## Abstract

We present a rigorous theoretical analysis of the two-wire waveguide. Obtaining the attenuation constant in terms of the dimensions of the waveguide analytically, we show that the absorption coefficient can be less than 0.01 cm^{−1}, with the appropriate values of the dimensions.

©2010 Optical Society of America

## 1. Introduction

One important component for advancing optical and microwave systems is the use of waveguides for channeling electromagnetic fields. The existence of practical and low-loss waveguides like fibers, coaxial cables and planar transmission lines drove generations of high-speed communication systems. Unfortunately, terahertz systems suffer from the lack of a low-loss well-behaved waveguide despite substantial research conducted on the subject.

Early terahertz waveguides include coplanar transmission lines with an attenuation constant of 20 cm^{−1} [1]. Microwave metal-pipe waveguides can have lower loss (about 1 cm^{−1}) but they are highly dispersive due to the existence of cutoff frequencies for propagating modes, and multi-mode propagation limits usable bandwidth [2]. Waveguides fabricated with dielectrics like sapphire or high-density polyethylene (HDPE), which have low loss in terahertz range, were reported with about 1 cm^{−1} loss [3], [4]. Parallel-plate waveguides solve the problem of dispersion by supporting TEM mode with 0.1 cm^{−1} absorption coefficient [5]. Dielectric fiber with sub-wavelength diameter for terahertz has been demonstrated [6]. The sub-wavelength fiber was designed to carry much of the terahertz wave’s energy in the air surrounding the fiber. Loss of 0.01 cm^{−1} at 0.3THz with a 200μm-diameter fiber made of polyethylene (PE) was reported. With a similar motivation to carry a terahertz wave in air, a stainless steel bare metal wire shows less than 0.03 cm^{−1} absorption at 1 THz [7]. However, it is difficult to couple an electromagnetic wave efficiently to the radially-polarized mode supported by this waveguide.

Two-wire waveguides [8] combine both low loss and efficient coupling properties. The field polarization of the TEM mode supported by this type of waveguide is very similar to the field emitted from a simple dipole, resulting in efficient coupling of the electromagnetic energy from typical terahertz sources into that mode. Also, for the TEM mode there is no cutoff frequency, and no group velocity dispersion. Thus, one can enjoy the low loss of the wire waveguide along with dispersion-free propagation of the TEM mode. Confining electromagnetic energy in a small area between the two wires is another important advantage of this waveguide that makes it more practical and more tolerant to bend loss [9].

In this paper we present a rigorous theoretical analysis of the two-wire terahertz waveguide for evaluating the absorption coefficient in terms of the dimensions of the waveguide. We calculate the field distribution for different dimensions of the waveguide and present limited design curves to choose wire radius and separation for a desired attenuation coefficient.

First, we explain the theory of estimating loss in terms of the field components. Then, we apply the theory to the two-wire waveguide. Finally, we present the results and draw conclusions from the results.

## 2. Theory

In a perfect conductor, the surface charge density should be

to give zero electric field inside the perfect conductor. The movement of the electric charges also produces a surface currentto give zero magnetic field inside the perfect conductor. Therefore, just outside the perfect conductor there are only normal electric field and tangential magnetic field. A good, but not perfect, conductor must exhibit approximately the same behavior. In [10], it is shown that the fields inside a good conductor are*ξ*is the normal coordinate inward into the conductor,

*H*is the tangential magnetic field outside the surface,

_{||}*E*and

_{c}*H*are the electric and magnetic field inside the conductor,

_{c}*μ*is the permeability of the conductor,

*δ*is the skin depth (Eq. (5) and

*σ*is the conductivity. Accordingly, the time-averaged power absorbed per unit area is

*A*is the surface area of the conductor. Equation (6) shows how the conductor loss changes with the frequency (or wavelength) and with the material of the conductor through the skin depth. To estimate the waveguide loss, Eq. (6) suggests we must obtain the tangential magnetic field just above the metal surface.

## 3. Two-wire waveguide

Obtaining fields and loss estimation calls for solving Maxwell's equations with the boundary conditions forced by the structure of the two-wire waveguide. As mentioned before, the two-wire waveguide can support a TEM mode as well as TE and TM modes. For the TEM mode, the problem of obtaining fields outside the conductor reduces from the wave equation to the 2D Laplace equation, a considerable simplification. One powerful mathematical method for solving the 2D Laplace equation is conformal mapping based on the properties of harmonic functions [11]. In this method, a complicated geometry is mapped to the simpler one through a complex analytic function. The Laplace equation is solved in the simpler region and the solution is transformed to the original geometry.

The following two functions map the cross section of the two-wire waveguide to two concentric circles like the cross section of a coaxial cable as shown in Fig. 1 .

The potential function in the region between the two concentric circles is

where*a*and

*b*are the radius of inner and outer circles, respectively. Substituting z in the potential according to the mapping functions results in the potential in the desired region, namely the cross section of the two-wire waveguide. The electric field can be derived from the gradient of the potential: Figure 2 shows the distribution of the amplitude of the electric field in the cross section of the two-wire waveguide obtained from the theoretical analysis. The field distribution is consistent with the simulation results reported in [8].

Figure 3 also illustrates the amplitude of the electric field in the x plane (Y = 0) between the two wires obtained from both the theory and the simulation. It is quite clear that the results are in good agreement, confirming the validity of the theory.

Figure 4
shows the field distribution for different values of the wire centre-to-centre distance *D*. The figure illustrates the wide variation in field distribution with change in *D*. When the wires are close, the field concentrates in the gap between the wires. When the wires are distant, the field more evenly surrounds the wires. Therefore, we expect that loss would be lower for larger distances between the wires. However, moving the wires farther apart makes the waveguide support higher modes.

The radius of the wires also has a large impact on the field distribution. As the radius *R* increases, the field is distributed over a larger area, suggesting lower loss for bigger wires.

For the TEM mode, the magnetic field can be derived easily from the electric field with the following relationships:

where*η*is the intrinsic impedance of air. Knowing the magnetic field just outside the surface of the conductor allows us to estimate the power loss per unit length of the waveguide according to

*ds*is the infinitesimal surface element of the conductor. Note that there is no attenuation due to dielectric loss because the two wires are surrounded by air. Another necessary parameter for determining attenuation constant is

*P*, the power flowing on the lossless line:

_{0}*S*is the surface element of the cross section of the waveguide. The attenuation constant is thenwhere

^{'}*P*is the power loss per unit length and

_{l}*L*is the length of the waveguide.

## 4. Results

Figure 5
illustrates the dependence of the attenuation constant on *D*, the distance between the wires, and *R*, the radius of the wires, at 1THz for a waveguide made out of gold. Attenuation constant decreases as *D* becomes larger, consistent with the conclusion we derived from the field distribution. Increasing *R* also results in a smaller attenuation constant, also inferred from the field distribution. All the curves corresponding to the different values of *R* show similar behavior, i.e. decreasing loss with increasing *D*. There is also a knee in each curve that corresponds to an appropriate value for *D*, because further increase in *D* would not reduce absorption considerably. A closer look at the difference between the curves shows that increasing *R* more than 300μm does not change the loss significantly. Therefore, this value is probably a good choice for *R* for a wide range of applications.

## 5. Conclusion

The two-wire waveguide offers some unique advantages like low loss, supporting a TEM mode, efficient coupling of the electromagnetic energy into the waveguide, and confining the field in the small area between the wires. In this paper, we present an analytical method for estimating the absorption coefficient for the two-wire waveguide in terms of its dimensions. Results enable the simple prediction of the waveguide performance and field configurations for a wide variety of physical conditions.

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