## Stimulated Brillouin scattering of pulses in optical fibers |

Optics Express, Vol. 22, Issue 11, pp. 13351-13365 (2014)

http://dx.doi.org/10.1364/OE.22.013351

Acrobat PDF (952 KB)

### 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

## 1. Introduction

3. M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, and 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.

*T*, that causes substantial deviations from Eq. (2) when

_{B}*τ*≤ Θ

*T*.

_{B}*L/υ*) is greater than the pulse duration

*τ*. Given these three assumptions, we have derived a simple equation, Eq. (14) below, that generalizes Eq. (2) and is valid for all pulse lengths.

*g*and the phonon lifetime

_{B}*T*were needed. These values depend on the composition of the fiber, and were not known in advance; we therefore fit our equations for the Brillouin threshold to the data, treating

_{B}*g*and

_{B}*T*as free parameters. The best fit values for

_{B}*g*and

_{B}*T*are similar to the values found for other fibers, and the results are shown in Fig. 1.

_{B}## 2. Brillouin threshold in long fibers

*τ*. As the pulse travels a distance

*z*for a time

*t*in the fiber, the envelope of the pulse is described by an amplitude function

*A*(

_{P}*z*,

*t*). In the undepleted pump approximation, the pulse propagates with group velocity

*υ*without changing its shape: The pump pulse generates a backward-propagating optical Brillouin (or Stokes) wave

*A*(

_{B}*z*,

*t*) and a phonon field

*Q*(

*z*,

*t*) according to the following equations [1]: where Γ

*is the inverse of the phonon lifetime, Γ*

_{B}*= 1/*

_{B}*T*. The constants

_{B}*κ*

_{1}and

*κ*

_{2}are proportional to the electrostrictive constant of the fiber, and their product is proportional to the Brillouin gain

*g*: where

_{B}*A*is the effective mode area of the fiber.

_{eff}*A*and

_{P}*A*are normalized so that |

_{B}*A*|

_{P}^{2}and |

*A*|

_{B}^{2}are the instantaneous powers of the pump and Brillouin pulses. In Eq. (5), a term describing the propagation of phonons (proportional to

*∂Q/∂z*) has been omitted, because the phonons decay before they can travel a significant distance [2].

*τ*that has elapsed since the leading edge of the pulse passed that point in the fiber. This is true no matter how far along the fiber he has traveled; therefore the Brillouin amplitude at the back of the pulse is independent of time (as long as the front of the pulse has not yet reached the exit end of the fiber). Another observer placed in the middle of the pulse would get a similar result. This invariance of the Brillouin amplitude for any co-moving observer can be expressed as an equation similar to Eq. (3):

*∂A*is eliminated: The derivative of this equation with respect to time gives where

_{B}/∂z*∂A*has been neglected, since as long as the coordinates

_{P}/∂t*z*and

*t*do not refer to the leading or trailing edge of the pump pulse,

*A*(

_{P}*z*,

*t*) is constant. We now use Eq. (5) to eliminate

*∂Q*from the above equation, and then Eq. (8) to eliminate

^{*}/∂t*Q*: We now have an equation for the optical Brillouin wave only, without reference to the phonon field

^{*}*Q*. The equation can be cast into a more convenient form using Eq. (6) to eliminate

*κ*

_{1}

*κ*

_{2}, defining the pump intensity

*I*= |

_{P}*A*|

_{P}^{2}/

*A*, and using Γ

_{eff}*= 1/*

_{B}*T*:

_{B}*z*is an amplitude that grows exponentially in time: with

*A*(

_{B}*z*,

*t*) as a function of both

*z*and

*t*, note that Eq. (7) requires that

*A*(

_{B}*z*,

*t*) be a function of

*t*−

*z/υ*. Therefore the complete solution is: where

*A*

_{B0}is a constant. This equation is valid whenever

*z*and

*t*are within the pump pulse, and the pulse has not yet reached the exit end of the fiber. Note that the beginning of the pump pulse occurs at

*t*−

*z/υ*= 0, and the back end of the pump pulse is found at

*t*−

*z/υ*=

*τ*.

*A*

_{B0}|

^{2}at the beginning of the pulse [9

9. 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]

*A*

_{B0}|

^{2}is calculated below in section 4.) The Brillouin threshold occurs when the Brillouin power at the back of the pulse is equal to the pump power [1]; that is, when The value of the threshold parameter Θ is about 22 for high peak power pulses, as shown below.

*L*at time

*t*=

*L/υ*, this would seem to require that

*τ*≤

*L/υ*. However, Eq. (14) is still valid for greater values of

*τ*, because of the time it takes for the information to propagate that that the front of the pulse has reached the end of the fiber. As long as the back of the pulse enters the fiber before it receives the information that the front of the pulse has left, Eq. (14) can still be used. That is, Eq. (14) is valid for pulse durations less than the round trip time in the fiber, or

*τ*≤ 2

*L/υ*.

*υ*appearing in Eq. (4) should not be just the group velocity of the material, but should include the effects of the Brillouin gain as well. However, it turns out that the correct group velocity in Eq. (4) is indeed the group velocity of the material only; all gain-dependent effects are automatically included in the solution to the equations.

## 3. Brillouin gain width

*ω*is equal to the difference between the pump and acoustic frequencies:

_{B}*ω*=

_{B}*ω*− Ω

_{P}*. To investigate the Brillouin gain bandwidth, we allow*

_{A}*ω*to vary, and look at the Brillouin gain at different frequencies. Defining Ω =

_{B}*ω*−

_{P}*ω*, Eq. (5) is replaced by [1]: Combining this equation with Eqs. (4) and (7), and following the same procedure as before, we find, in place of Eq. (9), where the frequency detuning parameter

_{B}*δ*is defined as:

*δ*= 2

*T*(Ω − Ω

_{B}*).*

_{A}*γ*is a dimensionless gain constant,

*β*+

*β*

^{*}. For low pump intensities,

*γ*≪ 1, and

*β*can be expanded in a Taylor series in

*γ*: This is the familiar Lorentzian profile, with half-width Δ

*δ*= 1, or For

*T*= 4 nsec, for example, this gives a half-width of Δ

_{B}*ν*= Δ

_{B}*ω*/(2

_{B}*π*) ≈ 20 MHz.

*γ*≫ 1), Eq. (18) implies that the gain

*β*+

*β*

^{*}has a half-width of

*I*= 100

_{P}*W/μm*

^{2}. Substituting this into Eq. (22), with

*υ*= 0.2 m/nsec,

*g*= 31

_{B}*μ*m

^{2}/(W-m), and

*T*= 4 nsec (see Section 7 for a discussion of these values), we find a half-width of:

_{B}10. V. I. Kovalev and R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. **31**, 161–163 (2006). [CrossRef] [PubMed]

11. Q. Yu, X. Bao, and L. Chen, “Strain dependence of Brillouin frequency, intensity, and bandwidth in polarization-maintaining fibers,” Opt. Lett. **29**, 1605–1607 (2004). [CrossRef] [PubMed]

*β*+

*β*

^{*}on resonance (when

*δ*= 0). We define the inverse of this quantity as the gain time Δ

*t*, the time it takes for the Brillouin radiation to increase

_{g}*e*-fold; Eq. (22) can then be rewritten: On the other hand, for low intensities, Eq. (21) can be written: We now recognize that the Brillouin gain width is a consequence of the uncertainty principle: the width is determined by a time that characterizes the phonon dynamics. For low intensities, this time is simply the phonon decay lifetime

*T*. For high intensities, however, the characteristic time is Δ

_{B}*t*, because the phonon population grows exponentially as exp(

_{g}*t*/Δ

*t*), and this rapid growth introduces Fourier components with a spectral width on the order of 1/Δ

_{g}*t*.

_{g}*ω*, as opposed to the gain width Δ

_{B}′*ω*. The gain width Δ

_{B}*ω*is larger than the width of the Brillouin pulse due to the spectral narrowing that is typical of amplified light. To calculate the final width Δ

_{B}*ω*of the Brillouin pulse, we approximate the power gain for high power pulses (

_{B}′*γ*≫ 1) as: where

*α*is the amplitude gain on resonance, as defined by Eq. (11). Substituting Eq. (23) into Eq. (20) and using the definition of

*δ*, we find that: Therefore the half-width of the Brillouin radiation (measured at the 1/

*e*point) is:

## 4. The threshold parameter Θ

9. 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]

*P*at the back of the pump pulse can be modeled as if it is built up from thermal noise at all frequencies at the front of the pump, amplified by the frequency-dependent gain discussed above [9

_{B}9. 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]

*n̄*is the average occupation number of phonons of frequency Ω

*at temperature*

_{A}*T: n̄*= {exp[

*ħ*Ω

*/(*

_{A}*kT*)] − 1}

^{−1}≈

*kT*/(

*ħ*Ω

*). Approximating*

_{A}*β*as in Eq. (23) and performing the integral, we find: where

*ω̄*is the Brillouin frequency on resonance:

_{B}*ω̄*=

_{B}*ω*− Ω

_{P}*. This equation can be made more convenient using the relation*

_{A}*ω̄*/Ω

_{B}*=*

_{A}*υ*/(2

*υ*), where

_{A}*υ*is the acoustic velocity, and

_{A}*υ*is (as usual) the velocity of light in the fiber [1].

*P*

_{B0}at the beginning of the pump pulse is (using Eq. (25)):

*P*; that is,

_{P}*υ*= 6 km/sec,

_{A}*υ*= 0.2 m/nsec,

*kT*= 4 × 10

^{−21}J,

*P*= 30 W,

_{P}*T*= 4 nsec,

_{B}*τ*= 20 nsec, and

*γ*= 50. The result is: Θ ≈ 22. Considering how different the present powers and Brillouin gain bandwidths are from those relevant to telecommunications, it is remarkable how close this result is to Smith’s result of 21.

## 5. Brillouin threshold in short fibers

*ẑ*and

*t̂*that are co-moving with the Brillouin radiation: The derivatives in terms of these new variables are: Using these variables, Eqs. (4) and (5) become: and

*t̂*of Eq. (33), and use Eq. (34) to eliminate

*∂q*. We end up with the equation: which can be rewritten as:

^{*}/∂t̂*γ*introduced in Eq. (19),

*γ*= 2

*g*; a dimensionless fiber length Λ, and a dimensionless time proportional to the amount by which the pulse duration exceeds the roundtrip time in the fiber,

_{B}I_{P}υT_{B}*τ̄*→ ∞), the first and second terms on the left hand side of Eq. (38) go to zero, and the third term becomes exp(

*γ*Λ/2). Therefore in this limit, Eq. (38) can be rewritten: which is the expected c.w. result, Eq. (1).

*τ̄*> 0, or

*τ*> 2

*L/υ*). The equation interpolates between the short pulse threshold given by Eq. (14) when

*τ*= 2

*L/υ*and the c.w. threshold when

*τ*→ ∞. Examples are shown in Fig. 1. The solid curve is the threshold for pulses shorter than the fiber round trip time, Eq. (14); the dashed curves are the thresholds for longer pulses (Eq. (38)) in fibers of length 1, 5, and 25 m. The circles and squares are data points from the experiments described below.

## 6. Experiment

*τ*> 20 nsec) we used a Hewlett Packard 8082A pulse generator; for short pulses, we used an Avtech AVMP-2-C-EPIA because of its shorter rise and fall times. A bias voltage was also provided to the Mach-Zehnder modulator to optimize the on/off extinction ratio at each repetition rate.

## 7. Results

*g*and the phonon lifetime

_{B}*T*were needed. In bulk silica,

_{B}*g*= 50

_{B}*μ*m

^{2}/(W-m) and

*T*= 5 nsec [1,2]. However, these values are different in optical fibers: The gain

_{B}*g*is decreased due to the imperfect overlap of the acoustic and optical modes [1], so that typical values of

_{B}*g*range from 10 to 30

_{B}*μ*m

^{2}/(W-m) [14

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

17. V. Lanticq, S. Jiang, R. Gabet, Y. Jaouën, F. Taillade, G. Moreau, and G. P. Agrawal, “Self-referenced and single-ended method to measure Brillouin gain in monomode optical fibers,” Opt. Lett. **34**, 1018–1020 (2009). [CrossRef] [PubMed]

*T*is also reduced in the fiber [14

_{B}14. M. Niklès, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. **15**, 1842–1851 (1997). [CrossRef]

18. V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. **27**, 2022–2024 (2002). [CrossRef]

*g*and

_{B}*T*for our fiber (Nufern PM-980 XP) were not known in advance, we fit them to the data. To obtain the gain

_{B}*g*, we used the two longest pulses (

_{B}*τ*= 300 and 400 nsec) measured in the 5 m fiber, since for these pulses, the steady state threshold equation (

*g*= Θ) holds to an excellent approximation. We used the average threshold intensity for these two pulse lengths, and calculated the Brillouin gain to be:

_{B}I_{P}L*g*= 31

_{B}*μ*m

^{2}/(W-m), close to the values reported above.

*τ*≤ 50 nsec) from the 5 m fiber. For this fit,

*g*was fixed at the above value of 31

_{B}*μ*m

^{2}/(W-m) and

*T*was determined by the least squares method. The result was:

_{B}*T*= 4.1 nsec, a little longer than the 2 to 3 nsec reported in other fibers, but less than the bulk silica value of 5 nsec.

_{B}*g*and

_{B}*T*were used to plot Eq. (14) and Eq. (38) with

_{B}*L*= 5 m on the same graph as the data, as shown in Fig. 1. The theory and experiment closely agree. For reference, the thresholds for 1 m and 25 m length fibers are also shown.

## Appendix: Riemann’s method

*L*, the Brillouin wave is described by Eq. (12) until the front of the pulse reaches the exit end of the fiber at

*z*=

*L*and

*t*=

*L/υ*, or, in terms of the hatted coordinates,

*ẑ*=

*L*and

*t̂*= 2

*L/υ*. As mentioned before, Eq. (12) continues to hold at later times

*t*for those parts of the pulse that have not yet received the news that the front of the pulse has reached the end of the fiber; that is, for all

*ẑ*provided that

*t̂*≤ 2

*L/υ*. We therefore have: In terms of the variable

*𝒜,*the first boundary condition is therefore that at

*t̂*= 2

*L/υ*,

*A*(

_{B}*ẑ*=

*L*,

*t̂*) =

*A*

_{B0}, or

*z̄*and

*t̄*that are scaled so they are dimensionless, and are chosen so that the boundary conditions occur more conveniently at

*z̄*= 0 and

*t̄*= 0. Since the first boundary condition, Eq. (39), occurs at

*t̂*= 2

*L/υ*, we set The second boundary condition (Eq. (40)) states that the Brillouin wave starts with thermal noise at

*ẑ*=

*L*; the Brillouin amplitude then grows in the negative

*ẑ*direction to reach its maximum at

*z*= 0. We invert the z-coordinate, and define so that the boundary condition occurs at

*z̄*= 0 and the Brillouin amplitude increases in the positive

*z̄*direction.

*γ*= 2

*g*. In terms of these new variables, Eq. (35) becomes The boundary conditions, Eqs. (39) and (40), can then be written: with

_{B}I_{P}vT_{B}*α*defined by Eq. (11), and

*w*(

*z̄*,

*t̄*) that has the following properties: first, it satisfies Eq. (44), Second,

*w*obeys the boundary conditions:

*B*(

*z̄*,

*t̄*) is: Although a derivation of this formula is beyond the scope of this paper, it can be checked that the function

*B*(

*z̄*,

*t̄*) defined by this equation indeed satisfies Eq. (44) and the necessary boundary conditions, due to the properties of the Riemann function given in Eqs. (47) and (48).

*τ*, the Brillouin amplitude

*A*should equal

_{B}*A*

_{B0}exp(Θ/2). To find the threshold condition in terms of

*B*(

*z̄*,

*t̄*),

*B*should be evaluated at the point

*z̄*=

*L*/(2

*υT*) ≡ Λ and

_{B}*t̄*= (

*τ*− 2

*L/υ*)/(2

*T*) ≡

_{B}*τ̄*. From the definitions of

*B*(Eq. (43)) and

*𝒜*(Eq. (32)), the threshold condition becomes: Substituting Eq. (52) into this equation yields the threshold condition, Eq. (38).

## Acknowledgments

## References and links

1. | G. P. Agrawal, |

2. | R. W. Boyd, |

3. | M. J. Leonardo, M. W. Byer, G. L. Keaton, D. J. Richard, F. J. Adams, K. Monro, J. L. Nightingale, S. Guzsella, and 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. |

4. | N. M. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. |

5. | D. Pohl and W. Kaiser, “Time-resolved investigations of stimulated Brillouin scattering in transparent and absorbing media: Determination of phonon lifetimes,” Phys. Rev. B. |

6. | H. Li and K. Ogusu, “Dynamic behavior of stimulated Brillouin scattering in a single-mode optical fiber,” Jpn. J. Appl. Phys. |

7. | V. P. Kalosha, E. A. Ponomarev, L. Chen, and X. Bao, “How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses,” Opt. Express |

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

9. | R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. |

10. | V. I. Kovalev and R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. |

11. | Q. Yu, X. Bao, and L. Chen, “Strain dependence of Brillouin frequency, intensity, and bandwidth in polarization-maintaining fibers,” Opt. Lett. |

12. | D. Williams, X. Bao, and L. Chen, “Characterization of high nonlinearity in Brillouin amplification in optical fibers with applications in fiber sensing and photonic logic,” Photon. Res. |

13. | V. Lecoeuche, D. J. Webb, C. N. Pannell, and D. A. Jackson, “Transient response in high-resolution Brillouin-based distributed sensing using probe pulses shorter than the acoustic relaxation time,” Opt. Lett. |

14. | M. Niklès, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. |

15. | V. I. Kovalev and R. G. Harrison, “Means for easy and accurate measurement of the stimulated Brillouin scattering gain coefficient in optical fiber,” Opt. Lett. |

16. | M. D. Mermelstein, “SBS threshold measurements and acoustic beam propagation modeling in guiding and anti-guiding single mode optical fibers,” Opt. Express |

17. | V. Lanticq, S. Jiang, R. Gabet, Y. Jaouën, F. Taillade, G. Moreau, and G. P. Agrawal, “Self-referenced and single-ended method to measure Brillouin gain in monomode optical fibers,” Opt. Lett. |

18. | V. I. Kovalev and R. G. Harrison, “Waveguide-induced inhomogeneous spectral broadening of stimulated Brillouin scattering in optical fiber,” Opt. Lett. |

19. | S. L. Sobolev, |

20. | G. Arfken, |

**OCIS Codes**

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(290.5900) Scattering : Scattering, stimulated Brillouin

**ToC Category:**

Fiber Optics

**History**

Original Manuscript: March 19, 2014

Revised Manuscript: May 3, 2014

Manuscript Accepted: May 5, 2014

Published: May 27, 2014

**Citation**

Gregory L. Keaton, Manuel J. Leonardo, Mark W. Byer, and Derek J. Richard, "Stimulated Brillouin scattering of pulses in optical fibers," Opt. Express **22**, 13351-13365 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13351

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### References

- 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.
- 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]
- 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]
- 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]
- V. P. Kalosha, E. A. Ponomarev, L. Chen, X. Bao, “How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses,” Opt. Express 14, 2071–2078 (2006). [CrossRef] [PubMed]
- 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. 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]
- V. I. Kovalev, R. G. Harrison, “Suppression of stimulated Brillouin scattering in high-power single-frequency fiber amplifiers,” Opt. Lett. 31, 161–163 (2006). [CrossRef] [PubMed]
- Q. Yu, X. Bao, L. Chen, “Strain dependence of Brillouin frequency, intensity, and bandwidth in polarization-maintaining fibers,” Opt. Lett. 29, 1605–1607 (2004). [CrossRef] [PubMed]
- D. Williams, X. Bao, L. Chen, “Characterization of high nonlinearity in Brillouin amplification in optical fibers with applications in fiber sensing and photonic logic,” Photon. Res. 2, 1–9 (2014). [CrossRef]
- V. Lecoeuche, D. J. Webb, C. N. Pannell, D. A. Jackson, “Transient response in high-resolution Brillouin-based distributed sensing using probe pulses shorter than the acoustic relaxation time,” Opt. Lett. 25, 156–158 (2000). [CrossRef]
- M. Niklès, L. Thévenaz, P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15, 1842–1851 (1997). [CrossRef]
- 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]
- 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). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- 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]
- 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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