Abstract

An experimental analysis of InGaAsP injection lasers shows an unexpected decrease of the differential quantum efficiency as a function of injected current when optical power is fed back into the active cavity of a diode inserted into a long transmission line. To investigate the response of laser diodes to optical feedback, we base our analysis on a microscopic model, resulting in a set of coupled equations that include the microscopic parameters that characterize the material and the device. This description takes into account the nonlinear dependence of the interband carrier lifetime on the level of optical feedback. Good agreement between the analytical description and experimental data is obtained for threshold current and differential quantum efficiency as functions of the feedback ratio.

© 1992 Optical Society of America

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References

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  1. R. Lang, K. Kobayashi, “External optical feedback on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
    [Crossref]
  2. H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
    [Crossref]
  3. C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
    [Crossref]
  4. D. Lenstra, B. H. Verbeek, A. J. den Boef, “Coherence collapse in single mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
    [Crossref]
  5. K. Petermann, Laser Diode Modulation and Noise (Kluwer, Tokyo, 1988).
    [Crossref]
  6. G. Wenke, R. Gross, P. Meissner, E. Patzak, “Characteristics of a compact three cavity laser configuration,” IEEE J. Lightwave Technol. LT-5, 608–615 (1987); N. A. Olsson, J. P. van der Ziel, “Performance characteristics of 1.5-μm external cavity semiconductor lasers for coherent optical communications,” IEEE J. Lightwave Technol. LT-5, 510–515 (1987).
    [Crossref]
  7. O. Hirota, Y. Suematsu, “Noise properties of injection lasers due to reflected waves,” IEEE J. Quantum Electron. QE-15, 142–148 (1980).
  8. W. Bludau, R. Rossberg, “Characterization of laser to fiber coupling techniques by their optical feedback,” Appl. Opt. 21, 1933–1939 (1982).
    [Crossref] [PubMed]
  9. G. Chiaretti, C. Reverdito, G. Sacchi, F. Brivio, M. Milani, “Separate external cavity feedback effects in optical communication systems,” in New Laser Technologies and Applications, A. A. Carabelas, T. Letardi, eds. (Editrice Compositori, Bologna, 1989), pp. 303–320.
  10. G. Chiaretti, C. Vaccarino, M. Milani, “Gain versus current in semiconductor injection lasers: a microscopic approach,” in Advances in Image Processing, A. J. Oosterlinck, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.804, 144–152 (1988).
  11. G. Chiaretti, M. Brambilla, M. Milani, “A microscopic approach to amplitude modulation with small signal of current,” in Semiconductor Lasers, G. A. Acket, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1025, 82–84 (1989).
  12. P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
    [Crossref]
  13. G. Chiaretti, D. Reichenbach, C. Vaccarino, M. Milani, “Analytical derivation of the structure of carrier lifetime and its nonlinear dependence on optical field intensity in semiconductor laser structures,” Appl. Opt. 28, 4556–4559 (1989).
    [Crossref] [PubMed]
  14. G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).
  15. J. E. Ripper, “Measurement of spontaneous carrier lifetime from stimulated emission delay semiconductor lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
    [Crossref]
  16. J. S. Blakemore, Semiconductor Statistics (Dover, New York, 1987).
  17. G. P. Bava, L. A. Lugiato, “Dynamical equations for optical bistability in multiple quantum well structures,” Opt. Commun. 78, 195–202 (1990).
    [Crossref]

1990 (1)

G. P. Bava, L. A. Lugiato, “Dynamical equations for optical bistability in multiple quantum well structures,” Opt. Commun. 78, 195–202 (1990).
[Crossref]

1989 (1)

1987 (1)

G. Wenke, R. Gross, P. Meissner, E. Patzak, “Characteristics of a compact three cavity laser configuration,” IEEE J. Lightwave Technol. LT-5, 608–615 (1987); N. A. Olsson, J. P. van der Ziel, “Performance characteristics of 1.5-μm external cavity semiconductor lasers for coherent optical communications,” IEEE J. Lightwave Technol. LT-5, 510–515 (1987).
[Crossref]

1986 (2)

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[Crossref]

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[Crossref]

1985 (1)

D. Lenstra, B. H. Verbeek, A. J. den Boef, “Coherence collapse in single mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[Crossref]

1982 (1)

1980 (2)

O. Hirota, Y. Suematsu, “Noise properties of injection lasers due to reflected waves,” IEEE J. Quantum Electron. QE-15, 142–148 (1980).

R. Lang, K. Kobayashi, “External optical feedback on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[Crossref]

1972 (1)

J. E. Ripper, “Measurement of spontaneous carrier lifetime from stimulated emission delay semiconductor lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
[Crossref]

1970 (1)

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Abeles, J. H.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[Crossref]

Bava, G. P.

G. P. Bava, L. A. Lugiato, “Dynamical equations for optical bistability in multiple quantum well structures,” Opt. Commun. 78, 195–202 (1990).
[Crossref]

Bird, J. W.

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Blakemore, J. S.

J. S. Blakemore, Semiconductor Statistics (Dover, New York, 1987).

Bludau, W.

Brambilla, M.

G. Chiaretti, M. Brambilla, M. Milani, “A microscopic approach to amplitude modulation with small signal of current,” in Semiconductor Lasers, G. A. Acket, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1025, 82–84 (1989).

Brivio, F.

G. Chiaretti, C. Reverdito, G. Sacchi, F. Brivio, M. Milani, “Separate external cavity feedback effects in optical communication systems,” in New Laser Technologies and Applications, A. A. Carabelas, T. Letardi, eds. (Editrice Compositori, Bologna, 1989), pp. 303–320.

Burnham, R. D.

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Chiaretti, G.

G. Chiaretti, D. Reichenbach, C. Vaccarino, M. Milani, “Analytical derivation of the structure of carrier lifetime and its nonlinear dependence on optical field intensity in semiconductor laser structures,” Appl. Opt. 28, 4556–4559 (1989).
[Crossref] [PubMed]

G. Chiaretti, C. Reverdito, G. Sacchi, F. Brivio, M. Milani, “Separate external cavity feedback effects in optical communication systems,” in New Laser Technologies and Applications, A. A. Carabelas, T. Letardi, eds. (Editrice Compositori, Bologna, 1989), pp. 303–320.

G. Chiaretti, C. Vaccarino, M. Milani, “Gain versus current in semiconductor injection lasers: a microscopic approach,” in Advances in Image Processing, A. J. Oosterlinck, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.804, 144–152 (1988).

G. Chiaretti, M. Brambilla, M. Milani, “A microscopic approach to amplitude modulation with small signal of current,” in Semiconductor Lasers, G. A. Acket, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1025, 82–84 (1989).

Dapkus, P. D.

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

den Boef, A. J.

D. Lenstra, B. H. Verbeek, A. J. den Boef, “Coherence collapse in single mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[Crossref]

Gross, R.

G. Wenke, R. Gross, P. Meissner, E. Patzak, “Characteristics of a compact three cavity laser configuration,” IEEE J. Lightwave Technol. LT-5, 608–615 (1987); N. A. Olsson, J. P. van der Ziel, “Performance characteristics of 1.5-μm external cavity semiconductor lasers for coherent optical communications,” IEEE J. Lightwave Technol. LT-5, 510–515 (1987).
[Crossref]

Henry, C. H.

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[Crossref]

Hirota, O.

O. Hirota, Y. Suematsu, “Noise properties of injection lasers due to reflected waves,” IEEE J. Quantum Electron. QE-15, 142–148 (1980).

Holonyak, N.

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Kazarinov, R. F.

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[Crossref]

Keune, D. L.

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Kobayashi, K.

R. Lang, K. Kobayashi, “External optical feedback on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[Crossref]

Lang, R.

R. Lang, K. Kobayashi, “External optical feedback on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[Crossref]

Lawley, K. L.

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Lenstra, D.

D. Lenstra, B. H. Verbeek, A. J. den Boef, “Coherence collapse in single mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[Crossref]

Logan, R. A.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[Crossref]

Lugiato, L. A.

G. P. Bava, L. A. Lugiato, “Dynamical equations for optical bistability in multiple quantum well structures,” Opt. Commun. 78, 195–202 (1990).
[Crossref]

Meissner, P.

G. Wenke, R. Gross, P. Meissner, E. Patzak, “Characteristics of a compact three cavity laser configuration,” IEEE J. Lightwave Technol. LT-5, 608–615 (1987); N. A. Olsson, J. P. van der Ziel, “Performance characteristics of 1.5-μm external cavity semiconductor lasers for coherent optical communications,” IEEE J. Lightwave Technol. LT-5, 510–515 (1987).
[Crossref]

Milani, M.

G. Chiaretti, D. Reichenbach, C. Vaccarino, M. Milani, “Analytical derivation of the structure of carrier lifetime and its nonlinear dependence on optical field intensity in semiconductor laser structures,” Appl. Opt. 28, 4556–4559 (1989).
[Crossref] [PubMed]

G. Chiaretti, M. Brambilla, M. Milani, “A microscopic approach to amplitude modulation with small signal of current,” in Semiconductor Lasers, G. A. Acket, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1025, 82–84 (1989).

G. Chiaretti, C. Vaccarino, M. Milani, “Gain versus current in semiconductor injection lasers: a microscopic approach,” in Advances in Image Processing, A. J. Oosterlinck, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.804, 144–152 (1988).

G. Chiaretti, C. Reverdito, G. Sacchi, F. Brivio, M. Milani, “Separate external cavity feedback effects in optical communication systems,” in New Laser Technologies and Applications, A. A. Carabelas, T. Letardi, eds. (Editrice Compositori, Bologna, 1989), pp. 303–320.

Olsson, N. A.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[Crossref]

Panish, M. B.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[Crossref]

Patzak, E.

G. Wenke, R. Gross, P. Meissner, E. Patzak, “Characteristics of a compact three cavity laser configuration,” IEEE J. Lightwave Technol. LT-5, 608–615 (1987); N. A. Olsson, J. P. van der Ziel, “Performance characteristics of 1.5-μm external cavity semiconductor lasers for coherent optical communications,” IEEE J. Lightwave Technol. LT-5, 510–515 (1987).
[Crossref]

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer, Tokyo, 1988).
[Crossref]

Reichenbach, D.

Reverdito, C.

G. Chiaretti, C. Reverdito, G. Sacchi, F. Brivio, M. Milani, “Separate external cavity feedback effects in optical communication systems,” in New Laser Technologies and Applications, A. A. Carabelas, T. Letardi, eds. (Editrice Compositori, Bologna, 1989), pp. 303–320.

Ripper, J. E.

J. E. Ripper, “Measurement of spontaneous carrier lifetime from stimulated emission delay semiconductor lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
[Crossref]

Rossberg, R.

Sacchi, G.

G. Chiaretti, C. Reverdito, G. Sacchi, F. Brivio, M. Milani, “Separate external cavity feedback effects in optical communication systems,” in New Laser Technologies and Applications, A. A. Carabelas, T. Letardi, eds. (Editrice Compositori, Bologna, 1989), pp. 303–320.

Suematsu, Y.

O. Hirota, Y. Suematsu, “Noise properties of injection lasers due to reflected waves,” IEEE J. Quantum Electron. QE-15, 142–148 (1980).

Temkin, H.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[Crossref]

Thompson, G. H. B.

G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).

Vaccarino, C.

G. Chiaretti, D. Reichenbach, C. Vaccarino, M. Milani, “Analytical derivation of the structure of carrier lifetime and its nonlinear dependence on optical field intensity in semiconductor laser structures,” Appl. Opt. 28, 4556–4559 (1989).
[Crossref] [PubMed]

G. Chiaretti, C. Vaccarino, M. Milani, “Gain versus current in semiconductor injection lasers: a microscopic approach,” in Advances in Image Processing, A. J. Oosterlinck, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.804, 144–152 (1988).

Verbeek, B. H.

D. Lenstra, B. H. Verbeek, A. J. den Boef, “Coherence collapse in single mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[Crossref]

Walline, R. E.

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Wenke, G.

G. Wenke, R. Gross, P. Meissner, E. Patzak, “Characteristics of a compact three cavity laser configuration,” IEEE J. Lightwave Technol. LT-5, 608–615 (1987); N. A. Olsson, J. P. van der Ziel, “Performance characteristics of 1.5-μm external cavity semiconductor lasers for coherent optical communications,” IEEE J. Lightwave Technol. LT-5, 510–515 (1987).
[Crossref]

Appl. Opt. (2)

IEEE J. Lightwave Technol. (1)

G. Wenke, R. Gross, P. Meissner, E. Patzak, “Characteristics of a compact three cavity laser configuration,” IEEE J. Lightwave Technol. LT-5, 608–615 (1987); N. A. Olsson, J. P. van der Ziel, “Performance characteristics of 1.5-μm external cavity semiconductor lasers for coherent optical communications,” IEEE J. Lightwave Technol. LT-5, 510–515 (1987).
[Crossref]

IEEE J. Quantum Electron. (5)

O. Hirota, Y. Suematsu, “Noise properties of injection lasers due to reflected waves,” IEEE J. Quantum Electron. QE-15, 142–148 (1980).

R. Lang, K. Kobayashi, “External optical feedback on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[Crossref]

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index guided InGaAsP lasers,” IEEE J. Quantum Electron. QE-22, 286–293 (1986).
[Crossref]

C. H. Henry, R. F. Kazarinov, “Instability of semiconductor lasers due to optical feedback from distant reflectors,” IEEE J. Quantum Electron. QE-22, 294–301 (1986).
[Crossref]

D. Lenstra, B. H. Verbeek, A. J. den Boef, “Coherence collapse in single mode semiconductor lasers due to optical feedback,” IEEE J. Quantum Electron. QE-21, 674–679 (1985).
[Crossref]

J. Appl. Phys. (2)

J. E. Ripper, “Measurement of spontaneous carrier lifetime from stimulated emission delay semiconductor lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
[Crossref]

P. D. Dapkus, N. Holonyak, R. D. Burnham, D. L. Keune, J. W. Bird, K. L. Lawley, R. E. Walline, “Spontaneous and stimulated carrier lifetime (77 K) in the high purity surface free GaAs epitaxial layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[Crossref]

Opt. Commun. (1)

G. P. Bava, L. A. Lugiato, “Dynamical equations for optical bistability in multiple quantum well structures,” Opt. Commun. 78, 195–202 (1990).
[Crossref]

Other (6)

J. S. Blakemore, Semiconductor Statistics (Dover, New York, 1987).

G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).

K. Petermann, Laser Diode Modulation and Noise (Kluwer, Tokyo, 1988).
[Crossref]

G. Chiaretti, C. Reverdito, G. Sacchi, F. Brivio, M. Milani, “Separate external cavity feedback effects in optical communication systems,” in New Laser Technologies and Applications, A. A. Carabelas, T. Letardi, eds. (Editrice Compositori, Bologna, 1989), pp. 303–320.

G. Chiaretti, C. Vaccarino, M. Milani, “Gain versus current in semiconductor injection lasers: a microscopic approach,” in Advances in Image Processing, A. J. Oosterlinck, A. G. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.804, 144–152 (1988).

G. Chiaretti, M. Brambilla, M. Milani, “A microscopic approach to amplitude modulation with small signal of current,” in Semiconductor Lasers, G. A. Acket, ed. Proc. Soc. Photo-Opt. Instrum. Eng.1025, 82–84 (1989).

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Figures (7)

Fig. 1
Fig. 1

Actual laboratory setup. The inset shows the separate external cavity.

Fig. 2
Fig. 2

Separate external cavity effects. Emitted power versus external cavity length: (upper) from the monitor PIN (μA versus μm), (lower) from the transmitted light detector (μW versus μm).

Fig. 3
Fig. 3

Role of the feedback on the characteristic curve of the device. Emitted power versus injected current at different values of feedback ratio (device 1: T = 30°C).

Fig. 4
Fig. 4

Normalized threshold current (ratio between the threshold current in the presence of feedback and the threshold current in the absence of feedback) versus feedback ratio.

Fig. 5
Fig. 5

Differential quantum efficiency (dP/dI) versus feedback ratio measured at a bias level of twice the laser threshold.

Fig. 6
Fig. 6

Anomalous behavior of the differential quantum efficiency as a function of injected current just above threshold.

Fig. 7
Fig. 7

Three classes of behavior of the characteristic P–I curve (arbitrary units) of the device in the presence of optical feedback, compared with the reference curve obtained without optical feedback.

Equations (21)

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

d d t A 2 = ( 4 χ 2 / γ ) A 2 D - 2 k A 2 ,
d d t D = - ( 4 χ 2 / γ ) A 2 D - γ ( D - D 0 ) ,
τ s = 1 / [ γ - 1 ( 1 + b n ph ) ] ;             b = 4 χ 2 / γ γ ,
a = q b R ˜ ;             R ˜ = R ˜ ( k , k ) .
d d t n ph = ( 4 χ 2 / γ ) n c n ph - 2 k n ph + 2 k n ph ,
d d t n c = - ( 4 χ 2 / γ ) n c n ph + I / e - γ n c - a γ n c n ph ,
d d t n ph = ( 4 χ 2 / γ ) n c n ph - 2 k n ph ,
d d t n c = - ( 4 χ 2 / γ ) n c n ph + I / e - γ n c .
I th F = ( e / 2 χ 2 ) ( k - k ) γ γ ,
η D F = 1 / [ 2 e ( k - k ) ( 1 + a ) ] ,
n ph , st = I / [ 2 e ( k - k ) ( 1 + a ) ] ,
n c , st = γ ( k - k ) / 2 χ 2 ,
I th = ( e / 2 χ 2 ) k γ γ ,
η D = 1 / ( 2 e k ) .
I th F < I th ,
η D F η D .
I th F ( a = 0 ) < I th ,
η D F ( a = 0 ) > η D .
F = a ( k - k ) - k , F < 0             if             a < k / ( k - k ) ,
F = 0             if             a = k / ( k - k ) ,
F > 0             if             a > k / ( k - k ) .

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