximate expression for κ_{a}. Then the corresponding approximate expression u_{a} for u is denoted by

We make the first-order Taylor expansion f_{a}(u) for the function f(u) at the neighborhood of u_{r}, which is given by Eq. (11). Accordingly, the value f_{a}(u_{a}) at the point u_{a} can be written as

_{r}) is given by

Replacing the function f(u) and κ_{a} in Eq. (7) by f_{a}(u_{a}) and κ_{aa}, respectively, one can get

Comparing Eq. (19) with Eq. (10), we obtain:

Substituting Eq. (17) into Eq. (20), we finally obtain

_{r}), u

_{r}, and κ

_{ar}are given by Eq. (18), Eq. (11), and Eq. (14), respectively. To this end, we derive an approximately explicit formula for κ

_{aa}. The corresponding approximate value n

_{effa}for the effective index can be further obtained from the relation

To test the validity of the approximate solution, we compare it with the corresponding exact result, which is obtained by numerically solving Eq. (1). The metal, the frequency, and the range of radius are chosen as those in Section 2. The comparisons between the approximate solution κ_{aa} and the exact solution κ_{a} are shown in Fig. 4
. Similarly, the comparisons between the approximate solution n_{effa} and the exact solution n_{eff} are shown in Fig. 5
. One can see that, in the chosen range of 0.01mm≤R≤10^{4}mm, the maximum relative deviations of κ_{aa} and of n_{effa} are only about 2% and 3%, respectively. These deviations are much lower than those in the rough solution.

In addition, to see whether the approximate solution is valid in the whole range of terahertz radiation, we further make more numerical tests on other bands of THz, especially on the two ends. The results of n_{effa} for copper on 0.1THz and 10THz are shown Fig. 6
and Fig. 7
. From them, we can see that the approximate solution performs well in the whole range of terahertz radiation, especially on the higher frequency. The reason is that a higher frequency leads to a larger absolute value of u and better results.

To further test the applicability of our formula on more metals, we make numerical tests on other nonmagnetic metals mentioned in Ref [33]. We find that our formula performs also well for all other nonmagnetic metals of Al, Ag, Au, Mo, W, Pd, Ti, Pb, Pt, V. The maximum relative deviation of the effective index for all 11 tested nonmagnetic metals is smaller than 5% in the whole spectral region of THz wave when the radius of metal wire is in the wide range from 10μm to 10^{4} mm. We do not test the magnetic metals of Co, Fe and Ni because they are beyond the scope of this paper.

It should be pointed out that the approximate solution is actually valid for the wide radius range from 10μm to infinity, though its validity is only directly shown for the radius range of 10μm≤R≤10^{4} mm in Figs. 2-7. In each case, the accuracy of the approximate solution at a radius larger than 10^{4} mm is higher than that at the radius of 10^{4} mm.

## 4. Conclusions

In conclusion, we have obtained an explicit formula for metal wire plasmon of terahertz wave. This formula is valid for all the tested 11 kinds of nonmagnetic metals, for the whole spectral region of terahertz wave, and for the wide radius range from 10μm to infinity. For all the numerical tests, the relative deviation of the approximate formula for the effective index is smaller than 5%. The obtained formula can be used for the fast analyses and designs of metal wire plasmon of terahertz wave.

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