1. Introduction

Techniques for the generation and detection of pulsed terahertz radiation have been of considerable interest for several years [1–3]. While these techniques are still being refined and novel ones are being developed, several issues have limited the development of terahertz technology for security applications [4], biomedical imaging [5, 6], trace gas detection [7], and quality control [8]. One significant issue is the complexity associated with the use of terahertz spectroscopy systems. The use of free-space optics to guide and manipulate terahertz beams requires advanced experience with optical techniques. In addition, when free-space optics are used, the sample or area of interest must provide direct line-of-sight access. This prevents the use of terahertz radiation in certain real-world applications such as the biomedical field as there is currently no terahertz equivalent of the fiber-optic medical endoscope. These issues motivate the search for a terahertz waveguide with low loss and low dispersion.

There have been significant efforts devoted to the development of terahertz waveguides [9–12]. One principal difficulty has been the lack of materials well suited for guided propagation of radiation at these frequencies. Generally, materials that are transparent at terahertz frequencies are crystalline, which precludes their practical use in waveguide structures. Other materials such as glasses and polymers exhibit unacceptable absorption losses at terahertz frequencies. Consequently, many terahertz waveguide techniques rely on the terahertz pulse propagating through air inside of a confined geometry such as a metal tube. While waveguide structures of this type exhibit low losses, they often suffer from significant pulse reshaping due to group velocity dispersion [9]. A notable exception is the parallel-plate metal waveguide, which has been shown to exhibit dispersionless propagation and very low losses [13, 14].

We have recently described another option, based on the propagation of a surface wave on the exterior surface of a metal cylinder. A simple metal wire can be employed as a terahertz waveguide, exhibiting very low loss and negligible dispersion [15]. This guiding structure possesses the appealing feature of structural simplicity, which has for example permitted the first report of a terahertz endoscope [16]. However, compared to the earlier reported case of the parallel plate waveguide, the cylindrical symmetry of the wire waveguide poses a significant challenge. The typical linearly polarized terahertz wave has a very poor spatial overlap with the radially polarized guided mode. As a result, direct end-fire input coupling to a wire waveguide is not feasible using a linearly polarized wave.

In the earlier experiments, a scattering mechanism was employed in order to couple linearly polarized terahertz radiation to the radially polarized surface wave that propagates along the metal wire [15, 16]. In this letter, we present Finite Element Method (FEM) simulations of this coupling mechanism and use these results to estimate the coupling efficiency of this scattering mechanism. A photoconductive terahertz antenna with radial symmetry is introduced, and additional FEM simulations show that coupling efficiency is significantly enhanced when the polarization profile of the terahertz source is well matched to the dominant mode of the waveguide [9]. Finally, the radially symmetric photoconductive antenna is experimentally evaluated and compared to the FEM data.

2. Efficiency of dual-wire THz coupling experiment

In our previous work, the radial mode of the metal wire waveguide is excited via a scattering mechanism. Horizontally polarized terahertz pulses are focused onto the 0.9 mm diameter stainless steel wire waveguide. An identical stainless steel wire is placed perpendicular to the waveguide near the focal spot of the beam. A small portion of the scattered radiation excites the radially polarized surface wave that propagates along the metal wire. A graphical depiction of the experiment can be found in the previously published work [15, 16]. This radially polarized guided wave is known as a Sommerfeld wave [17, 18] and can also be described as an azimuthally polarized surface plasmon [19].

The experimental results from the coupling configuration described above indicated that less than 1% of the power of the incident terahertz beam is coupled to the wire waveguide by this scattering process. Due to experimental difficulties, it has not been possible to obtain a more accurate estimate of the coupling efficiency. Here, we develop a numerical model of the dual-wire THz coupler, using the finite element method (FEM), that demonstrates the nature of the coupling and that also provides a method to more accurately quantify the coupling efficiency.

The wire waveguide is modeled in three dimensions using a commercially available FEM package. Its outer boundary is defined as a perfect electrical conductor (PEC). It should be noted that a more exact model could consider the surface impedance of the wire’s outer boundary and the Drude conductivity of the metal. This would be necessary in order to correctly model the ohmic loss of the wire waveguide [15]. However, it is our intention to model only the coupling efficiency of the experiment and not to consider the waveguide’s inherent loss.

A second wire, the coupler wire, is placed in the vicinity of the wire waveguide, but in a direction perpendicular to it. The coupler wire has the same diameter as the waveguide and is also modeled as a PEC. The distance between the closest outer surfaces of the two wires is 0.5 mm. The linearly polarized terahertz pulse is modeled as a plane wave whose k-vector is incident at a 45 degree angle from the axis of the wire waveguide. The plane wave, 2 cm in diameter, is directed such that its center is incident at the center of the gap between the waveguide and coupler wires. The size of the plane wave is chosen to mirror the size of the loosely focused terahertz beam used in the actual scattering experiment. The linear input polarization is chosen such that its direction lies in the same plane (x-z plane) occupied by the long axis of the wire waveguide, again mirroring the experiment. The simulation domain is bounded by a box of air confining the waveguide and coupler wires and the input plane wave. A low-reflecting boundary condition is selected for the outer walls of the domain. This condition is chosen such that any back reflections of the EM waves in the simulation are minimized.

The 3-D simulation domain is bounded by enclosing the waveguide and the coupler wire in a rectangular box 20 cm by 2.5 cm by 2.5 cm. A cylinder (2 cm diameter, 4 cm length) is placed such that its central axis lies at the center of the gap between the waveguide and coupler wires. Part of the cylinder extends outside of the rectangular box, as shown in Fig. 1. In the simulation, the plane wave is excited on the circular face of the cylinder that lies outside of the box. Prior to solving, the 3-D simulation domain is discretized into approximately 1.8 million tetrahedral mesh elements yielding a computational model consisting of 2.1 million degrees of freedom. The large number of mesh elements is due to the need for at least 3, but generally more, mesh elements per wavelength of the propagating radiation [20]. The experiment is simulated using a time-harmonic solver, so only one input frequency is considered at a time. The model problem is solved using an iterative Generalized Minimal Residual (GMRES) iterative solver with Symmetric Successive Overrelaxation (SSOR) matrix preconditioning [21, 22]. For a waveguide 15 cm long, a workstation with dual 64-bit processors and 14 GB of RAM arrives at a solution in less than 20 hours.

 

Fig. 1 (a) FEM Simulation result of 0.1 THz wave coupling to a wire waveguide using a dual-wire coupling configuration (E_x shown here). The top inset shows a close-up view of the electric field (xz-plane) in the coupling region between the two wires, while the right inset shows the electric field at the end of the wire (xy-plane). Note that the majority of the incident wave scatters into free space, and only 0.4% percent of the incident power is coupled to the guided radial mode. Fig. 1(b) shows the x-component of the electric field along a line 300 microns above the wire and parallel to the wire axis. The inset shows a 0.1 THz sine wave fit to the extracted simulation data. This demonstrates that the simulation is discretized with a mesh density fine enough to resolve the oscillating electric field.

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Simulation results for a frequency of 0.1 THz can be seen in Fig. 1. The incident plane wave can be seen scattering at the gap between the wire coupler and waveguide. It can be seen from the surface plot that the majority of the plane wave propagates unimpeded and only a small amount of the incident radiation is coupled to the radial mode of the waveguide. A plot of the x-component of the guided wave’s electric field at the end of the wire is also shown highlighting the radial nature of the propagating mode. At the same location, the y-component of the field is similar to the aforementioned plot, but rotated by 90 degrees. Fig. 1(b) is a plot of the x-component of the electric field 300 microns above the wire and along its z-axis. The large peak at z=0 is part of the incident input wave. It can be seen that both forward and backward propagating modes are excited. Recalling that the wires are modeled as PEC’s, we expect little attenuation once the mode is excited, which is consistent with the results of the calculation. A sine wave with a frequency of 0.1 THz can be easily fit to the oscillating electric field along the waveguide, demonstrating that the model is simulating wave propagation.

These simulations clearly demonstrate the propagation of a surface wave along the cylindrical wire, over a distance of 15 cm. On first glance, this may be a surprising result, since it is well known that a perfect conductor cannot support surface waves [17, 23]. In our simulations, the boundary condition chosen for the surface of the waveguide is that of a perfect electric conductor (PEC), which means that the electric field component parallel to the surface (E_z) must vanish at the wire surface. As expected, the analytic Sommerfeld model predicts that, under this condition, the wave is unbounded, with amplitude extending infinitely far from the wire surface. We note, however, that a mode with infinite lateral extent would be impossible to excite using a finite-sized input field. We therefore expect that the simulated field is not single mode, but instead consists of a superposition of many modes, all possessing azimuthal symmetry. Of course, each of these modes has a distinct attenuation spectrum, so extracting accurate attenuation coefficients from these simulations is obviously not possible using the PEC condition. Nevertheless, these simulations do demonstrate the coupling of terahertz radiation into a radially polarized mode of the wire waveguide, and the propagation of this mode over several tens of cm.

It is quite simple to calculate the coupling efficiency at 0.1 THz from the simulation results shown in Fig. 1. The incident linearly polarized wave is defined on the base of a cylinder in our model. The outline of the side of the cylinder can be seen in Fig. 1. The coupled input power is defined by integrating the time-averaged power over the area defined by the circular base of the cylinder, where the input wave is introduced into the model. Likewise, the coupled power is determined by integrating the time-averaged power over a circular area normal to the z-axis at the end of the waveguide (15 cm from the point of excitation). These calculations produce a simulated power coupling efficiency for the dual wire coupling configuration of 0.42%, comparable to the estimate generated from experimental data [15]. It should be noted that this number is probably the upper limit on any expected coupling efficiency. The wires in this simulation were modeled as PEC’s, so no loss due to finite conductivity was considered. Finally, while the efficiency is expected to vary with wavelength, wire separation, etc., there is no reason to expect it to increase greatly, since the polarization of the waveguide mode and the incident wave are so poorly mismatched.

3. Enhanced THz wire waveguide coupling

3.1 Photoconductive terahertz antenna with radial symmetry

The very low coupling efficiency associated with the aforementioned dual-wire coupling scheme results mainly from the polarization mismatch between the terahertz source and the primary mode of the wire waveguide. Whether the method of terahertz pulse generation is via a photoconductive antenna or optical rectification in a NLO crystal, the generated pulse is linearly polarized [24]. While others have sought to address the coupling problem via modification of the waveguide itself, our efforts have focused on the source of the terahertz radiation [25]. We have proposed a novel photoconductive terahertz antenna with radial symmetry [26]. Recently, this concept was demonstrated by Jeon and Grischkowsky [18]. While traditional photoconductive antennas use a linear dipole configuration or similar, resulting in linearly polarized pulses, our design (Fig. 2(a)) uses a cylindrical symmetry to produce a “radial array” of dipoles. Our previous work showed using both analytical methods and FEM simulations that an idealized radial antenna can produce a radially polarized terahertz beam (Fig. 2(b)). The idealized radial antenna ignores the effects of the feed electrode and the break in the outer electrode shown in Fig. 2(a). As the ideal antenna lies in the x-y plane, the radial polarization of the generated radiation is demonstrated by the presence of the “donut” mode propagating along the z-axis. The cylindrical symmetry can be seen in that the radiation pattern is symmetric about the z-axis. Here we present FEM simulation results of the actual radial antenna design, for which the radial symmetry is broken by a feed line to the inner electrode. These may be compared with an idealized design with perfect radial symmetry.

 

Fig. 2 (a) Photoconductive antenna with radial symmetry (b) FEM simulation of the power emitted by an “ideal” radial antenna in free space (i.e., no substrate) at 0.5 THz. The antenna sits at the center of the sphere in the xy-plane. The emitted field is zero along the z-axis, as expected for a ‘donut’ mode.

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The radially symmetric photoconductive antenna was simulated with the same three-dimensional electromagnetic wave FEM package used to model the dual-wire THz waveguide coupler. The center of the antenna consists of a circular electrode 5 μm in diameter, fed by an electrode 1 μm wide that approaches the center from the negative y-axis direction. The separation between the center electrode and the 10 μm wide outer electrode is 100 μm. At the bottom of the antenna where the outer ring electrode approaches the feed electrode, there is a 7.5 μm gap on each side of the feed electrode. To simplify the model, the boundaries of the electrodes are defined as PEC’s. In order to model an actual experimental configuration, the antenna is placed at the interface between a section of air and a 0.5 mm thick dielectric substrate, with a dielectric constant of 13, roughly equal to that of GaAs. As is the case for standard THz-TDS [27], a substrate-matched aplanatic hyperhemispherical (not collimating) silicon dome (2 mm radius) is placed on the other side of the GaAs substrate in order to couple the beam into free space and collimate it. For comparison, an idealized radial antenna (with no feed electrode and no gap in the outer circular electrode) with a 100 μm radius is also placed on the same GaAs substrate with the same silicon dome configuration.

For the modeling of the antenna configuration, we use a two step approach, in which we model first the DC fields present in the antenna and use those results to simulate the generation of terahertz radiation [21]. In the first step, the outer electrode is grounded and a potential is applied to the center electrode. A charge density with a Gaussian distribution and a 1/e width of 40 μm is placed at the center of the antenna. This mimics the charge carriers generated in the GaAs by the optical pump pulse in a typical photoconductive generation scheme. The electrostatic fields are computed using the FEM solvers and then those fields are used as the time-varying input fields for the electromagnetic wave propagation model. In all, these models typically consisted of approximately 90,000 mesh elements and were run in less than 4 hours on Pentium PC with 2 GB of RAM.

The results of the radial antenna simulations are presented in Fig. 3. All four plots show the generated radiation fields at 0.1 THz in the x-y plane immediately after the silicon dome, which is visible in addition to the GaAs substrate. Figures 3(a) and 3(b) show the horizontal and vertical components of the generated electric field from the idealized radial antenna. The generated field from the idealized antenna is perfectly radial. The polarization of the generated field from the actual radial antenna structure (Fig. 3(c) and 3(d)) is not perfectly radially oriented. The lack of radial symmetry in the field generated by the actual antenna is caused by the lack of symmetry in the actual antenna design. Both the break in the outer electrode and the presence of the feed electrode create asymmetry in the lower half of the antenna structure. This results in the y-component of the electric field being stronger in amplitude than the x-component and the strongest parts of the x-component of the generated field not being centered on the antenna.

 

Fig. 3. FEM simulation results of the radial antenna in a typical THz configuration. (a) & (b) Plots of the x and y component of the electric field for the idealized radial antenna. (c) & (d) Plots of the x and y component of the electric field for the actual radial antenna.

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3.2 Simulation of enhanced wire waveguide coupling

While it is clear from Fig. 3 that the actual radial antenna design does not generate a perfectly radial terahertz beam, the results indicate that the beam is in fact largely radially polarized. By calculating the spatial overlap of the fields between these two cases (idealized antenna vs. actual antenna), we find that approximately 60% of the power generated by the actual antenna emerges in the form of a radial mode. In order to test the effectiveness of the new antenna design, we further employ FEM simulation tools to model the coupling of the radial antenna output to a wire waveguide. The same models used in the previous section are modified to include the addition of a wire waveguide directly end-coupled to the center and exterior of the silicon domes. The wire waveguide’s dimensions, 0.9 mm diameter, are identical to the waveguides used in the simulations reported in Section 2 of this work and in previously published results [15, 16]. For simulations involving the idealized antenna, the wire waveguide is 2.75 cm long, whereas those with the actual radial antenna design the waveguide is 1.75 cm long. This discrepancy is a result of our computer capability, which constrains the simulations to a maximum number of mesh elements. As the geometry with the actual radial antenna is much more complex, the antenna structure requires more mesh elements than the idealized antenna. Consequently, with a reduced number of mesh elements available, the wire waveguide is shorter. In both cases, the simulated domain is 6 mm by 6 mm in the tramsverse dimensions. As with the simulation results presented in Section 2, the wire waveguide is modeled as a PEC.

 

Fig. 4. FEM simulation model of radial antenna coupling to a wire waveguide (a) Plot of E_x at the end of a waveguide coupled to an ideal radial antenna. (b) Plot of E_y at the end of a waveguide coupled to an ideal radial antenna. (c) Plot of E_x at the end of a waveguide coupled to an actual radial antenna. (d) Plot of E_y at the end of a waveguide coupled to an actual radial antenna.

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FEM simulations at 0.1 THz of the idealized and actual radial antennas coupled to wire waveguides can be seen in Fig. 4. Plots of the field at the end of the wire from the idealized antenna results show that the idealized antenna is capable of exciting the low-order radial mode of the wire waveguide. Likewise, while less power is coupled to the waveguide, the actual antenna design is also shown to be capable of exciting the radial mode. A more quantitative observation can be obtained by performing a coupling efficiency calculation similar to the one performed in Section 2. The power available to be coupled into the waveguide is calculated by integrating the time-averaged power over a boundary in the x-y plane immediately after the silicon dome. This is the same plane where the electric field is plotted in Fig. 3(c) and 3(d). The power coupled into the waveguide is determined in a similar manner. The time-averaged power is integrated over a boundary at the end of the waveguide and normal to it. The areas integrated over in both cases are identical. For the actual radial antenna design, this calculation yields a coupling efficiency of approximately 56%, an improvement of more than 2 orders of magnitude over the 0.4 % coupling efficiency obtained via the scattering mechanism discussed earlier. Simulations with the waveguide removed were also performed. For both the case of the idealized and actual antenna structures, only a very tiny amount of power propagated to the end of the simulation domain.

 

Fig. 5. Setup for experimental testing of the radially symmetric terahertz emitter antenna. The emitter was pumped with a free-space beam. A 0.9 mm diameter, 27 cm long wire waveguide was end-coupled to the silicon dome of the emitter. THz radiation was detected at the end of the waveguide by a fiber-coupled LT-GaAs detector sensitive to only the horizontal polarization component.

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It should be noted here that our definition of efficiency is not the same as the overall efficiency of the system. If one were to consider the efficiency of the ultrafast laser system and the overall photoconductive generation of the terahertz in addition to the coupling efficiency, the reported numbers would be severely reduced, due to the low efficiency of the photoconductive generation process. An important note though is that our scheme fully utilizes the vast majority of the power generated by the radial photoconductive antenna. It is well known that for a linear dipole antenna located at an air-dielectric interface, most of the power generated by the dipole is transmitted into the dielectric [28]. Our previous results predict that for a radial antenna at an air-dielectric interface, over 98% percent of the power is radiated into the dielectric [26]. The conventional substrate + matched silicon dome configuration permits most of the power available to be coupled to the waveguide. Others have reported experimental results using a radial antenna similar in design to ours, but their configuration involved optically pumping the antenna through the dielectric substrate [18]. In this case, there is a significant drawback in that the amount of power available to be coupled to the waveguide is significantly lower because most of the power is lost into the high dielectric substrate. Another method for exciting the radial mode of a wire at terahertz frequencies involves coupling via a series of periodically spaced grooves milled into the waveguide [25]. This method is quite versatile, in that it permits the facile manipulation of the terahertz pulse shape, although no estimates of coupling efficiency have been advanced.

4. Experimental results

Following the FEM simulation efforts, photoconductive antennas with radial symmetry were fabricated and tested. The antennas were photolithographically defined on a 500 μm thick semi-insulating GaAs substrate. Subsequent to device fabrication, a matched hyperhemispherical silicon dome was mounted on the opposite side of the GaAs substrate. The results reported here use antennas consisting of an 8 μm diameter inner electrode separated from the outer electrode by 75 μm. A 20 V DC bias was applied to the antenna with the center circular electrode serving as the anode. The antenna was pumped with a free space 800 nm, 100 fs laser pulse with an average power of 25 mW. The pump beam was focused on the center of the anode such that only a portion of the region between the electrodes was illuminated. A 0.9 mm diameter, 27 cm long, stainless steel wire waveguide was end-coupled to the emitter by placing it in contact with the silicon dome at its apex (see Fig. 6). The terahertz pulse was detected at the end of the wire with a fiber-coupled photoconductive receiver. The receiver used here was a standard bowtie antenna, sensitive to only vertical or horizontal polarization components depending on the orientation of the antenna. The described setup is depicted in Fig. 5. Figure 6 is a picture of the wire waveguide end-coupled to the silicon dome mounted on the opposite side of the GaAs substrate from the radial antenna. The experimental configuration of the end-coupling of the waveguide to the antenna assembly mirrored the configuration simulated in the previous section.

 

Fig. 6. The wire waveguide end-coupled to the silicon dome, which is mounted on the opposite side of the GaAs substrate from the radial antenna.

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The primary advantage of the fiber-coupled terahertz receiver is that it permits detection at various positions close to the end of the wire waveguide. As reported in our previous work, there is only a very small amount of THz radiation present on the center axis at the end of the wire waveguide [15]. The strongest part of the guided THz field is typically located at a vertical or horizontal offset of a few mm from the center axis. Figure 7 shows plots of the experimental results from the testing of the coupling capability of the radial antenna. The data shown here are measurements of the horizontal component of the radially polarized propagating mode. It can be seen in Fig. 7(a), that only a very small signal is detected when the detector is placed at the center axis at the end of the wire. At a horizontal offset of +3 mm from the axis, a strong terahertz signal is present. A strong signal is present as well when the receiver antenna is placed at a horizontal offset of -3 mm. While these signals are virtually equal in magnitude, there is a clear polarity reversal between the two measurement positions. The equivalent magnitude of both signals and the reversal of polarity between them demonstrate the radial polarization of the guided mode, consistent with simulation results.

The increased coupling capability offered by the use of the radial antenna and the utility of the waveguide is further demonstrated in Fig. 7(b). When the wire waveguide is removed and measurements are made with the receiver at the same offset positions described above, the generated terahertz pulse is considerably smaller at a distance of 27 cm. In addition, we note that the polarity reversal is absent, as would be expected for a diverging spherical wave. A similar effect was also seen in the simulations reported in the previous section when the waveguide was removed. In fact, any standard terahertz spectroscopy or imaging setup in which the beam propagates through a distance this large would normally employ several lenses or mirrors to confine and guide the beam. The alignment of such beam-guiding optics can be difficult and often results in signal attenuation and bandwidth reduction. Our results show the utility of the radial antenna-waveguide configuration.

 

Fig. 7. Time-domain measurements of the terahertz pulse detected 27 cm away from the radial emitter. Measurements have been performed with the wire waveguide both present (7a) and absent (7b). All other experimental parameters are unchanged. The terahertz pulse exhibits a polarity reversal when measured at opposite locations at the end of the waveguide (upper and lower curves in (a)), a hallmark of the guided mode’s radial polarization.

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A qualitative determination of the coupling efficiency of the radial antenna-wire waveguide configuration can be obtained by comparing the two sets of results shown in Fig. 7. At the +3 mm horizontal offset, the peak-to-peak amplitude of the terahertz signal measured at the end of the waveguide is approximately 20 times larger than that for the pulse measured at the same position when the waveguide is not present. While that number does not represent the actual power coupling efficiency since these waveforms were measured at only a single point in space, it provides some insight into the considerable advantages of using the radial antenna-wire waveguide configuration.

4. Conclusion

In conclusion, we have performed finite element method simulations of terahertz experiments in which terahertz beams were scattered at an intersection of two wires. We simulate the same phenomenon observed experimentally, namely that the scattering process excites the low order radial mode of the wire waveguide. Calculations indicate that the coupling efficiency in this simulation, 0.4%, is comparable to earlier experimental estimates. We have shown the importance of spatial overlap between the polarization of the input terahertz beam and the radial mode of the wire waveguide. Simulation results showed that a novel photoconductive antenna with radial symmetry can be used in place of the standard linear dipole antennas to achieve coupling efficiencies greater than 50%. Integrating our antenna design into the standard substrate + silicon dome configuration also uses the vast majority of the generated terahertz radiation. Such antennas have been fabricated and tested. The largely radially polarized beam coupled to a wire waveguide effectively. A radially polarized mode was excited and guided over the length of the 27 cm wire waveguide. The magnitude of the peak terahertz electric field detected at the end of the waveguide was twenty times larger than the radiation detected at the same position with the waveguide removed. This is an important demonstration of the effectiveness of using a radially symmetric photoconductive terahertz antenna end-coupled to a metallic wire waveguide for convenient and efficient guiding of terahertz radiation.