Abstract

We investigate the problem of total transmission at the interface separating an isotropic regular material and an indefinite medium [Phys. Rev. Lett. 90, 077405 (2003) ] in which not all of the principal elements of the permeability and permittivity tensors have the same sign, for TE- and TM-polarized electromagnetic waves. We make a detailed investigation on the existence conditions of total transmission and the corresponding Brewster’s angles when an electromagnetic wave is incident on the interface from an isotropic regular material. We show that the propagation characteristics of electromagnetic waves at such interfaces are quite different from those at regular interfaces. For both TE and TM waves, total transmission is possible at the interfaces containing an indefinite medium; in particular, normally incident total transmission and omnidirectional total transmission are also allowed, provided that suitable physical parameters for the two materials across the interface are chosen.

© 2006 Optical Society of America

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ϵ and μ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
    [CrossRef]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  5. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3999 (2000).
    [CrossRef] [PubMed]
  6. L. Venema, "A lens less ordinary," Nature 420, 119-120 (2002).
    [CrossRef] [PubMed]
  7. C. Seife, "Offbeat lenses promise perfect fidelity," Science 290, 1066 (2000).
    [CrossRef]
  8. D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef] [PubMed]
  9. I. L. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media-media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
    [CrossRef]
  10. L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
    [CrossRef]
  11. X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
    [CrossRef]
  12. L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, "Optomagnetic composite medium with conducting nanoelements," Phys. Rev. B 66, 155411 (2002).
    [CrossRef]
  13. N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
    [CrossRef]

2005 (1)

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

2003 (1)

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

2002 (4)

L. Venema, "A lens less ordinary," Nature 420, 119-120 (2002).
[CrossRef] [PubMed]

L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, "Optomagnetic composite medium with conducting nanoelements," Phys. Rev. B 66, 155411 (2002).
[CrossRef]

2001 (2)

I. L. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media-media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (3)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3999 (2000).
[CrossRef] [PubMed]

C. Seife, "Offbeat lenses promise perfect fidelity," Science 290, 1066 (2000).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ϵ and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Chen, J.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Chui, S. T.

L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

Ding, J. P.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Fan, Y. X.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Grigorenko, A. N.

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, "Optomagnetic composite medium with conducting nanoelements," Phys. Rev. B 66, 155411 (2002).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Hu, L. B.

L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

Hu, X. H.

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Ilvonen, S.

I. L. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media-media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Jia, W. L.

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Li, Y. Z.

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Lindell, I. L.

I. L. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media-media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Liu, X. H.

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Makhnovskiy, D. P.

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, "Optomagnetic composite medium with conducting nanoelements," Phys. Rev. B 66, 155411 (2002).
[CrossRef]

Ming, N. B.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Nikoskinen, K. I.

I. L. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media-media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Panina, L. V.

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, "Optomagnetic composite medium with conducting nanoelements," Phys. Rev. B 66, 155411 (2002).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3999 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

Seife, C.

C. Seife, "Offbeat lenses promise perfect fidelity," Science 290, 1066 (2000).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shen, N. H.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Smith, D. R.

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Tian, Y. J.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Tretyakov, S. A.

I. L. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media-media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Venema, L.

L. Venema, "A lens less ordinary," Nature 420, 119-120 (2002).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ϵ and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Wang, H. T.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Wang, Q.

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Wang, X.

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Xu, C.

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Zi, J.

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

X. Wang, X. H. Hu, Y. Z. Li, W. L. Jia, C. Xu, X. H. Liu, and J. Zi, "Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures," Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

I. L. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media-media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Nature (1)

L. Venema, "A lens less ordinary," Nature 420, 119-120 (2002).
[CrossRef] [PubMed]

Phys. Rev. B (3)

L. B. Hu and S. T. Chui, "Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials," Phys. Rev. B 66, 085108 (2002).
[CrossRef]

L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, "Optomagnetic composite medium with conducting nanoelements," Phys. Rev. B 66, 155411 (2002).
[CrossRef]

N. H. Shen, Q. Wang, J. Chen, Y. X. Fan, J. P. Ding, H. T. Wang, Y. J. Tian, and N. B. Ming, "Optically uniaxial left-handed materials," Phys. Rev. B 72, 153104 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

D. R. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3999 (2000).
[CrossRef] [PubMed]

Science (2)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

C. Seife, "Offbeat lenses promise perfect fidelity," Science 290, 1066 (2000).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ϵ and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of the interface between media 1 and 2 and the coordinate system.

Fig. 2
Fig. 2

Existence regions of total transmission at the interface between an isotropic regular medium 1 and an isotropic left-handed medium 2, for TE and TM waves, on the parameter plane ( ϵ 2 ϵ 1 , μ 2 μ 1 ) ; the solid straight line and the solid circle correspond to the zero Brewster’s angle and omnidirectional total transmission, respectively.

Fig. 3
Fig. 3

Total transmission for TE waves at the interface between an isotropic regular medium 1 and an indefinite medium 2 of the first special category. Existence regions of total transmission for TE waves (a) for the case ϵ 2 μ 2 < ϵ 1 μ 1 , with the solid straight line corresponding to the zero Brewster’s angle, (b) for the case ϵ 2 μ 2 > ϵ 1 μ 1 , with the solid straight line corresponding to the zero Brewster’s angle, and (c) for the case ϵ 2 μ 2 = ϵ 1 μ 1 on the parameter plane ( ϵ 2 ϵ 1 , μ 2 μ 1 ) , with the solid straight line representing omnidirectional total transmission. The hollow circle needs eliminating.

Fig. 4
Fig. 4

Total transmission for TM waves at the interface between an isotropic regular medium 1 and an indefinite medium 2 of the first special category. Existence regions of total transmission for TM wave (a) for the case ϵ 2 μ 2 < ϵ 1 μ 1 , with the solid straight line corresponding to the zero Brewster’s angle, (b) for the case ϵ 2 μ 2 > ϵ 1 μ 1 , with the solid straight line corresponding to the zero Brewster’s angle, and (c) for the case ϵ 2 μ 2 = ϵ 1 μ 1 on the parameter plane ( ϵ 2 ϵ 1 , μ 2 μ 1 ) , with the solid straight line representing omnidirectional total transmission. The hollow circle needs eliminating.

Fig. 5
Fig. 5

Total transmission for the TE waves at the interface between an isotropic regular medium 1 and an indefinite medium 2 of the second special category. Existence regions of total transmission for TE waves on the parameter plane ( ϵ 2 ϵ 1 , μ 2 μ 1 ) . The solid straight line represents the zero Brewster’s angle. The solid circle represents omnidirectional total transmission, and the hollow circles need eliminating.

Fig. 6
Fig. 6

Total transmission for TM waves at the interface between an isotropic regular medium 1 and an indefinite medium 2 of the second special category. Existence regions of total transmission for TM waves on the parameter plane ( ϵ 2 ϵ 1 , μ 2 μ 1 ) . The solid straight line represents the zero Brewster’s angle. The solid circle represents omnidirectional total transmission, and the hollow circles need eliminating.

Tables (12)

Tables Icon

Table 1 Existence Conditions of Brewster’s Angles for TE Waves at Interface of Isotropic Regular Medium 1 and Indefinite Medium 2

Tables Icon

Table 2 Total Transmission (Brewster’s Angle) for TE Waves at Interface of Isotropic Regular Medium 1 and Indefinite Medium 2

Tables Icon

Table 3 Existence Conditions of Brewster’s Angles for TM Waves at Interface of Isotropic Regular Medium 1 and Indefinite Medium 2

Tables Icon

Table 4 Total Transmission (Brewster’s Angle) for TM Waves at Interface of Isotropic Regular Medium 1 and Indefinite Medium 2

Tables Icon

Table 5 Total Transmission (Brewster’s Angle) at Interface of Isotropic Regular Material and Isotropic Left-handed Material

Tables Icon

Table 6 Existence Conditions of Total Transmission for TE Waves at Interface of Isotropic Regular Material and Indefinite Medium with Symmetric Principal Axes of ϵ ¯ ¯ 2 and μ ¯ ¯ 2 Coincident Normal to Interface

Tables Icon

Table 7 Existence Conditions of Total Transmission for TM Waves at Interface of Isotropic Regular Material and Indefinite Medium with Symmetric Principal Axes of ϵ ¯ ¯ 2 and μ ¯ ¯ 2 Coincident Normal to Interface

Tables Icon

Table 8 Total Transmission (Brewster’s Angle) at Interface of Isotropic Regular Material and Indefinite Medium with Symmetric Principal Axes of ϵ ¯ ¯ 2 and μ ¯ ¯ 2 Coincident Normal to Interface

Tables Icon

Table 9 Existence Conditions of Total Transmission for TE Waves at Interface of Isotropic Regular Material and Indefinite Medium with Symmetric Principal Axes of ϵ ̿ 2 and μ ̿ 2 Coincident Parallel to Interface

Tables Icon

Table 10 Existence Conditions of Total Transmission for TM Waves at Interface of Isotropic Regular Material and Indefinite Medium with Symmetric Principal Axes of ϵ ̿ 2 and μ ̿ 2 Coincident Parallel to Interface

Tables Icon

Table 11 Total Transmission (Brewster’s Angle) for TE Waves at Interface of Isotropic Regular Material and Indefinite Medium with Symmetric Principal Axes of ϵ ̿ 2 and μ ̿ 2 Coincident Parallel to Interface

Tables Icon

Table 12 Total Transmission (Brewster’s Angle) for TM Waves at Interface of Isotropic Regular Material and Indefinite Medium with Symmetric Principal Axes of ϵ ̿ 2 and μ ̿ 2 Coincident Parallel to Interface

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

ϵ ¯ ¯ 2 = [ ϵ 2 x 0 0 0 ϵ 2 y 0 0 0 ϵ 2 z ] , μ ¯ 2 = [ μ 2 x 0 0 0 μ 2 y 0 0 0 μ 2 z ] .
E i = E 0 e ̂ y exp ( j k 1 z z ) ,
H i = E 0 ω μ 0 μ 1 ( k 1 z e ̂ x + k x e ̂ z ) exp ( j k 1 z z ) ,
E r = r E 0 e ̂ y exp ( j k 1 z z ) ,
H r = r E 0 ω μ 0 μ 1 ( k 1 z e ̂ x + k x e ̂ z ) exp ( j k 1 z z ) ,
E t = t E 0 e ̂ y exp ( j k 2 z z ) ,
H t = t E 0 ω μ 0 ( k 2 z μ 2 x e ̂ x + k x μ 2 z e ̂ z ) exp ( j k 2 z z ) ,
k x 2 = ( ω 2 c 2 ) ϵ 1 μ 1 sin 2 θ 1 ,
k x 2 ϵ 2 y μ 2 z + k 2 z 2 ϵ 2 y μ 2 x = ω 2 c 2
k x 2 ϵ 2 z μ 2 y + k 2 z 2 ϵ 2 x μ 2 y = ω 2 c 2
k 1 z 2 = ( ω 2 c 2 ) ϵ 1 μ 1 k x 2 .
r TE = μ 2 x k 1 z μ 1 k 2 z μ 2 x k 1 z + μ 1 k 2 z ,
t TE = 2 μ 2 x k 1 z μ 2 x k 1 z + μ 1 k 2 z .
r TM = ϵ 2 x k 1 z ϵ 1 k 2 z ϵ 2 x k 1 z + ϵ 1 k 2 z ,
t TM = 2 ϵ 2 x k 1 z ϵ 2 x k 1 z + ϵ 1 k 2 z .
μ 2 x k 1 z = μ 1 k 2 z
ϵ 2 x k 1 z = ϵ 1 k 2 z
k 2 z 2 = ( ω c ) 2 [ ϵ 2 y μ 2 x ( μ 2 x μ 2 z ) ϵ 1 μ 1 sin 2 θ 1 ] .
sin 2 θ 1 B TE = ( ϵ 1 μ 2 x ϵ 2 y μ 1 ) μ 2 z ( μ 2 x μ 2 z μ 1 2 ) ϵ 1 .
0 < ( ϵ 1 μ 2 x ϵ 2 y μ 1 ) μ 2 z ( μ 2 x μ 2 z μ 1 2 ) ϵ 1 < 1 .
( μ 2 x μ 2 z μ 1 2 ) ( ϵ 1 μ 2 x ϵ 2 y μ 1 ) > 0 ,
( μ 2 x μ 2 z μ 1 2 ) ( ϵ 2 y μ 2 z ϵ 1 μ 1 ) > 0 ,
( μ 2 x μ 2 z μ 1 2 ) ( ϵ 1 μ 2 x ϵ 2 y μ 1 ) < 0 ,
( μ 2 x μ 2 z μ 1 2 ) ( ϵ 2 y μ 2 z ϵ 1 μ 1 ) > 0 ,
ϵ 2 y μ 2 z < ϵ 1 μ 1 ,
( μ 2 x μ 2 z μ 1 2 ) ( ϵ 2 y μ 2 z ϵ 1 μ 1 ) > 0 ,
sin 2 θ 1 B TM = ( ϵ 2 x μ 1 ϵ 1 μ 2 y ) ϵ 2 z ( ϵ 2 x ϵ 2 z ϵ 1 2 ) μ 1 .
sin 2 θ 1 B TE = ( ϵ 1 μ 2 ϵ 2 μ 1 ) μ 2 ( μ 2 2 μ 1 2 ) ϵ 1 .
ϵ 2 μ 2 < ϵ 1 μ 1 ϵ 2 μ 2 > ϵ 1 μ 1
ϵ 2 μ 2 > ϵ 1 μ 1 ϵ 2 μ 2 < ϵ 1 μ 1 .
sin 2 θ 1 B TM = ( ϵ 2 μ 1 ϵ 1 μ 2 ) ϵ 2 ( ϵ 2 2 ϵ 1 2 ) μ 1 ,
ϵ 2 μ 2 < ϵ 1 μ 1 ϵ 2 μ 2 < ϵ 1 μ 1
ϵ 2 μ 2 > ϵ 1 μ 1 ϵ 2 μ 2 > ϵ 1 μ 1 .
ϵ ¯ ¯ 2 = [ ϵ 2 0 0 0 ϵ 2 0 0 0 ϵ 2 ] , μ ¯ ¯ 2 = [ μ 2 0 0 0 μ 2 0 0 0 μ 2 ] ,
sin 2 θ 1 B TE = ( ϵ 1 μ 2 ϵ 2 μ 1 ) μ 2 ( μ 2 μ 2 μ 1 2 ) ϵ 1
sin 2 θ 1 B TM = ( ϵ 2 μ 1 ϵ 1 μ 2 ) ϵ 2 ( ϵ 2 ϵ 2 ϵ 1 2 ) μ 1
ϵ ¯ ¯ 2 = [ ϵ 2 0 0 0 ϵ 2 0 0 0 ϵ 2 ] , μ ¯ ¯ 2 = [ μ 2 0 0 0 μ 2 0 0 0 μ 2 ] ,
sin 2 θ 1 B TE = ( ϵ 1 μ 2 ϵ 2 μ 1 ) ϵ 2 ( μ 2 2 μ 1 2 ) ϵ 1
sin 2 θ 1 B TM = ( ϵ 2 μ 1 ϵ 1 μ 2 ) ϵ 2 ( ϵ 2 2 ϵ 1 2 ) μ 1

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