Abstract

We propose a simple iterative method for calculating the dynamic behavior of ring resonators with fast and slow (cumulative) optical nonlinearities when an optical pulse with an arbitrary-shaped envelope is incident into them. In the case of a slow nonlinearity, the nonlinear phase shift and nonlinear absorption are temporally-integrated over the incident pulse. In this paper, we consider two types of single-ring resonators made out of As2Se3 chalcogenide glass with high nonlinearity and investigate the dynamic properties (especially the effect of the cumulative nonlinearity on optical bistability) using known nonlinear material parameters. It is found that the cumulative nonlinearity suppresses overshoot and ringing after switching, decreases the width of the hysteresis loop between the input and output powers, and shifts its center corresponding to the operating point. The obtained results are useful in developing chalcogenide-based bistable optical devices and the proposed approach is applicable to modeling of a variety of nonlinear optical devices.

© 2011 OSA

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References

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    [CrossRef]
  5. T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
    [CrossRef]
  6. T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
    [CrossRef] [PubMed]
  7. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004).
    [CrossRef] [PubMed]
  8. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006).
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  9. D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. 6(11), 552–553 (1981).
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  10. J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994).
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  11. K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995).
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  13. H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998).
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    [CrossRef] [PubMed]
  16. K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
    [CrossRef] [PubMed]
  17. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
    [CrossRef]
  18. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
    [CrossRef]
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  22. K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009).
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  23. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979).
    [CrossRef]
  24. A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997).
    [CrossRef]
  25. K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. 31(7), 909–911 (2006).
    [CrossRef] [PubMed]
  26. A. Yariv, Optical Electronics (Holt, Rinehart and Winston, New York, 1985), p. 151.

2009 (2)

K. Ogusu and K. Shinkawa, “Optical nonlinearities in As2Se3 chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express 17(10), 8165–8172 (2009).
[CrossRef] [PubMed]

K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009).
[CrossRef]

2008 (2)

K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express 16(19), 14780–14791 (2008).
[CrossRef] [PubMed]

K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008).
[CrossRef] [PubMed]

2007 (1)

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

2006 (2)

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006).
[CrossRef] [PubMed]

K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. 31(7), 909–911 (2006).
[CrossRef] [PubMed]

2004 (4)

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004).
[CrossRef] [PubMed]

O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters–A tutorial overview,” J. Lightwave Technol. 22(5), 1380–1394 (2004).
[CrossRef]

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
[CrossRef] [PubMed]

2003 (1)

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

2002 (1)

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

1998 (2)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998).
[CrossRef]

1997 (1)

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997).
[CrossRef]

1996 (1)

K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. 32(9), 1537–1543 (1996).
[CrossRef]

1995 (1)

K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995).
[CrossRef] [PubMed]

1994 (1)

J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994).
[CrossRef]

1992 (1)

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

1981 (1)

D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. 6(11), 552–553 (1981).
[CrossRef] [PubMed]

1979 (1)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979).
[CrossRef]

Absil, P. P.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Almeida, V. R.

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004).
[CrossRef] [PubMed]

Amarnath, K.

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

Baker, N. J.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Calhoun, L. C.

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Capmany, J.

J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994).
[CrossRef]

Choi, D. Y.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Eggleton, B. J.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Finsterbusch, K.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Fraile-Pelaez, F. J.

J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994).
[CrossRef]

Fu, L.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Grover, R.

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Hagan, D. J.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Ho, P. T.

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

Ho, P.-P.

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Hoad, J. E.

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997).
[CrossRef]

Ibrahim, T. A.

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Ikeda, K.

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979).
[CrossRef]

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Johnson, F. G.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Kanakaraju, S.

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Kitao, M.

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
[CrossRef] [PubMed]

Kuo, L. C.

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

Lamont, M. R. E.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Li, H.

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998).
[CrossRef]

Lipson, M.

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006).
[CrossRef] [PubMed]

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004).
[CrossRef] [PubMed]

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Luther-Davies, B.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Lynch, S.

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997).
[CrossRef]

Madden, S.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Maeda, S.

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
[CrossRef] [PubMed]

Minakata, M.

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
[CrossRef] [PubMed]

Moss, D. J.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Muriel, M. A.

J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994).
[CrossRef]

Nguyen, H. C.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Nishio, H.

K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. 31(7), 909–911 (2006).
[CrossRef] [PubMed]

Oda, Y.

K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009).
[CrossRef]

Ogusu, K.

K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009).
[CrossRef]

K. Ogusu and K. Shinkawa, “Optical nonlinearities in As2Se3 chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express 17(10), 8165–8172 (2009).
[CrossRef] [PubMed]

K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express 16(19), 14780–14791 (2008).
[CrossRef] [PubMed]

K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008).
[CrossRef] [PubMed]

K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. 31(7), 909–911 (2006).
[CrossRef] [PubMed]

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
[CrossRef] [PubMed]

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998).
[CrossRef]

K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. 32(9), 1537–1543 (1996).
[CrossRef]

K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995).
[CrossRef] [PubMed]

Said, A. A.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Sarid, D.

D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. 6(11), 552–553 (1981).
[CrossRef] [PubMed]

Schwelb, O.

O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters–A tutorial overview,” J. Lightwave Technol. 22(5), 1380–1394 (2004).
[CrossRef]

Sheik-Bahae, M.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Shigekuni, H.

K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995).
[CrossRef] [PubMed]

Shinkawa, K.

K. Ogusu and K. Shinkawa, “Optical nonlinearities in As2Se3 chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express 17(10), 8165–8172 (2009).
[CrossRef] [PubMed]

K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express 16(19), 14780–14791 (2008).
[CrossRef] [PubMed]

K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008).
[CrossRef] [PubMed]

Steele, A. L.

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997).
[CrossRef]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Suzuki, K.

K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. 31(7), 909–911 (2006).
[CrossRef] [PubMed]

Ta’eed, V. G.

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

Van, V.

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Van Stryland, E. W.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Wang, J.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

Wei, T. H.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Xu, Q.

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006).
[CrossRef] [PubMed]

Yamasaki, J.

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
[CrossRef] [PubMed]

Yokota, Y.

K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995).
[CrossRef] [PubMed]

Young, J.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

IEEE J. Quantum Electron. (3)

J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994).
[CrossRef]

K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. 32(9), 1537–1543 (1996).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[CrossRef]

J. Lightwave Technol. (1)

O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters–A tutorial overview,” J. Lightwave Technol. 22(5), 1380–1394 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. 9(3), 405–414 (1992).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009).
[CrossRef]

Opt. Commun. (3)

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979).
[CrossRef]

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997).
[CrossRef]

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998).
[CrossRef]

Opt. Express (4)

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007).
[CrossRef] [PubMed]

K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express 16(19), 14780–14791 (2008).
[CrossRef] [PubMed]

K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008).
[CrossRef] [PubMed]

K. Ogusu and K. Shinkawa, “Optical nonlinearities in As2Se3 chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express 17(10), 8165–8172 (2009).
[CrossRef] [PubMed]

Opt. Lett. (7)

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29(3), 265–267 (2004).
[CrossRef] [PubMed]

K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995).
[CrossRef] [PubMed]

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004).
[CrossRef] [PubMed]

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004).
[CrossRef] [PubMed]

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006).
[CrossRef] [PubMed]

D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. 6(11), 552–553 (1981).
[CrossRef] [PubMed]

K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. 31(7), 909–911 (2006).
[CrossRef] [PubMed]

Other (3)

A. Yariv, Optical Electronics (Holt, Rinehart and Winston, New York, 1985), p. 151.

D. G. Rabus, Integrated ring resonators (Springer, Berlin, 2007).

A. Zakery, and S. R. Elliott, Optical nonlinearities in chalcogenide glasses and their applications (Springer, Berlin, 2007).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of single-ring resonators and the definition of electric fields for analysis. (a) A single-coupler ring resonator. (b) A double-coupler ring resonator.

Fig. 2
Fig. 2

Segmentation of the ring waveguide and numerical calculation over one round-trip for analyzing the nonlinear ring resonator shown in Fig. 1(a). The filled symbols indicate known data and the open symbols represent values to be determined.

Fig. 3
Fig. 3

Temporal change in the power transmitted from the output port when a Gaussian pulse with a peak power of 3 W and a pulse width of 500 ps is incident into the nonlinear single-coupler ring resonator with κ=0.1and Δϕ 0=−0.1π for two values of the coupler loss γ: (a) γ=0, (b) γ=0.1.

Fig. 4
Fig. 4

Input-output characteristics corresponding to Fig. 3.

Fig. 6
Fig. 6

Input-output characteristics of the nonlinear single-coupler ring resonator with γ=0.1, κ=0.1, and Δϕ 0=−0.1π for three values of pulse width τ P.

Fig. 5
Fig. 5

Input-output characteristics of the nonlinear single-coupler ring resonator with γ=0.1 and κ=0.1 for three values of initial detuning Δϕ 0. The incident pulse width is τ P=500 ps.

Fig. 7
Fig. 7

Temporal change in the power transmitted from the two output ports when a Gaussian pulse with a peak power of 3 W and a pulse width of 500 ps is incident in the nonlinear double-coupler ring resonator with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1 (b). For comparison, the results are given for two cases (ii) and (iii).

Fig. 8
Fig. 8

Input-output characteristics of the nonlinear double-coupler ring resonator with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1 (b) for two values of pulse width τ P. For comparison, the results are given for two cases (ii) and (iii).

Equations (23)

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n = n 0 + n 2 I = n 0 + n 2 n 0 2 η 0 | E | 2 = n 0 + n 2 P S eff ,
E out ( t ) = 1 γ ( 1 κ E in ( t ) j κ E c ( L , t ) ) ,
E c ( 0 , t ) = 1 γ ( j κ E in ( t ) + 1 κ E c ( L , t ) ) ,
E c ( L , t ) = E c ( 0 , t τ R ) exp ( α L / 2 ) exp [ j ( ϕ 0 + ϕ N ( t τ R ) ) ] ,
ϕ N ( t τ R ) = n 0 n 2 k 0 2 η 0 0 L | E c ( z , t τ R + n 0 z / c ) | 2 d z = n 0 n 2 k 0 2 η 0 | E c ( 0 , t τ R ) | 2 1 exp ( α L ) α ,
E out ( t ) = 1 γ { 1 κ E in ( t ) j κ E c ( 0 , t τ R ) exp ( α L / 2 ) exp [ j ( ϕ 0 + ϕ N ( t τ R ) ) ] } ,
E c ( 0 , t ) = 1 γ { j κ E in ( t ) + 1 κ E c ( 0 , t τ R ) exp ( α L / 2 ) exp [ j ( ϕ 0 + ϕ N ( t τ R ) ) ] } ,
E c ( 0 , t τ R ) = | E c ( 0 , t τ R ) | exp ( j θ 0 ( t τ R ) ) ,
E c ( L , t ) = | E c ( L , t ) | exp [ j ( θ 0 ( t τ R ) + ϕ 0 + ϕ N ( t τ R ) ) ] ,
d I ( z , t ) d z = ( α + β I ( z , t ) + σ ab N ( z , t ) ) I ( z , t ) ,
d ϕ N ( z , t ) d z = k 0 Δ n = k 0 ( n 2 I ( z , t ) + σ r N ( z , t ) ) ,
d N ( z , t ) d t = α I ( z , t ) ω N ( z , t ) τ ,
N ( z , t ) =     t α ω I ( z , t ) d t .
N ( 0 , t τ R ) = N ( 0 , t τ R Δ t ) + ( α Δ t / ω ) I ( 0 , t τ R Δ t ) I ( Δ z , t τ R + Δ t ) = I ( 0 , t τ R ) exp [ ( α + β I ( 0 , t τ R )                                                                       at       z = 0 ,                                                                                               + σ ab N ( 0 , t τ R ) ) Δ z ]
N ( Δ z , t τ R + Δ t ) = N ( Δ z , t τ R ) + ( α Δ t / ω ) I ( Δ z , t τ R ) I ( 2 Δ z , t τ R + 2 Δ t ) = I ( Δ z , t τ R + Δ t ) exp [ ( α + β I ( Δ z , t τ R + Δ t )                           at     z = Δ z ,                                                                                                               + σ ab N ( Δ z , t τ R + Δ t ) ) Δ z ]                                                                                                          
N ( L Δ z , t Δ t ) = N ( L Δ z , t 2 Δ t ) + ( α Δ t / ω ) I ( L Δ z , t 2 Δ t ) I ( L , t ) = I ( L Δ z , t Δ t ) exp [ ( α + β I ( L Δ z , t Δ t )                                         at       z = L Δ z ,                                                                                                                                                                                                                                       + σ ab N ( L Δ z , t Δ t ) ) Δ z ]
N ( L , t ) = N ( L , t Δ t ) + ( α Δ t / ω ) I ( L , t Δ t )                                                     at       z = L .
φ N ( t τ R ) = k 0 n 0 Δ z i = 0 M 1 I ( i Δ z , t τ R + i Δ t ) + I ( ( i + 1 ) Δ z , t τ R + ( i + 1 ) Δ t ) 2                                                                               + k 0 σ r Δ z i = 0 M 1 N ( i Δ z , t τ R + i Δ t ) + N ( ( i + 1 ) Δ z , t τ R + ( i + 1 ) Δ t ) 2           .
E r ( t ) = 1 γ ( 1 κ E in ( t ) j κ E c 2 ( L , t ) ) ,
E c 1 ( 0 , t ) = 1 γ ( j κ E in ( t ) + 1 κ E c 2 ( L , t ) ) ,
E t ( t ) = j 1 γ κ E c 1 ( L / 2 , t ) ,
E c 2 ( L / 2 , t ) = 1 γ 1 κ E c 1 ( L / 2 , t ) ,
τ c = n 0 L c [ 1 ( 1 γ ) ( 1 κ ) e α L ] n 0 L c κ       f o r       a < < 1       a n d         γ < < 1.

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