Abstract

An analytical model for propagation through and reflection from a discontinuity in coupled-cavity waveguides (CCWs) (also known as coupled-resonator optical waveguides—CROW) is developed. The theory is based on a modification of the tight-binding theory for propagation in nonuniform structures. Explicit analytic expressions for the reflection and transmission coefficients are obtained. These expressions resemble in form and structure the well-known Fresnel coefficients, with the traditional wave impedance parameter replaced by the device bandwidth. Matching of two uniform CCWs with the use of an intermediate serial section is also discussed, and an analogy to the well-known quarter-wavelength plate is pointed out.

© 2006 Optical Society of America

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  1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  2. M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
    [CrossRef] [PubMed]
  3. A. Boag and B. Z. Steinberg, "Narrow-band microcavity waveguides in photonic crystals," J. Opt. Soc. Am. A 18, 2799-2805 (2001).
    [CrossRef]
  4. B. Z. Steinberg, A. Boag, and R. Lisitsin, "Sensitivity analysis of narrow-band photonic crystal filters and waveguides to structure variations and inaccuracy," J. Opt. Soc. Am. A 20, 138-146 (2003).
    [CrossRef]
  5. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
    [CrossRef] [PubMed]
  6. D. N. Christodoulides and N. K. Efremidis, "Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals," Opt. Lett. 27, 568-570 (2002).
    [CrossRef]
  7. S. Mookherjea and A. Yariv, "Second-harmonic generation with pulses in a coupled-resonator optical waveguide," Phys. Rev. E 65, 026607 (2002).
    [CrossRef]
  8. S. Mookherjea and A. Yariv, "Optical pulse propagation and holographic storage in a coupled-resonator optical waveguide," Phys. Rev. E 64, 066602 (2001).
    [CrossRef]
  9. S. Mookherjea and A. Yariv, "Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion," Phys. Rev. E 65, 056601 (2002).
    [CrossRef]
  10. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B 21, 1665-1673 (2004).
    [CrossRef]
  11. B. Z. Steinberg, "Rotating photonic crystals: a medium for compact optical gyroscopes," Phys. Rev. E 71, 056621-7 (2005).
    [CrossRef]
  12. 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]
  13. T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, "High quality two-dimensional photonic crystal slab cavities," Appl. Phys. Lett. 79, 4289-4291 (2001).
    [CrossRef]
  14. B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537 (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. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  17. S. F. Mingaleev and Y. S. Kivshar, "Effective equations for photonic-crystal waveguides and circuits," Opt. Lett. 27, 231-233 (2002).
    [CrossRef]
  18. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley1975).
  19. B. Z. Steinberg, A. Boag, and O. Hershkoviz, "Substructuring approach to optimization of matching for photonic crystal waveguides," Microwave Opt. Technol. Lett. (to be published).
  20. A. Boag, B. Z. Steinberg, and O. Bushmakin, "Matching of narrow band photonic crystal filters and waveguides to free space and dielectric waveguides," Presented at 2003 IEEE AP-S International Symposium and USNC/CNC/URSI National Radio Science Meeting, Columbus, Ohio, 22-27 June 2003.
  21. 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]

2005 (2)

B. Z. Steinberg, "Rotating photonic crystals: a medium for compact optical gyroscopes," Phys. Rev. E 71, 056621-7 (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 (2)

2003 (2)

2002 (4)

S. F. Mingaleev and Y. S. Kivshar, "Effective equations for photonic-crystal waveguides and circuits," Opt. Lett. 27, 231-233 (2002).
[CrossRef]

D. N. Christodoulides and N. K. Efremidis, "Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals," Opt. Lett. 27, 568-570 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Second-harmonic generation with pulses in a coupled-resonator optical waveguide," Phys. Rev. E 65, 026607 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion," Phys. Rev. E 65, 056601 (2002).
[CrossRef]

2001 (3)

S. Mookherjea and A. Yariv, "Optical pulse propagation and holographic storage in a coupled-resonator optical waveguide," Phys. Rev. E 64, 066602 (2001).
[CrossRef]

T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, "High quality two-dimensional photonic crystal slab cavities," Appl. Phys. Lett. 79, 4289-4291 (2001).
[CrossRef]

A. Boag and B. Z. Steinberg, "Narrow-band microcavity waveguides in photonic crystals," J. Opt. Soc. Am. A 18, 2799-2805 (2001).
[CrossRef]

2000 (1)

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

1999 (2)

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]

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]

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 (2003).
[CrossRef] [PubMed]

Bayindir, M.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Boag, A.

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

A. Boag and B. Z. Steinberg, "Narrow-band microcavity waveguides in photonic crystals," J. Opt. Soc. Am. A 18, 2799-2805 (2001).
[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]

A. Boag, B. Z. Steinberg, and O. Bushmakin, "Matching of narrow band photonic crystal filters and waveguides to free space and dielectric waveguides," Presented at 2003 IEEE AP-S International Symposium and USNC/CNC/URSI National Radio Science Meeting, Columbus, Ohio, 22-27 June 2003.

B. Z. Steinberg, A. Boag, and O. Hershkoviz, "Substructuring approach to optimization of matching for photonic crystal waveguides," Microwave Opt. Technol. Lett. (to be published).

Bushmakin, O.

A. Boag, B. Z. Steinberg, and O. Bushmakin, "Matching of narrow band photonic crystal filters and waveguides to free space and dielectric waveguides," Presented at 2003 IEEE AP-S International Symposium and USNC/CNC/URSI National Radio Science Meeting, Columbus, Ohio, 22-27 June 2003.

Chen, H.

T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, "High quality two-dimensional photonic crystal slab cavities," Appl. Phys. Lett. 79, 4289-4291 (2001).
[CrossRef]

Christodoulides, D. N.

Deppe, D.

T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, "High quality two-dimensional photonic crystal slab cavities," Appl. Phys. Lett. 79, 4289-4291 (2001).
[CrossRef]

Efremidis, N. K.

Hershkoviz, O.

B. Z. Steinberg, A. Boag, and O. Hershkoviz, "Substructuring approach to optimization of matching for photonic crystal waveguides," Microwave Opt. Technol. Lett. (to be published).

Huang, Y.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley1975).

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Kivshar, Y. S.

Lee, R. K.

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]

Lisitsin, R.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mingaleev, S. F.

Mookherjea, S.

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
[CrossRef] [PubMed]

S. Mookherjea and A. Yariv, "Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion," Phys. Rev. E 65, 056601 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Second-harmonic generation with pulses in a coupled-resonator optical waveguide," Phys. Rev. E 65, 026607 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Optical pulse propagation and holographic storage in a coupled-resonator optical waveguide," Phys. Rev. E 64, 066602 (2001).
[CrossRef]

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 (2003).
[CrossRef] [PubMed]

Ozbay, E.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Painter, O.

Paloczi, G. T.

Poon, J. K. S.

Scherer, A.

Scheuer, J.

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 (2003).
[CrossRef] [PubMed]

Steinberg, B. Z.

B. Z. Steinberg, "Rotating photonic crystals: a medium for compact optical gyroscopes," Phys. Rev. E 71, 056621-7 (2005).
[CrossRef]

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

A. Boag and B. Z. Steinberg, "Narrow-band microcavity waveguides in photonic crystals," J. Opt. Soc. Am. A 18, 2799-2805 (2001).
[CrossRef]

B. Z. Steinberg, A. Boag, and O. Hershkoviz, "Substructuring approach to optimization of matching for photonic crystal waveguides," Microwave Opt. Technol. Lett. (to be published).

A. Boag, B. Z. Steinberg, and O. Bushmakin, "Matching of narrow band photonic crystal filters and waveguides to free space and dielectric waveguides," Presented at 2003 IEEE AP-S International Symposium and USNC/CNC/URSI National Radio Science Meeting, Columbus, Ohio, 22-27 June 2003.

Temelkuran, B.

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

Vuckovic, J.

T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, "High quality two-dimensional photonic crystal slab cavities," Appl. Phys. Lett. 79, 4289-4291 (2001).
[CrossRef]

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]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Xu, Y.

Yariv, A.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B 21, 1665-1673 (2004).
[CrossRef]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
[CrossRef] [PubMed]

S. Mookherjea and A. Yariv, "Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion," Phys. Rev. E 65, 056601 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Second-harmonic generation with pulses in a coupled-resonator optical waveguide," Phys. Rev. E 65, 026607 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Optical pulse propagation and holographic storage in a coupled-resonator optical waveguide," Phys. Rev. E 64, 066602 (2001).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

Yoshie, T.

T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, "High quality two-dimensional photonic crystal slab cavities," Appl. Phys. Lett. 79, 4289-4291 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

T. Yoshie, J. Vuckovic, A. Scherer, H. Chen, and D. Deppe, "High quality two-dimensional photonic crystal slab cavities," Appl. Phys. Lett. 79, 4289-4291 (2001).
[CrossRef]

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

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

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]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. E (4)

S. Mookherjea and A. Yariv, "Second-harmonic generation with pulses in a coupled-resonator optical waveguide," Phys. Rev. E 65, 026607 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, "Optical pulse propagation and holographic storage in a coupled-resonator optical waveguide," Phys. Rev. E 64, 066602 (2001).
[CrossRef]

S. Mookherjea and A. Yariv, "Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion," Phys. Rev. E 65, 056601 (2002).
[CrossRef]

B. Z. Steinberg, "Rotating photonic crystals: a medium for compact optical gyroscopes," Phys. Rev. E 71, 056621-7 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

M. Bayindir, B. Temelkuran, and E. Ozbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140-2143 (2000).
[CrossRef] [PubMed]

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]

Science (1)

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

Other (4)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley1975).

B. Z. Steinberg, A. Boag, and O. Hershkoviz, "Substructuring approach to optimization of matching for photonic crystal waveguides," Microwave Opt. Technol. Lett. (to be published).

A. Boag, B. Z. Steinberg, and O. Bushmakin, "Matching of narrow band photonic crystal filters and waveguides to free space and dielectric waveguides," Presented at 2003 IEEE AP-S International Symposium and USNC/CNC/URSI National Radio Science Meeting, Columbus, Ohio, 22-27 June 2003.

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Figures (9)

Fig. 1
Fig. 1

Two examples for a system of two different CCWs connected in series. (a) Realization in photonic crystal. Local defects are shown by solid circles. The CCWs may differ in the local defect form and intercavity spacing, thus differing in both central frequency and bandwidth. (b) Realization using an array of microring resonators.

Fig. 2
Fig. 2

Two different CCWs connected in series, matched by an intermediate section with M microcavities.

Fig. 3
Fig. 3

Two sections with identical central frequencies and different bandwidths. All the dielectric cylinders in the crystal are of radius 0.6 and ϵ r = 8.41 . The crystal spacing is a = 4 . The dielectric cylinder used to match the system output to free space is seen on the right, external to the structure. Its material is identical to that of the PhC. Its radius is 0.4 and its x coordinate is 3.3 to the right of the rightmost cylinder in the crystal.[19]

Fig. 4
Fig. 4

Reflection coefficient obtained for the configuration described in Fig. 3.

Fig. 5
Fig. 5

Two sections with different central frequencies and bandwidths. The parameters are the same as in the previous example, except for the shaded circles representing dielectric cylinders of the same material but of radius 0.64. The dielectric cylinder used to match the system output to free space is seen on the right. It has the same parameters as in the previous example.

Fig. 6
Fig. 6

Reflection coefficient obtained for the configuration described in Fig. 5.

Fig. 7
Fig. 7

Quarter-wavelength plate analog of two different CCWs connected in series, matched by an intermediate section of a single microcavity. The crystal parameters are the same as in the previous examples, except for the solid large circle and shaded small circles, representing dielectric cylinders of the same material but of radius 0.66 and 0.587, respectively.

Fig. 8
Fig. 8

Reflection coefficient obtained by using a single-cavity intermediate matching section.

Fig. 9
Fig. 9

Phase of the transmitted field in CCW 3 relative to the phase of the incident field in CCW 1 .

Equations (54)

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d n ( r ) = 1 ϵ d n ( r ) 1 ϵ b ( r ) ,
1 ϵ r ( r ) = 1 ϵ b ( r ) + n d n ( r ) .
Θ × 1 ϵ r ( r ) × , Θ b × 1 ϵ b ( r ) × ,
Θ n × d n ( r ) × .
( Θ b × Θ n ) H n ( r ) = ( ω n c ) 2 H n ( r ) .
Θ H ( r ) = ( Θ b + n Θ n ) H ( r ) = ( ω c ) 2 H ( r ) .
H ( r ) = n A n H n ( r ) ,
n { [ ( ω k c ) 2 ( ω c ) 2 ] I n k + T n , k } A n = 0 k ,
I n m H n , H m , T n , m H n , j m Θ j H m .
Ω 2 A k + 2 c 2 ω 0 δ ω k A k + T k 1 , k A k 1 + T k + 1 , k A k + 1 = 0 k ,
Ω 2 ( ω 0 2 ω 2 ) c 2 .
T n , m = j m Θ j H n , H m = j m , n Θ j H n , H m + Θ n H n , H m = j n Θ j H n , H m + Θ n H n , H m Θ m H n , H m = T m , n + Θ n H n , H m Θ m H m , H n ( H n , Θ j real and self - adjoint ) .
T n , m + Θ m H m , H n = T m , n + Θ n H n , H m .
T n , m = T m , n .
Ω 2 A k + τ ( A k 1 + A k + 1 ) = 0 k .
A k = e i β k ,
Ω 2 = 2 τ cos β .
Ω 2 = 1 c 2 ( ω 0 ω ) ( ω 0 + ω ) 2 ω 0 c 2 ( ω 0 ω ) ,
ω ω 0 = ( Δ ω 2 ) cos β ,
τ = ω 0 Δ ω 2 c 2 .
δ ω k = 0 k ,
T k 1 , k = { τ 1 k 0 τ 2 k 1 } .
T m , n = T n , m m , n 0 or m , n 0 ,
T k 1 , k = T k + 1 , k = { τ 1 k 1 τ 2 k 2 } .
A k = { e i β 1 k + Γ e i β 1 k , k 0 T e i β 2 k , k 1 } .
Ω 2 ( e i β 1 k + Γ e i β 1 k ) + 2 τ 1 cos β 1 ( e i β 1 k + Γ e i β 1 k ) = 0 , k 1 ,
Ω 2 T e i β 2 k + 2 τ 2 T cos β 2 e i β 2 k = 0 , k 2 ,
Ω 2 = 2 τ 1 cos β 1 , Ω 2 = 2 τ 2 cos β 2 .
ω ω 0 = ( Δ ω 1 2 ) cos β 1 ,
ω ω 0 = ( Δ ω 2 2 ) cos β 2 ,
Δ ω 1 , 2 = 2 c 2 τ 1 , 2 ω 0 .
Ω 2 A 0 + T 1 , 0 A 1 + T 1 , 0 A 1 = 0 ( k = 0 ) ,
Ω 2 A 1 + T 0 , 1 A 0 + T 2 , 1 A 2 = 0 ( k = 1 ) .
T = 1 + Γ , Γ ( ω ) = τ 1 e β 1 τ 2 e i β 2 τ 1 e i β 1 τ 2 e i β 2 = τ 1 e i β 1 τ 2 e i β 2 τ 1 e i β 1 τ 2 e i β 2 .
Γ ( ω ) = Δ ω 2 1 ( ω ω 0 Δ ω 2 2 ) 2 Δ ω 1 1 ( ω ω 0 Δ ω 1 2 ) 2 Δ ω 2 1 ( ω ω 0 Δ ω 2 2 ) 2 + Δ ω 1 1 ( ω ω 0 Δ ω 1 2 ) 2 .
Γ ( ω 0 ) = Δ ω 2 Δ ω 1 Δ ω 2 + Δ ω 1 .
δ ω k = { 0 , k 0 δ ω 0 = Δ 0 c 2 2 ω 0 , k 1 } ,
T 0 , 1 T 1 , 0 τ 2 .
Ω 2 ( e i β 1 k + Γ e i β 1 k ) + 2 τ 1 cos β 1 ( e i β 1 k + Γ e i β 1 k ) = 0 , k 1 ,
( Ω 2 + Δ 0 ) T e i β 2 k + 2 τ 2 T cos β 2 e i β 2 k = 0 , k 2 ,
Ω 2 = 2 τ 1 cos β 1 , Ω 2 + Δ 0 = 2 τ 2 cos β 2 .
ω ω 01 = ( Δ ω 1 2 ) cos β 1 ,
ω ω 02 = ( Δ ω 2 2 ) cos β 2 , ( ω 02 = ω 01 + δ ω 0 ) ,
T = T 0 , 1 τ 2 ( 1 + Γ ) , Γ ( ω ) = τ 1 e i β 1 α τ 2 e i β 2 τ 1 e i β 1 α τ 2 e i β 2 ,
α = T 1 , 0 T 0 , 1 τ 2 2 .
Γ ( ω ) = 2 α ( ω 02 ω ) 2 ( ω 01 ω ) + i [ Δ ω 1 s 1 Δ ω 2 α ω 02 ω 01 s 2 ] 2 α ( ω 02 ω ) 2 ( ω 01 ω ) i [ Δ ω 1 s 1 + Δ ω 2 α ω 02 ω 01 s 2 ] ,
s 1 , 2 = 1 ( ω ω 01 , 2 Δ ω 12 2 ) 2 .
A k i = e i β 1 k , A k r = R e i β 1 k , k 0 ,
R = Γ 12 + T 12 T 21 Γ 23 e 2 i β 2 M n = 0 [ e 2 i β 2 M Γ 23 Γ 21 ] n .
R = Γ 12 Γ 23 [ Γ 12 Γ 21 T 12 T 21 ] e 2 i β 2 M 1 Γ 23 Γ 21 e 2 i β 2 M .
Γ 12 = Γ 23 [ Γ 12 Γ 21 T 21 T 21 ] e 2 i β 2 M .
Γ 12 = Γ 23 e 2 i β 2 M .
M = 2 n + 1 with n = 0 , 1 , 2 , , Δ ω 2 = Δ ω 1 Δ ω 3 .
[ 1 1 e i β e i β e i 2 β e i 2 β : : e i N β e i N β ] [ a + ( β ) a ( β ) ] = [ E 0 E 1 : E N ] .

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