## 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

Stimulated Brillouin scattering is one of the principal nonlinear processes that affect optical fiber systems [1, 2]. It occurs when narrow bandwidth light guided in an optical fiber interacts with the fiber itself to produce a density grating that backscatters the light. The stimulated Brillouin wave grows exponentially and, if the intensity of the initial beam is greater than a certain threshold value, the Brillouin scattering depletes most of the light from the beam.

The Brillouin threshold presents an enormous obstacle to transmitting high-power, single-frequency pulses in a fiber. Pulsed fiber lasers and amplifiers such as those used for micro-machining [3] achieve peak powers of tens of kilowatts or more, and when the pulse duration is greater than about one nanosecond, the Brillouin threshold puts a firm upper limit to the power that can be delivered.

Until now, however, there has been no easy way to calculate the Brillouin threshold intensity for a pulse in a single mode fiber. By contrast, the Brillouin threshold intensity *I _{P}* for a continuous (c.w.) pump beam is simply related to the Brillouin gain

*g*, the fiber length

_{B}*L*, and a threshold parameter Θ by the formula [1]:

*L*in the above equation with an interaction length related to the pulse duration

*τ*. The Brillouin and pump waves travel in opposite directions at speed

*υ*, so the longest overlap time is

*τ*/2, and the effective interaction length should be

*υτ*/2. This reasoning suggests the formula:

This equation is accurate when the pulse is longer than about 100 nsec, but it severely underestimates the Brillouin threshold for shorter pulses. The reason is that there is a time scale relevant to Brillouin scattering, the phonon lifetime *T _{B}*, that causes substantial deviations from Eq. (2) when

*τ*≤ Θ

*T*.

_{B}Relatively few authors have investigated Brillouin scattering at such short time scales (known as transient Brillouin scattering). Some studies pertain to small interaction cells that are much shorter than the pulse length [2,4,5], the limit opposite to the one relevant for fiber optics. Other papers look at the Brillouin scattering of pulses in optical fibers, but use numerical integration [6, 7] or experiments [8] to obtain results.

None of these authors give an elementary equation analogous to Eq. (2) for the Brillouin threshold for a pulse in fiber. One of the goals of this paper is to provide such an equation, relevant to the high-powered nanosecond pulses produced in fiber laser systems today. Such pulses typically have peak intensities greater than 10 W/*μ*m^{2}, and the fibers used to deliver these pulses are typically less than 10 meters in length. We consider only Brillouin scattering in passive fiber; our results would be modified somewhat for fibers with gain, such as fiber lasers and amplifiers.

We have found that under three assumptions, it is possible to derive an algebraic expression for the Brillouin threshold of a pulse in passive fiber. The assumptions are, first, that the pump pulse is square (that is, it has negligible rise and fall times, and the body of the pulse has constant intensity); second, that a negligible amount of power is depleted from the pump pulse; and finally, that the round trip time in the fiber (2*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.

The second assumption, the undepleted pump approximation, is valid as long as (1) fiber losses are negligible, which is the case for the short lengths of fiber used for high power pulse delivery, and (2) the equations are not used to model the pulse after it reaches the Brillouin threshold. This second limitation does not affect our ability to calculate the Brillouin threshold itself, however, and this threshold is the most important quantity to predict, since pulses beyond the threshold are generally too noisy for their intended application (such as material processing).

Extending our analysis, we have found that the Brillouin gain width for high intensity pulses is significantly greater than the narrow width usually associated with Brillouin scattering. This result is presented in Section 3. We have also found that when the third assumption above is relaxed—that is, when relatively short fibers are used—we can still find an analytic solution to the differential equations governing Brillouin scattering, though the resulting expression for the Brillouin threshold, Eq. (38), requires numerical root-finding techniques to evaluate.

Finally, we have conducted experiments to test these predictions. To compare the experiment and the theory, accurate values of the Brillouin gain *g _{B}* and the phonon lifetime

*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

We begin with the fundamental equations for the Brillouin scattering of a square pump pulse with duration *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:

*A*(

_{B}*z*,

*t*) and a phonon field

*Q*(

*z*,

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

*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}The amplitudes *A _{P}* and

*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].

The above equations are difficult to solve because the pump and Brillouin pulses travel in opposite directions, and the phonons are essentially stationary. Therefore three different frames of reference are important, and there is apparently no way to describe all three waves simply in a single frame of reference.

However, when the fiber is longer than the pulse and when the pump is undepleted, a simplification is possible. The basic insight is that to an observer that is co-moving with the pump pulse, the Brillouin amplitude at the observer’s position does not change. To see this, consider an observer positioned at the trailing edge of the pump pulse: at any given instant he sees a Brillouin wave that has built up over the time *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):

Equation (7) is the key simplification that makes it possible to derive an algebraic expression for the Brillouin threshold of the pulse. Adding Eq. (7) to Eq. (4), the spatial derivative *∂A _{B}/∂z* is eliminated:

*∂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,

^{*}/∂t*Q*:

^{*}*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}Equation (9) is an ordinary differential equation that can easily be solved. The solution at a given coordinate *z* is an amplitude that grows exponentially in time:

To find the complete behavior of *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:

*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*.

The Brillouin radiation grows exponentially from thermal noise of power |*A*_{B0}|^{2} at the beginning of the pulse [9]. (The noise power |*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

Equations (11) and (13) and can now be solved for the pump intensity at threshold:

*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*.

A possible concern about the above approach is that the group velocity of the Brillouin wave is substantially smaller than the group velocity of the pump, due to the narrowness of the Brillouin gain bandwidth [1]. This leads to the objection that perhaps the velocity *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.

For short pulses (*τ* ≪ Θ*T _{B}*), the threshold intensity in Eq. (14) grows rapidly, inversely proportional to

*τ*

^{2}. On the other hand, for long pulses (

*τ*≫ Θ

*T*), Eq. (14) reduces to

_{B}*υτ*/2, as expected.

## 3. Brillouin gain width

Brillouin scattering is well known to have a very narrow gain bandwidth, on the order of tens of MHz [1, 2]. However, we have found that at high intensitites, the Brillouin bandwidth becomes much larger, on the order of 1 GHz or more.

To see why, we return to Eq. (5) and generalize it slightly. Equation (5) is valid only when the Brillouin wave is exactly on resonance; that is, when the Brillouin optical frequency *ω _{B}* is equal to the difference between the pump and acoustic frequencies:

*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]:

_{B}*δ*is defined as:

*δ*= 2

*T*(Ω − Ω

_{B}*).*

_{A}The solution to Eq. (16) is an exponentially growing wave similar to before:

where and*γ*is a dimensionless gain constant,

The Brillouin power at the back of the pump pulse is:

We can therefore determine the Brillouin gain bandwidth by examining the frequency dependence of the quantity*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

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

_{B}*ν*= Δ

_{B}*ω*/(2

_{B}*π*) ≈ 20 MHz.

On the other hand, for high intensity pump pulses (*γ* ≫ 1), Eq. (18) implies that the gain *β* + *β*^{*} has a half-width of
$\mathrm{\Delta}\delta \approx \sqrt{\gamma}$, or

*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}This is an important result, because it means that Brillouin radiation is more difficult to suppress than previously thought: Some methods of reducing Brillouin scattering rely on changing the frequency of the Brillouin gain peak by more than the gain bandwidth before the Brillouin radiation can reach threshold. (These methods include introducing a temperature [10] or strain [11] gradient.) Given the increased gain bandwidth at high powers, this change will have to be much larger than expected.

Other authors have noted that the Brillouin gain width increases under certain circumstances. For example, the width can increase due to saturation effects [12] or due to the Fourier transform limit of short pulses [7,13]. However, the width given by Eq. (22) is caused by neither of these, since it occurs in the unsaturated regime (the pump is undepleted), and does not depend on the pulse duration.

What, then, is the intuitive explanation for this dramatic increase of gain width at high powers? The right hand side of Eq. (22) is approximately equal to the peak gain *β* + *β*^{*} on resonance (when *δ* = 0). We define the inverse of this quantity as the gain time Δ*t _{g}*, the time it takes for the Brillouin radiation to increase

*e*-fold; Eq. (22) can then be rewritten:

*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}For the next section, we need to calculate the width of the actual Brillouin radiation, Δ*ω _{B}′*, as opposed to the gain width Δ

*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:

*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:

*e*point) is:

## 4. The threshold parameter Θ

The value of the threshold parameter Θ has been calculated to be about 21 by Smith [9] for powers and fiber lengths appropriate to telecom applications. However, in view of the substantial changes introduced at high powers, it is worth revisiting this calculation.

The Brillouin power *P _{B}* 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]:

*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:

*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].

From Eq. (27), we can infer that the effective Brillouin power *P*_{B0} at the beginning of the pump pulse is (using Eq. (25)):

The threshold parameter Θ is defined as the logarithmic gain when the final Brillouin power is equal to the pump power *P _{P}*; that is,

To calculate the value of Θ to be used for the experiments described below, we use the following values: *υ _{A}* = 6 km/sec,

*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

Once the front of the pump pulse exits the fiber, the invariance condition, Eq. (7), no longer holds, and so the key assumption behind Eq. (14) is no longer valid. Therefore, to find the Brillouin threshold in fibers whose round trip time is shorter than the pulse duration, we must then return to Eqs. (4) and (5) and begin again. To solve these equations, we first change to coordinates *ẑ* and *t̂* that are co-moving with the Brillouin radiation:

We define new phonon and optical Brillouin variables,

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

*∂q*. We end up with the equation:

^{*}/∂t̂We have therefore reduced the two equations describing the Brillouin and phonon waves to the single equation above. This equation can be solved using Riemann’s method, the details of which are given in the Appendix. The solution is most easily written in terms of dimensionless variables: the dimensionless gain *γ* introduced in Eq. (19), *γ* = 2*g _{B}I_{P}υT_{B}*; a dimensionless fiber length Λ,

Using these dimensionless variables, the Brillouin threshold condition for pulses longer than the roundtrip time in the fiber can be written in terms of the modified Bessel function *I*_{0} (see Appendix):

*τ̄*) and fiber length (proportional to Λ), Eq. (38) can be solved numerically for

*γ*to obtain the intensity

*I*at the Brillouin threshold.

_{P}For long pulse lengths (*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:

Equation (38) is valid whenever the pulse is longer than the roundtrip time in the fiber (*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

To test this analysis, we measured the Brillouin thresholds for different pulse durations and fiber lengths. The experimental setup is shown in Fig. 2. The c.w. output of a seed diode was modulated to create pulses; the pulses were amplified by a series of fiber amplifiers, then sent through a length of passive fiber to induce Brillouin scattering.

The seed diode was a DFB laser operating at 1060 nm, such as is available from QPC Lasers or Eagleyard Photonics. The output was formed into a train of pulses by a Mach-Zehnder modulator from Photline Technologies (model NIR-MX-LN10) driven by a pulse generator. For long pulses (*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.

The pulses were amplified, first in a two-stage fiber pre-amplifier, then in a two-stage main fiber amplifier, the last stage of which used a cladding-pumped photonic crystal fiber available from NKT Photonics (DC-200/40-PZ-Yb). The output was put through an isolator, then coupled into polarization maintaining passive fiber (PM 980 XP from Nufern) that was angle-cleaved at the far end. A half-wave plate was used to align the polarization of the beam with one of the principal axes of the fiber. (In practice, this was done by rotating the half-wave plate until the Brillouin scattering was maximized.)

The output of the fiber was then collimated and its average power measured by a power meter. To examine the shapes of the pulses, a glass pickoff was placed in the output beam, reflecting in p-polarization, so that less than 2% of the power was deflected toward a fiber-coupled fast detector (ThorLabs SIR5) whose output was viewed on a 1 GHz oscilloscope.

To measure the Brillouin threshold, the average power of the pump beam was increased by slowly increasing the power of the main amplifier. The pulses were monitored on the oscilloscope for any sign of Brillouin scattering. When Brillouin scattering started to occur, the trailing edge of the pulse became noisy and generally reduced in intensity. The power was then further increased until the pulses reached Brillouin threshold.

First, however, an experimental criterion for the Brillouin threshold had to be defined. Mathematically, the Brillouin threshold occurs when the Brillouin intensity equals the pump intensity in the undepleted pump approximation. However, this is an idealized calculation; in reality, the pump is depleted as the Brillouin wave grows, so the Brillouin intensity never actually reaches the pump intensity, but asymptotically approaches it.

Therefore, the Brillouin threshold must be defined in practice as when the Brillouin intensity equals some fraction of the pump intensity. The exact value of this fraction is somewhat arbitrary; but fortunately, it is not too important since the threshold value only weakly depends on the fraction chosen. For example, in the c.w. case, if the Brillouin intensity at threshold is selected to be 20%, as opposed to 50%, of the pump intensity, the threshold pump power changes by only 6%.

For experimental convenience, the Brillouin threshold was defined near the visible onset of Brillouin scattering, when the back of the pump pulse was reduced by 20%. At all pulse widths, this introduced only a small loss in the pump pulse energy (estimated to be < 3%), so the launched pump power at threshold was then approximated as being equal to the power exiting the fiber. Although the threshold measurement had some uncertainty due to the noise inherent in Brillouin scattering, it was reproducible to within a few percent. An example is shown in Fig. 3.

Brillouin thresholds were measured for 50 m and 5 m lengths of fiber, with measurement errors estimated to be on the order of ±10%. The pulse repetition rate was 500 kHz or 1 MHz, which left plenty of time for the last of the Brillouin radiation to escape the fiber and for the phonons to decay back down to their thermal level before the next pulse arrived.

## 7. Results

The results of the experiments are plotted in Fig. 1. To compare the experiment to the theory, the values of the Brillouin gain *g _{B}* and the phonon lifetime

*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) [14–17]. The lifetime

*T*is also reduced in the fiber [14, 18], with typical values in the range of 2–3 nsec when the optical wavelength is 1060 nm.

_{B}Since the values of *g _{B}* and

*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.

Equation (14) was then fit to the short pulse data. This included all of the pulses measured in the 50 m fiber, and the shorter pulses (*τ* ≤ 50 nsec) from the 5 m fiber. For this fit, *g _{B}* was fixed at the above value of 31

*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}Finally, the fitted values of *g _{B}* and

*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.

In conclusion, the equation for the Brillouin threshold for short pulses in long fibers, Eq. (14), has proved accurate as well as simple. The more complicated expression for the Brillouin threshold for long pulses in short fibers, Eq. (38), also agrees with experiment. Furthermore, the surprising finding that the Brillouin gain bandwidth increases dramatically for high powered pulses, Eq. (22), will be important when considering methods to prevent Brillouin scattering. All of these results will be useful when designing pulsed fiber systems.

## Appendix: Riemann’s method

This appendix describes the solution of Eq. (35), subject to the appropriate boundary conditions. To find the boundary conditions, we note that for a fiber of length *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:

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

*t̂*= 2

*L/υ*,

The second boundary condition is that the Brillouin power at the far end of the fiber is due to thermal noise. That is, *A _{B}*(

*ẑ*=

*L*,

*t̂*) =

*A*

_{B0}, or

We now introduce one more change of variables: we define new distance and time coordinates *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

*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.

We also define a dimensionless Brillouin amplitude

and use the dimensionless gain defined in Eq. (19):*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

The problem defined by the three equations above can be solved by Riemann’s method [19]. The method consists of first finding a so-called Riemann function *w*(*z̄*, *t̄*) that has the following properties: first, it satisfies Eq. (44),

*w*obeys the boundary conditions:

Assuming that such a function can be found, then according to Riemann’s method, the solution for *B*(*z̄*, *t̄*) is:

*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).

The only remaining step is to find the Riemann function *w*(*z̄*, *t̄*). This can be done by expanding the function in a power series:

*c*(

_{n}*t̄*): Using the boundary conditions, Eq. (48), and reasoning by induction, we find that and therefore

*I*

_{0}[20], so the Riemann function can be expressed as:

Equation (51) can now be substituted into Eq. (49), along with the boundary conditions (Eqs. (45) and (46)), to obtain:

*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

We would like to thank Jim Morehead for helpful conversations.

## References and links

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