Abstract

We analyze the transmission of light through coupled-resonator optical waveguides in the form of evanescently coupled Fabry–Perot resonator arrays. We develop a transfer matrix method to calculate the amplitude and phase responses of the arrays. We also discuss the inclusion of optical gain in the system to compensate for losses in these structures. Owing to the compact length along the propagation direction in evanescently coupled arrays, large slowing factors of the order of 102103 can be achieved even with a weak index contrast of 0.1%. The large slowing factor, coupled with weak index contrast, makes this structure a promising candidate for artificial slow light system.

© 2007 Optical Society of America

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References

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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. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electron. 37, 525-532 (2001).
    [CrossRef]
  3. S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. , E85C, 181-189 (2002).
  4. H. Altug and J. Vuckovic, "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005).
    [CrossRef]
  5. F. N. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
    [CrossRef]
  6. M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
    [CrossRef]
  7. S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Osterle, "Miniband transmission in a photonic crystal waveguide coupled-resonator optical waveguide," Opt. Lett. 26, 1019-1051 (2001).
    [CrossRef]
  8. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
    [CrossRef]
  9. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, "Transmission and group delay in microring coupled-resonator optical waveguides," Opt. Lett. 31, 456-458 (2006).
    [CrossRef] [PubMed]
  10. A. Scherer, O. Painter, B. D'Urso, R. Lee, and A. Yariv, "InGaAsP photonic band gap crystal membrane microresonators," J. Vac. Sci. Technol. B 16, 3906-3910 (1998).
    [CrossRef]
  11. U. Peschel, O. Egorov, and F. Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909-1911 (2004).
    [CrossRef] [PubMed]
  12. D. Botez and D. R. Scifres, Diode Laser Arrays (Cambridge U. Press, 1994).
    [CrossRef]
  13. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997).
  14. 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]
  15. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
    [CrossRef] [PubMed]
  16. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, 1999).
  17. M. Sumetsky and B. J. Eggleton, "Modeling and optimization of complex photonic resonant cavity circuits," Opt. Express 11, 381-391 (2003).
    [CrossRef] [PubMed]
  18. P. Chak and J. E. Sipe, "Minimizing finite-size effects in artificial resonance tunneling structures," Opt. Lett. 13, 2568-2570 (2006).
    [CrossRef]
  19. S. Mookherjea, "Using gain to tune the dispersion relation of coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 715-717 (2006).
    [CrossRef]

2007

F. N. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
[CrossRef]

2006

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, "Transmission and group delay in microring coupled-resonator optical waveguides," Opt. Lett. 31, 456-458 (2006).
[CrossRef] [PubMed]

P. Chak and J. E. Sipe, "Minimizing finite-size effects in artificial resonance tunneling structures," Opt. Lett. 13, 2568-2570 (2006).
[CrossRef]

S. Mookherjea, "Using gain to tune the dispersion relation of coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 715-717 (2006).
[CrossRef]

2005

H. Altug and J. Vuckovic, "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

2004

U. Peschel, O. Egorov, and F. Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909-1911 (2004).
[CrossRef] [PubMed]

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]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

2003

2002

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. , E85C, 181-189 (2002).

2001

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Osterle, "Miniband transmission in a photonic crystal waveguide coupled-resonator optical waveguide," Opt. Lett. 26, 1019-1051 (2001).
[CrossRef]

1999

1998

A. Scherer, O. Painter, B. D'Urso, R. Lee, and A. Yariv, "InGaAsP photonic band gap crystal membrane microresonators," J. Vac. Sci. Technol. B 16, 3906-3910 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop filters in photonic crystals," Opt. Express 3, 4-11 (1998).
[CrossRef] [PubMed]

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Altug, H.

H. Altug and J. Vuckovic, "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

Bayindir, M.

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

Benisty, H.

Botez, D.

D. Botez and D. R. Scifres, Diode Laser Arrays (Cambridge U. Press, 1994).
[CrossRef]

Chak, P.

P. Chak and J. E. Sipe, "Minimizing finite-size effects in artificial resonance tunneling structures," Opt. Lett. 13, 2568-2570 (2006).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

DeRose, G. A.

D'Urso, B.

A. Scherer, O. Painter, B. D'Urso, R. Lee, and A. Yariv, "InGaAsP photonic band gap crystal membrane microresonators," J. Vac. Sci. Technol. B 16, 3906-3910 (1998).
[CrossRef]

Eggleton, B. J.

M. Sumetsky and B. J. Eggleton, "Modeling and optimization of complex photonic resonant cavity circuits," Opt. Express 11, 381-391 (2003).
[CrossRef] [PubMed]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Egorov, O.

Fan, S.

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Haus, H. A.

Houdre, R.

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Huang, Y.

Ishikawa, H.

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. , E85C, 181-189 (2002).

Joannopoulos, J. D.

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Krauss, T.

Lan, S.

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. , E85C, 181-189 (2002).

Lederer, F.

Lee, R.

A. Scherer, O. Painter, B. D'Urso, R. Lee, and A. Yariv, "InGaAsP photonic band gap crystal membrane microresonators," J. Vac. Sci. Technol. B 16, 3906-3910 (1998).
[CrossRef]

Lee, R. K.

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, 1999).

Mookherjea, S.

S. Mookherjea, "Using gain to tune the dispersion relation of coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 715-717 (2006).
[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]

Nishikawa, S.

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. , E85C, 181-189 (2002).

Olivier, S.

Osterle, U.

Ozbay, E.

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

Painter, O.

A. Scherer, O. Painter, B. D'Urso, R. Lee, and A. Yariv, "InGaAsP photonic band gap crystal membrane microresonators," J. Vac. Sci. Technol. B 16, 3906-3910 (1998).
[CrossRef]

Paloczi, G. T.

Peschel, U.

Poon, J. K. S.

Rattier, M.

Scherer, A.

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]

A. Scherer, O. Painter, B. D'Urso, R. Lee, and A. Yariv, "InGaAsP photonic band gap crystal membrane microresonators," J. Vac. Sci. Technol. B 16, 3906-3910 (1998).
[CrossRef]

Scheuer, J.

Scifres, D. R.

D. Botez and D. R. Scifres, Diode Laser Arrays (Cambridge U. Press, 1994).
[CrossRef]

Seiferth, F.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Sekaric, L.

F. N. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Sipe, J. E.

P. Chak and J. E. Sipe, "Minimizing finite-size effects in artificial resonance tunneling structures," Opt. Lett. 13, 2568-2570 (2006).
[CrossRef]

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Smith, C.

Sumetsky, M.

Tanriseven, S.

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Van, V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Villeneuve, P. R.

Vlasov, Y.

F. N. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Vuckovic, J.

H. Altug and J. Vuckovic, "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

Wada, O.

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. , E85C, 181-189 (2002).

Weisbuch, C.

Xia, F. N.

F. N. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Xu, Y.

Yariv, A.

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, 1999).

Zhu, L.

Appl. Phys. A

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one-dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

Appl. Phys. Lett.

H. Altug and J. Vuckovic, "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

IEEE J. Quantum Electron.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical filters," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

IEEE Photon. Technol. Lett.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

S. Mookherjea, "Using gain to tune the dispersion relation of coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 715-717 (2006).
[CrossRef]

IEICE Trans. Electron.

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, "Engineering photonic crystal impurity bands for waveguides, all-optical switches and optical delay lines," IEICE Trans. Electron. , E85C, 181-189 (2002).

J. Vac. Sci. Technol. B

A. Scherer, O. Painter, B. D'Urso, R. Lee, and A. Yariv, "InGaAsP photonic band gap crystal membrane microresonators," J. Vac. Sci. Technol. B 16, 3906-3910 (1998).
[CrossRef]

Nat. Photonics

F. N. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Other

D. Botez and D. R. Scifres, Diode Laser Arrays (Cambridge U. Press, 1994).
[CrossRef]

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997).

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, 1999).

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Figures (4)

Fig. 1
Fig. 1

Schematic of (a) waveguide laser and (b) DFB laser arrays in a planar geometry as implementations of CROWs. The input–output can be (c) side coupled or (d) end coupled into–out of the array. Λ is the period of the CROW. The slanted lines represent reflectors that define each resonator. The arrows indicate the field propagation direction inside each resonator or waveguide. An optical pulse in the structure propagates in the direction of periodicity of the resonators.

Fig. 2
Fig. 2

Schematic illustrating the role of the additional resonance or boundary condition in y. On the left, the resonance condition β ( s ) L = m π selects the resonance frequencies from the dispersion relations of the waveguide array. These frequencies correspond to particular values of s π ( N + 1 ) on the right.

Fig. 3
Fig. 3

(a) Transmission spectrum at the through port and (b) the transmission and reflection spectra at the input and drop ports for the side-coupled array.

Fig. 4
Fig. 4

(a) Transmission and (b) phase responses of a resonator array for various gain values. The input is end coupled into the first element of the array.

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

ϵ ( r ) = ϵ ¯ ( r ) + n = 1 N Δ ϵ ( r n Λ z ̂ ) ,
E ( r ) = n = 1 N c n ( y ) E n ( x , z ) exp ( i β 0 y ) ,
E ( y ) [ c 1 ( y ) e i β 0 y c 2 ( y ) e i β 0 y c N ( y ) e i β 0 y ] [ E 1 ( y ) E 2 ( y ) E N ( y ) ] .
d E d y = C E ,
C = i [ β 0 + M l κ l 0 0 0 0 κ l β 0 + M l κ l 0 0 0 κ l β 0 + M l ] .
κ l = ω ϵ 0 4 E n * ( r ) [ ϵ ( r ) Δ ϵ ( r n Λ ) ] E n + 1 ( r ) d r ,
M l = ω ϵ 0 4 E n * ( r ) [ ϵ ( r ) Δ ϵ ( r n Λ ) ] E n ( r ) d r ,
β 0 2 ω μ E m * ( r ) E n * ( r ) d r = δ m , n .
( C + i β I ) E = 0 ,
β ( s ) = β 0 + M l + 2 κ l cos ( s π N + 1 ) ,
a n ( s ) = sin ( n s π N + 1 ) ,
ω ( s ) = Ω n eff ( Ω ) n eff ( ω ( s ) ) [ 1 M l L m π 2 κ l L m π cos ( s π N + 1 ) ] .
lim N s π ( N + 1 ) K Λ .
ω ( K ) = Ω [ 1 M m π 2 κ m π cos ( K Λ ) ] ,
S = c v g max = n L 2 κ Λ = n 2 κ l Λ .
U ( y ) = [ E 1 ( + ) ( y ) E 2 ( + ) ( y ) E N ( + ) ( y ) ] T ,
D ( y ) = [ E 1 ( ) ( y ) E 2 ( ) ( y ) E N ( ) ( y ) ] T ,
[ U ( L ) D ( L ) ] = S ( 2 ) Q S ( 1 ) [ U ( 0 ) D ( 0 ) ] .
S ( q ) = [ S 11 ( q ) S 12 ( q ) S 21 ( q ) S 22 ( q ) ] = [ P 11 ( q ) P 12 ( q ) F 11 ( q ) F 12 ( q ) F 11 ( q ) F 12 ( q ) P 11 ( q ) P 12 ( q ) P 21 ( q ) P 22 ( q ) F 21 ( q ) F 22 ( q ) F 21 ( q ) F 22 ( q ) P 21 ( q ) P 22 ( q ) ] ,
[ E n ( + ) ( l 1 ) E n ( ) ( l 1 ) ] = [ F 11 ( 1 ) F 12 ( 1 ) F 21 ( 1 ) F 22 ( 1 ) ] [ E n ( + ) ( 0 ) E n ( ) ( 0 ) ] ,
[ E n ( + ) ( L ) E n ( ) ( L ) ] = [ F 11 ( 2 ) F 12 ( 2 ) F 21 ( 2 ) F 22 ( 2 ) ] [ E n ( + ) ( l 2 ) E n ( ) ( l 2 ) ] .
d d y [ U ( y ) D ( y ) ] = [ C C ] [ U ( y ) D ( y ) ] .
[ U ( l 2 ) D ( l 2 ) ] = [ Q Q ] [ U ( l 1 ) D ( l 1 ) ] Q [ U ( l 1 ) D ( l 1 ) ] ,
Q = exp ( C L ) .
[ U ( L ) D ( L ) ] = [ S 11 ( 2 ) S 12 ( 2 ) S 21 ( 2 ) S 22 ( 2 ) ] [ Q Q ] [ S 11 ( 1 ) S 12 ( 1 ) S 21 ( 1 ) S 22 ( 1 ) ] [ U ( 0 ) D ( 0 ) ] [ G 11 G 12 G 21 G 22 ] [ U ( 0 ) D ( 0 ) ] ,
U ( L ) = ( G 11 G 12 G 22 1 G 21 ) U ( 0 ) ,
D ( 0 ) = ( G 22 1 G 21 ) U ( 0 ) ,
R 1 = D 1 ( 0 ) U 1 ( 0 ) , R N = D N ( 0 ) U 1 ( 0 ) ,
T 1 = U 1 ( L ) U 1 ( 0 ) , T N = U N ( L ) U 1 ( 0 ) ,
ϵ ( r ) = ϵ ¯ ( r ) + n = 1 N [ Δ ϵ ( r n Λ ) + i 2 ϵ eff β 0 Δ γ ( r n Λ ) ] ,
κ ̃ l = κ l + i ω ϵ 0 ϵ eff 2 β 0 j = 1 N E n * ( r ) Δ γ ( r j Λ ) E n + 1 ( r ) d r κ l + i κ l ,
M ̃ l = M l + i ω ϵ 0 ϵ eff 2 β 0 j = 1 N E n * ( r ) Δ γ ( r j Λ ) E n ( r ) d r M l + i M l ,
ω ( K ) = Ω [ 1 M m π 2 κ m π cos ( K R Λ ) cosh ( K I Λ ) 2 κ m π sin ( K R Λ ) sinh ( K I Λ ) ] .
γ ( K ) = M l + 2 κ l cos ( K R Λ ) cosh ( K I Λ ) 2 κ l sin ( K R Λ ) sinh ( K I Λ ) .
coth ( K I Λ ) = κ l κ l tan ( K R Λ ) ,

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