Abstract

The dependence of spontaneous carrier lifetime τs on the electromagnetic field in the laser cavity is discussed in the framework of a microscopic approach to laser dynamics. An analytical relationship between τs and the number of photons in the laser cavity is derived. The structure of τs suggests new experiments for the determination of microscopic parameters of the active region material (e.g., the intraband carrier lifetime τin). This relationship is nonlinear and does not exhibit any kind of singularity or discontinuity, as sometimes reported in the literature. The comparison between theory and experimental results is discussed.

© 1989 Optical Society of America

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References

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  1. H. Haken, Laser Theory (Springer-Verlag, Berlin1984).
  2. E. Scholl, P. T. Landsberg, “Nonequilibrium Kinetics of Coupled Photons and Electrons in Two-Level Systems of the Laser Type,” J. Opt. Soc. Am. 73, 1197–1206 (1983);J. L. Oudar, “Transient Nonlinear Optical Effects in Semiconductors,” in Nonlinear Optics: Materials and Devices, C. Flytzanis, J. L. Oudar, Eds. (Springer-Verlag, Berlin, 1986), p. 91.
    [CrossRef]
  3. M. J. Adams, M. Osinski, “Longitudinal Mode Competition in Semiconductor Lasers—Rate Equations Revisited,” IEE Proc. 129 Pt. I, No. 6271–274 (1982).
  4. R. Salathe, C. Voumard, H. Weber, “Rate Equation Approach for Diode Lasers Part I—Steady State Solution for a Single Diode,” Opto-electronics 6, 451–456 (1974).
  5. G. Chiaretti, C. Vaccarino, M. Milani, “Gain Versus Current in Semiconductor Injection Laser: a Microscopic Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 864, 144–152 (1988).
  6. G. H. B. Thompson, Physics of Semiconductor Laser Devices (Wiley, New York, 1980).
  7. P. M. Boers, M. T. Vlaardingerbroek, M. Danielsen, “Dynamic Behaviour of Semiconductor Lasers,” Electron. Lett. 11, 206–208 (1975).
    [CrossRef]
  8. K. Konnerth, C. Lanza, “Delay Between Current Pulse and Light Emission of a GaAs Injection Laser,” Appl. Phys. Lett. 4, 120–121 (1964).
    [CrossRef]
  9. J. E. Ripper, “Measurement of Spontaneous Carrier Lifetime from Stimulated Emission Delays in Semiconductor Lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
    [CrossRef]
  10. P. D. Dapkus et al., “Spontaneous and Stimulated Carrier Lifetime (77°K) in a High-Purity, Surface Free GaAs Epitaxial Layer,” J. Appl. Phys. 41, 4194–4199 (1970).
    [CrossRef]
  11. L. A. Lugiato, “Theory of Optical Bistability,” Prog. Opt. 21, 71–216 (1984).
  12. M. Milani, R. Bonifacio, A. Lugiato, “A Model for Bistability in a Coherently Driven Josephson Junction,” Lett. Nuovo Cimento 26, 35–39 (1979).
    [CrossRef]
  13. S. M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1969).
  14. Y. Horikoski, “Temperature Dependence of Laser Threshold Current,”in GalnAsP Alloy Semiconductors, T. P. Pearsall, Ed. (Wiley, New York, 1982).

1988 (1)

G. Chiaretti, C. Vaccarino, M. Milani, “Gain Versus Current in Semiconductor Injection Laser: a Microscopic Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 864, 144–152 (1988).

1984 (1)

L. A. Lugiato, “Theory of Optical Bistability,” Prog. Opt. 21, 71–216 (1984).

1983 (1)

1982 (1)

M. J. Adams, M. Osinski, “Longitudinal Mode Competition in Semiconductor Lasers—Rate Equations Revisited,” IEE Proc. 129 Pt. I, No. 6271–274 (1982).

1979 (1)

M. Milani, R. Bonifacio, A. Lugiato, “A Model for Bistability in a Coherently Driven Josephson Junction,” Lett. Nuovo Cimento 26, 35–39 (1979).
[CrossRef]

1975 (1)

P. M. Boers, M. T. Vlaardingerbroek, M. Danielsen, “Dynamic Behaviour of Semiconductor Lasers,” Electron. Lett. 11, 206–208 (1975).
[CrossRef]

1974 (1)

R. Salathe, C. Voumard, H. Weber, “Rate Equation Approach for Diode Lasers Part I—Steady State Solution for a Single Diode,” Opto-electronics 6, 451–456 (1974).

1972 (1)

J. E. Ripper, “Measurement of Spontaneous Carrier Lifetime from Stimulated Emission Delays in Semiconductor Lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
[CrossRef]

1970 (1)

P. D. Dapkus et al., “Spontaneous and Stimulated Carrier Lifetime (77°K) in a High-Purity, Surface Free GaAs Epitaxial Layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[CrossRef]

1964 (1)

K. Konnerth, C. Lanza, “Delay Between Current Pulse and Light Emission of a GaAs Injection Laser,” Appl. Phys. Lett. 4, 120–121 (1964).
[CrossRef]

Adams, M. J.

M. J. Adams, M. Osinski, “Longitudinal Mode Competition in Semiconductor Lasers—Rate Equations Revisited,” IEE Proc. 129 Pt. I, No. 6271–274 (1982).

Boers, P. M.

P. M. Boers, M. T. Vlaardingerbroek, M. Danielsen, “Dynamic Behaviour of Semiconductor Lasers,” Electron. Lett. 11, 206–208 (1975).
[CrossRef]

Bonifacio, R.

M. Milani, R. Bonifacio, A. Lugiato, “A Model for Bistability in a Coherently Driven Josephson Junction,” Lett. Nuovo Cimento 26, 35–39 (1979).
[CrossRef]

Chiaretti, G.

G. Chiaretti, C. Vaccarino, M. Milani, “Gain Versus Current in Semiconductor Injection Laser: a Microscopic Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 864, 144–152 (1988).

Danielsen, M.

P. M. Boers, M. T. Vlaardingerbroek, M. Danielsen, “Dynamic Behaviour of Semiconductor Lasers,” Electron. Lett. 11, 206–208 (1975).
[CrossRef]

Dapkus, P. D.

P. D. Dapkus et al., “Spontaneous and Stimulated Carrier Lifetime (77°K) in a High-Purity, Surface Free GaAs Epitaxial Layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[CrossRef]

Haken, H.

H. Haken, Laser Theory (Springer-Verlag, Berlin1984).

Horikoski, Y.

Y. Horikoski, “Temperature Dependence of Laser Threshold Current,”in GalnAsP Alloy Semiconductors, T. P. Pearsall, Ed. (Wiley, New York, 1982).

Konnerth, K.

K. Konnerth, C. Lanza, “Delay Between Current Pulse and Light Emission of a GaAs Injection Laser,” Appl. Phys. Lett. 4, 120–121 (1964).
[CrossRef]

Landsberg, P. T.

Lanza, C.

K. Konnerth, C. Lanza, “Delay Between Current Pulse and Light Emission of a GaAs Injection Laser,” Appl. Phys. Lett. 4, 120–121 (1964).
[CrossRef]

Lugiato, A.

M. Milani, R. Bonifacio, A. Lugiato, “A Model for Bistability in a Coherently Driven Josephson Junction,” Lett. Nuovo Cimento 26, 35–39 (1979).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, “Theory of Optical Bistability,” Prog. Opt. 21, 71–216 (1984).

Milani, M.

G. Chiaretti, C. Vaccarino, M. Milani, “Gain Versus Current in Semiconductor Injection Laser: a Microscopic Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 864, 144–152 (1988).

M. Milani, R. Bonifacio, A. Lugiato, “A Model for Bistability in a Coherently Driven Josephson Junction,” Lett. Nuovo Cimento 26, 35–39 (1979).
[CrossRef]

Osinski, M.

M. J. Adams, M. Osinski, “Longitudinal Mode Competition in Semiconductor Lasers—Rate Equations Revisited,” IEE Proc. 129 Pt. I, No. 6271–274 (1982).

Ripper, J. E.

J. E. Ripper, “Measurement of Spontaneous Carrier Lifetime from Stimulated Emission Delays in Semiconductor Lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
[CrossRef]

Salathe, R.

R. Salathe, C. Voumard, H. Weber, “Rate Equation Approach for Diode Lasers Part I—Steady State Solution for a Single Diode,” Opto-electronics 6, 451–456 (1974).

Scholl, E.

Sze, S. M.

S. M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1969).

Thompson, G. H. B.

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

Vaccarino, C.

G. Chiaretti, C. Vaccarino, M. Milani, “Gain Versus Current in Semiconductor Injection Laser: a Microscopic Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 864, 144–152 (1988).

Vlaardingerbroek, M. T.

P. M. Boers, M. T. Vlaardingerbroek, M. Danielsen, “Dynamic Behaviour of Semiconductor Lasers,” Electron. Lett. 11, 206–208 (1975).
[CrossRef]

Voumard, C.

R. Salathe, C. Voumard, H. Weber, “Rate Equation Approach for Diode Lasers Part I—Steady State Solution for a Single Diode,” Opto-electronics 6, 451–456 (1974).

Weber, H.

R. Salathe, C. Voumard, H. Weber, “Rate Equation Approach for Diode Lasers Part I—Steady State Solution for a Single Diode,” Opto-electronics 6, 451–456 (1974).

Appl. Phys. Lett. (1)

K. Konnerth, C. Lanza, “Delay Between Current Pulse and Light Emission of a GaAs Injection Laser,” Appl. Phys. Lett. 4, 120–121 (1964).
[CrossRef]

Electron. Lett. (1)

P. M. Boers, M. T. Vlaardingerbroek, M. Danielsen, “Dynamic Behaviour of Semiconductor Lasers,” Electron. Lett. 11, 206–208 (1975).
[CrossRef]

IEE Proc. (1)

M. J. Adams, M. Osinski, “Longitudinal Mode Competition in Semiconductor Lasers—Rate Equations Revisited,” IEE Proc. 129 Pt. I, No. 6271–274 (1982).

J. Appl. Phys. (2)

J. E. Ripper, “Measurement of Spontaneous Carrier Lifetime from Stimulated Emission Delays in Semiconductor Lasers,” J. Appl. Phys. 43, 1762–1763 (1972).
[CrossRef]

P. D. Dapkus et al., “Spontaneous and Stimulated Carrier Lifetime (77°K) in a High-Purity, Surface Free GaAs Epitaxial Layer,” J. Appl. Phys. 41, 4194–4199 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

Lett. Nuovo Cimento (1)

M. Milani, R. Bonifacio, A. Lugiato, “A Model for Bistability in a Coherently Driven Josephson Junction,” Lett. Nuovo Cimento 26, 35–39 (1979).
[CrossRef]

Opto-electronics (1)

R. Salathe, C. Voumard, H. Weber, “Rate Equation Approach for Diode Lasers Part I—Steady State Solution for a Single Diode,” Opto-electronics 6, 451–456 (1974).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

G. Chiaretti, C. Vaccarino, M. Milani, “Gain Versus Current in Semiconductor Injection Laser: a Microscopic Approach,” Proc. Soc. Photo-Opt. Instrum. Eng. 864, 144–152 (1988).

Prog. Opt. (1)

L. A. Lugiato, “Theory of Optical Bistability,” Prog. Opt. 21, 71–216 (1984).

Other (4)

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

S. M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1969).

Y. Horikoski, “Temperature Dependence of Laser Threshold Current,”in GalnAsP Alloy Semiconductors, T. P. Pearsall, Ed. (Wiley, New York, 1982).

H. Haken, Laser Theory (Springer-Verlag, Berlin1984).

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Figures (4)

Fig. 1
Fig. 1

Plot of the curve τs = τs(P) for different values of the parameters β ¯ ( β ¯ = β 10 3 ): a, β ¯ = 1.1 × 10 1 (normalization point A); b, β ¯ = 1.1 × 10 2 (normalization point B); c, β ¯ = 1.1 × 10 2 (normalization point A). Normalization point A: (P = 300 mW; τs = 0.6 ns). Normalization point B: (P = 119 mW; τs = 1.0 ns). See text for the definition of the normalization point and procedures.

Fig. 2
Fig. 2

Comparison between the experimental data and fit given in Ref. 10 with the curve τs = τs(P) for β ¯ = 1.1 × 10 2.

Fig. 3
Fig. 3

Family of curves τs= τs(P) is plotted. The curves are calculated for different values of β ¯ and normalized at the same point N(P = 900 mW; τs= 0.28 ns): a, β ¯ = 1.1 × 10 3; b, β ¯ = 5.5 × 10 3; c, β ¯ = 1.1 × 10 2; d, β ¯ = 4.4 × 10 2.

Fig. 4
Fig. 4

Family of curves τs = τs(P) is plotted as a function of the parameter β and for a given γ 1 = τ s ( P = 0 ) ..

Equations (19)

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d d t D = 4 χ 2 γ A 2 D γ ( D D 0 ) ,
d d t A 2 = 4 χ 2 γ A 2 D 2 k A 2 ,
d d t n c = 4 χ 2 γ n ph n c γ n c + I e ,
d d t n ph = 4 χ 2 γ n ph n c 2 k n ph ,
d d t n c = J e d n c τ s g n ph ,
d d t n ph = g n ph n ph τ ph + b n c τ s .
d d t n c = a I n c τ ,
τ = τ ( n c ) ,
d d t n c = a I n c τ ( n c ) .
τ s γ 1 .
τ s = γ 1 ( 1 + 4 χ 2 γ γ n ph ) 1 ,
τ s = γ 1 ( 1 + β P ) 1 ,
n ph = P E ph τ ph ,
β = β * τ ph E ph ; β * 4 χ 2 γ γ .
β = 21 s J 1 ( β is expressed in SI units ) .
I th α exp ( T / T 0 ) ( see Ref . 14 ) .
χ 2 = ω μ 2 2 V ,
n c = 2 k γ 4 χ 2
I th = γ γ k e 2 χ 2 = 2 β * k e [ see Eq . ( 9 ) ] .

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