Abstract

We first use the nonlocalized, fluctuating source model for the stimulated Brillouin scattering to get the exact spectrum of the Stokes wave in optical fibers with attenuation loss. A new relation for the evaluation of the critical pump power (or Brillouin threshold) depending on the fiber length is then introduced, which should be more precise than the well-known Smith formula. Furthermore, we give for the first time, to the best of our knowledge, an approximate solution for standard steady-state Brillouin equations, which consists of two simple relations.

© 2003 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, London, 1995).
  2. R. W. Boyd, K. Rzazewski, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
    [CrossRef] [PubMed]
  3. A. L. Gaeta, R. W. Boyd, “Stochastics dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3210 (1991).
    [CrossRef] [PubMed]
  4. S. Le Floch, P. Cambon, “Study of Brillouin gain spectrum in standard single-mode optical fiber at low temperatures (1.4 K to 370 K) and high hydrostatic pressures (1 to 250 bars),” Opt. Commun. (to be published).
  5. M. Niklès, L. Thévenaz, Ph. Robert, “Brillouin gain spectrum characterization in single-mode fibers” J. Lightwave Technol. 15, 1842–1851 (1997).
    [CrossRef]
  6. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
    [CrossRef] [PubMed]
  7. L. Chen, X. Bao, “Analytical and numerical solutions for steady-state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
    [CrossRef]
  8. C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” Appl. Phys. 37, 2945–2955 (1966).
    [CrossRef]
  9. A. Küng, “Laser emission in stimulated Brillouin scattering in optical fibers,” Ph.D. dissertation No. 1740 (Ecole Polytechnique Féderale de Lausanne, Lausanne, Switzerland, 1997).

1998 (1)

L. Chen, X. Bao, “Analytical and numerical solutions for steady-state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

1997 (1)

M. Niklès, L. Thévenaz, Ph. Robert, “Brillouin gain spectrum characterization in single-mode fibers” J. Lightwave Technol. 15, 1842–1851 (1997).
[CrossRef]

1991 (1)

A. L. Gaeta, R. W. Boyd, “Stochastics dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3210 (1991).
[CrossRef] [PubMed]

1990 (1)

R. W. Boyd, K. Rzazewski, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

1972 (1)

1966 (1)

C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” Appl. Phys. 37, 2945–2955 (1966).
[CrossRef]

Agrawal, G. P.

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

Bao, X.

L. Chen, X. Bao, “Analytical and numerical solutions for steady-state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Boyd, R. W.

A. L. Gaeta, R. W. Boyd, “Stochastics dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3210 (1991).
[CrossRef] [PubMed]

R. W. Boyd, K. Rzazewski, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

Cambon, P.

S. Le Floch, P. Cambon, “Study of Brillouin gain spectrum in standard single-mode optical fiber at low temperatures (1.4 K to 370 K) and high hydrostatic pressures (1 to 250 bars),” Opt. Commun. (to be published).

Chen, L.

L. Chen, X. Bao, “Analytical and numerical solutions for steady-state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Gaeta, A. L.

A. L. Gaeta, R. W. Boyd, “Stochastics dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3210 (1991).
[CrossRef] [PubMed]

Küng, A.

A. Küng, “Laser emission in stimulated Brillouin scattering in optical fibers,” Ph.D. dissertation No. 1740 (Ecole Polytechnique Féderale de Lausanne, Lausanne, Switzerland, 1997).

Le Floch, S.

S. Le Floch, P. Cambon, “Study of Brillouin gain spectrum in standard single-mode optical fiber at low temperatures (1.4 K to 370 K) and high hydrostatic pressures (1 to 250 bars),” Opt. Commun. (to be published).

Niklès, M.

M. Niklès, L. Thévenaz, Ph. Robert, “Brillouin gain spectrum characterization in single-mode fibers” J. Lightwave Technol. 15, 1842–1851 (1997).
[CrossRef]

Robert, Ph.

M. Niklès, L. Thévenaz, Ph. Robert, “Brillouin gain spectrum characterization in single-mode fibers” J. Lightwave Technol. 15, 1842–1851 (1997).
[CrossRef]

Rzazewski, K.

R. W. Boyd, K. Rzazewski, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

Smith, R. G.

Tang, C. L.

C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” Appl. Phys. 37, 2945–2955 (1966).
[CrossRef]

Thévenaz, L.

M. Niklès, L. Thévenaz, Ph. Robert, “Brillouin gain spectrum characterization in single-mode fibers” J. Lightwave Technol. 15, 1842–1851 (1997).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” Appl. Phys. 37, 2945–2955 (1966).
[CrossRef]

J. Lightwave Technol. (1)

M. Niklès, L. Thévenaz, Ph. Robert, “Brillouin gain spectrum characterization in single-mode fibers” J. Lightwave Technol. 15, 1842–1851 (1997).
[CrossRef]

Opt. Commun. (1)

L. Chen, X. Bao, “Analytical and numerical solutions for steady-state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Phys. Rev. A (2)

R. W. Boyd, K. Rzazewski, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef] [PubMed]

A. L. Gaeta, R. W. Boyd, “Stochastics dynamics of stimulated Brillouin scattering in an optical fiber,” Phys. Rev. A 44, 3205–3210 (1991).
[CrossRef] [PubMed]

Other (3)

S. Le Floch, P. Cambon, “Study of Brillouin gain spectrum in standard single-mode optical fiber at low temperatures (1.4 K to 370 K) and high hydrostatic pressures (1 to 250 bars),” Opt. Commun. (to be published).

A. Küng, “Laser emission in stimulated Brillouin scattering in optical fibers,” Ph.D. dissertation No. 1740 (Ecole Polytechnique Féderale de Lausanne, Lausanne, Switzerland, 1997).

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

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Figures (5)

Fig. 1
Fig. 1

Critical gain versus fiber length: dashed lines, Smith’s model; solid curve, relation (25).

Fig. 2
Fig. 2

Pump power threshold versus fiber length: dashed curve, Smith’s model; solid curve, relation (25).

Fig. 3
Fig. 3

Backscattered Stokes power versus incident pump power: dashed curve, nondepleted pump model; dots, numerical solution of the Brillouin coupled equations; solid curve, Eq. (32) and relation (37).

Fig. 4
Fig. 4

Transmitted pump power versus incident pump power: dashed line, nondepleted pump model; dots, numerical solution of the Brillouin coupled equations; solid curve, Eq. (32) and relation (37).

Fig. 5
Fig. 5

Reflectivity as a function of the gain factor G: dashed curve, nondepleted pump model; dots, relation (37) with ln[IP(L)/IS(L)]=-10.5; solid curve, numerical solution of the steady-state coupled equations.

Equations (46)

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EL(z, t)z+ncEL(z, t)t+α2 EL(z, t)
=igEρ(z, t)ES(z, t), 
ES(z, t)z-ncES(z, t)t-α2 ES(z, t)
=-igEρ*(z, t)EL(z, t),
ρ(z, t)t+Γ2 ρ(z, t)=igAES*(z, t)EL(z, t)+f(z, t),
f(z, t)=0,
f(z, t)f*(z, t)=Qδ(z-z)δ(t-t).
ES(z, τ)z
=-igEρ*(z, τ)EL(0)exp(-αz/2)+α2 ES(z, τ),
ρ(z, τ)τ+Γ2 ρ(z, τ)
=igAES*(z, τ)EL(0)exp(-αz/2)+f(z, τ).
PLcrit(0)21 Aeffg0Leff.
ILz=-g0ISIL-αIL,
ISz=-g0ISIL+αIS,
E˜S(z, Δω)z=-igEρ˜*(z, Δω)EL(0)×exp(-αz/2)+α2 E˜S(z, Δω),
ρ˜(z, Δω)=1Γ/2-iΔω [igAE˜S*(z, Δω)EL(0)×exp(-αz/2)+f˜(z, Δω)].
E˜Sz=-gAgE|EL(0)|2exp(-αz)Γ/2+iΔω E˜S-igEEL(0)exp(-αz/2)Γ/2+iΔω f˜*+α2 E˜S.
g0=4π2γ2ncλ02ρ0vΓ.
E˜Sz=-g0IL(0)exp(-αz) Γ/4Γ/2+iΔω E˜S-igEEL(0)exp(-αz/2)Γ/2+iΔω f˜*+α2 E˜S.
E˜S(z, Δω)=igEEL(0)exp(αz/2)Γ/2+iΔω×zLdzexp(-αz)f˜*(z, Δω)×expg0IL(0) Γ/4Γ/2+iΔω×exp(-αz)-exp(-αz)α.
f˜(z, Δω)f˜*(z, Δω)=Qδ(z-z).
I˜S(0, Δω)=kTνSAeffνBexp[GLor(Δω)]×exp(-αL)+1GαLor(Δω)-1+1GαLor(Δω),
G=g0IL(0)Leff
I˜S(0, Δω)=kTνSAeffνB {exp[g0IL(0)LLor(Δω)]-1}.
I˜S(0, Δω)=kTνSAeffνBexp(-αL)Lor(Δω)G,
I˜S(0, Δω)=kTνsAeffνBexp(-αL)×{exp[g0IL(0)LeffLor(Δω)]-1}.
IS(0)=kTΓ4Aeffν0νB G exp(G/2)[I0(G/2)-I1(G/2)],
IS(0)kTΓ4Aeffν0νBexp(-αL)G×exp(G/2)[I0(G/2)-I1(G/2)],
Rsp=g0kTΓ4Aeffν0νB Leffexp(-αL).
R=Rspexp(G/2)[I0(G/2)-I1(G/2)].
R=Rspπ1/2exp(G)G3/2.
Gcritln4AeffνBG¯3/2π1/2g0kTΓν0Leff.
Gcrit(L+ΔL)=Gcrit(L)-ln1-exp(-αL)1-exp[-α(L+ΔL)].
Gcrit0ln4αAeffνBG¯3/2π1/2g0kTΓν0.
Σ=IL+IS2,Δ=IL-IS2,
X=ILIS,Y=ILIS,
Δz=-αΣ,Σz=-αΔ-2g0X,
 ln Xz=-g0Σ, ln Yz=-2α+g0Δ.
ln X-g0α Δz=0;
IL(z)IS(z)
=IL(0)IS(0)expg0α [IP(z)-IP(0)+IS(0)-IS(z)].
 ln(IS)z=-g0IL+α,
Δz=-αΔ,
ΔΔ0exp(-αz).
YY0exp(g0Δ0zeff-2αz)
IL(z)IS(z)IL(0)IS(0)exp(-2αz)exp{g0[IL(0)-IS(0)]zeff}.

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