Abstract

The steady-state light-vs-pumping characteristic for lasers is modeled by a rate-equation and Fabry-Perot approach. It is found that the two models predict a similar dependence of the output power on pumping rate in the limit of vanishingly small gain and loss. It is also discovered that the Fabry-Perot result contains multiplicative factors which arise from an explicit consideration of the resonator. These factors are missing in the rate-equation analysis but can be included by redefining the equivalent cavity loss and scaling the spontaneous emission factor and gain in an appropriate fashion. The results of this investigation indicate that, for diode lasers, the Fabry-Perot approach to modeling the steady-state behavior should be preferred.

© 1983 Optical Society of America

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References

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  1. J. C. Goodwin, B. K. Garside, IEEE J. Quantum Electron. QE-18, 1264 (1982).
    [CrossRef]
  2. D. Kato, IEEE J. Quantum Electron. QE-14, 563 (1978).
    [CrossRef]
  3. Y. Suematsu, S. Akiba, T. Hong, IEEE J. Quantum Electron. QE-13, 596 (1977).
    [CrossRef]
  4. G. Arnold, P. Russer, K. Petermann, in Semiconductor Devices for Optical Communication, H. Kressel, Ed. (Springer, Berlin, 1980).
  5. M. J. Adams, Opto electronics 5, 201, 1973.
    [CrossRef]
  6. To achieve a quadratic relationship, the gain and spontaneous emission are assumed to depend linearly on the pumping rate.
  7. W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-18, 1918 (1982).
    [CrossRef]
  8. B. Zee, IEEE J. Quantum Electron. QE-14, 727 (1978).
    [CrossRef]
  9. E. I. Gordon, Bell Syst. Tech. J. vol. 43, 507 (1964).
  10. D. Marcuse, IEEE J. Quantum Electron. QE-19, 63 (1983).
    [CrossRef]
  11. D. T. Cassidy, submitted to IEEE J. Quantum Electron., June1983.
  12. B. W. Hakki, T. W. Paoli, J. Appl. Phys. 46, 1299 (1975).
    [CrossRef]

1983 (1)

D. Marcuse, IEEE J. Quantum Electron. QE-19, 63 (1983).
[CrossRef]

1982 (2)

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-18, 1918 (1982).
[CrossRef]

J. C. Goodwin, B. K. Garside, IEEE J. Quantum Electron. QE-18, 1264 (1982).
[CrossRef]

1978 (2)

D. Kato, IEEE J. Quantum Electron. QE-14, 563 (1978).
[CrossRef]

B. Zee, IEEE J. Quantum Electron. QE-14, 727 (1978).
[CrossRef]

1977 (1)

Y. Suematsu, S. Akiba, T. Hong, IEEE J. Quantum Electron. QE-13, 596 (1977).
[CrossRef]

1975 (1)

B. W. Hakki, T. W. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

1973 (1)

M. J. Adams, Opto electronics 5, 201, 1973.
[CrossRef]

1964 (1)

E. I. Gordon, Bell Syst. Tech. J. vol. 43, 507 (1964).

Adams, M. J.

M. J. Adams, Opto electronics 5, 201, 1973.
[CrossRef]

Akiba, S.

Y. Suematsu, S. Akiba, T. Hong, IEEE J. Quantum Electron. QE-13, 596 (1977).
[CrossRef]

Arnold, G.

G. Arnold, P. Russer, K. Petermann, in Semiconductor Devices for Optical Communication, H. Kressel, Ed. (Springer, Berlin, 1980).

Burnham, R. D.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-18, 1918 (1982).
[CrossRef]

Cassidy, D. T.

D. T. Cassidy, submitted to IEEE J. Quantum Electron., June1983.

Garside, B. K.

J. C. Goodwin, B. K. Garside, IEEE J. Quantum Electron. QE-18, 1264 (1982).
[CrossRef]

Goodwin, J. C.

J. C. Goodwin, B. K. Garside, IEEE J. Quantum Electron. QE-18, 1264 (1982).
[CrossRef]

Gordon, E. I.

E. I. Gordon, Bell Syst. Tech. J. vol. 43, 507 (1964).

Hakki, B. W.

B. W. Hakki, T. W. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

Hong, T.

Y. Suematsu, S. Akiba, T. Hong, IEEE J. Quantum Electron. QE-13, 596 (1977).
[CrossRef]

Kato, D.

D. Kato, IEEE J. Quantum Electron. QE-14, 563 (1978).
[CrossRef]

Marcuse, D.

D. Marcuse, IEEE J. Quantum Electron. QE-19, 63 (1983).
[CrossRef]

Paoli, T. W.

B. W. Hakki, T. W. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

Petermann, K.

G. Arnold, P. Russer, K. Petermann, in Semiconductor Devices for Optical Communication, H. Kressel, Ed. (Springer, Berlin, 1980).

Russer, P.

G. Arnold, P. Russer, K. Petermann, in Semiconductor Devices for Optical Communication, H. Kressel, Ed. (Springer, Berlin, 1980).

Scifres, D. R.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-18, 1918 (1982).
[CrossRef]

Streifer, W.

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-18, 1918 (1982).
[CrossRef]

Suematsu, Y.

Y. Suematsu, S. Akiba, T. Hong, IEEE J. Quantum Electron. QE-13, 596 (1977).
[CrossRef]

Zee, B.

B. Zee, IEEE J. Quantum Electron. QE-14, 727 (1978).
[CrossRef]

Bell Syst. Tech. J. (1)

E. I. Gordon, Bell Syst. Tech. J. vol. 43, 507 (1964).

IEEE J. Quantum Electron. (6)

D. Marcuse, IEEE J. Quantum Electron. QE-19, 63 (1983).
[CrossRef]

J. C. Goodwin, B. K. Garside, IEEE J. Quantum Electron. QE-18, 1264 (1982).
[CrossRef]

D. Kato, IEEE J. Quantum Electron. QE-14, 563 (1978).
[CrossRef]

Y. Suematsu, S. Akiba, T. Hong, IEEE J. Quantum Electron. QE-13, 596 (1977).
[CrossRef]

W. Streifer, D. R. Scifres, R. D. Burnham, IEEE J. Quantum Electron. QE-18, 1918 (1982).
[CrossRef]

B. Zee, IEEE J. Quantum Electron. QE-14, 727 (1978).
[CrossRef]

J. Appl. Phys. (1)

B. W. Hakki, T. W. Paoli, J. Appl. Phys. 46, 1299 (1975).
[CrossRef]

Opto electronics (1)

M. J. Adams, Opto electronics 5, 201, 1973.
[CrossRef]

Other (3)

To achieve a quadratic relationship, the gain and spontaneous emission are assumed to depend linearly on the pumping rate.

G. Arnold, P. Russer, K. Petermann, in Semiconductor Devices for Optical Communication, H. Kressel, Ed. (Springer, Berlin, 1980).

D. T. Cassidy, submitted to IEEE J. Quantum Electron., June1983.

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Figures (6)

Fig. 1
Fig. 1

Fabry-Perot model of a laser showing the equivalent input fluxes b± and the circulating electric field E z ±. The end mirrors have an intensity reflectance of R. The single-pass intensity gain is G, so that | E z + | 2 = G | E 0 + | 2.

Fig. 2
Fig. 2

Total power and intensities of modes 0, 1, and 2 (measured relative to the gain peak) as predicted by the Fabry-Perot model with linear gain, Eq. (9). The spontaneous emission factor β = 10−3 for this example.

Fig. 3
Fig. 3

Total power and mode intensities as predicted by a traveling wave analysis (solid curves). The same parameters as used in the calculations for Fig. 2 were assumed. Note that, by scaling the axes of Fig. 3 relative to that of Fig. 2, the total-power-vs-pumping curves can be made similar to each other, but the distribution of energy between the modes is quite different in each case. To appreciate the magnitude of the scale change, the broken lines are the data of Fig. 2 replotted.

Fig. 4
Fig. 4

Plot of single-pass intensity gain G and total output power as a function of pumping. The curves were calculated in the small gain approximation, Eq. (8).

Fig. 5
Fig. 5

Experimentally determined single-pass intensity gain and output power. The axes are scaled arbitrarily to provide easier comparison to the theoretical curves.

Fig. 6
Fig. 6

Plots of single-pass intensity gain G and total output power as a function of the pumping rate. The curves were calculated using a traveling wave analysis.11 Note the similarity in shape between the gain curves of Fig. 5 (experiment) and this figure.

Tables (1)

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Table I Definition of Variables Used in the Analysis

Equations (11)

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d n d t = N n τ 2 n j σ j I j ,
d I m d t = β n τ m + n σ m I m α I m .
n = N S + 2 j σ j I j ,
I m = N σ m α S 2 j σ j I j S 1 + ( N σ m α S 2 j σ j I j S 1 ) 2 + 8 β N σ m α S 2 τ m 4 σ m / S .
| E z + | 2 = | b | 2 ( 1 + R G ) ( 1 R G ) 2 + 4 R G sin 2 ϕ .
I m = B m ( 1 R G m ) ,
B m | G m = 1 = β n τ m = β N / τ m 2 j σ j I j + S .
G m = a + a n σ m ,
I m = R N a σ m ( 1 R a ) S 2 j σ j I j S 1 + ( R N a σ m ( 1 R a ) S 2 j σ j I j S 1 ) 2 + 8 β N a σ m ( 1 R a ) S 2 τ m 4 σ m / S .
E z ±
j

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