Distributed Brillouin Sensing with Sub-meter Spatial Resolution: Modeling and Processing

A general analytic solution for Brillouin distributed sensing in optical fibers with sub-meter spatial resolution is obtained by solving the acoustical-optical coupled wave equations by a perturbation method. The Brillouin interaction of a triad of square pump pulses with a continuous signal is described, covering a wide range of pumping schemes. The model predicts how the acoustic wave, the signal amplitude and the optical gain spectral profile depend upon the pumping scheme. Sub-meter spatial resolution is demonstrated for bright-, dark-and π-shifted interrogating pump pulses, together with disturbing echo effects, and the results compare favorably with experimental data. This analytic solution is an excellent tool not only for optimizing the pumping scheme but also for post-processing the measured data to remove resolution degrading features. 1. L. Thévenaz, " Brillouin distributed time-domain sensing in optical fibers: state of the art and perspectives, " Front. Comparison of distributed fiber optic sensing methods for localisation and quantity information measurements, " Opt. Characterization of the Brillouin-loss spectrum of single-mode fibers by use of very short (<10-ns) pulses, " Opt. Transient response in high-resolution Brillouin-based distributed sensing using probe pulses shorter than the acoustic relaxation time, " Opt. Signal processing technique for distributed Brillouin sensing at centimeter spatial resolution, " IEEE Sens. Brillouin spectral deconvolution method for centimeter spatial resolution and high-accuracy strain measurement in Brillouin sensors, " Opt. Differential pulse-width pair BOTDA for high spatial resolution sensing, " Opt. How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses, " Opt. Effect of optical phase on a distributed Brillouin sensor at centimeter spatial resolution, " Opt. Using pulse with a dark base to achieve high spatial and frequency resolution for the distributed Brillouin sensor, " Opt. Numerical study of transient stimulated Brillouin scattering, " J. High spatial and spectral resolution long-range sensing using Brillouin echoes, " J. General analytical model for distributed Brillouin sensors with sub-meter spatial resolution " , Theoretical evaluation of the Brillouin threshold and the steady-state Brillouin equations in standard single-mode optical fibers, " J. Steady-state noise analysis of spontaneous and stimulated Brillouin scattering in optical fibers, " J. Differential pulse-width pair BOTDA for high spatial resolution sensing, " Opt. Subpeaks in the Brillouin loss spectra of distributed fiber-optic sensors, " Opt. Effect of the finite extinction ratio of an electro-optic modulator on the performance of distributed probe-pump Brillouin sensor systems, " Opt.


Introduction
During the past decade interesting observations and innovative configurations have been proposed to realize distributed measurements with sub-meter spatial resolution using stimulated Brillouin scattering (SBS) in a pump-probe configuration [1].The Brillouin Optical Time Domain Analysis (BOTDA) configuration based, on a signal pulse and a counterpropagating continuous pump, can achieve 1 meter spatial resolution over more than 30 km together with 1MHz accuracy on the determination of the local Brillouin shift [2].In the SBS process the signal gain spectrum corresponds to the convolution between the pump pulse spectrum and the natural Brillouin spectrum.As a result, the spatial resolution of BOTDA technique was limited to 1m (10ns) [3].However, it was observed [4] that, for pulses substantially shorter than the acoustic lifetime in silica, the measured gain profiles abandon their expected spectral broadening and the linewidth was shown to gradually return to the natural value, as determined by the acoustic lifetime.This astonishing feature was first observed using pulses superimposed on a continuous pump level [4], then later using dark pulses [5] and eventually using π-phase pulses [6].It was soon explained that this behavior results from the sharp modified reflection of the pump wave from a pre-existing acoustic wave, formerly built up by a continuous component in the pump wave [7].The passage of very short pulses turns out to have a negligible effect on the acoustic wave amplitude and the changes in the signal amplitude are entirely dominated by the modified reflection of the pump wave from the steady acoustic wave during the pulse duration.Since the acoustic wave is essentially created by continuous waves, it will respond to frequency detuning according to the natural gain spectral profile but the distance range is limited by the pump depletion, the interaction taking place all over the fiber length.To reduce this effect, a new scheme based on combination of signal gain and loss has been recently demonstrated [8].Also, signal processing techniques have been developed to measure strain at resolution better than the Brillouin linewidth limit [9].By fitting each Brillouin gain spectrum along the fiber and processing the compound spectrum, two small successive sensor sections can be detected [10][11][12].However, the improvement of the spatial resolution is limited by the Brillouin gain.In fact, for sections smaller than 10cm, the Brillouin gain is very low and pump waveform must be optimized.For example, a technique based on two intensity pump pulse with different pulse widths has been developed to increase the spatial resolution to 10cm without spectral broadening [13].It turns out that no clear physical description of the experimental situation has been reported and these observations were so far justified by numerically solving the set of three coupled equations governing the Brillouin interaction [14][15][16][17].Note that an interesting model describing the Brillouin interaction between a pulsed pump wave with a CW component, and a continuous probe wave has been recently proposed [18].The authors formulated a simple integral expression relating the Brillouin signal for an arbitrary pump pulse and the computation times is greatly reduced.The main problem arising from numerical justification is the difficulty to gain physical insight into the involved processes and their subsequent effect on the sensor response, and also the difficulty to optimize the different parameters of the sensor using an approach more efficient than a pure random choice that is gradually refined.
In this paper, we formulate an analytic equation for signal gain under some hypotheses that are valid in practical cases.By a simple perturbation method and under realistic assumptions we could analytically solve the coupled equations governing the interactions for a pump waveform, comprising three square parts, having arbitrary complex amplitudes.The general analytic solution expressed in the time domain matches the observed sensor response for all configurations proposed to date [4,6,19] and for any pump-signal frequency detuning.It completes our efforts to realize an optimized high-spatial resolution distributed sensor based on a solid physical description [20,21], by deriving the exact and full response of the system under general pump conditions.While classical steady-state analytical equations [22] fail in dealing with pulses on the order of the acoustic lifetime, the proposed model is general and does not have such limitations.Furthermore, the model can be advantageously used to predict the minimum pulse width that can be accurately treated by the classical steady-state approach.Using this tool we can easily determine the sensor configuration providing the best response, and also anticipate and describe all unwanted contributions to the signal.We also present experimental results on the Brillouin echo distributed sensor (BEDS) configuration and we compare with analytic model.Finally, we demonstrate a signal processing algorithm, using inverse filtering to suppress all unwanted contributions to the signal.

Analytic model
Let ˆ( , ) p E z t describe the field of a pump laser light, with center angular frequency ω p and wavenumber k p , that is injected at the input of the fiber in the direction of positive z.The Stokes signal light ˆ( , ) S E z t , characterized by center angular frequency ω s and wavenumber k s , is assumed to propagate in the negative z direction (see Fig. 1 (a)).Finally, let ˆ( , )  Q z t denote the acoustic (density) wave, which results from the interaction between the pump and signal fields.Thus, its acoustic temporal and spatial angular frequencies are given by p s      and, q = k p + k s , respectively, which are related by where L V denotes the longitudinal sound velocity in the medium.Denoting the corresponding complex amplitudes by ( , )

and ( , )
Q A z t , the above mentioned three waves can be written as (cc stands for Complex Conjugate): The three coupled equations governing these three interacting complex amplitudes can be derived from Maxwell's equations for the electric fields and Navier-Stokes equations for the acoustic field [22].Neglecting the transverse field variations and using the slowly varying envelope approximation, the Brillouin coupled equations are expressed as: The frequency detuning parameter is where Ω Β /2π and Ω/2π are the Stokes resonance frequency (i.e., the Brillouin shift) and the pump-signal frequency difference, respectively, at a given z.The acoustic damping constant, B  , is related to the full width of half maximum (FWHM) of the spontaneous Brillouin gain spectrum , where A  is the energy decay time of the acoustic wave and is equal to 6ns in standard silica fibers.
e SP g z z nc  is the mean density of the medium, n is its refractive index at ω = ω S,P and c is the light velocity in vacuum).In Eq. (2)c) the acoustic velocity is assumed to be so much smaller than the light velocity (V g ), that the acoustic field, generated by the optical waves at a given z, can be considered a local, timedependent but non-propagating disturbance.Traditionally, Eq. (2)c) is solved analytically under steady state conditions, for which / 0 Q t    .This is only justified for continuous (CW) pump and signal waves, or for optical pulses much longer than the acoustic lifetime [23,24], and this assumption won't be considered hereafter.
Being interested in high spatial resolution sensing, where the pump or signal pulse widths are comparable to or smaller than τ A , a transient analysis of the set of Eqs.(2) is required.Below, a general analytic solution is obtained, subject to a few reasonable approximations.First, the effect of linear attenuation is neglected over the region where the waves interact.Second, the interaction is supposed weak enough that pump depletion can be neglected (A p (z) = Constant).Consequently, there is no need to solve Eq. (2)a), the solution being trivial.Third, the only restriction imposed on the signal field is the small gain approximation during the interaction, which is well justified for very short pump pulses for which Brillouin amplification is realized over sub-meter interaction lengths.Therefore, the signal amplitude A s (z,t) can be expressed as the sum of a continuous constant wave 0 S A and a small varying term a s (z,t) resulting from the Brillouin gain.Consequently, the signal power P s after the propagation (from higher z values to z = 0) shows a linear dependence on the amplitude of the time varying Brillouin signal: In addition, A s can be considered constant in Eq. ( 2)a-2c), the constant term 0 S A being widely dominating, except when A s is differentiated, as in the left term of Eq. (2)b), where only a s (z,t) is present.Using these assumptions, Eqs.(2)a-2c) reduce to:

Analytic solution
In the undepleted pump approximation, the pump envelope, A p (z,t), propagates freely through the fiber, being only a function of the combined variable t-z/V g .We assume a three-section pump pulse of the form (Fig. 1(b)): where the dimensionless α is real without loss of generality, while β and γ can be complex to take into account step phase changes.Here  Assuming infinite rise and fall times, and CW input signal ( 0 S A = constant), Eqs.(4)(a-b) were solved for the pulse of Eq. ( 5) using Laplace transforms, and the general analytic solution is detailed in Appendix A.
While very useful, the general solution, Eq. (A12), does not easily lend itself to intuitive physical interpretations.To gain insight into the nature of the solution to Eq. ( 4), let's consider a very short homogenous (constant B  and B  ) fiber segment, extending from position z 0 to z 0 + Δz, being much shorter than both the distance over which light propagates during the pump pulse length T and the acoustic time τ A .The rest of the fiber is assumed to be inactive, Brillouin-wise, so that: and Eq.(A12) reduces to Eq. (A15).Further simplification can be achieved if one is interested only in those cases for which the  pulse of Fig. 1(b) is long enough to fully activate the acoustic field ( The solution, then, is given by Eq. (A16), repeated here: , and represents the pump intensity.
, and then exponentially approaching a constant value of *  .Note that for α = 0, β = 0 and γ = 0, all three terms vanish..In Fig. 2, we illustrate the Brillouin gain,  , as predicted by Eq. ( 6) for four different pump pulse encodings of the same source CW power, representing the most used distributed Brillouin sensor configurations.In all cases, we only consider the response at the Brillouin resonance ( and a pump  pulse duration of 5ns (within the approximations leading to Eq. ( 6), the actual length of the short segment only appears in the scale factor preceding the {} brackets).In the case of a classical intensity pulse ( 0     , β = 1), shown in Fig. 2(a), only the second term in Eq. ( 6) is of importance.In the absence of the  pulse, the inertial behavior of the acoustic field requires a duration of 3 A   for the pump and signal to fully build the density grating, responsible for the Brillouin interaction.Thus, for short pulses, T<τ A , high spatial resolution is obtained, but at the expense of weak Brillouin gains caused by the inertial behavior of the acoustic wave.After the pulse passage at ns 25 0   t t , the pump is turned off and the interaction stops.When the acoustic wave is pre-activated, Fig. 2(b-d), the changes in the pump level, brought about by the  pulse, are instantaneously translated by the already fully developed acoustic field into abrupt Brillouin gain adjustments, which are of key importance for high spatial resolution distributed measurements.Note, however, that the presence of the  pulse, along with its new pump level an adjustment of the acoustic field, resulting in a rising Brillouin gain for the bright pulse, Fig. 2(b), but a decaying gain for the πphase pulse, Fig. 2(d) (the acoustic field also decays for the dark pulse but since 0 


, there is no Brillouin interaction during the  pulse, Fig. 2(c)).In all these three cases, at the end of the  pulse, the pump level resumes its previous value (    ), but due to the preceding influence of the  pulse, the incoming  pulse meets an acoustic field, not optimized for the current magnitudes of the pump and signal.It takes a few A  's for the acoustic field and the associated Brillouin gain to exponentially restore their steady state values.The relatively slow changes, experienced by the Brillouin gain during and immediately following the  pulse, while becoming negligibly small for very short  pulses, may compromise the otherwise high spatial resolution made possible by the pre-activation of the acoustic field.
It should be also noted that among the three pre-activation techniques described in Fig. 2 the πphase shift one is by far the most efficient coding (for a given CW pump source power prior to pulse carving).This conclusion is entirely supported by experimental observations [21].
In spite of the limited direct usability of Eqs.(A15-16) ( must be two orders of magnitude smaller than both A  and T ), these equations are still very useful.In the Appendix we show how the Brillouin signal from a real fiber, when viewed as a concatenation of many very short segments, and subject to our approximations, is the sum of the individual contributions of the many short segments, as computed from Eqs. (A15) or (A16).Also, in Sec.6, these equations are used in a new post-processing technique to alleviate the detrimental effects accompanying the long trailing response due to the slow changes of the acoustic wave.

A spectral study
Following the in-resonance, time-domain analysis of Sec. 3, Fig. 2, we now use in Eq. ( 3) the general solution of Eq. (A12), applicable to a segment of arbitrary length, to study the Brillouin spectral response for the different pump pulse schemes of Fig. 2. A uniform fiber length of 1m was assumed, having a fixed Brillouin resonance frequency of 11 GHz, and a linewidth of 27 MHz ( ) 2 /( 1 ), representing standard values for a pump wavelength of 1550 nm.0 t was again assumed to be much longer than the acoustic lifetime, and the pump  pulse width was set to T = 1ns.The time scale was arbitrarily chosen so that the Brillouin amplified signal first appears at the fiber entrance (z = 0) at t = 10ns.We plot in Fig. 3(a) the signal spectrum for the classical intensity pulse configuration (α = γ = 0 and β = 1) for a frequency range of 2 GHz in steps of 1 MHz, while the red line in Fig. 3(b) represents a frequency cut through Fig. 3(a) at t = 11ns, i.e., right after the whole pulse entered the segment (as observed at the fiber entrance).We clearly observe the classical response of a widely broadened spectral profile for short pump intensity pulses, offering very poor spectral (i.e., strain and/or temperature) resolution.Pre-activation of the acoustic field indeed significantly suppresses this broadening, as seen in Fig. 3(b): for low backgrounds (α = γ = 0.1) a non-zero pedestal continues to accompany the growing peak; for large backgrounds (α = γ = 0.99) the pedestal almost disappears, at the expense of the gain contrast, and optimum performance can be analytically proved to be obtained for α = γ = 0.5β.Note that for all preactivated cases discussed below, results are shown after the background signal, introduced by the nonzero α or γ, has been removed by the subtraction of the time-average of the signal at each frequency point.In the dark pulse configuration (α = γ = 1 and β = 0) [5], the pump is turned off for a time interval T, so that neither acoustic wave generation nor Brillouin amplification take place in the fiber section covered by the β pulse and the gain drops.For a short enough dark pulse, the slightly decaying acoustic wave keeps most of its inertial vibrations, and Brillouin amplification is immediately restored when the pump light is turned on again (γ = 1).In Fig. 4(a), the general analytic solution, Eq. (A12), is used to calculate the 3D signal amplitude for a dark pump pulse of 1 ns on a 1m fiber length.We clearly see the two sharp amplitude changes at t = 10 ns and t = 20 ns corresponding to a 1 m long fiber section.The narrowness Brillouin linewidth of the spectrum is clearly observed.Note that due to the analytic nature of the solution, this diagram is obtained almost instantaneously, without the need to numerically integrate over the history of the propagating pulse.
Several time-domain plots of the Brillouin gain appear in Fig. 4(b) for different values of Ω B Ω. Once the dark (β) pulse enters the fiber, the Brillouin gain drops from its value, set by the preceding α pulse, with a fall time equal to the pulse width.Unlike the case of a fiber segment much shorter than the dark pulse, Fig. 2(c), where the gain drops to zero, here the 1ns dark pulse covers only a small fraction of the 1m segment, so the rest of the segment, still under the influence of the α pulse, continues to provide gain.As the dark pulse continues its travel through the segment, the gain continues to decrease, in spite of the fact that the dark pulse is followed by the γ pulse.This decrease is due to the fact that the incoming γ pulse meets a weaker acoustic wave, that turns slightly decayed from its original value during the passage of the dark pulse.The same argument explains the presence of the second long trailing amplitude recovery following the exit of the dark pulse [20].
and the effect of this imaginary part on Eq. ( 4) and on its analytic solution Eq. (A12).Since the homogeneous solution of Eq. ( 4)b) has now a natural frequency at B   , the sudden absence of acoustic generation during the dark pulse gives rise to an oscillating gain component at exactly B   , as shown in Fig. 4(b), which decays with a time constant on the order of τ A , and therefore, no oscillations appear on the α and γ steady-state pedestals.
In Fig. 5 we plot the general analytic solution, Eq. (A12) in (3), for a 1 ns π-phase pulse modulating a CW pump wave (α = γ = 1, β = 1).The fiber segment is again 1 m long.The resulting Brillouin gain distribution is very similar to the dark pulse configuration for the same conditions as shown in Fig. 4(a), but it is twice stronger, as shown in Fig. 5(b), offering a 3 dB dynamic range improvement.

Modeling experimental results for the π-phase shift technique
Experimental results, based on Brillouin echoes, were obtained using a simple modification of a classical pump and probe setup [20].The sensing fiber consisted of two different fibers with different core doping concentrations to create a step difference in Brillouin frequency.A 50 cm section of a G652A fiber (Fiber 2: Brillouin shift 10.87 GHz) was spliced in the middle of a 20 m G652D fiber (Fiber 1 and Fiber 3: Brillouin shift 10.73 GHz). Figure 6(a) shows the 3-D experimentally obtained distribution of the Brillouin gain, where the short 50 cm section is clearly observed (time was converted to distance using: Distance / 2 g V t  ) .The calculated Brillouin gain for the three concatenated fiber sections, as obtained from the full analytic solution, Eqs.(A10), (A12), appears in Fig. 6(b).Since the experimental acquisition involved high-pass filtering of the Brillouin signal to remove the large DC background, the simulation results were similarly processed by subtracting the time average from each distance-domain cut in the 2D plot of Fig. 6(b).Very good comparison is observed against the real measurements, Fig. 6(a).A less favorable feature in both Figs.6(a) and (b) is the appearance of stray gain in the 50cm section at 10.73 GHz.To better understand the source of this stray gain, which may be detrimental for the frequency resolution of the Brillouin distributed sensor, we show in Fig. 7 a distance-domain plot of the Brillouin gain at 10.73 GHz, as obtained from the full analytic solution (average was not subtracted).
The trace starts towards the end of the α pulse, the observed gain (235.8 a.u.) has been accumulated by the probe during its journey through the two 10m sections, which are both covered by the long α pulse.The 50 cm section, being very short and off-resonance, contributes very little gain at 10.73 GHz.Being of reverse polarity, the entering π phase shift pulse immediately reduces the gain; see Drop I in Fig. 7.But it also slightly disrupts the acoustic wave, readjusting its phase.Thus, the γ pulse, which follows the π phase shift pulse, meets a suboptimal acoustic wave, and initially provides lower Brillouin amplification than that provided by the previous α pulse, resulting in further dropping of the accumulated gain, Drop II in the figure.This gain decrease continues until the acoustic field fully recovers to optimally match the γ pulse (on the order of a few τ A ).For a long enough fiber section and a short enough pulse, the gain then reaches a steady state, having negative contributions from the region covered by the π phase shift pulse (~V g T long), as well as from a region (a few V g τ A long) of suboptimal acoustic wave, and a positive contribution from the rest of the fiber (including the last 10m section), where the γ or α pulses are fully effective.As the π phase shift pulse clears Fiber1 (the first 10m) and enters the off-resonance 50 cm section, its negative effect on the gain of Fiber1 at 10.73 GHz disappears and we witness an immediate increase, Increase I, similar in size to Drop I.If Fiber2 section were considerably longer than 50 cm, the gain would have increased to its maximum value (235.8 a.u.).In our case, however, it takes only 5 ns for the π phase shift pulse to leave the short 50 cm section and to enter the second 10 m section (Fiber3), allowing for only a small Increase II, and introducing Drop III.While similar in nature to Drop II, the final Drop IV reveals some oscillations, reflecting the small contribution of the off-resonance 50 cm section.The departure of the π phase shift pulse from the second 10m section is accompanied by the fast Increase III and slower Increase IV, which are the reverse analogs of Drop I and II.
Figure 8(a) shows (after inversion and average subtraction) the experimentally obtained distance dependence of the signal gain at 10.73 GHz (blue line), fully corroborating the above discussion.The experimentally obtained frequency dependence of the Brillouin signal in the middle of the 50 cm section is described by the blue line of Fig. 8(b).Instead of having a single peak at 10.87 GHz, an even stronger unwanted peak appears at 10.73 GHz due to the trailing slow recovery of the acoustic wave in Fiber1.A successful solution to this problem has been demonstrated using a differential pulsed pump scheme approach [6,20,25].In the next section we present a post-processing alternative.

Post processing the measured signal data
The full knowledge of the time domain response of a given distributed sensor configuration based on Brillouin echoes (BEDS) is decisively helpful to deconvolve the interfering effects, Fig. 7, that may seriously screen the real Brillouin response of the system.In Eq. ( 6), we clearly see that the trailing decay after each phase change is due to exponential terms related to the slow inertial acoustic response.As a consequence, we can extract the information on experimental results if we assume that data is the convolution between impulse response and real data, Eq. (A17), thereby the filter function simply processes the data according to the ztransform.No frequency detuning is considered and the filter response for π-phase shift configuration (α = γ = 1, β = 1) simply becomes: , Figure 9(a) represents the 3D graph of the processed data in time and frequency domain shown in Fig. 6(a); the edges of the different fiber sections are now much more visible, which is very helpful for high spatial resolution sensor.The green line in Fig. 8 represents the processed signal gain at 10.73 GHz (a) and the gain spectrum at 10.5 m (b).The study in the time domain clearly shows that unwanted contributions are much suppressed.For example, no residual gain of Fiber1 is observed at the location of Fiber2.Consequently, the section of Fiber2 is more visible in the 3D plot (Fig. 9(a)) and the small Fiber2 section (50 cm) is correctly measured as shown in Fig. 9(b).Elsewhere [20], we have demonstrated an experimental configuration using a double pulse of the pump wave to suppress the unwanted contributions and a 5 cm spatial resolution over a 5 km fiber length has been demonstrated.

Conclusions
We have presented an analytic model for the three coupled stimulated Brillouin scattering equations in a distributed fiber sensor configuration that accurately describes the evolution of the acoustic and signal waves in both the time and frequency domains.We have obtained a perturbation-based general solution for the Brillouin response of a concatenation of uniform fiber sections for all pump pulse coding waveforms, currently reported in the literature.Since the model is not based on numerical integration and gives an explicit analytic function for the sensor response, it is a very powerful tool to understand the details of the physical response of each component of the pump waveform, and thus, to optimize the sensor configuration and the data processing.The general model was validated by comparing its results with experimental data, where a very good agreement has been observed.A deconvolution-based signal processing technique was successfully applied to real data to suppress all unwanted contributions to the signal in a Brillouin echoes configuration, where the pump wave is modulated by a π-phase shift pulse.This approach can be straightforwardly applied to any other pulse coding scheme fitted by our model.We expect that the understanding and optimization, brought by this model, will help the development of Brillouin echo-based systems for future practical high spatial resolution distributed sensing systems.

Appendix
The analytic solution of Eqs.4(a-b) is derived here, starting with the acoustic field.First, Laplace transformation (over time) is applied to both Eq.(4b) and Eq. ( 5) to obtain: where the "~" hat denotes the Laplace transform, and s is the Laplace variable.Substituting Eq. (A2) in (A1), and performing the inverse Laplace transformation, the time-domain solution for the acoustic amplitude becomes: Next, in order to Laplace transform Eq. (4a), its right hand side must be handled first. where: Assuming ( , 0) 0 s a z t   , the Laplace transform of Eq. ( 4a) is (LT stands for Laplace Transform): But from the form of Eq. (A4) and the properties of the Laplace transform it follows that Eq. (A6) can be written as: where , the solution of Eq. ( A6) is readily obtained as: , ( ) Hs , ( )e x p ( ) .
Integration is performed in the direction of the Brillouin signal from: ( )H s, ( ) exp( ) .
For a known distribution of ) (z g and ( ) A z  along a fiber segment ] [ The uniform section solution Assume now a Brillouin-homogeneous fiber segment of length z  , occupying the range 0 0 [ ,( )] z z z   , while the rest of the fiber is assumed to be Brillouin in-active, so that . The integration over  in Eq. (A8) is carried out only from 0 ( ) z z   to 0 z to obtain: , and performing the laborious but straightforward inverse Laplace transformation, we find the following expression for the Brillouin signal from a uniform fiber segment: where: .
The short uniform section solution A simpler, yet very useful, expression can be obtained for the special case of a very short segment.A close analysis of Eq. (A4) and the right-hand side of Eq. (A8), reveals that the Laplace transform of the latter contains information about the physical parameters of the problem, A  and the pulse width T , primarily for s values that are not much larger than A  and T / 1 .Thus, if z  is a very small fraction of both g V T and With both Eqs.(A12) and (A15-A16) at hand, we found that for Eqs.(A15-A16) to provide good approximations to Eq. (A12), z  must be small enough so that is two orders of magnitude smaller than both A  and T .

The convolutional solution
Let us divide an arbitrary long fiber segment, extending from [ ] z z can be expressed using Eqs.(A10) and (A15) as:


Heaviside unit step function.The first section of the pulse of height -high, T-wide middle section, ending with an infinitely long section of height 0 P A  .Most if not all current modulation techniques to achieve high spatial resolution can be described by the compound pulse of Fig. 1(b).

Fig. 1 .
Fig. 1.(a) Brillouin amplifier configuration in a fiber segment of length Δz.(b) Pump coding waveform where α is purely real and β,γ are complex.The pump pulse duration T determines the spatial resolution.
Brillouin-induced contribution of a very short segment z  , as observed at time , each corresponding to a different component of the pump pulse of Fig. 1(b).First a constant gain is observed prior to the appearance of the  pulse, represented by the factor  2 .Then, at with the  pulse appears at 0  z , with the onset of the second term, manifesting itself as an abrupt signal change to a value of )the  pulse duration T. Following the end of the  pulse, the second term stops contributing and the third term takes over, exhibiting an immediate jump to a magnitude of

Fig. 4 .
Fig. 4. (a) 3D gain signal diagram in a dark pulse configuration (α = γ = 1, β = 0) as calculated from the full analytic solution of Eq. (A12) for an interacting fiber segment Δz = 1m and T = 1ns.The pulse enters the fiber at t = 10ns.The background Brillouin signal, introduced by the nonzero α or γ has been removed in this 3D diagram by subtraction and the polarity of the spectrum was inverted for clarity.(b) Time-domain plots of the Brillouin gain for different frequency detunings, Eq. (A12) (Here, the background Brillouin signal, mentioned above was not removed and the true polarity of the signal is shown).The observed oscillation frequencies are equal to the detuning values.When the acoustic wave is pre-activated, off-resonance ( B S P        ) oscillations are observed in Fig. 4 [5,26,27].They originate from the non-zero imaginary part of A  , Eq. (2):

Fig. 6 .
Fig. 6.(a) Experimental gain distribution with 0.5ns πphase pump pulse.(b) Calculated gain distribution for the three concatentated fiber segments used in the experiment, as obtained from the full analytic solution in Eqs.(A10), (A12).

Fig. 7 .
Fig. 7.A distance-domain plot of the Brillouin gain at 10.73GHz, as obtained from the full analytic solution (average was not subtracted), see text for details.

Fig. 8 .
Fig. 8. (a) Experimental signal gain at 10.73 GHz and (b) signal spectrum at 10.7m.The blue and green lines correspond to acquired and processed data (Sec.6), respectively.The scale is linear.

Fig. 9 .
Fig. 9. (a) Experimental gain distribution after filtering by the impulse response of Eq. (8).The scale is linear.(b) Effect of filtering on the frequency distribution along the fiber.The green and blue line corresponds to the original measured data and processed data, respectively.
Inverse Laplace Transform (ILT), we finally get: we can assume that all