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.
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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  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 . 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) . 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 Hφ, and two electric field components Er and Ez. The only transverse magnetic field component Hφ 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  and using the continuities of Ez and Hφ at the interface, one can get the following eigen-equation [5,37,38]:
We first consider the first term of Eq. (1). Through a lot of numerical calculations, we find that the effective index neff is always about 1 provided that the radius R is not extremely small. That is to say, neff≈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.Eq. (2), one can get the following approximation
We now further consider the second term of Eq. (1). Unlike κm in the first term, the parameter κa in the second term changes apparently with the change of radius R, so does the ratio K1(k0κaR)/K0(k0κaR). In order to find an approximate expressions for κa, we need first to find a suitable approximate for K1(k0κaR)/K0(k0κaR). For convenience, we define the ratio K1(k0κaR)/K0(k0κaR) as a function f(u):
In terms of the parameter a and the function f(u), Eq. (1) can be approximately expressed as
We denote by fr(u) a rough expression for f(u). We choose fr(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 fr(u) approach 1 when u approaches infinity.
2) By use of the asymptotic properties of modified Bessel functions, one can prove that K1(u)~1/u, K0(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 fr(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 fr(u) and f(u) in the wide range from u = 0.001 to u = ∞, we let fr(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 fr(u) in the range of 0.001≤u≤10 are shown in Fig. 1 (a) . The corresponding relative deviation [f(u)-fr(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 fr(r) is only a rough expression for f(u). Obviously, fr(u) becomes invalid for smaller u, because the relative deviation becomes larger and larger in this case.
We denote by κar a rough expression for κa. Replacing the function f(u) and κa in Eq. (7) by fr(ur) and κar, respectively, one can get
Equation (12) has two roots in mathematics. However, only one of them has physical meaning. Because the field distribution of Hφ decays with the increase of radial coordinate r, κar should have a positive real part . Accordingly, one should choose the following rootEq. (4) and Eq. (13), respectively. The corresponding rough solution neffr for neff can be further obtained from the relation
To get an intuitive impression on the rough solution, we calculate the κar 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., λ0 = 0.6 mm). The corresponding εm value is εm = −6.3 × 105 + j2.77 × 106 according to a fitted Drude formula for copper . The scope of metal wire is chosen to be 0.01mm≤R≤104mm. For comparison, the exact values κa are also calculated by numerically solving Eq. (1). The values of κar and κa are shown in Fig. 2 (a) . The relative deviations for the real part and the imaginary part of κar are shown in Fig. 2 (b). The rough values neffr and the exact values neff are shown in Fig. 3 (a) . And the relative deviations for the real part and the imaginary part of neffr are shown in Fig. 3 (b).
From the above two figures, we can see that, in the chosen range of 0.01mm≤R≤104mm, the maximum relative deviation of κa is about 20%, and the maximum relative deviation of neffr 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 κaa the approximate expression for κa. Then the corresponding approximate expression ua for u is denoted by
We make the first-order Taylor expansion fa(u) for the function f(u) at the neighborhood of ur, which is given by Eq. (11). Accordingly, the value fa(ua) at the point ua can be written as
Replacing the function f(u) and κa in Eq. (7) by fa(ua) and κaa, respectively, one can get
Substituting Eq. (17) into Eq. (20), we finally obtainEq. (18), Eq. (11), and Eq. (14), respectively. To this end, we derive an approximately explicit formula for κaa. The corresponding approximate value neffa 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 κaa and the exact solution κa are shown in Fig. 4 . Similarly, the comparisons between the approximate solution neffa and the exact solution neff are shown in Fig. 5 . One can see that, in the chosen range of 0.01mm≤R≤104mm, the maximum relative deviations of κaa and of neffa 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 neffa 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 . 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 104 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≤104 mm in Figs. 2-7. In each case, the accuracy of the approximate solution at a radius larger than 104 mm is higher than that at the radius of 104 mm.
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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