## Abstract

We present a comprehensive analysis of the technique of Longitudinal-Mode-Filling (LMF) to reduce Stimulated Brillouin Scattering (SBS) limitations in Ytterbium Doped Fibre Amplifiers (YDFA), for the generation of nanosecond, temporally shaped pulses. A basic Master-Oscillator-Power-Amplifier (MOPA) system, comprising an output YDFA with 10µm-core active fibre, is experienced for benchmarking purposes. Input pulse-shaping is operated thanks to direct current modulation in highly multimode laser-diode seeds, either based on the use of Distributed Feed-Back (DFB) or of a Fibre Bragg Grating (FBG). These seeds enable wavelength control. We verify the effectiveness of the combination of LMF, with appropriate mode spacing, in combination with natural chirp effects from the seed to control the SBS threshold in a broad range of output energies, from a few to some tens of µJ. These variations are discussed versus all the parameters of the laser system. In accordance with the proposal of a couple of basic principles and with the addition of gain saturation effects along the active fibre, we develop a full-vectorial numerical model. Fine fits between experimental results and theoretical expectations are demonstrated. The only limitation of the technique arises from broadband beating noise, which is analysed thanks to a simplified, but fully representative description to discuss the signal-to-noise ratio of the amplified pulses. This provides efficient tools for application to the design of robust and cost-effective MOPAs, aiming to the generation of finely shaped and energetic nanosecond pulses without the need for any additional electro-optics.

© 2014 Optical Society of America

## Corrections

24 September 2019: A typographical correction was made to the author listing.

## 1. Introduction

The well-known phenomena of Stimulated-Brillouin-Scattering (*SBS*) takes place among the most deleterious limitations for the production of high energies [1–4] using all-fiber Master-Oscillator-Power-Amplifiers (*MOPA*). They occur in the backward direction of propagation, thus leading to power transmission issues, severe pulse distortions and possibly catastrophic damage effects in the optical components. The onset of *SBS* is characterized by the so-called *SBS* threshold (*P _{th_SBS}*), which is defined in terms of propagating laser intensity along a given length of fiber. Plus or minus efficient techniques may be used to cancel out or reduce

*SBS*effects, using Large-Mode-Area fibers. A very common option is based on the use of active phase-modulation [5,6], to be selected for single-frequency operation [2,3]. Other options [7–10] either consist of specific fiber core-clad designs or of the application of thermo-mechanical constraints by external means, when fibers are long enough. Fine in-line filtering [11] may also be chosen thanks to selective Fiber-Bragg-Gratings (

*FBG*). The option of phase-modulation is the most usual solution, but at the expense of the addition of bulky and electricity-hungry electro-optics devices.

In this paper we present another option of potential interest, which is based on the combination of properly controlled Longitudinal-Mode-Filling (*LMF*) with natural chirp from the seed of a *MOPA*. This ensures multimode seed injection, aiming to improved robustness and cost-efficient opportunities for system integration. A number of critical issues can be avoided this way, regarding the pulse-shaping performance and the reliability of the laser system which are inherent to phase-modulation techniques. In particular we can get free from Frequency-Modulation to Amplitude-Modulation transfer [4], modulation-failure or setup time delays. Among the huge number of already published studies on *SBS*, nowadays, only a few ones concern specific works dedicated to bandwidth-limited configurations in connection with broadband spectral behavior [12–14]. Furthermore, as far as we know, none of these works refers to the comprehensive analysis of *LMF* as an efficient process to cancel out *SBS* in the presence of controlled chirp from the seed. We investigate the technique thanks to operating semiconductor cavities in the regime of direct current modulation, in connection with arbitrary pulse-shaping. In this paper we consider time-shaped, 10 to 100ns long pulses at the wavelength *λ _{o}* = 1064nm, to be amplified in a

*MOPA*based on the use of Ytterbium-Doped-Fiber-Amplifiers (

*YDFA*) with saturated gain. The anticipation of gain-saturation along the active fiber in the final

*YDFA*takes part of our major concerns, due to the need of efficient output energy extraction. By comparison with single longitudinal-mode seeding techniques,

*LMF*implies unavoidable broadband temporal modulations. This results from complex beating phenomena and mode competition effects. Modulations occur in a broad band of frequencies, which depends on the spacing between adjacent longitudinal modes and on the number of modes contained in the spectrum of the seed. This is the reason why properly controlled

*LMF*also requires the management of a basic tradeoff, between the enhancement of threshold for the onset of

*SBS*and the specification of a given measurement bandwidth (

*MB*), for efficient output pulse-shaping. The selected pulse durations (

*T*) imply values of the

_{p}*MB*in the range of a few GHz up to some tens of GHz. This will be discussed. We study in detail the strategy to choosing appropriate multimode Laser-Diode (

*LD*) seeds, either externally coupled to a

*FBG*reflector or based on the use of a broadband Distributed-Feed-Back (

*DFB*) design, and optimizing their operating conditions.

The article is organized as follows. In the first section we characterize the *FBG* and *DFB* seeds to identify the basic spectral features of interest. In the second section we focus on the chirp phenomena, to be discriminated from pure *LMF* for the management of *SBS* limitations in the presence of input pulse-shaping with direct current modulation. Thirdly, we present a comprehensive numerical model to discriminate the relative impacts of *LMF* and of chirp on the effective Brillouin gain and thus on the onset of *SBS*. In a fourth step, we test the robustness of this model by comparing simulation results with several series of experimental data. Benchmarking measurements are made in the configuration of a two-stages *MOPA* configuration, comprising two *YDFA*s with 6µm and 10µm-Mode-Field-Diameters (*MFD*). In the final step, we develop a fully numerical process dedicated to the calculation of the Optical Signal-To-Noise-Ratio (*OSNR*) in the presence of multimodal beating noise effects, as a function of the characteristics of *DBF* and *FBG* seeds. This will help to optimize seed options regarding the above tradeoff between the *SBS* limitations and beating noise, to ensure the generation of smooth pulse profiles within a given *MB.*

## 2. Multimode laser-diode seeds

Prior starting the design of seeds, it is worth to specify the incoming experimental conditions. Our setup is based on the implementation of a standard scheme with two amplifier stages (Fig. 1). This optical design will enable appropriate benchmarking a simple way, thanks to fits with theoretical predictions, in in terms of *SBS* limitations coupled to in-line saturated amplification. All fibers are polarization maintaining. The first *YDFA* stage is operated in the regime of nearly linear gain, just aiming to produce the suitable value of input power for driving the second stage up to saturation, at a Pulse Repetition Frequency (*PRF*) of 10kHz. It enables a fine control of the constant value of the average power to be delivered downstream, up to 2mW for 10 to 100ns long input pulses.

The second *YDFA* consists of the stage of prime interest for benchmarking. It comprises a 1m long, 10µm-*MFD* active fiber. This was the commercial fiber with the larger *MFD*, among available products with a bulky core, to enable easy splicing with very short connections. Despite lower power handling than photonic-clad structures, with larger *MFD*, this helps to make sure of no risk of spurious *SBS* parasitic effect, possibly localized along intermediate interconnections. This *YDFA* is pumped in such a way that it is capable to deliver pulses with energies from 5 to nearly 100µJ and durations *T _{p}* = 10-100ns. A bi-directional tap is placed between the two amplifiers, to enable the monitoring of backward

*SBS*from the output of the

*MOPA.*Most applications require a precise definition of the peak output wavelength (

*λ*) and of the laser bandwidth, either to be defined in terms of wavelengths (

_{o}*Δλ*) or of frequency (

_{signal}*Δν*) versus the reference of interest. This is the reason why frequency-selective techniques must be used [15], when starting from the naturally broadband Spectral Density of Power (

_{signal}*SPD*) in

*LD*-chips (Fig. 2). The values of

*Δλ*and

_{signal}*λ*may be adjusted as required in a broad range of values [15–17] whatever the selection of a

_{o}*DFB*or of an

*FBG*. The former

*DFB*architecture is based on the implementation of an internal grating [18,19] on the chip itself (Fig. 2(a)). The latter

*FBG*involves the coupling of an external grating, which is operated as a frequency-selective reflector to be positioned at an adjustable distance (

*d*) from the chip. At the opposite of the

*DFB*, this enables the construction of long optical cavities, giving access to a very large number of longitudinal modes whatever the selected value of

*Δλ*.

_{signal}The longitudinal mode spacing [16] refers to the Free Spectral Range *(Δν _{FSR}*) of the cavity. Given

*n*the refractive index of silica and the basic relationship$\Delta {\nu}_{FSR}=\frac{c}{2{n}_{o}d}$, typical values for

_{o}*FBG*designs with

*d*= 10-30cm lead to

*Δν*= 0.3-1GHz. Then the

_{FSR}*Δν*may be lowered as required, simply increasing

_{FSR}*d*, to take advantage of easily upgradable cavities for highly efficient

*LMF*. Due to geometrical considerations, the

*FSR*of

*DFB*s is in tens of GHz. In order to meet the generic needs of laser amplification we will make use of values of

*Δλ*in the range of some fractions of nm, typically 0.15 to 0.5nm. In view of a comprehensive study of

_{signal}*LMF*, we need to vary the number of seeding modes (

*M*) nearly continuously, from a just few units up to several tens.

Commercially available *DFB* seeds (Oclaro^{TM}) at λ_{o} = 1064 nm, which can be referenced as broadband designs, and standard *LD*-chips (3S-Photonics^{TM}) coupled to short lengths of polarization maintaining fiber and external *FBG*s will be experienced. Our grating of reference is narrowband, with a peak reflectivity of about 15% and a transmission bandwidth *Δλ _{signal}* = 0.2nm. Test results are presented in Fig. 3, considering direct current modulation with flat-top pulses as representative operating conditions. As expected the optimization of the spectral contrast over the transmission bandwidth of the grating requires a careful adjustment of the temperature of the semiconductor chip, to ensure proper positioning of the peak of the fluorescence (Fig. 3(a)). Depending upon the value of the peak modulation current (

*I*), this enables values of the contrast in the range of 15-18dB. Second, as expected even though not shown in the figure, it must be noted that the lasing bandwidth and the location of the peak wavelength remain quite stable whatever the average and peak current values. This is not the case with the

_{peak}*DFB*(Fig. 3(b)). We rather obtain

*Δλ*= 0.2-0.5nm, which corresponds to

_{signal}*M*= 3-10 modes. A gradual shift of the central lasing wavelength is experienced, towards longer wavelength as

*I*is increased. This corresponds to the gradual activation of more and more longitudinal modes inside, from 1 to 2 near the threshold up to about 6-8. As shown the

_{peak}*FSR*is slightly less than 100pm, i.e. about ~30GHz in terms of frequency spacing. Furthermore, varying the values of

*I*and the average power (Fig. 3(c)) by a factor of 50 by increasing the

_{peak}*PRF*and the length of the driving pulse, it is worth to underline gradual and very significant broadening effects in the lasing bandwidth. Comparing the spectra in Fig. 3(b) and Fig. 3(c), where selected mode positions have been superimposed for the sake of visualization, gradual mode filling and non-adiabatic effects are clearly evidenced. The maximum number of longitudinal modes does not exceed

*M*= 6 for a peak current of up to

*I*= 1A with

_{peak}*PRF*= 10kHz and

*T*= 100ns. But

_{p}*M*nearly doubles in Fig. 3(c) when increasing the average power by a factor of 40 with

*PRF*= 100kHz and

*T*= 500ns, while modal characteristics still remain unchanged at low current values. This reveals that the value of spectral broadening with

_{p}*Δλ*= 500pm at the maximum value of

_{signal}*I*mainly originates from transient chirp, just containing a very small adiabatic contribution. The precision of these measurements remains limited by the minimum resolution of the spectrometer, about 20pm, but this provides representative orders of magnitude for values of the chirp rate to be used further. Under the same operating conditions, it is worth to notice that the

_{peak}*DFB*gives access to a larger spectral contrast than the

*FBG*, nearly 40dB up to the peak driving current. On the other step, the definition of seed conditions also implies the specification of relevant input pulse-shapes (Fig. 4).

Their implementation requires specific electronic means for full Arbitrary Waveform Generation (*AWG*). Here we only consider three characteristic pulse-shapes, in order to take advantage of significant variations of the chirp. This selection comprises 10ns-50ns long, flat-top pulses, 25-50ns long ramps and exponential pulse-shapes, to recover all the suitable operating conditions required for the analysis of the phenomena involved. Our purpose is a comprehensive validation of the effectiveness of the combination of *LMF* and chirp for total or partial cancellation of SBS limitations in the 10µm-*MFD* fiber. This gives access to various regimes of total and transient chirp. This also aims to account for gain saturation effects along the active fiber of the *YDFA*. Ramp-like and exponential shapes then enable upstream compensation, in order to minimize the discrepancy of the amplified pulse-shape with a super-gaussian. We will demonstrate that this takes part of the optimization process to minimize global *SBS* effects. But the noise contributions of the photo-detector and of the real-time oscilloscope, and despite the presence of unavoidable beating noise to be discussed further, these seed envelopes look rather smooth. The *MB* is DC-3GHz, which is consistent with the management of optical transition times down to about 150ps, if required for shaping. Our setup is based on the coupling of a standard *AWG* to a fast current driver (OPM^{,} inc.), thanks to cost effective and easy to use driving electronics.

It is now time to analyze chirp in more details, as a parameter of major concern in the management of *SBS*. This takes place in the well-known characteristics of semiconductors subjected to direct current modulation, in connection with pulse-shaping conditions.

## 3. Chirp effects

In the following we consider chirp effects [20–22] as a full contribution to the search of proper seed conditions to optimize the output performance, in combination with *LMF*. They must be discriminated, to determine an effective value of the Brillouin gain as a function of this combination. Starting from basic rules, this implies a three-step process:

- • define transient chirp due to pulse-shaping effects with propagation along an
*YDFA*of short fiber length. - • estimate the effective Brillouin gain, to be reduced under these conditions.
- • integrate the total chirp effects, as the sum of transient and adiabatic chirp contributions, for the sake of relevant calculations versus global seed conditions.

Transient chirp effects originate from current transients, while adiabatic chirp depends upon actual thermal loading due to the average operating power. Former effects are due to the rapid changes in the refractive index of the semiconductor material induced by the variations of carrier density. This leads to a rapid shift of the instantaneous frequency deviation and results in a noticeable broadening of the integrated spectral bandwidth. The amount of shift is governed by pulse-shaping conditions and by the peak driving current. Denoting *P _{in}(t)* the input power from the seed, we can refer to single-mode operation (

*M*= 1) and make use of [14]:

*Δν*, $\raisebox{1ex}{$d{P}_{in}(t)$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.$,

_{L}(t)*α*and

*κ*figure the instantaneous chirp (Hz), the temporal gradients of input power, the non-adiabatic transient chirp coefficient (dimensionless) and the adiabatic chirp (Hz/W) coefficient, respectively. The determination of

*α*and of

*κ*allows discriminating between two very different phenomena. For application to

*LMF*, proportional variations of

*α*and

*κ*may be assumed with

*M*. Respective numbers from previous works [22,23], of concern with common

*LD*-chips, lie in the typical ranges of

*α*= 1 to 3 and

*κ*= 5.10

^{12}to 10

^{13}Hz/W. Normalizing the total chirp (

*Δν*) to the length of the pulse, a huge variety of chirp rate coefficients (

_{tot}*V*, Hz/sec) may be experienced. Typical numbers range from some 10

_{ch}^{12}to more than 10

^{17}Hz/sec. Because they may vary a lot versus the operating regime and versus the nature of the

*LD*-chip, the actual values must be verified more precisely prior application to

*SBS*modelling. The integrated value

*Δν*refers to an average value of spectral broadening, mode per mode in the

_{tot}*SPD,*as a function of the modulation current. To appreciate the relative contribution of current transients using the results from Fig. 3, it is possible to extract an approximate value of

*Δν*by using a three-step Gaussian-fitting process. The local intensity

_{tot}*I*(W/Hz

_{tot _J}^{1/2}) is approximated by means of a weighted sum of Gaussian-like components, as a function of the maximum driving current (

*J*, A), in the following form:

*λ*is the central wavelength of the “i” mode,

_{i}*M(J)*and

*Δλ*figure the variable number of contributing modes and the spectral width of each Gaussian component versus the actual value of

_{i}(J)*J*, respectively. The fitting procedure starts with the determination of a first Gaussian fit of the single clearly resolved peak at low-current, J just above lasing threshold. This gives

*Δλ*Then the number of Gaussian curves used in the fit can be increased step by step to calculate the discrepancy,

_{1}(J).*ΔΙ(λ)*, between the measured

*SPD*and its initial approximation

*I*. The latter optimization helps discriminating pure chirp due to transient current over the variations of

_{tot_J}(λ)*M(J).*Finally, we adjust

*Δλ*to minimize the quantity

_{i}(J)*ΔΙ(λ)*over the whole bandwidth.

Fit results obtained this way indicate that the value of local *Δν _{tot}*, as defined mode per mode, varies up to about 40GHz when

*J*ranges from 0.2 to 1A, whatever the situation with 100 or 500ns long pulses. Despite the resolution limit of 20pm in our analyzer (6GHz in terms of frequency bandwidth), this also shows that

*Δν*does not undergo a linear increase with

_{tot}*J*. Spectral broadening effects exhibit rapid variations up to

*J*= 0.3 - 0.44A. But they stabilize gradually when reaching the current range of 0.8-0.9A. To conclude with experimental data which can make sense at this step, it is worth to keep in mind an average value of

*Δν*= 20GHz in the middle of the current dynamics range. For 100ns long pulses this corresponds to an average value of the chirp rate

_{tot}*V*= 2.10

_{ch}^{17}Hz/sec. It is worth noting that this elevated value still remains consistent with large modulation currents, as involved here, but that no confusion must be made with total spectral broadening effects due to pure

*LMF.*This refers to much larger values, typically up to

*Δν*= 400GHz with

_{tot}*Δν*= 30GHz (Fig. 3(c)) or even more. Equation (1) provides a simple tool to make use of or verify experimental data, in connection with the study of variable chirp rate effects versus current modulation and pulse-shaping conditions (Fig. 4).

_{FSR}Considering median orders of magnitude [22], *α* = 2.5 and *κ* = 6.10^{12} Hz/W and the different pulse-shapes in Fig. 4, we can verify the consistency of computed values of $\Delta {\nu}_{L}(t)$and $\Delta {\nu}_{tot}={\displaystyle \underset{0}{\overset{{T}_{pulse}}{\int}}\Delta {\nu}_{L}(t)}dt$with former experimental estimates of *V _{ch}*, and

*Δν*(Fig. 5). It is worth noting the strong variations of

_{tot}*Δν*with those of the local slope. For 50ns long pulses with about 300mW peak power,

_{tot}*V*ranges over 10

_{ch}^{14}Hz/s while

*Δν*increases from 80 to nearly 250GHz (Fig. 5(a)). Furthermore, 400ns long pulses may lead to

_{tot}*Δν*in excess of 400GHz (Fig. 5(b)).

_{tot}## 4. SBS modelling process with gain saturation

As mentioned earlier, we will focus on short length *YDFA*s. It is important to notice that in the case of such short length of fiber the pulse does not fit entirely in the temporal domain. The transit times (τ) for *L* = 1-3m range from 5 to 15ns, which determines the useful interaction length for computations. Thus, in order to discuss relevant values of *P _{th_SBS}*, we need to account for right time-space overlap between the pulse and the doped fiber. Just as a reminder and for definition of the notations, the basic description of

*SBS*in the usual single-frequency configuration where

*M*= 1 refers to the following set of parameters: the spectral distribution

*g*of the Brillouin gain, of which the peak value

_{B}(ν)*g*is spaced by

_{Bo}*Δυ*≈15GHz from the central frequency ${\nu}_{L}=c/{\lambda}_{L}$of the so-called pump wavelength, and the spectral bandwidth of

_{B}*g*,

_{B}(ν)*Γ*.

_{B}*Γ*is closed to 50MHz in most silica fibers. The single-frequency distribution

_{B}*g*is given in a Lorentzian form:

_{B}(ν)Accordingly, the Stokes frequency denoted *ν _{B}* is equal to${\nu}_{B}={\nu}_{L}-\Delta {\nu}_{B}$. The gain value of reference in silica fibers for

*M*= 1 is g

_{Bo}= 5.10

^{−11}m/W. Even though expression (3) assumes pure homogeneous spectral broadening over a broad range of frequencies, previous works [23] have already demonstrated negligible phenomena of frequency-coupling down to frequency gaps as low as some times the value of

*Γ*. This will be verified further for the implementation of

_{B}*LMF*. To account for the aforementioned pulse-fiber (or time-space) overlap, we must pay attention to the z-integral of

*g*$\underset{0}{\overset{L}{\int}}{g}_{b}(z).dz$. This involves the calculation of an integrated form of

_{B, }*g*over the length of the amplifier, as in [22] where White et al. studied the initiating conditions and the situation of ‘worst cases scenario’ for the onset of

_{B}*SBS*. This corresponds to the frequency offset between the resonance at the peak of

*SBS*and the pump frequency, which enables the largest value of the effective Brillouin gain along

*L*. Denoting

*v*the group velocity along the fiber of the

_{g}*YDFA*, it can be shown that:

The coefficient *γ* in Eq. (4) determines the actual ratio for the reduction of the efficiency of net Brillouin amplification with a short *L*, assuming a moderate longitudinal gradient of the pump intensity along. It is worth to notice that the value of *γ* may undergo huge variations in the range 0-1, depending upon the value of *V _{ch}*. The transition area from larger to lower values is located over

*V*= 5.10

_{ch}^{15}Hz/sec. This means that the determination of the chirp rate remains quite critical. Merging the effects of

*LMF*with those of chirp, it is possible to account for both of them at the same time thanks to an effective value (

*g*, m/W) of the Brillouin gain:

_{B_Eff}This expression takes into account all the operating conditions: namely the chirp rate effects from (1, 4) and related total chirp governed by the pulse duration involved, z-integral limitations and *LMF*. It will be shown later that satisfactory fits are always permitted using (5) with operating conditions as those involved here, for rather strong chirp rate effects. The corresponding values of the dimensionless constant *γ* then lie in the range of 2.10^{−2}-2 10^{−3}, for values of *V _{ch}* in excess of 10

^{17}Hz/sec. Equations (1)–(5) remain valid whatever the mode of operation, single-frequency or

*LMF*, as far as the mode spacing remains consistent with the limitations due to homogeneous broadening effects [24–26]. It is also assumed that the reduction of the Brillouin gain is similar for all the modes.

Let us go on with the modelling process. Seed features being determined in terms of *LMF* and chirp specifications as a result of the conditions for direct current modulation, it is now possible to focus on our main topic of interest. This concerns development of a representative numerical model, dedicated to the prediction of the *SBS* threshold and of the related location along the active fiber of the output *YDFA*. The source term which governs the onset of *SBS*, in the presence of non-zero thermo-mechanical noise, is the amplified signal (pump) wave. Furthermore, due to the generic need of efficient energy extraction, this model must take into account the saturation of the gain along the active fiber [27]. This implies plus or minus significant pulse distortions, which primarily recover similar space-time locations as those where *SBS* starts to initiate. But some satisfactory results in particular cases [26], earlier works of reference suffered from some lack of generality. This is the reason why our model aims to combine, at the same time, all of the most important degrees of freedom to size a *MOPA*. The suggested modelling process provides an all-vectorial description for the sake of widespread use, considering each longitudinal mode of the seed individually with its respective spectral density. The chirp, on the other hand, is assumed to be identical for each mode in a first order approximation. Under these assumptions we can write a set of basic rate equations in the presence of *M* contributing modes, in the form of a system of (1 + 2x*M*) coupled nonlinear equations, as follows:

*v*is the group velocity;

_{g}*A*is the effective mode area, as defined by the value of the

*MFD*;

*G*and

_{p}*α*are respectively the linear gain and the loss coefficients of the of the gain fiber;

*Q*is the Langevin noise source for the initiation of the process of

_{o}*SBS*. Typical values for the above parameters are as follows:

*v*≈2.10

_{g}^{8}m/s, in silica;

*A*= 80µm

^{2}, given 10µm-

*MFD,*and

*α*= 0.1 to 5cm

_{p}^{−1}depending on the nature of the fiber. The value of the unsaturated gain coefficient

*G*(cm

_{p}^{−1}) must be adjusted as necessary, in order to match the net value of total gain in the saturated

*YDFA*along the selected

*L*. As already said, we assume in Eq. (6) that all contributors which take place in the family of

*M*longitudinal modes from the seed remain spaced by a minimum interval so that fully inhomogeneous

*SBS*amplification may be considered. Previous works [28,29] have shown that this remains consistent with spacing intervals in the order of a couple of hundred MHz. Then, splitting the input seed

*SPD*into

*M*active modes also implies the need to deal with

*M*Stokes waves and

*M*acoustic waves. Global energy conservation can be verified in Eq. (6), just upgrading more usual single-mode procedures. The actual values of

*M*and of modal intensities are determined thanks to a fit of the seed

*SPD*with a series of

*M*weighted modes, as introduced in Fig. 3, giving the best match. Each pump mode is supposed to give rise to a single acoustic mode

*Q*. Each corresponding Stokes wave is then emitted as a single component.

_{i}The resolution of such a system is straightforward, but the computation time can be quite long. Large values of *M*, in the range of some tens, may necessitate long run times and appropriate memory depth. Anyway satisfactory results are obtained with regular computers for values of *M* up to 30. We stress that the size of this system can vary by a factor of 5 (and up to 10) depending on its application to *DFB* or *FBG* seeds. The sequence of calculation is based on the discretization of the active fiber into segments of equal length (Fig. 6). Typically up to 8 or 12 segments, depending on the saturation rate of the amplifier and on the precision requirements. This process helps to reveal the onset of *SBS* effects alongside the gradual deformation of the propagating pulse during complete amplification, up reaching *P _{th_SBS}* and slightly over in the last segment. Up to

*P*, the computation method is as follows: since the chirp has a fundamental impact of the growth of

_{th_SBS}*SBS*, a temporal chirp function is applied to the relevant temporal portion of the pulse prior to entering the next segment of fiber. In the meantime we also calculate the saturation input parameter for the next segment. We then use this input field to solve Eq. (6) for the next segment of fiber, in order to evaluate the

*SBS*across the space-time computational domain in the presence of relevant limit and boundary conditions. This loop is repeated until the end of the fiber or until

*P*is reached. Prior looking at the combination of gradual pulse amplification in connection to

_{th_SBS}*SBS*along the whole length of fiber, we can focus on the complete data related to

*SBS*interactions across the output section of fiber, using a more finely meshed space-time domain. This enables more insight into

*SBS*interactions near the threshold. The computational results in Fig. 7 depict space-time variations of the transmitted pump, of the Stokes, and of the acoustic waves, for the sake of illustration. The example plotted corresponds to a

*DFB*seed with

*M*= 6 modes, and for

*I*= 0.5A, leading to

_{DFB}*g*= 8.10

_{Beff}^{−12}m/W. Other parameters considered in Eqs. (1), (2), (4), and (5) are

*Δν*= 150GHz and ${\rho}_{pulse-shaping}\gamma $ = 2.5.10

_{tot}^{−3}for

*V*= 5.10

_{ch}^{17}Hz/sec.

Then we present in Fig. 8 and Fig. 9 global calculation results along the complete space-time domain, as well in the presence of very significant as of low gain saturation effects, respectively, for 80ns long flat-top pulses and for 50ns long exponential pulse-shapes as those depicted in Fig. 4. Given a constant value of the peak input power in both cases, the input energy of the flat-top pulse is 0.8µJ. But it reduces down to 0.1µJ for the exponential pulse. In both cases we keep *MFD* = 10µm, *M* = 6 and we consider a peak input power of 40W. The weighting law considered follows the corresponding spectral envelope, as presented in Fig. 3(b). We assume that the values of *Δν _{tot}* and of

*g*are 180GHz and 6-8pm/W, respectively, which will be shown to be consistent with further operating conditions. Saturation effects are accounted for using the standard model from Frantz and Nodvik [27], as part of the numerical process depicted in Fig. 6. This approximation remains acceptable at 1064nm. We make use of a saturation fluence of about 62J/cm

_{Beff}^{2}. Computations make use of appropriate values for the small-signal gain, to be consistent with experimental observations. A double convergence criterion is required. We need to select the appropriate peak power to enable a 12 to 15dB saturated gain, while taking advantage of optimized conditions to size value of

*L*in the

*YDFA*. This means that the position where

*P*is crossed remains located near the end of the fiber, in correspondence with the actual value of

_{th_SBS}*L*, about 1m. Figures 8(a) and 9(a) then illustrate the combination of

*SBS*effects with gradual, plus or minus significant deformation of the pulse-shape during the amplification process.

The distortions of the pulse are taken into account thanks to computing the local normalization ratio of the relative output power to the relative input power, and indexing the location involved along the length of fiber. This can be done thanks to preliminary calculations, out of any *SBS*, just assuming pure saturation as shown in Fig. 8(a) in the case of the flat-top pulse. The data for upcoming in-line correction, in the presence of *SBS*, are determined in each of the selected fiber sections, along *L*. Due to unavoidable numerical approximations in the correction of gradual distortions, space-time connections between adjacent segments of fiber remain un-perfect. This is can be seen in Fig. 8(b) and Fig. 9(a). Also note that the normalization ratio may increase up to more than 60% for the flat-top pulse, depending upon the position along the fiber, but that it remains below 10% for the exponential pulse (not shown in Fig. 9). The results presented seem to show that the attainable output energy in the case of exponential pulses, here less than 3µJ for 50ns pulse duration, is much lower than the attainable limit with flat-top seeding. This will be confirmed experimentally. Modelling capabilities then give access to the analysis of relevant trends, to predict the variations of the value of *P _{th_SBS}* versus the architecture of the

*MOPA*and to determine the most suited seeding conditions. This will be useful for optimization. More especially, ranging

*Δν*and

_{tot}*M*independently in a broad range of values a consistent way with the value of

*γ*provided by Eq. (4) as a function of

*V*, we can put numbers of their relative contributions in the enhancement of

_{ch}*P*. This helps to verify that in most of the actual situations,

_{th_SBS}*LMF*always remains the major contributor, but that the contribution of chirp alone to the reduction of

*SBS*may reach 30%.

## 5. Experimental validation of theoretical predictions for SBS

The model is validated over three phases:

- • The analysis of typical optical pulse-shapes near
*P*and slightly above, thanks to real time monitoring of the transmitted pump and the Stokes waves in the forward and backward directions. This implies a fine control of the pump and of the seed power, together with a statistical analysis due to the spike-like nature of_{th_SBS}*SBS*. Our purpose here is to verify the precision of our numerical calculations, which indicate rather narrow (≈10ns) Stokes pulse-envelopes. - • The discrimination of transient chirp effects, as a complementary validation of the generic trends, versus pulse-shaping conditions and peak current in the seed. We intend here to highlight the chirp contribution to the overall
*SBS*reduction and also to crosscheck the predicted values of the total chirp*Δν*._{tot} - • The comparison of the variations of
*P*versus seeding conditions, for a flat-top seed profile of reference with variable pulse duration. This will provide the necessary data required for benchmarking and for optimizing the design of the_{th_SBS}*MOPA*.

Figure 10 shows real time measurements of the transmitted pump and of the backward Stokes pulse envelopes in the case of large values of *M*, typically from 10 to 20, with strong pump depletion. More specifically, Fig. 10(a) describes the raw Stokes pulses corresponding to a series of transmitted pulses down the *YDFA* when varying the input pulse energy up to nearly 50% depletion of the pump wave. Figure 10(b) is the same, but normalization with respect to the respective peak powers to get free from gain effects, and help for visualization. The gradual increase of the input energy leads to a progressive pump depletion due to the re-amplified Stokes wave. These measurements illustrate the characteristic depletion behavior of *FBG*s with a rather short value of Δν_{FSR}. The total amount of energy transferred from the pump towards the Stokes wave depends on the size of the temporal portion of the pump which gets depleted, always starting from the trailing edge of the incoming pulse. This must be reminded in connection with former data. These observations are consistent with the theoretical predictions illustrated in Fig. 6 where *LMF* dominates the chirp effects. Further normalization of the intensity distribution of each Stokes spike and of the corresponding depleted pump waves helps gaining more insight into the sequence of energy transfers caused by the onset of *SBS* in a broad range of Stokes energies. We stress that proper post-synchronization has been accounted for in Fig. 10, to compensate for the propagation delays along the various fiber segments between for the forward and backward pulses, has been ensured. We move on, then, to the analysis of transient chirp in the presence of lower values of *M,* and more specifically of its influence on the temporal location along the pump pulse of the onset point of *SBS*. This can be made more easily with moderate values of *M*, i.e. using *DFB* seed.

Our three specific pulse-shapes are launched at the input of the 10µm-*MFD YDFA* (Fig. 11), to investigate the effect of very different values of the chirp rate (see Eq. (1) and Fig. 5) and of total chirp. In each situation the gain of the *YDFA* is gradually increased, up to the value where *SBS* just starts to initiate.

The corresponding, nearly un-depleted output pulses (solid lines) are superimposed with a series of related backward Stokes spikes (dotted lines). This helps to evidence significant variations of the onset point and of the corresponding value of *P _{th_SBS}*. At the time of appearance of the Stokes spikes we measure

*P*= 90-750-950W, respectively, for the 100ns long, slowly increasing exponential (solid black line) for the 100ns long (solid blue line) and then for the 10ns long flat-top (solid red line) flat-top pulses. These numbers correspond to very different values of the attainable output pulse energy,

_{th_SBS}*E*= 7µJ-13µJ-36µJ, respectively. Our interpretation of these results is as follows: the lower the chirp rate and the longer the delay for initiation along the duration of the pulse, which appear as typical characteristics of the exponential profile, the lowest value of

_{out}*P*. This also goes with lowered values of

_{th_SBS}*M*in the left area. The evidence of a reduced value for

*E*then comes from the combination of un-advantageous situations for

_{out}*LMF*and chirp in this particular case. This value is about four times lower than the one observed with a flat-top input of which the length is the same, i.e. 100ns long, while the corresponding

*P*is increased by a factor of nearly ten. This obviously comes from much more advantageous conditions with the flat-top, in terms of

_{th_SBS}*LMF*and chirp. In addition, we notice here the early onset of Stokes spikes along the temporal window of the pulse. These observations are consistent with basic expectations above, as already illustrated thanks to the comparison of blue and black curves in the right side of Fig. 5(a). More especially they make sense when considering the additional contribution of gradual distortion effects along the pulse-shapes involved, due to gain saturation from the

*YDFA*. But excessive saturation, which is out of concern here, distortions then appear stronger with flat-top pulses, given similar energy levels. The numerical model provided help to verify that initial considerations make sense. A higher value of

*V*in Eq. (4) leads to a higher

_{ch}*γ*and lower

*g*. It has been validated that the introduction of rapid changes in the chirp rate defined by (1) and of the appropriate sequence of values for

_{Beff}*M*during the step-by-step process presented in Fig. 6 actually lead to significant variations in the time of onset of

*SBS*along the pulse.

This provides an appropriate description of the impact of seed conditions. This is the best way to explain the reason why in some situations, as opposed to those of *FBG* seed where depletion usually starts from the trailing edge of the pulse (Fig. 7 and Fig. 10), *SBS* can be initiated from different locations along the amplified pulse. Anyway this always implies rather low - or highly variable - values of *M* along the pulse, in connection with moderate changes of the chirp rate. It is now time to benchmark the effectiveness of the technique thanks to a variety of flat-top input pulses, of which the duration and the peak power are scanned from 20 to 100ns and 100-300mW, respectively (Fig. 12). This enables the variation of relative contributions of the chirp and *LMF.* Huge variations of the peak attainable output energy (*E _{out}*) are evidenced, up to crossing

*P*. Whatever the configuration of

_{th_SBS}*DFB*or of

*FBG*seed,

*E*starts to undergo a nearly linear increase with the pulse duration as soon as it exceeds the range

_{out}*T*= 40-50ns. This indicates that, for such pulses, the more relevant parameter to put numbers on the definition of

_{p}*P*remains the peak power in the amplified pulse. Anyway, more especially in the case of

_{th_SBS}*DFB*seed, shorter pulses rather imply some negative variation of

*E*with

_{out}*T*(Fig. 12(a)). This is consistent with the previous analysis of the situation of predominant chirp effects coupled to a low

_{p}*M*, referring to Eq. (4) and to the dependency of the coefficient

*γ*with

*L*. Indeed, under these conditions, shorter and shorter pulses from the

*DFB*go with the larger and larger spatial overlap along the length of fiber (see z-integral of

*g*). Then, the fractional energy from the total pulse which contributes to the onset of

_{B}*SBS*evidences a noticeable increase. The variations of

*E*are not of the same sign and magnitude as those involved with longer pulses. This can apply as far as the increase of

_{out}(T_{p})*T*does not lead to a significant enlargement of

_{p}*M*, related to broader

*Δν*. For the sake of validation, a series of spots have been superimposed on the plots which figure test results, together with the data involved in the model for computation. In the situation of 50ns long pulses (Fig. 12(a), right area), for example, we considered

_{tot}*Δν*= 180GHz, for

_{tot}*M*= 6, and${\rho}_{pulse-shaping}\gamma $ = 2.5.10

^{−3}as already involved above in Fig. 7 for a chirp rate of 5.10

^{17}Hz/W. Under these conditions, it can be verified that computational results lead to the same trends and orders of magnitude as the experimental data provided in Fig. 12(b). This validates the correlation between actual values of

*M*and

*E*, across the full range of permitted values, of interest for the management of input pulse-shaping by direct current modulation. As expected, in addition to the gradual increase observed as

_{out}*M*varies from 1 to 6,

*P*logically increases at a faster rate for 100ns pulses (Fig. 12(b), right hand side) than for 50ns pulses (Fig. 12(b), left hand side). This results from the stronger impact of the chirp rate for shorter values of the seed pulse duration. Rather good fits are obtained, both for

_{th_SBS}*DFB*and

*FGB*seeds and in a broad range of pulse lengths, which is not so surprising due to the variety of adjustment options enabled with Eqs. (5) and (6). So far, we just determined the criteria for the management of seed to cancel out

*SBS*. We did not still pay any attention to possible tradeoffs to be discussed when using

*LMF*, aiming to the generation of properly controlled and smooth output pulse-shapes. This will be the topic of the next section.

## 6. Noise limitations versus spectral and pulse-shaping seed conditions

The only drawback to the implementation of *LMF* consists of the addition of unavoidable beating noise [15], due to the co-existence of multiple longitudinal modes from the seed. This is the reason why it is worth to look at the penalty due to consequent temporal modulations in the *MOPA*, within a given *MB* to be consistently specified for the definition of the pulse-shapes. Too much noise being possibly deleterious in applications which imply smooth and properly controlled pulse-shapes, this must be done versus *M, Δν _{FSR}* and

*Δν*. Referring to the basic rules in signal processing, upper boundary limits for the

_{signal}*MB*of interest for our finely shaped, 10-50-100ns long pulses with minimum transition times in the ns or sub-ns range (Fig. 4) do not need to exceed

*ν*= 30-50 GHz. Potentially, beating noise involves 0-100% ultra-broadband fluctuations of the optical power. Within plus or minus broad values of the

_{max}*MB*, according to the expected value of

*Δν*, the preservation of satisfactory values for the

_{signal}*OSNR*implies some tradeoffs. These tradeoffs may be determined in terms of affordable values for

*M*and for the

*FSR*from the seed. This is the reason why we now need to investigate the characteristics of beating noise and put numbers on the value of the

*OSNR*, referring to the specified

*MB*. Regarding pulse-shaping and related requirements in terms of

*OSNR*, we only need to focus on the lower-frequency band of the

*SPD*of beating noise. The upper-frequency band, which cannot be characterized very easily, does not exhibit any particular interest. Because any of two beating modes considered alone generate beating phenomena at the sum-frequency and difference-frequency of their own frequencies, lower beating frequencies must be expected with reduced frequency spacing. Then it seems clear that the

*SPD*of noise should extend towards lower and lower frequencies, as

*Δν*decreases. This also goes naturally for increased values of

_{FSR}*M*increases with

*FBG*seeds. But it is worth to put more precise numbers on the variations of the

*OSNR*thanks to the implementation of representative numerical calculations. Since all these notions may not be so intuitive, a dedicated model has been developed in the form of fairly simple but consistent numerical calculations. Calculations make use of straightforward temporal sampling, along the time-base determined by a consistent fraction of the optical path in the seed cavity. For appropriate sampling, a minimum of 10-30 locations must be selected along the elementary period defined by the central wavelength, in such a way that any elementary wavelength of any active mode may be summed to the others. Then we make use of a statistical calculation process in the form of the coherent sum of the total number

*M*of longitudinal modes contained in the lasing bandwidth Δν

_{signal},${E}_{tot}(t)={\displaystyle \sum _{i=1}^{M}{E}_{i}}\mathrm{exp}(j2\pi ({\nu}_{o}-i\frac{c}{2{n}_{o}d})t+{\phi}_{i})$, with ${\phi}_{i}$a random phase-shift between any of two modes involved among the modal family. For calculations, we select a time window of which the length recovers several 10

^{5}modal periods, each period being sampled using a minimum of 20-30 positions. These numbers help to satisfy computational requirements defined by the ratio of the central optical frequency to

*Δν*, in the order of 10

_{FSR}^{4}, for a relevant description of the beating effects in terms of frequency differences. Because these calculations just aim to a simplified description of beating effects [16], we do not care with possible spatial hole-burning effects in the

*LD*-chip. Previous works in the field showed that this is quite complex and that, in any case, there always remain a number of uncertainties in the results. Dispersion effects due to the propagation and thermal fluctuations are also ignored. Nevertheless these limiting assumptions still enable a satisfactory description in our case.

As expected, computational results evidence nearly 0-100% modulations, of which the *SPD* comprises larger and larger low-frequency contents for reduced values of the *FSR* and of *M*. When considering a unique temporal window of 5ps for the sake of consistent resolution, this is clearly shown in Fig. 13. Statistically, the temporal modulations from the *DFB* (Fig. 13(a)) undergo slower variations than those from the *FBG* (Fig. 13(b)). Then, ranging *M* from 2 to 10 in the *DFB* as previously shown in Fig. 3(b) in the presence of various values of the driving current, we also verify the generation of shorter and shorter spikes. The same way as in mode-locked lasers, but free from any phase locking effects, larger values of *M* can be related to broader lasing bandwidths. Similar phenomena are evidenced with the *FBG*, with more and more broadening of the lasing bandwidth due to narrower mode spacing. The comparison of Fig. 13(a) with Fig. 13(b) also helps to illustrate the reason why *DFB* seeds do not seem as noisy as *FBG* seeds, given a specified *MB*. This actually results from their broader *FSR*. In order to figure the dependency of noticeable noise with the *MB*, we just still have to implement numerical filtering and vary the cut-off frequency (Fig. 14).

For example (Fig. 14(a)), in the situation of *FBG* seed with *M* = 30 modes, variations of the *MB* from DC-1GHz to DC-10GHz correspond to an increase of the peak-to-peak modulations from a few percent to more than +/− 20% of the average pulse power. *M* being increased from 10 to 30 (Fig. 14(b)), the *RMS* noise varies from 10 to 15% if the *MB* extends from DC to *ν _{max}* = 5GHz, and from 15 to about 20% up to

*ν*= 10GHz, respectively. The consistency of our model is easily verified in two steps, thanks to first-order analytical considerations in terms of generic trends and thanks to the fit with experimental data. First, analytical considerations are as follows: under the approximation of broadband

_{max}*SPD*s in the normalized forms of flat-top distributions, with a constant value $\frac{1}{(M-1)\Delta {\nu}_{FSR}}$ from

*Δν*to

_{FSR}*M*x

*Δ ν*for the total noise, and with a constant value $\frac{1}{M}$ from 0 to

*ν*for the

_{max}*MB*, an estimate of the measurable noise power is provided by the spectral recovery of the two distributions.

Dividing the average signal power by the result provided with the spectral product, we get:

But the limited validity of assumptions involved, Eq. (7) helps to verify the effectiveness of basic trends. Broadening the *MB* always implies some decrease in the *OSNR*, as expected, as well as a reduction of *M*. Second, we can evidence satisfactory fits with noise measurements. The experimental data presented in Fig. 15 show the actual amount of beating noise produced by the two kinds of seeds, when operated with a flat-top current pulse and the peak operating current.

The selected transition times in the current pulse are on the order of 500ps, which is fast enough to investigate the complete bandwidth of resonance of the *LD*-chips and discriminate with possible effects due to additional resonances. This explains the short gain-switch pulse [18] to be noticed in the front part of the flat-top envelope. Because we do not evidence any additional significant contribution of this nature, beating noise effects actually consist of the major noise contribution. The comparison of measured noise fluctuations in Fig. 15 with the theoretical predictions from Fig. 14(a) shows a rather good fit, when subtracting the external noise contribution due to the broadband photodiode and real time oscilloscope. The appearance of a lower temporal noise in the case of the *DFB* can be clearly verified. Typically, when defined by the ratio of the peak envelope to the *RMS* fluctuations and when the *MB* broadens from DC-350MHz up to DC-30GHz, the *OSNR* gradually decreases from about 50 to 7-10 in the case of the *DFB*. Using an *FBG* with a short length fiber (*d* = 10cm), it decreases from 50 to less than 1.5. But narrow values below 2GHz, the variations of *RMS* noise are approximately proportional to the selected *MB*. This goes for a given value of *M*, when ranged from 10 to 30, i.e. thinking about the peak and the standard values of interest with typical *DFB* and *FBG* seeds. These quasi linear variations may be accounted for as a constant-rate coefficient ($\Gamma $, GHz^{−1}). The lower is *M*, as shown in Fig. 14(b), the higher is the value of this coefficient. A relevant order of magnitude for the dependency of $\Gamma $with *M* is $\Gamma $ = 1.3.GHz^{−1}*M*^{−1}, in the range of values of interest for *M* and the *MB*. Then, more especially for narrow values of the lasing bandwidth, multimodal noise limitations need to be considered in terms of a tradeoff between and attainable anti-*SBS* capabilities when using *LMF*. The former case of the *FBG* with *Δν _{signal}* = 0.25nm and

*L*= 20cm provides a good example of affordable possibilities in the reduction of

*Δν*. The generation of smooth pulses of which the length ranges over some tens of nanoseconds, within reasonable values of the

_{FSR}*MB*, is not permitted when the fiber length exceeds 30-40cm. Despite the interest of rapidly increased values for

*M*with slightly longer fibers to cancel

*SBS*, this must be underlined.

## 7. Conclusion

We reviewed the major topics of interest for *SBS*-free *MOPA*s, when seeded by means of shaped, highly multimode *LD*-seeds operated in the regime of direct current modulation. The seeds of interest may be based on two kinds of optical architectures, either *DFB* or *FBG*, leading to different performance limitations. We have demonstrated through a thorough analysis that the technique of *LMF*, also combined naturally with intrinsic chirp from the semiconductor chip involved, can be very efficient to overcome *SBS* limitations. This technique has been experienced at 1064nm with varied seed configurations, thanks to the amplification of 10-100ns long pulses in a 10µm-*MFD YDFA*. A comprehensive analysis has been provided with proper discrimination for *LMF* and chirp effects versus pulse-shaping conditions. For the sake of understanding and consistent validation, we developed a representative numerical model in a full-vectorial form. This description takes into account all of the contributing terms, due to *LMF* and chirp given input pulse-shaping conditions, together with gain saturation along the fiber of the output *YDFA* in the *MOPA* involved. To complete the analysis of ultimate performance limitations using *LMF*, we finally discussed the global tradeoff between the requirements to cancel out *SBS* and the need to account for the unavoidable beating noise effects of the technique. This implied the computation of statistical distributions of the beating noise under specified *LMF* conditions, in view of the generation of properly smoothed pulse-shapes within a specified measurement bandwidth.

The potential interest of this study does not only apply to the particular situation of 10µm-*MFD*, which was selected here for the sake of consistent benchmarking. Applications to more complex *MOPA*s or other laser systems, possibly comprising different fibers with different values of the *MFD* and intermediate fiber connections, should be straightforward. Our modelling process may provide a useful tool for the prediction of possible locations for the onset of *SBS* and for sizing, whatever the optical architecture. We proposed a new efficient tool to optimize the design of high-performance advanced laser systems. More especially, we refer here to the increasing number of *MOPA* applications using Large-Mode-Area fibers with Photonic-Clad structures. The development of new, robust and cost-effective fibered architectures for the production of highly energetic pulses, possibly using *LMF* with efficient semiconductor seeding, should exhibit some interest in an increasing number of emerging applications in medicine and in the industry. Some examples are micromachining, seeders for larger scale laser systems applied to material processing, and lightning systems.

## Acknowledgments

This work was carried out with the support of the Conseil Régional d’Aquitaine, within the frame of the cluster Route des Lasers, in order to support current projects and to help prepare prospective plans in the field of fibered laser systems. Many thanks also to Dr. C. Aguergaray from *ALPhANOV,* for useful reading and comments.

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