Abstract

We introduce the technique of perturbation of boundary condition to the problem of terahertz two-wire metallic waveguides with different radii. Based on the quasi-TEM analytical mode fields derived by use of Möbius transformation, a concise expression for the complex effective index is obtained analytically. The expression is in good agreement with the simulation result. Further, the dispersion and attenuation are obtained from the expression. In addition, we find a zero value point of the group velocity dispersion around 1.268 THz. The results show that the technique of perturbation of boundary condition is helpful in the analysis and design of terahertz metal waveguide.

© 2015 Optical Society of America

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References

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2014 (5)

2013 (5)

A. Markov and M. Skorobogatiy, “Two-wire terahertz fibers with porous dielectric support,” Opt. Express 21(10), 12728–12743 (2013).
[Crossref] [PubMed]

J. S. Jo, T.-I. Jeon, and D. R. Grischkowsky, “Prototype 250 GHz bandwidth chip to chip electrical interconnect, characterized with ultrafast optoelectronics,” IEEE Trans. THz Sci. Technol. 3(4), 453–460 (2013).

Z. Zheng, N. Kanda, K. Konishi, and M. Kuwata-Gonokami, “Efficient coupling of propagating broadband terahertz radial beams to metal wires,” Opt. Express 21(9), 10642–10650 (2013).
[Crossref] [PubMed]

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
[Crossref] [PubMed]

A. Markov, A. Mazhorova, and M. Skorobogatiy, “Planar porous THz waveguides for low-loss guidance and sensing applications,” IEEE Trans. THz Sci. Technol. 3(1), 96–102 (2013).

2012 (2)

L. Chusseau and J.-P. Guillet, “Coupling and propagation of Sommerfeld waves at 100 and 300 GHz,” J. Infrared Milli. Terahz Waves 33(2), 174–182 (2012).
[Crossref]

R. B. Zhong, J. Zhou, W. H. Liu, and S. G. Liu, “Theoretical investigation of a terahertz transmission line in bipolar coordinate system,” Sci. China Inf. Sci. 55(1), 35–42 (2012).
[Crossref]

2011 (3)

2010 (3)

2009 (3)

2008 (2)

H. Liang, S. Ruan, and M. Zhang, “Terahertz surface wave propagation and focusing on conical metal wires,” Opt. Express 16(22), 18241–18248 (2008).
[Crossref] [PubMed]

P. Smorenburg, W. Op’t Root, and O. Luiten, “Direct generation of terahertz surface plasmon polaritons on a wire using electron bunches,” Phys. Rev. B 78(11), 115415 (2008).
[Crossref]

2006 (4)

J. A. Deibel, K. Wang, M. D. Escarra, and D. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 (2006).
[Crossref] [PubMed]

Y. Chen, Z. Song, Y. Li, M. Hu, Q. Xing, Z. Zhang, L. Chai, and C.-Y. Wang, “Effective surface plasmon polaritons on the metal wire with arrays of subwavelength grooves,” Opt. Express 14(26), 13021–13029 (2006).
[Crossref] [PubMed]

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]

H. Xiao-Yong, C. Jun-Cheng, and F. Song-Lin, “Simulation of the propagation property of metal wires terahertz waveguides,” Chin. Phys. Lett. 23(8), 2066–2069 (2006).
[Crossref]

2005 (4)

Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13(2), 511–518 (2005).
[Crossref] [PubMed]

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]

M. Wächter, M. Nagel, and H. Kurz, “Frequency-dependent characterization of THz Sommerfeld wave propagation on single-wires,” Opt. Express 13(26), 10815–10822 (2005).
[Crossref] [PubMed]

M. Walther, M. R. Freeman, and F. A. Hegmann, “Metal-wire terahertz time-domain spectroscopy,” Appl. Phys. Lett. 87(26), 261107 (2005).
[Crossref]

2004 (2)

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004).
[Crossref] [PubMed]

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25(4), 348–355 (2004).

2003 (1)

M. C. Kemp, P. Taday, B. E. Cole, J. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[Crossref]

2002 (1)

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002).
[Crossref] [PubMed]

1996 (2)

G. Pelosi and P. Y. Ufimtsev, “The impedance boundary condition,” IEEE Antennas Propag. Mag. 38(1), 31–35 (1996).
[Crossref]

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[Crossref]

1995 (1)

1989 (1)

1985 (1)

Alexander, R. W.

Al-Naib, I.

Andrews, S. R.

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]

Bell, R. J.

Cao, Q.

Chai, L.

Chen, Y.

Chusseau, L.

L. Chusseau and J.-P. Guillet, “Coupling and propagation of Sommerfeld waves at 100 and 300 GHz,” J. Infrared Milli. Terahz Waves 33(2), 174–182 (2012).
[Crossref]

Clerici, M.

Cluff, J.

M. C. Kemp, P. Taday, B. E. Cole, J. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[Crossref]

Cole, B. E.

M. C. Kemp, P. Taday, B. E. Cole, J. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[Crossref]

Daneau, M.

Darcie, T. E.

Deibel, J. A.

Dupuis, A.

Escarra, M. D.

Exter, M.

Fattinger, C.

Ferrera, M.

Fitch, M. J.

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25(4), 348–355 (2004).

Fitzgerald, A. J.

M. C. Kemp, P. Taday, B. E. Cole, J. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[Crossref]

Freeman, M. R.

M. Walther, M. R. Freeman, and F. A. Hegmann, “Metal-wire terahertz time-domain spectroscopy,” Appl. Phys. Lett. 87(26), 261107 (2005).
[Crossref]

Gao, H.

García-Vidal, F. J.

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]

Gordon, R.

Grischkowsky, D.

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]

M. Exter, C. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. 14(20), 1128–1130 (1989).
[Crossref] [PubMed]

Grischkowsky, D. R.

J. S. Jo, T.-I. Jeon, and D. R. Grischkowsky, “Prototype 250 GHz bandwidth chip to chip electrical interconnect, characterized with ultrafast optoelectronics,” IEEE Trans. THz Sci. Technol. 3(4), 453–460 (2013).

Guerboukha, H.

Guillet, J.-P.

L. Chusseau and J.-P. Guillet, “Coupling and propagation of Sommerfeld waves at 100 and 300 GHz,” J. Infrared Milli. Terahz Waves 33(2), 174–182 (2012).
[Crossref]

Hegmann, F. A.

M. Walther, M. R. Freeman, and F. A. Hegmann, “Metal-wire terahertz time-domain spectroscopy,” Appl. Phys. Lett. 87(26), 261107 (2005).
[Crossref]

Heshmat, B.

Hu, B. B.

Hu, M.

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[Crossref]

Jadidi, M. M.

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kunmar, “Plasmonic terahertz waveguide based on anisotropically etched silicon substrate,” IEEE Trans. THz Sci. Technol. 4(4), 454–458 (2014).

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
[Crossref] [PubMed]

Jahns, J.

Jeon, T.-I.

J. S. Jo, T.-I. Jeon, and D. R. Grischkowsky, “Prototype 250 GHz bandwidth chip to chip electrical interconnect, characterized with ultrafast optoelectronics,” IEEE Trans. THz Sci. Technol. 3(4), 453–460 (2013).

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]

Jo, J. S.

J. S. Jo, T.-I. Jeon, and D. R. Grischkowsky, “Prototype 250 GHz bandwidth chip to chip electrical interconnect, characterized with ultrafast optoelectronics,” IEEE Trans. THz Sci. Technol. 3(4), 453–460 (2013).

Jun-Cheng, C.

H. Xiao-Yong, C. Jun-Cheng, and F. Song-Lin, “Simulation of the propagation property of metal wires terahertz waveguides,” Chin. Phys. Lett. 23(8), 2066–2069 (2006).
[Crossref]

Kanda, N.

Kemp, M. C.

M. C. Kemp, P. Taday, B. E. Cole, J. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[Crossref]

Konishi, K.

Kumar, G.

Kunmar, G.

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kunmar, “Plasmonic terahertz waveguide based on anisotropically etched silicon substrate,” IEEE Trans. THz Sci. Technol. 4(4), 454–458 (2014).

Kurz, H.

Kuwata-Gonokami, M.

Lalanne, P.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002).
[Crossref] [PubMed]

Lavertu, P. L.

Li, D.

Li, S.

Li, Y.

Liang, H.

Liu, S. G.

R. B. Zhong, J. Zhou, W. H. Liu, and S. G. Liu, “Theoretical investigation of a terahertz transmission line in bipolar coordinate system,” Sci. China Inf. Sci. 55(1), 35–42 (2012).
[Crossref]

Liu, W. H.

R. B. Zhong, J. Zhou, W. H. Liu, and S. G. Liu, “Theoretical investigation of a terahertz transmission line in bipolar coordinate system,” Sci. China Inf. Sci. 55(1), 35–42 (2012).
[Crossref]

Long, L. L.

Luiten, O.

P. Smorenburg, W. Op’t Root, and O. Luiten, “Direct generation of terahertz surface plasmon polaritons on a wire using electron bunches,” Phys. Rev. B 78(11), 115415 (2008).
[Crossref]

Maier, S. A.

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]

Markov, A.

Martín-Moreno, L.

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]

Mazhorova, A.

Mbonye, M.

M. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[Crossref]

Mendis, R.

M. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[Crossref]

Mittleman, D.

Mittleman, D. M.

M. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009).
[Crossref]

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004).
[Crossref] [PubMed]

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[Crossref]

Morandotti, R.

Mridha, M. K.

Murphy, T. E.

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kunmar, “Plasmonic terahertz waveguide based on anisotropically etched silicon substrate,” IEEE Trans. THz Sci. Technol. 4(4), 454–458 (2014).

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kumar, “Terahertz surface plasmon polaritons on a semiconductor surface structured with periodic V-grooves,” Opt. Express 21(6), 7041–7049 (2013).
[Crossref] [PubMed]

Nagel, M.

Nuss, M. C.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[Crossref]

B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20(16), 1716–1718 (1995).
[Crossref] [PubMed]

Op’t Root, W.

P. Smorenburg, W. Op’t Root, and O. Luiten, “Direct generation of terahertz surface plasmon polaritons on a wire using electron bunches,” Phys. Rev. B 78(11), 115415 (2008).
[Crossref]

Ordal, M. A.

Osiander, R.

M. J. Fitch and R. Osiander, “Terahertz waves for communications and sensing,” Johns Hopkins APL Tech. Dig. 25(4), 348–355 (2004).

Pahlevaninezhad, H.

Peccianti, M.

Pelosi, G.

G. Pelosi and P. Y. Ufimtsev, “The impedance boundary condition,” IEEE Antennas Propag. Mag. 38(1), 31–35 (1996).
[Crossref]

Querry, M. R.

Razzari, L.

Reimer, C.

Ropagnol, X.

Rozé, M.

Ruan, S.

Shen, S.

Skorobogatiy, M.

Smorenburg, P.

P. Smorenburg, W. Op’t Root, and O. Luiten, “Direct generation of terahertz surface plasmon polaritons on a wire using electron bunches,” Phys. Rev. B 78(11), 115415 (2008).
[Crossref]

Song, Z.

Song-Lin, F.

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[Crossref]

Tannouri, P.

Teng, D.

Tribe, W. R.

M. C. Kemp, P. Taday, B. E. Cole, J. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
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Ung, B.

Vidal, F.

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[Crossref]

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Zhou, C.

Zhou, J.

R. B. Zhong, J. Zhou, W. H. Liu, and S. G. Liu, “Theoretical investigation of a terahertz transmission line in bipolar coordinate system,” Sci. China Inf. Sci. 55(1), 35–42 (2012).
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Appl. Opt. (1)

Appl. Phys. Lett. (3)

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
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[Crossref]

IEEE Antennas Propag. Mag. (1)

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A. Markov, A. Mazhorova, and M. Skorobogatiy, “Planar porous THz waveguides for low-loss guidance and sensing applications,” IEEE Trans. THz Sci. Technol. 3(1), 96–102 (2013).

S. Li, M. M. Jadidi, T. E. Murphy, and G. Kunmar, “Plasmonic terahertz waveguide based on anisotropically etched silicon substrate,” IEEE Trans. THz Sci. Technol. 4(4), 454–458 (2014).

J. S. Jo, T.-I. Jeon, and D. R. Grischkowsky, “Prototype 250 GHz bandwidth chip to chip electrical interconnect, characterized with ultrafast optoelectronics,” IEEE Trans. THz Sci. Technol. 3(4), 453–460 (2013).

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B. Ung, A. Mazhorova, A. Dupuis, M. Rozé, and M. Skorobogatiy, “Polymer microstructured optical fibers for terahertz wave guiding,” Opt. Express 19(26), B848–B861 (2011).
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M. Wächter, M. Nagel, and H. Kurz, “Frequency-dependent characterization of THz Sommerfeld wave propagation on single-wires,” Opt. Express 13(26), 10815–10822 (2005).
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J. Yang, Q. Cao, and C. Zhou, “An explicit formula for metal wire plasmon of terahertz wave,” Opt. Express 17(23), 20806–20815 (2009).
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J. Yang, Q. Cao, and C. Zhou, “Theory for terahertz plasmons of metallic nanowires with sub-skin-depth diameters,” Opt. Express 18(18), 18550–18557 (2010).
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A. Markov and M. Skorobogatiy, “Two-wire terahertz fibers with porous dielectric support,” Opt. Express 21(10), 12728–12743 (2013).
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H. Gao, Q. Cao, M. Zhu, D. Teng, and S. Shen, “Nanofocusing of terahertz wave in a tapered hyperbolic metal waveguide,” Opt. Express 22(26), 32071–32081 (2014).
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Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002).
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Proc. SPIE (1)

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[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Geometry of THz two-wire metallic waveguide with different radii embedded in a dielectric cladding. The permittivities of the metal and the dielectric cladding are εm and εd, respectively. The vectors n and τ are the unit normal from the metal and the unit tangent along the curve C in the cross section, respectively. The curve C consists of the left curve C1 and the right one C2. R1 and R2 are the radii of the left and right wires, respectively. The center to center distance is D. R1, R2 and D are much larger than the skin depth in the THz region.
Fig. 2
Fig. 2 The Möbius trasnformation of two circles side-by-side with a different radius ratio R1/R2 to a coaxial geometry. The distance of the two centers are D. The parameters a and b are the radii of the inner and outer circles of the conformal coaxial space.
Fig. 3
Fig. 3 The distributions and polarizations of normalized transverse fields in the gap region. (a) The analytical electric field from Eq. (22). (b) The electric field from simulation, and (c) the magnetic field from simulation with a 90° rotation, for the waveguide in the free space. And (d) the magnetic field from simulation with a 90° rotation, for the waveguide with a polystyrene foam cladding. The frequency is 0.5 THz. The metal is copper, whose permittivity εm = –6.3 × 105 + 2.77 × 106i from the fitted Drude model [41] at 0.5 THz, and the refraction index of the polystyrene foam nd = 1.0104 + 1.5059 × 10−4i at 0.5 THz according to the fitted formula and Fig. 8 in [34]. The geometric setting is as followed: D = 0.65 mm, R1 = 300 μm and R2 = 150 μm.
Fig. 4
Fig. 4 The normalized transverse magnetic fields with a 90° rotation, z×( Z 0 H t ), (a) along the left circle C1, and (b), along the right circle C2. (c) The fields far away from the wires along the midperpendicular of the line segment connecting the two opposite vertices [x = (D + R1-R2)/2], within the gap. Dashed line for simulation results, and solid line for analytical results. The settings of frequency, material and geometry are the same as the ones shown in Fig. 3.
Fig. 5
Fig. 5 Change of real and imaginary parts of n eff n d for the two-wire metallic waveguide with the radius ratio, ranging from 1 to 10, when keeping the gap (edge-to-edge distance of the two wires) constant, D-R1-R2 = 0.2 mm, and R2 = 150 μm. Black lines (solid for real parts and dashed for imaginary parts) are from simulation, and red marks (“×” for real parts and “+” for imaginary parts) are from the analytical calculations, Eq. (23). The waveguide is with a dielectric cladding nd = 1.0104 + 1.5059 × 10−4i. The settings of frequency, and the metal are the same as that shown in Fig. 3.
Fig. 6
Fig. 6 Change of the complex effective index n eff 1 with the frequency, from 0.1 to 2 THz, for the waveguide in the free space from analytical formula (red marks) and simulation (black lines). The solid line and red “×” show Re( n eff 1 ) ; dashed line and red “+” show Im( n eff 1 ) . The metal is copper, whose permittivity εm is from the fitted Drude model [41]. The two wires are identical, R1 = R2 = 150 μm, and the center-to-center distance D = 0.5 mm.
Fig. 7
Fig. 7 Group velocity dispersion (GVD) with respect to the frequency, nd = 1. The insert shows the zero value of GVD.
Fig. 8
Fig. 8 Distributions and polarizations of the transverse magnetic fields with a 90° rotation, z×( Z 0 H t ), from (a) simulation and (b) analytical formula, Eq. (22), in the gap region of the waveguide in the vaccum. The frequency is 10 THz, and εm = −3.0457 × 104 + 6.684 × 103i according to the fitted Drude model [41]. The transverse magnetic fields from simulation and analytical formula do not match at this frequency. R1 = R2 = 0.6 mm, and D = 3.2 mm.
Fig. 9
Fig. 9 Change of the complex effective index n eff 1 with the frequency, from 2 to 10 THz, for the waveguide in the free space (nd = 1) from analytical formula (red marks) and simulation (black lines). The solid line and red “×” show Re( n eff 1 ) ; dashed line and red “+” show Im( n eff 1 ) . The metal is copper, whose permittivity εm is from the fitted Drude model [41]. The two wires are identical, R1 = R2 = 150 μm and the center-to-center distance D = 0.5 mm. The analytical results (red marks) start to deviate from the simulated ones (black lines).

Equations (49)

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E 1 ε m n×( Z 0 H ).
E z | C 1 ε m Z 0 H τ | C ,
E τ | C 1 ε m Z 0 H z | C .
E t = i k 0 2 ( ε d n eff 2 ) [ k 0 n eff t E z k 0 z× t ( Z 0 H z ) ],
Z 0 H t = i k 0 2 ( ε d n eff 2 ) [ k 0 n eff t ( Z 0 H z )+ k 0 ε d z× t E z ].
[ t 2 k 0 2 ( n eff 2 ε d ) ]{ E z H z }=0.
n eff E z τ | C ( Z 0 H z ) n | C i k 0 ( ε d n eff 2 ) ε m Z 0 H z | C .
t E z =i k 0 [ n eff E t +z×( Z 0 H t ) ].
t 2 Φ 0 =0, Φ 0 | C 1 =1, Φ 0 | C 2 =0.
ε d S E t t Φ 0 ds = n eff S [ z×( Z 0 H t ) ] t Φ 0 ds .
S ( p t 2 q+ t p t q )ds= C p q n dl,
i k 0 ( n eff 2 ε d ) S [ z×( Z 0 H t ) ] t Φ 0 ds = ε d C E z Φ 0 n dl .
n eff 2 ε d = ε d k 0 ε m C Z 0 H τ Φ 0 n dl S [ z×( Z 0 H t ) ] t Φ 0 ds .
z×( Z 0 H t ) t Φ h .
n eff 2 ε d ε d k 0 ε m C Φ h n Φ 0 n dl C Φ h ( Φ 0 n )dl .
Φ h | C Φ 0 | C , Φ h n | C Φ 0 n | C .
n eff n d n d 2 k 0 ε m C | Φ 0 n | 2 dl C Φ 0 ( Φ 0 n )dl ,
w=f(ζ)= ζ- x 1 ζ- x 2 ,
x 1 , x 2 = 1 2D [ ( D 2 + R 1 2 R 2 2 ) ( D 2 + R 1 2 R 2 2 ) 2 4 R 1 2 D 2 ],
a=| R 1 + x 1 R 1 + x 2 |, b=| R 2 +D x 1 R 2 +D x 2 |.
Φ 0 = ln( b/ | w | ) ln( b/a ) .
E x = 1 ln( b/a ) [ x x 1 ( x x 1 ) 2 + y 2 x x 2 ( x x 2 ) 2 + y 2 ],
E y = 1 ln( b/a ) y( x 2 x 1 )( 2x x 1 x 2 ) [ ( x x 1 ) 2 + y 2 ][ ( x x 2 ) 2 + y 2 ] .
n eff n d n d 2 k 0 ε m 1 ( x 2 x 1 ) a+1/a +b+1/b ln( b/a ) ,
n eff 1 1+i 4 δ ( x 2 x 1 ) a+1/a +b+1/b ln( b/a ) .
n eff 2 ε d = κ d 2 = ε d k 0 ε m Φ h n | C 1 Φ h n | C 2 Φ h | C 1 Φ h | C 2 = ε d κ d ε m K 1 ( k 0 κ d R 1 ) K 0 ( k 0 κ d R 1 ) ,
× H m = i k 0 ε m Z 0 E m ,
× E m =i k 0 Z 0 H m ,
n ξ
n× H m ξ i k 0 ε m Z 0 E m ,
n× E m ξ i k 0 Z 0 H m .
2 H m ξ 2 + k 0 2 ε m H m 0,
n H m 0.
H m = H exp( i k 0 ε m ξ ),
E m n×( Z 0 H ) ε m exp( i k 0 ε m ξ ).
ε d t E t = n eff t [ z×( Z 0 H t ) ].
S ( Φ 0 t E t + t Φ 0 E t ) ds= C Φ 0 E n dl ,
S { Φ 0 t [ z×( Z 0 H t ) ]+ t Φ 0 [ z×( Z 0 H t ) ] } ds= C Φ 0 ( Z 0 H τ )dl ,
S { ε d t Φ 0 E t n eff t Φ 0 [ z×( Z 0 H t ) ] } ds= C Φ 0 [ n eff ( Z 0 H τ ) ε d E n ]dl .
ρ 1 = ε 0 ε d C 1 E n dl ,
I 1 = C 1 H τ dl ,
I 1 z + ρ 1 t =0,
C Φ 0 [ n eff ( Z 0 H τ ) ε d E n ]dl = C 1 [ n eff ( Z 0 H τ ) ε d E n ]dl =0,
| Φ 0 n | C 1 = | Φ 0 n w n w n ζ | C 1 = | Φ 0 n w / dζ dw | | w |=a ,
| dζ dw | | w |=a = | x 2 x 1 ( w1 ) 2 | | w |=a , | Φ 0 n w | | w |=a = 1 aln( b/a ) .
C 1 Φ 0 ( Φ 0 n )dl = C 1 | Φ 0 n || dζ | = C 1 | Φ 0 n w / dζ dw || dζ dw | | dw | = | w |=a | Φ 0 n w || dw | = 2πa aln( b/a ) = 2π ln( b/a ) ,
C 1 | Φ 0 n | 2 dl = | w |=a | Φ 0 n w | 2 / | dζ dw | 2 | dζ dw || dw | = | w |=a | Φ 0 n w | 2 / | dζ dw | | dw | = | w |=a 1 a 2 ln 2 ( b/a ) | ( w1 ) 2 x 2 x 1 || dw | = 0 2π | (a e iθ 1) 2 |adθ | x 2 x 1 | a 2 ln 2 ( b/a ) = 2π( a+1/a ) ( x 2 x 1 ) ln 2 ( b/a ) . ( x 2 > x 1 )
C 2 Φ 0 ( Φ 0 n )dl =0,
C 2 | Φ 0 n | 2 dl = 2π( b+1/b ) ( x 2 x 1 ) ln 2 ( b/a ) .

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