Abstract

We derive analytic expressions for the Brillouin thresholds of square pulses in optical fibers. The equations are valid for pulse durations in the transient Brillouin scattering regime (less than 100 nsec), as well for longer pulses, and have been confirmed experimentally. Our analysis also gives a firm theoretical prediction that the Brillouin gain width increases dramatically for intense pulses, from tens of MHz to one GHz or more.

© 2014 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 5thedition (Academic,Oxford, 2013).
  2. R. W. Boyd, Nonlinear Optics, 2nd edition (Academic, San Diego, 2003).
  3. M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. M. Niklès, L. Thévenaz, P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15, 1842–1851 (1997).
    [CrossRef]
  15. V. I. Kovalev, R. G. Harrison, “Means for easy and accurate measurement of the stimulated Brillouin scattering gain coefficient in optical fiber,” Opt. Lett. 33, 2434–2436 (2008).
    [CrossRef] [PubMed]
  16. M. D. Mermelstein, “SBS threshold measurements and acoustic beam propagation modeling in guiding and anti-guiding single mode optical fibers,” Opt. Express 17, 16225–16237 (2009).
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  17. V. Lanticq, S. Jiang, R. Gabet, Y. Jaouën, F. Taillade, G. Moreau, G. P. Agrawal, “Self-referenced and single-ended method to measure Brillouin gain in monomode optical fibers,” Opt. Lett. 34, 1018–1020 (2009).
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  18. V. I. Kovalev, R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27, 2022–2024 (2002).
    [CrossRef]
  19. S. L. Sobolev, Partial Differential Equations of Mathematical Physics (Dover, New York, 1989).
  20. G. Arfken, Mathematical Methods for Physicists, 3rd edition (Academic, Orlando, 1985).

2014

2009

2008

2006

2004

2002

2000

1999

H. Li, K. Ogusu, “Dynamic behavior of stimulated Brillouin scattering in a single-mode optical fiber,” Jpn. J. Appl. Phys. 38, 6309–6315 (1999).
[CrossRef]

1997

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

1972

1970

D. Pohl, W. Kaiser, “Time-resolved investigations of stimulated Brillouin scattering in transparent and absorbing media: Determination of phonon lifetimes,” Phys. Rev. B. 1, 31–43 (1970).
[CrossRef]

1965

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–43 (1965).
[CrossRef]

Adams, F. J.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Agrawal, G. P.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd edition (Academic, Orlando, 1985).

Bao, X.

Bigelow, M. S.

M. S. Bigelow, S. G. Lukishova, R. W. Boyd, M. D. Skeldon, “Transient stimulated Brillouin scattering dynamics in polarization maintaining optical fiber,” CLEO2001, paper CTuZ3.

Boyd, R. W.

M. S. Bigelow, S. G. Lukishova, R. W. Boyd, M. D. Skeldon, “Transient stimulated Brillouin scattering dynamics in polarization maintaining optical fiber,” CLEO2001, paper CTuZ3.

R. W. Boyd, Nonlinear Optics, 2nd edition (Academic, San Diego, 2003).

Byer, M. W.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Chen, L.

Gabet, R.

Guzsella, S.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Harrison, R. G.

Jackson, D. A.

Jaouën, Y.

Jiang, S.

Kaiser, W.

D. Pohl, W. Kaiser, “Time-resolved investigations of stimulated Brillouin scattering in transparent and absorbing media: Determination of phonon lifetimes,” Phys. Rev. B. 1, 31–43 (1970).
[CrossRef]

Kalosha, V. P.

Keaton, G. L.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Kovalev, V. I.

Kroll, N. M.

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–43 (1965).
[CrossRef]

Lanticq, V.

Lecoeuche, V.

Leonardo, M. J.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Li, H.

H. Li, K. Ogusu, “Dynamic behavior of stimulated Brillouin scattering in a single-mode optical fiber,” Jpn. J. Appl. Phys. 38, 6309–6315 (1999).
[CrossRef]

Lukishova, S. G.

M. S. Bigelow, S. G. Lukishova, R. W. Boyd, M. D. Skeldon, “Transient stimulated Brillouin scattering dynamics in polarization maintaining optical fiber,” CLEO2001, paper CTuZ3.

Mermelstein, M. D.

Monro, K.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Moreau, G.

Nightingale, J. L.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Niklès, M.

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

Ogusu, K.

H. Li, K. Ogusu, “Dynamic behavior of stimulated Brillouin scattering in a single-mode optical fiber,” Jpn. J. Appl. Phys. 38, 6309–6315 (1999).
[CrossRef]

Pannell, C. N.

Pohl, D.

D. Pohl, W. Kaiser, “Time-resolved investigations of stimulated Brillouin scattering in transparent and absorbing media: Determination of phonon lifetimes,” Phys. Rev. B. 1, 31–43 (1970).
[CrossRef]

Ponomarev, E. A.

Richard, D. J.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Robert, P. A.

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

Skeldon, M. D.

M. S. Bigelow, S. G. Lukishova, R. W. Boyd, M. D. Skeldon, “Transient stimulated Brillouin scattering dynamics in polarization maintaining optical fiber,” CLEO2001, paper CTuZ3.

Smith, R. G.

Smoliar, L.

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

Sobolev, S. L.

S. L. Sobolev, Partial Differential Equations of Mathematical Physics (Dover, New York, 1989).

Taillade, F.

Thévenaz, L.

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

Webb, D. J.

Williams, D.

Yu, Q.

Appl. Opt.

J. Appl. Phys.

N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36, 34–43 (1965).
[CrossRef]

J. Lightwave Technol.

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

Jpn. J. Appl. Phys.

H. Li, K. Ogusu, “Dynamic behavior of stimulated Brillouin scattering in a single-mode optical fiber,” Jpn. J. Appl. Phys. 38, 6309–6315 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Photon. Res.

Phys. Rev. B.

D. Pohl, W. Kaiser, “Time-resolved investigations of stimulated Brillouin scattering in transparent and absorbing media: Determination of phonon lifetimes,” Phys. Rev. B. 1, 31–43 (1970).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics, 5thedition (Academic,Oxford, 2013).

R. W. Boyd, Nonlinear Optics, 2nd edition (Academic, San Diego, 2003).

M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, L. Smoliar, “Versatile, nanosecond laser source for precision material processing,” presented at the 28th International Congress on Applications of Lasers and Electro-Optics (ICALEO), Orlando, Florida, 2–5 Nov.2009, paper #M103.

M. S. Bigelow, S. G. Lukishova, R. W. Boyd, M. D. Skeldon, “Transient stimulated Brillouin scattering dynamics in polarization maintaining optical fiber,” CLEO2001, paper CTuZ3.

S. L. Sobolev, Partial Differential Equations of Mathematical Physics (Dover, New York, 1989).

G. Arfken, Mathematical Methods for Physicists, 3rd edition (Academic, Orlando, 1985).

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Figures (3)

Fig. 1
Fig. 1

The Brillouin threshold for 1060 nm wavelength pulses. The solid curve is the threshold for pulses that are shorter than the roundtrip time in the fiber, Eq. (14). The dashed curves, calculated from Eq. (38), give the thresholds for longer pulses in 1, 5, and 25 m length fibers. The parameters used for these curves are: gB = 31 μm2/(W-m), TB = 4 nsec, υ = 0.2 m/nsec, and Θ = 22. The points represent data taken with 5 m (circles) and 50 m (squares) lengths of fiber.

Fig. 2
Fig. 2

The experimental setup. The output of a 1060 nm seed diode laser was modulated by a Mach-Zehnder interferometer to produce a train of pulses that were amplified by a series of fiber amplifiers. The resulting high power pulses were coupled into a passive fiber to induce Brillouin scattering, then analyzed with a fast detector and a thermal power meter.

Fig. 3
Fig. 3

A typical pump pulse at Brillouin threshold. The back of the pulse is noisy due to Brillouin scattering, and its intensity is reduced by approximately 20%.

Equations (72)

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g B I P L = Θ ,
g B I P ( υ τ 2 ) = Θ .
A P z + 1 υ A P t 0 .
A B z + 1 υ A B t = i κ 1 A P Q *
Q t + Γ B 2 Q = i κ 2 A P A B *
κ 1 κ 2 g B Γ B 4 A eff ,
A B z + 1 υ A B t = 0 .
2 υ A B t = i κ 1 A P Q * .
2 A B t 2 = i κ 1 υ 2 A P Q * t ,
2 A B t 2 = i κ 1 υ 2 A P ( i κ 2 A P * A B Γ B 2 Q * ) ,
2 A B t 2 = κ 1 κ 2 υ 2 | A P | 2 A B Γ B 2 A B t .
2 A B t 2 + 1 2 T B A B t g B υ I P 8 T B A B = 0 .
A B ( z , t ) = A B ( z , 0 ) e α t ,
α = 1 4 T B ( 1 + 1 + 2 g B I P υ T B ) .
A B ( z , t ) = A B 0 e α ( t z / υ )
2 α τ = ln [ | A P | 2 | A B 0 | 2 ] Θ .
I P = 2 Θ g B υ τ ( T B Θ τ + 1 )
g B I P ( υ τ 2 ) = Θ ,
Q t + [ Γ B 2 + i ( Ω A Ω ) ] Q = i κ 2 A P A B * .
2 A B t 2 + 1 2 T B ( 1 + i δ ) A B t g B υ I P 8 T B A B = 0 ,
A B ( z , t ) = A B 0 e β ( t z / v ) ,
β = 1 4 T B { ( 1 + i δ ) + ( 1 + i δ ) 2 + γ } ,
γ = 2 g B I P υ T B
| A B | 2 = | A B 0 | 2 e ( β + β * ) τ
β + β * γ 4 T B 1 1 + δ 2 .
Δ ω B = 1 2 T B ( γ 1 ) .
Δ ω B g B I P υ 2 T B ( γ 1 ) .
Δ ν B = Δ ω B 2 π 1.4 GHz .
Δ ω B Δ t g 1 .
Δ ω B T B 1 .
β + β * 2 α δ 2 4 T B 1 + γ
| A B | 2 = | A B 0 | 2 e 2 α τ exp { T B τ ( ω P ω B Ω A ) 2 1 + γ } .
Δ ω B = ( 1 + γ ) 1 / 4 T B τ .
P B = d ω B 2 π ω B n ¯ e [ β ( ω B ) + β * ( ω B ) ] τ .
P B k T 2 π ( ω ¯ B Ω A ) e 2 α τ π Δ ω B
P B 0 = | A B 0 | 2 = 1 4 π k T ( υ υ A ) ( 1 + γ ) 1 / 4 T B τ
Θ = ln [ P P P B 0 ] = ln { 4 π ( υ A υ ) P P k T T B τ ( 1 + γ ) 1 / 4 }
z ^ = z ,
t ^ = t + z / υ .
t = t ^ t t ^ + z ^ t z ^ = t ^ ,
z = t ^ z t ^ + z ^ z z ^ = 1 υ t ^ + z ^ .
A B z ^ = i κ 1 A P Q *
Q t ^ + 1 2 T B Q = i κ 2 A P A B * .
q = exp [ t ^ 2 T B ] Q
𝒜 = exp [ t ^ 2 T B ] A B
𝒜 z ^ = i κ 1 A P q *
q t ^ = i κ 2 A P 𝒜 * .
2 𝒜 z ^ t ^ = κ 1 κ 2 | A P | 2 𝒜 ,
2 𝒜 z ^ t ^ = g B I P 4 T B 𝒜
Λ = L 2 υ T B ;
τ ¯ = τ 2 T B 2 Λ .
I 0 ( 2 γ Λ τ ¯ ) e τ ¯ + e τ ¯ 0 Λ d y I 0 ( 2 γ ( Λ y ) τ ¯ ) ( 1 + γ 1 ) e ( 1 + γ 1 ) y + 0 τ ¯ d s I 0 ( 2 γ Λ s ) e s = e Θ / 2 .
g B I P L = Θ ,
A B ( z ^ , t ^ ) = A B 0 e α ( t ^ 2 z ^ / υ ) ( t ^ 2 L / υ ) ,
𝒜 ( z ^ , t ^ = 2 L / υ ) = A B 0 exp [ L υ T B ] exp [ 2 α υ ( L z ^ ) ]
𝒜 ( z ^ = L , t ^ ) = A B 0 exp [ t ^ 2 T B ]
t ¯ = 1 2 T B ( t ^ 2 L υ ) .
z ¯ = 1 2 υ T B ( L z ^ )
B = 𝒜 A B 0 e L / ( υ T B ) ,
2 B z ¯ t ¯ = + γ 2 B .
B ( z ¯ , 0 ) = e 4 α T B z ¯
B ( 0 , t ¯ ) = e t ¯ .
2 w z ¯ t ¯ = γ 2 w .
w ( z ¯ , 0 ) = w ( 0 , t ¯ ) = 1 .
B ( z ¯ , t ¯ ) = B ( 0 , 0 ) w ( z ¯ , t ¯ ) + 0 t ¯ d t ¯ w ( z ¯ , t ¯ , t ¯ ) B t ¯ ( 0 , t ¯ ) + 0 z ¯ d z ¯ w ( z ¯ z ¯ , t ¯ ) B z ¯ ( z ¯ , 0 ) .
w ( z ¯ , t ¯ ) = n = 0 c n ( t ¯ ) z ¯ n n ! .
d c n + 1 d t ¯ = γ 2 c n .
c n ( t ¯ ) = ( γ 2 ) n t ¯ n n ! ;
w ( z ¯ , t ¯ ) = n = 0 ( γ t ¯ z ¯ 2 ) n 1 ( n ! ) 2 .
w ( z ¯ , t ¯ ) = I 0 ( 2 γ t ¯ z ¯ ) .
B ( z ¯ , t ¯ ) = I 0 ( 2 γ t ¯ z ¯ ) + 0 t ¯ d t ¯ I 0 ( 2 γ ( t ¯ t ¯ ) z ¯ ) e t ¯ + 0 z ¯ d z ¯ I 0 ( 2 γ t ¯ ( z ¯ z ¯ ) ) 4 α T B e 4 α T B z ¯
e τ ¯ B ( Λ , τ ¯ ) = e Θ / 2

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