## Abstract

We calculate theoretically the coupling of a terahertz wave from a dipole into a two-wire waveguide. The field transmission and reflection are obtained using a Single Mode Matching (SMM) technique at the input port of the two-wire waveguide. The results show more than 70 percent coupling efficiency for the waveguide using 500μm radii wires with 2mm center-to-center separation and the exciting field cross section of 1mm × 1mm. The results also show good agreement with the full-wave numerical simulations using the Finite Element Method (FEM).

©2010 Optical Society of America

## 1. Introduction

It is anticipated that waveguides will form an important part of future terahertz systems. Using waveguides can be a way to avoid the alignment and loss challenges associated with the manipulation of terahertz waves by terahertz optics. Probes based on good terahertz waveguides can also have many biomedical applications.

Several microwave and optical waveguide structures have been used for terahertz applications like coplanar strip line (CPL) [1], metal pipes [2], dielectric fibers [3,4], etc. But they have either high attenuation or high dispersion that prevents terahertz wave or pulse transmission over long distances. The single-wire waveguide [5], however, shows very low loss and almost no dispersion, but it is hard to couple electromagnetic waves efficiently from common terahertz sources to the radially-polarized mode supported by this waveguide. Coupling efficiency of less than 1% to the single-wire waveguide was reported using the scattering method [6]. Radially-symmetric photoconductive antennas, however, can significantly enhance the coupling because the emitted electric field is more similar to the waveguide's mode [6]. Two-wire waveguides [7] prove tolerant to bend loss [8] and have low attenuation [9]. The field polarization of the TEM mode supported by this type of waveguide is also 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.

In this paper we evaluate the coupling efficiency from a typical photoconductive terahertz source into a two-wire waveguide quantitatively and show that the appropriate values of the dimensions can lead to a high coupling efficiency.

## 2. Approach in analytic framework

Dipole antennas are used widely in terahertz photoconductive sources [10–13]. The radiation pattern of a dipole on a dielectric substrate has been investigated in detail in [14–18]. Inside the substrate the terahertz wave is like a spherical wave. A silicon lens transforms this spherical wave approximately to a plane wave [19]. Coupling efficiency determines how much power can be coupled from the dipole source to the two-wire waveguide. Calculating the coupling efficiency is, in general, not a trivial problem, especially for a complicated structure like the dipole and the two-wire waveguide. However, given [19], we make the assumption that the far-field radiation of the dipole on the substrate can be approximated by a plane wave in the plane perpendicular to the dipole. This approach assumes that a plane wave with certain dimensions (depending on the directivity of the source) impinges on the input port of the waveguide. This approximation is valid when the waveguide cross section is comparable to or smaller than the far-field radiation pattern of the dipole, which is the case for a typical terahertz source [19–21].

A second approximation replaces the radiated plane wave with certain dimensions at the input port of the two-wire waveguide, by the TEM mode of a fictitious parallel-plate waveguide of the same dimensions, as shown in Fig. 1 . If the plane wave is a good approximation for the incident THz wave, the field incident on the two-wire waveguide is the same whether the incident wave is a plane wave or the TEM mode of a parallel-plate waveguide with the same dimensions. What matters in coupling to the two-wire waveguide is the distribution of the field at its input port, which is identical for both cases. This approximation enables us to use a simple mode-matching technique to calculate the coupling efficiency from the TEM mode of a parallel-plate waveguide into the TEM mode of a two-wire waveguide.

We assume that the TEM mode of a parallel-plate waveguide (with the cross section of *w × d* and with 1W power) corresponding to a plane wave, impinges on the input port of a two-wire waveguide. The coupling efficiency is the ratio of the transmitted power to the incident power [2].

## 3. The mode-matching technique to calculate the coupling

Transmitted power and reflected power are calculated by matching the transverse components of the electric and magnetic fields at the intersection of the two waveguides. The incident and reflected waves of a TEM mode on one side of the boundary are matched to a single TEM mode on the other side. The approximation in this method is to neglect the excitation of the higher-order modes including propagating, cutoff and evanescent modes on both sides at the boundary. This approach has proven to be accurate when the dimensions of the waveguides are small enough that the lowest-order mode dominates the behavior [22–24].

The physical modes supported by the waveguides form a complete, orthogonal basic set; therefore, the fields can be expressed in terms of the modes on both sides [25,26]:

*E*,

*H*are the electric and magnetic field,

*a*,

_{m}*b*are the field coefficients for forward and backward propagating waves,

_{m}*e*,

_{m}*h*are the normalized electric and magnetic fields of the

_{m}*m*th mode, and subscripts

*1*,

*2*are for the parallel-plate waveguide and the two-wire waveguide, respectively. The orthonormalization of the modes can be written [25,26]

*l*,

*m*are the mode subscripts, and

*δ*is the Kronecker delta. The boundary conditions at the interface of the two waveguides require:

_{lm}*t*indicates the transverse components of the fields. The single lowest-order TEM mode with 1W power is incident from the left side. Continuity of the transverse electric field expanded in terms of the orthogonal modes on both sides gives:

*r*,

*t*are the reflection and transmission coefficients of the fields for the TEM modes, respectively. Multiplying both sides of Eq. (6) by

*1/2(h*, integrating over the transverse plane (XY-plane), and using orthoganality relations for the parallel-plate waveguide's modes yields

_{1})_{TEM}*Similarly, continuity of the transverse magnetic field expanded in terms of the orthogonal modes on both sides gives

*1/2(e*, integrating over the transverse plane, and using the orthogonality relations for the two-wire waveguide's modes yields

_{2})_{TEM}*Neglecting the effects of the higher-order modes results in simple relations for the transmission and reflection coefficients:

where*|t|*is a measure of the coupling coefficient. Therefore, the coupling coefficient can be obtained by calculating

^{2}*κ*. The amplitude of

*κ*is a number between 0 and 1 and indicates how well the field distributions of the two TEM modes are matched, 1 corresponds to the perfectly-matched case and 0 when the modes are not matched at all.

## 4. Calculation of the *κ*

According to Eq. (8), calculation of *κ* calls for knowing the electric and magnetic fields of the TEM mode for the two waveguides. The TEM mode for the parallel-plate waveguide is [24]:

*η*is the intrinsic impedance of free space and

_{0}*A*is the normalization constant that is chosen so that the TEM mode carries 1W power:where

_{1}*w*are the width of the plates, and

*d*is the distance between the plates. The TEM mode for the two-wire waveguide is [9]

*C*and

_{1}*C*in Eq. (18), (19) depend on the radii of the wires (

_{2}*R*), and the distance between them (

*D*), by

*A*is the normalization constant that can be obtained from the equation below,

_{2}*S*is the surface specified with the pattern in Fig. 2 . Explained in the Appendix, integration over

_{1}*S*using Green's theorem gives

_{1}## 5. Results and discussion

To verify the method, we first consider the case where the edge-to-edge distance of the two wires is kept constant and equal to the parallel-plate separation. As shown in Fig. 2(a), if the radii of the wires increase, the two-wire waveguide becomes more like the parallel-plate waveguide. Obviously, this change should lead to a higher transmission and a lower reflection coefficients. Figure 2(b) shows the transmission and reflection coefficients of the power (*|t| ^{2}*,

*|r|*) versus

^{2}*R*, obtained from the proposed method for the same situation. They show the expected behavior, consistent with the intuitive conclusion.

Ideally, theoretical predictions would be validated experimentally. However, there are significant difficulties associated with the experimental measurement of the coupling. For instance, the absolute value of the radiated power from the source is difficult to measure. Also, other properties of the waveguide, like attenuation, vary with the change in the geometry of the waveguide, making it difficult to distinguish the impact of changes in each property. So for now, we present numerical simulations to confirm the theoretical results.

We compare the coupling obtained from theory with results from 3D, full-wave simulations with Finite Element Method (FEM) using the Ansoft HFSS frequency-domain solver. The simulations are applied to two-wire waveguides made out of gold with constant center-to-center distance of 400μm and different radii of the wires. Two cases are considered, both at 1THz. First, the waveguide is excited by a 0.5mm-long parallel-plate waveguide with 1mm × 0.4mm cross section, as shown in Fig. 3(a) . Second, the waveguide is excited by a dipole, as shown in Fig. 3(b). The simulations are bounded by a rectangular box with 1mm × 1mm × 2mm size with radiation boundaries assigned to the walls of the box to avoid reflection. Only the TEM mode of the parallel-plate waveguide is excited in x-direction, using a wave port as the input port of the parallel-plate waveguide. For the dipole case, the distance between the dipole source and the input port of the two-wire waveguide is 200μm and the direction of the dipole is in x-direction.

To make a correct comparison between the results from the theory and the simulations, the incident power at the input port of the two-wire waveguide should be the same for both cases (theory and simulations), so we first measure numerically the power after the 0.5mm-long parallel-plate waveguide and also the radiating power of the dipole at 200μm away from the source, in the absence of the two-wire waveguide, using the Field Calculator of HFSS. Then we normalize all the measured power to those values so that we have the same input power to the two-wire waveguide in each case (1W). Then, the coupling efficiency is the normalized power measured at the output port of the two-wire waveguide. Note that at the input port of the waveguide, many modes, both propagating and evanescent, must be superposed in order to keep the boundary conditions satisfied. This is responsible for the peculiar shape of the field near the input of the waveguide that is conspicuous in Fig. 3(b). Far away from the input port, however, the field is relatively simple, with only the TEM mode (provided the dimensions of the waveguide allow just single-mode propagation) propagating with appreciable amplitude. The cutoff modes have considerable amplitude just near the input port and they fade over distances [26]. We also choose the length of the two-wire waveguide to be small enough that the waveguide loss is negligible and does not affect the coupling. As a result of these considerations, the measured normalized power at the output port appears to provide a legitimate value for the coupling efficiency.

Figure 4
compares the coupling obtained from the theory with the one from the simulations. In the theoretical calculations, we choose *w* × *d* = 1mm × 0.5mm (equal to the parallel-plate dimensions) for the parallel-plate case, and *w* × *d* = 1mm × 1mm (equal to the cross section of the box) for the dipole one. At the lower limit, when the radii of the wires go to zero, there is no waveguide to support the TEM mode; thus, the coupling should go to zero. Coupling should also go to zero at the higher limit, when the radii of the wires become equal to *D/2* (the center-to-center distance of the wires, *D*, is constant and equal to 400μm) because the edges of the two wires touch each other and the waveguide cannot support the TEM mode. Therefore, there should be an optimum value for the radius corresponding to the peak value of the coupling between the limits. Figure 4 shows that the results are consistent with this expectation. The results from the theory and the simulations with the parallel-plate waveguide, depicted in Fig. 4(a), show good agreement. Figure 4(b) also illustrates the results from the theory and the simulations with the dipole source. They show the same overall behavior even though there are some discrepancies due to neglecting the excitation of radiating and higher-order modes at the input port in the theoretical calculations.

We also studied the case where a plane wave with the cross section of *w* × *d* = 1mm × 1mm impinges a two-wire waveguide with constant radii of the wires, 500μm, for different center-to-center distances. Figure 5
shows the coupling efficiency for this case. *D* starts from its minimum value, *2R*, where the two wires touch each other and the coupling is zero. As *D* increases, the waveguide starts to support the TEM mode that overlaps with the field of the plane wave, enhancing the coupling. But when the edges of the wires are too close, the TEM mode supported by the waveguide is mostly concentrated in a very small area between the wires, resulting in small coupling. However, as *D* becomes larger the field is more distributed on the surface of the wires [9], increasing the overlap area and, in turn, the coupling. This trend continues up to the point that the edge-to-edge distance of the wires is equal to the size of the plane wave. This is the point where the TEM mode can capture the most power from the impinging plane wave, as shown in Fig. 6(a)
. After that point, the field area with high amplitude escapes from the plane wave region as shown in Fig. 6(b), leaving small field' overlap and coupling. These results are consistent with the experimental and theoretical results reported in [8], [9].

The coupling efficiency is highly dependent on the dimensions of the aperture defining the plane wave impinging the waveguide, as well. If this is large or small compared to the waveguide, the waveguide cannot possibly pick up sizeable power, as clearly shown in Fig. 6(b). Given the size of the aperture, Eq. (24) and Eq. (25) can give optimum values for the dimensions of the two-wire waveguide, *R* and *D*. But, roughly speaking, the optimum point always happens when the size of the aperture is close to the edge-to-edge distance of the wires, like the case shown in Fig. 6(a). Note that the proposed method is only accurate when a single TEM mode dominates the behavior of the waveguide. Avoiding higher-order modes limits the dimensions of the waveguide. However, the fact that the linearly-polarized incident field is well-matched with the TEM mode supported by the waveguide alleviates that concern to some extent.

## 6. Conclusion

We estimate the coupling from a typical terahertz source with a linearly-polarized electric field into the TEM mode of the two-wire waveguide using some physically justifiable approximations and a simple mode-matching technique. The theoretical calculation can be used to obtain the optimum configuration of the two-wire waveguide for maximum coupling of the field. The maximum coupling corresponds to the case when the field distribution of the two-wire waveguide is best matched with the wave impinging the waveguide, confirmed by numerical simulations. For instance, 70% coupling can be obtained by 2mm distance and 500μm radius.

## Appendix

Green's theorem makes the calculation of the surface integration in Eq. (23) significantly easier.

Green's Theorem: Let *F _{1}* and

*F*be continuous differentiable functions on a simply connected domain

_{2}*S*, and let Γ be a positively oriented simple closed contour around

*S*. Then Eq. (26) is valid:

*x*component of the electric field in Eq. (8) and using Green's theorem yields

*S*and the contour Γ are shown in Fig. 7 . The integrations on the routes 1 and 12 are zero because

_{1}*y*is constant. So are the integrations on the routes 3 to 10 because the integrand is the even function of

*y*. Thus, the calculation of

*κ*reduces to the integrations on the routes 2, 11 and 13 that yield

Note that Eq. (30) is valid provided the cross section of the parallel-plate waveguide is larger than the one for the two-wire waveguide. This method can be applied to the other cases as well. Figure 8(b)
shows the contour for the case that the separation of the parallel plates is smaller than the edge-to-edge distance of the wires. The integrations on routes 2 and 4 are zero because *y* is constant. So the calculation of *κ* reduces to the integrations on routes 1 and 3 that result in the same equation as Eq. (30). However, the case shown in Fig. 8(a) results in a different relationship for *κ*. The integrations over 1 and 6 are zero because *y* is constant, and so are the integrations on routes 3, 4, 8, and 9 because the integrand is the even function of *y*. Therefore, the calculation of *κ* reduces to the integrations on the routes 2, 5, 7, and 10 that yield

## Acknowledgement

This work is supported by funding from the Natural Sciences and Engineering Research Council (NSERC) Canada.

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