Open Access
Add to BibSonomyAbstract
A theory for treating terahertz (THz) surface wave propagation and focusing on conical metal wire waveguides is presented. According to the theory, the surface wave propagation and focusing on the conical copper wire waveguide is investigated by the numerical calculation, and the results show the theory agrees with experimental measurement result obtained by Ji et al. The theory presented provides a basis for describing surface wave propagation and focusing on conical metal wire waveguides and, importantly, the theory suggests that a simple conical metal wire can be used for subwavelength energy concentration and superfocusing.
©2008 Optical Society of America
1. Introduction
Terahertz (THz) waveguide techniques have attracted extensive attention over the last several years [1–3]. Especially, Wang and Mittleman demonstrated that a single metal wire can be used as a THz waveguide with low propagation losses and negligible group velocity dispersion [4]. Subsequently, many researchers and scientists have studied wave propagation on a single metal wire in the THz frequency range [5–8]. The basic demonstration is based on the theory of a single wire presented by Sommerfeld in 1899 [9].
Recently, a conical wire tip was used to launch a THz pulse on a metal wire waveguide by many researchers such as Deibel [10], Jeon [5], Smorenburg [11] and so on. Ji et al investigated the propagation properties of the conical wire waveguide in the THz frequency range using simulations and experiments [12]. The measured THz pulse increased 4.5 times upon contact with the 30µm diameter conical wire tip compared with the THz pulse when a 500µm diameter cylindrical wire tip was used. The authors focused on the report of simulations and experimental observations, however, without a deep theoretical explanation.
In this paper, a theory for treating THz surface wave propagation and focusing on conical metal wire waveguides is presented, and the expression for describing surface wave propagation on this kind of waveguides is obtained based on the Sommerfeld surface wave model [13–15]. The results obtained agree with that of experimental measurement obtained by Ji et al [12]. What is more, we demonstrate that a simple conical metal wire waveguide can be used for subwavelength energy concentration and superfocusing. Compared with Maier’s periodically corrugated metal wire [16], this conical metal wire mainly has two interesting advantages. First, this wire can be used for subwavelength energy concentration and superfocusing of both narrowband and broadband THz wave, but Maier’s wire can be only used for subwavelength energy concentration and superfocusing of narrowband THz wave. Second, this wire, whose structure is simple, is much easier to be manufactured.
2. Surface wave propagation theory of conical metal wire waveguides
Figure 1 shows the configuration of a conical metal wire waveguide and the coordinate system. If the taper of the conical wire is small (namely θ is small in Fig.1), the conical wire can be considered as constitution of numberless cylinder wires. When the THz wave propagates in the positive z-direction, the field components can be written as [13,14]:
where Er is the electrical field of the radial direction, Ez is the electric field of the z direction, Hφ is the magnetic field of the angular direction, h(z) is the propagation constant of the mode, ω is the angular frequency of the THz wave. The wave vector k is
(ε=dielectric constant, µ=permeability, σ=metal conductivity). The subscript c refers to the interior of the conductor. The parameters γ and γc can be expressed as
The cylinder functions Z 0 and Z 1 are the Bessel functions J 0 and J 1 inside the conductor and are the Hankel functions H 0 (1) and H 1 (1)outside the conductor. From the boundary conditions, we can obtain that Ez and Hφ are continuous at the surface (r=az) of the conductor. Thus, γ(z), γc(z) and h(z) can be determined [13] and the field components can be acquired subsequently. If we obtain A(z), we can know the whole propagation characteristics of the conical metal wire waveguide.
The energy propagating through the area of the transverse plane for az < r < ∞ can be written as [14]
When the THz surface wave propagates on the metal wire waveguide, the attenuation is very low [4]. If the attenuation is neglected, N(z) is equivalent for different z, and so we can obtain the equation:
According to Eqs. (1), (2), (3) and (9), we obtain
A(z) is given by
According to Eqs. (1), (3) and (11), the ratios of the field components in the position which is a away from the surface of the conical metal wire waveguide can be written as:
, so Ez is usually neglected [14]. As , according to Eqs. (12) and (13), the ratios of the field components in the position which is a away from the surface of the conical metal wire waveguide can be written as:
h(0), h(z), γ(0) and γ(z) can be obtained by numerical calculation [13]. Surface wave propagation characteristics on conical metal wire waveguides can be obtained by Eqs. (1), (2), (3) and (14).
A conical copper wire, whose diameter gradually decreases from 500µm to 30µm, is adopted. Figure 2 shows the ratio of the E field component (the radial distance from the surface a=30µm) at the 30µm-diameter tip to that at the 500µm-diameter tip versus the frequency. As shown in Fig. 2, when the frequency changes from 0.1THz to 1.0THz, the ratio of the E field changes about from 5.4 to 5.0. When the frequency is 0.15, the ratio is about 5.4. The ratio obtained by experimental measurement is 4.5 [12]. The theory result is slightly larger than the experimental value. There are mainly two reasons responsible for the difference. First, the field intensity near the wire surface is very sensitive, but an exact 30µm separation between the tip’s surface and the dipole antenna gap can not be made in the experiment [12]. Second, the attenuation of the conical metal wire is neglected during the theory deduction above. Therefore, the theory agrees with the experimental measurement result obtained by Ji et al.
3. Surface wave propagation and focusing characteristics of conical metal wire waveguides
3.1 Relations between focusing characteristics and the tip diameter
A conical copper wire with 500µm beginning-diameter is adopted. The frequency of the THz wave adopted is 0.15THz. Figure 3 shows the ratio of the E field component (the radial distance from the surface a=0µm) at the end-tip to that at the beginning-tip versus the end-diameters. As shown in Fig. 3, the smaller the end-tip diameter is, the larger the ratio of the E field is. Therefore, the conical wire with smaller end-diameter is more helpful for energy concentration and focusing.
3.2 Relations between focusing characteristics and the THz wave frequency
A conical copper wire with 500µm beginning-diameter and 30µm end-diameter is adopted. Figure 4 shows the ratio of the E field component (a=0µm) at the end-tip to that at the beginning-tip versus the frequency. As shown in Fig. 4, the lower the THz wave frequency is, the larger the ratio of the E field is. However, the ratio variation is small, and so the effect of the THz wave frequency on the energy concentration and focusing is small.
4. Conical metal wires for subwavelength energy concentration and superfocusing
A conical copper wire with 500µm beginning-diameter and 10µm end-diameter is adopted. Figure 5 shows the normalized amplitude of E field versus the radial distance around the end-tip for the THz surface wave with 0.15THz frequency. There are mainly two interesting superfocusing characteristics for this conical wire. First, as shown in Fig. 5, the E field component in the position that is 0.02λ from the surface decreases to about 10% of that on the surface. Second, as shown in Fig. 3, the E field component on the surface of the end-tip, compare with that on the surface of the beginning-tip, increases about 41 times.
In order to compare the propagation characteristics of the conical wire with those of the cylindrical wire, we also calculate the E field component around the cylindrical copper wire with 500µm diameter. Figure 6 shows the normalized amplitude of E field versus the radial distance around the end-tip for the THz surface wave with 0.15THz frequency. As shown in Fig. 6, the E field component in the position that is λ from the surface decreases to about 10% of that on the surface. What is more, as the attenuation of the metal wire is very low [4], the field around the end-tip is nearly the same as that around the beginning-tip for a short cylindrical wire.
As shown in Fig.5 and Fig. 6, compared with the cylindrical wire, the conical wire has two advantages as follow: first, it can concentrate the energy of THz surface wave greatly; second, it can increase the field of THz wave on the wire surface about 41 times.
5. Conclusion
A theory for treating THz surface wave propagation and focusing on conical metal wire waveguides is presented. According to the theory, the surface wave propagation on the conical copper wire waveguide is investigated, and the results show the theory agrees with the experimental measurement result obtained by Ji et al. Furthermore, both the surface wave focusing properties of the conical copper wire waveguides versus the wire diameter and versus the THz wave frequency are studied by the numerical calculation. Finally, we demonstrate that a simple conical metal wire can be used for subwavelength energy concentration and superfocusing, and so this wire has great promise for guiding terahertz wave to subwavelength volumes for near-field imaging, spectroscopy, and sensing applications.
References and links
1. C. Yeh, F. Shimabukuro, P. Stanton, V. Jamnejad, W. Imbriale, and F. Manshadi, “Communication at millimeter- submillimetre wavelengths using a ceramic ribbon,” Nature 404, 584–588 (2000). [CrossRef] [PubMed]
2. R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, 846–848 (2001). [CrossRef]
3. H. Han, H. Park, M. Cho, and J. Kim, “Terahertz pulse propagation in a plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634–2636 (2002). [CrossRef]
4. K. Wang and D. M. Mittleman, “Metal wires for terahertz waveguiding,” Nature 432, 376–379 (2004). [CrossRef] [PubMed]
5. T. -I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86, 161904 (2005). [CrossRef]
6. K. Wang and D. M. Mittleman, “Guided propagation of terahertz pulses on metal wires,” J. Opt. Soc. Am. B 22, 2001–2008 (2005). [CrossRef]
7. M. Wächter, M. Nagel, and H. Kurz, “Frequency-dependent characterization of THz Sommerfeld wave propagation on single wires,” Opt. Express 13, 10815–10822 (2005). [CrossRef] [PubMed]
8. K. Wang and D. M. Mittleman, “Dispersion of surface plasmon polaritons on metal wires in the terahertz frequency range,” Phys. Rev. Lett. 96, 157401 (2006). [CrossRef] [PubMed]
9. Sommerfeld, “Ueber die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Ann. Phys. u. Chemie67, 233–290 (1899).
10. J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, “Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,” Opt. Express 14, 8772–8778 (2006). [CrossRef] [PubMed]
11. P. W. Smorenburg, W. P. E. M. O. Root, and O. J. Luiten, “Direct generation of terahertz surface plasmon polaritons on a wire using electron bunches,” Phys. Rev. B 78, 115415 (2008) [CrossRef]
12. Y. B. Ji, E. S. Lee, J. S. Jang, and T. -I. Jeon, “Enhancement of the detection of THz Sommerfeld wave using a conical wire waveguide,” Opt. Express 16, 271–278 (2008). [CrossRef] [PubMed]
13. G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21, 1119–1128 (1950). [CrossRef]
14. M. J. King and J. C. Wiltse, “Surface-wave propagation on coated and uncoated metal wires at millimeter wavelengths,” IRE Trans. Antennas Propag. 10, 246–254 (1962). [CrossRef]
15. X. He, J. Cao, and S. Feng, “Simulation of the propagation property of metal wires terahertz waveguides,” Chin. Phys. Lett. 23, 2066–2069 (2006). [CrossRef]
16. S. A. Maier, S. R. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97, 176805 (2006) [CrossRef] [PubMed]
