Abstract

Electrodynamics in rotating optical elements has attracted much interest due to its potential application to ultra-sensitive rotating sensing. And it is important to investigate the Sagnac effect in some novel photonic structures for it may lead to a variety of unusual manifestations. We propose a Finite-Difference Time-Domain (FDTD) method to model the Sagnac effect, which is based on the modified constitutive relation in rotating frame. The time-stepping expressions for the FDTD routine are derived and discussed, and the classical Sagnac phase shift along a waveguide is calculated. Further discussions about numerical dispersion, dielectric boundary condition and perfect matched layer (PML) absorbing boundary conditions in the rotating FDTD model are also presented respectively. The theoretical analysis and simulation results prove that the numerical algorithm can analyze the Sagnac effect effectively, and can be applied to general cases with various material properties and complex geometric structures. The proposed algorithm provides a promising systematic tool to study the properties of rotating optical elements, and to accurately analyze, design and optimize rotation sensitive optical devices.

© 2008 Optical Society of America

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  1. U. Leonhardt and P. Piwnitski, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A 62, 055801 (2000).
    [Crossref]
  2. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
    [Crossref]
  3. J. Scheuer and A. Yariv, “Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures,” Phys. Rev. Lett.. 96, 053901 (2006).
    [Crossref] [PubMed]
  4. C. Peng, Z. Li, and A. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express 15, 3864–3875 (2007).
    [Crossref] [PubMed]
  5. B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E. 71, 056621 (2005).
    [Crossref]
  6. B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. 24, 142–151 (2006).
    [Crossref]
  7. S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801 (2006).
    [Crossref]
  8. S. Sunada and T. Harayama, “Design of resonant microcavities: application to optical gyroscopes,” Opt. Express 15, 16245–16254 (2007).
    [Crossref] [PubMed]
  9. C. Peng, Z. Li, and A. Xu, “Rotation sensing based on a slow-light resonating structure with high group dispersion,” Appl. Opt. 46, 4125–4131 (2007).
    [Crossref] [PubMed]
  10. B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007).
    [Crossref]
  11. E. J. Post, “Sagnac Effect,” Rev. Mod. Phys. 39, 475–493(1967).
    [Crossref]
  12. B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
    [Crossref]
  13. T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
    [Crossref]
  14. J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. 181, 1765–1775 (1969).
    [Crossref]
  15. J. Van Bladel, “Relativity and Engineering”, Springer, Berlin, (1984)
  16. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic,” J. Computational Physics 114185–200 (1994).
  17. A. Taflove, “Computational Electrodynamics: The Finite-Difference Time-Domain Method,” Artech House, Boston, (1995).
  18. D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE,  4268–270, (1994).
    [Crossref]
  19. R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. 93, 143901 (2004).
    [Crossref] [PubMed]
  20. V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756–6769 (1997).
    [Crossref]

2007 (4)

2006 (4)

B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. 24, 142–151 (2006).
[Crossref]

S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801 (2006).
[Crossref]

B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[Crossref]

J. Scheuer and A. Yariv, “Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures,” Phys. Rev. Lett.. 96, 053901 (2006).
[Crossref] [PubMed]

2005 (1)

B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E. 71, 056621 (2005).
[Crossref]

2004 (2)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[Crossref]

R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. 93, 143901 (2004).
[Crossref] [PubMed]

2000 (1)

U. Leonhardt and P. Piwnitski, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A 62, 055801 (2000).
[Crossref]

1997 (1)

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

1994 (2)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic,” J. Computational Physics 114185–200 (1994).

D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE,  4268–270, (1994).
[Crossref]

1973 (1)

T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
[Crossref]

1969 (1)

J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. 181, 1765–1775 (1969).
[Crossref]

1967 (1)

E. J. Post, “Sagnac Effect,” Rev. Mod. Phys. 39, 475–493(1967).
[Crossref]

Anderson, J. L.

J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. 181, 1765–1775 (1969).
[Crossref]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic,” J. Computational Physics 114185–200 (1994).

Boag, A.

B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007).
[Crossref]

B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[Crossref]

B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. 24, 142–151 (2006).
[Crossref]

Harayama, T.

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[Crossref]

Katz, D. S.

D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE,  4268–270, (1994).
[Crossref]

Leonhardt, U.

U. Leonhardt and P. Piwnitski, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A 62, 055801 (2000).
[Crossref]

Li, Z.

Maleki, L.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[Crossref]

Mandelshtam, V. A.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[Crossref]

Peng, C.

Piwnitski, P.

U. Leonhardt and P. Piwnitski, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A 62, 055801 (2000).
[Crossref]

Post, E. J.

E. J. Post, “Sagnac Effect,” Rev. Mod. Phys. 39, 475–493(1967).
[Crossref]

Ryon, J. W.

J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. 181, 1765–1775 (1969).
[Crossref]

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[Crossref]

Scheuer, J.

Shamir, A.

B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[Crossref]

Shiozawa, T.

T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
[Crossref]

Steinberg, B. Z.

B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007).
[Crossref]

B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. 24, 142–151 (2006).
[Crossref]

B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[Crossref]

B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E. 71, 056621 (2005).
[Crossref]

Sunada, S.

Taflove, A.

D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE,  4268–270, (1994).
[Crossref]

A. Taflove, “Computational Electrodynamics: The Finite-Difference Time-Domain Method,” Artech House, Boston, (1995).

Taylor, H. S.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

Thiele, E. T.

D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE,  4268–270, (1994).
[Crossref]

Van Bladel, J.

J. Van Bladel, “Relativity and Engineering”, Springer, Berlin, (1984)

Wang, R.

R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. 93, 143901 (2004).
[Crossref] [PubMed]

Xu, A.

Yao, A.

R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. 93, 143901 (2004).
[Crossref] [PubMed]

Yariv, A.

J. Scheuer and A. Yariv, “Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures,” Phys. Rev. Lett.. 96, 053901 (2006).
[Crossref] [PubMed]

Zheng, Y.

R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. 93, 143901 (2004).
[Crossref] [PubMed]

Appl. Opt. (1)

J. Chem. Phys. (1)

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Opt. Soc. Am. B. (1)

B. Z. Steinberg and A. Boag “Splitting of microcavity degenerate modes in rotating photonic crystals-the miniature optical gyroscopes,” J. Opt. Soc. Am. B. 24, 142–151 (2006).
[Crossref]

Microwave and Guided Wave Letters, IEEE (1)

D. S. Katz, E. T. Thiele, and A. Taflove, “Validation and extension to three dimensions of the Berenger PM-Labsorbing boundary condition for FD-TD meshes,” Microwave and Guided Wave Letters, IEEE,  4268–270, (1994).
[Crossref]

Opt. Commun. (1)

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[Crossref]

Opt. Express (2)

Phys. Rev. (1)

J. L. Anderson and J. W. Ryon, “Electromagnetic radiation in accelerated systems,” Phys. Rev. 181, 1765–1775 (1969).
[Crossref]

Phys. Rev. A (2)

U. Leonhardt and P. Piwnitski, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A 62, 055801 (2000).
[Crossref]

S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801 (2006).
[Crossref]

Phys. Rev. E (1)

B. Z. Steinberg, A. Shamir, and A. Boag, “Two-dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[Crossref]

Phys. Rev. E. (1)

B. Z. Steinberg, “Rotating photonic crystals: A medium for compact optical gyroscopes,” Phys. Rev. E. 71, 056621 (2005).
[Crossref]

Phys. Rev. Lett. (1)

R. Wang, Y. Zheng, and A. Yao “Generalized Sagnac Effect,” Phys. Rev. Lett. 93, 143901 (2004).
[Crossref] [PubMed]

Phys. Rev. Lett.. (1)

J. Scheuer and A. Yariv, “Sagnac Effect in Coupled-Resonator Slow-Light Waveguide Structures,” Phys. Rev. Lett.. 96, 053901 (2006).
[Crossref] [PubMed]

Proc. IEEE (1)

T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
[Crossref]

Rev. Mod. Phys. (1)

E. J. Post, “Sagnac Effect,” Rev. Mod. Phys. 39, 475–493(1967).
[Crossref]

Other (3)

J. Van Bladel, “Relativity and Engineering”, Springer, Berlin, (1984)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic,” J. Computational Physics 114185–200 (1994).

A. Taflove, “Computational Electrodynamics: The Finite-Difference Time-Domain Method,” Artech House, Boston, (1995).

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Figures (3)

Fig. 1.
Fig. 1.

2D Yee lattice

Fig. 2.
Fig. 2.

The field diagram for Hz

Fig. 3.
Fig. 3.

The Sagnac phase shift vs. distance. The reference position is x=15 and y=25. The solid line is the theoretic result from Eq. 3.1, and the marked points are simulation results from the FDTD program.

Equations (65)

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D = ε E c 2 Ω × r × H
B = μ H + c 2 Ω × r × E
D t = × H J J = J s + σ E
B t = × E M M = M s + σ m H
H x t = 1 μ [ E y z E z y ( M sx + σ m H x ) c 2 x Ω E z t ]
H y t = 1 μ [ E z x E x z ( M sy + σ m H y ) c 2 y Ω E z t ]
H z t = 1 μ [ E x y E y x ( M sz + σ m H z ) + c 2 y Ω E z t + c 2 x Ω E x t ]
E x t = 1 ε [ H z y H y z ( J sx + σ E x ) + c 2 x Ω H z t ]
E y t = 1 ε [ H x y H z x ( J sy + σ E y ) + c 2 y Ω H z t ]
E z t = 1 ε [ H y x H x y ( J sz + σ E z ) c 2 y Ω H y t c 2 x Ω H x t ]
E x t + σ ε E x = 1 ε H z y + c 2 x Ω ε μ ( E x y E y x σ m H z )
E y t + σ ε E y = 1 ε H z x + c 2 y Ω ε μ ( E x y E y x σ m H z )
H z t + σ m μ H z = 1 μ E x y 1 μ E y x + c 2 y Ω ε μ ( H z x σ E y )
+ c 2 x Ω ε μ ( H z y σ E x )
H zx t + σ m μ H zx = 1 μ E y x + c 2 y Ω ε μ ( H z x σ E y )
H zy t + σ m μ H zy = 1 μ E x y + c 2 x Ω ε μ ( H z y σ E x )
E x i , j + 1 2 n + 1 2 = exp ( σ i , j + 1 2 Δ t ε i , j + 1 2 ) E x | i , j + 1 2 n 1 2 1 exp ( σ i , j + 1 2 Δ t ε i , j + 1 2 ) Δ y σ i , j + 1 2 H z | i , j + 1 2 n
+ 1 exp ( σ i , j + 1 2 Δ t ε i , j + 1 2 ) σ i , j + 1 2 c 2 x Ω μ i , j + 1 2 ( E x y E y x σ m H z ) i , j + 1 2 n
E y i + 1 2 , j n + 1 2 = exp ( σ i + 1 2 , j Δ t ε i + 1 2 , j ) E y | i + 1 2 , j n 1 2 1 exp ( σ i + 1 2 , j Δ t ε i + 1 2 , j ) Δ x σ i + 1 2 , j H z | i + 1 2 , 2 j n
1 exp ( σ i + 1 2 , j Δ t ε i + 1 2 , j ) σ i + 1 2 , j c 2 y Ω μ i + 1 2 , j ( E x y E y x σ m H z ) i + 1 2 , j n
H zx i , j n + 1 = exp ( σ m ; i , j Δ t μ i , j ) H zx | i , j n 1 exp ( σ m ; i , j Δ t μ i , j ) Δ x σ m ; i , j E y | i , j n + 1 2
1 exp ( σ m ; i , j Δ t μ i , j ) σ m ; i , j c 2 y Ω ε i , j ( H z x σ E y ) i , j n + 1 2
H zy i , j n + 1 = exp ( σ m ; i , j Δ t μ i , j ) H zy | i , j n 1 exp ( σ m ; i , j Δ t μ i , j ) Δ y σ m ; i , j E x | i , j n + 1 2
+ 1 exp ( σ m ; i , j Δ t μ i , j ) σ m ; i , j c 2 x Ω ε i , j ( H z y σ E x ) i , j n + 1 2
lim σ 0 = 1 exp ( σ Δ t ε ) σ = Δ t ε , lim σ m 0 1 exp ( σ m Δ t m u ) σ m = Δ t μ
Δ ϕ = 2 π v · Δ l c λ
E x t = 1 ε H z y + V y n 2 ( E x y E y x )
E y t = 1 ε H z x V x n 2 ( E x y E y x )
H z t = 1 μ E x y 1 μ E y x + V x n 2 H z x + V y n 2 H z y
E x | i , j + 1 2 n + 1 2 E x | i , j + 1 2 n 1 2 Δ t = 1 ε H z i , j + 1 n H z i , j n Δ y
+ V y n 2 ( E x | i , j + 1 n E x | i , j n Δ y E y | i + 1 2 , j + 1 2 n E x | i 1 2 , j + 1 2 n Δ x )
F | I , J n = F 0 exp [ j ( ω Δ t k ~ x I Δ x k ~ y J Δ y ) ] where F = E x , E y , H z
E x 0 = Δ t H z 0 ε Δ y · sin ( k ~ y Δ y 2 ) sin ( ω Δ t 2 ) Δ t V y E x 0 n 2 Δ y · sin ( k ~ y Δ y 2 ) sin ( ω Δ t 2 ) + Δ t V y E y 0 n 2 Δ x · sin ( k ~ x Δ y 2 ) sin ( ω Δ t 2 )
E y 0 = Δ t H z 0 ε Δ x · sin ( k ~ x Δ x 2 ) sin ( ω Δ t 2 ) + Δ t V x E x 0 n 2 Δ y · sin ( k ~ y Δ y 2 ) sin ( ω Δ t 2 ) Δ t V x E y 0 n 2 Δ x · sin ( k ~ x Δ x 2 ) sin ( ω Δ t 2 )
H z 0 = Δ t E x 0 μ Δ y · sin ( k ~ y Δ y 2 ) sin ( ω Δ t 2 ) + Δ t E y 0 μ Δ x · sin ( k ~ x Δ x 2 ) sin ( ω Δ t 2 )
Δ t V x H z 0 n 2 Δ x · sin ( k ~ x Δ x 2 ) sin ( ω Δ t 2 ) Δ t V y H z 0 n 2 Δ y · sin ( k ~ y Δ y 2 ) sin ( ω Δ t 2 )
[ n c Δ t sin ( ω Δ t 2 ) ] 2 = [ 1 G Δ x sin ( k ~ x Δ x 2 ) ] 2 + [ 1 G Δ y sin ( k ~ y Δ y 2 ) ] 2
where G = 1 + Δ t V x n 2 Δ x · sin ( k ~ x Δ x 2 ) sin ( ω Δ t 2 ) + Δ t V y n 2 Δ y · sin ( k ~ y Δ y 2 ) sin ( ω Δ t 2 )
F | I , J n = F 0 exp [ j ( ω Δ t k x I Δ x k y J Δ y Δ ϕ ) ]
Δ ϕ = 2 π λ c ( k k · V ) ( k k · Δ r )
= 1 n 2 ω ( k x V x + k y V y ) ( k x I Δ x + k y J Δ y )
k ~ x = G 0 k x , k ~ y = G 0 k y where G 0 = ( 1 + 1 n 2 ω k x V x + 1 n 2 ω k y V y )
( n ω c ) 2 = ( k ~ x G 0 ) 2 + ( k ~ y G 0 ) 2
n · ( D 2 D 1 ) = n · ( B 2 B 1 ) = 0
n × ( E 2 E 1 ) = 0 n × ( H 2 H 1 ) = K
n · D = n · ( ε E ) c 2 n · ( Ω × r × H )
n · B = n · ( μ H ) + c 2 n · ( Ω × r × E )
( Ω × r × H ) · d S = 0 ; ( Ω × r × E ) · d S = 0 ;
n · ( ε E 2 ε E 1 ) = n · ( μ H 2 μ H 2 ) = 0
j ω ε s y E ~ x = H ~ z y + j ω c 2 V y H ~ z
j ω ε s x E ~ y = H ~ z x j ω c 2 V x H ~ z
j ω μ s x * H zx ~ = E y ~ x j ω c 2 V x E y ~
j ω μ s y * H zy ~ = E x ~ y j ω c 2 V y E x ~
s w = 1 + σ w j ω ε ; s w * = 1 + σ w * j ω μ ; where w = x , y
1 s x s x * 2 H ~ z x 2 + 1 s y s y * 2 H ~ z y 2 + j ω c 2 V x s x s x * H ~ z x + j ω c 2 V y s y s y * H ~ z y + ω 2 ε μ H ~ z = 0
H ~ z = H 0 exp ( j s x s x * k x ~ x ) · exp ( j s y s y * k y ~ y )
k ~ x 2 + k ~ X 2 2 ω c 2 V x k ~ x s x s x * 2 ω c 2 V y k ~ y s y s y * = ( n ω c ) 2
H 2 z ~ = H 0 τ · exp ( j s 2 x s 2 x * k 2 x ~ x ) · exp ( j s y s y * k 2 y ~ y )
E 2 x ~ = H 0 τ [ k 2 y ~ ω ε 2 s 2 x * s 2 y + V y ε 2 s 2 y c 2 ] · exp ( j s 2 x s 2 x * k 2 x ~ x ) · exp ( j s 2 y s 2 y * k 2 y ~ y )
E 2 y ~ = H 0 τ [ k 2 x ~ ω ε 2 s 2 x * s 2 x V x ε 2 s 2 x c 2 ] · exp ( j s 2 x s 2 x * k 2 x ~ x ) · exp ( j s 2 y s 2 y * k 2 y ~ y )
H 1 z ~ = H 0 [ 1 + Γ exp ( 2 j k 1 x ) ] · exp ( j k 1 x x ) exp ( j k 1 y y )
E 1 x ~ = H 0 [ 1 + Γ exp ( 2 j k 1 x ) ] [ k 1 y ω ε 1 + V y ε 1 c 2 ] · exp ( 2 j k 1 x ) ] · exp ( j k 1 x x ) exp ( j k 1 y y )
E 1 y ~ = H 0 [ 1 + Γ exp ( 2 j k 1 x ) ] [ k 1 x ω ε 1 V x ε 1 c 2 ] · exp ( 2 j k 1 x ) ] · exp ( j k 1 x x ) exp ( j k 1 y y )
Γ = ( k 1 x ω ε 1 k 2 x ω ε 2 s x * s x + V x ε 2 s 2 x c 2 V x ε 1 c 2 ) ( k 1 x ω ε 1 + k 2 x ω ε 2 s x * s x V x ε 2 s 2 x c 2 + V x ε 1 c 2 ) 1
Γ = V x 2 c ( 1 s 2 1 )

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