Abstract

We theoretically analyze inhomogeneous plane waves propagating in a Rochon prism made from uniaxial absorbing material. Expressions for the propagation and attenuation directions of the ordinary and the extraordinary waves in the lossy Rochon prism are derived by complex ray-tracing method. As an application, the propagation characteristic of the inhomogeneous plane waves within the lossy Rochon prisms at normal incidence are analyzed and discussed.

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References

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  1. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, 1999).
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    [Crossref] [PubMed]
  7. W. Wu, F. Wu, M. Shi, F. Su, P. Han, and L. Ma, “A unit structure Rochon prism based on the extraordinary refraction of uniaxial birefringent crystals,” Opt. Express 21(11), 13162–13168 (2013).
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  8. M. Avendaño-Alejo, “Analysis of the refraction of the extraordinary ray in a plane-parallel uniaxial plate with an arbitrary orientation of the optical axis,” Opt. Express 13(7), 2549–2555 (2005).
    [Crossref] [PubMed]
  9. M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane-parallel uniaxial plate,” J. Opt. Soc. Am. A 23(4), 926–932 (2006).
    [Crossref] [PubMed]
  10. L. P. Mosteller and F. Wooten, “Optical properties and reflectance of uniaxial absorbing crystals,” J. Opt. Soc. Am. 58(4), 511–518 (1968).
    [Crossref]
  11. O. E. Piro, “Optical properties, reflectance, and transmittance of anisotropic absorbing crystal plates,” Phys. Rev. B Condens. Matter 36(6), 3427–3435 (1987).
    [Crossref] [PubMed]
  12. F. Bréhat and B. Wyncke, “Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals,” J. Phys. D Appl. Phys. 24(11), 2055–2066 (1991).
    [Crossref]
  13. T. Hasegawa, J. Umemura, and T. Takenaka, “Simple refraction law for uniaxial anisotropic absorbing media,” Appl. Spectrosc. 47(3), 338–340 (1993).
    [Crossref]
  14. R. Echarri and M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbing media,” Pure Appl. Opt. 3(3), 931–941 (1994).
    [Crossref]
  15. H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttg.) 114(3), 109–112 (2003).
    [Crossref]
  16. C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49(9), 1553–1566 (2002).
    [Crossref]
  17. J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernández, 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(6), 1767–1775 (2007).
    [Crossref] [PubMed]
  18. 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(10), 2476–2488 (2008).
    [Crossref] [PubMed]
  19. J. M. Diñeiro, C. Alberdi, B. Hernández, and C. Sáenz, “Uniaxial absorbing media: conditions for refraction in the direction of the optical axis,” J. Opt. Soc. Am. A 30(3), 385–391 (2013).
    [Crossref] [PubMed]
  20. Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25(3), 653–657 (2008).
    [Crossref] [PubMed]
  21. F. Parmigiani, “Some aspects of the reflection and refraction of an electromagnetic wave at an absorbing surface,” Am. J. Phys. 51(3), 245–247 (1983).
    [Crossref]
  22. M. A. Dupertuis, M. Proctor, and B. Acklin, “Generalization of complex Snell-Descartes and Fresnel laws,” J. Opt. Soc. Am. A 11(3), 1159–1166 (1994).
    [Crossref]
  23. P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3–4), 327–341 (2005).
    [Crossref]
  24. L. G. Guimarães and E. E. S. Sampaio, “A note on snell laws for electromagnetic plane waves in lossy media,” J. Quant. Spectrosc. Radiat. Transf. 109(11), 2124–2140 (2008).
    [Crossref]
  25. S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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(9), 1776–1784 (2004).
    [Crossref] [PubMed]

2013 (2)

2008 (3)

2007 (2)

2006 (1)

2005 (2)

M. Avendaño-Alejo, “Analysis of the refraction of the extraordinary ray in a plane-parallel uniaxial plate with an arbitrary orientation of the optical axis,” Opt. Express 13(7), 2549–2555 (2005).
[Crossref] [PubMed]

P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3–4), 327–341 (2005).
[Crossref]

2004 (1)

2003 (1)

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttg.) 114(3), 109–112 (2003).
[Crossref]

2002 (1)

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49(9), 1553–1566 (2002).
[Crossref]

1999 (1)

1995 (1)

1994 (2)

R. Echarri and M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbing media,” Pure Appl. Opt. 3(3), 931–941 (1994).
[Crossref]

M. A. Dupertuis, M. Proctor, and B. Acklin, “Generalization of complex Snell-Descartes and Fresnel laws,” J. Opt. Soc. Am. A 11(3), 1159–1166 (1994).
[Crossref]

1993 (1)

1991 (1)

F. Bréhat and B. Wyncke, “Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals,” J. Phys. D Appl. Phys. 24(11), 2055–2066 (1991).
[Crossref]

1987 (1)

O. E. Piro, “Optical properties, reflectance, and transmittance of anisotropic absorbing crystal plates,” Phys. Rev. B Condens. Matter 36(6), 3427–3435 (1987).
[Crossref] [PubMed]

1983 (1)

F. Parmigiani, “Some aspects of the reflection and refraction of an electromagnetic wave at an absorbing surface,” Am. J. Phys. 51(3), 245–247 (1983).
[Crossref]

1980 (1)

1968 (1)

Acklin, B.

Alberdi, C.

Alfonso, S.

Avendaño-Alejo, M.

Berrogui, M.

Bréhat, F.

F. Bréhat and B. Wyncke, “Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals,” J. Phys. D Appl. Phys. 24(11), 2055–2066 (1991).
[Crossref]

Chang, P. C. Y.

P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3–4), 327–341 (2005).
[Crossref]

Diñeiro, J. M.

Dupertuis, M. A.

Echarri, R.

R. Echarri and M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbing media,” Pure Appl. Opt. 3(3), 931–941 (1994).
[Crossref]

Garea, M. T.

R. Echarri and M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbing media,” Pure Appl. Opt. 3(3), 931–941 (1994).
[Crossref]

Gaylord, T. K.

Greiner, H.

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttg.) 114(3), 109–112 (2003).
[Crossref]

Guimarães, L. G.

L. G. Guimarães and E. E. S. Sampaio, “A note on snell laws for electromagnetic plane waves in lossy media,” J. Quant. Spectrosc. Radiat. Transf. 109(11), 2124–2140 (2008).
[Crossref]

Han, P.

Hasegawa, T.

Hernández, B.

Hopcraft, K. I.

P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3–4), 327–341 (2005).
[Crossref]

Larocca, P. A.

Liang, L.

Ma, L.

Montarou, C. C.

Moreno, I.

Mosteller, L. P.

Parmigiani, F.

F. Parmigiani, “Some aspects of the reflection and refraction of an electromagnetic wave at an absorbing surface,” Am. J. Phys. 51(3), 245–247 (1983).
[Crossref]

Piro, O. E.

O. E. Piro, “Optical properties, reflectance, and transmittance of anisotropic absorbing crystal plates,” Phys. Rev. B Condens. Matter 36(6), 3427–3435 (1987).
[Crossref] [PubMed]

Proctor, M.

Rosete-Aguilar, M.

Sáenz, C.

Sampaio, E. E. S.

L. G. Guimarães and E. E. S. Sampaio, “A note on snell laws for electromagnetic plane waves in lossy media,” J. Quant. Spectrosc. Radiat. Transf. 109(11), 2124–2140 (2008).
[Crossref]

Shi, M.

Simon, M. C.

Stavroudis, O.

Su, F.

Takenaka, T.

Umemura, J.

Walker, J. G.

P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3–4), 327–341 (2005).
[Crossref]

Wang, Y.

White, R.

Wooten, F.

Wu, F.

Wu, L.

Wu, W.

Wyncke, B.

F. Bréhat and B. Wyncke, “Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals,” J. Phys. D Appl. Phys. 24(11), 2055–2066 (1991).
[Crossref]

Xin, H.

Am. J. Phys. (1)

F. Parmigiani, “Some aspects of the reflection and refraction of an electromagnetic wave at an absorbing surface,” Am. J. Phys. 51(3), 245–247 (1983).
[Crossref]

Appl. Opt. (2)

Appl. Spectrosc. (1)

J. Mod. Opt. (1)

C. Alberdi, S. Alfonso, M. Berrogui, J. M. Diñeiro, C. Sáenz, and B. Hernández, “Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media,” J. Mod. Opt. 49(9), 1553–1566 (2002).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

S. Alfonso, C. Alberdi, J. M. Diñeiro, M. Berrogui, B. Hernández, 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(9), 1776–1784 (2004).
[Crossref] [PubMed]

J. M. Diñeiro, M. Berrogui, S. Alfonso, C. Alberdi, B. Hernández, 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(6), 1767–1775 (2007).
[Crossref] [PubMed]

M. Avendaño-Alejo, I. Moreno, and O. Stavroudis, “Minimum deviation angle in uniaxial prisms,” J. Opt. Soc. Am. A 24(8), 2431–2437 (2007).
[Crossref] [PubMed]

Y. Wang, L. Liang, H. Xin, and L. Wu, “Complex ray tracing in uniaxial absorbing media,” J. Opt. Soc. Am. A 25(3), 653–657 (2008).
[Crossref] [PubMed]

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(10), 2476–2488 (2008).
[Crossref] [PubMed]

J. M. Diñeiro, C. Alberdi, B. Hernández, and C. Sáenz, “Uniaxial absorbing media: conditions for refraction in the direction of the optical axis,” J. Opt. Soc. Am. A 30(3), 385–391 (2013).
[Crossref] [PubMed]

M. A. Dupertuis, M. Proctor, and B. Acklin, “Generalization of complex Snell-Descartes and Fresnel laws,” J. Opt. Soc. Am. A 11(3), 1159–1166 (1994).
[Crossref]

M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane-parallel uniaxial plate,” J. Opt. Soc. Am. A 23(4), 926–932 (2006).
[Crossref] [PubMed]

J. Phys. D Appl. Phys. (1)

F. Bréhat and B. Wyncke, “Reflectivity, transmissivity and optical constants of anisotropic absorbing crystals,” J. Phys. D Appl. Phys. 24(11), 2055–2066 (1991).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (2)

P. C. Y. Chang, J. G. Walker, and K. I. Hopcraft, “Ray tracing in absorbing media,” J. Quant. Spectrosc. Radiat. Transf. 96(3–4), 327–341 (2005).
[Crossref]

L. G. Guimarães and E. E. S. Sampaio, “A note on snell laws for electromagnetic plane waves in lossy media,” J. Quant. Spectrosc. Radiat. Transf. 109(11), 2124–2140 (2008).
[Crossref]

Opt. Express (2)

Optik (Stuttg.) (1)

H. Greiner, “Power splitting between refracted ordinary and extraordinary waves in uniaxial crystals with absorption,” Optik (Stuttg.) 114(3), 109–112 (2003).
[Crossref]

Phys. Rev. B Condens. Matter (1)

O. E. Piro, “Optical properties, reflectance, and transmittance of anisotropic absorbing crystal plates,” Phys. Rev. B Condens. Matter 36(6), 3427–3435 (1987).
[Crossref] [PubMed]

Pure Appl. Opt. (1)

R. Echarri and M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbing media,” Pure Appl. Opt. 3(3), 931–941 (1994).
[Crossref]

Other (2)

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

M. Born and E. Wolf, Principles of Optics (Pergamon, 1999).

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Figures (7)

Fig. 1
Fig. 1 Ordinary and extraordinary rays in a lossless Rochon prism. The optical axes of the Rochon prism are such arranged that in the first wedge both of the ordinary and the extraordinary rays propagate along the optical axis at normal incidence, and in the second wedge the light rays propagate perpendicular to the optical axis. The ordinary ray (o ray) in the first wedge will be converted into the extraordinary ray (oe ray) in the second wedge, and the extraordinary ray (e ray) will be converted into the ordinary ray (eo ray), conversely.
Fig. 2
Fig. 2 Ray tracing in a lossy Rochon prism. (a) Ordinary and extraordinary rays in the lossy Rochon prism. (b) Incident and refracted rays at the entrance face of the lossy Rochon prism. (c) Geometry of the interface between the two wedges in the lossy Rochon prism. (d) Incident and refracted rays at the exiting face of the lossy Rochon prism.
Fig. 3
Fig. 3 Plots of the apparent refractive indices ( n m and κ m ) and the refracted angles ( θ m and ψ m ) for various values of wedge angleβ with the complex refractive indices of n ˜ o =1.60.5i and n ˜ e =1.40.5i . Total reflection occurs at the angle of β=arcsin[ n e κ e / n o κ o ]=69 .3 .
Fig. 4
Fig. 4 Plots of the apparent refractive indices ( n m and κ m ) and the refracted angles ( θ m and ψ m ) for various values of wedge angleβ with the complex refractive indices of n ˜ o =1.61.0i and n ˜ e =1.41.0i . Total reflection occurs at the angle of β=arcsin[ n e κ e / n o κ o ]=69 .3 .
Fig. 5
Fig. 5 Plots of the apparent refractive indices ( n m and κ m ) and the refracted angles ( θ m and ψ m ) for various values of wedge angleβ with the complex refractive indices of n ˜ o =1.40.5i and n ˜ e =1.60.5i .
Fig. 6
Fig. 6 Plots of the apparent refractive indices ( n m and κ m ) and the refracted angles ( θ m and ψ m ) for various values of wedge angleβ with the complex refractive indices of n ˜ o =1.41.0i and n ˜ e =1.61.0i .
Fig. 7
Fig. 7 Plots of the apparent refractive indices ( n m and κ m ) and the refracted angles ( θ m and ψ m ) for various values of wedge angleβ with the refractive indices of n ˜ o =1.60.0i and n ˜ e =1.40.0i . Total reflection occurs at the angle of β=arcsin[ n e / n o ]=69 .3 .

Equations (57)

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n i sin θ i = n ˜ sin θ ˜ t ,
K ˜ = ω c n ˜ k ˜ ,
D ˜ = n ˜ c k ˜ × H ˜ ,
μ 0 H ˜ = n ˜ c k ˜ × E ˜ ,
D ˜ k ˜ =0,
H ˜ k ˜ =0,
D ˜ = ε ˜ E ˜ ,
F ˜ =Fexp(i θ F )exp[ iω( t n ˜ c r k ˜ ) ] u ˜ F =Fexp(i θ F )exp[ iω( t 1 c r(n k Re +κ k Im ) ) ]exp[ ω c r(n k Im κ k Re ) ] u ˜ F , =Fexp(i θ F ) P r A t u ˜ F
k Pr = n k Re +κ k Im n 2 k Re 2 + κ 2 k Im 2 ,
k At = n k Im κ k Re n 2 k Im 2 + κ 2 k Re 2 .
α k =arccos( k Pr k At ).
ε ˜ 0 [ n ˜ o 2 0 0 0 n ˜ o 2 0 0 0 n ˜ e 2 ],
k ˜ X 2 1 n ˜ 2 1 n ˜ o 2 + k ˜ Y 2 1 n ˜ 2 1 n ˜ o 2 + k ˜ Z 2 1 n ˜ 2 1 n ˜ e 2 =0.
n ˜ eff ( θ ˜ )= n ˜ o n ˜ e n ˜ e 2 cos 2 θ ˜ + n ˜ o 2 sin 2 θ ˜
S ˜ e = E ˜ e × H ˜ e = S e exp(i θ S e )exp[ i2ω( t n ˜ eff c r s ˜ e ) ] s ˜ e = S e exp(i θ S e )exp[ i2ω( t n ˜ eff cos α ˜ e c r k ˜ e ) ] s ˜ e ,
tan α ˜ e =tan( ξ ˜ θ ˜ )= ( n ˜ o 2 n ˜ e 2 )tan θ ˜ n ˜ e 2 + n ˜ o 2 tan 2 θ ˜ .
k ˜ o =(sin θ ˜ o ,0,cos θ ˜ o ) ,
k ˜ e =(sin θ ˜ e ,0,cos θ ˜ e ) ,
tan θ ˜ e = n ˜ e n i sin θ ˜ i n ˜ o n ˜ e 2 n i 2 sin 2 θ ˜ i
s ˜ e =(sin ξ ˜ e ,0,cos ξ ˜ e ) ,
θ ˜ i o,e = θ ˜ o,e +β ,
θ ˜ oe =arcsin[ n ˜ o n ˜ e sin( θ ˜ o +β)]
θ ˜ eo =arcsin[ n ˜ eff n ˜ o sin( θ ˜ e +β)],
k ˜ oe =(sin θ ˜ oe ,0,cos θ ˜ oe ),
k ˜ eo =(sin θ ˜ eo ,0,cos θ ˜ eo ).
γ ˜ oe =arcsin[ n ˜ e n i sin(β θ ˜ oe )],
γ ˜ eo =arcsin[ n ˜ o n i sin(β θ ˜ eo )],
θ ˜ oe =arcsin[ n ˜ o n ˜ e sin(β)].
Κ ˜ oe = ω c n ˜ e k ˜ oe = ω c n ˜ e ( sin θ ˜ oe 0 cos θ ˜ oe ) = ω c ( n m k Pr i κ m k At )= ω c n m ( sin θ m 0 cos θ m )i ω c κ m ( sin ψ m 0 cos ψ m ),
n m 2 κ m 2 = n e 2 κ e 2 ,
n m κ m cos α k = n e κ e .
n m 2 = ( n e 2 κ e 2 )+( n o 2 + κ o 2 ) sin 2 β 2 + [( n e 2 κ e 2 )( n o 2 κ o 2 ) sin 2 β] 2 +4 [ n e κ e n o κ o sin 2 β] 2 2 ,
κ m 2 = ( n o 2 + κ o 2 ) sin 2 β( n e 2 κ e 2 ) 2 + [( n e 2 κ e 2 )( n o 2 κ o 2 ) sin 2 β] 2 +4 [ n e κ e n o κ o sin 2 β] 2 2 ,
n m sin θ m = n o sinβ,
κ m sin ψ m = κ o sinβ.
A t oe =exp( ω c κ m d 2 cos α k ),
Κ ˜ oe t oe = ω c n i k ˜ oe t oe = ω c n i ( sin γ ˜ oe 0 cos γ ˜ oe ) = ω c ( n m t k Pr t i κ m t k Pr t )= ω c n m t ( sin θ m t 0 cos θ m t )i ω c κ m t ( sin ψ m t 0 cos ψ m t ),
n m t 2 κ m t 2 = n i 2 ,
ψ m t = π 2 + θ m t .
n m t 2 = n i 2 + n m 2 sin 2 (β θ m )+ κ m 2 sin 2 (β ψ m ) 2 + [ n i 2 n m 2 sin 2 (β θ m )+ κ m 2 sin 2 (β ψ m )] 2 +4 n m 2 κ m 2 sin 2 (β θ m ) sin 2 (β ψ m ) 2 ,
κ m t 2 = n m 2 sin 2 (β θ m )+ κ m 2 sin 2 (β ψ m ) n i 2 2 + [ n i 2 n m 2 sin 2 (β θ m )+ κ m 2 sin 2 (β ψ m )] 2 +4 n m 2 κ m 2 sin 2 (β θ m ) sin 2 (β ψ m ) 2 ,
n m t sin θ m t = n m sin(β θ m ),
κ m t sin ψ m t = κ o sin(β ψ m ).
n m = n e 2 + n o 2 sin 2 β 2 + | n e 2 n o 2 sin 2 β | 2 ={ n e , n e 2 > n o 2 sin 2 β n o sinβ, n e 2 n o 2 sin 2 β ,
κ m = n o 2 sin 2 β n e 2 2 + | n e 2 n o 2 sin 2 β | 2 ={ 0, n e 2 > n o 2 sin 2 β n o 2 sin 2 β n e 2 , n e 2 n o 2 sin 2 β ,
θ m ={ arcsin( n o n e sinβ), n e 2 > n o 2 sin 2 β π/2, n e 2 n o 2 sin 2 β ,
ψ m 0.
n m = ( n e 2 κ e 2 )+( n o 2 + κ o 2 ) sin 2 β 2 + | ( n e 2 κ e 2 )( n o 2 κ o 2 ) sin 2 β | 2 , ={ n e 2 κ e 2 + κ o 2 sin 2 β , ( n o 2 κ o 2 ) sin 2 β< n e 2 κ e 2 n o sinβ, ( n o 2 κ o 2 ) sin 2 β n e 2 κ e 2
κ m = ( n o 2 + κ o 2 ) sin 2 β( n e 2 κ e 2 ) 2 + | ( n e 2 κ e 2 )( n o 2 κ o 2 ) sin 2 β | 2 , ={ κ o sinβ, ( n o 2 κ o 2 ) sin 2 β< n e 2 κ e 2 n o 2 sin 2 β( n e 2 κ e 2 ) , ( n o 2 κ o 2 ) sin 2 β n e 2 κ e 2
θ m ={ arcsin[ n o sinβ n e 2 κ e 2 + κ o 2 sin 2 β ], ( n o 2 κ o 2 ) sin 2 β< n e 2 κ e 2 π/2, ( n o 2 κ o 2 ) sin 2 β n e 2 κ e 2 ,
ψ m ={ π/2, ( n o 2 κ o 2 ) sin 2 β< n e 2 κ e 2 arcsin[ κ o sinβ n o 2 sin 2 β( n e 2 κ e 2 ) ], ( n o 2 κ o 2 ) sin 2 β n e 2 κ e 2
{ n m = n o n e κ e / κ o κ m = n e 2 κ e 2 / κ o 2 ( n e 2 κ e 2 ) θ m π/2 ψ m =arcsin[ κ o 2 n e κ e / n o κ o n e 2 κ e 2 κ o 2 ( n e 2 κ e 2 ) ] ,
{ ( n o 2 κ o 2 ) n e κ e ( n e 2 κ e 2 ) n o κ o sinβ= n e κ e / n o κ o .
n m = n i 2 + n e 2 sin 2 (β θ m )+ 2 + | n i 2 n e 2 sin 2 (β θ m ) | 2 ={ n i , n i 2 > n e 2 sin 2 (β θ m ) | n e sin (β θ m ) |, n i 2 n e 2 sin 2 (β θ m ) ,
κ m = n e 2 sin 2 (β θ m ) n i 2 2 + | n i 2 n e 2 sin 2 (β θ m ) | 2 ={ 0, n i 2 > n e 2 sin 2 (β θ m ) n e 2 sin 2 (β θ m ) n i 2 n i 2 n e 2 sin 2 (β θ m ) ,
θ m ={ arcsin[ n e sin(β θ m ) n i ], n i 2 > n e 2 sin 2 (β θ m ) π/2, n i 2 n e 2 sin 2 (β θ m ) ,
ψ m 0.

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