Abstract

We analytically demonstrate that both single-mode and multimode instabilities may occur in a Gaussian-cavity-mode laser model with Gaussian pump profile. As a necessary condition, the ratio of the beam waist to the pump waist must exceed a given limiting value, which depends on the population decay rate. For an infinitely concentrated pump the plane-wave model instability thresholds are recovered, and there exists an optimum value of the waists ratio for which the second laser threshold is minimum.

© 1998 Optical Society of America

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References

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  1. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Weinheim, 1991).
  2. Ya. Khanin, Principles of Laser Dynamics (North-Holland, Amsterdam, 1995).
  3. F. T. Arecchi and R. G. Harrison, eds., Selected Papers on Optical Chaos, Proc. SPIE MS75 (1993).
  4. A. V. Uspenskii, Radio Eng. Electron. Phys. 8, 1145 (1963). 9, 605 (1964).
  5. A. Z. Grazyuk and A. N. Oraevskii, in Quantum Electronics and Coherent Light, P. A. Miles, ed. (Academic, New York, 1964).
  6. H. Haken, Z. Phys. 190, 327 (1966).
    [CrossRef]
  7. H. Risken, C. Schmidt, and W. Weidlich, Z. Phys. 194, 337 (1966).
    [CrossRef]
  8. H. Haken, Phys. Lett. A 53, 77 (1975).
    [CrossRef]
  9. E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
    [CrossRef]
  10. R. Risken and K. Nummedal, Phys. Lett. A 26, 275 (1968); J. Appl. Phys. 39, 4662 (1968).
    [CrossRef]
  11. R. Graham and H. Haken, Z. Phys. 213, 420 (1968).
    [CrossRef]
  12. For a study of the intimate relation existing between the single-mode and RNGH instabilities see, e.g., L. A. Lugiato and L. M. Narducci, Phys. Rev. A 32, 1576 (1985).
    [CrossRef] [PubMed]
  13. S. Ovadia and M. Sargent III, Opt. Commun. 49, 447 (1984).
    [CrossRef]
  14. C.-Z. Ning and H. Haken, Z. Phys. B 77, 157 (1989).
    [CrossRef]
  15. E. Roldán, G. J. de Valcárcel, and R. Vilaseca, Opt. Commun. 104, 85 (1993).
    [CrossRef]
  16. F. Prati, M. Brambilla, and L. A. Lugiato, Riv. Nuovo Cimento 17, 1 (1994) and references therein.
    [CrossRef]
  17. L. A. Lugiato and M. Milani, Opt. Commun. 46, 57 (1983).
    [CrossRef]
  18. S. Stuut and M. Sargent III, J. Opt. Soc. Am. B 1, 95 (1984).
    [CrossRef]
  19. L. A. Lugiato and M. Milani, J. Opt. Soc. Am. B 2, 15 (1985).
    [CrossRef]
  20. L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, Phys. Rev. A 30, 1366 (1984).
    [CrossRef]
  21. L. A. Lugiato and M. Milani, Z. Phys. B 50, 171 (1983).
    [CrossRef]
  22. C. P. Smith and R. Dykstra, Opt. Commun. 117, 107 (1995).
    [CrossRef]
  23. C. P. Smith and R. Dykstra, Opt. Commun. 129, 69 (1996).
    [CrossRef]
  24. E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
    [CrossRef]
  25. E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, Phys. Rev. A 56, 4086 (1997).
    [CrossRef]
  26. I. S. Gradshteyn and I. M. Solomonovich, Table of Integrals, Series and Products (Academic, New York, 1980).

1997 (2)

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, Phys. Rev. A 56, 4086 (1997).
[CrossRef]

1996 (1)

C. P. Smith and R. Dykstra, Opt. Commun. 129, 69 (1996).
[CrossRef]

1995 (1)

C. P. Smith and R. Dykstra, Opt. Commun. 117, 107 (1995).
[CrossRef]

1994 (1)

F. Prati, M. Brambilla, and L. A. Lugiato, Riv. Nuovo Cimento 17, 1 (1994) and references therein.
[CrossRef]

1993 (1)

E. Roldán, G. J. de Valcárcel, and R. Vilaseca, Opt. Commun. 104, 85 (1993).
[CrossRef]

1989 (1)

C.-Z. Ning and H. Haken, Z. Phys. B 77, 157 (1989).
[CrossRef]

1985 (2)

For a study of the intimate relation existing between the single-mode and RNGH instabilities see, e.g., L. A. Lugiato and L. M. Narducci, Phys. Rev. A 32, 1576 (1985).
[CrossRef] [PubMed]

L. A. Lugiato and M. Milani, J. Opt. Soc. Am. B 2, 15 (1985).
[CrossRef]

1984 (3)

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, Phys. Rev. A 30, 1366 (1984).
[CrossRef]

S. Stuut and M. Sargent III, J. Opt. Soc. Am. B 1, 95 (1984).
[CrossRef]

S. Ovadia and M. Sargent III, Opt. Commun. 49, 447 (1984).
[CrossRef]

1983 (2)

L. A. Lugiato and M. Milani, Opt. Commun. 46, 57 (1983).
[CrossRef]

L. A. Lugiato and M. Milani, Z. Phys. B 50, 171 (1983).
[CrossRef]

1975 (1)

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

1968 (1)

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

1966 (2)

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

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

1963 (1)

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

Bonfrate, G.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, Phys. Rev. A 56, 4086 (1997).
[CrossRef]

Brambilla, M.

F. Prati, M. Brambilla, and L. A. Lugiato, Riv. Nuovo Cimento 17, 1 (1994) and references therein.
[CrossRef]

Corbalán, R.

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

de Valcárcel, G. J.

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

E. Roldán, G. J. de Valcárcel, and R. Vilaseca, Opt. Commun. 104, 85 (1993).
[CrossRef]

Dykstra, R.

C. P. Smith and R. Dykstra, Opt. Commun. 129, 69 (1996).
[CrossRef]

C. P. Smith and R. Dykstra, Opt. Commun. 117, 107 (1995).
[CrossRef]

Fontana, F.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, Phys. Rev. A 56, 4086 (1997).
[CrossRef]

Gilmore, R.

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

Graham, R.

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

Haken, H.

C.-Z. Ning and H. Haken, Z. Phys. B 77, 157 (1989).
[CrossRef]

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

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

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

Horowicz, R. J.

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, Phys. Rev. A 30, 1366 (1984).
[CrossRef]

Lorenz, E. N.

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

Lugiato, L. A.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, Phys. Rev. A 56, 4086 (1997).
[CrossRef]

F. Prati, M. Brambilla, and L. A. Lugiato, Riv. Nuovo Cimento 17, 1 (1994) and references therein.
[CrossRef]

L. A. Lugiato and M. Milani, J. Opt. Soc. Am. B 2, 15 (1985).
[CrossRef]

For a study of the intimate relation existing between the single-mode and RNGH instabilities see, e.g., L. A. Lugiato and L. M. Narducci, Phys. Rev. A 32, 1576 (1985).
[CrossRef] [PubMed]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, Phys. Rev. A 30, 1366 (1984).
[CrossRef]

L. A. Lugiato and M. Milani, Opt. Commun. 46, 57 (1983).
[CrossRef]

L. A. Lugiato and M. Milani, Z. Phys. B 50, 171 (1983).
[CrossRef]

Martínez, V. J.

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

Milani, M.

L. A. Lugiato and M. Milani, J. Opt. Soc. Am. B 2, 15 (1985).
[CrossRef]

L. A. Lugiato and M. Milani, Opt. Commun. 46, 57 (1983).
[CrossRef]

L. A. Lugiato and M. Milani, Z. Phys. B 50, 171 (1983).
[CrossRef]

Narducci, L. M.

For a study of the intimate relation existing between the single-mode and RNGH instabilities see, e.g., L. A. Lugiato and L. M. Narducci, Phys. Rev. A 32, 1576 (1985).
[CrossRef] [PubMed]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, Phys. Rev. A 30, 1366 (1984).
[CrossRef]

Ning, C.-Z.

C.-Z. Ning and H. Haken, Z. Phys. B 77, 157 (1989).
[CrossRef]

Ovadia, S.

S. Ovadia and M. Sargent III, Opt. Commun. 49, 447 (1984).
[CrossRef]

Pessina, E. M.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, Phys. Rev. A 56, 4086 (1997).
[CrossRef]

Prati, F.

F. Prati, M. Brambilla, and L. A. Lugiato, Riv. Nuovo Cimento 17, 1 (1994) and references therein.
[CrossRef]

Risken, H.

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

Roldán, E.

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

E. Roldán, G. J. de Valcárcel, and R. Vilaseca, Opt. Commun. 104, 85 (1993).
[CrossRef]

Sargent III, M.

S. Ovadia and M. Sargent III, Opt. Commun. 49, 447 (1984).
[CrossRef]

S. Stuut and M. Sargent III, J. Opt. Soc. Am. B 1, 95 (1984).
[CrossRef]

Schmidt, C.

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

Smith, C. P.

C. P. Smith and R. Dykstra, Opt. Commun. 129, 69 (1996).
[CrossRef]

C. P. Smith and R. Dykstra, Opt. Commun. 117, 107 (1995).
[CrossRef]

Strini, G.

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, Phys. Rev. A 30, 1366 (1984).
[CrossRef]

Stuut, S.

Vilaseca, R.

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

E. Roldán, G. J. de Valcárcel, and R. Vilaseca, Opt. Commun. 104, 85 (1993).
[CrossRef]

Weidlich, W.

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

J. Atmos. Sci. (1)

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

J. Opt. Soc. Am. B (2)

Opt. Commun. (5)

C. P. Smith and R. Dykstra, Opt. Commun. 117, 107 (1995).
[CrossRef]

C. P. Smith and R. Dykstra, Opt. Commun. 129, 69 (1996).
[CrossRef]

S. Ovadia and M. Sargent III, Opt. Commun. 49, 447 (1984).
[CrossRef]

E. Roldán, G. J. de Valcárcel, and R. Vilaseca, Opt. Commun. 104, 85 (1993).
[CrossRef]

L. A. Lugiato and M. Milani, Opt. Commun. 46, 57 (1983).
[CrossRef]

Phys. Lett. A (1)

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

Phys. Rev. A (3)

For a study of the intimate relation existing between the single-mode and RNGH instabilities see, e.g., L. A. Lugiato and L. M. Narducci, Phys. Rev. A 32, 1576 (1985).
[CrossRef] [PubMed]

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, Phys. Rev. A 56, 4086 (1997).
[CrossRef]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, Phys. Rev. A 30, 1366 (1984).
[CrossRef]

Quantum Semiclassic. Opt. (1)

E. Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, R. Gilmore, and V. J. Martínez, Quantum Semiclassic. Opt. 9, 1 (1997).
[CrossRef]

Riv. Nuovo Cimento (1)

F. Prati, M. Brambilla, and L. A. Lugiato, Riv. Nuovo Cimento 17, 1 (1994) and references therein.
[CrossRef]

Z. Phys. (3)

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

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

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

Z. Phys. B (2)

C.-Z. Ning and H. Haken, Z. Phys. B 77, 157 (1989).
[CrossRef]

L. A. Lugiato and M. Milani, Z. Phys. B 50, 171 (1983).
[CrossRef]

Other (7)

I. S. Gradshteyn and I. M. Solomonovich, Table of Integrals, Series and Products (Academic, New York, 1980).

R. Risken and K. Nummedal, Phys. Lett. A 26, 275 (1968); J. Appl. Phys. 39, 4662 (1968).
[CrossRef]

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Weinheim, 1991).

Ya. Khanin, Principles of Laser Dynamics (North-Holland, Amsterdam, 1995).

F. T. Arecchi and R. G. Harrison, eds., Selected Papers on Optical Chaos, Proc. SPIE MS75 (1993).

A. V. Uspenskii, Radio Eng. Electron. Phys. 8, 1145 (1963). 9, 605 (1964).

A. Z. Grazyuk and A. N. Oraevskii, in Quantum Electronics and Coherent Light, P. A. Miles, ed. (Academic, New York, 1964).

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Figures (5)

Fig. 1
Fig. 1

(a) Dependence of the second to first laser thresholds, r, on the cavity losses, σ, for several values of the ratio of field intensity to pump waists, η (indicated in the figure). The dashed curve corresponds to the plane-wave model (Lorenz instability). (b) Dependence of the second threshold pump parameter value 2CHB on σ for several values of η. In both figures, b=1.

Fig. 2
Fig. 2

Dependence of the minimum (solid curve) and maximum (dashed curve) values of the cavity losses σ for instabilities to occur for the values of b indicated in the figure. Notice that for b=1 the upper bound on σ disappears.

Fig. 3
Fig. 3

Variation of the lowest (as a function of cavity losses) pump parameter value at the second laser threshold (2CHBmin) with η, for three values of the population decay rate b.

Fig. 4
Fig. 4

Dependence on the population decay rate b of the minimum η allowing instabilities (ηmin), of the value of η for which instabilities are more easily accessible (ηopt), and of the value of the optimum (minimum in σ and η) pump parameter value at the second laser threshold, 2CHBmin,opt (dashed curve).

Fig. 5
Fig. 5

Influence of η on the multimode instability for small population decay rate values. (a) Ratio of the second to first laser thresholds. (b) Value of the wave number of the most unstable mode normalized to its value corresponding to the plane-wave model (i.e., for η).

Equations (61)

Equations on this page are rendered with MathJax. Learn more.

czf(z, t)+tf(z, t)
=κ-f(z, t)+2C0dρ 4ρw02 ×exp[-(1+2η)ρ2/w02]P(ρ, z, t),
tP(ρ, z, t)
=γ[-P(ρ, z, t)+D(ρ, z, t)f(z, t)×exp(-ρ2/w02)],
tD(ρ, z, t)
=γ[-D(ρ, z, t)+1-P(ρ, z, t)f(z, t)×exp(-ρ2/w02)],
η=w02wp2.
f0=0,P0=0,D0=1,
P1(ρ)=f1 exp(-ρ2/w02)1+f12 exp(-2ρ2/w02),
D1(ρ)=11+f12 exp(-2ρ2/w02),
1r=F(η; f12)k=0 1+η1+η+k (-f12)k,
r2C1+η
r=f12ln(1+f12),(η0).
λ+ik=σ-1+1λ(λ+1+b) λ+2b-br λ+2λ+1 Fη; b(λ+1)(λ+b) f12,
λ3+(1+b+σ+ik)λ2+[b(σ+r)+ik(b+1)]λ
+ibkr+2bσ(r-1)=0,
ωHB2=2bσ(σ+1)σ-b-1(η)
rHB=σ(σ+b+3)σ-b-1(η),
σb+1,
r>rRNGH=5+3b+24+6b+2b2,
ωHB2=2σbσ+1 (rHB-1),
f1,HB2=2+η1+η σ(σ+3)(σ+1)2 (rHB-1)2rHB.
rHBFη; 2+η1+η σ(σ+3)(σ+1)2 (rHB-1)2rHB=1.
(η+1)2η(η+2) (σlim+1)2σlim(σlim+3)=1,
σlim±=η22+η-1±12 η(η+4)(η2-4).
kRNGH2=b6 (σ-3)2(rRNGH-1),
ωRNGH2=32 b(rRNGH-1),
f1,RNGH2=98 2+η1+η (rRNGH-1)2rRNGH.
rRNGHFη; 98 2+η1+η (rRNGH-1)2rRNGH=1.
λ+ik=σg(λ, r; b, η).
±iω0=σ0g(±iω0, r0; b0, η0).
k0(σ)σ-σ0σ0 ω0.
d Re(λ)dk{λ=iω0,k=k0(σ)}
=σ Imgλ0σ Imgλ02+σ Regλ0-12,
iω1=σ1g(iω1, r1; b0, η0),
iω1+idω=(σ1+dσ)g(iω1+idω, r1+dr; b0, η0)=σ1g(iω1, r1; b0, η0)+dσg(iω1, r1; b0, η0)+σ1igλ1dω+gr1dr,
Imgλ1=Q(r1, ω1)drdσ1,
Q(r1, ω1)=1ω1 Regλ1-σ1Regλ1 Regr1+Imgλ1 Imgr1,
czf(z, t)+tf(z, t)
=κ-f(z, t)+4Cπw02 00dxdyK(x, y)P(x, y, z, t),
tP(x, y, z, t)
=γ{-P(x, y, z, t)+D(x, y, z, t)f(z, t)×exp[-(x2+y2)/w02]},
tD(x, y, z, t)
=γ{-D(x, y, z, t)+1-P(x, y, z, t)f(z, t)×exp[-(x2+y2)/w02]},
K(x, y)=exp[-(1+2η)(x2+y2)/w02],
limαα2 exp[-α2π(x2+y2)]=δ(x, y),
K(x, y)πw022η δ(x, y),
czf(z, t)+tf(z, t)=κ-f(z, t)+2Cη P(0, 0, z, t).
tP(0, 0, z, t)=γ[-P(0, 0, z, t)+D(0, 0, z, t)f(z, t)],
tD(0, 0, z, t)=γ[-D(0, 0, z, t)+1-P(0, 0, z, t)f(z, t)].
12C=I(η; f12)0dρ 4ρw02 exp[-2(1+η)ρ2/w02]1+f12 exp(-2ρ2/w02).
I(η; f12)=01dζ ζη1+f12ζ,
I(η; f12)=11+η2F1(1, 1+η; 2+η; -f12)=11+η k=0 (1)k(1+η)k(2+η)k (-f12)kk!,
(s)k=Γ(s+k)Γ(s)
I(η; f12)=k=0 (-f12)k1+η+k.
F(η; f12)=2F1(1, 1+η; 2+η; -f12),
F(η; f12)=(1+η)01dζ ζη1+f12ζ=k=0 1+η1+η+k (-f12)k.
limηF(η; u)=11+u.
limuuF(η; u)=limu(1+η)01dζ uζη1+uζ=(1+η)01dζζη-1=η+1η.
u F(η; u)=(1+η) u 01dζ ζη1+uζ=-(1+η)01dζ ζη+1(1+uζ)2.
u F(η; u)=1+ηu 11+u-F(η; u).

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