Abstract

In this work we study the anomalous negative refraction when a plane wave is incident from an isotropic medium to a uniaxial absorbing medium. We study the influence of the anisotropy on the refractive indices and the anisotropy on the absorption coefficients in the negative refraction of these materials. Negative refraction can occur for the ordinary and extraordinary waves for a wide range of values of the angle of incidence. The allowed values of the angle of incidence that lead to negative refraction are analyzed in detail as a function of the orientation of the optic axis and the values of the refractive indices and absorption coefficients.

© 2010 Optical Society of America

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  1. Z. Liu, Z. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with an uniaxial media,” Phys. Rev. B 69, 115402 (2004).
    [Crossref]
  2. J.-P. Liu, H.-F. Yau, Z. Ye, C.-H. Kuo, and B. Ke, “Amphoteric refraction exhibited by anisotropic crystals,” Optik (Stuttgart) 118, 147–152 (2007).
    [Crossref]
  3. H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
    [Crossref]
  4. Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
    [Crossref]
  5. W. Q. Zhang and F. Yang, “Negative refraction at various crystal interfaces,” Opt. Commun. 281, 3081–3086 (2008).
    [Crossref]
  6. C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic–anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067–1073 (2006).
    [Crossref]
  7. Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and C.-T. Lin, “Negative refraction in a uniaxial absorbent dielectric material,” Eur. J. Phys. 30, 1381–1390 (2009).
    [Crossref]
  8. H. Luo and Z. Ren, “Polarization-sensitive propagation in an anisotropic metamaterial with double-sheeted hyperboloid dispersion relation,” Opt. Commun. 281, 501–507 (2008).
    [Crossref]
  9. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Experimental verification of a negative index of refraction,” Science 305, 788–792 (2004).
    [Crossref] [PubMed]
  10. D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
    [Crossref] [PubMed]
  11. P. Hillion, “Electromagnetic wave reflection from right- and left-handed anisotropic materials,” Phys. Scr. 80, 065015 (2009).
    [Crossref]
  12. Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25, 653–657 (2008).
    [Crossref]
  13. C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
    [Crossref]
  14. S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in absorbing isotropic media,” J. Opt. Soc. Am. A 21, 1776–1784 (2004).
    [Crossref]
  15. H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109–112 (2003).
    [Crossref]
  16. C. Alberdi, S. Alfonso, J. M. Diñeiro, B. Hernández, and C. Sáenz, “Interface between transparent isotropic and uniaxial absorbing dielectric media: equations for ray tracing and for the direction of propagation,” J. Opt. Soc. Am. A 25, 2476–2488 (2008).
    [Crossref]
  17. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  18. J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media,” J. Opt. Soc. Am. A 24, 1767–1775 (2007).
    [Crossref]

2009 (2)

Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and C.-T. Lin, “Negative refraction in a uniaxial absorbent dielectric material,” Eur. J. Phys. 30, 1381–1390 (2009).
[Crossref]

P. Hillion, “Electromagnetic wave reflection from right- and left-handed anisotropic materials,” Phys. Scr. 80, 065015 (2009).
[Crossref]

2008 (4)

Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25, 653–657 (2008).
[Crossref]

C. Alberdi, S. Alfonso, J. M. Diñeiro, B. Hernández, and C. Sáenz, “Interface between transparent isotropic and uniaxial absorbing dielectric media: equations for ray tracing and for the direction of propagation,” J. Opt. Soc. Am. A 25, 2476–2488 (2008).
[Crossref]

H. Luo and Z. Ren, “Polarization-sensitive propagation in an anisotropic metamaterial with double-sheeted hyperboloid dispersion relation,” Opt. Commun. 281, 501–507 (2008).
[Crossref]

W. Q. Zhang and F. Yang, “Negative refraction at various crystal interfaces,” Opt. Commun. 281, 3081–3086 (2008).
[Crossref]

2007 (2)

2006 (1)

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic–anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067–1073 (2006).
[Crossref]

2005 (2)

H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
[Crossref]

Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
[Crossref]

2004 (3)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Experimental verification of a negative index of refraction,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Z. Liu, Z. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with an uniaxial media,” Phys. Rev. B 69, 115402 (2004).
[Crossref]

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernandez, and C. Sáenz, “Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in absorbing isotropic media,” J. Opt. Soc. Am. A 21, 1776–1784 (2004).
[Crossref]

2003 (1)

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109–112 (2003).
[Crossref]

2002 (1)

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

2000 (1)

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[Crossref] [PubMed]

Alberdi, C.

Alfonso, S.

Bellver-Cebreros, C.

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic–anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067–1073 (2006).
[Crossref]

Berrogui, M.

Chui, S. T.

Z. Liu, Z. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with an uniaxial media,” Phys. Rev. B 69, 115402 (2004).
[Crossref]

Diñeiro, J. M.

Greiner, H.

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109–112 (2003).
[Crossref]

Hernandez, B.

Hernández, B.

Hillion, P.

P. Hillion, “Electromagnetic wave reflection from right- and left-handed anisotropic materials,” Phys. Scr. 80, 065015 (2009).
[Crossref]

Hu, W.

H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
[Crossref]

Jen, Y. -J.

Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and C.-T. Lin, “Negative refraction in a uniaxial absorbent dielectric material,” Eur. J. Phys. 30, 1381–1390 (2009).
[Crossref]

Ke, B.

J.-P. Liu, H.-F. Yau, Z. Ye, C.-H. Kuo, and B. Ke, “Amphoteric refraction exhibited by anisotropic crystals,” Optik (Stuttgart) 118, 147–152 (2007).
[Crossref]

Kroll, N.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[Crossref] [PubMed]

Kuo, C. -H.

J.-P. Liu, H.-F. Yau, Z. Ye, C.-H. Kuo, and B. Ke, “Amphoteric refraction exhibited by anisotropic crystals,” Optik (Stuttgart) 118, 147–152 (2007).
[Crossref]

Lakhtakia, A.

Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and C.-T. Lin, “Negative refraction in a uniaxial absorbent dielectric material,” Eur. J. Phys. 30, 1381–1390 (2009).
[Crossref]

Liang, L.

Lin, C. -T.

Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and C.-T. Lin, “Negative refraction in a uniaxial absorbent dielectric material,” Eur. J. Phys. 30, 1381–1390 (2009).
[Crossref]

Lin, Z.

Z. Liu, Z. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with an uniaxial media,” Phys. Rev. B 69, 115402 (2004).
[Crossref]

Liu, H.

H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
[Crossref]

Liu, J. -P.

J.-P. Liu, H.-F. Yau, Z. Ye, C.-H. Kuo, and B. Ke, “Amphoteric refraction exhibited by anisotropic crystals,” Optik (Stuttgart) 118, 147–152 (2007).
[Crossref]

Liu, Z.

Z. Liu, Z. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with an uniaxial media,” Phys. Rev. B 69, 115402 (2004).
[Crossref]

Lu, Y.

Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
[Crossref]

Luo, H.

H. Luo and Z. Ren, “Polarization-sensitive propagation in an anisotropic metamaterial with double-sheeted hyperboloid dispersion relation,” Opt. Commun. 281, 501–507 (2008).
[Crossref]

H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
[Crossref]

Ming, H.

Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
[Crossref]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Experimental verification of a negative index of refraction,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Ren, Z.

H. Luo and Z. Ren, “Polarization-sensitive propagation in an anisotropic metamaterial with double-sheeted hyperboloid dispersion relation,” Opt. Commun. 281, 501–507 (2008).
[Crossref]

Rodriguez-Danta, M.

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic–anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067–1073 (2006).
[Crossref]

Sáenz, C.

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Experimental verification of a negative index of refraction,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[Crossref] [PubMed]

Wang, P.

Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
[Crossref]

Wang, Y.

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Experimental verification of a negative index of refraction,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Wu, L.

Xie, J.

Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
[Crossref]

Xin, H.

Yang, F.

W. Q. Zhang and F. Yang, “Negative refraction at various crystal interfaces,” Opt. Commun. 281, 3081–3086 (2008).
[Crossref]

Yao, P.

Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yau, H. -F.

J.-P. Liu, H.-F. Yau, Z. Ye, C.-H. Kuo, and B. Ke, “Amphoteric refraction exhibited by anisotropic crystals,” Optik (Stuttgart) 118, 147–152 (2007).
[Crossref]

Ye, Z.

J.-P. Liu, H.-F. Yau, Z. Ye, C.-H. Kuo, and B. Ke, “Amphoteric refraction exhibited by anisotropic crystals,” Optik (Stuttgart) 118, 147–152 (2007).
[Crossref]

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yi, X.

H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
[Crossref]

Yu, C. -W.

Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and C.-T. Lin, “Negative refraction in a uniaxial absorbent dielectric material,” Eur. J. Phys. 30, 1381–1390 (2009).
[Crossref]

Zhang, W. Q.

W. Q. Zhang and F. Yang, “Negative refraction at various crystal interfaces,” Opt. Commun. 281, 3081–3086 (2008).
[Crossref]

Zhu, J.

H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
[Crossref]

Eur. J. Phys. (1)

Y.-J. Jen, A. Lakhtakia, C.-W. Yu, and C.-T. Lin, “Negative refraction in a uniaxial absorbent dielectric material,” Eur. J. Phys. 30, 1381–1390 (2009).
[Crossref]

J. Mod. Opt. (1)

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernandez, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49, 1553–1566 (2002).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

C. Bellver-Cebreros and M. Rodriguez-Danta, “Amphoteric refraction at the isotropic–anisotropic biaxial media interface: an alternative treatment,” J. Opt. A, Pure Appl. Opt. 8, 1067–1073 (2006).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Commun. (4)

H. Luo and Z. Ren, “Polarization-sensitive propagation in an anisotropic metamaterial with double-sheeted hyperboloid dispersion relation,” Opt. Commun. 281, 501–507 (2008).
[Crossref]

H. Luo, W. Hu, X. Yi, H. Liu, and J. Zhu, “Amphoteric refraction at the interface between isotropic and anisotropic media,” Opt. Commun. 254, 353–360 (2005).
[Crossref]

Y. Lu, P. Wang, P. Yao, J. Xie, and H. Ming, “Negative refraction at the interface of uniaxial anisotropic media,” Opt. Commun. 246, 429–435 (2005).
[Crossref]

W. Q. Zhang and F. Yang, “Negative refraction at various crystal interfaces,” Opt. Commun. 281, 3081–3086 (2008).
[Crossref]

Optik (Stuttgart) (2)

J.-P. Liu, H.-F. Yau, Z. Ye, C.-H. Kuo, and B. Ke, “Amphoteric refraction exhibited by anisotropic crystals,” Optik (Stuttgart) 118, 147–152 (2007).
[Crossref]

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttgart) 114, 109–112 (2003).
[Crossref]

Phys. Rev. B (1)

Z. Liu, Z. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with an uniaxial media,” Phys. Rev. B 69, 115402 (2004).
[Crossref]

Phys. Rev. Lett. (1)

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[Crossref] [PubMed]

Phys. Scr. (1)

P. Hillion, “Electromagnetic wave reflection from right- and left-handed anisotropic materials,” Phys. Scr. 80, 065015 (2009).
[Crossref]

Science (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Experimental verification of a negative index of refraction,” Science 305, 788–792 (2004).
[Crossref] [PubMed]

Other (1)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

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Figures (7)

Fig. 1
Fig. 1

A plane wave falls with an angle of incidence α o from an isotropic medium with index n o to a uniaxial absorbing medium with indices n ω and n ε : X m , Y m , and Z m are the coordinate system in the interface; X Y Z is the coordinate system of principal axis; and X π , Y π , and Z π are defined so that Y π Z π is the plane of incidence π.

Fig. 2
Fig. 2

Normal surfaces for the refraction in negative uniaxial media ( n ω > n ε ) with the optic axis contained in the plane of incidence: (a) The optic axis forms a negative angle; negative refraction is not possible. (b) The optic axis forms a positive angle.

Fig. 3
Fig. 3

Normal surfaces for the refraction in positive uniaxial media ( n ω < n ε ) with the optic axis contained in the plane of incidence: (a) The optic axis forms a negative angle. (b) The optic axis forms a positive angle; negative refraction is not possible.

Fig. 4
Fig. 4

The limiting angle α l as a function of the orientation of the optic axis γ for transparent media with n ω = 2 and different values of n ε in the range 1.2–2.8.

Fig. 5
Fig. 5

The limiting angle α l as a function of the orientation of the optic axis γ for absorbing media with n ω = 2 and n ε = 2.4 : (a) k ω = 0. 2 ̣ and k ε is between 0.3 and 1.5; (b) k ω = 1.5 and k ε is between 1.6 and 1.9.

Fig. 6
Fig. 6

The limiting angle α l as a function of the orientation of the optic axis γ for absorbing media with n ω = 2 and n ε = 1.6 : (a) k ε = 0. 2 ̣ and k ω is between 0.3 and 1.5; (b) k ε = 1.5 and k ω is between 1.6 and 1.9.

Fig. 7
Fig. 7

The limiting angle α l as a function of the orientation of the optic axis: (a) n ω = 2 , n ε = 1.6 , k ω = 0. 2 ̣ , and k ε is between 0.3 and 1.5; (b) n ω = 2 , n ε = 2.4 , k ε = 0.2 , and k ω is between 0.3 and 1.5.

Equations (30)

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( v x v y v z ) = ( cos   γ 0 sin   γ 0 1 0 sin   γ 0 cos   γ ) ( v x m v y m v z m ) .
sin   α 1 = n 1 n o ( n ω 2 n ε 2 ) sin   γ   cos   γ n ω 4 cos 2 γ + n ε 4 sin 2 γ .
F ̃ = F u ̃ F   exp [ i ω ( t n ̃ c r s ̃ ) ] exp ( i θ F ) = F u ̃ F   exp [ ω c n r ( χ s R s I ) ] exp [ i ω ( t n c r ( s R + χ s I ) ) ] exp ( i θ F ) = F u ̃ F   exp ( i θ F ) A t P r .
s o = ( sin   α o   sin   δ , sin   α o cos   δ , cos   α o ) .
s ̃ 1 , 2 = ( n o n ̃ 1 , 2 sin   α o   sin   δ , n o n ̃ 1 , 2 sin   α o   cos   δ , 1 n o 2 n ̃ 1 , 2 2 sin 2 α o ) ,
( s R + χ s I ) 1 , 2 = ( n o n ̃ 1 , 2 sin   α o   sin   δ , n o n ̃ 1 , 2 sin   α o   cos   δ , a ) ,
a = 1 2 { [ 1 χ 1 , 2 2 n o 2 sin 2 α o n 1 , 2 2 ] 2 + 4 χ 1 , 2 2 + [ 1 χ 1 , 2 2 n o 2 sin 2 α o n 1 , 2 2 ] } .
s o = ( sin   α o , 0 , cos   α o ) ,
( s R + χ s I ) 1 , 2 = ( n o ñ 1 , 2 sin   α o   sin   δ , 0 , a ) .
Io = Re [ exp ( i β ω ) ( s ̃ x y [ u R x y u x i u I x y u y ] + s z ( u R x y 2 + u I x y 2 ) u z ) ] ,
Ie = Re [ 1 n ̃ 1 ( { s ̃ z n ̃ ω 2 } { u R x y 2 + u I x y 2 } u z + u ̃ xy { s x y n ̃ ε 2 } ) ] ,
s ̃ = s ̃ x y u ̃ xy + s ̃ z u z .
1 n ̃ 1 2 = 1 s ̃ 1 z 2 n ̃ ε 2 + s ̃ 1 z 2 n ̃ ω 2 = 1 n ̃ ε 2 ,
1 n ̃ 2 2 = 1 n ̃ ω 2 .
s ̃ = s ̃ xy = u xy ,
s ̃ R = u Rxy = u R x y u x ,     s ̃ I = u Ixy = u I x y u y ,
Io = Re [ exp ( i β ω ) ( s ̃ x y [ u R x y u x i u I x y u y ] ) ] = Re [ n ω | n ̃ ω | ( 1 i χ ω ) ( [ u R x y u x i u I x y u y ] ) ] = n ω | n ̃ ω | [ u R x y u x χ ω u I x y u y ] = n ω | n ̃ ω | ( s R χ ω s I ) ,
Ie = Re [ 1 n ̃ 1 ( u ̃ xy { 1 n ̃ ε 2 } ) ] = 1 | n ̃ ε | 4 Re [ exp ( i β ε ) s ̃ ] = n ε | n ̃ ε | 4 ( s R + χ ε s I ) ,
n ̃ ε , ω = | n ̃ ε , ω | exp ( i β ε , ω ) = n ε , ω ( 1 i χ ε , ω ) .
s ̃ 1 , 2 = ( cos   γ n o n ̃ 1 , 2 sin   α o + sin   γ 1 n o 2 n ̃ 1 , 2 2 sin 2 α o , 0 , sin   γ n o n ̃ 1 , 2 sin   α o cos   γ 1 n o 2 n ̃ 1 , 2 2 sin 2 α o ) ,
s ̃ x, y 1 , 2 = ( cos   γ n o n ̃ 1 , 2 sin   α o + sin   γ 1 n o 2 n ̃ 1 , 2 2 sin 2 α o , 0 , 0 ) .
u I x y 1 , 2 = 0 ,
u R x y 1 , 2 = ( 1 , 0 , 0 ) = u x ,
s ̃ x , y 1 , 2 = ( cos   γ n o n ̃ 1 , 2 sin   α o + sin   γ 1 n o 2 n ̃ 1 , 2 2 sin 2 α o , 0 , 0 ) = s x 1 , 2 ,
s ̃ z 1 , 2 = sin   γ n o n ̃ 1 , 2 sin   α o cos   γ 1 n o 2 n ̃ 1 , 2 2 sin 2 α o = s ̃ z 1 , 2 ,
Io = Re [ exp ( i β ω ) ( s ̃ x y u x s ̃ z u z ) ] = Re [ exp ( i β ω ) s ̃ ] = n ω | n ̃ ω | ( s R + χ ω s I ) ,
Ie = Re [ 1 n ̃ 1 s ̃ z ( n ̃ ω 2 ) ] u z + Re [ 1 n ̃ 1 s ̃ x ( n ̃ ε 2 ) ] u x .
I o y = n ω | n ̃ ω | ( s R χ ω s I ) y = n ω | n ̃ ω | ( 1 χ ω 2 ) n o   sin   α o 1 + χ ω 2 .
Ie = ( cos   γ   Re [ 1 n ̃ 1 s ̃ x ( n ̃ ε 2 ) ] + sin   γ   Re [ 1 n ̃ 1 s ̃ z ( n ̃ ω 2 ) ] , 0 , sin   γ   Re [ 1 n ̃ 1 s ̃ x ( n ̃ ε 2 ) ] + cos   γ   Re [ 1 n ̃ 1 s ̃ z ( n ̃ ω 2 ) ] ) .
cos   γ   Re [ 1 n ̃ 1 s ̃ x ( n ̃ ε 2 ) ] + sin   γ   Re [ 1 n ̃ 1 s ̃ z ( n ̃ ω 2 ) ] = 0.

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