Abstract

Microring tower resonators, which are a chain of microring resonators stacked on top of each other, are of great interest for nonlinear optics due to their unique features such as very high compactness, coupling efficiency and quality factor. In this research, we investigate the optical bistability in microring tower (MRT) with Kerr nonlinearity by using the coupled mode theory, and demonstrate how a proper defect into the structure can lead to low threshold bistability. In particular, we observed optical bistability in nonlinear defect modes with switching power as low as 165 μ Wthrough numerical calculations in a structure with a overall loss on the order of 0.01mm−1 . In addition, we also develop an analytical model that excellently gives the position of defect modes in linear regime.

©2010 Optical Society of America

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References

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  1. H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, 1985).
  2. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
    [Crossref] [PubMed]
  3. A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams, “Power and wavelength polarization bistability with very wide hysteresis cycles in a 1550 nm-VCSEL subject to orthogonal optical injection,” Opt. Express 17(26), 23637–23642 (2009).
    [Crossref]
  4. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30(19), 2575–2577 (2005).
    [Crossref] [PubMed]
  5. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
    [Crossref]
  6. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
    [Crossref]
  7. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  8. M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005).
    [Crossref] [PubMed]
  9. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009).
    [Crossref] [PubMed]
  10. D. N. Christodoulides and E. D. Eugenieva, “Minimizing bending losses in two-dimensional discrete soliton networks,” Opt. Lett. 26(23), 1876–1878 (2001).
    [Crossref]
  11. K. Okamoto, Fundamentals of Optical Waveguides, (Elsevier, 2006), Chap. 4.
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    [Crossref]
  13. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13(9), 794–796 (1988).
    [Crossref] [PubMed]
  14. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
    [Crossref]
  15. A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
    [Crossref]
  16. A. Shinya, S. Matsuo, T. Yosia, E. Tanabe, T. Kuramochi, T. Sato, Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express 16(23), 19382–19387 (2008).
    [Crossref]
  17. H. Zhang, V. Gauss, P. Wen, and S. Esener, “Observation of wavelength and multiple bistabilities in 850nm Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs),” Opt. Express 15(18), 11723–11730 (2007).
    [Crossref] [PubMed]
  18. E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007).
    [Crossref]
  19. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005).
    [Crossref] [PubMed]
  20. N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008).
    [Crossref] [PubMed]
  21. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B 21(9), 1665–1673 (2004).
    [Crossref]
  22. M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004).
    [Crossref] [PubMed]
  23. F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express 15(12), 7888–7893 (2007).
    [Crossref] [PubMed]
  24. L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express 17(23), 21108–21117 (2009).
    [Crossref] [PubMed]

2009 (3)

2008 (2)

2007 (3)

2005 (4)

2004 (2)

2002 (1)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
[Crossref]

2001 (1)

1998 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

1993 (1)

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
[Crossref]

1988 (1)

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18(10), 1580–1583 (1982).
[Crossref]

Adams, M. J.

Agha-Bolorizadeh, M.

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Baets, R.

Bogaerts, W.

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Brambilla, G.

Broderick, N. G. R.

Cassette, S.

E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007).
[Crossref]

Christodoulides, D. N.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Combri’e, S.

E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007).
[Crossref]

de Rossi, A.

E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007).
[Crossref]

Dumon, P.

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Esener, S.

Eugenieva, E. D.

Fink, Y.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
[Crossref]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Gauss, V.

Haret, L. D.

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Horak, P.

Hurtado, A.

Ibanescu, M.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
[Crossref]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18(10), 1580–1583 (1982).
[Crossref]

Joannopoulos, J. D.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
[Crossref]

Johnson, S. G.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
[Crossref]

Joseph, R. I.

Kakitsuka,

Khanzadeh, M.

Kira, G.

Kuramochi, E.

Kuramochi, T.

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Lin, C.-H.

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
[Crossref]

Lin, H.-H.

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
[Crossref]

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Matsuo, S.

Mitsugi, S.

Moghaddam, R. F.

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Morthier, G.

Notomi, M.

Pesquera, L.

Poon, J. K. S.

Priem, G.

Quirce, A.

Sato, T.

Scheuer, J.

Shafiei, M.

Shinya, A.

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Soljacic, M.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
[Crossref]

Stegeman, G.

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
[Crossref]

Sumetsky, M.

Tanabe, E.

Tanabe, T.

Tran, N.

E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007).
[Crossref]

Valle, A.

Van Thourhout, D.

Villeneuve, A.

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
[Crossref]

Weidner, E.

E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007).
[Crossref]

Wen, P.

Xu, F.

Xu, Y.

Yang, C.

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
[Crossref]

Yariv, A.

Yosia, T.

Zhang, H.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

A. Villeneuve, C. Yang, G. Stegeman, C.-H. Lin, and H.-H. Lin, “Nonlinear refractive-index and two photon-absorption near half the band gap in AlGaAs,” Appl. Phys. Lett. 62(20), 2465–2467 (1993).
[Crossref]

E. Weidner, S. Combri’e, A. de Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101–118 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18(10), 1580–1583 (1982).
[Crossref]

J. Lightwave Technol. (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Micro-ring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

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

Opt. Express (10)

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
[Crossref] [PubMed]

M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005).
[Crossref] [PubMed]

M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004).
[Crossref] [PubMed]

L. D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express 17(23), 21108–21117 (2009).
[Crossref] [PubMed]

A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams, “Power and wavelength polarization bistability with very wide hysteresis cycles in a 1550 nm-VCSEL subject to orthogonal optical injection,” Opt. Express 17(26), 23637–23642 (2009).
[Crossref]

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005).
[Crossref] [PubMed]

F. Xu, P. Horak, and G. Brambilla, “Optical microfiber coil resonator refractometric sensor,” Opt. Express 15(12), 7888–7893 (2007).
[Crossref] [PubMed]

H. Zhang, V. Gauss, P. Wen, and S. Esener, “Observation of wavelength and multiple bistabilities in 850nm Vertical-Cavity Semiconductor Optical Amplifiers (VCSOAs),” Opt. Express 15(18), 11723–11730 (2007).
[Crossref] [PubMed]

N. G. R. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008).
[Crossref] [PubMed]

A. Shinya, S. Matsuo, T. Yosia, E. Tanabe, T. Kuramochi, T. Sato, Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express 16(23), 19382–19387 (2008).
[Crossref]

Opt. Lett. (3)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 055601–055604 (2002).
[Crossref]

Phys. Rev. Lett. (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Other (3)

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

H. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, 1985).

K. Okamoto, Fundamentals of Optical Waveguides, (Elsevier, 2006), Chap. 4.

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Figures (8)

Fig. 1
Fig. 1 a) Sketch of MRT with N rings. b) Schematic of an infinite periodic VMR structure with a defect.

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Fig. 2
Fig. 2 Defect modes B d e f f versus δ B d e f for two different defect modes: (right) positive and (left) negative defects. The Curves denote the model results assuming l = 68 ; and K = 0.8. Filled circles denote the numerical results where the number of rings is 9 and the coupling parameter between the input waveguide and its adjacent ring, K 0 , is 0.8.

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Fig. 3
Fig. 3 dependence of the Q factor a) on the δ β d e f , and b) on the K0 and K.

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Fig. 4
Fig. 4 a) Detailed spectrum of a resonant mode at different input powers for a lossless uniform MRT with N = 9, K 0 = 0.2 , and K = 2. b) Output power ( P o u t ) versus input power ( P i n ) for various detuning parameters.

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Fig. 5
Fig. 5 nonlinear response of the structure of Fig. 4 for another mode near B = 425.389 for three different detuning parameters.

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Fig. 6
Fig. 6 a) A section of transmission spectrum of a lossless MRT ( N = 9 , K = K 0 = 0.8 , and δ B d e f e c t = 1.6 ) for low input powers, a sharp dip shows a defect mode near B = 424.998 . b) Nonlinear transmission for different input powers. c) - d) Input/Output characteristic for two different δ .

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Fig. 7
Fig. 7 a) Nonlinear transmission response of a) a lossless MRT with parameters: N = 11 , K = 0.6 , K 0 = 0.4 , and δ B d e f e c t = 1.3 , at different input powers. b) A modified design, where the dimensionless propagation constant of first and last rings are changed by the amount of Δ B = + 0.85.

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Fig. 8
Fig. 8 a) Filled circles show the calculated quality factors with different losses for the microcavity of the Fig. 7(a). The solid line is the curve 8.51 / ( 0.13 α + 1 ) . b) The relationship between bistability threshold and loss. The solid line is the curve c × α 2 with appropriate constant c.

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Equations (13)

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

i d A n / d s + κ n − 1 , n A n − 1 e i ( β n − 1 − β n ) s + κ n , n + 1 A n + 1 e i ( β n + 1 − β n ) s = 0   , n = 0 , ± 1 , ± 2 , ⋅ ⋅ ⋅
A n ( 0 ) = A n ( L ) exp ( i β n L ) ,
{ i d a 1 / d s + κ ′ a 0 + κ a 2 = 0 i d a 0 / d s + δ β d e f a 0 + κ ′ ( a − 1 + a 1 ) = 0 i d a − 1 / d s + κ a 0 + κ ′ a − 2 = 0   .
a n = { e i μ s ( e i ξ z n + r e − i ξ z n )    ​ ,    n < 0 q e i μ s                                    ,    n = 0 t e i μ s e i ξ z n                          ​ ,   n > 0
r = t − 1 , q = κ t / κ ′ ,
t = 2 i κ ′ 2 sin ξ d 2 ( κ ′ 2 − κ 2 ) cos ξ d + 2 i κ ′ 2 sin ξ d + κ δ β d e f .
B = 2 π l − 2 K cos ( ξ d ) ,
t = − K ′ 2 [ ( B − 2 π l ) 2 − 4 K 2 ] 1 / 2 ( K ′ 2 − K 2 ) ( 2 π l − B ) − K ′ 2 [ ( B − 2 π l ) 2 − 4 K 2 ] 1 / 2 + K 2 δ B d e f ,
B d e f = 2 π l − δ B d e f ( K ′ 2 − K 2 ) ± K ′ 2 δ B d e f 2 − 4 ( K 2 − 2 K ′ 2 ) K 2 − 2 K ′ 2 .
i d A m d s + κ m , m − 1 A m − 1 e i ( β m − 1 − β m ) s + γ | A m | 2 A m = 0 i d A n d s n + κ n , n − 1 A n − 1 e i ( β n − 1 − β n ) s + κ n + 1 , n A n + 1 e i ( β n + 1 − β n ) s + γ | A n | 2 A n = 0   i d A − m d s − m + κ − m + 1 , − m A − m + 1 e i ( β − m + 1 − β − m ) s + γ | A − m | 2 A − m = 0
A n ( 0 ) = A n ( L ) exp ( i β n L − α L ) ,                   n = 0 , ± 1 , … , ± ( m − 1 ) ,
A ± ( m + 1 ) ( o u t ) = ( 1 − ρ ) 1 / 2 [ cos ( K 0 ) A ± ( m + 1 ) ( i n ) + i sin ( K 0 ) A ± m ( L ) ] A ± m ( 0 ) = ( 1 − ρ ) 1 / 2 [ i sin ( K 0 ) A ± ( m + 1 ) ( i n ) + cos ( K 0 ) A ± m ( L ) ]
A ⋅ x = b

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