Open Access
Add to BibSonomyAbstract
Bare metal wires have recently been demonstrated as waveguides for transporting terahertz (THz) radiation, where the guiding mode is radially polarized surface Sommerfeld waves. In this study, we demonstrate high-efficiency coupling of a broadband radially polarized THz pulsed beam, which is generated with a polarization-controlled beam by a segmented half-wave-plate mode converter, to bare copper wires. A total coupling efficiency up to 16.8% is observed, and at 0.3 THz, the maximum coupling efficiency is 66.3%. The results of mode-overlap calculation and numerical simulation support the experimental data well.
©2013 Optical Society of America
1. Introduction
At present, most terahertz (THz) technologies rely on free propagation of waves rather than on waveguide transportation. THz waveguides provide a promising approach for overcoming the limitations of the system that use these technologies, such as large diffraction and bulky volumes. However, in the THz range, efficient waveguiding is challenging owing to high losses from the finite conductivity of metals or high absorption of dielectric materials in this spectral range. Moreover, to enjoy the wide bandwidth (0.1–4 THz) of typical time domain measurements, it is necessary to find a guiding scheme without obvious dispersion; the wide band range required for typical time domain measurements makes it difficult to find a guiding scheme without obvious dispersion. Until now, dielectric structures such as planar dielectric waveguides [1], dielectric fibers [2], and polymer tubes [3] as well as metallic structures such as rectangular waveguides [4], slit waveguides [5], and parallel plate waveguides [6] have shown to support THz wave propagation. Bare metal wires, which support a plasmonic mode (known as Sommerfeld wave [7]) on their surfaces in THz range, are one of the best candidates for guiding THz waves because of their very low losses, negligible dispersion propagation, and structural simplicity [8].
Compared to the other THz waveguides, the cylindrical symmetry of metal wire waveguides poses a significant challenge because the typical linearly polarized THz wave has a very poor spatial overlap with the Sommerfeld mode that is radially polarized. A large mismatch between the waveguide mode and the free propagating mode results in very low in-coupling efficiency. In early experiments of THz waveguiding with bare metal wire, the scattering in-couple method was employed [8], but the measured coupling efficiency was lower than 0.5% [9,10]. To increase the coupling efficiency, radially symmetric antennas [9,11], mode filters [12,13], plasmonic in-couplers [14,15], or directly generating radially polarized modes on metal wires [16] were tried. Most of these methods use antenna-based mechanisms for THz generation [9,11,17,18] and hence only a narrow bandwidth, typically under 0.5 THz [18], can be achieved. Until now, to the best of our knowledge, efficient coupling of broadband propagating THz waves to bare metal wires has not been realized, and experimental evaluations of the increased coupling efficiency are not performed.
Previously, in [9], Deibel et al. estimated theoretically a high coupling efficiency for a radially polarized THz emission from a radially symmetric photoconductive antenna with a hyperhemispherical lens to a metal wire. The method was demonstrated without experimental determination of efficiency. Recently, as described in [19], we have developed a new convenient method to generate a broadband THz cylindrical vector beam [20], which is also a promising component in realizing efficient coupling to bare metal wires. Applying this generation method in this study, we demonstrate efficient coupling to metal wires from radially polarized THz beams propagating in free space by focusing with conventional lenses. The coupling efficiency is experimentally evaluated, and mode-overlap calculation and numerical simulation are also performed to confirm the validity of the experimental data.
2. Experimental setup
When a nonlinear crystal possessing threefold rotational symmetry is illuminated by linearly polarized optical pulses, the emitted THz pulses are linearly polarized and the amplitude of the THz pulses remains unchanged when the azimuthal direction φ of incident linearly polarized optical pulses rotate [21]. In this case, when φ changes by Δφ, the azimuthal angle of radiated THz wave θ changes by Δθ = −2Δφ [19]. This property of threefold rotation symmetrical optical rectification crystals offers the possibility of manipulating the polarization state of a THz wave without introducing amplitude variations.
A predesigned mode converter, consisting of eight pieces of half-wave plate (HWP), was made for controlling the spatial polarization mode of the incident laser beam. The fast axis of each piece of HWP is arranged as shown in Fig. 1(a). When linearly polarized laser beams have a polarization oriented along the same direction as the fast axis direction of the top marked piece [Fig. 1(b)], the beams are converted to the polarization-spatial-variant mode for THz cylindrical vector (CV) beam generation [Fig. 1(a)]. The mode overlaps between an ideal CV beam and the quasi-CV beam generated using such an eight-segmented HWP is evaluated to be as large as 93% [22].
Fig. 1 (a) Schematic of the function of the segmented HWP mode converter. The red arrow indicates the polarization direction of the incident beam. The blue arrows show the fast axis orientation for each piece of the wave-plate. The green arrows indicate the polarization direction inside the polarization-spatial-variant mode. (b) A photograph of segmented HWP mode converter. (c) Intensity distribution images for the THz radial beams. The arrows indicate the polarization directions. The leftmost image was obtained without WGP.
The THz CV beam was generated by exciting a threefold rotation symmetrical nonlinear crystal with the polarization-spatial-variant mode, as shown in Fig. 1. Previously, we used a GaP(111) crystal for THz generation [19], but in this study we use a ZnTe(111) crystal to achieve a higher amplitude of the generated THz wave in the case of 800-nm excitation. Figure 1(c) shows intensity distribution images of the generated THz radial beam obtained at the focal point using a THz camera (IRV-T0831HS, NEC Corp.) with a wire grid polarizer (WGP) in several orientations. Two lobes aligned along the polarization direction are clearly observed. These results clearly indicate that THz radial beams were successfully generated. Switching between radially and azimuthally polarized modes was achieved by simply rotating the orientation of the axis of the crystal by 90° [19].
It should be noted that in comparison with our previous method [19], a segmented HWP has replaced a segmented nonlinear crystal. The reason for this is that a technique to cut and combine HWPs is commercially available, although obtaining large ZnTe(111) wafers is difficult. Principally, the generated radial THz mode is the same using either the wave-plate or the crystal.
The overall experimental setup is shown in Fig. 2. A THz time-domain spectroscopy (THz-TDS) setup was employed for this investigation. The light source was a regeneratively amplified Ti:Sapphire laser system (Hurricane, Spectra-Physics Lasers, Inc.) with 1 kHz repetition rate, wavelength centered at 800 nm, 100 fs pulse width, pulse energy of 0.8 mJ, and polarized along x axis in Fig. 2. The laser beam was divided into two by a beam sampler; a part with 99% of the total power served as a pump beam and passed through the segmented HWP mode converter to generate a radially polarized THz beam, which was then focused by a THz lens with a 50 mm focal length (Tsurupica, Pax Corp.). The beam size was 10 mm in diameter on the segmented HWP. A straight copper wire with a length of 20 cm and a radius of 0.5 mm was fixed in two polytetrafluoroethylene (PTFE) holders, with one end being placed at the focal point of the lens. At the other end of the copper wire, a photo-conductive antenna was placed for THz detection and the other part of the laser beam (1% of the total power) was introduced as a probe beam. By changing the time delay between the pump pulse and the probe pulse, THz waveforms in the time domain were obtained. In the experiments, a commercial photo-conductive antenna with a 5 micron gap was used. With the gap lying along x or y directions, x or y electric fields of THz wave were detected, respectively. That allowed the detection of all in-plane electric field components of the THz field by changing the in-plane direction of the antenna. In the setup, the probe beam was expanded to a spot with about 20 mm diameter, so the antenna can be moved in that area to scan the THz electric field distribution. Here, a 21 × 21 point area is scanned with a resolution of 0.25 mm × 0.25 mm. The experimental setup was purged with dry nitrogen gas to prevent absorption of the beams by water vapor.
To experimentally investigate the in-coupling efficiency, antenna scanning was also performed at the in-couple plane (Fig. 2), with the copper wire removed from the setup. Information about the coupling efficiency is obtained from these two scanning results.
3. Results and analysis
Figure 3(a) shows the distribution of THz intensity, which is integrated in time domain (I(x,y) = ∫[E2x(t) + E2y(t)]dt). This image is obtained at the output plane (Fig. 2) where a donut-shaped electric field distribution is clearly observed. Examples of THz waveforms measured at offsets in y axis of + 1.2 mm and −1.2 mm from the central dark point (marked as stars in Fig. 3(a)) are shown in Fig. 3(b). It should be noted that the phases are opposite at these two places, confirming that the guiding mode is radially polarized [8].
Fig. 3 (a) Intensity distribution at the output plane. (b) Waveforms at y offset positions of + 1.2 mm and −1.2 mm from the center (marked as stars in Fig. 3(a)).
In Fig. 4, waveforms (electric fields as functions of time) and spectra in the THz frequency domain of the in-couple plane (black) and output plane (red) are shown. These four graphs are normalized at the peak values. The obtained output plane spectrum [red line in Fig. 4(a)] is observed over a frequency range of up to about 1.5 THz, which is about three times broader than the previously reported results in which the maximum frequency was only about 0.5 THz [8,10,13,18,23].
Fig. 4 Normalized waveforms (a) and spectra(b) (Upper (black): In-couple plane; Lower (red): Output plane).
To further prove that a radial guiding mode is present in the copper wire, a comparison between phase information at the in-couple plane and the output plane was performed. The phase information was extracted from the waveforms shown in Fig. 4(a) (see methods described in [10,23]), and subsequently the phase velocity and group velocity of the guided wave were calculated, as shown in Fig. 5. In contrast to the reported results [10,23], our data extend to as high as 2.0 THz in spectral range. From Fig. 5, in the low-frequency range (0.1–0.5 THz), our results well reproduce the reported results. In the 0.5–2.0 THz frequency range, the low dispersion propagation of Sommerfeld waves is experimentally observed and the phase velocity is consistent with theoretical predictions [23].
To classify the advantage of the combination of a metal wire with a THz radial beam, two reference measurements were undertaken, as shown in Fig. 6. In Fig. 6(a), at the same position at the output plane, the black line shows the THz waveform with the copper wire waveguide, while the red line shows the waveform with the copper wire removed from the setup. In Fig. 6(b), the black line represents the maximum electric field waveform with a radially polarized beam in-coupled, and the red line shows the maximum electric field waveform with an azimuthally polarized beam in-coupled (with the ZnTe(111) crystal rotated by 90°). Comparing these data, the following can be concluded:
Fig. 6 Black lines: waveforms of a guided THz electric field at the output plane. Red lines: (a) waveform measured with the copper wire removed from the setup, (b) waveform with the azimuthally polarized mode in-coupled.
- 1. Only the radially polarized mode is supported on the copper wire. The azimuthally polarized beam cannot be coupled to the wire.
- 2. The freely propagating THz electric field is one order of magnitude smaller than the guiding field.
To determine the in-coupling efficiency of our method, we meshed both the in-coupled and output planes into 11 × 11 grids; in order to reduce the scanning time, we decreased the scanning grid number here in comparison with 21 × 21 grids used for measuring the field distribution. Then, at each grid, the THz field Ei,o(x, y, t) was measured (subscript i and o represent in-coupled and output planes, respectively), and a Fourier transformation was performed as follows: . Subsequently, the frequency dependence of the in-coupled and output energies were determined by integrating the in-coupled and output planes, respectively: Ii,o(ω) = ∫∫dxdy|Ei,o(x, y, ω)|2. Then, the frequency dependence of the coupling efficiency was calculated using η (ω) = Io (ω)/Ii (ω), the results of which are shown in Fig. 7 (black solid line). From Fig. 7, it can be seen that the maximum coupling efficiency is as high as 66.3% at approximately 0.3 THz, while a total efficiency (η (ω) = ∫Io(ω)dω/∫Ii(ω)dω) of 16.8% is found when the propagating THz radial mode energy is coupled to the copper wire. This maximum value is about two orders of magnitude larger than the coupling efficiency measured using the scattering coupling method [10], and slightly higher than the estimated coupling efficiency determined using the antenna-coupling method described in [9].
Fig. 7 Frequency dependence of the coupling efficiencies obtained by experiment, mode-overlap calculation and numerical simulation
To confirm the validity of the experimental values of coupling efficiency, we calculate the mode-overlap coefficient between the in-coupling focused radially polarized field and the guiding Sommerfeld mode. The in-plane electric field of the focused radially polarized beam can be expressed in a cylindrical coordinate system as [24,25]
where ρ is the distance from the optical axis, θ is the angle that the ray makes with the axis, θmax is the maximum value of θ that corresponds to half of the aperture angle determined by the numerical aperture (NA) of the lens (in our case, NA = 0.1), E0(θ) is the electric field distribution before focusing lens, k0 = ω/c is the wavenumber, and J1(x) is a Bessel function of the first kind. Because in-plane component is much larger than the longitudinal component Ez in this case, only in-plane component is considered here.Sommerfeld wave guiding on the copper wire can be described by a Hankel function H1(1)(γρ), where γ is defined in terms of the propagation constant k of the field outside the wire according to γ 2 = (ω /c)2 − k2 [10] and inside the metal, because the relative permittivity of metal is far larger than 1. Hence, the modal field will decay very fast; in fact, it can penetrate into the metal side for only 1 μm or less [26]. Therefore, the modal field on the copper wire is a hollow-cored mode. The size of the hollow core is determined by the radius of the metal wire R, which can be expressed as follows:
Then, the mode-overlap coefficient is defined by the integration [27,28]:and the calculated result is shown in Fig. 7 (blue dotted line).Numerical simulations performed by a commercially available program package (CST Micro Wave Studio) were also applied to determine the coupling efficiency. In the simulation, a model replicating the experimental setup was built. The NA of the focusing lens and the metal wire radius were set to 0.1 and 0.5 mm, respectively, in a frequency range of 0.1–2 THz using a 0.1 THz step size. The field distributions were inspected and the energies passing through the input and output planes of the metal wire were integrated; the ratio of the incident energy and the output energy gives the coupling efficiency. The results of this simulation are shown by the red dashed line in Fig. 7. One can see that the numerical simulation result reproduces the experiment data well. The mode-overlap calculation also qualitatively supported the experimental and numerically simulated results, but some deviations were observed, especially in the higher frequency region. The main factors contributing to this discrepancy would be the nearly 10% propagation loss in the frequency range higher than 0.5 THz in the 20 cm copper wire [8, 10] and losses at the PTFE holder surfaces (reported to be <3%).
Moreover, in the calculation of mode-overlap coefficient, it should be noted that the frequency dependence of the coupling efficiency η(ω) is indeed dependent on two parameters: the NA of focusing lens and the metal wire radius R. By changing one of these two parameters while leaving the other unchanged, the relationship between the η(ω) and the parameters R and NA was investigated; some results are shown in Fig. 8. In the case where the metal wire radius is constant and set to R = 0.5 mm [Fig. 8(a)], when NA becomes smaller, the peak frequency of coupling efficiency shifts to higher frequencies, while the bandwidth of the coupling efficiency obviously broadens. In the case where the NA of the focusing lens is held at a constant value of 0.1 [Fig. 8(b)], as the wire radius decreases, the peak coupling efficiency starts to decrease and shifts to higher frequencies when the radius becomes smaller than approximately 0.03 mm.
Fig. 8 Frequency dependence of mode-overlap calculation, (a) wire radius R = 0.5 mm, NA = 0.2, 0.1, and 0.05; (b) NA = 0.1, wire radius R = 0.5 mm, 0.2 mm, 0.05 mm, 0.03 mm, and 0.01 mm.
Considering these results, it is possible to tune the frequency dependence of the coupling efficiency by choosing an appropriate NA. In particular, it is effective to use a small NA in order to achieve efficient coupling in a higher and broader frequency range, which could be important for many applications of THz technology, such as biomedical imaging [29,30], THz tomography, and quality control [31].
4. Conclusion
To employ a convenient method for generating broadband and stable THz vector beams, we demonstrated directly focusing and coupling THz radial beams to a bare copper wire with high efficiency. By applying an antenna scanning THz-TDS setup, the radially polarized guiding Sommerfeld wave was confirmed. The guiding spectra had a wide bandwidth, extending as high as 1.5 THz, which is about four times wider than the previously reported results [8,10,13,18,23]. The frequency dependence of the energy coupling efficiency was determined by a comparison between the input and output energies, and a sufficient coupling efficiency of 66.3% at 0.3 THz was obtained; this represents an improvement of two orders of magnitude in comparison with the reported experimental results. In total, 16.8% of the input energy is coupled to the copper wire. Mode-overlap calculation and numerical simulation of coupling a propagating radially polarized mode of the THz beam to a metal wire were performed, and the calculated coupling efficiency matched well with the experimental data. It was shown via mode-overlap calculation that the magnitude and bandwidth of the in-coupled frequency could be greatly increased by optimizing the parameters of the focusing lens, where using smaller NA lens led to vastly broadened bandwidths. This coupling method is a promising technique for the development of new THz manipulation technologies, which is beneficial for the development of various THz applications [32–34].
Acknowledgments
This research was supported by the Photon Frontier Network Program, KAKENHI (20104002), the Special Coordination Funds for Promoting Science and Technology of MEXT, Japan, and by the JSPS through its FIRST Program.
References and links
1. R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88(7), 4449–4451 (2000). [CrossRef]
2. L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun, “Low-loss subwavelength plastic fiber for terahertz waveguiding,” Opt. Lett. 31(3), 308–310 (2006). [CrossRef] [PubMed]
3. D. Chen and H. Chen, “A novel low-loss Terahertz waveguide: Polymer tube,” Opt. Express 18(4), 3762–3767 (2010). [CrossRef] [PubMed]
4. D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by terahertz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1122–1135 (2000). [CrossRef]
5. M. Wächter, M. Nagel, and H. Kurz, “Low-loss terahertz transmission through curved metallic slit waveguides fabricated by spark erosion,” Appl. Phys. Lett. 92(16), 161102 (2008). [CrossRef]
6. S.-H. Kim, E. S. Lee, Y. B. Ji, and T.-I. Jeon, “Improvement of THz coupling using a tapered parallel-plate waveguide,” Opt. Express 18(2), 1289–1295 (2010). [CrossRef] [PubMed]
7. G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21(11), 1119–1128 (1950). [CrossRef]
8. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004). [CrossRef] [PubMed]
9. J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 (2006). [CrossRef] [PubMed]
10. K. Wang and D. M. Mittleman, “Guided propagation of terahertz pulses on metal wires,” J. Opt. Soc. Am. B 22(9), 2001–2008 (2005). [CrossRef]
11. S. Winnerl, R. Hubrich, M. Mittendorff, H. Schneider, and M. Helm, “Universal phase relation between longitudinal and transverse fields observed in focused terahertz beams,” New J. Phys. 14(10), 103049 (2012). [CrossRef]
12. T. Grosjean, F. Baida, R. Adam, J.-P. Guillet, L. Billot, P. Nouvel, J. Torres, A. Penarier, D. Charraut, and L. Chusseau, “Linear to radial polarization conversion in the THz domain using a passive system,” Opt. Express 16(23), 18895–18909 (2008). [CrossRef] [PubMed]
13. L. Chusseau and J.-P. Guillet, “Coupling and propagation of Sommerfeld waves at 100 and 300 GHz,” J. Infrared Millim. Terahz Waves. 33(2), 174–182 (2012). [CrossRef]
14. H. Cao and A. Nahata, “Coupling of terahertz pulses onto a single metal wire waveguide using milled grooves,” Opt. Express 13(18), 7028–7034 (2005). [CrossRef] [PubMed]
15. A. Agrawal and A. Nahata, “Coupling terahertz radiation onto a metal wire using a subwavelength coaxial aperture,” Opt. Express 15(14), 9022–9028 (2007). [CrossRef] [PubMed]
16. W. Zhu, A. Agrawal, H. Cao, and A. Nahata, “Generation of broadband radially polarized terahertz radiation directly on a cylindrical metal wire,” Opt. Express 16(12), 8433–8439 (2008). [CrossRef] [PubMed]
17. S. Winnerl, B. Zimmermann, F. Peter, H. Schneider, and M. Helm, “Terahertz Bessel-Gauss beams of radial and azimuthal polarization from microstructured photoconductive antennas,” Opt. Express 17(3), 1571–1576 (2009). [CrossRef] [PubMed]
18. T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005). [CrossRef]
19. R. Imai, N. Kanda, T. Higuchi, Z. Zheng, K. Konishi, and M. Kuwata-Gonokami, “Terahertz vector beam generation using segmented nonlinear optical crystals with threefold rotational symmetry,” Opt. Express 20(20), 21896–21904 (2012). [CrossRef] [PubMed]
20. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]
21. T. Higuchi, N. Kanda, H. Tamaru, and M. Kuwata-Gonokami, “Selection rules for light-induced magnetization of a crystal with threefold symmetry: the case of antiferromagnetic NiO,” Phys. Rev. Lett. 106(4), 047401 (2011). [CrossRef] [PubMed]
22. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef] [PubMed]
23. K. Wang and D. M. Mittleman, “Dispersion of surface plasmon polaritons on metal wires in the terahertz frequency range,” Phys. Rev. Lett. 96(15), 157401 (2006). [CrossRef] [PubMed]
24. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000). [CrossRef]
25. K. S. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]
26. Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13(2), 511–518 (2005). [CrossRef] [PubMed]
27. M. Fadhali, J. Saktioto, Y. Zainal, J. Munajat, Ali, and R. Abdul Rahman, “Mode matching for efficient laser diode to single mode fiber coupling,” Proc. SPIE 6793, 67930G, 67930G-8 (2008). [CrossRef]
28. S. Sarkar, K. Thyagarajan, and A. Kumar, “Gaussian approximation of the fundamental mode in single mode elliptic core fibers,” Opt. Commun. 49(3), 178–183 (1984). [CrossRef]
29. D. Crawley, C. Longbottom, V. P. Wallace, B. Cole, D. Arnone, and M. Pepper, “Three-dimensional terahertz pulse imaging of dental tissue,” J. Biomed. Opt. 8(2), 303–307 (2003). [CrossRef] [PubMed]
30. R. M. Woodward, V. P. Wallace, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulsed imaging of skin cancer in the time and frequency domain,” J. Biol. Phys. 29(2/3), 257–259 (2003). [CrossRef] [PubMed]
31. S. Wang and X.-C. Zhang, “Pulsed terahertz tomography,” J. Phys. D Appl. Phys. 37(4), R1–R36 (2004). [CrossRef]
32. A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE J. Sel. Top. Quantum Electron. 14(6), 1515–1521 (2008). [CrossRef]
33. S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97(17), 176805 (2006). [CrossRef] [PubMed]
34. A. I. Fernández-Domínguez, C. R. Williams, F. J. García-Vidal, L. Martín-Moreno, S. R. Andrews, and S. A. Maier, “Terahertz surface plasmon polaritons on a helically grooved wire,” Appl. Phys. Lett. 93(14), 141109 (2008). [CrossRef]
