Abstract

We show that the instability threshold for a laser in which both polarizations are active can be much lower than the Haken second threshold and does not require the bad-cavity limit.

© 1987 Optical Society of America

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References

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  1. H. Haken, Z. Phys. 190, 327 (1966); H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1968).
    [CrossRef]
  2. C. O. Weiss, J. Brock, Phys. Rev. Lett. 57, 2804 (1986); N. B. Abraham, D. Dangoisse, P. R. Glorieux, P. Mandel, J. Opt. Soc. Am. B 2, 408 (1985); C. O. Weiss, W. Klische, P. S. Ering, M. Cooper, Opt. Commun. 52, 408 (1985).
    [CrossRef] [PubMed]
  3. M. A. Dupertuis, R. R. E. Salomaa, M. R. Siegrist, Opt. Commun. 57, 410 (1986); N. M. Lawandy, D. V. Plant, Opt. Commun. 59, 55 (1986).
    [CrossRef]
  4. G. L. Lippi, N. B. Abraham, G. P. Puccioni, F. T. Arecchi, J. R. Tredicce, submitted to Phys. Rev. Lett.
  5. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).
  6. M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addision-Wesley, Reading, Mass., 1974).
  7. H. Haken, Synergetics: An Introduction (Springer-Verlag, Berlin, 1977); Advanced Synergetics (Springer-Verlag, Berlin, 1983).

1986 (3)

C. O. Weiss, J. Brock, Phys. Rev. Lett. 57, 2804 (1986); N. B. Abraham, D. Dangoisse, P. R. Glorieux, P. Mandel, J. Opt. Soc. Am. B 2, 408 (1985); C. O. Weiss, W. Klische, P. S. Ering, M. Cooper, Opt. Commun. 52, 408 (1985).
[CrossRef] [PubMed]

M. A. Dupertuis, R. R. E. Salomaa, M. R. Siegrist, Opt. Commun. 57, 410 (1986); N. M. Lawandy, D. V. Plant, Opt. Commun. 59, 55 (1986).
[CrossRef]

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).

1966 (1)

H. Haken, Z. Phys. 190, 327 (1966); H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1968).
[CrossRef]

Abraham, N. B.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).

G. L. Lippi, N. B. Abraham, G. P. Puccioni, F. T. Arecchi, J. R. Tredicce, submitted to Phys. Rev. Lett.

Arecchi, F. T.

G. L. Lippi, N. B. Abraham, G. P. Puccioni, F. T. Arecchi, J. R. Tredicce, submitted to Phys. Rev. Lett.

Bandy, D. K.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).

Brock, J.

C. O. Weiss, J. Brock, Phys. Rev. Lett. 57, 2804 (1986); N. B. Abraham, D. Dangoisse, P. R. Glorieux, P. Mandel, J. Opt. Soc. Am. B 2, 408 (1985); C. O. Weiss, W. Klische, P. S. Ering, M. Cooper, Opt. Commun. 52, 408 (1985).
[CrossRef] [PubMed]

Dupertuis, M. A.

M. A. Dupertuis, R. R. E. Salomaa, M. R. Siegrist, Opt. Commun. 57, 410 (1986); N. M. Lawandy, D. V. Plant, Opt. Commun. 59, 55 (1986).
[CrossRef]

Haken, H.

H. Haken, Z. Phys. 190, 327 (1966); H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1968).
[CrossRef]

H. Haken, Synergetics: An Introduction (Springer-Verlag, Berlin, 1977); Advanced Synergetics (Springer-Verlag, Berlin, 1983).

Lamb, W. E.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addision-Wesley, Reading, Mass., 1974).

Lippi, G. L.

G. L. Lippi, N. B. Abraham, G. P. Puccioni, F. T. Arecchi, J. R. Tredicce, submitted to Phys. Rev. Lett.

Lugiato, L. A.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).

Narducci, L. M.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).

Puccioni, G. P.

G. L. Lippi, N. B. Abraham, G. P. Puccioni, F. T. Arecchi, J. R. Tredicce, submitted to Phys. Rev. Lett.

Salomaa, R. R. E.

M. A. Dupertuis, R. R. E. Salomaa, M. R. Siegrist, Opt. Commun. 57, 410 (1986); N. M. Lawandy, D. V. Plant, Opt. Commun. 59, 55 (1986).
[CrossRef]

Sargent, M.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addision-Wesley, Reading, Mass., 1974).

Scully, M. O.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addision-Wesley, Reading, Mass., 1974).

Siegrist, M. R.

M. A. Dupertuis, R. R. E. Salomaa, M. R. Siegrist, Opt. Commun. 57, 410 (1986); N. M. Lawandy, D. V. Plant, Opt. Commun. 59, 55 (1986).
[CrossRef]

Tredicce, J. R.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).

G. L. Lippi, N. B. Abraham, G. P. Puccioni, F. T. Arecchi, J. R. Tredicce, submitted to Phys. Rev. Lett.

Weiss, C. O.

C. O. Weiss, J. Brock, Phys. Rev. Lett. 57, 2804 (1986); N. B. Abraham, D. Dangoisse, P. R. Glorieux, P. Mandel, J. Opt. Soc. Am. B 2, 408 (1985); C. O. Weiss, W. Klische, P. S. Ering, M. Cooper, Opt. Commun. 52, 408 (1985).
[CrossRef] [PubMed]

Opt. Commun. (1)

M. A. Dupertuis, R. R. E. Salomaa, M. R. Siegrist, Opt. Commun. 57, 410 (1986); N. M. Lawandy, D. V. Plant, Opt. Commun. 59, 55 (1986).
[CrossRef]

Phys. Rev. (1)

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, D. K. Bandy, Phys. Rev. 33A, 1842 (1986).

Phys. Rev. Lett. (1)

C. O. Weiss, J. Brock, Phys. Rev. Lett. 57, 2804 (1986); N. B. Abraham, D. Dangoisse, P. R. Glorieux, P. Mandel, J. Opt. Soc. Am. B 2, 408 (1985); C. O. Weiss, W. Klische, P. S. Ering, M. Cooper, Opt. Commun. 52, 408 (1985).
[CrossRef] [PubMed]

Z. Phys. (1)

H. Haken, Z. Phys. 190, 327 (1966); H. Risken, K. Nummedal, J. Appl. Phys. 39, 4662 (1968).
[CrossRef]

Other (3)

G. L. Lippi, N. B. Abraham, G. P. Puccioni, F. T. Arecchi, J. R. Tredicce, submitted to Phys. Rev. Lett.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addision-Wesley, Reading, Mass., 1974).

H. Haken, Synergetics: An Introduction (Springer-Verlag, Berlin, 1977); Advanced Synergetics (Springer-Verlag, Berlin, 1983).

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Figures (4)

Fig. 1
Fig. 1

The three-level atom considered in the model. The upper level is threefold degenerate, but only two of these substates interact with the lower level.

Fig. 2
Fig. 2

Instability threshold for different values of γc and γ|| = 0.01. The limit for γc → ∞ gives the Haken second threshold (all values normalized to γ).

Fig. 3
Fig. 3

Instability threshold obtained separately from the two matrices for γ|| = 0.01 and k = 3 (all values normalized to γ); the dashed line corresponds to the 3 × 3 matrix and the solid line to the 4 × 4.

Fig. 4
Fig. 4

Output power obtained from numerical integration of Eqs. (4) for γ|| = 0.01, γc = 1, k = 0.5, σ = 10 (γ||, γc, and k normalized to γ). Transients have been discarded.

Equations (14)

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ρ ˙ 11 = γ | | ρ 11 + i ( V 10 * ρ 10 - c . c . ) + λ 1 , ρ ˙ - 1 - 1 = - γ | | ρ - 1 - 1 + i ( V - 10 * ρ - 10 - c . c . ) + λ 1 , ρ ˙ 00 = - γ | | ρ 00 + i ( V 10 ρ 10 * + V - 10 ρ - 10 * - c . c . ) + λ 0 , ρ ˙ 10 = - ( i ω 0 + γ ) ρ 10 + i [ V 10 ( ρ 11 - ρ 00 ) + V - 10 ρ 1 - 1 ] , ρ ˙ - 10 = - ( i ω 0 + γ ) ρ - 10 + i [ V - 10 ( ρ - 1 - 1 - ρ 00 ) + V 10 ρ 1 - 1 * ] , ρ ˙ 1 - 1 = - γ c ρ 1 - 1 + i ( V - 10 * ρ 10 - V 10 ρ - 10 * ) ,
E ˙ R = - k R E R - i ν R E R - i g * P R , P ˙ R = - ( i ω 0 + γ ) P R + i g D R E R + i g C E L , D ˙ R = - γ | | ( D R - σ ) + i [ ( 2 g * E R * P R - g E L P L * ) - c . c . ] , E ˙ L = - k L E L - i ν L E L - i g * P L , P ˙ L = - ( i ω 0 + γ ) P L + i g D L E L + i g C * E R , D ˙ L = - γ | | ( D L - σ ) + i [ ( 2 g * E L * P L - g E R P R * ) - c . c . ] , C ˙ = - γ c C + i g * E L * P R - i g E R P L * ,
E = γ | | γ 2 g E ˜ ,             P = i k γ | | γ 2 g 2 P ˜ , D = k γ g 2 D ˜ ,             C = k γ g 2 C ˜
E ˙ R = - k ( E R - P R ) , P ˙ R = - γ ( P R - E R D R - E L C ) , D ˙ R = - γ | | ( D R - σ ) - γ | | ( E R P R + ½ E L P L ) , E ˙ L = - k ( E L - P L ) , P ˙ L = - γ ( P L - E L D L - E R C ) , D ˙ L = - γ | | ( D L - σ ) - γ | | ( E L P L + ½ E R P R ) , C ˙ = - γ c C - γ | | 4 ( E L P R + E R P L ) .
A + = A R + A L ,             A - = A R - A L             ( A = E , P , D ) ,
E ˙ + = - k ( E + - P + ) , P ˙ + = - γ P + + ½ γ ( E + D + + E - D - ) + γ E + C , D ˙ + = - γ | | ( D + - 2 σ ) - ¾ γ | | ( E + P + + E - P - ) , E ˙ - = - k ( E - - P - ) , P ˙ - = - γ P - + ½ γ ( E + D - + E - D + ) - γ E - C , D ˙ - = - γ | | D - - / 4 1 γ | | ( E + P - + E - P + ) , C ˙ = - γ c C - γ | | ( E + P + - E - P - ) ,
C = E + = E - = P + = P - = D - = 0 ,             D + = 2 σ ,
E + = E - = P + = P - = σ - 1 ,             D + = ( 3 + σ ) / 2 , D - = ( 1 - σ ) / 2 ,             C = 0 ,
E + = P + = 2 2 γ c ( σ - 1 ) 3 γ c + γ | | ,             D + = 2 + 2 γ | | ( σ - 1 ) 3 γ c + γ | | , E - = P - = D - = 0 ,             C = - γ | | ( σ - 1 ) 3 γ c + γ | | .
λ 4 + ( k + γ + γ c + γ | | ) λ 3 + [ ( k + γ ) ( γ | | + γ c ) + γ | | γ c + γ | | γ 4 E 2 ] λ 2 + [ ( k + γ ) γ | | γ c + γ | | γ 8 × E 2 ( 6 k - γ | | + 3 γ c ) ] λ + 3 γ c k γ | | γ 4 E 2 = 0 ,
λ 3 + ( k + γ | | + γ ) λ 2 + [ k γ | | + γ | | γ + γ | | γ 8 γ c × E 2 ( γ c - 2 k ) ] λ + k γ γ | | 4 γ c E 2 ( γ c - γ | | ) = 0 ,
γ | | / γ c > 1 ,
σ > [ ( k + γ ) 2 + γ | | ( k + γ ) ] ( 3 γ c + γ | | ) γ ( 2 k 2 + 2 k γ + k γ c - γ | | γ c - γ c γ ) + 1.
Ω = [ 2 γ | | k ( γ c - γ | | ) ( k + γ ) 2 k 2 + k γ c + 2 k γ | | - γ γ c - γ | | γ c ] 1 / 2 .

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