Quite recently, it was found that metal wires can effectively guide terahertz radiation. Based on the fact that the absolute values of the relative permittivities of metals in the spectral region of terahertz radiation are huge, we here analyse the properties of this kind of waveguide and explain the related experimental results. In particular, we show that the observed waveguiding is due to the propagation of an azimuthally polarized surface plasmon along the wire. Some related aspects, such as the choice of metal and the slowly decaying modal field, are also discussed. In particular, we show that, if a copper wire with a radius of 0.45 mm is used, the attenuation coefficient is smaller than 2×10-3 cm-1 in the whole range of 0.1~1 THz.
© 2005 Optical Society of America
The terahertz (THz) region of the electromagnetic spectrum is located between microwave and optical frequencies, normally defined as the range from 0.1 to 10 THz (or correspondingly, from 30 µm to 3 mm in wavelength). In recent years, THz radiation has attracted a lot of interests [1–9], because it offers significant scientific and technological potential in many fields, such as in sensing, in imaging, and in spectroscopy. However, waveguiding in this intermediate spectral region is still a challenge [10–16]. Both of the conventional metal waveguides for microwave radiation and the dielectric fibers for visible radiation cannot be used to effectively guide THz radiation. The obstacles come from the high loss from the finite conductivity of metals or the high absorption coefficient of dielectric materials in this spectral range. Quite recently, Wang and Mittleman  found that a simple metal wire can be used as effective THz waveguide. This finding paves the way for a wide range of new applications for THz sensing and imaging. In Ref. , the authors focused on the report of experimental observations, however, without a deep theoretical explanation.
In this paper, we present the needed theory and explain the related experimental results. In particular, we show that the observed waveguiding effect is due to the propagation of an azimuthally polarized surface plasmon (APSP) along the wire. In addition, we also discuss some related problems. In particular, we suggest the use of copper wires instead of stainless steel wires as more effective THz waveguides.
2. Explanations for THz metal wire waveguides
It is well known that there exists an electromagnetic bound state at a flat metal-dielectric interface. This bound state can only exist for the TM polarization and is called surface plasmon (SP) [18, 19]. SPs can be excited by periodic structures like metal gratings , and can propagate along flat metal-dielectric interfaces . Similar to THz radiation, SPs have also attracted much attention in recent years .
It is relatively less well known that SP can also exist at a cylindrical metal-dielectric interface [22–24]. For an SP at a flat metal-dielectric interface, there are one magnetic field component Hy, and two electrical field components Ex and Ez. The only transverse magnetic field component Hy indicates the TM polarization. For an SP at a cylindrical metal-dielectric interface, however, there are one magnetic field component Hϕ and two electric field components Er and Ez. The only transverse magnetic field component Hϕ indicates the TM polarization. The latter kind of SP can be reasonably called APSP, because the only magnetic field component Hϕ is angular.
Consider the eigenproblem of a cylindrical metal wire surrounded by air. We denote by ε and µ the relative permittivity and the relative magnetic permeability, respectively. For the metal, we use εm and µm. And for air, ε=µ=1. We denote by λ0, c, and R the wavelength in free space, the light speed in free space, and the radius of the metal wire, respectively. We assume a temporal factor exp(-jωt), where ω=2πc/λ0, is the angular frequency.
We are interested in axially symmetrical eigenmodes. For them, the relations ∂E/∂ϕ=0 and ∂H/∂ϕ=0 hold in the cylindrical coordinates. By use of these relations, one can separate the electromagnetic field into two families. One is the TE polarization, and the other is the TM polarization. For the former, there are three components Eϕ, Hr and Hz. For the latter, there are another three components Hϕ, Er and Ez. The two families of TE and TM polarizations are decoupling in this case. We now focus on the TM polarization. By substituting the relations ∂E/∂ϕ=0 and ∂H/∂ϕ=0 into Maxwell’s equations , one obtains
All of Hϕ, Er and Ez are functions of the two variables r and z. For the formulation of the eigenproblem, they can be written as Hϕ(r,z)=Hϕ(r)exp(jk0neffz), Er(r,z)=Er(r)exp(jk0neffz), and Ez(r,z)=Ez(r)exp(jk0neffz), respectively, where neff is the effective index of the eigenmode. Then the operator ∂/∂z can be replaced by jk0neff. In the remainder of this paper, we refer to Hϕ, Er and Ez specifically as Hϕ(r), Er(r) and Ez(r), respectively. By using the relation ∂/∂z=jk0neff in Eq. (3), one gets
where ρ=k0[(neff)2-µε]1/2r. The suitable solution for Ez has the form I0(k0κmr) in the metal and K0(k0κar) in air, because this solution approximately decays exponentially from the interface, where I0(.) and K0(.) are the generalized Bessel functions , κm=[(neff)2-µmεm]1/2 and κa=[(neff)2-1]1/2. Substituting this kind of solution into Eq. (5), and using the relations  dI0(k0κmr)/dr=k0κmI1(k0κmr) and dK0(k0κar)/dr=-k0κaK1(k0κar), one can determine the solution form of Hϕ. By using the continuities of Ez and Hϕ at the interface r=R, one can obtain the following eigenvalue equation
The existence of the solution of Eq. (7) results from the negative real part of εm. This negative value is due to the electron plasma contribution when the frequency is lower than the plasma frequency. Accordingly, this eigenmode is traditionally called SP. In particular, when R→∞, the relations I1(k0κmR)/I0(k0κmR)=1 and K1(k0κaR)/K0(k0κaR)=1 hold. By use of these properties, one can obtain, for very large R,
Eq. (8) is exactly valid for an SP at a flat metal-air interface. It can be expected that Eq. (8) is also approximately valid for an APSP at the interface of a very thick metal wire. In particular, for nonmagnetic metals with µm=1, Eq. (8) reduces to neff=[εm/(1+εm)]1/2. Eq. (8) is already enough for some qualitative analyses, though the exact neff value of the APSP should be obtained by numerically solving Eq. (7).
We now briefly discuss the dimensions of the APSP modal field. For simplicity, we discuss the only magnetic field component Hϕ(r). The Hϕ(r) field is proportional to K1(k0κar)/K0(k0κaR) outside the metal. As we shall point out below, neff is approximately equal to 1. As a consequence, κa is very small. Because κa is very small, then the field distribution K1(k0κar)/K0(k0κaR) decays very slowly with the radial coordinate r. Typically, the modal field has a width of about several tens times of the radius R outside the metal. In the metal, the Hϕ(r) field is proportional to I1(k0κmr)/I0(k0κmR). As we shall see below, |εm| is far larger than 1 in the THz spectral region. Using the relations |εm|≫1 and neff≈1, one can get the relation κm≈(-εm)1/2. As a consequence, the real part of κm is quite large. Typically, it has the order of magnitude of 1.0×103 or larger. This property leads to the fast decay of the modal field in the metal. Typically, the modal field can exist in the metal for only about 1 µm or less. Therefore, the modal field of an APSP has a very large beam width but with a hollow center.
To get an intuitive impression, we calculate the Hϕ(r) field distribution of an APSP of a copper wire with a radius of 0.45 mm. The frequency is 0.5 THz (the corresponding wavelength is 0.6 mm). The related calaulation method used here will be stated in Section 3.
From Fig. 1(a), one can see that the modal field outside the metal has a long tail of about 20 mm. In contrast, the modal field decays very fast in the metal. From Fig. 1(b), one can see that the penetration depth in the metal is smaller than 0.5 µm.
We now attribute the observed waveguiding of a metal wire  to the propagation of an APSP along the wire. In the following, we try to use this model to explain the related experimental results of Ref. .
1). It is well known that the absolute values of εm of metals are huge in the spectral region of THz radiation. This property can be deduced from the Drude model [25, 27]. Typically , -Re(εm) is larger than 104 and Im(εm) is larger than 105, where Re(.) and Im(.) indicate the real and the imaginary parts, respectively. By using the properties |εm|≫1 and |εm|≫µm in Eq. (8), one can find that neff≈1. This result explains the low loss of metal wire waveguides because Im(neff) is very small.
2). The relation neff≈1 holds for a very wide bandwidth, because |εm| is huge in the whole range of THz radiation. This property explains the very low dispersion of metal wire waveguides.
3). It can be proved that the ration |Ez/Er| is in the order of magnitude of |κa/neff| in air. By using the relation neff≈1, one can find that |Ez|≪|Er|. This property implies that the longitudinal field component Ez is negligible in air. As a result, the modal field is an approximate TEM field in air. Note that it is no longer true in the metal. In the metal, |Er|≪|Ez|. It is worth mentioning that, it is not the radial field component Er but the longitudinal field component Ez that leads to the attenuation of the eigenmode.
4). Because the longitudinal field Ez is negligible, the remaining only electric field component in air is Er. Then it looks like that the electric field is radially polarized.
5). The electrical field of the light source is linearly polarized, but the electric field of the APSP is approximately radially polarized. The polarization mismatch between them leads to a very low coupling efficiency, though some additional setup was used to help the coupling.
6). As we state above, the modal field is quite large. As a result, the guided propagation can be easily coupled between two waveguides that are in contact with each other.
7). As stated above, the modal field is slowly decaying outside the metal. As a result, almost all of the energy is propagated in air whose refractive index is 1. Also, neff≈1. These two properties imply that the Fresnel reflectivity at the end of metal wire is negligible. Therefore, the mode can propagate off the end of the waveguide.
8). The metal wire is axially symmetric. For axially symmetric electromagnetic fields, TM and TE polarizations are decoupling. Thus, the radial polarization of the electric field can be maintained during propagation, even after propagating off the end of the metal wire.
9). The beam width of the modal field is quite large because it decays slowly outside the metal. As a result, the waveguide can be only slightly curved. Otherwise, emission due to waveguide bending may happen.
Clearly, these excellent explanations confirm that the waveguide effect of a metal wire is indeed due to the propagation of an APSP. It is worth mentioning that, besides the zero-order SP (i.e., APSP) mentioned above, there also exist higher-order SPs . However, they are not really guided modes. In fact, those higher-order modes were called virtual radiative SPs in Ref. , in contrast to APSP, which is called real nonradiative SP in Ref. . Therefore, the existence of those virtual radiative SPs does not affect our analyses.
3. Further discussions
In Section 2 we have explained the experimental results of Ref. . In this Section we discuss some further problems.
As we state above, the low loss and the very low dispersion of an APSP come from the huge absolute values |εm|. In the spectral region of THz radiation, different metals still have different |εm|  though those values are all huge. For example, the value |εm| of copper is more than 10 times higher than that of iron . Generally speaking, the larger the value |εm|, the better the waveguide. In addition, the relative magnetic permeability µm also plays a role.
By use of a Taylor expansion of Eq. (8) for huge εm, one can prove that neff≈1-µm/(2εm). From this expression, one can deduce that, for the same εm, a magnetic metal will increase the loss by µm times. Therefore, the use of ferromagnetic metal [25, 28], such as iron and nickel, should be avoided. We suggest the use of nonmagnetic metals with very large |εm|. Copper, silver and gold are good candidates. Copper, of course, would be preferrable because it is much cheaper than silver and gold. We notice that stainless steel wires are used in the experiments of Ref. . Stainless steels can be magnetic or “nonmagnetic” [29, 30], depending on the kinds. In Ref. , there is no report about the µm value of the used stainless steel. Therefore, it is difficult to concretely calculate the waveguide parameters of the stainless steel wires used there . However, the µm value should be not smaller than 1 . Also, the |εm| value of a stainless steel should be smaller than that of copper, because the |εm| value of iron, which is the main chemical composition of stainless steel, is at least 10 times smaller than that of copper . Therefore, a copper wire waveguide is superior to a stainless steel wire waveguide.
As we state above, the exact neff value can only be obtained by numerically solving the eigenvalue equation (7), though Eq. (8) is very useful for qualititative analyses. It is known that both the functions I1(.) and I0(.) approximately increase exponentially with the increase of the real part of the argument. Unfortunately, the real part of the argument k0κmR is quite large in the investigated case. For example, in the example of Fig. 1, the argument k0κmR is as large as 6.21×103-j4.95×103. Because the argument k0κmR is quite large, numerical overflow will happen if one directly calculates the functions I1(k0κmR) and I0(k0κmR) that appear in Eq. (7). To solve this problem, we use the asymptotic expression 
where ν is the order and u is the argument. By using this relation in the ratio I1(k0κmR)/I0(k0κmR), the diverging factor exp(k0κmR) is removed and the overflow is overcome. Then the eigenvalue equation (7) can be solved by complex-root search methods. Local search methods would be effective. We use the Newton method to solve the eigenvalue problem. Once obtaining the eigenvalue neff, one can immediately obtain the complete modal field.
To see the influence of the finite radius of the metal wire, we calculate the effective indices neff of the APSPs at the interfaces of copper wires with different R. The frequency is chosen to be 0.5 THz (i.e., λ0=0.6 mm). The εm value is determined to be εm=-6.3×105+j2.77×106 according to a fitted Drude formula for copper . The relation µm=1 is used. The calculated neff values, as a function of R, are shown in Fig. 2. One can see that, just as expected, neff approach to the value determined by Eq. (8) for very large R. One can also see that, both the values of Im(neff) and Re(neff)-1 increase with the decrease of the radius R. Thick metal wires are sometimes inconvenient for certain applications, but with lower attenuation. Therefore, one should balance between the size and the attenuation when designing a concrete metal wire waveguide.
To see the dependency on the frequency, we calculate the neff values, as a function of frequency. The metal is chosen to be copper, and the radius R is chosen to be 0.45 mm. The frequency-dependent εm of copper is obtained from the corresponding Drude formula of Ref. . The calculated neff, as a function of frequency is shown in Fig. 3(a). By use of the relation α=k0Im(neff), one can further obtain the attenuate coefficient α. The calculated α value is shown in Fig. 3(b). One can see that the attenuation is smaller than 2×10-3 cm-1 in the range of 0.1~1THz. This result explicitly shows the superiority of a copper wire waveguide.
One may find that the attenuations shown in Fig. 3(b) here are smaller than the measured values of Ref. . Also, in our calculations, the attenuation increases with increasing frequency, but an opposite trend was observed in Ref. . These differences indicate that the dominant loss does not result from the attenuation of the eigenmode (i.e., APSP) itself but from other sources. We guess some other metal components that are not far away from the metal wire are probably the loss sources. As we state above, the modal field of an APSP is quite large. If some other metal components (especially magnetic metal components, note that optical tables are normally magnetic) are in the range of the modal field, then the loss will increase. In addition, through numerical calculations, we found that, for a certain metal wire, the width of the modal field increases with the decrease of the frequency. Then, the APSPs for low frequencies will have higher losses if some other metal components are not far from the metal wire, because larger modal fields have higher losses in this case.
As we state above, the beam width of an APSP is quite large, though the metal wire is thin. This property is important for the focusing application. One should be aware of the large beam width when choosing a focusing lens. If the lens is not large enough, the truncation of the modal field may happen and lead to the loss of a lot of energy. It is worth mentioning that APSPs are related to radially polarized ligh beams [31–33]. Radially polarized light beams can be used for many applications, such as microscopy, optical tweezer, and optical data storage. Clearly, a metal wire waveguide also represents a viable method to generate radially polarized light beams with high purity. The drwaback is the low efficiency.
We have attributed the observed THz metal wire waveguide to the propagation of an APSP, and explained the related experimental results of Ref. . Some further problems, such as the choice of metal and the influence of the finite radius of metal wire, have been also discussed. In particular, we have predicted that the attenuation coefficient is smaller than 2×10-3 cm-1 in the whole range of 0.1~1 THz if a copper wire with a radius of 0.45 mm is used. The work presented in this paper not only provides a theoretical basis for THz metal wire waveguides, but also hopefully further paves the way for a wide range of new applications for THz sensing and imaging.
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