Abstract

Gain saturation is treated by formulating a model that accounts for the dependence of gain on irradiance level in distributed-feedback lasers. Solutions of the coupled-mode equations of the model are obtained numerically and are presented for both overcoupled and undercoupled laser structures (KL in the 10−13–10 range). Parasitic losses that are distributed uniformly within the structure are found to influence the output power characteristics of distributed-feedback lasers. Also the optimum coupling strength that results in the most efficient conversion of pump power to useful laser output is determined. The optimum coupling strength for any distributed-feedback laser is dependent on the magnitude of the parasitic losses and on the required output-power level.

© 1975 Optical Society of America

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  1. H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
    [CrossRef]
  2. C. V. Shank, J. E. Bjorkholm, H. Kogelnik, Appl. Phys. Lett. 18, 394 (1971).
    [CrossRef]
  3. I. P. Kaminow, H. P. Weber, E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
    [CrossRef]
  4. R. L. Fork, K. R. German, E. A. Chandross, Appl. Phys. Lett. 20, 139 (1972).
    [CrossRef]
  5. P. Zory, Appl. Phys. Lett. 22, 125 (1973).
    [CrossRef]
  6. K. O. Hill, A. Watanabe, Opt. Commun. 5, 389 (1972).
    [CrossRef]
  7. K. O. Hill, A. Watanabe, Appl. Opt. 12, 430 (1973).
    [CrossRef] [PubMed]
  8. J. E. Bjorkholm, C. V. Shank, IEEE J. Quantum Electron. QE. 8, 133 (1972).
  9. J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
    [CrossRef]
  10. D. P. Schinke, R. G. Smith, E. G. Spencer, M. F. Galvin, Appl. Phys. Lett. 21, 494 (1972).
    [CrossRef]
  11. S. Wang, S. Sheem, Appl. Phys. Lett. 22, 460 (1973).
    [CrossRef]
  12. M. Nakamura, A. Yariv, H. W. Yen, S. Somekh, Appl. Phys. Lett. 22, 515 (1973).
    [CrossRef]
  13. M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
    [CrossRef]
  14. D. R. Scifres, R. D. Burnham, W. Streifer, Post-deadline paper presented at VIII International Quantum Electronics Conference, San Francisco, Calif. (10–13 June 1974).
  15. H. Kogelnik, C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
    [CrossRef]
  16. D. Marcuse, IEEE J. Quantum Electron. QE-8, 661 (1972).
    [CrossRef]
  17. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]
  18. K. Sakuda, A. Yariv, Opt. Commun. 8, 1 (1973).
    [CrossRef]
  19. H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
    [CrossRef]
  20. S. Wang, J. Appl. Phys. 44, 767 (1973).
    [CrossRef]
  21. S. R. Chinn, IEEE J. Quantum Electron. QE-9, 574 (1973).
    [CrossRef]
  22. R. E. DeWames, W. F. Hall, Appl. Phys. Lett. 23, 28 (1973).
    [CrossRef]
  23. S. Wang, IEEE J. Quantum Electron. QE-10, 413 (1974).
    [CrossRef]
  24. R. F. Cordero, S. Wang, Appl. Phys. Lett. 24, 474 (1974).
    [CrossRef]
  25. A. Yariv, H. W. Yen, Opt. Commun. 10, 120 (1974).
    [CrossRef]
  26. S. R. Chinn, P. L. Kelly, Opt. Commun. 10, 123 (1974).
    [CrossRef]
  27. R. Shubert, J. Appl. Phys. 45, 209 (1974).
    [CrossRef]
  28. M. Nakamura, A. Yariv, Opt. Commun. 11, 18 (1974).
    [CrossRef]
  29. See, for example, A. Yariv, Quantum ElectronicsWiley, New York, 1967) or A. Maitland, M. H. Dunn, Laser Physics (American Elsevier, New York, 1969).
  30. J. P. Wittke, RCA Rev. 33, 674 (1972).
  31. K. O. Hill, J. Opt. Soc. Am. 64, 525 (1974).
    [CrossRef]

1974 (7)

S. Wang, IEEE J. Quantum Electron. QE-10, 413 (1974).
[CrossRef]

R. F. Cordero, S. Wang, Appl. Phys. Lett. 24, 474 (1974).
[CrossRef]

A. Yariv, H. W. Yen, Opt. Commun. 10, 120 (1974).
[CrossRef]

S. R. Chinn, P. L. Kelly, Opt. Commun. 10, 123 (1974).
[CrossRef]

R. Shubert, J. Appl. Phys. 45, 209 (1974).
[CrossRef]

M. Nakamura, A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

K. O. Hill, J. Opt. Soc. Am. 64, 525 (1974).
[CrossRef]

1973 (12)

P. Zory, Appl. Phys. Lett. 22, 125 (1973).
[CrossRef]

K. O. Hill, A. Watanabe, Appl. Opt. 12, 430 (1973).
[CrossRef] [PubMed]

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

S. Wang, S. Sheem, Appl. Phys. Lett. 22, 460 (1973).
[CrossRef]

M. Nakamura, A. Yariv, H. W. Yen, S. Somekh, Appl. Phys. Lett. 22, 515 (1973).
[CrossRef]

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

K. Sakuda, A. Yariv, Opt. Commun. 8, 1 (1973).
[CrossRef]

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

S. Wang, J. Appl. Phys. 44, 767 (1973).
[CrossRef]

S. R. Chinn, IEEE J. Quantum Electron. QE-9, 574 (1973).
[CrossRef]

R. E. DeWames, W. F. Hall, Appl. Phys. Lett. 23, 28 (1973).
[CrossRef]

1972 (7)

H. Kogelnik, C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

D. Marcuse, IEEE J. Quantum Electron. QE-8, 661 (1972).
[CrossRef]

D. P. Schinke, R. G. Smith, E. G. Spencer, M. F. Galvin, Appl. Phys. Lett. 21, 494 (1972).
[CrossRef]

J. E. Bjorkholm, C. V. Shank, IEEE J. Quantum Electron. QE. 8, 133 (1972).

K. O. Hill, A. Watanabe, Opt. Commun. 5, 389 (1972).
[CrossRef]

R. L. Fork, K. R. German, E. A. Chandross, Appl. Phys. Lett. 20, 139 (1972).
[CrossRef]

J. P. Wittke, RCA Rev. 33, 674 (1972).

1971 (3)

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, Appl. Phys. Lett. 18, 394 (1971).
[CrossRef]

I. P. Kaminow, H. P. Weber, E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

J. E. Bjorkholm, C. V. Shank, IEEE J. Quantum Electron. QE. 8, 133 (1972).

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, Appl. Phys. Lett. 18, 394 (1971).
[CrossRef]

Burnham, R. D.

D. R. Scifres, R. D. Burnham, W. Streifer, Post-deadline paper presented at VIII International Quantum Electronics Conference, San Francisco, Calif. (10–13 June 1974).

Chandross, E. A.

R. L. Fork, K. R. German, E. A. Chandross, Appl. Phys. Lett. 20, 139 (1972).
[CrossRef]

I. P. Kaminow, H. P. Weber, E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[CrossRef]

Chinn, S. R.

S. R. Chinn, P. L. Kelly, Opt. Commun. 10, 123 (1974).
[CrossRef]

S. R. Chinn, IEEE J. Quantum Electron. QE-9, 574 (1973).
[CrossRef]

Cordero, R. F.

R. F. Cordero, S. Wang, Appl. Phys. Lett. 24, 474 (1974).
[CrossRef]

DeWames, R. E.

R. E. DeWames, W. F. Hall, Appl. Phys. Lett. 23, 28 (1973).
[CrossRef]

Fork, R. L.

R. L. Fork, K. R. German, E. A. Chandross, Appl. Phys. Lett. 20, 139 (1972).
[CrossRef]

Galvin, M. F.

D. P. Schinke, R. G. Smith, E. G. Spencer, M. F. Galvin, Appl. Phys. Lett. 21, 494 (1972).
[CrossRef]

Garmire, E.

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

Garvin, H. L.

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

German, K. R.

R. L. Fork, K. R. German, E. A. Chandross, Appl. Phys. Lett. 20, 139 (1972).
[CrossRef]

Hall, W. F.

R. E. DeWames, W. F. Hall, Appl. Phys. Lett. 23, 28 (1973).
[CrossRef]

Hill, K. O.

K. O. Hill, J. Opt. Soc. Am. 64, 525 (1974).
[CrossRef]

K. O. Hill, A. Watanabe, Appl. Opt. 12, 430 (1973).
[CrossRef] [PubMed]

K. O. Hill, A. Watanabe, Opt. Commun. 5, 389 (1972).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, H. P. Weber, E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[CrossRef]

Kelly, P. L.

S. R. Chinn, P. L. Kelly, Opt. Commun. 10, 123 (1974).
[CrossRef]

Kogelnik, H.

H. Kogelnik, C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, Appl. Phys. Lett. 18, 394 (1971).
[CrossRef]

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. QE-8, 661 (1972).
[CrossRef]

Nakamura, M.

M. Nakamura, A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

M. Nakamura, A. Yariv, H. W. Yen, S. Somekh, Appl. Phys. Lett. 22, 515 (1973).
[CrossRef]

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

Sakuda, K.

K. Sakuda, A. Yariv, Opt. Commun. 8, 1 (1973).
[CrossRef]

Schinke, D. P.

D. P. Schinke, R. G. Smith, E. G. Spencer, M. F. Galvin, Appl. Phys. Lett. 21, 494 (1972).
[CrossRef]

Scifres, D. R.

D. R. Scifres, R. D. Burnham, W. Streifer, Post-deadline paper presented at VIII International Quantum Electronics Conference, San Francisco, Calif. (10–13 June 1974).

Shank, C. V.

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

J. E. Bjorkholm, C. V. Shank, IEEE J. Quantum Electron. QE. 8, 133 (1972).

H. Kogelnik, C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, Appl. Phys. Lett. 18, 394 (1971).
[CrossRef]

Sheem, S.

S. Wang, S. Sheem, Appl. Phys. Lett. 22, 460 (1973).
[CrossRef]

Shubert, R.

R. Shubert, J. Appl. Phys. 45, 209 (1974).
[CrossRef]

Smith, R. G.

D. P. Schinke, R. G. Smith, E. G. Spencer, M. F. Galvin, Appl. Phys. Lett. 21, 494 (1972).
[CrossRef]

Somekh, S.

M. Nakamura, A. Yariv, H. W. Yen, S. Somekh, Appl. Phys. Lett. 22, 515 (1973).
[CrossRef]

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

Sosnowski, T. P.

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

Spencer, E. G.

D. P. Schinke, R. G. Smith, E. G. Spencer, M. F. Galvin, Appl. Phys. Lett. 21, 494 (1972).
[CrossRef]

Stoll, H.

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

Streifer, W.

D. R. Scifres, R. D. Burnham, W. Streifer, Post-deadline paper presented at VIII International Quantum Electronics Conference, San Francisco, Calif. (10–13 June 1974).

Wang, S.

S. Wang, IEEE J. Quantum Electron. QE-10, 413 (1974).
[CrossRef]

R. F. Cordero, S. Wang, Appl. Phys. Lett. 24, 474 (1974).
[CrossRef]

S. Wang, J. Appl. Phys. 44, 767 (1973).
[CrossRef]

S. Wang, S. Sheem, Appl. Phys. Lett. 22, 460 (1973).
[CrossRef]

Watanabe, A.

K. O. Hill, A. Watanabe, Appl. Opt. 12, 430 (1973).
[CrossRef] [PubMed]

K. O. Hill, A. Watanabe, Opt. Commun. 5, 389 (1972).
[CrossRef]

Weber, H. P.

I. P. Kaminow, H. P. Weber, E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[CrossRef]

Wittke, J. P.

J. P. Wittke, RCA Rev. 33, 674 (1972).

Yariv, A.

M. Nakamura, A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

A. Yariv, H. W. Yen, Opt. Commun. 10, 120 (1974).
[CrossRef]

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

M. Nakamura, A. Yariv, H. W. Yen, S. Somekh, Appl. Phys. Lett. 22, 515 (1973).
[CrossRef]

K. Sakuda, A. Yariv, Opt. Commun. 8, 1 (1973).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

See, for example, A. Yariv, Quantum ElectronicsWiley, New York, 1967) or A. Maitland, M. H. Dunn, Laser Physics (American Elsevier, New York, 1969).

Yen, H. W.

A. Yariv, H. W. Yen, Opt. Commun. 10, 120 (1974).
[CrossRef]

M. Nakamura, A. Yariv, H. W. Yen, S. Somekh, Appl. Phys. Lett. 22, 515 (1973).
[CrossRef]

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

Zory, P.

P. Zory, Appl. Phys. Lett. 22, 125 (1973).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (12)

H. Kogelnik, C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).
[CrossRef]

C. V. Shank, J. E. Bjorkholm, H. Kogelnik, Appl. Phys. Lett. 18, 394 (1971).
[CrossRef]

I. P. Kaminow, H. P. Weber, E. A. Chandross, Appl. Phys. Lett. 18, 497 (1971).
[CrossRef]

R. L. Fork, K. R. German, E. A. Chandross, Appl. Phys. Lett. 20, 139 (1972).
[CrossRef]

P. Zory, Appl. Phys. Lett. 22, 125 (1973).
[CrossRef]

J. E. Bjorkholm, T. P. Sosnowski, C. V. Shank, Appl. Phys. Lett. 22, 132 (1973).
[CrossRef]

D. P. Schinke, R. G. Smith, E. G. Spencer, M. F. Galvin, Appl. Phys. Lett. 21, 494 (1972).
[CrossRef]

S. Wang, S. Sheem, Appl. Phys. Lett. 22, 460 (1973).
[CrossRef]

M. Nakamura, A. Yariv, H. W. Yen, S. Somekh, Appl. Phys. Lett. 22, 515 (1973).
[CrossRef]

M. Nakamura, H. W. Yen, A. Yariv, E. Garmire, S. Somekh, H. L. Garvin, Appl. Phys. Lett. 23, 224 (1973).
[CrossRef]

R. E. DeWames, W. F. Hall, Appl. Phys. Lett. 23, 28 (1973).
[CrossRef]

R. F. Cordero, S. Wang, Appl. Phys. Lett. 24, 474 (1974).
[CrossRef]

IEEE J. Quantum Electron. (5)

S. Wang, IEEE J. Quantum Electron. QE-10, 413 (1974).
[CrossRef]

S. R. Chinn, IEEE J. Quantum Electron. QE-9, 574 (1973).
[CrossRef]

J. E. Bjorkholm, C. V. Shank, IEEE J. Quantum Electron. QE. 8, 133 (1972).

D. Marcuse, IEEE J. Quantum Electron. QE-8, 661 (1972).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

J. Appl. Phys. (3)

H. Kogelnik, C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[CrossRef]

S. Wang, J. Appl. Phys. 44, 767 (1973).
[CrossRef]

R. Shubert, J. Appl. Phys. 45, 209 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

K. O. Hill, J. Opt. Soc. Am. 64, 525 (1974).
[CrossRef]

Opt. Commun. (6)

M. Nakamura, A. Yariv, Opt. Commun. 11, 18 (1974).
[CrossRef]

A. Yariv, H. W. Yen, Opt. Commun. 10, 120 (1974).
[CrossRef]

S. R. Chinn, P. L. Kelly, Opt. Commun. 10, 123 (1974).
[CrossRef]

K. Sakuda, A. Yariv, Opt. Commun. 8, 1 (1973).
[CrossRef]

H. Stoll, A. Yariv, Opt. Commun. 8, 5 (1973).
[CrossRef]

K. O. Hill, A. Watanabe, Opt. Commun. 5, 389 (1972).
[CrossRef]

RCA Rev. (1)

J. P. Wittke, RCA Rev. 33, 674 (1972).

Other (2)

See, for example, A. Yariv, Quantum ElectronicsWiley, New York, 1967) or A. Maitland, M. H. Dunn, Laser Physics (American Elsevier, New York, 1969).

D. R. Scifres, R. D. Burnham, W. Streifer, Post-deadline paper presented at VIII International Quantum Electronics Conference, San Francisco, Calif. (10–13 June 1974).

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Figures (19)

Fig. 1
Fig. 1

The normalized irradiance distributions (solid lines) and the normalized amplitude distributions of the positive-going waves (dashed lines) as a function of normalized distance in lossless laser for N = 1 modes at threshold (P = 0) and with (a) KL = 10−5, (b) KL = 1, (c) KL = 10.

Fig. 2
Fig. 2

The phase distributions (in degrees) of the positive-going waves for the three cases shown in Fig. 1.

Fig. 3
Fig. 3

The normalized irradiance and amplitude distributions for the same conditions as given in Fig. 1, except that P = 30

Fig. 4
Fig. 4

The normalized irradiance (solid lines) and amplitude (dashed lines) distributions for N = 2 modes at threshold with (a) KL = 10−5, (b) KL = 1, (c) KL = 10 in a lossless laser.

Fig. 5
Fig. 5

The phase distributions (in degrees) of the positive-going waves for the three cases shown in Fig. 4.

Fig. 6
Fig. 6

The normalized irradiance (solid lines) and amplitude (dashed lines) distributions for N = 2 modes for an output-power level P = 30 and with (a) KL = 10−5, (b) KL = 1, (c) KL = 10 in a lossless laser.

Fig. 7
Fig. 7

The unsaturated-gain eigenvalue αuL of the N = 1 mode as a function of the frequency-shift eigenvalue δL for several values of output-power level P in a lossless laser. The dashed lines connect points of equal coupling strengths KL.

Fig. 8
Fig. 8

The unsaturated-gain eigenvalue αuL as a function of δL for the N = 2 mode of a lossless laser.

Fig. 9
Fig. 9

The dependence of the unsaturated-gain eigenvalue on the coupling constant KL for several values of the output-power level P of the N = 1 mode of a lossless laser.

Fig. 10
Fig. 10

The unsaturated-gain eigenvalue αuL as a function of KL for the N = 2 mode of a lossless laser.

Fig. 11
Fig. 11

The dependence of the output-power level P on the unsaturated-gain eigenvalue αuL for several values of the coupling constant KL. The results are given for the N = 1 mode of a lossless laser.

Fig. 12
Fig. 12

The normalized irradiance (solid lines) and the normalized amplitude of the positive-going waves (dashed lines) as a function of normalized distance in the laser for N = 1 modes with P = 10 and KL = 10−5 and parasitic losses (a) αLL = 0, (b) αLL = 5.

Fig. 13
Fig. 13

The normalized irradiance and amplitudes for the same conditions as in Fig. 12, except that KL = 1.

Fig. 14
Fig. 14

The dependence of the unsaturated-gain eigenvalue αuL on the parasitic-loss constant αLL for several values of the output-power level P of the N = 1 mode of a critically coupled laser.

Fig. 15
Fig. 15

The dependence of αuL on αLL of the N = 1 mode of an overcoupled laser (KL = 10).

Fig. 16
Fig. 16

The unsaturated-gain eigenvalue as a function of the coupling constant at several values of the output-power level of the N = 1 mode for a laser with moderate parasitic losses (αLL = 0.5).

Fig. 17
Fig. 17

The unsaturated-gain eigenvalue αuL as a function of KL for the N = 1 mode of a laser with high parasitic losses (αLL = 5).

Fig. 18
Fig. 18

The eigenvalue diagram for the N = 1 mode of a laser with moderate parasitic losses (αLL = 0.5).

Fig. 19
Fig. 19

The dependence of the output-power level P on the unsaturated-gain eigenvalue αuL for the N = 1 mode of overcoupled and critically coupled lossy lasers with αLL = 0.5.

Equations (13)

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

α = α u / ( 1 + I ) ,
I = | a + exp ( i k ζ ) + a exp ( i k ζ ) | 2 = | a + | 2 + | a | 2 + 2 a + a cos ( 2 k ζ ) .
d C N + / d ζ = i K exp ( 2 i δ ζ ) C N + ( α α L ) C N + , d C N / d ζ = i K * exp ( 2 i δ ζ ) C N + ( α α L ) C N ,
α = α u / ( 1 + | C N + | 2 + | C N | 2 )
C N + ( 0 ) C N ( L ) } = 0 ,
{ K δ α α L } { K L δ L α L α L L }
C N + ( 0 ) C N ( 1 ) } = 0.
C + ( Z n + 1 ) = { A 1 exp [ ( i δ L + η ) Z n + 1 ] + B 1 exp [ ( i δ L η ) Z n + 1 ] } C + ( Z n ) + { A 2 exp [ ( i δ L + η ) Z n + 1 ] + B 2 exp [ ( i δ L η ) Z n + 1 ] } C ( Z n ) , C ( Z n + 1 ) = { A 1 ( i δ L + η α 0 L ) i K L exp [ ( i δ L + η ) Z n + 1 ] + B 1 ( i δ L η α 0 L ) i K L exp [ ( i δ L η ) Z n + 1 ] } C + ( Z n ) + { A 2 ( i δ L + η α 0 L ) i K L exp [ ( i δ L + η ) Z n + 1 ] + B 2 ( i δ L η α 0 L ) i K L exp [ ( i δ L η ) Z n + 1 ] } C ( Z n ) ,
A 1 = [ ( i δ L + η + α 0 L ) / 2 η ] exp [ ( i δ L η ) Z n ] , B 1 = [ ( i δ L + η α 0 L ) / 2 η ] exp [ ( i δ L + η ) Z n ] , A 2 = [ ( i K L ) / 2 η ] exp [ ( i δ L η ) Z n ] , B 2 = [ ( i K L ) / 2 η ] exp [ ( i δ L + η ) Z n ]
α 0 L = ( α α L ) L , η = ( | K | 2 + α 0 2 2 i δ α 0 δ 2 ) 1 / 2 · L .
D = [ C + ( 1 ) ] e C + ( 1 )
D δ L = D / [ ( δ L ) ] , D α u L = D / [ ( α u L ) ] .
Δ D = D δ L · Δ ( δ L ) + D α u L · Δ ( α u L ) = D

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