Abstract

An analysis of the quantum noise in distributed-feedback lasers based on the Fokker–Planck equation has been developed to take into account the nonorthogonal nature of the laser modes. We have obtained numerical results of the steady-state solution of the single-mode operation that reveal the difference between the standard approach orthogonal laser modes and the realistic model (mode nonorthogonality included) for a distributed-feedback laser with nonvanishing end reflectivity and the complex coupling coefficient.

© 1994 Optical Society of America

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References

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  1. K. Pertmann, IEEE J. Quantum Electron. QE-15, 566 (1979).
  2. A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
    [CrossRef] [PubMed]
  3. R. Graham, H. Haken, Z. Phys. 243, 289 (1971).
    [CrossRef]
  4. M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677 (1978).
    [CrossRef]
  5. H. Risken, The Fokker–Planck Equation, Methods of Solution and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989), Chap. 12, p. 374.
  6. P. Szczepański, A. Kujawski, Opt. Commun. 87, 259 (1992).
    [CrossRef]
  7. P. Szczepański, Appl. Opt. 24, 2574 (1985).
    [CrossRef]
  8. A. Tyszka-Zawadzka, P. Szczepański, W. Woliński, IEEE J. Quantum Electron. 29, 2873 (1993).
    [CrossRef]

1993 (1)

A. Tyszka-Zawadzka, P. Szczepański, W. Woliński, IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

1992 (1)

P. Szczepański, A. Kujawski, Opt. Commun. 87, 259 (1992).
[CrossRef]

1989 (1)

A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

1985 (1)

P. Szczepański, Appl. Opt. 24, 2574 (1985).
[CrossRef]

1979 (1)

K. Pertmann, IEEE J. Quantum Electron. QE-15, 566 (1979).

1978 (1)

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677 (1978).
[CrossRef]

1971 (1)

R. Graham, H. Haken, Z. Phys. 243, 289 (1971).
[CrossRef]

Graham, R.

R. Graham, H. Haken, Z. Phys. 243, 289 (1971).
[CrossRef]

Haken, H.

R. Graham, H. Haken, Z. Phys. 243, 289 (1971).
[CrossRef]

Kujawski, A.

P. Szczepański, A. Kujawski, Opt. Commun. 87, 259 (1992).
[CrossRef]

Mandel, L.

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677 (1978).
[CrossRef]

M-Tehrani, M.

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677 (1978).
[CrossRef]

Pertmann, K.

K. Pertmann, IEEE J. Quantum Electron. QE-15, 566 (1979).

Risken, H.

H. Risken, The Fokker–Planck Equation, Methods of Solution and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989), Chap. 12, p. 374.

Siegman, A. E.

A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

Szczepanski, P.

A. Tyszka-Zawadzka, P. Szczepański, W. Woliński, IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

P. Szczepański, A. Kujawski, Opt. Commun. 87, 259 (1992).
[CrossRef]

P. Szczepański, Appl. Opt. 24, 2574 (1985).
[CrossRef]

Tyszka-Zawadzka, A.

A. Tyszka-Zawadzka, P. Szczepański, W. Woliński, IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

Wolinski, W.

A. Tyszka-Zawadzka, P. Szczepański, W. Woliński, IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

Appl. Opt. (1)

P. Szczepański, Appl. Opt. 24, 2574 (1985).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Tyszka-Zawadzka, P. Szczepański, W. Woliński, IEEE J. Quantum Electron. 29, 2873 (1993).
[CrossRef]

K. Pertmann, IEEE J. Quantum Electron. QE-15, 566 (1979).

Opt. Commun. (1)

P. Szczepański, A. Kujawski, Opt. Commun. 87, 259 (1992).
[CrossRef]

Phys. Rev. A (2)

A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

M. M-Tehrani, L. Mandel, Phys. Rev. A 17, 677 (1978).
[CrossRef]

Z. Phys. (1)

R. Graham, H. Haken, Z. Phys. 243, 289 (1971).
[CrossRef]

Other (1)

H. Risken, The Fokker–Planck Equation, Methods of Solution and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989), Chap. 12, p. 374.

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Figures (6)

Fig. 1
Fig. 1

Steady-state distribution function pst versus the normalized laser intensity Ĩ for the amplitudes of the end reflectivity r = 0.1 (solid curves) and r = 0.5 (dashed curves), with the phase of the end reflectivity φr as a parameter. The normalized pump parameter is ã = 2, and the coupling coefficient |κL| = 5. The curve with asterisks was obtained for orthogonal modes.

Fig. 2
Fig. 2

Mean normalized laser intensity 〈Ĩ〉 as a function of the normalized pump parameter ã for |κL| = 0.1 and the amplitude of the end reflectivity r = 0.1 (solid curves) and for |κL| = 5 and r = 0.5 (dashed curves), with φr as a parameter. The curve with asterisks was obtained for orthogonal modes.

Fig. 3
Fig. 3

Variation of the intensity fluctuation σ with the normalized pump parameter ã for r = 0.1 (solid curves) and r = 0.5 (dashed curves), with φr as a parameter. The coupling coefficient is |κL| = 5. The curve with asterisks was obtained for orthogonal modes.

Fig. 4
Fig. 4

Steady-state distribution function pst versus the normalized laser intensity Ĩ for |κL| = 5 (solid curves) and |κL| = 1 (dashed curves), with the phase of the complex coupling coefficient φ as a parameter. The normalized pump parameter ã = 2. The curve with asterisks was obtained for orthogonal modes.

Fig. 5
Fig. 5

Mean normalized laser intensity 〈Ĩ〉 as a function of the normalized pump parameter ã for |κL| = 1, with φ as a parameter. The curve with asterisks was obtained for orthogonal modes.

Fig. 6
Fig. 6

Variation of the intensity fluctuation σ with the normalized pump parameter ã for |κL| = 5 (solid curves) and |κL| = 1 (dashed curves), with φ as a parameter. The curve with asterisks was obtained for orthogonal modes.

Equations (10)

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d E d t = ( a ˜ - β E 2 ) E + q ( t ) , d E * d t = ( a ˜ - β E 2 ) E * + q * ( t ) ,
q ( t ) q ( t ) = q * ( t ) q * ( t ) = 0 , q ( t ) q * ( t ) = 2 K δ ( t - t ) ,
K = v d s d z Φ N ( s , z ) Φ N * ( s , z ) ,
P t = - E ( A p ) - E * ( A * p ) + 4 D i j 2 p E E * ,
A = ( a ˜ - β E 2 ) E , A * = ( a ˜ - β E 2 ) E *
p ( I ˜ ) = Q - 1 exp [ ( 1 2 a ˜ I ˜ - 1 4 β I ˜ 2 ) K - 1 ] ,
σ = I ˜ 2 - I ˜ 2 = ( 1 / β ) [ 2 - 2 a ˜ / ( K Q ) - 4 / Q 2 ] ,
Q = π K / β exp [ a ˜ 2 / ( 4 K β ) ] erfc ( - a ˜ / 2 K β ) .
Φ N ( z ) = [ S N ( z ) exp ( - i β z ) R N ( z ) exp ( i β z ) ] ,
R N = sinh γ N ( z + L / 2 ) - R sinh γ N ( z - L / 2 ) , S N = - sinh γ N ( z - L / 2 ) + R sinh γ N ( z + L / 2 ) ,

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