Abstract

The modes in a circular metallic waveguide loaded with a high permittivity dielectric rod may possess similar dispersion relations to the modes in the left-handed metamaterial (LHM) waveguide. Therefore such dielectric-loaded metallic waveguide may also support slow light with parameters properly selected. The slow light in the GaAs-rod-loaded metallic waveguide is numerically studied. The results show that the wavelength of slow light varies with the parameters of the waveguide. A linearly tapered waveguide and other realizable simple structures are proposed accordingly to realize the “trapped rainbow” phenomena. Moreover, the practical lossy tapered waveguide is also investigated in the terahertz region. It is shown that the slow light with low loss can be achieved in a realistic GaAs-loaded metallic waveguide.

© 2010 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2010 (2)

2009 (4)

Q. Gan, Y. J. Ding, F. J. Bartoli, ““Rainbow” trapping and releasing at telecommunication wavelengths,” Phys. Rev. Lett. 102(5), 056801 (2009).
[CrossRef] [PubMed]

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

T. Jiang, J. Zhao, Y. Feng, “Stopping light by an air waveguide with anisotropic metamaterial cladding,” Opt. Express 17(1), 170–177 (2009).
[CrossRef] [PubMed]

T. Jiang, Q. Zhang, Y. Feng, “Compensating loss with gain in slow-light propagation along slab waveguide with anisotropic metamaterial cladding,” Opt. Lett. 34(24), 3869–3871 (2009).
[CrossRef] [PubMed]

2008 (5)

J. He, Y. Jin, Z. Hong, S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008).
[CrossRef] [PubMed]

Q. Gan, Z. Fu, Y. J. Ding, F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,” Phys. Rev. Lett. 100(25), 256803 (2008).
[CrossRef] [PubMed]

Y. J. Huang, W. T. Lu, S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77(6), 063836 (2008).
[CrossRef]

A. Reza, M. M. Dignam, S. Hughes, “Can light be stopped in realistic metamaterials?” Nature 455(7216), E10–E11 (2008).
[CrossRef]

K. L. Tsakmakidis, A. D. Boardman, O. Hess, “Can light be stopped in realistic metamaterials? Reply,” Nature 455, E11–E12 (2008).
[CrossRef]

2007 (1)

K. L. Tsakmakidis, A. D. Boardman, O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

2006 (2)

J. He, S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
[CrossRef]

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89(20), 201103 (2006).
[CrossRef]

2005 (1)

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004).
[CrossRef] [PubMed]

2003 (2)

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5 Pt 2), 057602 (2003).
[CrossRef] [PubMed]

A. Melloni, F. Morichetti, M. Martinelli, “Optical slow wave structures,” Opt. Photon. News 14(11), 44–48 (2003).
[CrossRef]

2001 (2)

K. Lee, N. M. Lawandy, “Optically induced pulse delay in a solid-state Raman amplifier,” Appl. Phys. Lett. 78(6), 703–705 (2001).
[CrossRef]

M. D. Lukin, A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413(6853), 273–276 (2001).
[CrossRef] [PubMed]

1975 (1)

G. N. Tsandoulas, “Propagation in Dielectric-Lined Square Waveguides,” IEEE Trans. Microw. Theory Tech. 23(5), 406–410 (1975).
[CrossRef]

1963 (1)

P. J. B. Clarricoats, A. B. Birtles, “Circular Waveguide Backward-wave Experiments,” J. Electron Control 15, 325–330 (1963).
[CrossRef]

Aryal, D. P.

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89(20), 201103 (2006).
[CrossRef]

Ballato, J.

Barbieri, S.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Bartoli, F. J.

Q. Gan, Y. J. Ding, F. J. Bartoli, ““Rainbow” trapping and releasing at telecommunication wavelengths,” Phys. Rev. Lett. 102(5), 056801 (2009).
[CrossRef] [PubMed]

Q. Gan, Z. Fu, Y. J. Ding, F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,” Phys. Rev. Lett. 100(25), 256803 (2008).
[CrossRef] [PubMed]

Birtles, A. B.

P. J. B. Clarricoats, A. B. Birtles, “Circular Waveguide Backward-wave Experiments,” J. Electron Control 15, 325–330 (1963).
[CrossRef]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, O. Hess, “Can light be stopped in realistic metamaterials? Reply,” Nature 455, E11–E12 (2008).
[CrossRef]

K. L. Tsakmakidis, A. D. Boardman, O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

Burka, L.

Christodoulides, D. N.

Clarricoats, P. J. B.

P. J. B. Clarricoats, A. B. Birtles, “Circular Waveguide Backward-wave Experiments,” J. Electron Control 15, 325–330 (1963).
[CrossRef]

Davies, A. G.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Dhillon, S. S.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Dignam, M. M.

A. Reza, M. M. Dignam, S. Hughes, “Can light be stopped in realistic metamaterials?” Nature 455(7216), E10–E11 (2008).
[CrossRef]

Ding, Y. J.

Q. Gan, Y. J. Ding, F. J. Bartoli, ““Rainbow” trapping and releasing at telecommunication wavelengths,” Phys. Rev. Lett. 102(5), 056801 (2009).
[CrossRef] [PubMed]

Q. Gan, Z. Fu, Y. J. Ding, F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,” Phys. Rev. Lett. 100(25), 256803 (2008).
[CrossRef] [PubMed]

Engelen, R. J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

Feng, Y.

Fink, Y.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004).
[CrossRef] [PubMed]

Foy, P.

Fu, Z.

Q. Gan, Z. Fu, Y. J. Ding, F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,” Phys. Rev. Lett. 100(25), 256803 (2008).
[CrossRef] [PubMed]

Gan, Q.

Q. Gan, Y. J. Ding, F. J. Bartoli, ““Rainbow” trapping and releasing at telecommunication wavelengths,” Phys. Rev. Lett. 102(5), 056801 (2009).
[CrossRef] [PubMed]

Q. Gan, Z. Fu, Y. J. Ding, F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,” Phys. Rev. Lett. 100(25), 256803 (2008).
[CrossRef] [PubMed]

Gersen, H.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

Hawkins, T.

He, J.

J. He, Y. Jin, Z. Hong, S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008).
[CrossRef] [PubMed]

J. He, S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
[CrossRef]

He, S.

J. He, Y. Jin, Z. Hong, S. He, “Slow light in a dielectric waveguide with negative-refractive-index photonic crystal cladding,” Opt. Express 16(15), 11077–11082 (2008).
[CrossRef] [PubMed]

J. He, S. He, “Slow propagation of electromagnetic waves in a dielectric slab waveguide with a left-handed material substrate,” IEEE Microw. Wirel. Compon. Lett. 16(2), 96–98 (2006).
[CrossRef]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, O. Hess, “Can light be stopped in realistic metamaterials? Reply,” Nature 455, E11–E12 (2008).
[CrossRef]

K. L. Tsakmakidis, A. D. Boardman, O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[CrossRef] [PubMed]

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89(20), 201103 (2006).
[CrossRef]

Hong, Z.

Huang, Y. J.

Y. J. Huang, W. T. Lu, S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77(6), 063836 (2008).
[CrossRef]

Hughes, S.

A. Reza, M. M. Dignam, S. Hughes, “Can light be stopped in realistic metamaterials?” Nature 455(7216), E10–E11 (2008).
[CrossRef]

Ibanescu, M.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004).
[CrossRef] [PubMed]

Imamoglu, A.

M. D. Lukin, A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413(6853), 273–276 (2001).
[CrossRef] [PubMed]

Jamois, C.

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89(20), 201103 (2006).
[CrossRef]

Jiang, T.

Jin, Y.

Joannopoulos, J. D.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004).
[CrossRef] [PubMed]

Johnson, S. G.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004).
[CrossRef] [PubMed]

Jukam, N.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Karle, T. J.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

Khanna, S. P.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5 Pt 2), 057602 (2003).
[CrossRef] [PubMed]

Klaedtke, A.

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, O. Hess, “Single-mode operation in the slow-light regime using oscillatory waves in generalized left-handed heterostructures,” Appl. Phys. Lett. 89(20), 201103 (2006).
[CrossRef]

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

Krauss, T. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

Kuipers, L.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94(7), 073903 (2005).
[CrossRef] [PubMed]

Lawandy, N. M.

K. Lee, N. M. Lawandy, “Optically induced pulse delay in a solid-state Raman amplifier,” Appl. Phys. Lett. 78(6), 703–705 (2001).
[CrossRef]

Lee, K.

K. Lee, N. M. Lawandy, “Optically induced pulse delay in a solid-state Raman amplifier,” Appl. Phys. Lett. 78(6), 703–705 (2001).
[CrossRef]

Linfield, E. H.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Lu, W. T.

Y. J. Huang, W. T. Lu, S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77(6), 063836 (2008).
[CrossRef]

Lukin, M. D.

M. D. Lukin, A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413(6853), 273–276 (2001).
[CrossRef] [PubMed]

Luo, C.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004).
[CrossRef] [PubMed]

Madeo, J.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Manquest, C.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Martinelli, M.

A. Melloni, F. Morichetti, M. Martinelli, “Optical slow wave structures,” Opt. Photon. News 14(11), 44–48 (2003).
[CrossRef]

McMillen, C.

Melloni, A.

A. Melloni, F. Morichetti, M. Martinelli, “Optical slow wave structures,” Opt. Photon. News 14(11), 44–48 (2003).
[CrossRef]

Morichetti, F.

A. Melloni, F. Morichetti, M. Martinelli, “Optical slow wave structures,” Opt. Photon. News 14(11), 44–48 (2003).
[CrossRef]

Oustinov, D.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Podila, R.

Rao, A. M.

Reppert, J.

Reza, A.

A. Reza, M. M. Dignam, S. Hughes, “Can light be stopped in realistic metamaterials?” Nature 455(7216), E10–E11 (2008).
[CrossRef]

Rice, R. R.

Roundy, D.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92(6), 063903 (2004).
[CrossRef] [PubMed]

Salandrino, A.

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5 Pt 2), 057602 (2003).
[CrossRef] [PubMed]

Sirtori, C.

N. Jukam, S. S. Dhillon, D. Oustinov, J. Madeo, C. Manquest, S. Barbieri, C. Sirtori, S. P. Khanna, E. H. Linfield, A. G. Davies, J. Tignon, “Terahertz amplifier based on gain switching in a quantum cascade laser,” Nat. Photonics 3(12), 715–719 (2009).
[CrossRef]

Sridhar, S.

Y. J. Huang, W. T. Lu, S. Sridhar, “Nanowire waveguide made from extremely anisotropic metamaterials,” Phys. Rev. A 77(6), 063836 (2008).
[CrossRef]

Sukhorukov, A. A.

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Other (2)

J. He, Centre for THz Research, China Jiliang University, Hangzhou 310018, China, and Z. Hong et al. are preparing a manuscript to be called “Backward coupling of modes in a left-handed metamaterial tapered waveguide.”

G. Grüner, Millimeter and Submillimeter Wave Spectroscopy of Solids (Springer, Berlin, 1998), Chap. 3.

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Figures (5)

Fig. 1
Fig. 1

(a) Dispersion curve for the ideal GaAs-rod-loaded cylindrical metallic waveguide as shown in the inset. For the waveguide, R 0 = 2R 1 = 0.4mm. The permittivity of GaAs is 12.9, and the dielectric between the rod and hollow metallic waveguide is air. The dashed line corresponds to the complex mode. (b) The time average Poynting vector along the waveguide for the forward mode corresponding to circle point A in Fig. 1(a). (c) The time average Poynting vector for the backward mode B.

Fig. 2
Fig. 2

(a) The group velocities of modes in the GaAs-rod-loaded metallic waveguide with various R 1. The R 0 is 0.4mm and wavelength of modes is 0.214mm. (b) The wavelength, λcp , of mode at the critical point for different R 1 of waveguide. R 0 = 0.4mm. (c) The λcp for different R 0 of waveguide. R 1 = 0.2mm.

Fig. 3
Fig. 3

(a) Schematic diagram of a tapered waveguide with R 0 = 0.4mm, R 1 = 0.23mm, R 2 = 0.21mm and L = 12mm. (b) The electric field distribution of the wave coupled from left port of the waveguide. (c)-(f) the absolute electric field (|E|) distribution at the plane of y = 0 for four different incident waves in the tapered waveguide, respectively. (c) λ = 2.16mm, (d) λ = 2.14mm, (e) λ = 2.12mm, and (f) λ = 2.10mm.

Fig. 4
Fig. 4

The wavelength at the critical point varies with permittivity of dielectric rod of the waveguide (R 0 = 2R 1 = 0.4mm).

Fig. 5
Fig. 5

(a) The propagation constants of modes as R 1 of lossy and lossless waveguide varies. The wavelength of the guided mode equals to 2.14mm. For both waveguides, R 0 = 0.4mm. The black and gray solid curves correspond to the lossy case, and the green thick lines and red dotted lines correspond to the lossless case. (b) The |E| distribution at plane of y = 0 for incident wave of λ = 2.14mm in the lossy waveguide as shown in Fig. 3(a).

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