Photonics, Silicon Valley Technical Center, 110 Pioneer Way, Suite A, Mountain View, CA, 94041, USA 2 JDS Uniphase Corporation, 430 N. McCarthy Blvd., Milpitas, CA, 95035, USA ∗ [email protected]

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 OCIS codes: (290.5900) Scattering, stimulated Brillouin; (190.4370) Nonlinear optics, fibers.

References and links 1. G. P. Agrawal, Nonlinear Fiber Optics, Fifth Edition (Academic, Oxford, 2013). 2. R. W. Boyd, Nonlinear Optics, Second 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, 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. 36, 34–43 (1965). 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. 1, 31–43 (1970). 6. H. Li and K. Ogusu, “Dynamic behavior of stimulated Brillouin scattering in a single-mode optical fiber,” Jpn. J. Appl. Phys. 38, 6309–6315 (1999). 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 14, 2071–2078 (2006). 8. M. S. Bigelow, S. G. Lukishova, R. W. Boyd, and M. D. Skeldon, “Transient stimulated Brillouin scattering dynamics in polarization maintaining optical fiber,” CLEO 2001, 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. 11, 2489–2494 (1972). 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). 11. Q. Yu, X. Bao, and L. Chen, “Strain dependence of Brillouin frequency, intensity, and bandwidth in polarizationmaintaining fibers,” Opt. Lett. 29, 1605–1607 (2004). 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. 2, 1–9 (2014). 13. V. Lecoeuche, D. J. Webb, C. N. Pannell, and D. A. Jackson, “Transient response in high-resolution Brillouinbased distributed sensing using probe pulses shorter than the acoustic relaxation time,” Opt. Lett. 25, 156–158 (2000). 14. M. Nikl`es, L. Th´evenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15, 1842–1851 (1997). 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. 33, 2434–2436 (2008).

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Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13351

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). 17. V. Lanticq, S. Jiang, R. Gabet, Y. Jaou¨en, F. Taillade, G. Moreau, and G. P. Agrawal, “Self-referenced and singleended method to measure Brillouin gain in monomode optical fibers,” Opt. Lett. 34, 1018–1020 (2009). 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). 19. S. L. Sobolev, Partial Differential Equations of Mathematical Physics (Dover, New York, 1989). 20. G. Arfken, Mathematical Methods for Physicists, Third Edition (Academic, Orlando, 1985).

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, singlefrequency pulses in a fiber. Pulsed fiber lasers and amplifiers such as those used for micromachining [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 IP for a continuous (c.w.) pump beam is simply related to the Brillouin gain gB , the fiber length L, and a threshold parameter Θ by the formula [1]: gB IP L = Θ,

(1)

where Θ is a pure number, approximately equal to 21. To determine the threshold for a pulse, one might try to replace the fiber length 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: υτ = Θ. (2) gB IP 2 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 TB , that causes substantial deviations from Eq. (2) when τ ≤ Θ TB . 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/µm2 , 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. #208546 - $15.00 USD (C) 2014 OSA

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13352

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 (2L/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 gB and the phonon lifetime TB 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 gB and TB as free parameters. The best fit values for gB and TB are similar to the values found for other fibers, and the results are shown in Fig. 1. 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 AP (z,t). In the undepleted pump approximation, the pulse propagates with group velocity υ without changing its shape: ∂ AP 1 ∂ AP + ≈ 0. ∂z υ ∂t

(3)

The pump pulse generates a backward-propagating optical Brillouin (or Stokes) wave AB (z,t) and a phonon field Q(z,t) according to the following equations [1]: −

∂ AB 1 ∂ AB + = iκ1 AP Q∗ ∂z υ ∂t

(4)

∂ Q ΓB + Q = iκ2 AP A∗B (5) ∂t 2 where ΓB is the inverse of the phonon lifetime, ΓB = 1/TB . The constants κ1 and κ2 are proportional to the electrostrictive constant of the fiber, and their product is proportional to the Brillouin gain gB : g B ΓB κ1 κ2 = , (6) 4Aeff where Aeff is the effective mode area of the fiber. #208546 - $15.00 USD (C) 2014 OSA

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13353

The amplitudes AP and AB are normalized so that |AP |2 and |AB |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): ∂ AB 1 ∂ AB + = 0. ∂z υ ∂t

(7)

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 ∂ AB /∂ z is eliminated: 2 ∂ AB = iκ1 AP Q∗ . (8) υ ∂t The derivative of this equation with respect to time gives ∂ 2 AB iκ1 υ ∂ Q∗ = AP , ∂t 2 2 ∂t where ∂ AP /∂t has been neglected, since as long as the coordinates z and t do not refer to the leading or trailing edge of the pump pulse, AP (z,t) is constant. We now use Eq. (5) to eliminate ∂ Q∗ /∂t from the above equation, iκ1 υ ΓB ∗ ∂ 2 AB ∗ = A −iκ A A − Q , P 2 P B ∂t 2 2 2 and then Eq. (8) to eliminate Q∗ : ∂ 2 AB κ1 κ2 υ ΓB ∂ AB = |AP |2 AB − . 2 ∂t 2 2 ∂t 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 κ1 κ2 , defining the pump intensity IP = |AP |2 /Aeff , and using ΓB = 1/TB : ∂ 2 AB 1 ∂ AB gB υIP + − AB = 0. 2 ∂t 2TB ∂t 8TB

(9)

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: AB (z,t) = AB (z, 0)eαt , #208546 - $15.00 USD (C) 2014 OSA

(10)

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13354

with α=

p 1 −1 + 1 + 2gB IP υTB . 4TB

(11)

To find the complete behavior of AB (z,t) as a function of both z and t, note that Eq. (7) requires that AB (z,t) be a function of t − z/υ. Therefore the complete solution is: AB (z,t) = AB0 eα(t−z/υ)

(12)

where AB0 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 |AB0 |2 at the beginning of the pulse [9]. (The noise power |AB0 |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 |AP |2 ≡ Θ. (13) 2ατ = ln |AB0 |2 The value of the threshold parameter Θ is about 22 for high peak power pulses, as shown below. Equations (11) and (13) and can now be solved for the pump intensity at threshold: 2 Θ TB Θ IP = +1 (14) gB υτ τ This equation is the first main result of this paper, and is shown as the solid curve in Fig. 1. It gives the Brillouin threshold for a pulse whose duration is less than the relevant transit time in the fiber. Since the front of the pulse exits a fiber of length 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 ≤ 2L/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 TB ), the threshold intensity in Eq. (14) grows rapidly, inversely proportional to τ 2 . On the other hand, for long pulses (τ Θ TB ), Eq. (14) reduces to υτ gB IP = Θ, 2 which is Eq. (2), the c.w. Brillouin threshold for an interaction length of 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.

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Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13355

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 ωB is equal to the difference between the pump and acoustic frequencies: ωB = ωP − ΩA . To investigate the Brillouin gain bandwidth, we allow ωB to vary, and look at the Brillouin gain at different frequencies. Defining Ω = ωP − ωB , Eq.(5) is replaced by [1]: ΓB ∂Q + + i(ΩA − Ω) Q = iκ2 AP A∗B . (15) ∂t 2 Combining this equation with Eqs. (4) and (7), and following the same procedure as before, we find, in place of Eq. (9), ∂ 2 AB 1 ∂ AB gB υIP + (1 + iδ ) − AB = 0, 2 ∂t 2TB ∂t 8TB

(16)

where the frequency detuning parameter δ is defined as: δ = 2TB (Ω − ΩA ). The solution to Eq. (16) is an exponentially growing wave similar to before:

where

1 β= 4TB

AB (z,t) = AB0 eβ (t−z/v) ,

(17)

q 2 −(1 + iδ ) + (1 + iδ ) + γ ,

(18)

and γ is a dimensionless gain constant, γ = 2gB IP υTB

(19)

The Brillouin power at the back of the pump pulse is: |AB |2 = |AB0 |2 e(β +β

∗ )τ

(20)

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 : γ 1 . β +β∗ ≈ · 4TB 1 + δ 2 This is the familiar Lorentzian profile, with half-width ∆δ = 1, or ∆ωB =

1 2TB

(γ 1).

(21)

For TB = 4 nsec, for example, this gives a half-width of ∆νB = ∆ωB /(2π) ≈ 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 ∆δ ≈ γ, or r gB IP υ ∆ωB ≈ (γ 1). (22) 2TB This is the second main result of this paper: the Brillouin gain width for high pump intensities. This gain width can be considerably larger than at low intensities. For example, a typical pump pulse traveling in a large mode area fiber might have an intensity of IP = 100 W /µm2 .

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Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13356

Substituting this into Eq. (22), with υ = 0.2 m/nsec, gB = 31 µm2 /(W-m), and TB = 4 nsec (see Section 7 for a discussion of these values), we find a half-width of: ∆νB =

∆ωB ≈ 1.4 GHz . 2π

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 ∆tg , the time it takes for the Brillouin radiation to increase e-fold; Eq. (22) can then be rewritten: ∆ωB ∆tg ≈ 1. On the other hand, for low intensities, Eq. (21) can be written: ∆ωB TB ≈ 1. 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 TB . For high intensities, however, the characteristic time is ∆tg , because the phonon population grows exponentially as exp(t/∆tg ), and this rapid growth introduces Fourier components with a spectral width on the order of 1/∆tg . For the next section, we need to calculate the width of the actual Brillouin radiation, ∆ωB 0 , as opposed to the gain width ∆ωB . The gain width ∆ωB 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 ∆ωB 0 of the Brillouin pulse, we approximate the power gain for high power pulses (γ 1) as: β + β ∗ ≈ 2α −

δ2 √ 4TB 1 + γ

(23)

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: TB τ(ωP − ωB − ΩA )2 2 2 2ατ √ |AB | = |AB0 | e exp − . (24) 1+γ Therefore the half-width of the Brillouin radiation (measured at the 1/e point) is: ∆ωB 0 =

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(1 + γ)1/4 √ . TB τ

(25)

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13357

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 PB 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]: Z

PB =

∗ dωB h¯ ωB n¯ e[β (ωB )+β (ωB )]τ . 2π

(26)

Here n¯ is the average occupation number of phonons of frequency ΩA at temperature T : n¯ = {exp[¯hΩA /(kT )] − 1}−1 ≈ kT /(¯hΩA ). Approximating β as in Eq. (23) and performing the integral, we find: kT ω B 2ατ √ π ∆ωB 0 (27) e PB ≈ 2π ΩA where ω B is the Brillouin frequency on resonance: ω B = ωP − ΩA . This equation can be made more convenient using the relation ω B /ΩA = υ/(2υA ), where υA is the acoustic velocity, and υ is (as usual) the velocity of light in the fiber [1]. From Eq. (27), we can infer that the effective Brillouin power PB0 at the beginning of the pump pulse is (using Eq. (25)): 1 (1 + γ)1/4 υ √ PB0 = |AB0 |2 = √ kT υA 4 π TB τ The threshold parameter Θ is defined as the logarithmic gain when the final Brillouin power is equal to the pump power PP ; that is, √ √ υA PP TB τ PP Θ = ln = ln 4 π · (28) PB0 υ kT (1 + γ)1/4 To calculate the value of Θ to be used for the experiments described below, we use the following values: υA = 6 km/sec, υ = 0.2 m/nsec, kT = 4 x 10−21 J, PP = 30 W, TB = 4 nsec, 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 zˆ and tˆ that are co-moving with the Brillouin radiation: zˆ = z, tˆ = t + z/υ. The derivatives in terms of these new variables are: ∂ tˆ ∂ ∂ zˆ ∂ ∂ ∂ = + = , ∂t ∂t ∂ tˆ ∂t ∂ zˆ ∂ tˆ ∂ ∂ tˆ ∂ ∂ zˆ ∂ 1 ∂ ∂ = + = + . ˆ ˆ ∂z ∂z ∂t ∂ z ∂ zˆ υ ∂t ∂ zˆ #208546 - $15.00 USD (C) 2014 OSA

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13358

Using these variables, Eqs. (4) and (5) become: − and

∂ AB = iκ1 AP Q∗ ∂ zˆ

1 ∂Q + Q = iκ2 AP A∗B . ∂ tˆ 2TB

We define new phonon and optical Brillouin variables, tˆ Q q = exp 2TB and

(30)

(31)

tˆ AB 2TB

(32)

∂A = iκ1 AP q∗ ∂ zˆ

(33)

A = exp

(29)

in terms of which Eqs. (29) and (30) become: − and

∂q = iκ2 AP A ∗ . (34) ∂ tˆ We now take the derivative with respect to tˆ of Eq. (33), and use Eq. (34) to eliminate ∂ q∗ /∂ tˆ. We end up with the equation: ∂ 2A = κ1 κ2 |AP |2 A , − ∂ zˆ ∂ tˆ which can be rewritten as: ∂ 2A gB IP =− A (35) ∂ zˆ ∂ tˆ 4TB We 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 = 2gB IP υTB ; a dimensionless fiber length Λ, L Λ= ; (36) 2υTB and a dimensionless time proportional to the amount by which the pulse duration exceeds the roundtrip time in the fiber, τ − 2Λ. (37) τ¯ = 2TB 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 I0 (see Appendix): Z p ¯ −τ¯ + e−τ¯ I0 ( 2γΛτ)e

Λ

0

√ p p ¯ dy I0 ( 2γ (Λ − y)τ)( 1 + γ − 1)e( 1+γ−1)y

(38) Z τ¯

+ 0

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p ds I0 ( 2γΛs)e−s = eΘ/2 .

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13359

Equation (38) is the third main result of this paper, the Brillouin threshold in short fibers. Unfortunately, this equation is not as simple as the threshold in long fibers, Eq. (14). However, for ¯ and fiber length (proportional to Λ), Eq. (38) can be solved a given pulse duration (related to τ) numerically for γ to obtain the intensity IP at the Brillouin threshold. For long pulse lengths (τ¯ → ∞), the first and second terms on the left hand side of Eq. (38) go to zero, and the third term becomes exp(γΛ/2). Therefore in this limit, Eq. (38) can be rewritten: gB IP L = Θ, which is the expected c.w. result, Eq. (1). Equation (38) is valid whenever the pulse is longer than the roundtrip time in the fiber (τ¯ > 0, or τ > 2L/υ). The equation interpolates between the short pulse threshold given by Eq. (14) when τ = 2L/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.

Brillouin threshold intensity, W/µm2

1000

100

10

1m

1

5m 0.1

25 m 0.01

L > !"/2 0.001

1

5

10

50

100

500

1000

Pulse duration, nsec 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.

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

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Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13360

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. Pulse generator

Seed diode

MachZehnder modulator

preamplifier

Isolator

main amplifier

Isolator

HWP Power meter

Passive PM fiber

Pickoff

Lens

Oscilloscope

Lens

Fast detector

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.

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 #208546 - $15.00 USD (C) 2014 OSA

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13361

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.

70 nsec

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%.

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 gB and the phonon lifetime TB were needed. In bulk silica, gB = 50 µm2 /(W-m) and TB = 5 nsec [1,2]. However, these values are different in optical fibers: The gain gB is decreased due to the imperfect overlap of the acoustic and optical modes [1], so that typical values of gB range from 10 to 30 µm2 /(W-m) [14–17]. The lifetime TB 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. Since the values of gB and TB for our fiber (Nufern PM-980 XP) were not known in advance, #208546 - $15.00 USD (C) 2014 OSA

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13362

we fit them to the data. To obtain the gain gB , we used the two longest pulses (τ = 300 and 400 nsec) measured in the 5 m fiber, since for these pulses, the steady state threshold equation (gB IP L = Θ) holds to an excellent approximation. We used the average threshold intensity for these two pulse lengths, and calculated the Brillouin gain to be: gB = 31 µm2 /(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, gB was fixed at the above value of 31 µm2 /(W-m) and TB was determined by the least squares method. The result was: TB = 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. Finally, the fitted values of gB and TB were used to plot Eq. (14) and Eq. (38) with 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, zˆ = L and tˆ = 2L/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 zˆ provided that tˆ ≤ 2L/υ. We therefore have: AB (ˆz, tˆ) = AB0 eα(tˆ−2ˆz/υ)

(tˆ ≤ 2L/υ),

In terms of the variable A , the first boundary condition is therefore that at tˆ = 2L/υ, 2α L ˆ exp (L − zˆ) A (ˆz, t = 2L/υ) = AB0 exp υTB υ

(39)

The second boundary condition is that the Brillouin power at the far end of the fiber is due to thermal noise. That is, AB (ˆz = L, tˆ) = AB0 , or tˆ A (ˆz = L, tˆ) = AB0 exp (40) 2TB We now introduce one more change of variables: we define new distance and time coordinates z¯ and t¯ that are scaled so they are dimensionless, and are chosen so that the boundary conditions occur more conveniently at z¯ = 0 and t¯ = 0. Since the first boundary condition, Eq. (39), occurs at tˆ = 2L/υ, we set 1 2L ¯t = ˆt − . (41) 2TB υ The second boundary condition (Eq. (40)) states that the Brillouin wave starts with thermal noise at zˆ = L; the Brillouin amplitude then grows in the negative zˆ direction to reach its maxi#208546 - $15.00 USD (C) 2014 OSA

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13363

mum at z = 0. We invert the z-coordinate, and define z¯ =

1 (L − zˆ) 2υTB

(42)

so that the boundary condition occurs at z¯ = 0 and the Brillouin amplitude increases in the positive z¯ direction. We also define a dimensionless Brillouin amplitude B=

A −L/(υTB ) e , AB0

(43)

and use the dimensionless gain defined in Eq. (19): γ = 2gB IP vTB . In terms of these new variables, Eq. (35) becomes ∂ 2B γ = + B. (44) ∂ z¯∂ t¯ 2 The boundary conditions, Eqs. (39) and (40), can then be written:

with α defined by Eq. (11), and

B(¯z, 0) = e4αTB z¯

(45)

B(0, t¯) = et¯.

(46)

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(¯z, t¯) that has the following properties: first, it satisfies Eq. (44), ∂ 2w γ = w. (47) ∂ z¯∂ t¯ 2 Second, w obeys the boundary conditions: w(¯z, 0) = w(0, t¯) = 1.

(48)

Assuming that such a function can be found, then according to Riemann’s method, the solution for B(¯z, t¯) is: B(¯z, t¯) = B(0, 0)w(¯z, t¯) +

Z t¯

dt¯0 w(¯z, t¯ − t¯0 )

0

∂B (0, t¯0 ) + ∂ t¯

Z z¯ 0

d z¯0 w(¯z − z¯0 , t¯)

∂B 0 (¯z , 0). ∂ z¯

(49)

Although a derivation of this formula is beyond the scope of this paper, it can be checked that the function B(¯z, t¯) 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(¯z, t¯). This can be done by expanding the function in a power series: cn (t¯)¯zn . n! n=0 ∞

w(¯z, t¯) =

∑

Substituting this power series into Eq. (47) gives a condition on the functions cn (t¯): dcn+1 γ = cn . dt¯ 2 Using the boundary conditions, Eq. (48), and reasoning by induction, we find that γ n t¯n cn (t¯) = ; 2 n! #208546 - $15.00 USD (C) 2014 OSA

Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13364

and therefore

∞

w(¯z, t¯) =

∑

n=0

γ t¯z¯ 2

n

1 . (n!)2

(50)

It turns out that this is related to the power series for the modified Bessel function I0 [20], so the Riemann function can be expressed as: p w(¯z, t¯) = I0 ( 2γ t¯z¯). (51) Equation (51) can now be substituted into Eq. (49), along with the boundary conditions (Eqs. (45) and (46)), to obtain: Z t¯ Z z¯ p p p 0 0 t¯0 0 ¯ ¯ ¯ ¯ ¯ B(¯z, t ) = I0 ( 2γ t z¯) + dt I0 ( 2γ (t − t )¯z)e + d z¯0 I0 ( 2γ t¯(¯z − z¯0 ))4αTB e4αTB z¯ (52) 0

0

The Brillouin threshold condition is that, at the beginning of the fiber at time τ, the Brillouin amplitude AB should equal AB0 exp(Θ/2). To find the threshold condition in terms of B(¯z, t¯), B ¯ From the should be evaluated at the point z¯ = L/(2υTB ) ≡ Λ and t¯ = (τ − 2L/υ)/(2TB ) ≡ τ. definitions of B (Eq. (43)) and A (Eq. (32)), the threshold condition becomes: ¯ = eΘ/2 e−τ¯ B(Λ, τ)

(53)

Substituting Eq. (52) into this equation yields the threshold condition, Eq. (38). Acknowledgment We would like to thank Jim Morehead for helpful conversations.

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Received 19 Mar 2014; revised 3 May 2014; accepted 5 May 2014; published 27 May 2014 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013351 | OPTICS EXPRESS 13365