1. Introduction

In the late nineteenth century, Sommerfeld showed theoretically that a single cylindrical conductor of finite conductivity could support a guided wave mode [1]. The dominant propagating mode is an azimuthally symmetric transverse electromagnetic wave, often referred to as a Sommerfeld wave. Propagation of this mode on a metal wire has been shown to allow for low loss, low distortion guided-wave propagation of broadband THz pulses [2–5]. In addition to the excellent propagation characteristics, a single wire, and variations of it, have elicited interest recently for applications including spectroscopic analyses of materials [6,7] and terahertz (THz) near-field imaging [8,9]. With regard to the latter application, it has recently been predicted using numerical simulations that wires, either corrugated [10, 11] or uncorrugated [12] along the shaft, may allow for unique capabilities including that of subwavelength focusing of THz radiation.

A significant issue that arises in the development of these applications is associated with coupling broadband or narrowband THz radiation onto the wire. In the simplest case of an uncorrugated cylindrical metal wire, the dominant mode is radially polarized, which does not match the mode properties of conventional THz emitters. This limitation has lead to the development of a number of interesting coupling techniques. In the original demonstration of guided wave propagation, an orthogonally oriented wire placed in close proximity to the wire waveguide was used to effect coupling [2,5]. Subsequent work used a radially symmetric photoconductive antenna to directly generate the requisite field pattern [3, 13], which is expected to allow for enhanced coupling efficiency. Radiation was coupled by placing the wire in contact with the center conductor of the antenna. In each of these cases, broadband THz pulses were coupled to the waveguide. Recently, we demonstrated a simple, flexible approach for coupling freely propagating broadband THz radiation to either a broadband or narrowband bound-wave propagating mode on a single cylindrical metal wire using grooves fabricated directly into the wire [14]. In that demonstration, we showed that each groove was able to couple the incident THz radiation to a propagating surface wave. Thus, by varying the number of grooves and the groove separation, both the linewidth and center frequency could be straightforwardly manipulated. However, a limitation of this approach is that the wire presents a relatively small cross-section to the incident focused THz beam. This, combined with the excitation geometry, may ultimately limit the coupling efficiency.

In this submission, we demonstrate a simple technique for coupling freely propagating THz radiation onto a cylindrical metal wire. While the approach may be used with either broadband or narrowband incident radiation, here we demonstrate the coupling of broadband freely propagating THz radiation to multi-cycle THz pulses on the metal wire. The basic approach involves inserting the end of a tapered metal wire into the center of a circular subwavelength aperture fabricated on a freestanding planar metallic sheet. The subwavelength aperture may be surrounded by a number of concentric annular grooves. The technique involves two important characteristics. First, we have previously demonstrated the ability to generate arbitrary shaped THz pulses by observing the transmission through a subwavelength aperture surrounded by annular grooves on a planar metallic sheet (bullseye structures) [15–18]. In that work, each annular groove couples a fraction of incident broadband THz radiation to a surface wave pulse and focuses it towards the central aperture. By using multiple grooves, multiple oscillations, delayed in time from one another in accordance with the groove separation, coherently superpose at the aperture. Second, by inserting a tapered wire into the center of the subwavelength aperture, we fundamentally alter the transmission characteristics of the aperture, since the transmission properties of the hole are altered from that of a circular subwavelength aperture to that of a coaxial waveguide. It should be noted that the transmission properties of a coaxial waveguide surrounded by annular grooves has been studied theoretically by Baida et al. [19] and demonstrated experimentally by Caglayan et al. [20] in the context of its enhanced optical transmission properties.

2. Experimental details

We fabricated two separate bullseye structures, one having 25 concentric grooves about the central subwavelength aperture and the other having 2 grooves. Figure 1(a) shows a photograph of the 25 groove bullseye structure. The structures were fabricated by chemical etching in a freestanding 150 μm thick stainless steel foil. The bullseye structure consisted of 500 μm wide annular grooves periodically spaced by 1 mm and a typical annular groove depth of 100 μm. The circular aperture had a diameter of 490 μm and was milled into the center of each bullseye structure. The 1 mm annular groove periodicity corresponds to a transmission resonance at approximately 0.3 THz. Based on our recent work on enhanced terahertz transmission through a single subwavelength aperture, it should be noted that the shape of the THz pulses radiated from the aperture is dependent on the geometry and position of individual grooves as well as the number of grooves around the aperture [16–19]. As we demonstrate below, this shaped THz pulse exiting the aperture on the metal foil is directly coupled onto the cylindrical wire. On the same metal foils, we also fabricated 490 μm diameter bare apertures with no surrounding metal structures.

 

Fig. 1. (a). Photograph of a typical bullseye structure used in the experiment. (b) Schematic diagram of the experimental setup. The THz radiation was normally incident on the structured metal foil. The photoconductive detector was placed approximately 7 cm from the metal foil and offset from the center of the wire. The wire length was approximately 6 cm.

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The wires used for waveguiding were 6 cm long with a diameter of 700 μm, in which one end of the wire was sharply tapered to a diameter of approximately 250 μm (commercially available sewing pin). As noted above, the tapered ends of the wires were inserted into the center of the circular aperture of the bullseye structure or the bare aperture fabricated in a metal foil. For reference purposes, we used an identical tapered wire that was only 1 cm long. The purpose of this last wire sample was to determine the extent to which freely propagating THz radiation exiting the annular groove contributed to the observed time-domain waveforms. The wires were suspended in air by puncturing a 250 μm thick Teflon sheet and care was taken to ensure the wires did not contact the planar metal foils.

The experimental setup for generating, coupling, and detecting THz radiation is shown in Fig. 1(b). A mode-locked Ti:sapphire laser operating at 820 nm with a repetition rate of 89 MHz was used as the optical source. The optical pump beam, with an average power of ~100 mW, was used to generate THz pulses from a photoconductive emitter. The linearly polarized THz electric field was normally incident on the bullseye structure. The photoconductive detector was situated approximately 7 cm from the metal foil in all cases, with the receiver offset from the axis of the wire by approximately 3 mm. As discussed below, the detector was placed in different positions surrounding the wire to ensure that the measured wave was radially polarized, which would correspond to the Sommerfeld wave.

3. Results and discussion

Before measuring the coupling of THz pulses to cylindrical wire using a bullseye structure, we measured the properties of the incident THz radiation. The measurement was performed with the Teflon sheet in place, but without the bullseye structure and wire in the terahertz beam path. Figure 2(a) shows the measured time-domain THz waveform in this experimental configuration, demonstrating that a single cycle THz pulse was incident on the wire. Figure 2(b) shows the corresponding amplitude spectrum. Then we placed the single aperture structures and the wire in the setup as shown in Fig. 1 and measured the resulting time-domain waveforms corresponding to the THz radiation coupled onto the wire and radiated from the end of the cylindrical wire.

 

Fig. 2. (a). Measured terahertz waveform incident on the bullseye structure. The measurement was performed with the Teflon sheet in place, but without the bullseye structure or the wire in the THz beam path. (b). The corresponding amplitude spectra.

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Figure 3(a) shows the observed time-domain THz waveforms radiated from the wires inserted into the center of the bare aperture and bullseye structure. Note that the magnitude of these waveforms cannot be compared to the waveform of Fig. 2(a), because of the different experimental parameters. Several features are readily apparent from these waveforms. In comparing the upper (red) trace and the lower (black) trace, it is clear that freely propagating THz radiation transmitted through the coaxial aperture does not contribute to the observed waveforms. This is not surprising, given that the absolute transmission through the annular aperture is small and the radiated THz pulse would be strongly divergent. In comparing the red, green, and blue waveforms, it is also apparent that the annular aperture serves to couple the incident THz bipolar waveform on the wire and that the annular grooves of the bullseye structure couple a series of periodically delayed replicas of the initial bipolar waveform.

We have shown previously that each groove on a bullseye pattern serves to couple a large fraction of the incident THz pulse in the form of a surface wave oscillation, such that there is a one-to-one correspondence between the number of grooves and the number of oscillations in the measured time-domain waveform [16]. When multiple grooves are periodically spaced, this corresponds to a narrowband THz pulse. The center frequency and linewidth of this excitation depend upon the groove spacing and number of grooves, respectively. Figure 3(b) shows the amplitude spectra computed from the corresponding to the time-domain waveforms in Fig. 3(a). The bare aperture (blue traces) leads to coupling of broadband radiation. With the bullseye, however, the 1 mm period spacing between the grooves corresponds to coupling of radiation that is peak about approximately 0.3 THz. This peak is increasingly sharpened with the number of annular grooves, as evidenced by comparing the data for the 2 groove and 25 groove bullseye structures. It is worth noting that the broad background surrounding this resonance feature arises from the initial bipolar waveform.

 

Fig. 3. (a). Experimentally observed time-domain waveforms for a tapered 6 cm long wire inserted into the center of the bullseye and the two-ring structure (red and green trace respectively), a tapered 6 cm long wire inserted into the center of the bare aperture (blue trace), and a tapered 1 cm long wire inserted into the center of the bullseye structure (black trace). The waveforms have been offset from the origin for clarity. Note that the magnitude of these waveforms may be compared against one another, but not with the waveform of Fig. 2(a). (b). Corresponding amplitude spectra using the same color scheme noted above. The additional black dashed spectrum is the reference spectrum from Fig. 2(b), appropriately scaled for purposes of comparison. The spectra have been offset from the origin for clarity.

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We noted above that the dominant propagating mode on the wire is radially polarized. In order to demonstrate this, we measured the time-domain properties of the THz waveform for two different detector positions: 3 mm above the wire center and 3 mm below the wire center. These two waveforms are shown in Fig. 4. The two waveforms are clearly related through a simple sign inversion, demonstrating that the polarization direction is reversed between these two positions. This is consistent with earlier results reported by Wang and Mittleman [2].

Finally, it should be pointed out that when the 6 cm long wire is moved so that the tip of the wire lies just outside of the aperture (< 50 μm from the exit surface of the metal foil), such that the aperture no longer appears as a coaxial waveguide, we observe no coupled THz radiation (i.e. the measured waveform is essentially identical to the lower (black) trace (not shown)). Thus, a coaxial waveguide geometry appears critical for coupling THz radiation to the wire. To better understand why the annular aperture exhibits significantly improved coupling characteristics than the circular aperture, it is important to identify the mode properties and cutoff frequencies associated with these two geometries.

 

Fig. 4. Measured time-domain THz waveforms for THz pulses coupled to a tapered 700 μm diameter stainless steel wire for two different detection points. The upper trace shows the waveform measured with the photoconductive detector located 3 mm above the wire center, while the lower trace corresponds to the observed waveform taken with the detector placed 3 mm below the wire center. The inversion of the observed waveform with the change in the detector position demonstrates clearly the radial polarization of the wave.

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The lowest order mode for a circular subwavelength aperture is the TE₁₁ mode. For a perfectly conducting metal, which is an appropriate approximation for the behavior of metals at THz frequencies, the cutoff frequency, f_c, is given by [21]

where c is the speed of light in vacuum, b is the aperture radius, and m is a constant associated with the mode for the cylindrical waveguide. For the lowest order TE₁₁ mode, m = 1.841. The diameter of the bare aperture 2b, used in this study was 490μm, corresponding to a cutoff frequency of ~0.36 THz. This value is larger than the resonance frequency associated with the bullseye structure. Thus, this narrowband feature, as well as a large fraction of the surrounding broadband spectral information, will evanescently propagate through the aperture.

On the other hand, inserting a metal post, in the form of a tapered wire, at the center of the circular aperture forms a coaxial waveguide. The two lowest order modes for this structure are the TEM mode, which does not have an associated cutoff frequency, and the TE₁₁ mode, which for a perfectly conducting metal, exhibits a cutoff frequency, f_c, given by [21]

where a and b are the inner and outer radii of the annular opening, respectively. The coaxial aperture used in our study has an external diameter, 2b, of 490μm, while the internal diameter, 2a, of the aperture is determined by the extent to which the tapered end of the metal wire is inside the 490μm diameter aperture. We estimate that the average value of 2a was approximately 300μm, resulting in a cutoff frequency of ~0.24 THz. It should be noted that the cutoff frequency of the coaxial waveguide could be varied by either modifying the diameter of the subwavelength circular aperture or by changing the extent to which the tapered end of the cylindrical wire is inserted into the aperture. The cutoff frequency associated with this mode is smaller than the resonance frequency of the bullseye structure. Thus, whether coupling through the coaxial waveguide is via the TEM or the TE₁₁ mode, the attenuation properties of the coaxial waveguide are reduced relative to the cylindrical waveguide, leading to improved coupling characteristics of the former aperture.

The electric field distribution of the TEM mode in a coaxial waveguide is radially polarized. Thus, a linearly polarized incident beam is not expected to be able to efficiently excite this mode. On the other hand, since a linearly polarized incident beam should be capable of efficiently exciting the TE₁₁ mode, we would expect it to be the dominant mode. However, the experimental data, shown in Fig. 3(b), does not seem to support this conclusion. If the broad spectral content associated with THz pulses coupled to the cylindrical wire from the coaxial waveguide [blue and red spectra in Fig. 3(b)] is compared to the spectrum of the incident THz radiation [dashed black spectrum in Fig. 3(b)], they are nearly identical. If the TE₁₁ mode were dominant, we would expect that the spectral content associated with the two wire measurements (blue, green and red spectra) to be increasing suppressed with decreasing frequency below ~0.3 THz. The fact that this is not the case suggests that we may indeed be exciting the TEM mode. Although it is beyond the scope of the present work, further investigation is necessary to fully understand the details of the coupling mechanism.

It is not possible to measure experimentally what fraction of the incident THz beam is coupled to the wire. This is due primarily to the differences in the collection and detection efficiencies for the incident beam versus the radially polarized wave that radiated from the wire. However, as with previous attempts to estimate experimentally the coupling efficiency [13], we can qualitatively describe the benefit of using the coaxial waveguide geometry to increase the coupling efficiency. First, in comparing the results of Fig. 3, it is apparent that the use of an annular aperture offers dramatic improvement in coupling efficiency over a subwavelength circular aperture. In addition, we can compare the present approach to our previously reported approach [14] in which grooves were fabricated directly into the cylindrical wire. Such a comparison can be made because although the details of the coupling mechanism differ somewhat, in both cases, we used the same emitter and the same detection setup. Furthermore, the dimensions of the wires in the two experiments are different. However, the low-loss nature of guided THz waves mitigates this issue [2,3]. Using a wire that had 3 grooves milled into it [14] and an equivalent bullseye structure, we find that the relative coupling efficiency of the bullseye approach described here is approximately a factor of 10 greater that for the wire with milled grooves. This is not surprising given the orthogonal excitation geometry that we previously used.

4. Conclusion

In summary, we have demonstrated a flexible approach for coupling freely propagating THz radiation to a single cylindrical metal wire. In the case of a single subwavelength aperture, we are able to couple broadband THz radiation. By fabricating concentric annular grooves about the aperture, narrow band THz radiation may be coupled to the wire. This approach offers significant flexibility in terms of the spectral content that may be coupled to the wire as well as the type of emitter that may be used. In the present study, we used periodically spaced grooves that had identical cross-sections. However, by using bullseye structures in which the spacing and cross-section of the annular grooves can be arbitrarily designed [17], we expect this approach to allow for coupling of THz pulses that are more arbitrarily shaped. We note that although we used a linearly polarized source in our demonstration, a broader range of emitters may be used. For example, radially polarized sources [3,13,22] may enable more efficient coupling, since the polarization would better match the lowest order TEM mode of the annular aperture. Also, by using very broadband THz sources, one could couple narrowband THz radiation onto the wire at frequencies much higher than might be possible using photoconductive emitters.