Abstract

A numerical study on the complex propagation constants of the surface plasmon polariton (SPP) rectangular hollow waveguide by the method of lines (MoL) is performed. New cut-off conditions are proposed for the SPP waveguide. A SPP rectangular hollow waveguide constructed of gold is first considered. The dependences of complex propagation constants on the sizes of the waveguide and on the wavelength are investigated. Fundamental and unusual characteristics of the SPP waveguide are revealed. The validity and limitations of effective index method (EIM) are examined by comparing the numerical results obtained by the MoL with the approximate results obtained by EIM. The differences in the propagation characteristics among the various metals are then shown.

© 2008 Optical Society of America

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  5. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
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    [CrossRef] [PubMed]
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    [CrossRef]
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2008 (1)

2007 (2)

S. Collin, F. Pardo, and J.-L. Pelouard, "Waveguiding in nanoscale metallic apertures," Opt. Express 15, 4310-4320 (2007).
[CrossRef] [PubMed]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316,430-432 (2007).
[CrossRef] [PubMed]

2006 (4)

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization," Phys. Rev. B 73, 035407 (2006).
[CrossRef]

F. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B  74, 153411-1-4 (2006).
[CrossRef]

R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Trans. Nanotech. 5, 291-294 (2006).
[CrossRef]

2005 (2)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

R. Gordon and A. G. Brolo, "Increased cut-off wavelength for a subwavelength hole in a real metal," Opt. Express 13, 1933-1938 (2005).
[CrossRef] [PubMed]

2004 (2)

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Comm. 239, 61-66 (2004).
[CrossRef]

2003 (2)

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2001 (2)

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

T. Yatsui, M. Kourogi, and M. Ohtsu, "Plasmon waveguide for optical far/near-field conversion," Appl. Phys. Lett. 79, 4583-4585 (2001).
[CrossRef]

2000 (1)

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

1997 (1)

1996 (1)

P. Berini and K. Wu, "Modeling lossy anisotropic dielectric waveguides with the method of lines," IEEE Trans. Microwave Theory Tech. 44, 749-759 (1996).
[CrossRef]

1993 (1)

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).
[CrossRef]

1972 (1)

Johnson and Christy, "Optical constants of the nobel metals," Phys. Rev. B 12, 4370-4379 (1972).
[CrossRef]

Atwater, H. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316,430-432 (2007).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization," Phys. Rev. B 73, 035407 (2006).
[CrossRef]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef] [PubMed]

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Berini, P.

P. Berini, "Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics," Opt. Express 7, 329-335 (2000).
[CrossRef] [PubMed]

P. Berini and K. Wu, "Modeling lossy anisotropic dielectric waveguides with the method of lines," IEEE Trans. Microwave Theory Tech. 44, 749-759 (1996).
[CrossRef]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Brolo, A. G.

R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Trans. Nanotech. 5, 291-294 (2006).
[CrossRef]

R. Gordon and A. G. Brolo, "Increased cut-off wavelength for a subwavelength hole in a real metal," Opt. Express 13, 1933-1938 (2005).
[CrossRef] [PubMed]

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Collin, S.

Degiron, A.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Comm. 239, 61-66 (2004).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316,430-432 (2007).
[CrossRef] [PubMed]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef] [PubMed]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization," Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Comm. 239, 61-66 (2004).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Enoch, S.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Garcia-Vidal, F.

F. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B  74, 153411-1-4 (2006).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Gordon, R.

R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Trans. Nanotech. 5, 291-294 (2006).
[CrossRef]

F. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B  74, 153411-1-4 (2006).
[CrossRef]

R. Gordon and A. G. Brolo, "Increased cut-off wavelength for a subwavelength hole in a real metal," Opt. Express 13, 1933-1938 (2005).
[CrossRef] [PubMed]

Kik, P. G.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Kobayashi, T.

Koerkamp, K. J. K.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Kourogi, M.

T. Yatsui, M. Kourogi, and M. Ohtsu, "Plasmon waveguide for optical far/near-field conversion," Appl. Phys. Lett. 79, 4583-4585 (2001).
[CrossRef]

Kuipers, L.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Kumar, A.

Kumar, L. K.

F. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B  74, 153411-1-4 (2006).
[CrossRef]

Kumar, L. K. S.

R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Trans. Nanotech. 5, 291-294 (2006).
[CrossRef]

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316,430-432 (2007).
[CrossRef] [PubMed]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef] [PubMed]

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Comm. 239, 61-66 (2004).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Maier, S. A.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Martin-Moreno, L.

F. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B  74, 153411-1-4 (2006).
[CrossRef]

Meltzer, S.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Moreno, E.

F. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B  74, 153411-1-4 (2006).
[CrossRef]

Morimoto, A.

Ohtsu, M.

T. Yatsui, M. Kourogi, and M. Ohtsu, "Plasmon waveguide for optical far/near-field conversion," Appl. Phys. Lett. 79, 4583-4585 (2001).
[CrossRef]

Pardo, F.

Pelouard, J.-L.

Pregla, R.

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).
[CrossRef]

Requicha, A. A. G.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Rogge, U.

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).
[CrossRef]

Segerink, F. B.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Srivastava, T.

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization," Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Takahara, J.

Taki, H.

Tanaka, K.

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

Tanaka, M.

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

van Hulst, N. F.

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Wu, K.

P. Berini and K. Wu, "Modeling lossy anisotropic dielectric waveguides with the method of lines," IEEE Trans. Microwave Theory Tech. 44, 749-759 (1996).
[CrossRef]

Yamagishi, S.

Yamamoto, N.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Comm. 239, 61-66 (2004).
[CrossRef]

Yatsui, T.

T. Yatsui, M. Kourogi, and M. Ohtsu, "Plasmon waveguide for optical far/near-field conversion," Appl. Phys. Lett. 79, 4583-4585 (2001).
[CrossRef]

Adv. Mater. (1)

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A route to nanoscale optical devices," Adv. Mater. 13, 1501-1505 (2001).
[CrossRef]

Appl. Phys. Lett. (2)

T. Yatsui, M. Kourogi, and M. Ohtsu, "Plasmon waveguide for optical far/near-field conversion," Appl. Phys. Lett. 79, 4583-4585 (2001).
[CrossRef]

K. Tanaka and M. Tanaka, "Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide," Appl. Phys. Lett. 82, 1158-1160 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. Berini and K. Wu, "Modeling lossy anisotropic dielectric waveguides with the method of lines," IEEE Trans. Microwave Theory Tech. 44, 749-759 (1996).
[CrossRef]

IEEE Trans. Nanotech. (1)

R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal film," IEEE Trans. Nanotech. 5, 291-294 (2006).
[CrossRef]

J. Lightwave Technol. (1)

U. Rogge and R. Pregla, "Method of lines for the analysis of dielectric waveguides," J. Lightwave Technol. 11, 2015-2020 (1993).
[CrossRef]

Nano Lett. (1)

J. A. Dionne, H. J. Lezec, and H. A. Atwater, "Highly confined photon transport in subwavelength metallic slot waveguides," Nano Lett. 6, 1928-1932 (2006).
[CrossRef] [PubMed]

Nature (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Comm. (1)

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Comm. 239, 61-66 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. B (1)

F. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. Kumar, and R. Gordon, "Transmission of light through a single rectangular hole in a real metal," Phys. Rev. B  74, 153411-1-4 (2006).
[CrossRef]

Phys. Rev. B (2)

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization," Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Johnson and Christy, "Optical constants of the nobel metals," Phys. Rev. B 12, 4370-4379 (1972).
[CrossRef]

Phys. Rev. Lett. (2)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, "Channel plasmon-polariton guiding by subwavelength metal grooves," Phys. Rev. Lett. 95, 046802 (2005).
[CrossRef] [PubMed]

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef] [PubMed]

Science (1)

H. J. Lezec, J. A. Dionne, and H. A. Atwater, "Negative refraction at visible frequencies," Science 316,430-432 (2007).
[CrossRef] [PubMed]

Other (2)

D. W. Lynch and W. R. Hunter, "Aluminum (Al) and nickel (Ni)," in Handbook of optical constants of solids, E. D. Palik, ed. (Academic, New York, 1985).

R. Pregla and W. Pascher "The method of lines," in Numerical techniques for microwave and millimeter-wave passive structures T. Itoh, Ed. (New York: Wiley, 1989).

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Figures (11)

Fig.1.
Fig.1.

Geometry of the problem. A rectangular hollow waveguide with width a and height b is made in metallic material with a complex relative permittivity ε 1. The space in the waveguide is a vacuum with relative permittivity ε 0.

Fig. 2.
Fig. 2.

Dependences of the normalized phase constant β and the attenuation constant α on the waveguide-height b/λ for the TE01 mode with waveguide-width a/λ as a parameter. The open symbols denote the results for the EIM, and the bold black curves denote the results for the PEC waveguide (constructed of gold).

Fig. 3.
Fig. 3.

Dependences of the normalized phase constant β and the attenuation constant α on the waveguide-height b/λ for the TE02 mode with waveguide-width a/λ as a parameter. The black bold curve denotes the result for the PEC waveguide. The PEC waveguide is in the cut-off condition TE02 mode in the range of waveguide sizes shown in this figure (constructed of gold).

Fig. 4.
Fig. 4.

Dependences of normalized phase constant β and attenuation constant α on the waveguide width a/λ with waveguide height b/λ as a parameter for the TE01 mode. The open symbols are the results for the EIM (constructed of gold).

Fig. 5.
Fig. 5.

Dependences of normalized phase constant β and attenuation constant α on the waveguide-width a/λ with waveguide-height b/λ as a parameter for the TE02 mode (constructed of gold).

Fig. 6.
Fig. 6.

Cut-off wavelength of TE01 mode as a function of waveguide width a for a silver rectangular hollow waveguide with b=270 nm. Orange solid circles show the results by the MoL. Solid curve is the result by analytical methods [17]. Red points show the results by EIM [14]. Broken line shows the result of PEC case.

Fig. 7.
Fig. 7.

Dependences of normalized phase constant β on the wavelength with waveguide-size as a parameter for TE01 mode. The bold black curves represent the results for the PEC waveguide (constructed of gold).

Fig. 8.
Fig. 8.

Dependences of normalized attenuation constant α on the wavelength with waveguide-size as a parameter for TE01 mode. The bold black curves represent the results for the PEC waveguide (constructed of gold).

Fig. 9.
Fig. 9.

Typical distributions of electric field components (a) Re[E x(x,y)], (b) Re[E y(x,y)], and (c) Re[E z(x,y)] of the TE01 mode and (d) Re[E x(x,y)], (e) Re[E y(x,y)], and (f) Re[E z(x,y)] of the TE02 mode.

Fig. 10.
Fig. 10.

Dependences of normalized attenuation constant α and phase constant β of the TE01 mode on the wavelength for the gold, silver, and copper. The bold black curves represent the results for the PEC waveguide.

Fig. 11.
Fig. 11.

Dependences of normalized attenuation constant α and phase constant β of the TE01 mode on the wavelength for the aluminum and nickel. The bold black curves represent the results for the PEC waveguide.

Tables (2)

Tables Icon

Table 1. Numerical values of α=β in the cut-off condition calculated by MoL and values calculated by EIM in Fig. 2.

Tables Icon

Table 2. Numerical values of b/λ in the cut-off condition calculated by MoL and by EIM in Fig. 2.

Equations (18)

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E = ε r 1 ( x ) × × Π e j k 0 × Π h
η 0 H = j k 0 × Π e + × × Π h
Π e , h = k 0 2 ψ e , h e j k z z i x
[ Y ¯ ( k z ) ] E ¯ = 0
E x ( x , y , z ) cos ( k x x ) cos ( k y y ) ex p ( j k z z ) ( TE 01 )
cos ( k x x ) sin ( k y y ) ex p ( j k z z ) . ( TE 02 )
k z = ( k 0 2 k x 2 k y 2 ) 1 2 .
α = 0 , β = [ 1 n 2 ( 2 b λ ) 2 ] 1 2 for b λ 1 2
α = [ n 2 ( 2 b λ ) 2 1 ] 1 2 , β = 0 for b λ < 1 2
E x ( x , y , z ) cosh ( k x x ) cos ( k y y ) exp ( j k z z )
k z ( k 0 2 + k x 2 k y 2 ) 1 2
α + j β j [ 1 + k x 2 k 0 2 k y 2 k 0 2 ] 1 2 .
α I m ( k y 2 k 0 2 k x 2 k 0 2 ) ( 2 β )
β [ 1 + R e ( k x 2 k 0 2 ) R e ( k y 2 k 0 2 ) + α 2 ] 1 2
α [ R e ( k y 2 k 0 2 ) 1 R e ( k x 2 k 0 2 ) + β 2 ] 1 2
β I m ( k y 2 k 0 2 k x 2 k 0 2 ) ( 2 α )
1 + R e ( k x 2 k 0 2 ) R e ( k y 2 k 0 2 ) = 0
α β 1 2 1 2 [ I m ( k x 2 k 0 2 ) I m ( k y 2 k 0 2 ) ] 1 2

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