## Abstract

We introduce the technique of perturbation of boundary condition to the problem of terahertz two-wire metallic waveguides with different radii. Based on the quasi-TEM analytical mode fields derived by use of Möbius transformation, a concise expression for the complex effective index is obtained analytically. The expression is in good agreement with the simulation result. Further, the dispersion and attenuation are obtained from the expression. In addition, we find a zero value point of the group velocity dispersion around 1.268 THz. The results show that the technique of perturbation of boundary condition is helpful in the analysis and design of terahertz metal waveguide.

© 2015 Optical Society of America

## 1. Introduction

Terahertz (THz) waves, typically defined as electromagnetic radiation in the frequency range of 0.1–10 THz (or from 30 *μ*m to 3 mm in wavelength), has attracted a lot of interest owing to its potential applications in spectroscopy, imaging, communications, sensing, security and defense [1–5]. In particular, THz waveguide, one important branch of terahertz technology, has become an active research area. Different types of THz waveguides have been proposed including highly-doped silicon plasmonic waveguides [6,7], porous dielectric waveguides [8,9], and metallic waveguides [10–36]. In 2004, Wang and Mittleman reported that a simple single metal wire can effectively guide THz wave [10]. Since then, many interesting works on the metal wire waveguides have been carried out [11–26]. The THz waveguide effect of the single metal wire is referred to as azimuthally polarized surface plasmons [11], cylindrical surface plasmons [12,13], or Sommerfeld wave [14]. Surface plasmons confine to and travel along flat metal-dielectric interfaces have been known for decades [37], which can be excited by periodic structures like metal gratings [38]. Unlike surface plasmons at flat interface, azimuthally polarized surface plasmons have radially polarized electric fields and are difficult to be coupled from the commonly used linearly polarized THz sources by direct end-fire input coupling [25]. Thus, efforts have been made to enhance the coupling efficiency, for example by use of radially symmetric antennas [25], and by a segmented half-wave-plate mode converter with a polarization-controlled beam [26].

The two-wire metallic waveguides with identical radii combines low loss and efficient coupling properties [27–29]. The approximately linearly polarized mode supported by this type of waveguides is very similar to the field emitted from the commonly used linearly polarized source, resulting in efficient coupling of the electromagnetic energy from the THz source into the mode [29]. Two-wire metallic waveguide have been used to connect two optoelectronic chips with prototype 250 GHz bandwidth [30], and to enhance the throughput by directly generating a THz field within the waveguide [31]. Also a tapered dual elliptical plasmon waveguide [32] has been suggested as a high efficient coupler from an approximate plate waveguide to a two-wire waveguide for THz waves. Low loss foams and porous dielectric fibers are proposed to support the two-wire waveguides [33,34].

In order to analyze the dispersion and attenuation of the THz waveguide, complex effective index *n _{eff}*, whose definition can be found in [32], needs to be obtained. The real part Re(

*n*) of the complex effective index is directly related to the dispersion and the imaginary part Im(

_{eff}*n*) to the attenuation. For two-wire metallic waveguide, even with identical radii, it is a big challenge to obtain

_{eff}*n*. However, owing to the fact that the absolute values of the relative permittivities of metals in the THz region are large, the two-wire metallic surface plasmon has quasi-TEM mode field. The field of the surface plasmon outside the metals can be well approximated by the TEM mode field just like the metals are perfect conductors, which is a great simplification by involving only the 2D Laplace equation [28]. Based on the mode field distribution, one can approximately obtain the attenuation, by use of energy conservation [28,35]. The method is direct and has intuitive appeal, but fails to yield the value of Re(

_{eff}*n*), thus unable to estimate the dispersion analytically. Therefore, an explicit expression for the complex effective index is welcome.

_{eff}In this paper, we are able to solve the above problem by use of the technique called perturbation of boundary conditions, for not only the THz two-wire metallic waveguides with identical radii, but also their general form: THz two-wire metallic waveguides with different radii. The paper is organized as follows. In Section 2, we shall introduce the perturbation of boundary conditions and provide an integral solution of the *n _{eff}* for THz two-wire metallic waveguide with different radii. The whole waveguide is embedded in one dielectric cladding. In Section 3, we shall further transform the integral solution to a simple approximate analytical one for the case of THz two-wire metallic waveguides. It comes from the fact that the mode fields of the waveguide can be approximated by the TEM mode fields derived by the Möbius transformation. In Section 4, the analytical results (both mode fields and complex effective indices

*n*) for the waveguides with or without a dielectric cladding are compared with the simulation ones. In Section 5, we calculate the group velocity dispersion (GVD) and the attenuation. In Section 6, we discuss the validity range of our theoretical model and then we conclude this paper in Section 7. In this paper, we consider the single mode propagation of monochromatic harmonic waves. The time factor is

_{eff}*e*and can be omitted by replacing ∂/∂

^{-iωt}*t*by –

*iω*, and the gradient along

*z*axis ∂/∂

*z*can be replaced by

*ik*, where parameter

_{z}*i*is the imaginary unit,

*ω*is the angular frequency, and

*k*=

_{z}*k*

_{0}

*n*is the complex propagation wavenumber. The wavenumber in the free space

_{eff}*k*

_{0}=

*ω*/

*c*= 2π/

*λ*, where

*c*is the speed of light and

*λ*is the wavelength in the free space. For simplicity, we only discuss the nonmagnetic case, namely the relative permeabilities of the metal and the cladding are always 1.

## 2. Perturbation of boundary conditions

We now introduce the perturbation of boundary conditions. The method of perturbation of boundary conditions [39] is capable of, at least in principle, obtaining answers to any desired degree of accuracy, although we apply it only to the lowest order. The basic idea is as follows: First, one consider the (unperturbed) scalar potential *ψ*_{0} for case of perfectly conducting wall. The solution *ψ*_{0} is usually solvable. Second, one should then apply the perturbed boundary condition by substituting the scalar potential *ψ* and ∂*ψ*/∂*n* at the real metal wall with the initial unperturbed solutions *ψ*_{0} and ∂*ψ*_{0}/∂*n*. Finally, one could derive the complex effective index *n _{eff}*. For the TEM mode in the two-wire metallic waveguide, we set a modified scalar potential Φ

*to be*

_{h}*ψ*and set the scalar potential Φ

_{0}to be

*ψ*

_{0}.

Let us consider the problem of THz two-wire metallic waveguiding. In order to make it more general, we assume that two wires with different radii are embedded in a dielectric cladding, as shown in Fig. 1. The THz wave propagates along the *z* axis, whose unit vector is **z**. The vector **n** and **τ** are the unit normal from the metal and the unit tangent along the curve *C* in the cross section, respectively. The vectors **z**, **n**, and **τ** satisfy the right-handed rule, namely **z** = **n** × **τ**. The permittivity of the metal wires *ε _{m}* is in the second quadrant for the time factor

*e*, that is Re(

^{-iωt}*ε*)<0, Im(

_{m}*ε*)>0.

_{m}*ε*is the relative permittivity of the cladding and the permeability

_{d}*μ*is 1. Then the index of the cladding

_{d}*n*satisfies ${n}_{d}^{2}$=

_{d}*ε*. The radii of the wires, and the distance between the two wires are much larger than the skin depth. They are not smaller than 10

_{d}*μ*m in our case. The above selection of the parameters is to guarantee the validity of the impedance boundary condition (see Appendix A).

The tangential electric field, **E**_{||} = *E _{τ}*|

_{C}**τ**+

*E*|

_{z}

_{C}**z**, on the metal wall is exactly zero when the metal is perfect conductor. The subscript

*z*denotes the longitudinal part of the field, and

*τ*denotes the tangential part of the field on the curve

*C*in the cross section. The subscript

*C*denotes the field on the curve. However, for real metal,

**E**

_{||}is not zero, although it is still much smaller than the normal one on the wall just outside the metal. The small tangential electric field

**E**

_{||}is the key to the problem of the two-wire metallic waveguiding, and is difficult to be carried out by directly solving the Helmholtz equation. However, for the metal with very large absolute permittivity |

*ε*|, there is a simple approximate relation between the tangential electric field

_{m}**E**

_{||}and the tangential magnetic field,

**H**

_{||}=

*H*|

_{τ}

_{C}**τ**+

*H*|

_{z}

_{C}**z**, on the metal wall (see Appendix A):

*μ*

_{0}and

*ε*

_{0}are the permittivity and permeability of the free space, respectively. Equation (1) is usually referred to as the impedance boundary condition (IBC) [39,40]. However, we choose permittivity

*ε*rather than conductivity

_{m}*σ*to describe the metal for convenience. It is more convenient to do that, because

*ε*of the metal can be estimated from the fitted Drude model in the THz region [41]. With the right-handed rule,

_{m}**z**=

**n**×

**τ**, one can easily get: For our quasi-TEM case,

**E**

*and*

_{n}**H**

*are the dominant field components on*

_{τ}*C*. That means |

*H*|

_{τ}*>>|*

_{C}*H*|

_{z}*. This results in |*

_{C}*E*|

_{z}*>>|*

_{C}*E*|

_{τ}*according to Eqs. (2) and (3), although |*

_{C}*E*|

_{z}*is already quite small compared with |*

_{C}*E*|

_{n}*. That is |*

_{C}*E*|

_{n}*>>|*

_{C}*E*|

_{z}*>>|*

_{C}*E*|

_{τ}*.*

_{C}For single mode propagation, in the dielectric cladding, the transverse components of the electric and magnetic fields can be determined by their longitudinal parts [39]:

*t*denotes the transverse parts of the fields and the operators. The longitudinal parts of

*E*and

_{z}*H*satisfy the 2D scalar Helmholtz equation:

_{z}*H*in the field. However in the two-wire waveguide,

_{z}*H*usually cannot be ignored. We operate dot product by

_{z}**τ**on both sides of Eq. (4). With Eq. (3), one can obtain a relation among ∂

*E*/∂

_{z}*τ*, ∂

*H*/∂

_{z}*n*, and

*H*on

_{z}*C*:

*E*can be expressed asFor the TEM mode with perfect conductor walls,

_{z}*E*=

_{z}*H*= 0. Then from Eq. (8), one can find that ${n}_{eff}{E}_{t}+z\times \left({Z}_{0}{H}_{t}\right)=0,$ and ${n}_{eff}{E}_{t}=-{\nabla}_{t}{\Phi}_{0},$ where Φ

_{z}_{0}is the scalar potential and satisfies the 2D scalar Laplace equation:The subscripts

*C*

_{1}and

*C*

_{2}denote the left and the right curves.

For the real metal walls with huge |*ε _{m}*|,

*E*and ${n}_{eff}{E}_{t}+z\times \left({Z}_{0}{H}_{t}\right)$ are not zero, and the 2D gradient of

_{z}*E*[Eq. (8)] is quite unlike $-{\nabla}_{t}{\Phi}_{0}$. However the absolute value of $\left[{n}_{eff}{E}_{t}+z\times \left({Z}_{0}{H}_{t}\right)\right]$ is very small compared with that of

_{z}*n*

_{eff}**E**

*and -*

_{t}**z**× (

*Z*

_{0}

**H**

*). That is to say, although the distributions of*

_{t}**E**

*and -*

_{t}**z**× (

*Z*

_{0}

**H**

*) are very similar to $-{\nabla}_{t}{\Phi}_{0}$, they are not exactly the same. In fact, there is a relation between*

_{t}**E**

*and*

_{t}**H**

*in the dielectric cladding (see Appendix B):*

_{t}**n**. Firstly, we substitute

*p*with

*E*and

_{z}*q*with Φ

_{0}, and apply the 2D scalar Laplace equation of Φ

_{0}shown in Eq. (9). The first term in left hand side of Eq. (11) is zero. Then by substituting ${\nabla}_{t}{E}_{z}$ with Eq. (8) and use of Eq. (10), We can get

**H**

*is almost zero for the quasi-TEM mode. Then -*

_{t}**z**× (

*Z*

_{0}

**H**

*) can be regarded as a 2D irrotational field, which means that ${\nabla}_{t}\times \left[-z\times \left({Z}_{0}{H}_{t}\right)\right]\approx 0.$ Therefore, -*

_{t}**z**× (

*Z*

_{0}

**H**

*) can be approximated byHere, we shall refer to Φ*

_{t}*as the modified scalar potential. Then*

_{h}*H*|

_{τ}*can be expressed from the normal gradient of Φ*

_{C}*. By use of Eq. (14), Laplace equation of Φ*

_{h}_{0}[Eq. (9)], and Green’s first identity [Eq. (11)], Eq. (13) can be reduced to

*n*. In order to obtain the solution, the value of the modified scalar potential Φ

_{eff}*and its normal gradient along*

_{h}*C*should be known first.

Since the electric and magnetic fields of the quasi-TEM mode of the two-wire real metal waveguide are very similar to that of the TEM mode of the two-wire perfect metal waveguide [28], we can apply the method of perturbation of boundary condition to the lowest order. That is to say, the modified scalar potential Φ* _{h}* can be well approximated by the unperturbed field Φ

_{0}. Φ

*can be expressed as*

_{h}## 3. Analytical expressions of the effective indices

We now focus on the potential Φ_{0} of the TEM mode. The conformal mapping as an important mathematical method is very useful for solving these problems [28,36]. By choosing an appropriate mapping, one can transform the inconvenient geometry into a much more convenient one. As one of the conformal mapping, the Möbius transformation [42] can directly maps the cross section of the two-wire waveguide to a simpler geometry of two concentric circles, similar to the cross section of a coaxial cable, as shown in Fig. 2.

The complex expression is as follows:

where*R*

_{1}and

*R*

_{2}denote the radii of the two wires, respectively, and

*D*denotes the distance of the two centers. The parameters

*a*and

*b*are the radii of the inner and outer circles of the coaxial cable cross section, respectively.

The scalar potential Φ_{0} in the gap region between the two concentric circles is given by

*w*in Eq. (21) from Eq. (18), one can get the potential Φ

_{0}in the cross section of the two-wire waveguide, and thus the transverse electric field can be derived from the gradient of the potential:

*w*space, the analytical expression of

*n*can be given (see Appendix C) by

_{eff}The first product factor in the right hand side of Eq. (23) indicates the contribution of the metal, cladding material and the wavelength. The second factor indicates that *n _{eff}*-

*n*is in inverse proportion to the geometry size. It means that smaller geometry size leads to higher loss, and vice versa. Thus, one needs to balance between geometry size and losses. The third factor is dimensionless and indicates the contribution of the geometry shape. For the two-wire perfect metal waveguide in the free space, |

_{d}*ε*| is infinity and

_{m}*n*=

_{eff}*n*= 1. The exact TEM mode without any loss and dispersion can exist in the waveguide. For the case with real metal, |

_{d}*ε*| is finite but still huge, |

_{m}*ε*| of copper for example is ~10

_{m}^{6}in the THz region, and it causes the low loss and dispersion of the mode in the waveguide. Generally speaking, the larger the value |

*ε*|, the better waveguide with lower loss. In addition, for the two-wire waveguide in the free space (

_{m}*n*= 1), one can also see from Eq. (23) that if the permittivity of metal

_{d}*ε*is a negative number, Im(

_{m}*n*) is zero. Then, the wave in the waveguide has no loss. The last two factors in the right hand side of Eq. (23) do not affect the zero result of Im(

_{eff}*n*) when the imaginary part of permittivity is zero. This zero result of Im(

_{eff}*n*) is irrelevant to the geometric size, which means one can achieve small mode field size with no loss by scaling down the geometric size.

_{eff}Equation (23) can be extended to other frequencies such as RF frequencies. In the RF frequencies, the permittivities of the metal *ε _{m}* are almost pure imaginary. Then for the waveguide in the free space (

*n*= 1), Eq. (23) becomes

_{d}Dielectric claddings are however needed to hold the two wire waveguide, yet they often have absorption losses in the THz region, which imply that the refractive index *n _{d}* in Eq. (23) is complex valued.

## 4. Comparison of the analytical results with the simulations

The mode fields and the effective indices *n _{eff}* of the two-wire waveguides of different radii have also been calculated by full-wave simulation with a commercial software (COMSOL Multiphysics 4.3b, RF Module) as a comparison to the analytical results. The eigenvalue solver was used to find modes of the two-wire waveguide in the simulation. The frequency is chosen to be 0.5 THz. For the real metal, copper, the relative permittivity

*ε*= –6.3 × 10

_{m}^{5}+ 2.77 × 10

^{6}

*i*according to a fitted Drude model [41]. In order to support the two wire waveguide, dielectric cladding is needed. As an example, we use the low loss material, polystyrene foam, as the cladding, whose complex index of refraction is

*n*= 1.0104 + 1.5059 × 10

_{d}^{−4}

*i*at 0.5 THz according to the fitted formula and Fig. 8 in [34]. The permeabilities of the materials are always 1.

In our study, we fix the radius of the right wire (*R*_{2}) at 150 *μ*m and vary the radius of the left wire (*R*_{1}) in the range: 150-1500 *μ*m, while keeping the width of the inter-wire gap (*D*-*R*_{1}-*R*_{2}) fixed at 200 *μ*m.

Just as an example, we describe the field distribution and polarization in the gap region of the waveguide, whose radius of the left wire *R*_{1} = 300 *μ*m, in the box with an area of 2 mm × 1.8 mm. Figure 3 shows the analytical normalized transverse electric field,$-{\nabla}_{t}{\Phi}_{0}$[Fig. 3(a)], simulated normalized transverse electric field, **E*** _{t}* [Fig. 3(b)], and the simulated normalized transverse magnetic field with a 90° rotation$-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$[Fig. 3(c)] for the two-wire waveguide in the free space. Figure 3(d) shows the simulated normalized transverse magnetic field also with a 90° rotation $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ for the waveguide with the polystyrene foam. The electric field in Fig. 3(b) is perpendicular to the metal surfaces, and approximately linearly polarized in the gap region. The distributions of the fields shown in Fig. 3 are almost identical. Likewise, the polarizations of the fields shown in Fig. 3 are almost identical. These results imply the validity of approximation of Eqs. (14) and (16). Figs. 3(c) and 3(d) show the rotated magnetic field in order to compare$-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$with $-{\nabla}_{t}{\Phi}_{0}.$ One can imagine that the distributions of the real magnetic fields are exactly the same as shown in Figs. 3(c) and 3(d), but the polarization with a 90° rotation. That is the magnetic fields are parallel to the metal surfaces, not perpendicular to them. It is worth mentioning that, as we state in Section 2, although $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ is very similar to

**E**

*, they are not exactly the same, even for the field distribution. However, the difference between $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ and*

_{t}**E**

*can be neglected compared with $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ or*

_{t}**E**

*.*

_{t}In order to get an intuitive impression, we describe the field distribution in a quantitative way. The rotated magnetic field $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ from simulation for the waveguide in the free space and the analytical expression $-{\nabla}_{t}{\Phi}_{0}$ of Eq. (22) are shown in Fig. 4. Figure 4(a) shows the normal components of $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ and of $-{\nabla}_{t}{\Phi}_{0}$ along the left circle *C*_{1}, and Fig. 4(b) shows the ones along the right circle *C*_{2}. The absolute values of relative deviations between the normal components are less than 3%. The analytical results are in good agreements with the simulation results, which confirms the validity of the approximation, expression of Eqs. (14) and (16).

It is worth mentioning that the magnetic field distribution of the analytical result far away from the wires is quite different from that of the simulation result, shown in Fig. 4(c) [dashed line for $-z\times \left({Z}_{0}{{\rm H}}_{t}\right),$ and solid lines for $-{\nabla}_{t}{\Phi}_{0},$ along the midperpendicular of the line segment connecting the two opposite vertices, *x* = (*D* + *R*_{1}-*R*_{2})/2]. The 90° rotated magnetic field of simulation $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ in Fig. 4(c) decays exponentially with the distance far away from the two wires. This exponentially decaying field results from the none-zero value of ${n}_{eff}-1$, like the field far away from the single metal wire waveguide [11]. However, the analytical field,$-{\nabla}_{t}{\Phi}_{0},$ decays by 1/*r*^{2}. As most of the energy is restricted within the air gap, the field energy far away from the two wires can be neglected. Figure 4 only shows the fields of the waveguide in the free space. And for the waveguide with polystyrene foam cladding, there are similar behaviors.

As we can see from Fig. 5, the real and the imaginary parts of analytical *n _{eff}* agree well with the numerical simulations for the waveguides with the polystyrene foams claddings. We keep the gap (edge-to-edge distance) constants,

*D-R*

_{1}

*-R*

_{2}= 200

*μ*m, and set

*R*

_{2}= 150

*μ*m. The radius ratio

*R*

_{1}/

*R*

_{2}varies from 1 to 10. We can see that the values of Im(

*n*-

_{eff}*n*) in Fig. 5 are very small, with a magnitude of about 10

_{d}^{−4}. This is attributed to the large value of |

*ε*|, or the small ratio of the skin depth

_{m}*δ*(~72.5 nm) to the geometric size

*x*

_{2}-

*x*

_{1}(~0.5 mm) as stated below Eq. (24).

Here, it is necessary to clarify that one can either hold *R*_{2} or *R*_{1} constant while changing another. Due to the commutability of the geometry, we just show one case.

To see the dependency on the frequency, we calculate the analytical and the simulated values of *n _{eff}* for the two-wire waveguide in the free space (

*n*= 1). The metal is chosen to be copper, the two wires are identical, and

_{d}*R*

_{1}=

*R*

_{2}= 150

*μ*m. The center-to-center distance

*D*= 0.5 mm. The frequency-dependent

*ε*of copper is obtained from the corresponding Drude model [41]. The analytical and simulated ${n}_{eff}-1$ in the range of 0.1-2 THz are shown in Fig. 6. The absolute values of relative deviations of $\mathrm{Re}\left({n}_{eff}-1\right)$ and $\mathrm{Im}\left({n}_{eff}-1\right)$ are less than 1.42% and 2.95%, respectively. Again, both the real and the imaginary parts of the analytical ${n}_{eff}-1$ are in good agreements with the simulations.

_{m}The good agreements between the analytical and simulated results indicate that our simple approximate explicit formulas, Eqs. (22) and (23), are able to describe fields and the complex effective index *n _{eff}* of the THz two-wire metallic waveguides with different radii.

## 5. Group velocity dispersion and attenuation

With the expression of Eq. (23), one can describe the group velocity dispersion (GVD) and the attenuation of the two-wire metallic waveguide with *n _{d}* = 1 . We adopt the same parameters in Fig. 6. The GVD can be obtained by

*d*

^{2}

*β*/

*dω*

^{2}, where

*β =*Re(

*k*

_{0}

*n*). Figure 7 gives the GVD with respect to frequency from 0.1~2 THz, and the GVD is small in this range. Especially, in the range of 1.15~1.4 THz, the absolute value of GVD is below 1.0 × 10

_{eff}^{−3}ps

^{2}/m as shown in the insert of Fig. 7. We note that GVD has a zero value around 1.268 THz. One can deduce from Eq. (23) that this zero value point is only related to the permittivity of the metal and irrelevant to the geometric parameters.

The attenuation constant can be simply obtained by *α = k*_{0}Im(*n _{eff}*), where Im(

*n*) is ~10

_{eff}^{−4}. The attenuation constant is ~10

^{−2}cm

^{−1}in the frequency range of 0.1~2 THz.

## 6. Discussions

The coupling efficiency is obtained by a 3D simulation with COMSOL, in which we directly couple a Gaussian beam from the free space into the two-wire waveguide. The waist of the incident Gaussian beam is one *λ* (*λ* = 600 *μ*m), and the edge-to-edge distance of the two wires is set to be 200 *μ*m. We obtain that the coupling efficiency is 17.3% for *R*_{1} = *R*_{2} = 150 *μ*m, and 16.7% for *R*_{1} = 1500 *μ*m, *R*_{2} = 150 *μ*m. Although *R*_{1}/*R*_{2} increases to 10, the coupling efficiency has a little decrease. Actually, the edge-to-edge distance plays a more important role in the coupling instead of *R*_{1}/*R*_{2}. Further, one could improve the coupling efficiency by selecting other coupling schemes [32,43].

As an exception, when the mode is decoupled, and the field is split to each wire, the hybrid mode becomes the angular independent TM mode, as shown in Fig. 8(a). The radii of the wires are identical, and *R*_{1} = *R*_{2} = 0.6 mm. The center-to-center distance *D* = 3.2 mm. The frequency is chosen to be 10 THz. The relative permittivity of the copper *ε _{m}* = –3.0457 × 10

^{4}+ 6.684 × 10

^{3}

*i*according to the fitted Drude model [41]. The simulated field $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$in Fig. 8(a) is quite different from the analytical one [Eq. (22)] shown in Fig. 8(b). It means that the approximation [Eq. (16)] is not sufficient in this case, and Eqs. (17) and (23) are not able to describe the propagation in the waveguide. However, Eqs. (13)-(15) are still valid. We note that in this case,

*H*is zero and $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ is only the gradient of

_{z}*E*according to Eq. (5). From the definition of Φ

_{z}*[Eq. (14)], one can see that Φ*

_{h}*is now in proportion to*

_{h}*E*. And

_{z}*E*satisfies the scalar Helmholtz equation and is also angular independent. The expression of

_{z}*E*along the radial direction outside the metal can be seen in [11]. Then Eq. (15) changes to

_{z}*K*

_{0}(.) and

*K*

_{1}(.) are the 0th and 1st order decaying modified Bessel functions, and

*dK*

_{0}(

*k*

_{0}

*κ*)/

_{d}r*dr*= -

*k*

_{0}

*κ*

_{d}K_{1}(

*k*

_{0}

*κ*). It is the eigenvalue equation of the azimuthally polarized surface plasmons of the single metal wire with large radius [11,17].

_{d}rAt the higher frequencies, typically higher than 2 THz, the analytical expression Eq. (23) starts to deviate from the simulated result as shown in Fig. 9. We keep *R*_{1} = *R*_{2} = 150 *μ*m, *D* = 0.5 mm, and vary the frequency from 2 THz to 10 THz in Fig. 9. The frequency dependent relative permittivity of the copper is obtained from the fitted Drude model [41]. It can be seen in Fig. 9 that the deviation becomes evident in the high frequency. The maximum absolute values of relative deviations of $\mathrm{Re}\left({n}_{eff}-1\right)$and $\mathrm{Im}\left({n}_{eff}-1\right)$ are 15.6% and 24.7%, respectively. The higher the frequency, the greater the deviation. This is because in higher frequencies, the absolute values of relative permittivity |*ε _{m}*| gets smaller, the field of the single metal wire azimuthally polarized surface plasmons with a

*K*

_{1}(.) distribution starts to affect the original quasi-electrostatic field near the metal surface. That is the distribution of $-z\times \left({Z}_{0}{{\rm H}}_{t}\right)$ starts to deviate from $-{\nabla}_{t}{\Phi}_{0},$ like the case shown in Fig. 8. However, we note that in most circumstances, especially in low frequencies (0.1-2 THz), the strong coupled field in the gap of two adjacent wires can be well approximated by the electrostatic field, as demonstrated in Section 4 and in [28]. Therefore, our theoretical results, Eqs. (22) and (23), are able to give a simple, fast, and efficient description of the field and the complex effective index

*n*of THz wave in the two-wire metallic waveguide with different radii.

_{eff}In addition, we need to state that the invalidation of the analytic expression at high frequency is not due to the simulation parameters. In the COMSOL simulations, a convergency analysis is implemented by varying the computational regions and the mesh sizes. The real and imaginary parts of *n _{eff}* vary by less than 1%.

## 7. Conclusions

We develop a theoretical procedure to analyze the field and propagation of THz wave in the two-wire metallic waveguide with different radii. Approximate explicit formulas of the field and complex effective index *n _{eff}* have been given for the waveguides. The analytical results are in good agreements with the simulations. Besides, we find a zero value point of the group velocity dispersion around 1.268 THz. It can be expected that these results are helpful for future design of two-wire metallic waveguides capable of low dispersion and loss propagation in THz, and lower frequencies. These results also indicate that the technique of perturbation of boundary condition could be helpful in the analysis and design of other THz metal waveguide.

## Appendix A

#### Impedance boundary condition

The Maxwell curl equation in the nonmagnetic metal is

considering that the permeability of metal*μ*= 1. The field strongly damped into the metal within the skin depth

_{m}*δ*, and

*δ*is far smaller than the geometry size, e.g.

*δ*= 72.5 nm at 0.5 THz. Then the gradient operator can be written aswhere

**n**is the unit normal from the metal and

*ξ*is the normal coordinate inward into the metal. Then Eqs. (26a) and (26b) become

**H**

*inside the metal is parallel to the surface. We define the tangential*

_{m}**H**

*=*

_{||}*H*|

_{τ}

_{C}**τ**+

*H*|

_{z}

_{C}**z**is the magnetic field right on the metal surface. The solution of Eq. (29a) is thenin consideration of that the field will be damped into the metal and

*ε*is in the second quadrant, that is Re(

_{m}*ε*)<0, and Im(

_{m}*ε*)>0. By combining Eq. (28a) and the solution Eq. (30), the electric field in the metal can be approximated:

_{m}**E**

*and*

_{m}**H**

*gives the well known fact that*

_{m}**E**and

**H**rapidly exponentially decay in the metal. The magnetic field inside the metal is much larger than the electric field, and the tangential part of the electric field dominates over the normal one, as shown in [11]. Finally, from the continuity of the tangential

**E**and

**H**combined with Eq. (31), one can obtain the small tangential electric field on the metal surface in the dielectric cladding region as shown in Eq. (1).

One may find these steps in [39]. However, we use *ε _{m}* rather than conductivity

*σ*to describe the metal. It is more convenient, because

*ε*of the metal can be estimated from the fitted Drude model in the THz region [41].

_{m}## Appendix B

#### Conservation of charge

From Eqs. (4) and (5), one can easily get

**n**. Equations (32) and (33) then give

*C*

_{1}as an example. The line charge density along the longitudinal direction

*ρ*

_{1}, and the total current in the longitudinal section

*I*

_{1}of the left wire can be expressed as according to Gauss’s law and Ampere’s law. By applying the charge conservation,and replacing ∂/∂

*t*by –

*iω*, ∂/∂

*z*by

*ik*

_{0}

*n*, we can get

_{eff}*C*

_{2}, one may find that the right hand side of Eq. (34) is zero, and then Eq. (10) can be obtained.

## Appendix C

#### Expression of Complex Effective Index for THz Two-Wire Waveguide with Different Radii

Although the field in the gap region of the two wires in Eq. (22) is given, it is difficult to solve Eq. (17) in the original two-wire *ζ* space. The solution is much simpler in the conformal coaxial space *w*. The absolute normal derivative of potential Φ_{0} on *C*_{1} can be expressed in the *w* space by

_{0}/∂

*n*is the inward (from the circle to the gap region) normal derivative of potential on the circle in the conformal coaxial space

_{w}*w*. By use of Eqs. (18) and (21), one can get:

*C*

_{2}and combining Eq. (9), one can getand

## Acknowledgment

This work was financially supported by the National Natural Science Foundation of China with Grant Number 61275103.

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