Abstract

The self-pulsing instability in inhomogeneously broadened unidirectional traveling-wave single-mode lasers is discussed in detail. The analytic linear stability analysis of the stationary solutions shows that, in the case of a good-cavity limit and a large inhomogeneous linewidth, no self-pulsing instability occurs. On the other hand, in the case of a bad-cavity limit, there are two instabilities. At the same time, in the intermediate-cavity case two self-pulsing instabilities also exist; one corresponds to a lower threshold value, and the other to a higher one.

© 1985 Optical Society of America

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References

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  1. H. Haken, Z. Phys. 190, 327 (1966).
    [CrossRef]
  2. H. Risken, C. Schmid, W. Weidlich, Z. Phys. 194, 337 (1966).
    [CrossRef]
  3. H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1966).
    [CrossRef]
  4. R. Graham, H. Haken, Z. Phys. 213, 420 (1968).
    [CrossRef]
  5. H. Haken, H. Ohno, Opt. Commun. 16, 205 (1976).
    [CrossRef]
  6. M. Mayr, H. Risken, H. D. Vollmer, Opt. Commun. 36, 480 (1981).
    [CrossRef]
  7. H. Haken, Phys. Lett. 53A, 77 (1975).
  8. E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
    [CrossRef]
  9. L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
    [CrossRef]
  10. L. W. Casperson, Phys. Rev. A 21, 911 (1980).
    [CrossRef]
  11. S. T. Hendow, M. Sargent, Opt. Commun. 40, 385 (1982).
    [CrossRef]
  12. J. Bentley, N. B. Abraham, Opt. Commun. 41, 52 (1982).
    [CrossRef]
  13. P. Mandel, Opt. Commun. 44, 400 (1983).
    [CrossRef]
  14. P. Mandel, Opt. Commun. 45, 269 (1983).
    [CrossRef]
  15. L. A. Lugiato, L. M. Narducci, D. K. Bandy, N. B. Abraham, Opt. Commun. 46, 115 (1983).
    [CrossRef]
  16. H. Haken, in Encyclopedia of Physics, Laser Theory (Springer-Verlag, Berlin, 1983), Vol. XXV/2c.
  17. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  18. H. Haken, Advanced Synergetics (Springer-Verlag, Berlin, 1983).

1983

P. Mandel, Opt. Commun. 44, 400 (1983).
[CrossRef]

P. Mandel, Opt. Commun. 45, 269 (1983).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, N. B. Abraham, Opt. Commun. 46, 115 (1983).
[CrossRef]

1982

S. T. Hendow, M. Sargent, Opt. Commun. 40, 385 (1982).
[CrossRef]

J. Bentley, N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

1981

M. Mayr, H. Risken, H. D. Vollmer, Opt. Commun. 36, 480 (1981).
[CrossRef]

1980

L. W. Casperson, Phys. Rev. A 21, 911 (1980).
[CrossRef]

1978

L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

1976

H. Haken, H. Ohno, Opt. Commun. 16, 205 (1976).
[CrossRef]

1975

H. Haken, Phys. Lett. 53A, 77 (1975).

1968

R. Graham, H. Haken, Z. Phys. 213, 420 (1968).
[CrossRef]

1966

H. Haken, Z. Phys. 190, 327 (1966).
[CrossRef]

H. Risken, C. Schmid, W. Weidlich, Z. Phys. 194, 337 (1966).
[CrossRef]

H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1966).
[CrossRef]

1963

E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
[CrossRef]

Abraham, N. B.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, N. B. Abraham, Opt. Commun. 46, 115 (1983).
[CrossRef]

J. Bentley, N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Bandy, D. K.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, N. B. Abraham, Opt. Commun. 46, 115 (1983).
[CrossRef]

Bentley, J.

J. Bentley, N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

Casperson, L. W.

L. W. Casperson, Phys. Rev. A 21, 911 (1980).
[CrossRef]

L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

Graham, R.

R. Graham, H. Haken, Z. Phys. 213, 420 (1968).
[CrossRef]

Haken, H.

H. Haken, H. Ohno, Opt. Commun. 16, 205 (1976).
[CrossRef]

H. Haken, Phys. Lett. 53A, 77 (1975).

R. Graham, H. Haken, Z. Phys. 213, 420 (1968).
[CrossRef]

H. Haken, Z. Phys. 190, 327 (1966).
[CrossRef]

H. Haken, Advanced Synergetics (Springer-Verlag, Berlin, 1983).

H. Haken, in Encyclopedia of Physics, Laser Theory (Springer-Verlag, Berlin, 1983), Vol. XXV/2c.

Hendow, S. T.

S. T. Hendow, M. Sargent, Opt. Commun. 40, 385 (1982).
[CrossRef]

Lorenz, E. N.

E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, N. B. Abraham, Opt. Commun. 46, 115 (1983).
[CrossRef]

Mandel, P.

P. Mandel, Opt. Commun. 45, 269 (1983).
[CrossRef]

P. Mandel, Opt. Commun. 44, 400 (1983).
[CrossRef]

Mayr, M.

M. Mayr, H. Risken, H. D. Vollmer, Opt. Commun. 36, 480 (1981).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, N. B. Abraham, Opt. Commun. 46, 115 (1983).
[CrossRef]

Nummedal, K.

H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1966).
[CrossRef]

Ohno, H.

H. Haken, H. Ohno, Opt. Commun. 16, 205 (1976).
[CrossRef]

Risken, H.

M. Mayr, H. Risken, H. D. Vollmer, Opt. Commun. 36, 480 (1981).
[CrossRef]

H. Risken, C. Schmid, W. Weidlich, Z. Phys. 194, 337 (1966).
[CrossRef]

H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1966).
[CrossRef]

Sargent, M.

S. T. Hendow, M. Sargent, Opt. Commun. 40, 385 (1982).
[CrossRef]

Schmid, C.

H. Risken, C. Schmid, W. Weidlich, Z. Phys. 194, 337 (1966).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Vollmer, H. D.

M. Mayr, H. Risken, H. D. Vollmer, Opt. Commun. 36, 480 (1981).
[CrossRef]

Weidlich, W.

H. Risken, C. Schmid, W. Weidlich, Z. Phys. 194, 337 (1966).
[CrossRef]

IEEE J. Quantum Electron.

L. W. Casperson, IEEE J. Quantum Electron. QE-14, 756 (1978).
[CrossRef]

J. Appl. Phys.

H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1966).
[CrossRef]

J. Atmos. Sci.

E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
[CrossRef]

Opt. Commun.

H. Haken, H. Ohno, Opt. Commun. 16, 205 (1976).
[CrossRef]

M. Mayr, H. Risken, H. D. Vollmer, Opt. Commun. 36, 480 (1981).
[CrossRef]

S. T. Hendow, M. Sargent, Opt. Commun. 40, 385 (1982).
[CrossRef]

J. Bentley, N. B. Abraham, Opt. Commun. 41, 52 (1982).
[CrossRef]

P. Mandel, Opt. Commun. 44, 400 (1983).
[CrossRef]

P. Mandel, Opt. Commun. 45, 269 (1983).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, N. B. Abraham, Opt. Commun. 46, 115 (1983).
[CrossRef]

Phys. Lett.

H. Haken, Phys. Lett. 53A, 77 (1975).

Phys. Rev. A

L. W. Casperson, Phys. Rev. A 21, 911 (1980).
[CrossRef]

Z. Phys.

R. Graham, H. Haken, Z. Phys. 213, 420 (1968).
[CrossRef]

H. Haken, Z. Phys. 190, 327 (1966).
[CrossRef]

H. Risken, C. Schmid, W. Weidlich, Z. Phys. 194, 337 (1966).
[CrossRef]

Other

H. Haken, in Encyclopedia of Physics, Laser Theory (Springer-Verlag, Berlin, 1983), Vol. XXV/2c.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

H. Haken, Advanced Synergetics (Springer-Verlag, Berlin, 1983).

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Figures (12)

Fig. 1
Fig. 1

Threshold pump parameter ζζc versus cavity loss δ.

Fig. 2
Fig. 2

Critical self-pulsing frequency Ω versus cavity loss δ.

Fig. 3
Fig. 3

Threshold pump parameter ζζc versus cavity loss δ.

Fig. 4
Fig. 4

Critical self-pulsing frequency Ω versus cavity loss δ.

Fig. 5
Fig. 5

Threshold pump parameter ζ′ζ′c versus cavity loss δ.

Fig. 6
Fig. 6

Critical self-pulsing frequency Ω′ versus cavity loss δ.

Fig. 7
Fig. 7

Threshold pump parameter ζ′ ζ′c versus cavity loss δ.

Fig. 8
Fig. 8

Critical self-pulsing frequency Ω′ versus cavity loss δ.

Fig. 9
Fig. 9

Threshold pump parameter versus cavity loss. Solid curve denotes the second threshold, and dashed curve denotes the third one.

Fig. 10
Fig. 10

Critical self-pulsing frequency versus cavity loss. Solid curve denotes the second threshold, and dashed curve denotes the third one.

Fig. 11
Fig. 11

Threshold pump parameter versus cavity loss. Solid curve denotes the second threshold, and dashed curve denotes the third one.

Fig. 12
Fig. 12

Critical self-pulsing frequency versus cavity loss. Solid curve denotes the second threshold, and dashed curve denotes the third one.

Equations (79)

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d b d t = ( i ω κ ) b i g μ α μ ,
d α μ d t = ( i ω μ γ ) α μ + i g μ * σ μ b ,
d σ μ d t = γ ( d 0 μ σ μ ) 2 i ( g μ α μ b * g μ * α μ * b ) ,
g μ = g exp ( i k x μ ) ,
g μ * g μ = g 2 .
b = γ 2 g E exp ( i ϕ ) ,
α μ = κ γ 2 g 2 ( P + i Q ) exp ( i ϕ ) exp ( i k x μ ) ,
σ μ = κ γ g 2 D ,
d 0 μ = κ γ g 2 D ¯ f ,
t = τ γ .
d E d τ = κ γ E + κ γ Q d ω ¯ ,
d ϕ d τ = ω γ + κ γ E P d ω ¯ ,
d P d τ = P + ( ω ¯ γ d ϕ d τ ) Q ,
d Q d τ = Q + D E ( ω ¯ γ d ϕ d τ ) P ,
d D d τ = γ γ ( D ¯ f D ) E Q .
d ϕ d τ = Ω γ
E = P = Q = 0 ,
D = D ¯ f ,
ϕ = Ω γ τ ,
E = [ γ γ ( π D ¯ 2 γ 2 α 2 Re 2 W ( x + i y ) 1 ) ] 1 / 2 ,
P = γ N a D ¯ E ( ω ¯ Ω ) f ,
Q = γ N a D ¯ E f ,
D = 1 N a [ γ 2 + ( ω ¯ Ω ) 2 ] D ¯ f ,
ϕ = Ω γ τ ,
f = 1 α π exp [ ( ω ¯ ω 0 ) 2 α 2 ] ,
W ( x + i y ) = exp [ ( x + i y ) 2 ] erfc [ i ( x + i y ) ] ,
x = Ω ω 0 α ,
y = a α ,
a 2 = γ 2 + γ 3 γ E 2 ,
N a = ( ω ¯ Ω ) 2 + a 2 .
X ¯ = X + X e λ τ
( λ + κ γ ) E = κ γ Q d ω ¯ ,
λ E ϕ = ω Ω γ E + κ γ P d ω ¯ ,
( λ + 1 ) P = ω ¯ Ω γ Q λ Q ϕ ,
( λ + 1 ) Q = ω ¯ Ω γ P + D E + E D + λ P ϕ ,
( λ + γ λ ) D = E Q Q E .
γ λ + κ κ γ 2 ( λ + 1 ) + E 2 λ ( λ + 1 ) π D α a ( 1 + E 2 λ ( λ + 1 ) ) π D α b = 0 ,
1 + κ γ π D ( λ + 1 ) α a + κ a λ ( λ + 1 ) π D α γ ( λ + 2 ) κ γ π D ( λ + 1 ) α b κ a 2 λ ( λ + 1 ) π D γ α b = 0 ,
b 2 = γ 2 E 2 + γ 2 ( λ + 1 ) 2 .
γ = γ = γ ,
Ω = ω 0 ,
a 2 α 2 1 , b 2 α 2 1 .
i = 1 5 C i λ i = 0 ,
C 0 = 2 δ 2 ( ζ 1 ) 2 ,
C 1 = ζ 2 ( 2 δ δ 2 ) + ζ ( 4 δ 2 2 δ ) 3 δ 2 ,
C 2 = 6 ζ δ 4 δ ,
C 3 = ζ ( 1 + 2 δ δ 2 ) + δ 2 + 2 δ ,
C 4 = 2 δ + 2 ,
C 5 = 1 .
δ = κ / γ ,
ζ = π D ¯ 2 γ 2 / α 2 .
( λ 2 + Ω 2 ) n = 0 3 b n λ n = 0 .
Ω 2 = a ζ 2 b ζ + c a ζ b ,
ζ 4 + a 3 ζ 3 + a 2 ζ 2 + a 1 ζ + a 0 = 0 ,
λ 3 + b 2 λ 2 + b 1 λ + b 0 = 0 ,
a = δ 3 2 δ ,
b = 4 δ 3 + 4 δ 2 2 δ ,
c = 3 δ 3 + 4 δ 2 ,
a = δ 3 δ 2 1 ,
b = δ 3 + 3 δ 2 + 4 δ ,
b 0 = 1 Ω 2 [ 2 δ 2 ( ζ 1 ) 2 ] ,
b 1 = 1 Ω 2 [ ζ 2 ( 2 δ δ 2 ) + ζ ( 4 δ 2 2 δ ) 3 δ 2 ] ,
b 2 = 2 δ + 2 ,
b 3 = 1 ,
a 0 = 1 B ( 16 δ 5 + 24 δ 6 + 9 δ 7 + δ 8 ) ,
a 1 = 1 B ( 32 δ 4 + 2 δ 5 43 δ 6 17 δ 7 4 δ 8 ) ,
a 2 = 1 B ( δ 2 + 16 δ 3 43 δ 4 28 δ 5 + 27 δ 6 + 3 δ 7 + 6 δ 8 ) ,
a 3 = 1 B ( 20 δ 3 + 16 δ 4 + 10 δ 5 13 δ 6 + 9 δ 7 4 δ 8 ) ,
B = δ 2 + 4 δ 3 5 δ 4 + 5 δ 6 4 δ 7 + δ 8 .
i = 1 3 A i λ i = 0 ,
A 0 = 2 ζ δ ,
A 1 = ζ ( 2 δ δ 2 ) + ( δ + 1 ) 2 ,
A 2 = 2 δ + 2 ,
A 3 = 1 .
( λ 2 + Ω 2 ) ( λ h ) = 0 .
Ω 2 = ( δ + 1 ) 2 δ 2 δ 1 ,
ζ = ( δ + 1 ) 3 δ ( δ 2 δ 1 ) ,
h = 2 δ 2 .
X ( τ ) = X n exp ( i n Ω τ ) .

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