An explicit formula for metal wire plasmon of terahertz wave

Jie Yang; Qing Cao; Changhe Zhou

doi:10.1364/OE.17.020806

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]:

\frac{ε_{m}}{κ_{m}} \frac{I_{1} (k_{0} κ_{m} R)}{I_{0} (k_{0} κ_{m} R)} + \frac{1}{κ_{a}} \frac{K_{1} (k_{0} κ_{a} R)}{K_{0} (k_{0} κ_{a} R)} = 0

where κ_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. I₀(.), K₀(.), I₁(.) and K₁(.) are modified Bessel functions. k₀ = 2π/λ₀, where λ₀ and k₀ 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.

κ_{m} \approx \sqrt{1 - ε_{m}} .

By use of Eq. (2), one can get the following approximation

\frac{ε_{m}}{κ_{m}} \frac{I_{1} (k_{0} κ_{m} R)}{I_{0} (k_{0} κ_{m} R)} \approx \frac{ε_{m}}{\sqrt{1 - ε_{m}}} \frac{I_{1} (k_{0} R \sqrt{1 - ε_{m}})}{I_{0} (k_{0} R \sqrt{1 - ε_{m}})} .

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:

a = \frac{ε_{m}}{\sqrt{1 - ε_{m}}} \frac{I_{1} (k_{0} R \sqrt{1 - ε_{m}})}{I_{0} (k_{0} R \sqrt{1 - ε_{m}})} .

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 K₁(k₀κ_aR)/K₀(k₀κ_aR). In order to find an approximate expressions for κ_a, we need first to find a suitable approximate for K₁(k₀κ_aR)/K₀(k₀κ_aR). For convenience, we define the ratio K₁(k₀κ_aR)/K₀(k₀κ_aR) as a function f(u):

f (u) = \frac{K_{1} (u)}{K_{0} (u)},

where

u = k_{0} κ_{a} R .

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

a + \frac{f (u)}{κ_{a}} = 0.

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

f_{r} (u) = 1 + \frac{α}{u},

where α is a constant that needs to be determined. The reasons for such a choice are as follows:

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 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 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

α = 0.2018.

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.

Fig. 1 (a) The comparison between f (u) and f_r (u) in the range of u from 0.001 to 10. The red curve is f(u), and the black curve is f_r(u). (b) The relative deviation between f (u) and f_r (u) in the range of u from 0.001 to 10.

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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

a + \frac{f_{r} (u_{r})}{κ_{a r}} = 0,

where

u_{r} = k_{0} κ_{a r} R .

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

a κ_{a r}^{2} + κ_{a r} + c = 0,

where

c = \frac{0.2018}{k_{0} R} .

Equation (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

κ_{a r} = \frac{- 1 - \sqrt{1 - 4 a c}}{2 a},

where a and c are explicitly given by Eq. (4) and Eq. (13), respectively. The corresponding rough solution n_effr for n_eff can be further obtained from the relation

n_{e f f r} = \sqrt{κ_{a r}^{2} + 1} .

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., λ₀ = 0.6 mm). The corresponding ε_m value is ε_m = −6.3 × 10⁵ + j2.77 × 10⁶ according to a fitted Drude formula for copper [40]. The scope of metal wire is chosen to be 0.01mm≤R≤10⁴mm. 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).

Fig. 2 (a) The comparison between the rough values κar and the exact values κa, for metal copper and 0.5 THz. The red curves are the exact values, and the black curves are the rough solutions. The dashed curves are Im(κ_a) and Im(κ_ar), and the solid curves are Re(κ_a) and Re(κ_ar). (b) The relative deviation of κar. The solid curve represents the relative deviation of Re(κar), and the dashed curve is the relative deviation of Im(κar).

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Fig. 3 (a) The comparison between the rough values n_effr and the exact values n_eff, for metal copper and 0.5 THz. The red curves are the exact values n_eff, and the black curves are the rough solutions n_effr. The dashed curves are Im(n_effr) and Im(n_eff), and the solid curves are Re(n_effr)-1 and Re(n_eff)-1. (b) The relative deviation of n_effr. The solid curve represents the relative deviation of Re(n_effr)-1, and the dashed curve is the relative deviation of Im(n_effr).

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From the above two figures, we can see that, in the chosen range of 0.01mm≤R≤10⁴mm, 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

u_{a} = k_{0} κ_{a a} R .

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

f_{a} (u_{a}) = f (u_{r}) + f^{'} (u_{r}) (u_{a} - u_{r}),

where the first-order derivative f '(u_r) is given by

f^{'} (u {}_{r}) = \frac{K_{1}^{2} (u_{r})}{K_{0}^{2} (u_{r})} - \frac{K_{1} (u_{r})}{K_{0} (u_{r}) u_{r}} - 1.

Replacing the function f(u) and κ_a in Eq. (7) by f_a(u_a) and κ_aa, respectively, one can get

a + \frac{f_{a} (u_{a})}{κ_{a a}} = 0.

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

\frac{1}{κ_{a a}} f_{a} (u_{a}) = \frac{1}{κ_{a r}} f_{r} (u_{r}) .

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

κ_{a a} = κ_{a r} \frac{f (u_{r}) - f^{'} (u_{r}) u_{r}}{f_{r} (u_{r}) - f^{'} (u_{r}) u_{r}},

where f '(u_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

n_{e f f a} = \sqrt{κ_{a a}^{2} + 1} .

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≤10⁴mm, 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.

Fig. 4 (a) The comparison between the accurate values κ_aa and the exact values κ_a, for metal copper and 0.5 THz. The dashed curve is Im(κ_a) and the signs “+” show Im(κ_aa). The solid curve is Re(κ_a) and the signs “o” show Re(κ_aa). (b) The relative deviation of κ_aa. The solid curve represents the relative deviation of Re(κ_aa), and the dashed curve is the relative deviation of Im(κ_aa).

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Fig. 5 (a) The comparison between the approximate values n_effa and the exact values n_eff, for metal copper and 0.5 THz. The dashed curve is Im(n_eff) and the signs “+” show Im(n_effa). The solid curve is Re (n_eff)-1 and the signs “o” show Re(n_effa)-1. (b) The relative deviation of n_effa. The solid curve represents the relative deviation of Re(n_effa)-1, and the dashed curve is the relative deviation of Im(n_effa).

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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.

Fig. 6 The comparison between the approximate values n_effa and the exact values n_eff, for metal copper and 0.1 THz. The dashed curve is Im(n_eff) and the signs “+” show Im(n_effa). The solid curve is Re(n_eff)-1 and the signs “o” show Re(n_effa)-1. (b) The relative deviation of n_effa. The solid curve represents the relative deviation of Re(n_effa)-1, and the dashed curve is the relative deviation of Im(n_effa).

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Fig. 7 The comparison between the approximate values n_effa and the exact values n_eff, for metal copper and 10 THz. The dashed curve is Im(n_eff) and the signs “+” show Im(n_effa), and the solid curve is Re(n_effa)-1 and the signs “o” show Re(n_eff)-1. (b) The relative deviation of n_effa. The solid curve represents the relative deviation of Re(n_effa)-1, and the dashed curve is the relative deviation of Im(n_effa).

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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 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.

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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An explicit formula for metal wire plasmon of terahertz wave

Abstract

1. Introduction

2. A rough solution to the eigen-equation

3. Transform from the rough solution to an approximate solution

4. Conclusions

References

Cited By

Figures (7)

Equations (22)

Optics Express