Abstract

The threshold for multilongitudinal-mode emission in inhomogeneously broadened ring lasers is analytically investigated. In the homogeneous limit the multimode instability corresponds to the classical Risken–Nummedal–Graham–Haken instability. It is found that by increasing the inhomogeneous linewidth, the instability threshold is decreased and the growth of high-frequency side modes is favored. In the limit where the population-inversion decay rate γ is much smaller than the polarization decay rate γ (class B lasers), analytical expressions for the instability threshold are found, which are then generalized to three-level lasers for a comparison with experimental results obtained with erbium-doped fiber lasers. It is shown that even in class B lasers a full Maxwell–Bloch description (in opposition to a rate-equations approach) is necessary when the free spectral range of the cavity is less than (γ/γ)1/4γ.

© 2001 Optical Society of America

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References

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  1. H. Risken and K. Nummedal, “Instability of off resonance modes in lasers,” Phys. Lett. 26A, 275–276 (1968).
    [CrossRef]
  2. H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
    [CrossRef]
  3. R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
    [CrossRef]
  4. E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
    [CrossRef]
  5. B. Segard and B. Macke, “Self-pulsing in intrinsic optical bistability with two-level molecules,” Phys. Rev. Lett. 60, 412–415 (1988).
    [CrossRef] [PubMed]
  6. B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
    [CrossRef] [PubMed]
  7. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).
  8. P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge University, Cambridge, UK, 1997).
  9. L. M. Narducci and N. B. Abraham, Laser Physics and Laser Instabilities (World Scientific, Singapore, 1988).
  10. L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
    [CrossRef] [PubMed]
  11. Ya. I. Khanin, Principles of Laser Dynamics (Elsevier, Amsterdam, 1995).
  12. E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
    [CrossRef]
  13. T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” Proc. SPIE 4429, 112–115 (2001).
    [CrossRef]
  14. E. Roldán and G. J. de Valcárcel, “On the Observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped Fiber Lasers,” Europhys. Lett. 43, 255–260 (1998).
    [CrossRef]

2001

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

1999

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

1998

E. Roldán and G. J. de Valcárcel, “On the Observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped Fiber Lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

1997

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

1989

B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
[CrossRef] [PubMed]

1988

B. Segard and B. Macke, “Self-pulsing in intrinsic optical bistability with two-level molecules,” Phys. Rev. Lett. 60, 412–415 (1988).
[CrossRef] [PubMed]

1986

L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

1968

H. Risken and K. Nummedal, “Instability of off resonance modes in lasers,” Phys. Lett. 26A, 275–276 (1968).
[CrossRef]

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

Abraham, N. B.

L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

Bandy, D. K.

L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

Bonfrate, G.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

Brambilla, M.

B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
[CrossRef] [PubMed]

de Valcárcel, G. J.

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

E. Roldán and G. J. de Valcárcel, “On the Observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped Fiber Lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

Fontana, F.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

Graham, R.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

Haken, H.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

Lenz, M. O.

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Lugiato, L. A.

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
[CrossRef] [PubMed]

L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

Macke, B.

B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
[CrossRef] [PubMed]

B. Segard and B. Macke, “Self-pulsing in intrinsic optical bistability with two-level molecules,” Phys. Rev. Lett. 60, 412–415 (1988).
[CrossRef] [PubMed]

Mitschke, F.

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Narducci, L. M.

L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

Nummedal, K.

H. Risken and K. Nummedal, “Instability of off resonance modes in lasers,” Phys. Lett. 26A, 275–276 (1968).
[CrossRef]

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Pessina, E. M.

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

Prati, F.

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
[CrossRef] [PubMed]

Redondo, J.

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

Risken, H.

H. Risken and K. Nummedal, “Instability of off resonance modes in lasers,” Phys. Lett. 26A, 275–276 (1968).
[CrossRef]

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Roldán, E.

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

E. Roldán and G. J. de Valcárcel, “On the Observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped Fiber Lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

Segard, B.

B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
[CrossRef] [PubMed]

B. Segard and B. Macke, “Self-pulsing in intrinsic optical bistability with two-level molecules,” Phys. Rev. Lett. 60, 412–415 (1988).
[CrossRef] [PubMed]

Tredice, J. R.

L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

Voigt, T.

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Europhys. Lett.

E. Roldán and G. J. de Valcárcel, “On the Observability of the Risken–Nummedal–Graham–Haken multimode instability in erbium-doped Fiber Lasers,” Europhys. Lett. 43, 255–260 (1998).
[CrossRef]

J. Appl. Phys.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Phys. Lett.

H. Risken and K. Nummedal, “Instability of off resonance modes in lasers,” Phys. Lett. 26A, 275–276 (1968).
[CrossRef]

Phys. Rev. A

L. M. Narducci, J. R. Tredice, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode–mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986).
[CrossRef] [PubMed]

E. M. Pessina, F. Prati, J. Redondo, E. Roldán, and G. J. de Valcárcel, “Multimode instability in ring fiber lasers,” Phys. Rev. A 60, 2517–2528 (1999).
[CrossRef]

E. M. Pessina, G. Bonfrate, F. Fontana, and L. A. Lugiato, “Experimental observation of the Risken–Nummedal–Graham–Haken multimode laser instability,” Phys. Rev. A 56, 4086–4093 (1997).
[CrossRef]

B. Segard, B. Macke, L. A. Lugiato, F. Prati, and M. Brambilla, “Multimode instability in optical bistability,” Phys. Rev. A 39, 703–722 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett.

B. Segard and B. Macke, “Self-pulsing in intrinsic optical bistability with two-level molecules,” Phys. Rev. Lett. 60, 412–415 (1988).
[CrossRef] [PubMed]

Proc. SPIE

T. Voigt, M. O. Lenz, and F. Mitschke, “Risken–Nummedal–Graham–Haken instability finally confirmed experimentally,” Proc. SPIE 4429, 112–115 (2001).
[CrossRef]

Z. Phys.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–450 (1968).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge University, Cambridge, UK, 1997).

L. M. Narducci and N. B. Abraham, Laser Physics and Laser Instabilities (World Scientific, Singapore, 1988).

Ya. I. Khanin, Principles of Laser Dynamics (Elsevier, Amsterdam, 1995).

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Figures (6)

Fig. 1
Fig. 1

Multimode-emission threshold for γ=1, σ=0.01, and for the indicated values of the inhomogeneous gain linewidth u. The single-mode solution is unstable to the right of each curve. α is the scaled frequency of a side mode, which must be an integer multiple of the free spectral range α˜, and r is the pump parameter scaled to the laser threshold.

Fig. 2
Fig. 2

(a) Minimum instability threshold rc and (b) instability threshold at the spatial frequency α=10 as functions of the inhomogeneous gain linewidth u for two values of γ.

Fig. 3
Fig. 3

Comparison between the exact instability threshold (solid curves) and the instability threshold predicted by Eq. (21) in the pure spectral-hole-burning limit (dashed curves), for the same γ and σ of Fig. 1, and for the indicated values of the inhomogeneous gain linewidth u.

Fig. 4
Fig. 4

Instability threshold associated with the amplitude instability (solid curves) and the phase instability (dashed curves) for the indicated values of γ. σ=0.01, u=2.

Fig. 5
Fig. 5

Comparison between the exact instability threshold in a class B laser (diamonds) and the approximated curves valid in the limit α1 (dotted curve) and α=O(1) (solid curve). The parameters are γ=0.001, σ=0.1, and u=4.

Fig. 6
Fig. 6

Multimode-emission threshold in an erbium-doped fiber laser with G=60 for the indicated values of the inhomogeneous gain linewidth u.

Equations (87)

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(τ+ζ)F(ζ, τ)=σ-F+A-+dωL(ω)P,
τP(ω, ζ, τ)=γ-1[-(1+iω)P+FD],
τD(ω, ζ, τ)=γ[1-D-(FP*+F*P)/2].
τ=γγt,ζ=2πz/(α˜Lm),
α˜=2πc/(Lcγγ)
σ=κ/γγ,γ=γ/γ,
FSR=2πc/Lc=(γα˜)γ;
L(ω)=1πuu2+ω2,
F(0, τ)=F(2π/α˜, τ),
F=Iα exp[i(αζ-Ωτ)],
P(ω)=pα(ω)exp[i(αζ-Ωτ)],
D(ω)=dα(ω),
pα(ω)=1-i(ω-γΩ)1+Iα+(ω-γΩ)2 Iα,
dα(ω)=1+(ω-γΩ)21+Iα+(ω-γΩ)2,
A=Iα+1(Iα+1+u)×1+γαIα+1(1+γσ)+u2,
Ω=αIα+1+uIα+1(1+γσ)+u,
Ath(α)=(1+u)[1+(γα)2/(1+γσ+u)2].
A0Ath(0)=1+u.
R=1+I0,
r=A/A0=R(R+u)/(1+u),
rth(α)=1+(γα)2/(γσ+1+u)2.
λ+σ+iα=σ(1+γλ)J2+1-γR2-1γ+λJ0,
1+iα/λ=γσ[J2-ΓJ0/(1+γλ)],
Γ=(1+γλ)[1+γλ+γ(R2-1)/(γ+λ)],
Jn=A-+dωL(ω)ωn(R2+ω2)(Γ+ω2).
J0=R+u+ΓΓ(R+Γ)(u+Γ),
J2=Ru(R+Γ)(u+Γ).
0=γλ3+(1+γσ+γ2+iαγ)λ2+[γ(r+γσ)+iα(1+γ2)]λ+γ[2σ(r-1)+irα],
0=γλ2+γ(1+γσ+iαγ)λ+iα,
rc=5+3γ2+22(γ2+1)(γ2+2)
αc=3(rc-1)-γ22 1-σγ1+γ2rc-1+γ23(rc-1)-γ2.
λ4+(σ+2+iα)λ3+(R2-2σuR+σ+2iα)λ2
+[2σ(R2-uR-1)+iαR2]λ-2σuR(R2-1)=0
ω4-(3R2+2uR-4)ω2+2R(R3+uR2-R-2u)=0,
α=[(2+σ)ω2-2σ(R2-uR-1)]/(2ω).
Rc4+4uRc3-16Rc2+16=0,
ωc2=3Rc2/2+uRc-2.
rc=14.928-27.715u+O(u2),
αc=4.515-0.826σ-(2.338-1.777σ)u+O(u2).
f(α, γ)
=γ4α6+γ2(5+γ4)α4+(4+5γ4)α2+4γ2γ2α6+(1+3γ2+γ4)α4+(6-4γ2)α2-8.
r1+(α/u)2f(α, γ).
f(α, 1)=(α2+1)2/(α4+α2-2).
f(α, 0)=4α2/(α4+6α2-8).
λ=-iα+σu+γλ(Ru+1)γλ(u+γλ)-1.
λ=-iα+σ[-1+Ru/(u+γλ)].
λ0=σ[-1+Ru/(u+γλ)]λ=-iα=σRu2u2+γ2α2-1+iRuγαu2+γ2α2.
r1+(γα/u)2,
λ=-iα+σR+u+(1+γλ)(Ru+1)(R+1+γλ)(u+1+γλ)-1,
λ0=σR+u+(1+γλ)(Ru+1)(R+1+γλ)(u+1+γλ)-1λ=-iα.
Re λ0=-σγ2α2D-1[γ2α2+H(R, u)],
H(R, u)=R2(1-u)+R(1-u)(u+2)+(u+1)2.
γ2α2+H(R, u)<0.
R=γα/u-1+O(1).
λ=-iα+σ[-1+Ru/(R+γλ)]
λ0=σ[-1+Ru/(R+γλ)]λ=-iα;
Re λ0=-[γ2α2+R2(1-u)]/(γ2α2+R2),
r1+(γα)2/(1+u)2+2R(R+1)/(1+u)2rth(α)|γ1,
λ1=iσα/(1+u)×1-uR-1R+1-2(R-1)(R+u+1)α2,
Re λ2=-σPamp/[α(R+1)(u+1)]2,
Pamp=H(R, u)α4-3(R2-1)×[(R+u+1)2-Ru]α2+R(R2-1)(R+1)2(R+u)(u+2).
Pamp<0,
λ1=-iσα[u(R-1)/(R+1)-1]/(1+u),
Re λ2=-σPphase/[(R+1)(u+1)]2,
Pphase=H(R, u)α2+Ru(R2-1)(R+u+2),
Pphase<0,
(2u+1)2R5+(2u+1)2(2u+3)R4
+(4u4+16u3+27u2+10u-6)R3
+(10u3+9u2-26u-26)R2
+(u+1)2(5u2-8u-27)R-9(u+1)4=0,
αc2=3(Rc2-1)[(Rc+u+1)2-Rcu]/2H(Rc, u).
rc=9-36u+O(u2),
αc2=12-33.75u+O(u2),
rc=1+32/(17u2)+O(u-3),
αc2=8/3+80/(17u2)+O(u-3).
R=u+221+4(u+1)2(u-1)(u+2)2-1,
f1(u)=3R2-1u-1R2+(u+1)2+(u+2)R2R+u+2,
f2(u)=[u(u-1)(8+7u+u2)]-1/2.
s=(2R+u)2[f1(u)+γ2α4f2(u)]2/[α2(1+u)]2.
γ2αcoh4=f1(u)/f2(u).
Lcoh=πc(γ/γ)1/4γ-1.
A=G(W-γ)/(W+γ),
W=γ(G+A)/(G-A)=[G+R(R+u)]/[G-R(R+u)],
W0=γ(G+1+u)/(G-1-u),
WW0=G+R(R+u)G-R(R+u)G-1-uG+1+u.
ρ=2G/(G-A)=2G/[G-R(R+u)].
WW0=G+R(R+u)G-R(R+u)G-1-uG+1+u

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