Abstract

A manifestation of the Sagnac effect in a rotating photonic crystal that contains a microcavity with degenerate modes is explored. It is shown that generally rotation can cause the resonance frequency to split into M different frequencies, where M is the order of the stationary-system mode degeneracy. The results are derived using a new rotation-induced eigenvalue theory that holds for any two-dimensional or three-dimensional rotating microcavity with mode degeneracy. Comparison with exact numerical simulations of the rotating system is provided. Miniature optical gyroscopes are discussed.

© 2006 Optical Society of America

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  1. B. Z. Steinberg, "Rotating photonic crystals: a medium for compact optical gyroscopes," Phys. Rev. E 71, 056621 (2005).
    [CrossRef]
  2. B. Z. Steinberg, A. Boag, and R. Lisitsin, "Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy," J. Opt. Soc. Am. A 20, 138-146 (2003).
    [CrossRef]
  3. 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]
  4. E. J. Post, "Sagnac effect," Rev. Mod. Phys. 39, 475-493 (1967).
    [CrossRef]
  5. O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am. B 16, 275-285 (1999).
    [CrossRef]
  6. M. Loncar, M. Hochberg, A. Scherer, and Y. Qiu, "High quality factors and room-temperature lasing in a modified single-defect photonic crystal cavity," Opt. Lett. 29, 721-723 (2004).
    [CrossRef] [PubMed]
  7. T. Shiozawa, "Phenomenological and electron-theoretical study of the electrodynamics of rotating systems," Proc. IEEE 61, 1694-1702 (1973).
    [CrossRef]
  8. J. L. Anderson and J. W. Ryon, "Electromagnetic radiation in accelerated systems," Phys. Rev. 181, 1765-1775 (1969).
    [CrossRef]
  9. H. J. Arditty and H. C. Lefevre, "Sagnac effect in fiber gyroscopes," Opt. Lett. 6, 401-403 (1981).
    [CrossRef] [PubMed]
  10. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).
  11. R. F. Harrington, Field Computation by Moment Methods (Krieger, 1982).
  12. Y. Leviatan and A. Boag, "Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model," IEEE Trans. Antennas Propag. 35, 1119-1127 (1987).
    [CrossRef]
  13. A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
    [CrossRef]
  14. B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1542 (2003).
    [CrossRef] [PubMed]
  15. B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
    [CrossRef]
  16. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
    [CrossRef] [PubMed]
  17. J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
    [CrossRef] [PubMed]
  18. B. Z. Steinberg, J. Scheuer, and A. Boag, "Slow-light waveguides with mode degeneracy: rotation-induced super structures and optical gyroscopes," in Proceedings of Slow and Fast Light OSA Topical Meeting (Optical Society of America, 2006), paper MB3.
  19. B. Z. Steinberg, J. Scheuer, and A. Boag, "Rotation-induced super structure in slow-light waveguides with mode degeneracy," submitted to J. Opt. Soc. Am. B.

2006 (2)

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 (2)

B. Z. Steinberg, "Rotating photonic crystals: a medium for compact optical gyroscopes," Phys. Rev. E 71, 056621 (2005).
[CrossRef]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

2004 (1)

2003 (3)

B. Z. Steinberg, A. Boag, and R. Lisitsin, "Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy," J. Opt. Soc. Am. A 20, 138-146 (2003).
[CrossRef]

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1542 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

1999 (1)

1988 (1)

A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
[CrossRef]

1987 (1)

Y. Leviatan and A. Boag, "Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model," IEEE Trans. Antennas Propag. 35, 1119-1127 (1987).
[CrossRef]

1981 (1)

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]

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

Anderson, J. L.

J. L. Anderson and J. W. Ryon, "Electromagnetic radiation in accelerated systems," Phys. Rev. 181, 1765-1775 (1969).
[CrossRef]

Arditty, H. J.

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Asano, T.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1542 (2003).
[CrossRef] [PubMed]

Boag, 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]

B. Z. Steinberg, A. Boag, and R. Lisitsin, "Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy," J. Opt. Soc. Am. A 20, 138-146 (2003).
[CrossRef]

A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
[CrossRef]

A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
[CrossRef]

Y. Leviatan and A. Boag, "Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model," IEEE Trans. Antennas Propag. 35, 1119-1127 (1987).
[CrossRef]

B. Z. Steinberg, J. Scheuer, and A. Boag, "Rotation-induced super structure in slow-light waveguides with mode degeneracy," submitted to J. Opt. Soc. Am. B.

B. Z. Steinberg, J. Scheuer, and A. Boag, "Slow-light waveguides with mode degeneracy: rotation-induced super structures and optical gyroscopes," in Proceedings of Slow and Fast Light OSA Topical Meeting (Optical Society of America, 2006), paper MB3.

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Krieger, 1982).

Hochberg, M.

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Lancaster, P.

P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).

Lefevre, H. C.

Leviatan, Y.

A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
[CrossRef]

Y. Leviatan and A. Boag, "Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model," IEEE Trans. Antennas Propag. 35, 1119-1127 (1987).
[CrossRef]

Lisitsin, R.

Loncar, M.

Noda, S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1542 (2003).
[CrossRef] [PubMed]

Painter, O.

Post, E. J.

E. J. Post, "Sagnac effect," Rev. Mod. Phys. 39, 475-493 (1967).
[CrossRef]

Qiu, Y.

Ryon, J. W.

J. L. Anderson and J. W. Ryon, "Electromagnetic radiation in accelerated systems," Phys. Rev. 181, 1765-1775 (1969).
[CrossRef]

Scherer, A.

Scheuer, J.

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

B. Z. Steinberg, J. Scheuer, and A. Boag, "Rotation-induced super structure in slow-light waveguides with mode degeneracy," submitted to J. Opt. Soc. Am. B.

B. Z. Steinberg, J. Scheuer, and A. Boag, "Slow-light waveguides with mode degeneracy: rotation-induced super structures and optical gyroscopes," in Proceedings of Slow and Fast Light OSA Topical Meeting (Optical Society of America, 2006), paper MB3.

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]

Song, B. S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1542 (2003).
[CrossRef] [PubMed]

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Steinberg, B. Z.

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]

B. Z. Steinberg, A. Boag, and R. Lisitsin, "Sensitivity analysis of narrowband photonic crystal filters and waveguides to structure variations and inaccuracy," J. Opt. Soc. Am. A 20, 138-146 (2003).
[CrossRef]

B. Z. Steinberg, J. Scheuer, and A. Boag, "Rotation-induced super structure in slow-light waveguides with mode degeneracy," submitted to J. Opt. Soc. Am. B.

B. Z. Steinberg, J. Scheuer, and A. Boag, "Slow-light waveguides with mode degeneracy: rotation-induced super structures and optical gyroscopes," in Proceedings of Slow and Fast Light OSA Topical Meeting (Optical Society of America, 2006), paper MB3.

Tismenetsky, M.

P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).

Vahala, K. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Vuckovic, J.

Yariv, A.

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

Y. Leviatan and A. Boag, "Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model," IEEE Trans. Antennas Propag. 35, 1119-1127 (1987).
[CrossRef]

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

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

B. Z. Steinberg, J. Scheuer, and A. Boag, "Rotation-induced super structure in slow-light waveguides with mode degeneracy," submitted to J. Opt. Soc. Am. B.

O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am. B 16, 275-285 (1999).
[CrossRef]

Nat. Mater. (1)

B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005).
[CrossRef]

Nature (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Opt. Lett. (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. E (2)

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]

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]

Radio Sci. (1)

A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
[CrossRef]

Rev. Mod. Phys. (1)

E. J. Post, "Sagnac effect," Rev. Mod. Phys. 39, 475-493 (1967).
[CrossRef]

Science (1)

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1542 (2003).
[CrossRef] [PubMed]

Other (3)

P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).

R. F. Harrington, Field Computation by Moment Methods (Krieger, 1982).

B. Z. Steinberg, J. Scheuer, and A. Boag, "Slow-light waveguides with mode degeneracy: rotation-induced super structures and optical gyroscopes," in Proceedings of Slow and Fast Light OSA Topical Meeting (Optical Society of America, 2006), paper MB3.

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Figures (6)

Fig. 1
Fig. 1

Electric field magnitudes on a decibel scale of a doubly degenerate TM microcavity ( M = 2 ) , in a two-dimensional hexagonal photonic crystal (PhC). The crystal is made of dielectric cylinders, outlined by black circles. (a) E 0 ( 1 ) . (b) E 0 ( 2 ) . These modes are nonorthogonal, and E 0 ( 2 ) is a π 3 -rotated replica of E 0 ( 1 ) . (c) The linear combination E 0 ( 1 ) E 0 ( 1 ) + E 0 ( 2 ) . (d) The linear combination E 0 ( 2 ) E 0 ( 1 ) E 0 ( 2 ) . These modes are orthogonal.

Fig. 2
Fig. 2

PhC structure under study.

Fig. 3
Fig. 3

Splitting of the degenerate cavity resonance frequency due to rotation.

Fig. 4
Fig. 4

The integrand in Eq. (2.18c) used to compute the elements of B . The mode functions shown in Figs. 1c, 1d are used.

Fig. 5
Fig. 5

Intensity of the field inside the rotating cavity versus excitation wavelength, for various values of the angular velocity Ω.

Fig. 6
Fig. 6

Splitting of the degenerate cavity resonance frequency due to rotation, for the slab PhC (TE polarization).

Equations (49)

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Ω = z ̂ Ω .
× E = i ω B , B = 0 ,
× H = i ω D , D = 0 .
D = ϵ E c 2 Ω × r × H ,
B = μ H + c 2 Ω × r × E .
D × E = i ω μ H ,
D × H = i ω ϵ E ,
D i k β ( r ) , k = ω c , β ( r ) = c 1 Ω × r .
Θ H Ω ( r ) = k 2 H Ω ( r ) + i k L Ω H Ω ( r ) .
Θ × 1 ϵ r ( r ) × ,
L Ω H = × β ( r ) ϵ r ( r ) × H + β ( r ) ϵ r ( r ) × × H ,
Θ H 0 ( m ) ( r ) = k 0 2 H 0 ( m ) ( r ) , k 0 = ω 0 c , m = 1 , 2 , , M .
F , G F G ¯ d 3 r ,
Θ H 0 ( m ) , H Ω = k 0 2 H 0 ( m ) , H Ω ,
Θ H Ω , H 0 ( m ) = k 2 H Ω , H 0 ( m ) + i k L Ω H Ω , H 0 ( m ) ,
( k 2 k 0 2 ) H Ω , H 0 ( m ) + i k L Ω H Ω , H 0 ( m ) = 0 , m = 1 , 2 , M .
H Ω ( r ) = n = 1 M a n H 0 ( n ) ( r ) .
( k 2 k 0 2 ) n = 1 M a n A m n = i k n = 1 M a n L Ω H 0 ( n ) , H 0 ( m ) ,
m = 1 , 2 , M ,
A m n = H 0 ( n ) , H 0 ( m ) .
L Ω H 0 ( n ) , H 0 ( m ) = i c 1 Ω ω 0 B m n ,
B m n = ϵ 0 z ̂ × r , H ¯ 0 ( n ) × E 0 ( m ) + H 0 ( m ) × E ¯ 0 ( n ) .
Ω A 1 B a = ω 2 ω 0 2 ω ω 0 a , a = ( a 1 a 2 a M ) ,
ω = ω 0 + δ ω ,
ω 2 ω 0 2 ω ω 0 2 δ ω ω 0 .
C a = δ ω ω 0 Ω a , C = ( 1 2 ) A 1 B .
δ ω j ( Ω ) = Ω ω 0 Λ j , j = 1 , , M .
A m n = A ¯ n m , B m n = B ¯ n m ,
B m m = 2 ϵ 0 z ̂ × r , Re S 0 ( m ) .
B m n = ϵ 0 V u m n d 3 r = ϵ 0 V r ϕ ̂ ( H 0 ( n ) × E ¯ 0 ( m ) + H ¯ 0 ( m ) × E 0 ( n ) ) d 3 r .
A = H 0 ( 1 ) 2 I ,
C = 1 2 H 0 ( 1 ) 2 [ 0 B 12 B 13 B 1 M B ¯ 12 0 B 23 B 2 M B ¯ 13 B ¯ 23 0 B 3 M B ¯ 1 M B ¯ 2 M 0 ] i Γ ,
Λ 1 , 2 = ± B 12 2 H 0 ( 1 ) 2 ,
( a ( 1 , 2 ) ) 1 = i ( a ( 1 , 2 ) ) 2 ,
E 0 ( 1 , 2 ) = z ̂ η H 0 exp ( ± i k 0 n R ϕ ) ,
H 0 ( 1 , 2 ) = ± ρ ̂ H 0 exp ( ± i k 0 n R ϕ ) ,
B 11 ϵ 0 R ϕ ̂ , ϕ ̂ H ¯ 0 E 0 ϕ ̂ H 0 E ¯ 0 = 2 ϵ 0 R H 0 2 η .
C = R n c [ 1 0 0 1 ] ,
δ ω 12 = ω 0 Ω Λ 12 = ± ω 0 Ω R ( n c ) ,
R i eff = Λ i n c .
G Ω = I i 4 m = J m ( k 0 n γ m ρ < ) H m ( 1 ) ( k 0 n γ m ρ > ) exp [ i m ( θ θ 1 ) ] I i 4 H 0 ( 1 ) exp [ i Ω ω k 0 2 ( y x x y ) ] ,
Ω min = Δ ω ω 0 max j Λ j 1 Q max j Λ j ,
( × β ϵ r × H 0 ( n ) ) H ¯ 0 ( m ) = [ ( β ϵ r × H 0 ( n ) ) × H ¯ 0 ( m ) ] + ( β ϵ r × H 0 ( n ) ) ( × H ¯ 0 ( m ) ) ,
( β ϵ r × × H 0 ( n ) ) H 0 ( m ) = ( β ϵ r × H ¯ 0 ( m ) ) ( × H 0 ( n ) ) .
V [ ( β ϵ r × H 0 ( n ) ) × H ¯ 0 ( m ) ] d 3 x = S = V [ ( β ϵ r × H 0 ( n ) ) × H ¯ 0 ( m ) ] d s 0 .
L Ω H 0 ( n ) , H 0 ( m ) = β ϵ r × H 0 ( n ) , × H 0 ( m ) × H 0 ( n ) , β ϵ r × H 0 ( m ) .
L Ω H 0 ( n ) , H 0 ( m ) = β ϵ r , H ¯ 0 ( n ) × × H 0 ( m ) β ϵ r , H 0 ( m ) × × H ¯ 0 ( n ) .
L Ω H 0 ( n ) , H 0 ( m ) = c 1 Ω z ̂ × r ϵ r , H ¯ 0 ( n ) × × H 0 ( m ) H 0 ( m ) × × H ¯ 0 ( n ) .
L Ω H 0 ( n ) , H 0 ( m ) = i c 1 Ω ω 0 ϵ 0 z ̂ × r , H ¯ 0 ( n ) × E 0 ( m ) + H 0 ( m ) × E ¯ 0 ( n ) .

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