In this work, measurements and numerical field simulations highlighting the characteristic propagation behavior of THz surface-wave pulses along bare and dielectrically coated metal wires are presented. An optoelectronic time-domain measurement setup with a freely-positionable probe-tip is used for detection of electrical field transients after different propagation lengths along the wires. Frequency-dependent attenuation and dispersion parameters are determined in the range of 0.02 THz to 0.4 THz. Our results are in good agreement with numerical field simulations considering the propagation of an axial Sommerfeld surface-wave with metallic and dielectric losses. We discuss the influence of wire radius on wave propagation behavior and the application of THz single-wires for sensing.
©2005 Optical Society of America
The interest in THz technology has strongly increased in the last years with diverse applications in the fields of biotechnology [1,2], spectroscopy , imaging [4,5] and security . The majority of THz systems are based on transmission of free-space THz radiation. Comparably efficient guided THz transmission solutions are still under investigation. Alternative to free-space propagation experiments, guided-wave concepts have a significant potential for further applications. Surprisingly, a very old idea – the single-wire waveguide – is currently rediscovered as one of the most promising THz waveguide solutions with extremely low attenuation and dispersion characteristics . Indeed, the history of wave propagation along single metallic wires can be traced back to the 19th century. In 1888, Hertz made first investigations on wave propagation on a conductive wire . Eleven years later, it was Sommerfeld who found a first rigorous solution of Maxwell’s equations for the boundary value problem for wave propagation on a single-wire . Consequently, this type of waveguide is often called Sommerfeld wire and the guided axial surface-wave is a Sommerfeld wave. It then took about fifty years until first measurements demonstrated feasible excitation and detection of surface-waves on conducting plain or cylindrical interfaces and dispelled widespread doubts about their existence. In 1962, King and Wiltse theoretically analyzed surface-waves on single-wires in the range between 100 GHz and 1 THz , one year after the first published measurements in the mid millimeter wave range at 140 GHz by Sobel et al. .
Despite of its superior transmission behavior, the single-wire waveguide did not find the same broad application as other metal-based waveguide concepts like e.g. hollow , planar , or coaxial [14,15] waveguides. The electric field propagating along a single-wire largely extends into the surrounding air limiting its practical application especially at microwave (MW) frequencies. For example, a wire of 1 cm diameter carries 75% of its transmitted power at 10 GHz confined to a circular area of 40 cm radius around the wire axis . In order to confine the electromagnetic field closer to the surface, the wire can be coated with a dielectric . Depending on the thickness and the dielectric properties of the coating material this so called Goubau line shows additional dielectric loss and dispersive behavior. The field extension into air, however, decreases with frequency thus making singles-wires attractive waveguides at THz frequencies: The 75% power radius of a 100 μm diameter wire can be reduced to less than 1 mm at 1 THz.
The experimental identification of the significant loss mechanisms of THz single-wire waveguides is challenging at the extremely low attenuation levels these lines exhibit. Recent work [7,17] strikingly demonstrated negligible dispersion and very low attenuation of single-wires in the frequency range from approximately 20 GHz to 600 GHz, but failed to render the qualitative frequency behavior of the attenuation consistent with theory. An increasing attenuation for longer wavelengths, as observed by Wang et al. , can neither be explained with present analytic models [9,10] nor with numerical field simulations . A so far unintensified loss mechanism has been invoked to explain this discrepancy.
In this work, we experimentally characterize the frequency-dependent transmission behavior of coated and uncoated single-wires with an optimized measurement approach. The results are in good agreement with theoretical results from numerical field simulations considering the propagation of an axial Sommerfeld surface-wave with metallic and dielectric losses.
2. Experimental setup
The experimental setup is based on a standard THz time-domain pump/probe scheme. Sub-ps laser pulses at 800 nm wavelength and 78 MHz repetition rate are used for photoconductive excitation and detection of THz signals. For excitation and detection of THz pulses, photoconductive (PC) switches are fabricated. The switches are integrated into planar transmitter and probe-tip structures. The transmitter chip consists of a ring-shaped low-temperature (LT) grown GaAs layer as the PC-material, which is enclosed by two concentric gold electrodes on a sapphire substrate (see Fig. 1(a)). The inner electrode is biased at +100 V with respect to the outer one. Upon optical excitation, the transmitter generates a radial field mode that excites a radial surface-wave on the wire. The PC-probe tip is depicted in Fig. 1(b). It consists of a strip-line structure with an integrated PC-switch to probe the transient electrical field on the wire surface. The carrier substrate of the PC-probe tip is an optically transparent flexible polypropylene film with a thickness of 35 μm. It has a low permittivity of εr = 2.35 in the THz frequency range which allows efficient coupling to the probed field transients. The PC-probe tip is placed in a distance of a few micrometers to the wire surface.
Optical excitation pulses are coupled into a fiber and directed to the PC-switch of the transmitter chip with an average power of 15 mW. A pair of diffraction gratings is added to the excitation-beam path to compensate for the dispersion of the fiber. Time-domain measurements of propagating surface-wave signals are performed by changing the relative time delay between the excitation and the detection pulse with a mechanical delay-stage. The excitation beam is chopped at 12 kHz for lock-in detection. The metal wire used in this experiment has a radius r = 25 μm and is based on an alloy consisting of Cu, Mn, and Ni with 86, 12, and 2 mass-percent, respectively. The minimum distance of any metallic objects to the wire axis in transversal direction is approximately 5 cm. The wire is equipped with a 10-μm-thick dielectric coating of Polyurethane that can be removed by chemical wet-etching in H2SO4 : H2O2 (10:1). The wire is clamped between two PMMA disks of 3 mm thickness. The assembly of the wire, excitation chip and fiber is mounted on motorized translation stages for precise and reproducible positioning (Fig. 1(c)). To increase the reliability of measurement results of the determination of frequency-dependent attenuation, we make several optimizations: As recently demonstrated by Jeon et al.  – due to optimized mode-matching conditions – the transmitted field of a radial antenna structure can be coupled much more efficiently to a wire than a planar wave focused onto an aperture. However, different from Ref.  LT-GaAs is used instead of ion-implanted silicon-on-sapphire to increase further the amplitude of the emitted field . Additionally, the freely-positionable probe-tip allows for consecutive measurements of field transients at any position without requiring any readjustments of the setup. This not only reduces the probability of measurement errors due to misalignment, but also speeds up the whole characterization process. Measurements at different positions are used for error reduction by averaging over multiple data sets.
The accuracy of frequency-dependent attenuation determination increases with the overall attenuation caused by the wire. Higher attenuation is generally obtained by either increasing the length of the wire or decreasing the diameter of the wire . Increasing the length of the wire to more than 30 cm, however, would require impractically long time-delay stages. Therefore, we choose a thinner wire (radius 18 times smaller compared to the wire used in Ref. ), which is optionally coated with a 10-μm-thick dielectric coating based on polyurethane. These wires prospectively show higher losses compared to the wires used in former studies [7,17]. However, the wire radius used in this study is still large in comparison to skin-depth. Hence, the fundamental loss mechanisms that have to be considered are still the same as for thicker wires . THz wires and tubes with radii smaller than skin-depth have been theoretically considered, recently .
3. Field simulations
A semi-analytic description of surface-wave propagation on coated and uncoated metal wires at THz frequencies can be found in Ref. . In our work, all electromagnetic solutions are calculated with the commercially available numerical field simulation software HFSS (Ver. 9.2) from Agilent Technologies which is based on the finite element method and adaptive mesh optimization. The applied model for theoretical characterizations of propagation parameters of coated and uncoated single-wires is shown in Fig. 2(a). Electromagnetic fields are solved in the cylindrical model volume with radius R = 8 mm and length L = 3 mm except of the inner volume of the metal wire with radius r = 25 μm. The model accounts for frequency-dependent metallic loss by using a finite conductivity boundary condition for the wire surface based on the dc conductivity of copper (σ = 58 × 106 S/m), which is the main material component of the wire. This simplification is appropriate since the wire radius is much larger than the skin-depth in the considered frequency range. Dielectric loss is considered by including a frequency-independent loss factor tan δ = 0.001 and a relative permittivity εr = 2.1 for the Polyurethane coating with a thickness d = 10 μm. For the uncoated wire we set d = 0 μm. The dielectric environment of the wire is assumed as air with εr = 1.0006. The parallel circular faces of the cylindrical model are defined as wave ports which are assumed by HFSS as semi-infinitely long matched extensions of the waveguide. The two-dimensional solutions of the wave ports serve as boundary conditions for the three-dimensional model. They also directly yield the complex frequency-dependent propagation constant γ(f) = α(f) + iβ(f) with α describing the attenuation and β the dispersion of the considered TM01 fundamental mode of the axial surface-wave. The nonzero field components of this mode include electric fields in radial and longitudinal direction as well as azimuthal magnetic fields. Higher order modes are not considered, since they are known to exhibit drastically increased attenuation compared to the fundamental mode . In the experiment, a propagation distance of 10 mm from the wire input is applied to allow efficient decay of possible higher-order modes. Meshes are generated for 100 GHz and 300 GHz. Using these meshes port solutions are calculated with discrete frequency sweeps from 20 GHz to 200 GHz and from 200 GHz to 400 GHz.
Surface-waves propagating on a cylindrical surface without any discontinuity are not radiated . However, introducing a discontinuity to the wire, e.g. a corrugation  or a bending , causes radiation of a guided surface-wave or can conversely be used to promote coupling of free-space radiation for the excitation of a guided surface-wave. Despite of the non-radiative character of our transmission problem, we nevertheless define the cylindrical surface of the solution volume as a radiation boundary that does not reflect energy. In this way the approximation error encountered by solving the infinitely expanded surface-wave field within a finite solution volume can be estimated. For a sufficiently large radius R the influence of the limited solution volume to the complex propagation constant becomes negligible, as demonstrated in Fig. 2(b) for the amplitude attenuation α(R). R has to be adapted to the solution frequency and the radius of the wire: it has to be increased for decreasing frequency or increasing wire radius. A comparison of our numerical data with the semi-analytical model from Ref.  is depicted in the inset of Fig. 2(b). The data shows the frequency-dependent amplitude attenuation for a 1 mm radius Cu wire and R = 30 mm. With these dimensions, we achieve good agreement (11% deviation at 0.02 THz and 3% deviation at 0.4 THz) between the results given in Ref.  and our simulation. On the other hand, a decreased R causing considerably clipping of the guided field mode, can be used to monitor the frequency-dependent tendency of the wire for parasitic coupling to neighboring objects as demonstrated in Ref. .
4. Results and discussion
As described above, we use a radial antenna for the excitation of guided surface-waves. To avoid any ambiguity due to superposition of guided and radiated THz signal components we first measure the transmission of pure free-space radiation in the absence of a guiding structure. Measured free-space transmission signals for transmitter to probe distances between 0.05 mm and 6.05 mm are depicted in Fig. 3. Based on these data we can conclude that contributions of radiated THz signal components are negligible for transmission distances larger than 6 mm.
The propagation of surface-waves on coated and uncoated single-wires is investigated with the following measurements. Transient electric fields are probed in steps of 2 cm in the range from 1 cm to 9 cm along the coated wire and from 1 cm to 25 cm along the uncoated wire after excitation. The set of time-domain data probed on the coated wire is plotted in Fig. 4(a). The detected THz signals are characterized by a distinct initial peak, followed by an oscillating signal with strongly reduced amplitude. The oscillations originate from the transmitter and the coupling configuration between the transmitter and the wire-tip. The thin wires used for our experiments are simply cut with a pair of scissors. With this method a tapered shape of the wire ending is achieved, which is however, not very reproducible. Measurements at differently formed wire-tips exhibited a strong influence on the pulse shape. Therefore, a better technique for making uniformly pointed wires is expected to further improve coupling.
For the determination of transmission parameters of the waveguide, however, the spectral bandwidth of the excited surface-wave signal is much more important than its exact temporal shape. The usable bandwidth of our system is approximately 0.4 THz given by the temporal width of 1.8 ps (FWHM) of the initial peak probed at 1 cm propagation distance on both wires. As can be seen from the data in Fig. 4a, the pulsed THz signal is considerably attenuated while propagating along the coated wire. The effect of group velocity dispersion is clearly visible in Fig. 4(b), where the signal shapes probed after 1 cm and 9 cm propagation are compared. The signals are normalized and shifted in time for clarity. While the pulse on the coated wire is broadened to 3.8 ps (FWHM), the pulse on the bare wire stays almost unchanged. In contrast to former results  however, we observe a small (0.2 ps) but evident increase in pulse width. This already indicates a low-pass frequency transmission character of the wire.
The detailed frequency-dependent propagation parameters in terms of attenuation α(f) and effective permittivity εr,eff(f) are discussed in the following. Frequency-domain propagation parameters are derived from the Fourier transformed time-domain data sets using a procedure described in Ref.  based on linear regression fits. εr,eff(f) is calculated from β(f) by employing the relation εr,eff(f) = (βc/2πf)2, with c being the velocity of light. In Fig. 5(a) measured and simulated results for the amplitude attenuation α(f) are displayed. The measured data exhibit very good agreement with the simulations showing a monotonic increase of α with frequency. Even at this small radius, the attenuation of the uncoated wire remains low. Conductor loss scales roughly with f ½, whereas dielectric loss is proportional to f·tan δ. As a result, the influence of dielectric loss at the coated wire increases with frequency.
The coated wire exhibits a one or two order of magnitude higher attenuation in comparison to the bare wire. For completeness, the transmission parameters of copper-based single-wires with r = 450 μm – uncoated and coated (d = 10 μm) – are calculated as well. In this case the contribution of dielectric loss of the coated wire becomes negligibly small at lower frequencies. The attenuation of the uncoated wire with r = 450 μm is only 0.002 dB/mm at 400 GHz. Fig. 5(b) shows the measured and simulated values for the effective permittivity εr,eff(f) of the investigated wires. As expected, the uncoated wire with r = 25 μm is almost dispersion-free in the frequency range monitored. The slight increase of εr,eff observed for f → 0 is caused by skin-effect. This kind of implication on εr,eff(f) is mainly observed at metal-based waveguides with lateral dimensions much smaller than λ [21,24]. For larger radii this effect will be very difficult to detect as demonstrated by the simulation data for r = 450 μm. The dispersion behavior of the coated wire, on the other hand, is clearly dominated by the influence of the dielectric coating – even in the case of small wire radius and low frequency where the metallic dispersion effect contribution is maximal. With increasing frequency the decay length of the surface-wave field decreases, which causes the increase of εr,eff. The dispersion of the Goubau line is strongly reduced by choosing d ≪ r, as displayed in Fig. 5(b) for r = 450 μm.
Recently, a very interesting sensing application of single-wires has been suggested by Planken et al. : the monitored transmission signal can be used for detection of thin films at the wire surface. In this reference a thick wire with r = 0.5 mm has been considered. However, as can be seen from the calculated data in Fig. 5(b), phase retardation at identically thick coated wires is much larger for smaller r. Therefore, thinner wires appear favorable for sensing applications.
In this work, the THz transmission behavior of Sommerfeld waves on coated and uncoated metal single-wires has been investigated. Frequency-dependent attenuation and dispersion parameters have been determined experimentally up to 0.4 THz and compared with numerical field simulations. Our results confirm the Sommerfeld axial surface-wave model as still valid even at THz frequencies. Agreement is achieved between experimental results and numerical calculations if metallic and dielectric losses of these axial Sommerfeld are taken into account. Recent observations of increasing attenuation for f → 0 , however, are not confirmed by our results. Implications for sensing applications are noted.
This work was financially supported by the European Commission through the framework VI integrated project teranova. We also acknowledge helpful discussion with J. Gómez Rivas and H. M. Heiliger.
References and links
1 . P. H. Siegel , “ Terahertz Technology in Biology and Medicine ,” IEEE Trans. Microw. Th. and Tech. 52 , 2438 – 2447 ( 2004 ). [CrossRef]
2 . M. Nagel , P. Haring Bolivar , M. Brucherseifer , and H. Kurz , “ Integrated THz technology for label-free genetic diagnostics ,“ Appl. Phys. Lett. 80 , 154 – 156 ( 2002 ). [CrossRef]
5 . M. M. Awad and R. A. Cheville , “ Transmission terahertz waveguide-based imaging below the diffraction limit ,” Appl. Phys. Lett. 86 , 221107 /1–3 ( 2005 ). [CrossRef]
6 . D. L. Woolard , E. R. Brown , M. Pepper , and M. Kemp , “ Terahertz Frequency Sensing and Imaging: A Time of Reckoning Future Applications? ,” Proc. IEEE 93 , 1722 – 1743 ( 2005 ). [CrossRef]
8 . H. Hertz , Gesammelte Werke ( J. A. Barth , Leipzig, 1894 ).
9 . A. Sommerfeld , “ Ueber die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes ,“ Ann. Phys. u. Chemie 67 , 233 – 290 ( 1899 ).
10 . M. J. King and J. C. Wiltse , “ Surface-Wave Propagation on Coated or Uncoated Metal Wires at Millimeter Wavelengths ,” IEEE Trans. Ant. and Prop. 10 , 246 – 254 ( 1962 ). [CrossRef]
11 . F. Sobel , F. Wentworth , and J. C. Wiltse , “ Quasi-Optical Surface Waveguide and Other Components for the 100- to 300-Gc Region ,” IEEE Trans. MTT 9 , 512 – 518 ( 1961 ). [CrossRef]
12 . G. Gallot , S. P. Jamison , R. W. McGowan , and D Grischkowsky , “ Terahertz Waveguides ,” J. Opt. Soc. Am. B 17 , 851 – 863 ( 2000 ). [CrossRef]
13 . R. K. Hoffmann , Integrated Microwave Circuits ( Springer, Berlin 1983 ).
14 . N. Marcuvitz , “ Coaxial Waveguides ” in Waveguide Handbook ( McGraw-Hill, New York, 1951 ), pp. 72 – 80 .
15 . P. W. Hawkes , The Physics of Transmission Lines at High and Very High Frequencies ( Academic Press, 1970 ).
16 . G. Goubau , “ Surface Waves and Their Application to Transmission Lines ,” J. Appl. Phys. 21 , 1119 – 1128 ( 1950 ). [CrossRef]
17 . T. Jeon , J. Zhang , and D. Grischkowsky , “ THz Sommerfeld wave propagation on a single metal wire ,” Appl. Phys. Lett. 86 , 161904 /1–3 ( 2005 ). [CrossRef]
18 . Y. Xu and R.G. Bosisio , “ Study of Goubau lines for submillimetre wave and terahertz frequency applications ,” IEE Proc.-Microw. Antennas Propag. 151 , 460 – 464 ( 2004 ). [CrossRef]
19 . F. E. Donay , D. Grischkowsky , and C.-C. Chi , “ Carrier lifetime versus ion-implantation dose in silicon on sapphire ,” Appl. Phys. Lett. 50 , 460 – 462 ( 1987 ). [CrossRef]
21 . M. J. Hagmann , “ Isolated Carbon Nanotubes as High-Impedance Transmission Lines for Microwave Through Terahertz Frequencies ,” IEEE Trans. Nanotech. 4 , 289 – 296 ( 2005 ). [CrossRef]
22 . G. Goubau Waves on interfaces. IRE Trans. on Ant. and. Prop. 7 , 140 – 146 ( 1959 ). [CrossRef]
24 . H.-M. Heiliger , M. Nagel , H. G. Roskos , H. Kurz , F. Schnieder , W. Heinrich , R. Hey , and K. Ploog , “ Low-dispersion thin-film microstrip lines with cyclotene (benzocyclobutene) as dielectric medium ,“ Appl. Phys. Lett. 70 , 2233 – 2235 ( 1997 ). [CrossRef]
25 . N. C. J. van der Valk and P. C. M. Planken , “ Effect of a dielectric coating on terahertz surface plasmon polaritons on metal wires ,” Appl. Phys. Lett. 87 , 071106 /1–3 ( 2005 ). [CrossRef]