Gain dependence of the linewidth of Brillouin amplification in optical fibers

We report a novel, pump-power dependent, linewidth broadening effect in stimulated Brillouin amplification of a continuous-wave probe by a pulsed pump. This behavior is different from the case of two interacting continuous-wave pump and probe fields, where the shape of the logarithmic Brillouin gain spectrum is independent of the pump power. Studying this effect numerically and experimentally and also analytically, we find that for a given width of the pump pulse the Brillouin linewidth grows linearly with the Brillouin logarithmic gain with a slope, which inversely depends on the pulse width. Thus, for example, in a standard single-mode fiber, a 15ns pump pulse, strong enough to generate a gain of 0.5dB, broadens the logarithmic lineshape by ~1.5MHz, while a 45ns pulse, providing the same gain, increases the linewidth by only ~0.5MHz. Since the rising and falling slopes of the shape of the Brillouin gain spectrum are also gain dependent, this effect might challenge the calibration of Brillouin distributed slopeassisted sensing techniques. ©2014 Optical Society of America OCIS codes: (060.2370) Fiber optics sensors; (290.5830) Scattering, Brillouin; (330.1880) Detection; (190.0190) Nonlinear optics. References and links 1. R. W. Boyd, Nonlinear Optics, 3rd ed. (Acedemic, 2008). 2. L. Thévenaz, Advanced Fiber Optics Concepts and Technology (EPFL, 2011). 3. X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors 12(7), 8601–8639 (2012). 4. L. Thévenaz, S. F. Mafang, and J. Lin, “Effect of pulse depletion in a Brillouin optical time-domain analysis system,” Opt. Express 21(12), 14017–14035 (2013). 5. S. M. Foaleng, F. Rodríguez-Barrios, S. Martin-Lopez, M. González-Herráez, and L. Thévenaz, “Detrimental effect of self-phase modulation on the performance of Brillouin distributed fiber sensors,” Opt. Lett. 36(2), 97– 99 (2011). 6. S. Foaleng and L. Thévenaz, “Impact of Raman scattering and modulation instability on the performances of Brillouin sensors,” in 21st International Conference on Optical Fibre Sensors (OFS21), International Society for Optics and Photonics (2011). 7. Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011). 8. A. Motil, O. Danon, Y. Peled, and M. Tur, “Pump-power-independent double slope-assisted distributed and fast Brillouin fiber-optic sensor,” IEEE Photon. Technol. Lett. 26(8), 797–800 (2014). 9. J. Urricelqui, A. Zornoza, M. Sagues, and A. Loayssa, “Dynamic BOTDA measurements based on Brillouin phase-shift and RF demodulation,” Opt. Express 20(24), 26942–26949 (2012). 10. A. David and M. Horowitz, “Low-frequency transmitted intensity noise induced by stimulated Brillouin scattering in optical fibers,” Opt. Express 19(12), 11792–11803 (2011). 11. L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152(1–3), 65–70 (1998). 12. J.-C. Beugnot, M. Tur, S. F. Mafang, and L. Thévenaz, “Distributed Brillouin sensing with sub-meter spatial resolution: modeling and processing,” Opt. Express 19(8), 7381–7397 (2011). 13. Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express 20(8), 8584–8591 (2012). 14. G. L. Keaton, M. J. Leonardo, M. W. Byer, and D. J. Richard, “Stimulated Brillouin scattering of pulses in optical fibers,” Opt. Express 22(11), 13351–13365 (2014). #221849 $15.00 USD Received 27 Aug 2014; revised 7 Oct 2014; accepted 8 Oct 2014; published 29 Oct 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.027535 | OPTICS EXPRESS 27535


Introduction
In the process of Stimulated Brillouin Amplification (SBA) in optical fibers two counterpropagating pump and probe waves generate an acoustic wave, which then transfers optical power from the pump to the probe if the latter frequency, Probe ν , is downshifted from that of the pump, Pump ν , by an amount dictated by energy and momentum conservation rules [1].This amplification process is characterized by the Brillouin Gain Spectrum (BGS), which peaks at a frequency difference, B ν , called the Brillouin Frequency Shift (BFS), where the three involved waves are phase-matched.When , where A τ is the acoustic phonon lifetime [1].For standard singlemode optical fibers at around 1550nm, B ν ~11GHz and , B CW ν Δ ~30MHz [2].
Stimulated Brillouin amplification has found extremely important applications in distributed sensing of strain and temperature [2,3].In the common implementation, called Brillouin Optical Time Domain Analysis (BOTDA) [2], a pulsed pump propagates against a CW probe wave.By temporally studying the power of the emerging probe wave for different values of the frequency difference between the two counter-propagating signals, a frequencydistance map is generated showing the Brillouin gain spectrum, and consequently the Brillouin frequency shift, at each location along the fiber.Since the Brillouin frequency shift is sensitive to the local strain/temperature, a quantitative map of these properties is generated [2].Much like in radar, the spatial resolution depends on the width of the pump pulse, T .In the presence of short pump pulses the linewidth of the Brillouin gain spectrum is that of the convolution of the natural (CW) Brillouin gain spectrum, with the spectrum of the pump pulse [2].Thus, for pump pulses shorter than ~50ns the Brillouin gain spectrum begins to broaden beyond its natural width of 30MHz, and approaches an inverse dependence on T for T < 20ns (e.g., ).As mentioned above, for the CW case both theory and experiment clearly indicate that in the absence of depletion and non-linear effects [4][5][6] neither the shape nor the width of the logarithmic Brillouin gain spectrum, depend on the pump power.Contrary to the common assumption that these power-independent properties also hold for the case of a pulsed pump, as in BOTDA, we show here, for the first time to our knowledge, that for a pump pulse of duration T and power P both the linewidth, ( , ) B T P ν Δ and shape of the logarithmic Brillouin gain spectrum depend on the pump power level.We demonstrate, both theoretically and experimentally, that this logarithmic Brillouin gain spectrum broadening, ( ) grows linearly with the Brillouin gain with a gradient, which is approximately inversely proportional to the pump pulse width, T .Since, consequentially, the rising and falling slopes of the Brillouin gain spectrum shape are also gain dependent, these findings might challenge the calibration of dynamic, slope-assisted sensing methods [7][8][9].Sections 2 and 3, respectively, describe the simulation and experimental techniques used to obtain the results of Sec. 4. Sec. 5 provides an approximate analytical expression for the linear dependence of the gain-induced broadening on the Brillouin gain, as a function of the pump pulse width.The paper concludes with a short summary.

The governing differential equations and their solution procedure
Ignoring birefringence effects and the spontaneous generation of acoustic phonons, as well as fiber loss, the scalar stimulated Brillouin amplification process is commonly described by a set of three coupled equations [1]: Pump Probe A z t A z t are the slowly varying complex envelopes of the pump and probe, propagating in the z + and z − directions, respectively, and ( , ) Q z t represents the envelope of the acoustic wave.c is the speed of light in vacuum, n is the refractive index, assumed identical for the pump and probe, A Γ is the frequency detuning factor and B Γ is the inverse g and 2 g represent the electrostrictive and elasto-optic effects, underlying the operation of stimulated Brillouin amplification.For the steady state, CW case, Eqs. ( 3) can be readily solved to obtain the Lorentzian logarithmic gain of Eq. ( 1) with Note that no optical nonlinear effect other than Brillouin is assumed in Eqs.(3).Under pulsed conditions the solution of Eqs.(3) require the use of numerical techniques.Here we used the well-known Finite Difference Time Domain (FDTD) method [10], with fine grids in the z and t domains, obeying ( / ) / 2 z c n t Δ = Δ .When the pump pulse of width T and height P enters the fiber section of length L at ( 0, 0) , is determined.Pump depletion was avoided by judicious choice of the probe input power.The logarithmic Brillouin gain spectrum for a given combination of pump pulse width and power was obtained by solving Eqs.(3) for 600 values of the frequency, ν , spaced 0.4MHz apart, and spanning the range of 240MHz centered on the fibers Brillouin frequency shift.
The numerical procedure was validated in two regimes: (i) Results for very long pulses ( A T τ >> ), i.e., the CW case, were in excellent agreement with both the well-known analytical solution for the undepleted case [2], as well as with the semi-analytical solution for the depleted pump regime [11]; (ii) For arbitrarily short pump pulses but under conditions where the perturbation solution [12] of Eqs.(3) is valid, results of the numerical procedure of this paper were identical to those obtained from [12].Results obtained from the numerical solution of Eqs.(3) for the dependence of the linewidth of the logarithmic gain on the pump width and power are presented in Sec. 4.

The experimental setup
The setup of Fig. 1 was used to experimentally investigate the broadening of the Brillouin gain spectrum under strong pump power.It is basically a classical BOTDA setup, where the probe frequency is scanned against the fixed pump frequency using a computer controlled Arbitrary Waveform Generator (AWG) rather than a tunable RF synthesizer, as in the Fast BOTDA implementation [13].A 10mW probe signal was injected into a 10m of SMF-28 single mode fiber.In order to ensure that all pump-power-dependent effects occur only in the test fiber, the power level of the pump pulse was controlled by the adjustment of a polarization controller (PC2), followed by a polarizer, both placed after the high-power EDFA2 and in very close proximity (1m) to the test fiber.The Brillouin interaction at the interrogated point of interest was maximized by adjusting the probe state of polarization using polarization controller PC1.A frequency range of 200MHz centered around the Brillouin frequency shift of the test fiber (10.867GHz) was then scanned using 400 frequencies spaced 0.5MHz apart and 1024 time averages were used for each frequency setting.In this way the logarithmic gain spectrum was obtained for any given combination of pump pulse width T and power P , from which the −3dB spectral width was experimentally determined.

Results
Figure 2 shows the experimentally obtained normalized logarithmic Brillouin gain spectrum for a 45ns wide pump pulse at different values of the Brillouin gain.During the measurement, the 45ns square pump pulse peak power was varied between 0.08W (to achieve a Brillouin gain of 0.2dB) to 1.78W (gain = 4.4dB).The −3dB linewidth, ( , , is clearly seen to grow as the Brillouin logarithmic gain increases from 0.1dB to ~4dB, while the Brillouin gain spectrum lineshape remains symmetrical around its low pump power center.Theoretical and experimental results for the pump-power-induced differential increase of the −3dB linewidth, ( ) 2), as a function of the Brillouin gain, appear in Fig. 3 for a few values of the pump pulse width T .Clearly, the shorter the pump pulse the larger ( ) solely represents this newly reported effect of the additional linewidth broadening, which depends on the power of the pump pulse., decrease with T , as shown in Fig. 4 on a log-log scale in units of MHz/dB.Clearly, this linewidth broadening effect exists even for pulse widths much longer than the phonon life time, although the predicted broadening becomes increasingly insignificant.The X-marked experimental data in Fig. 4, while not in full quantitative agreement with the numerically calculated ones, especially for increasing values of T , do show the same trend.Both the theoretical and experimental results of Figs.2-4 show no dependence on the interrogation coordinate z along the fiber, provided enough time was allocated for the propagating pump pulse to fully pass that point.In particular, the experimental observation of this broadening effect at the pump entry point 0 z = and at as early as t T = attests to the fact that it is the Brillouin, rather than any other optical nonlinear effect, which is responsible for the pump-power-induced additional broadening of the Brillouin gain spectrum.Finally, note that (i) all experimental data reported here were obtained for a 10m test fiber, longer than the longest pulse tested, but the broadening effect itself was first observed on a much longer fiber (77m); and (ii) as T decreases, the gain range which can be covered by our experimental setup also decreases, since a decreasing pump pulse width requires more pump power to achieve a given Brillouin gain.

Discussion
Upon completion of this work, we noted that a recently published [14] analytical study of Eqs.
(3) for the threshold of stimulated Brillouin scattering in optical fibers under strong pump pulses, also describes an increase of the linewidth of the relevant gain spectrum.Adapting the approximate analysis of [14] to our scenario, subject to their assumptions and insights, we start from their Eq.(20) for the Brillouin logarithmic gain induced by a square pump pulse of duration T (our T g is g in [14]): ,max ,max Here we see two contributions to the Brillouin linewidth: (i) the natural width, 1/ 2 A πτ , and (ii) a broadening term that increases with the Brillouin gain with a slope of 3 / (4 ) T π [in Hz/Neper] (or 0.055 / [in Hz/dB] T ).Note that the analysis of [14] does not reproduce the linewidth broadening due to the spectral contents of the pump pulse.
We can now compare our results of Sec. 4 for ( ) B δ ν Δ with the second term of Eq. ( 5) since both deal only with the gain induced broadening of the Brillouin linewidth.Indeed, adding a plot of the T-dependence of the slope of the second term in Eq. ( 5) to Fig. 4 (the dashed line) results in a nice match to our numerical solutions.
Our experimental results agree quite well with the numerical solution of Eqs.(3) for T = 15ns but then displays systematic deviation as T increases.This deviation may be due to the fact that the test fiber was a standard weakly birefringent fiber, while the analytical model was based on a polarization maintaining fiber, where both the pump and probe are launched with their polarizations aligned along the same fiber axis.
This newly reported broadening of the Brillouin linewidth under pulsed operation of stimulated Brillouin amplification is rather small for low gains and, therefore, has little effect on Brillouin frequency shift measurements in classical BOTDA applications, where very low gains (a few tenths of 1dB or less) and full frequency scanning is used to determine the Brillouin frequency shift.However, in recently introduced [7-9] dynamic implementations of BOTDA, which utilize information from the slopes of either the magnitude or phase of the Brillouin gain spectrum, this pump power dependence of the shape of the Brillouin gain spectrum might affect the measurement calibration, leading to errors.Moreover, dynamic applications often go hand in hand with short sensing ranges and short pump pulses for increased spatial resolution.Under these circumstances relatively high gain values are both allowed and deemed useful, thereby further aggravating the problem, see Fig. 3.
In summary, this paper reports a novel pump-power-dependent linewidth broadening effect in stimulated Brillouin amplification of a continuous-wave probe by a pump pulse.The effect is predicted by numerical solution of the governing equations of the Brillouin amplification process, as well as by their approximate analytical solution, and is also experimentally observed.For a given width of the pump pulse the Brillouin linewidth grows linearly with the Brillouin logarithmic gain with a gradient which inversely depends on the pulse width.This linewidth broadening may be of concern to Brillouin distributed sensing techniques which rely on the slopes of the magnitude or phase of the Brillouin gain spectrum.Its influence on these techniques, as well as its source and full analytical evaluation are currently under further investigations.

Fig. 3 .
Fig. 3.The Brillouin-gain-dependent broadening of the Brillouin Gain Spectrum, ( ) ( , ) ( , 0) B B B T P T P δ ν ν ν Δ ≡Δ −Δ → , Eq. (2), for pump pulses having different widths.Solid lines represent the numerical solutions of Eqs.(3) while the X's stand for experimentally obtained results.The reported gains for the 45, 30 and 15ns pump pulses were obtained using peak powers of up to 1.78, 3, and 4.5W, respectively.The observed ( )B δ ν Δ appears to have linear dependence on the Brillouin logarithmic gain.Furthermore, the slopes of the lines in Fig.3, [( )]/ [ ] B d dG a i n δ ν Δ, decrease with T , as shown in Fig.4on a log-log scale in units of MHz/dB.Clearly, this linewidth broadening effect exists even for pulse widths much longer than the phonon life time, although the predicted broadening becomes increasingly insignificant.The X-marked experimental data in Fig.4, while not in full quantitative agreement with the numerically calculated ones, especially for increasing values of T , do show the same trend.Both the theoretical and experimental results of Figs.2-4show no dependence on the interrogation coordinate z along the fiber, provided enough time was allocated for the propagating pump pulse to fully pass that point.In particular, the experimental observation of this broadening effect at the pump entry point 0 z = and at as early as t T = attests to the fact that it is the Brillouin, rather than any other optical nonlinear effect, which is responsible for the pump-power-induced

Fig. 4 .
Fig. 4. A Logarithmic scale plot of the dependence of the slopes, [ ( )] / [ ] B d dG a i n δ ν Δ , of the lines in Fig. 3 on the pump pulse width, T .The black circles represent the gradients of the numerically calculated curves of Fig. 3 while the black X's represent experimentally obtained values.The dashed line describes the 0.055•10 −6 /T (in units of MHz/dB) dependence of the gain-slope of Eq. (6), Sec. 5.
both probe and pump are CW waves, the