Abstract

By using a true phasor approach and slowly varying envelope approximations along with Maxwell’s equations and the constitutive relations in their respective domains (time and frequency, respectively), we derive expressions for the allowable propagation vector(s), electromagnetic fields, Poynting vectors, phase, energy, and group velocities in a medium where independent material parameters such as permittivity, permeability, and chirality are frequency dependent, including not only the carrier (e.g., optical) frequency but excursions around the carrier. One definition of negative index, viz., contradirection of the propagation and Poynting vector, demands a large value for the chirality parameter, which may not be physically attainable. We show that by incorporating dispersions in these (independent) material parameters, it may be possible to achieve negative index as defined through contradirected phase and group velocities for a range of carrier frequencies that are lower than the resonant frequency for the aforementioned material parameters as described through the Lorenz (for permittivity and permeability) and Condon (for chirality) models, and without violating the upper bound on the chirality. This has the added advantage that losses will be minimal, and further justifies our approach of using real functions for the material parameters.

© 2009 Optical Society of America

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References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  2. J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57, 37-43 (2004).
    [CrossRef]
  3. S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
    [CrossRef]
  4. 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]
  5. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
    [CrossRef] [PubMed]
  6. M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

2007 (2)

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

P. P. Banerjee and G. Nehmetallah, “Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials,” J. Opt. Soc. Am. B 24, A69-A76 (2007).
[CrossRef]

2006 (3)

A. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

J. Q. Shen, “Negative refractive index in gyrotropically magnetoelectric media,” Phys. Rev. B 73, 045113 (2006).
[CrossRef]

P. P. Banerjee and G. Nehmetallah, “Linear and nonlinear propagation in negative index materials,” J. Opt. Soc. Am. B 23, 2348-2355 (2006).
[CrossRef]

2005 (2)

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

2004 (3)

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57, 37-43 (2004).
[CrossRef]

T. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E 69, 026602 (2004).
[CrossRef]

T. J. Cui and J. A. Kong, “Time domain electromagnetic energy in a frequency-dispersive left handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

2003 (1)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

2001 (1)

C.-C. Huang and Y.-Z. Zhang, “Poynting vector, energy density and energy velocity in anomalous dispersion medium,” Phys. Rev. A 65, 015802 (2001).
[CrossRef]

2000 (3)

A. Bers, “Note on group velocity and energy propagation,” Am. J. Phys. 68, 482-485 (2000).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[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]

1994 (1)

1993 (1)

1990 (1)

M. S. Kluskens and E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448-1455 (1990).
[CrossRef]

Akyurtlu, A.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

Anugula, P. R.

M. R. Chatterjee, P. P. Banerjee, and P. R. Anugula, “Investigation of negative refractive index in reciprocal chiral materials,” in The Nature of Light: Light in Nature, K.Creath, Ed., Proc. SPIE 6285, 628504-1-6 (2006).

Aylo, R.

Banerjee, P. P.

Bers, A.

A. Bers, “Note on group velocity and energy propagation,” Am. J. Phys. 68, 482-485 (2000).
[CrossRef]

Bloemer, M. J.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Boardman, A.

A. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

Chatterjee, M. R.

M. R. Chatterjee, P. P. Banerjee, and P. R. Anugula, “Investigation of negative refractive index in reciprocal chiral materials,” in The Nature of Light: Light in Nature, K.Creath, Ed., Proc. SPIE 6285, 628504-1-6 (2006).

Cui, T. J.

T. J. Cui and J. A. Kong, “Time domain electromagnetic energy in a frequency-dispersive left handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

D'Aguanno, G.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Dong, Q.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

Engheta, N.

Fleischhauer, M.

S. Yelin, J. Kastel, M. Fleischhauer, and R. L. Walsworth, “Negative refraction and electromagnetically induced chirality,” in Proceedings of the Nonlinear Optics Conference (2007), Paper WA3.

Goodhue, W. D.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Haus, J. W.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Huang, C.-C.

C.-C. Huang and Y.-Z. Zhang, “Poynting vector, energy density and energy velocity in anomalous dispersion medium,” Phys. Rev. A 65, 015802 (2001).
[CrossRef]

Kastel, J.

S. Yelin, J. Kastel, M. Fleischhauer, and R. L. Walsworth, “Negative refraction and electromagnetically induced chirality,” in Proceedings of the Nonlinear Optics Conference (2007), Paper WA3.

Kluskens, M. S.

M. S. Kluskens and E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448-1455 (1990).
[CrossRef]

Kong, J. A.

T. J. Cui and J. A. Kong, “Time domain electromagnetic energy in a frequency-dispersive left handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

Kontenbah, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Lakhtakia, A.

T. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E 69, 026602 (2004).
[CrossRef]

Li, J.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Lu, I.-T.

Mackay, T.

T. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E 69, 026602 (2004).
[CrossRef]

Marinov, K.

A. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

Marx, K. A.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

Mattiucci, N.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Nehmetallah, G.

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]

Newman, E. H.

M. S. Kluskens and E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448-1455 (1990).
[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]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57, 37-43 (2004).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Qiu, R. C.

Ramakrishna, S. A.

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Scalora, M.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Schultz, S.

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]

Shen, J. Q.

J. Q. Shen, “Negative refractive index in gyrotropically magnetoelectric media,” Phys. Rev. B 73, 045113 (2006).
[CrossRef]

Smith, D. R.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57, 37-43 (2004).
[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]

Tanielian, M. H.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

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]

Walsworth, R. L.

S. Yelin, J. Kastel, M. Fleischhauer, and R. L. Walsworth, “Negative refraction and electromagnetically induced chirality,” in Proceedings of the Nonlinear Optics Conference (2007), Paper WA3.

Wongkasem, N.

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

Yelin, S.

S. Yelin, J. Kastel, M. Fleischhauer, and R. L. Walsworth, “Negative refraction and electromagnetically induced chirality,” in Proceedings of the Nonlinear Optics Conference (2007), Paper WA3.

Zablocky, P. G.

Zhang, Y.-Z.

C.-C. Huang and Y.-Z. Zhang, “Poynting vector, energy density and energy velocity in anomalous dispersion medium,” Phys. Rev. A 65, 015802 (2001).
[CrossRef]

Zheltikov, A. M.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Am. J. Phys. (1)

A. Bers, “Note on group velocity and energy propagation,” Am. J. Phys. 68, 482-485 (2000).
[CrossRef]

Appl. Phys. B (1)

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, “Negative refraction of ultra-short pulses,” Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

M. S. Kluskens and E. H. Newman, “Scattering by a chiral cylinder of arbitrary cross section,” IEEE Trans. Antennas Propag. 38, 1448-1455 (1990).
[CrossRef]

N. Wongkasem, A. Akyurtlu, K. A. Marx, Q. Dong, J. Li, and W. D. Goodhue, “Development of chiral negative refractive index metamaterials for the terahertz frequency regime,” IEEE Trans. Antennas Propag. 55, 3052-3062 (2007).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (3)

Phys. Rev. A (1)

C.-C. Huang and Y.-Z. Zhang, “Poynting vector, energy density and energy velocity in anomalous dispersion medium,” Phys. Rev. A 65, 015802 (2001).
[CrossRef]

Phys. Rev. B (3)

T. J. Cui and J. A. Kong, “Time domain electromagnetic energy in a frequency-dispersive left handed medium,” Phys. Rev. B 70, 205106 (2004).
[CrossRef]

A. Boardman and K. Marinov, “Electromagnetic energy in a dispersive metamaterial,” Phys. Rev. B 73, 165110 (2006).
[CrossRef]

J. Q. Shen, “Negative refractive index in gyrotropically magnetoelectric media,” Phys. Rev. B 73, 045113 (2006).
[CrossRef]

Phys. Rev. E (1)

T. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E 69, 026602 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[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]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's Law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Phys. Today (1)

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57, 37-43 (2004).
[CrossRef]

Rep. Prog. Phys. (1)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Other (2)

M. R. Chatterjee, P. P. Banerjee, and P. R. Anugula, “Investigation of negative refractive index in reciprocal chiral materials,” in The Nature of Light: Light in Nature, K.Creath, Ed., Proc. SPIE 6285, 628504-1-6 (2006).

S. Yelin, J. Kastel, M. Fleischhauer, and R. L. Walsworth, “Negative refraction and electromagnetically induced chirality,” in Proceedings of the Nonlinear Optics Conference (2007), Paper WA3.

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Figures (4)

Fig. 1
Fig. 1

Variation of phase refractive index n p with normalized frequency for various values of β.

Fig. 2
Fig. 2

Variation of group refractive index n g with normalized frequency for various values of β.

Fig. 3
Fig. 3

Variation of relative permittivity with normalized frequency for various values of β.

Fig. 4
Fig. 4

Variation of difference between relative permittivity (which equals relative permeability) and the modulus of the chirality parameter with normalized frequency for various values of β.

Equations (71)

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

D ̃ = ε ̃ E ̃ j κ ̃ μ 0 ε 0 H ̃ ,
B ̃ = j κ ̃ μ 0 ε 0 E ̃ + μ ̃ H ̃ ,
[ D ( r ¯ , t ) B ( r ¯ , t ) E ( r ¯ , t ) H ( r ¯ , t ) ] = Re { [ D p ( r ¯ , t ) B p ( r ¯ , t ) E p ( r ¯ , t ) H p ( r ¯ , t ) ] e j ω 0 t } ,
[ D ̃ ( ω ) B ̃ ( ω ) E ̃ ( ω ) H ̃ ( ω ) ] = 1 2 { [ D ̃ p ( ω ω 0 ) + D ̃ p * ( ω ω 0 ) B ̃ p ( ω ω 0 ) + B ̃ p * ( ω ω 0 ) E ̃ p ( ω ω 0 ) + E ̃ p * ( ω ω 0 ) H ̃ p ( ω ω 0 ) + H ̃ p * ( ω ω 0 ) ] } .
[ ε ( t ) μ ( t ) κ ( t ) ] = { [ ε p ( t ) μ p ( t ) κ p ( t ) ] e j ω 0 t + c.c. } ,
[ ε ̃ ( ω ) μ ̃ ( ω ) κ ̃ ( ω ) ] = { [ ε ̃ p ( ω ω 0 ) + ε ̃ p * ( ω ω 0 ) μ ̃ p ( ω ω 0 ) + μ ̃ p * ( ω ω 0 ) κ ̃ p ( ω ω 0 ) + κ ̃ p * ( ω ω 0 ) ] } .
[ ε ̃ p ( ω ω 0 ) μ ̃ p ( ω ω 0 ) κ ̃ p ( ω ω 0 ) ] { [ ε ̃ p ( ω 0 ) + ( ω ω 0 ) ( ε ̃ p ω ) ω 0 μ ̃ p ( ω 0 ) + ( ω ω 0 ) ( μ ̃ p ω ) ω 0 κ ̃ p ( ω 0 ) + ( ω ω 0 ) ( κ ̃ p ω ) ω 0 ] } { [ ε ̃ p 0 + Ω ε ̃ p 0 μ ̃ p 0 + Ω μ ̃ p 0 κ ̃ p 0 + Ω κ ̃ p 0 ] } .
D ̃ = 1 2 [ E ̃ p ( ω ω 0 ) ε ̃ p ( ω ω 0 ) + E ̃ p * ( ω ω 0 ) ε ̃ p * ( ω ω 0 ) ] 1 2 j μ 0 ε 0 [ H ̃ p ( ω ω 0 ) κ ̃ p ( ω ω 0 ) + H ̃ p * ( ω ω 0 ) κ ̃ p * ( ω ω 0 ) ] ,
D ̃ p ( Ω ) = [ E ̃ p ( Ω ) ε ̃ p ( Ω ) j μ 0 ε 0 H ̃ p ( Ω ) κ ̃ p ( Ω ) ] = [ E ̃ p ( Ω ) ( ε ̃ p 0 + Ω ε ̃ p 0 ) j μ 0 ε 0 H ̃ p ( Ω ) ( κ ̃ p 0 + Ω κ ̃ p 0 ) ] .
D p ( t ) = [ ( ε ̃ p 0 j ε ̃ p 0 t ) E p ( t ) j μ 0 ε 0 ( κ ̃ p 0 j κ ̃ p 0 t ) H p ( t ) ] .
k ¯ × E ̃ p ( Ω ) = ω [ j κ ̃ p ( Ω ) μ 0 ε 0 E ̃ p ( Ω ) + μ ̃ p ( Ω ) H ̃ p ( Ω ) ] .
k ¯ × H ̃ p ( Ω ) = ω [ ε ̃ p ( Ω ) E ̃ p ( Ω ) j κ ̃ p ( Ω ) μ 0 ε 0 H ̃ p ( Ω ) ] .
j α ̃ p E ̃ p x + k ̃ z E ̃ p y + ω μ ̃ p H ̃ p x = 0 ,
k ̃ z E ̃ p x j α ̃ p E ̃ p y ω μ ̃ p H ̃ p y = 0 ,
ω ε ̃ p E ̃ p x j α ̃ p H ̃ p x k ̃ z H ̃ p y = 0 ,
ω ε ̃ p E ̃ p y + k ̃ z H ̃ p x j α ̃ p H ̃ p y = 0 ,
k ̃ z 1 = + ω κ ̃ p μ 0 ε 0 + ω μ ̃ p ε ̃ p ,
k ̃ z 2 = + ω κ ̃ p μ 0 ε 0 ω μ ̃ p ε ̃ p ,
k ̃ z 3 = ω κ ̃ p μ 0 ε 0 + ω μ ̃ p ε ̃ p ,
k ̃ z 4 = ω κ ̃ p μ 0 ε 0 ω μ ̃ p ε ̃ p ,
E ̃ p x arbitrary ,
E ̃ p y = ω 2 μ ̃ p ε ̃ p + α ̃ p 2 + k ̃ z 2 2 j α ̃ p k ̃ z E ̃ p x ,
H ̃ p x = k ̃ z 2 + ω 2 μ ̃ p ε ̃ p + α ̃ p 2 2 j α ̃ p ω μ ̃ p E ̃ p x ,
H ̃ p y = α ̃ p 2 + k ̃ z 2 + ω 2 μ ̃ p ε ̃ p 2 ω μ ̃ p k ̃ z E ̃ p x .
E ̃ p x ( Ω ) arbitrary ,
E ̃ p y ( Ω ) = j ( A 1 + B 1 Ω ) E ̃ p x ( Ω ) ,
H ̃ p x ( Ω ) = j ( A 2 + B 2 Ω ) E ̃ p x ( Ω ) ,
H ̃ p y ( Ω ) = ( A 3 + B 3 Ω ) E ̃ p x ( Ω ) .
E p x ( t ) arbitrary ,
E p y ( t ) = j A 1 E p x ( t ) + B 1 E p x ( t ) t ,
H ̃ p x ( t ) = j A 2 E p x ( t ) + B 2 E p x ( t ) t ,
H ̃ p y ( t ) = A 3 E p x ( t ) j B 3 E p x ( t ) t ,
[ ε ̃ p ( Ω ) μ ̃ p ( Ω ) α ̃ p ( Ω ) ] = [ ε ̃ p ( Ω ) μ ̃ p ( Ω ) ω μ 0 ε 0 κ ̃ p ( Ω ) ] = { [ ε ̃ p 0 + Ω ε ̃ p 0 μ ̃ p 0 + Ω μ ̃ p 0 ω μ 0 ε 0 ( κ ̃ p 0 + Ω κ ̃ p 0 ) ] } ,
E ̃ p y = ω 2 μ ̃ p ε ̃ p + α ̃ p 2 + k ̃ z 2 2 j α ̃ p k ̃ z E ̃ p x 2 α ̃ p 2 2 α ̃ p ω μ ̃ p ε ̃ p 2 j α ̃ p k ̃ z E ̃ p x = j E ̃ p x .
A 13 = 1 , B 13 = 0 .
H ̃ p x = j ε ̃ p μ ̃ p E ̃ p x ,
A 23 = ε ̃ p 0 μ ̃ p 0 , B 23 = 1 2 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 μ ̃ p 0 μ ̃ p 0 ) .
A 33 = A 23 = ε ̃ p 0 μ ̃ p 0 , B 33 = B 23 = 1 2 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 μ ̃ p 0 μ ̃ p 0 ) .
A 11 = A 13 , B 11 = B 13 ,
A 21 = A 23 , B 21 = B 23 ,
A 31 = A 33 , B 31 = B 33 .
S a v ( t ) = 1 2 [ E p ( t ) × H p * ( t ) ] = 1 2 [ E p x ( t ) H p y * ( t ) E p y ( t ) H p x * ( t ) ] a ̂ z .
S a v 3 ( t ) = 1 2 [ E p x ( t ) { A 23 E p x * ( t ) j B 23 E p x * ( t ) t } j E p x ( t ) { j A 23 E p x * ( t ) + B 23 E p x * ( t ) t } ] a ̂ z = { A 23 E p x 2 ( t ) j ( B 23 2 ) E p x 2 ( t ) t } a ̂ z ,
S ̃ a v 3 ( Ω ) = { A 23 + B 23 Ω 2 } I t [ E p x 2 ( t ) ] a ̂ z ,
w e ( t ) = 1 4 D p * ( t ) E p ( t ) , w m ( t ) = 1 4 B p ( t ) H p * ( t ) .
w e 3 ( t ) = 1 4 [ ( ε ̃ p 0 j ε ̃ p 0 t ) ( E p x * E p x + E p y * E p y ) + j μ 0 ε 0 ( κ ̃ p 0 j κ ̃ p 0 t ) ( H p x * E p x + H p y * E p y ) ] = 1 2 [ ( ε ̃ p 0 j ε ̃ p 0 t ) E p x 2 + μ 0 ε 0 ( κ ̃ p 0 j κ ̃ p 0 t ) ( A 23 E p x 2 + j ( B 23 2 ) E p x 2 t ) ] ,
w ̃ e 3 ( Ω ) = 1 2 [ ( ε ̃ p 0 + μ 0 ε 0 A 23 κ ̃ p 0 ) { μ 0 ε 0 ( B 23 2 ) κ ̃ p 0 ( ε ̃ p 0 + μ 0 ε 0 A 23 κ ̃ p 0 ) } Ω ] I t [ E p x 2 ( t ) ] .
w ̃ m 3 ( Ω ) = 1 2 [ ( μ ̃ p 0 A 23 2 + μ 0 ε 0 A 23 κ ̃ p 0 ) { μ 0 ε 0 ( B 23 2 ) κ ̃ p 0 ( μ ̃ p 0 A 23 2 + μ 0 ε 0 A 23 κ ̃ p 0 ) } Ω ] I t [ E p x 2 ( t ) ] .
w ̃ t 3 ( Ω ) = w ̃ e 3 ( Ω ) + w ̃ m 3 ( Ω ) = [ ( ε ̃ p 0 + μ 0 ε 0 A 23 κ ̃ p 0 ) { μ 0 ε 0 ( B 23 2 ) κ ̃ p 0 1 2 ( ε ̃ p 0 + 2 μ 0 ε 0 A 23 κ ̃ p 0 + A 23 2 μ ̃ p 0 ) } Ω ] I t [ E p x 2 ( t ) ] .
v ̃ e 3 ( Ω ) S ̃ a v 3 ( Ω ) w ̃ t 3 ( Ω ) = A 23 + ( B 23 2 ) Ω [ ( ε ̃ p 0 + μ 0 ε 0 A 23 κ ̃ p 0 ) { μ 0 ε 0 ( B 23 2 ) κ ̃ p 0 1 2 ( ε ̃ p 0 + 2 μ 0 ε 0 A 23 κ ̃ p 0 + A 23 2 μ ̃ p 0 ) } Ω ] a ̂ z .
v ̃ e 3 ( Ω ) = 1 [ { ε ̃ p 0 μ ̃ p 0 μ 0 ε 0 κ ̃ p 0 } + { 1 4 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 μ ̃ p 0 μ ̃ p 0 ) + 1 2 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 + μ ̃ p 0 μ ̃ p 0 ) μ 0 ε 0 κ ̃ p 0 } Ω ] a ̂ z .
v ̃ e 1 ( Ω ) 1 [ { ε ̃ p 0 μ ̃ p 0 + μ 0 ε 0 κ ̃ p 0 } + { 1 4 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 μ ̃ p 0 μ ̃ p 0 ) + 1 2 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 + μ ̃ p 0 μ ̃ p 0 ) + μ 0 ε 0 κ ̃ p 0 } Ω ] a ̂ z .
v ̃ p 3 ( Ω ) 1 [ { ε ̃ p 0 μ ̃ p 0 μ 0 ε 0 κ ̃ p 0 } + { 1 2 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 + μ ̃ p 0 μ ̃ p 0 ) μ 0 ε 0 κ ̃ p 0 } Ω ] a ̂ z .
v g 3 = 1 k ̃ z 3 ω = 1 k ̃ z 3 Ω ,
k ̃ z 3 = ( ω 0 + Ω ) κ ̃ p μ 0 ε 0 + ( ω 0 + Ω ) μ ̃ p ε ̃ p ,
v ̃ g 3 ( Ω ) 1 ε ̃ p μ ̃ p + ( ω 0 + Ω ) Ω ε ̃ p μ ̃ p κ ̃ p μ 0 ε 0 ( ω 0 + Ω ) μ 0 ε 0 κ ̃ p Ω a ̂ z .
v ̃ g 3 ( Ω ) = 1 [ { ε ̃ p 0 μ ̃ p 0 μ 0 ε 0 κ ̃ p 0 + ω 0 2 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 + μ ̃ p 0 μ ̃ p 0 ) ω 0 μ 0 ε 0 κ ̃ p 0 } + { ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 + μ ̃ p 0 μ ̃ p 0 ) 2 μ 0 ε 0 κ ̃ p 0 ω 0 4 ε ̃ p 0 μ ̃ p 0 ( ε ̃ p 0 ε ̃ p 0 μ ̃ p 0 μ ̃ p 0 ) 2 } Ω ] a ̂ z .
v ̃ p 3 ( Ω ) = ω k z 3 a ̂ z 1 ε ̃ p μ ̃ p κ ̃ p μ 0 ε 0 a ̂ z 1 ( ε ̃ p 0 μ ̃ p 0 κ ̃ p 0 μ 0 ε 0 ) + { ε ̃ p 0 μ ̃ p 0 2 ( ε ̃ p 0 ε ̃ p 0 + μ ̃ p 0 μ ̃ p 0 ) μ 0 ε 0 κ ̃ p 0 } Ω a ̂ z .
ε ̃ p r ( ω ) = 1 + ω p 2 ω c 2 ω 2 ,
μ ̃ p r ( ω ) = 1 + ω m 2 ω c 2 ω 2 ,
ξ ̃ p r ( ω ) = α c ω ω c 2 ω 2 ,
n p = n e = c v p 3 = c v e 3 = ε ̃ p r 0 μ ̃ p r 0 κ ̃ p r 0 ,
n g = c v g 3 = c v p 3 + ω 0 [ ε ̃ p r 0 κ ̃ p r 0 ] ,
ε ̃ p r 0 = μ ̃ p r 0 = 1 + β 2 1 ω n 2 ,
κ ̃ p r 0 = α [ ( 1 + β 2 ) ω n ω n 3 ] ( 1 ω n 2 ) 2 ,
ε ̃ p r 0 = 2 ω n β 2 ( 1 ω n 2 ) 2 ,
κ ̃ p r 0 = α [ ( 1 + β 2 ) + 3 β 2 ω n 2 ω n 4 ] ( 1 ω n 2 ) 3 .
ε ̃ p r 0 1 + β 2 ,
κ ̃ p r 0 α ω n ( 1 + β 2 ) ,
ε ̃ p r 0 0 ,
κ ̃ p r 0 α ( 1 + β 2 ) .

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