S Boscolo

L Kh Mouradian

C Finot

Enhanced nonlinear spectral compression in fibre by external sinusoidal phase modulation

Keywords: Spectral compression, nonlinear propagation, optical fibre, phase modulation

We propose a new, simple approach to enhance the spectral compression process arising from nonlinear pulse propagation in an optical fibre. We numerically show that an additional sinusoidal temporal phase modulation of the pulse enables efficient reduction of the intensity level of the side lobes in the spectrum that are produced by the mismatch between the initial linear negative chirp of the pulse and the self-phase modulation-induced nonlinear positive chirp. Remarkable increase of both the extent of spectrum narrowing and the quality of the compressed spectrum is afforded by the proposed approach across a wide range of experimentally accessible parameters.

Introduction

The effect of optical nonlinearity on an ultra-short pulse propagating in an optical fibre is ordinarily associated with broadening of the pulse spectrum. In certain regimes, however, the fibre nonlinearity can induce the opposite effect, resulting in spectral compression of the pulse [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF]. For example, a fundamental soliton propagating in an anomalous dispersion-increasing fibre can experience narrowing of its spectral width as a result of its adiabatic adaptation to the slowly varying fibre dispersion [START_REF] Nishizawa | Wideband spectral compression of wavelength-tunable ultrashort soliton pulse using comb-profile fibre[END_REF]. The working principle is the reverse operation of the well-known adiabatic soliton temporal compression in a dispersion-decreasing fibre [START_REF] Tamura | 54-fs, 10-GHz soliton generation from a polarizationmaintaining dispersion-flattened dispersion-decreasing fiber pulse compressor[END_REF]. Additionally, ultra-short laser pulses can be spectrally compressed in the regime of soliton self-frequency shift induced by the Raman effect in a highly nonlinear fibre [START_REF] Fedotov | Spectral compression of frequency-shifting solitons in a photoniccrystal fiber[END_REF]. Owing to the anomalous dispersion of the highly nonlinear fibre, laser pulses evolve toward solitons and experience a continuous frequency downshift. A lowering frequency and increasing dispersion of a redshifting soliton dictate spectral narrowing. Spectral narrowing can also be realized in an optical fibre by using self-phase modulation (SPM) of pulses with an initial negative frequency modulation (chirp) [START_REF] Zohrabian | Compression of the spectrum of picosecond ultrashort pulses[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF][START_REF] Planas | Spectral narrowing in the propagation of chirped pulses in single-mode fibers[END_REF], such as the one imparted by an anomalously dispersive component. Indeed, the intensity-dependent nonlinear phase shift induced by SPM results, in silica, in a frequency downshift in the leading edge of the pulse and an upshift in the trailing edge. Thus the effect of SPM is dependent on the sign of the initial chirp. Negatively chirped pulses, where the long and the short wavelengths are in the trailing and the leading edges, respectively, are spectrally compressed, since both the long and the short wavelengths are shifted toward the center wavelength. This method of spectral compression has been implemented using various types of fibres [START_REF] Washburn | Transform-limited spectral compression due to self-phase modulation in fibers[END_REF][START_REF] Andresen | Spectral compression of femtosecond pulses in photonic crystal fibers[END_REF][START_REF] Sidorov-Biryukov | Spectral narrowing of chirp-free light pulses in anomalously dispersive, highly nonlinear photonic-crystal fibers[END_REF][START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude shaped pulses with negative chirp[END_REF], optical gain fibres, and photonic crystal fibres, and is suitable for a very large range of wavelengths including Ti:sapphire wavelengths [START_REF] Washburn | Transform-limited spectral compression due to self-phase modulation in fibers[END_REF][START_REF] Andresen | Spectral compression of femtosecond pulses in photonic crystal fibers[END_REF], the widely used 1-m [START_REF] Rusu | All-fiber picosecond laser source based on nonlinear spectral compression[END_REF][START_REF] Limpert | SPM-induced spectral compression of picosecond pulses in a singlemode Yb-doped fiber amplifier[END_REF] and 1.55-m [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF] windows and the emerging 2-m band [START_REF] Bao | Spectral compression of a dispersion-managed mode-locked Tm:fiber laser at 1.9 um[END_REF]. The process can also sustain simultaneous amplification of the pulse, and has been reported for different types of fiber amplifiers [START_REF] Rusu | All-fiber picosecond laser source based on nonlinear spectral compression[END_REF][START_REF] Limpert | SPM-induced spectral compression of picosecond pulses in a singlemode Yb-doped fiber amplifier[END_REF][START_REF] Bao | Spectral compression of a dispersion-managed mode-locked Tm:fiber laser at 1.9 um[END_REF][START_REF] Limpert | High-average-power picosecond Yb-doped fiber amplifier[END_REF]. Therefore, spectral compression by SPM of negatively chirped pulses provides an attractive solution to convert ultra-short pulses delivered by femtosecond oscillators into powerful, near-transform-limited picosecond pulses, and to counteract the spectrum expansion that usually occurs with the direct amplification of picosecond structures. The concept can be extended to multistage architectures [START_REF] Korobko | Multistage fiber preamplifier comprising spectral compression for generation of high-energy laser pulses[END_REF] and fibre laser cavities, where in-cavity nonlinear spectral compression in a mode-locked fibre laser has been recently demonstrated numerically [START_REF] Boscolo | Amplifier similariton fiber laser with nonlinear spectral compression[END_REF]. However, the main limitation of SPM-driven spectral compression in the nonlinearity-dominant regime of propagation, in which fibre dispersion is of little importance, is the presence of residual side lobes in the compressed spectrum. These side lobes stem from the fact that in general an input pulse with a negative linear chirp cannot be compressed to the Fourier transform limit, because the cancellation of linear and nonlinear phases cannot in general be made complete. In order to overcome this limitation and, thus, approach a Fourier-transform-limited compression, several strategies have been proposed and experimentally demonstrated, including the use of complex phase modulation of the initial pulse to balance the (higher-order) dispersion and chirp [START_REF] Washburn | Transform-limited spectral compression due to self-phase modulation in fibers[END_REF] or the use of a pre-shaped input pulse profile such as a parabolic waveform [START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude shaped pulses with negative chirp[END_REF][START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. Another strategy to enhance the quality of the compressed pulse spectrum is to select a dispersive nonlinear regime of propagation in which the combined action of normal group-velocity dispersion (GVD) and SPM results in a deformation of the temporal profile of the pulse tending to acquire a rectangular shape while nearly complete compensation of the pulse chirp occurs [START_REF] Kutuzyan | Dispersive regime of spectral compression[END_REF][START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF].

In this paper, we present a new method for enhancing the performance characteristics of the spectral compression by SPM of conventional laser pulses in a fibre, which is based on imposing an additional sinusoidal temporal phase modulation on the propagating pulse. We provide a thorough characterization of the compression process based on extensive numerical simulations. The optimal operational conditions are identified, which enables us to provide general design rules for spectralcompression fibre schemes over the parameter space that is typically accessible experimentally. Remarkably, our results show that high-quality compression with efficient reduction of the intensity level of spectral substructures is possible even for a spectrum that has split into a complex pattern. First, we introduce the situation being studied and the metrics used to quantify the performance of the compression process. Next we recall the results that can be obtained in the purely nonlinear propagation regime, and then we describe the effects of an additional temporal phase modulation of the pulse. Finally, we discuss the influence of the initial stretching ratio on the performances.

Situation under investigation

Our numerical simulations of the spectral compression process in a fibre system are based on the standard nonlinear Schrödinger equation (NLSE) [START_REF] Agrawal | Nonlinear Fiber Optics[END_REF]:

2 2 2 2 1 0, 2 i z t            (1)
where ψ(z,t) is the complex electric field envelope, z is the propagation distance, t is the reduced time, β 2 is the GVD parameter and γ is the coefficient of cubic nonlinearity of the fibre. This equation neglects the effect of fibre loss, as well as higher-order linear and nonlinear effects. Although these effects can have noticeable impact on pulses shorter than 1ps, here we neglect them as the leading-order behavior is well approximated by Eq. ( 1).

For the purpose of illustration, we consider a transform-limited pulse with a Gaussian intensity profile

22 0 0 0 ( ) exp( / 2 ) t P t T  
as the initial condition for our study. The parameters T 0 and P 0 are a characteristic temporal value (the half-width at 1/e-intensity point in the case of a Gaussian-shaped pulse) and the peak power of the initial pulse, respectively. The linear negative temporal chirp required for spectral compression to take place can be imprinted onto the initial pulse by stretching the pulse in the temporal domain in an anomalously dispersive medium, such as a pair of diffraction gratings [START_REF] Andresen | Spectral compression of femtosecond pulses in photonic crystal fibers[END_REF][START_REF] Limpert | High-average-power picosecond Yb-doped fiber amplifier[END_REF], a prism pair [START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF][START_REF] Washburn | Transform-limited spectral compression due to self-phase modulation in fibers[END_REF], a fibre Bragg grating, or a segment of hollow core [START_REF] Rusu | All-fiber picosecond laser source based on nonlinear spectral compression[END_REF] or standard [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF] fibre with anomalous GVD (and very low nonlinearity). As a result of GVD [described by Eq. ( 1) with  = 0], the different spectral components of the pulse acquire a frequency dependent delay. Even though such phase changes do not affect the pulse spectrum, they lead to temporal broadening of the pulse and a time dependence of the pulse phase. If A denotes the temporal broadening factor, the peak power of the pulse will drop by the same factor (P 1 = P 0 /A) owing to the conservation of energy (in the absence of losses). In the far-field regime (i.e., when A >> 1), the temporal phase becomes parabolic, so that the pulse envelope at the exit of the dispersive element takes the form

    2 1 0, | | exp( ) t t ibt   , where   2 2 2 1 0 0 | | exp( / 2 ) t P A t A T  
is the stretched amplitude profile and

2 0 0 1 2 b A T 
is the chirp parameter. This negatively linearly chirped pulse is then spectrally compressed in a nonlinear fibre. Hereinafter, we consider the situation when the nonlinearity-dominant regime of propagation is applicable, which is typical of various demonstrations of spectral compression due to SPM in fibres reported so far [START_REF] Andresen | Spectral compression of femtosecond pulses in photonic crystal fibers[END_REF][START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. In this regime, the dispersion term in Eq. ( 1) plays a relatively minor role and can be neglected. The narrowing of the pulse spectrum is quantified with the spectral compression factor C fwhm (C rms ) defined as the ratio of the spectral full-width at half maximum (FWHM) [root-mean-square (rms)] width at some distance in the fibre and the entrance of the fibre. In order to assess the quality of the spectral compression, we use the Strehl ratio S defined as the ratio of the maximum spectral brilliance of the actual pulse to the spectral brilliance obtained assuming a flat temporal phase of the pulse [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. Therefore, S is comprised between 0 and 1, with 1 defining an ideal compression.

Pulse evolution in a purely nonlinear medium of a Gaussian pulse

The nonlinear term in the pulse-propagation equation (1), in the limit β 2 = 0, imposes a temporal intensity distribution-dependent phase shift on the initial pulse, so that after a propagation length z:

22 2 1 | ( )| | (0, )| 1 ( , ) (0, ) | ( ) | i t z i b t i t z z t e t e t        (2)
It is seen that the instantaneous frequency of the pulse,

2 1 0 2 22 0 11 ( | ( ) | ) 22 2 2 ( , ) exp 1 NL zt zt AT A d d t t z b t dt dt T A L                           (4)
cannot in general be made equal to zero for all times, which expresses the fact that an input Gaussian pulse (and by extension any other non-parabolic pulse [START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude shaped pulses with negative chirp[END_REF][START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]) with a negative linear chirp cannot be spectrally compressed to the Fourier-transform limit. In Eq. ( 3), L NL = 1/( P 0 ) is the nonlinear length associated with the pulse at the input to the system. It is useful here to expand the Gaussian function of the pulse profile in a Taylor series about the pulse center, namely,

2 4 6 3 5 7 2 2 2 2 2 0 0 0 0 2 2 1 1 1 ( , ) 1 .. 3 1 . 2 NL NL NL NL z z z z z t t t t t A T A L A L A T A L A T A L A T                                             (4)
The equation above shows that the linear part of the instantaneous frequency yields zero for all t at the propagation distance z 1 given by [START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF]:

2 1 2 NL z A L  (5)
However, the translation of the well-known time-space analogy [START_REF] Kolner | Space-time duality and the theory of temporal imaging[END_REF] into the spectral domain [START_REF] Andresen | Spectral Analog of the Gouy Phase Shift[END_REF] indicates that the higher-order chirp terms in (4) cause spectral aberrations.

An example of the pulse evolution along the nonlinear fibre is presented in Fig. 1 for the initial stretching factor A = 20. Significant spectral compression occurs in the fibre, followed by splitting of the pulse spectrum into an increasing number of substructures [Fig. 1(a)]. The evolutions of parameters C fwhm , C rms and S are given in Fig. 2. It is seen that the propagation distance at which the Strehl ratio reaches its maximum value (denoted by z 2 ) differs from the distance z 1 of chirp cancellation near the pulse center given by Eq. ( 5) [START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF]. We can also infer from Fig. 2 that the highest brilliance in the central region of the pulse spectrum is achieved at the point of maximum Strehl ratio. While compression factors above 10 can be achieved in terms of FWHM spectral width, the rms-width compression factor attains moderate values below 2. This is a signature of the presence of non-negligible pedestals in the pulse spectrum. These pedestals degrade the quality of the compression, as confirmed by the Strehl ratio's values that do not exceed 0.4. The abrupt drop of C fwhm after the distance at which it is at its maximum is due to an increase in the intensity level of spectral satellites above the -3 dB threshold used for its computation. The pulse spectrum splits up shortly after the distance of maximum C fwhm . In Fig. 1(b) we can see that at the distance z 1 foretold by Eq. ( 5) the pulse spectrum is effectively conveniently compressed, and the spectral profile does not display any oscillations, which is in agreement with the monotonic temporal variation of the frequency chirp at this distance (Fig. 3): the chirp continuously decreases across the pulse, and has a point of inflexion at the pulse center. At the other operational distances, the spectrum features an oscillating structure, which results from interference between different pulse parts having the same instantaneous frequency. This is confirmed by the nonmonotonic temporal variation of the chirp. Such a constructive interaction can have a beneficial impact on the spectral compression process, as it can increase the peak spectral intensity of the pulse with consequent enhancement of spectral brightness, as shown by the pulse spectrum obtained at the optimum compression point in terms of Strehl ratio z 2. However, for propagation in the fibre beyond z 2 , as a result of this interference phenomenon the pulse develops strong and detrimental oscillations in the wings together with strong spectral side lobes, as it appears from the spectrum at the distance z 3 = 750 L NL . On the other hand, at z 1 , while the spectrum narrowing in terms of FWHM spectral width is suboptimal, the spectrum exhibits significantly lower substructures than at further propagation distances. We note that this scenario is similar to the situation when the B-integral accumulation in chirped pulse amplification systems is not exactly compensated for: as a result of the distortion of the linear chirp by SPM, temporal structure unavoidably appears in the recompressed pulse, even for relatively small values of B [START_REF] Perry | Self-phase modulation in chirped-pulse amplification[END_REF].

Use of an additional external phase modulation

Now we describe the impact of an additional temporal phase modulation of the pulse on the spectral compression performances in the pure SPM case. Equation ( 5) gives the propagation distance at which the quadratic temporal phase (linear chirp) of the pulse is canceled, and Fig. 3 confirms at this distance the chirp is zero in the central part of the pulse. To compensate for the residual chirp, one could think of canceling the quartic and higher-order terms in the nonlinear phase stemming from the Taylor series expansion of the Gaussian profile (cubic and higher-order terms in Eq. ( 4)) by means of an external phase modulation. However, in practice, it may be quite difficult to synthetize such a complex phase profile in the temporal domain. Further, the validity of the Taylor expansion is rather limited to the central part of the pulse: for instance, an expansion to o(t 6 ) would noticeably deviate from the actual phase of the pulse at times |t|/(AT 0 ) greater than 0.5. On the other hand, we can see in Fig. 3 that at the distance z 3 , the chirp features strong oscillations with a close-to-sinusoidal variation in the central region of the pulse. This observation forms the basis of the simple idea proposed here to offset the chirp: apply to the pulse a corrective sinusoidal temporal phase modulation, ϕ a (t) = -a c cos(2 c t), producing the chirp  a (t) = a c  c sin(2 c t). Such a modulation can be imparted by a lithium-niobate phase modulator driven by a synchronized microwave signal [START_REF] Godil | Picosecond time-lenses[END_REF][START_REF] Plansinis | Spectral changes induced by a phase modulator acting as a time lens[END_REF] or by cross-phase modulation of the pulse with a sinusoidal signal [START_REF] Varlot | Experimental generation of optical flaticon pulses[END_REF]. The amplitude and frequency of the modulation can be intuitively chosen as a c = Δν ΔT and Figure 4 illustrates this approach at the distance z 3 . The chirp is indeed made linear and flat over most of the pulse (for |t|/T 0 < 30). On the contrary, the compensation of the linear chirp at z 1 leads to a significantly narrower region of chirp cancellation restricted to |t|/T 0 < 10. The resulting spectrum at z 3 (Fig. 5) highlights the striking enhancement of the spectral compression afforded by the corrective approach: even a very degraded spectrum can be corrected and changed to a high-quality compressed spectrum. Most of the pulse energy is concentrated in the central lobe of the spectrum while the intensity level of the side lobes is significantly lower than that occurring at z 1 or z 2 in the absence of phase correction [Fig. 5(b)], which entails a remarkable increase in spectral brightness or compression quality. The enhancement of the spectral compression performance characteristics enabled by the corrective phase modulation method is in fact noticeable at any propagation distance in the fibre beyond z 2 . Figures 6 and7 summarize the results obtained using this approach. It is seen that the frequency of the corrective modulation decreases monotonically with increasing propagation distance, whereas the amplitude increases. The sinusoidal phase correction has little impact on the performances for propagation distances between z 1 and z 2 , where the chirp remains nearly linear and flat near the pulse center. The strength of the approach increases with increasing propagation distance beyond z 2 as the pulse chirp develops an increasingly stronger sinusoidal-like temporal variation. The compression ratio is significantly increased with this strategy both in terms of FWHM and rms widths. The FWHM compression factor can now exceed 15. Remarkably, the rms compression factor and the Strehl ratio feature up to threefold and more than twofold increase, respectively, with respect to their maximum value in the absence of phase correction, confirming the efficient reduction of spectral wings and enhanced quality of the compression. We also studied the effect of the initial temporal broadening factor A on the performance characteristics of the compression process. In the maps presented in Fig. 8, we summarize the results obtained using stretching values that are typical of experimental realizations [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. We can note some common performance trends between the two cases of nonlinear propagation in the absence and in the presence of phase correction: larger A results in a larger maximum FWHM compression factor, while the overall quality of the process rather deteriorates with increasing A, as indicated by the moderate decrease of the Strehl ratio [START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF]. A larger A also leads to longer optimum propagation distances in terms of both compression factors and Strehl ratio. On the other hand, the application of a corrective sinusoidal phase modulation to the propagating pulse remarkably improves the compression process in both the achievable extent of spectrum narrowing and the quality of the compressed spectrum for all A values being studied. Further, the use of the corrective approach enables significant broadening of the range of operational parameters that support performances near to the optimum, thus providing higher flexibility in the design of the spectral compression scheme.

 c = 1/

Conclusion

We have presented a new, simple approach to enhance the spectral compression of negatively chirped pulses that occurs upon nonlinear propagation in a Kerr medium. We have numerically shown that in the nonlinearity-dominant regime of propagation, the intensity level of the spectral side lobes resulting from the mismatch between the initial linear chirp of the pulse and the SPM-induced nonlinear chirp can be efficiently reduced by an additional sinusoidal temporal phase modulation applied to the pulse. The required phase modulation can be readily accomplished by a lithium-niobate phase modulator or by employing cross-phase modulation with a sinusoidal signal. The proposed phase correction method enables notable improvement in the performance characteristics of the spectral compression process across a wide range of parameters that are readily available for experiments using picosecond pulses. For example, for an input picosecond pulse stretched by a factor 20 and with a Watt-level peak-power (after stretching), optimal spectral compression can be achieved after propagation in a km-long highly nonlinear fibre. The required modulating sinusoid would have a frequency of a few tens of gigaherts and a amplitude, complying with modulations routinely provided by telecommunications devices.

Figure 1 -

 1 Figure 1 -Spectral compression of an initial Gaussian pulse with the stretching factor A = 20 in a purely nonlinear fibre: (a) Longitudinal evolution of the spectral intensity profile. (b) Spectral intensity profile at different propagation distances in the fibre: the point z 1 , predicted by Eq. (5) (red), the point z 2 of maximum Strehl ratio (green), and the point z 3 (blue). Also shown is the initial pulse profile (black).

Figure 2 -

 2 Figure 2 -Spectral compression of an initial Gaussian pulse with the stretching factor A = 20 in a purely nonlinear fibre: longitudinal evolution of (a), (b) the FWHM and rms spectral compression factors, and (c) the Strehl ratio.

Figure 3 -

 3 Figure 3 -Temporal intensity and chirp profiles of an initial Gaussian pulse with the stretching factor A = 20 at different propagation distances in a purely nonlinear fibre: the point z 1 , predicted by Eq. (5) (red), the point z 2 of maximum Strehl ratio (green), and the point z 3 (blue). Also shown are the initial pulse profiles (black).

  2ΔT, where ΔT represents the time separation between the two local extrema of the chirp in the central region of the pulse at a given propagation distance z > z 1 , and Δν is the difference between the corresponding chirp values.

Figure 4 -

 4 Figure 4 -Spectral compression of an initial Gaussian pulse with the stretching factor A = 20 in a purely nonlinear fibre: temporal chirp profiles at the propagation distance z 3 before (dotted blue) and after (solid blue) phase correction. The corrective chirp δν a is plotted with a dotted black line. Also shown are the pulse characterizations at the entrance of the fibre (black), at the distance z 1 , predicted by Eq. (5) (red) in the absence of phase correction.

Figure 5 -

 5 Figure 5 -Impact of phase correction on the spectral compression of an initial Gaussian pulse with the stretching factor A = 20 in a purely nonlinear fibre: spectral intensity profiles plotted on (a) linear and (b) logarithmic scales at the propagation distance z 3 before (dotted blue) and after (solid blue) phase correction. Also shown are the profiles at the entrance of the fibre (black), at the distance z 1 , predicted by Eq. (5) (red), and the distance z 2 of maximum Strehl ratio (green), in the absence of phase correction.

Figure 6 -

 6 Figure 6 -Spectral compression of an initial Gaussian pulse with the stretching factor A = 20 in a purely nonlinear fibre with phase correction: longitudinal evolution of (a) the spectral intensity profile of the pulse, and (b), (c) the frequency  c and amplitude a c of the corrective phase modulation.

Figure 7 -

 7 Figure 7 -Spectral compression of an initial Gaussian pulse with the stretching factor A = 20 in a purely nonlinear fibre with phase correction: longitudinal evolution of (a), (b) the FWHM and rms spectral compression factors, and (c) the Strehl ratio.Also shown are the results obtained without correction (dotted grey).

Figure 8 -

 8 Figure 8 -Evolution of (a), (b) the FWHM and rms compression factors, and (c) the Strehl ratio versus stretching factor A and normalized propagation distance for an initial Gaussian pulse propagating in a purely nonlinear fibre without (subplots 1) and with (subplots 2) phase correction.

Acknowledgement

The authors would like to acknowledge support by the Région Bourgogne (Pari Photcom) and the Agence Nationale de la Recherche (Labex Action ANR-11-LABX-01-01).


Sonia Boscolo

Christophe Finot
email: christophe.finot@u-bourgogne.fr

Enhanced spectral compression in nonlinear optical fibres

Keywords: nonlinear fibre optics, spectral compression, nonlinear pulse shaping

We propose two new approaches to enhance the spectral compression process arising from nonlinear pulse propagation in an optical fibre. We numerically show that an additional sinusoidal temporal phase modulation of the pulse enables efficient reduction of the intensity level of side lobes in the spectrum. Another strategy is to select a regime of propagation in which normal group-velocity dispersion reshapes the initial stretched pulse to a near-Fourier-transform-limited rectangular waveform.

INTRODUCTION

The effect of self-phase modulation (SPM) on an optical pulse propagating in a nonlinear optical fibre is ordinarily associated with broadening of the pulse spectrum [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF]. However, this is not always the case with chirped pulses, where pulses with an initial negative chirp (such as that imparted by an anomalously dispersive element) in the normal group-velocity dispersion (GVD) region can experience spectral narrowing [START_REF] Zohrabian | Compression of the spectrum of picosecond ultrashort pulses[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF]. Indeed, as a result of SPM, the propagating pulse will have the frequencies of the leading edge downshifted and the frequencies of the trailing edge upshifted. Therefore, in the case of an initially negatively chirped pulse, both the trailing and leading edges of the pulse will experience a spectral shift in the direction of the central frequency. It is worth noting here that the anomalous GVD enhances the effect of SPM rather than compensating it. However, in general an input pulse with a negative linear chirp cannot be compressed to the Fourier transform limit, because the cancellation of linear and nonlinear phases cannot in general be made complete. In order to overcome this limitation, several strategies have been proposed and experimentally demonstrated, based on the use of complex phase modulation of the initial pulse [START_REF] Washburn | Transform-limited spectral compression due to self-phase modulation in fibers[END_REF] or the use of a pre-shaped input pulse profile such as a parabolic waveform [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. In this paper, we present two new approaches to the enhancement of the performance characteristics of the spectral compression process. First, we numerically show that an additional sinusoidal temporal phase modulation of the pulse enables efficient reduction of the intensity level of spectral side lobes. Then we show that the progressive reshaping of the initial pulse toward a rectangular waveform that results from the combination of SPM and normal GVD facilitates the full compensation of the initial chirp [START_REF] Kutuzyan | Dispersive regime of spectral compression[END_REF][START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF].

SPECTRAL COMPRESSION IN THE NONLINEAR REGIME

First, we briefly recall the results that can be obtained in the nonlinearity-dominant regime of propagation, in which the fibre dispersion is of little importance and, thus, can be neglected. For the purpose of illustration, at the entrance of the fibre we take a Gaussian pulse that has been temporally stretched by a factor of 20 in an anomalously dispersive linear element. We use distance and time variables normalized to L N =1/(γP 0 ) and T 0 , respectively, where P 0 and T 0 are the peak power and the characteristic temporal value of the pulse at the input to the linear element, and γ is the Kerr nonlinearity coefficient of the fibre. The narrowing of the pulse spectrum is quantified with the spectral compression factor C fwhm (C rms ) defined as the ratio of the spectral full-width at halfmaximum (FWHM) [root-mean-square (rms)] width at some distance in the fibre and the entrance of the fibre. In order to assess the quality of the spectral compression, we use the Strehl ratio S defined as the ratio of the maximum spectral brilliance of the actual pulse to the spectral brilliance obtained assuming a flat temporal phase of the pulse [START_REF] Fatome | All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping[END_REF]. The longitudinal evolution of the pulse spectrum in Fig. 1(a) clearly shows that significant spectral compression occurs in the fibre, followed by splitting of the spectrum into an increasing number of substructures. At the propagation distance z 1 , where the linear and nonlinear phases cancel each other in the central part of the pulse, the pulse spectrum is effectively conveniently compressed, and the spectral profile does not display any oscillations [Fig. 1(b)]. At the distance of optimum compression in terms of Strehl ratio, z 2 , owing to interference between different pulse parts having the same instantaneous frequency, the spectrum is narrower and features an increased brightness [START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF], but it suffers from residual side lobes. For further propagation in the fibre, as a result of this interference phenomenon the pulse develops strong and detrimental oscillations in the wings together with strong spectral side lobes, as it appears from the spectrum at the distance z 3 . The evolutions of parameters S and C are given in Fig. 1(c). The moderate values of S and C rms at z 2 are a signature of the oscillatory structure of the spectrum.

USE OF AN ADDITIONNAL PHASE MODULATION

In this section, we describe the impact of an additional phase modulation of the pulse on the spectral compression performances in the pure SPM case. We can see in Fig. 2(a) that at the propagation distance z 1 the chirp of the pulse is zero near the pulse centre. At the distance z 3 , the chirp features strong oscillations with a close-to-sinusoidal variation in the central region of the pulse. This observation forms the basis of the simple idea proposed here to offset the chirp: impart to the pulse a corrective sinusoidal temporal phase modulation with suitably chosen frequency and amplitude and opposite sign to the chirp to compensate for. By use of this method, the chirp is indeed made linear and flat over most of the pulse. The resulting spectrum [Fig. 2(b)] highlights the significant enhancement of the spectral compression afforded by the corrective approach: most of the pulse energy is concentrated in the central lobe of the spectrum while the intensity level of the side lobes is strikingly lower, which entails a significant increase in spectral brilliance. Figure 3 summarizes the results obtained using the corrective phase modulation method. The strength of the approach increases with increasing propagation distance in the fibre beyond z 2 as the pulse chirp develops a closer-to-sinusoidal temporal variation. The compression ratio is significantly increased with this strategy both in terms of FWHM and rms widths. Remarkably, the rms compression factor and the Strehl ratio feature up to threefold and twofold increase, respectively, with respect to their maximum value in the absence of phase correction, confirming the efficient reduction of spectral wings and enhanced quality of the compression.

Figure 3. Spectral pulse compression in a purely nonlinear fibre with phase correction: longitudinal evolutions of (a) the spectral intensity profile, and (b) the compression ratios C fwhm , C rms and the Strehl ratio S (solid). Also

shown are the results obtained without correction (dashed).

EVOLUTION IN A NORMALLY DISPERSIVE FIBRE

The performance characteristics of the spectral compression process can also be enhanced by taking advantage of the progressive reshaping of the initial pulse that occurs upon propagation in a highly nonlinear fibre with normal dispersion. We have performed a detailed analysis of the spectral compression process taking into account the influence of GVD, based on numerical simulation of the nonlinear Schrödinger equation propagation model. In this context, for a given value of the initial stretching factor, finding the best system parameters results in a two-dimensional optimization problem in the plane (ξN), where ξ=z/L D , parameter N is defined as N 2 =L NL /L D , and L D =T 0 2 / 2 is the dispersion length ( 2 is the fibre GVD coefficient) associated with the pulse at the input to the linear element. The results obtained for an initial stretching factor of 20 are summarized in Fig. 4(a). Optimum compression occurs at the point (ξ opt =13.6N opt =4.4), and the Strehl ratio is about 0.9 at this point. The shape of the resulting spectrum [Fig. 4(b)] indicates that the pulse is very close to the Fouriertransform limit. The GVD significantly changes the evolution of the pulse spectrum in the fibre [Fig. 4(c)]: contrary to the pure SPM case [Fig. 1(a)], the spectrum does not split into multiple sidebands after the stage of optimum compression. The concomitant temporal dynamics are illustrated in Fig. 4(d): the combination of SPM and GVD make the initial Gaussian pulse reshape into an almost rectangular pulse form at the point of optimum compression, while nearly complete compensation of the pulse chirp occurs. The nearly rectangular pulse shape explains the sine-cardinal-like shape of the spectrum. Figure 4(d) also shows that the pulse acquires an almost parabolic intensity profile before the point of optimum compression (at ξ=9). Contrary to the parabolic pulses that can be generated through progressive reshaping of initially un-chirped pulses upon nonlinear propagation in a normally dispersive fibre [START_REF] Finot | Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device[END_REF], the pulse formed in the parameter region of optimum spectral compression features a very low chirp value over its central part. ). The intensity profile at ξ=9 is compared to a parabolic fit (blue circles).

CONCLUSIONS

We have presented two new approaches to enhance the spectral compression of negatively chirped pulses that occurs upon nonlinear propagation in a fibre. In the purely nonlinear regime of propagation, the intensity level of the spectral side lobes can be efficiently reduced by an additional sinusoidal temporal phase modulation imparted to the pulse. Another strategy is to operate in a propagation regime in which normal dispersion reshapes the initial stretched pulse to a near-Fourier-transform-limited rectangular waveform.

Figure 1 .

 1 Figure 1. Spectral pulse compression in a purely nonlinear fibre. (a) Longitudinal evolution of the spectral intensity profile. (b) Spectra at different distances in the fibre: the entrance of the fibre (black), the point z 1 of chirp cancellation near the pulse centre (red), the point z 2 of maximum Strehl ratio (green), and the point z 3 (blue). (c) Longitudinal evolutions of the compression ratios C fwhm , C rms and the Strehl ratio S.

Figure 2 .

 2 Figure 2. Impact of phase correction on the spectral pulse compression in a purely nonlinear fibre. (a) Temporal chirp profiles at: z=0 (black), z=z 1 (green), and z=z 3 before (solid red) and after (blue) phase correction. Also shown are the corrective phase modulation applied at z 3 (dashed red), and the initial temporal intensity profile. (b),(c) Spectral intensity profiles plotted on linear and logarithmic scales at: z=0 (black), z=z 1 (red), z=z 2 (green), and z=z 3 before (dashed blue) and after (solid blue) phase correction.

Figure 4 .

 4 Figure 4. Spectral pulse compression in a nonlinear dispersive fibre. (a) Evolution of the Strehl ratio versus ξ and N. (b) Spectral intensity profiles of a pulse with the optimum power parameter N opt =4.4 at different distances in the fibre plotted on linear and logarithmic scales. The spectrum the entrance of the fibre (black) is compared with the spectrum at the distance of optimum compression ξ opt =13.6 (solid red), and the Fouriertransform-limited spectrum expected at this distance (dashed red). Also shown is the spectrum at the distance ξ=9 where the temporal profile is close to a parabola (blue). (c) Longitudinal evolution of the spectral intensity profile for N=N opt . (d) Temporal intensity and chirp profiles at different stages of propagation (same colors as subplot b1). The intensity profile at ξ=9 is compared to a parabolic fit (blue circles).


Tatiane Guimarães

Communities of practice of innovative startups Cooperation or competition: is this the question?

Keywords: communities of practice, innovation process, coopetition, startup

The communities of practice literature offer the potential cooperation space for the development of the open innovation, principally to the organization likes the startups that suffer from lack of resources to deal with the risks and uncertainties related to their project. This cooperation logic however, can meet the competition logic confronting this population. In this paper, we investigate how communities of practice of innovative startups manage this paradoxical logic related to the coopetition. The multilevel analysis of the communities of practice of San Pedro Valley in Brazil allows us to understand the effects of coopetition on the community functioning but also at the innovation ecosystem and across the city in which this evolves.

Introduction

In open innovation environment [START_REF] Chesbrough | The era of Open Innovation[END_REF], most companies aim to develop cooperation in innovation networks more or less formal [START_REF] Loilier | Innovations, Coopérations et Proximités, Quelles configurations pour les réseaux innovateurs ?[END_REF]. This is indeed a way to build new knowledge, optimize mobilized resources or spread innovation [START_REF] Callon | La gestion stratégique de la recherche et de la technologie[END_REF] in an extremely competitive environment. In the particular case of innovative startups, the innovation, and the cooperation become "an inseparable pair" (Fréchet, 2004). Due to their agility and relative independence, these organizational structures can indeed be free of constraints and the inherent inertia of larger companies and so more quickly develop new products or services (Basu & Phelps, 2009). However, they confronts to a lack of resources and a required to cooperate to better explore the novelty and accelerate the innovation process. In this context, the establishment of partnership relations may represent a difficulty faced the immaturity of these enterprises (Rispal, 1993) and threats that may represent the most experienced competitors. A "handicap of novelty" (Baum & Silverman, 2004, p. 411) can be part of the inter-organizational social reality, especially when startups are too far away of the capabilities and of the other potential partners of innovation (Hill & Birkinshaw 2008).

The appropriate logics for cooperation must then deploys to ensure their survival in volatile and highly competitive environments, particularly in the high technology field [START_REF] Teece | Competition, cooperation, and innovation: Organizational arrangements for regimes of rapid technological progress[END_REF]. In this context, the communities of practice (CoP) are such as a component of the innovation ecosystem to the extent they offer a more secure space of cooperation to the startups. The participation of entrepreneurs at a CoP allows startups to remain open to all sources of knowledge.

Analyzes of CoPs in the Silicon Valley support this consideration [START_REF] Dibiaggio | Communautés de pratique et réseaux sociaux dans la dynamique de fonctionnement des clusters de hautes technologies[END_REF].

These authors show that cooperation between peers in CoP (researchers, lawyers, consultants, etc.) alters their representation systems and enables the production of common frameworks better adapted to innovation context. While the authors highlight the ability of CoP supports the innovation ecosystem by creating stronger links between individuals of the same expertise, they do not analyze the effects of competition between the experts involved in these spaces traditionally devoted to cooperation.

In a CoP of innovative startups, the dimension related to the competition seems nevertheless important. One of the peculiarities of this population relates to their ability to interact and cooperate with each other while they are the poles of interest partially congruent and often need access to the same market (Brandenburger & Nalebuff, 2011). Beyond cooperation, the entrepreneurs must ensure that their enterprises are commercially viable by engaging in the business logics of the sector. This feature being even little explored in the literature, we resort the neologism of coopetition (Brandenburger & Nalebuff, 2011;[START_REF] Bengtsson | Cooperation and Competition in Relationships Between Competitors in Business Networks[END_REF] to study the paradoxical logic related to cooperate and to compete. Our focus is to lead a multilevel analysis (at the community, the innovation ecosystem, and the city) taking into account the logic of coopetition of entrepreneurs of innovative startups as part of the same Cop. To do this, we structure this article into three parts. The first theoretical part presents the notion of communities of practice and a network of practices as a space of cooperation where coopetition can emerge. The second part presents the qualitative methodology and the specificities of a field research in Brazil. Thus, in the third part, the research results are presented and subsequently feed the discussion outcome of this research.

THE COOPETITION IN INTER-ORGANIZATIONAL COMMUNITIES OF PRACTICES

The literature on communities of practice highlights the specificities of these spaces for cooperation among peers. Crossing this with the literature on coopetition, we identify dimensions to explore the existence and integration of the competition logic when communities become in inter-organizational networks.

From communities to networks of practice: a cooperation area for the development of innovations

The communities of practice (CoP) appear as a special form of an informal network of individuals who share the same interest and a strong identity dimension (Wenger et al. 2002, p. 4). They are defined as "groups of people who share a concern, a set of problems, or a passion about a topic, and who deepen their knowledge and expertise in this area by interacting on an ongoing basis" and "a group of people informally bound together by shared expertise and passion for a joint enterprise" (Wenger & Snyder 2000, p. 139). The group acts as a forum to share what they know, to learn from one another regarding some aspects of their work and to provide a social context for their work. This environment has been directly linked to knowledge creation and sharing processes. The extensive literature on CoPs (Brown & Duguid, 1991;[START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF]Wenger, , 2002) ) expressed nowadays a shared recognition of the potential of these informal Organizations to benefit from cooperation to foster collective learning within and beyond the borders of the organizations [START_REF] Brown | The Social Life of Information[END_REF]Jacob et al. 2009).

At the inter-organizational level, CoPs represent an area in which entrepreneurs can face the challenges collectively and carry out managerial and innovation actions related to their own businesses (Dupouët & Barlatier, 2011). From this perspective, Dibiaggio and Ferrari (2003) observed the formation of different CoP into the ecosystem of innovation in Silicon Valley. The cooperation between peers has created common sense deal with ambiguities and discrepancies related in particular to new social links between partners. This vision provides an understanding of tensions between the simple reproduction of knowledge and routines and the emergence of new practices to solve problems.

Thus, for innovative entrepreneurs, engage in a community can support the operation of technological ambiguities, partnerships and commercial innovation-related. The cooperation practices in CoP can be a learning source of multiple behaviors and activities that entrepreneurs must develop to survive (innovate, organize and manage) (Burger-Helmchen, 2008). They can also be an exploration process or a knowledge creation related to the new products and services [START_REF] Harvey | Knowing Communities in the Front End of Innovation[END_REF]. The common field of knowledge, the intensity of relations, the voluntary commitment of members, the share norms, and values, and engaged a dialog with each other at a professional level of CoP foster trust between their members [START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF]. These "protected areas" of cooperation constitute an intermediate level between the individual and the organization and becomes a creation of conditions for new knowledge and technologies [START_REF] Constant | The social locus of technological practice: Community, System, or Organization?[END_REF].

At the scale of a city CoPs will tend to geographically disperse and become practice networks (NoP) according to Agterberg et al. (2010). In spite of this dispersion, the community dimension maintains within the NoP by embedding three types of actors in the network (Agterberg et al., 2010):

(1) Embeddedness in practice "when the knowledge shared concerns common practices and is perceived as relevant";

(2) Social embeddedness: "aware of who knows what in the network and the more the network is characterized by strong social ties";

(3) Organizational embeddedness: "the more organizations are involved in the network".

For these authors, "more embeddedness is better than less for knowledge sharing". (Goglio-Primard, 2014, p.4). These 'lock-in' effects are often associated with communities of practice and their boundaries (Agterberg et al., 2010). Reticular connections are thus articulated with common social identity to the community to be the privileged spaces of learning and innovation [START_REF] Wenger | Promoting and assessing value creation in communities and networks: a conceptual framework[END_REF]. At the city level, these informal networks are collective learning generators contributing to the development of territorial intelligence (Joyal, 2008).

CoP and NoP are then characterized by the member´s cooperation relationship whereas they have inter-organizational network structures, then the competition relationship may also exist.

Traditional approaches tend to oppose cooperation and competition. While the cooperation logics is inherent to community functioning, the entrepreneurs present in the industry must choose between the two logics or they can use both? To lead this reflection the literature use the notion of coopetition to understand how these two approaches can co-exist [START_REF] Bengtsson | Cooperation and Competition in Relationships Between Competitors in Business Networks[END_REF].

The coopetition logics for innovation development

Research on coopetition strategy has been growing in the last number of years.

Following an approach by networks (formal and informal), [START_REF] Bengtsson | Coopetition'' in business networks: to cooperate and compete simultaneously[END_REF] define coopetition as a "dyadic and paradoxical relationship that emerges when two firms cooperate in some activities, and at the same time compete with each other in other activities" (Bengtsson & Kock, 2000, p. 412). By adopting a strategy of coopetition, firms can:

I)
develop competitive relationships in certain markets and cooperation relations on other; II) cooperating on some elements of the value chain and compete on others; III) work together on R&D; IV) pooling of resources in the promotion of a common brand or in the case of supply of goods [START_REF] Bengtsson | Cooperation and Competition in Relationships Between Competitors in Business Networks[END_REF]. Authors in this field of study (e.g. [START_REF] Bengtsson | Cooperation and Competition in Relationships Between Competitors in Business Networks[END_REF], 2000;[START_REF] Cunningham | Competition and competitive groupings: An exploratory study in information technology markets[END_REF] suggest analysis of activities in networks, at individual and collective levels, and the various roles played by actors (Bengtsson et al. 2013).

The logic of coopetition management requires resources and different skills (Walley, 2007).

Pellegrin-Boucher (2010) identified in the computer industry three managerial approaches of coopetition: spontaneous approach, the compartmentalized approach, the integrated approach.

Table 1 presents the objectives and instruments used by these three approaches.

Table 1 -Managerial approaches to coopetition (Pellegrin-Boucher, 2010)

Although the principles of cooperation have been studied for over two decades, some concepts are still to develop (Bengtsson & Kock, 2014). Despite their advantages, coopetitive dynamics produces a vulnerability to certain firms, which need to carefully balance the dissemination and protection of knowledge [START_REF] Baumard | An asymmetric perspective on coopetitive strategies[END_REF]. For the success of coopetition, a friendly mindset is required in the network and hostility tends to diverge firms looking to maximize their own benefit [START_REF] Bengtsson | Coopetition'' in business networks: to cooperate and compete simultaneously[END_REF]. As a result, multidimensional (Allah Raza et al., 2014) and multi-level tensions (inter-organizational, intra-organizational and interindividual) can generates. According to Fernandez et al. (2014), it is necessary to incorporate a paradoxical vision to manage these tensions. Active management define "what to share, with whom, when

and under what conditions" (Levy et al., 2003, p. 642).

Despite this growing interest in the last twenty years (Bengtsson et al, 2013;Boucken el al, 2015) the primarily researches on coopetition appears from the perspective of large firms. The little-explored angle of SMEs, family businesses, and startups seems to be a promising way to further this research topic (Gast el al. 2015). Regarding the level of analysis, Boucken et al.

( 2015) show that the most explored is the level of inter-firms or organizational. Few works explore the dynamics of interactions at the network level (e.g. [START_REF] Peng | The coexistence of competition and cooperation between networks: implications from two Taiwanese healthcare networks[END_REF].

The essential skill for startups in the early stages is the capability to assemble and utilize complementary assets in spite of the limitations inherent in being a startup (Paradkar et al. 2015). To ensure a sustainable competitive advantage, startups need to obtain strategic resources that serve as entry barriers for rivals. Coopetition can be advantageous for startup, by alliances with external partners, as suppliers, customers or also competitors [START_REF] Teece | Competition, cooperation, and innovation: Organizational arrangements for regimes of rapid technological progress[END_REF]. Bouncken et al. (2015, p.19) argue that future research should aim to analyze to "what extent coopetition can really solve the liabilities of smallness and newness to the growth and success of these firms; when, how and why SME and young firms should engage in coopetition; to what degree coopetition can be linked to entrepreneurial orientation and innovation".

However, this coopetition logic can function in areas traditionally governed by cooperation?

Cooperate or compete? That may be the question of entrepreneur's members of these informal networks.

Framework: coopetition and community functioning at innovation context

Dynamic networks models (Dagnino et al. 2007) contest traditional visions about interorganizational dynamics between firms, customers, suppliers and competitors. The challenge is to capture the dynamic interaction within these networks of actors and identify the different roles they play and the effects that it produces. The coopetition thus becomes a singular search object that requires further theoretical examination, methodological and practical specific to provide a new approach to business dynamics (Bengtsson & Kock, 2014). Our conceptual framework takes account of these considerations and highlights the specificities of CoP and CoP, where relations of coopetition are studied. We analyze them through the dynamics of participation/reification [START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF] to understand how competition between their members can change the community-operating environment in which they operate.

Participation demonstrates the intensity of the mutual commitment of the members as well as the content and actions of trade within the CoP [START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF]. The analyze in this research make the link between the level of competition of the startups (level of competition in the market, the profile of the partners, relational risk and sector governance structure) with the model of participation of entrepreneurs in the CoP inter-organizational. The level of embedding of potentially competing startups in this informal network, when it is geographically disperses, may also check the intensity of their community involvement (Agterberg et al. 2010[START_REF] Goglio-Primard | Cracking the network boundaries: How Networks of Practice and Communities of Practice contribute to knowledge sharing in ICT sector[END_REF]). We will thus examine the object of exchanges between members of the CoP to identify the presence of the three types of embeddedness (in practice, social and organizational).

The concept of reification, stresses the result of these processes of participation, represented by tangible elements (concepts, symbols, stories, rituals, practices) that contribute to synthesize and simplify the shared message [START_REF] Wenger | La théorie des communautés de pratique[END_REF]. A broader reification analysis can also investigate the effects that this process produces in innovation ecosystem and in the city where the Cop is located. The coopetition logics of interaction thus mark the development of these ties. They are both the means and the result of this dynamic.

RESEARCH DESIGN

This study is a exploratory case study approach (Yin, 2002). The coopetitive relationships in an inter-organizational innovative startup community in a cluster of innovation on the Brazil explore new points of the coopetition and the functioning of inter-organizational CoP. This constitutes a phase of a larger research project focusing on competition and cooperation between community actors.

Methods of collection and analysis of data

To identify different types of coopetitive relationships between firms in SPV different actors have been selected to participate in a campaign of semi-structured interviews. Table 3 shows the profile of the interviewed as well as the codes used for the verbatim in the case study.

Table 3 -Sampling of research and codes used to build the analysis results

We also collected additional primary data. We conducted a non-participant observation during meetings conducted in the community ecosystem of events relating to a broader ecosystem of startups. A research journal updated was used to notes the contents of these exchanges. In addition, we have employed a "netnography" method research (Kozinets, 1997(Kozinets, , 2002) ) to investigate the virtual environments of firms at SPV (web site, Facebook page, Slack platform of communication). Messages from the platform of the community with its 441 members interacting on 29 different subjects (technology, business management, law, Seed, social responsibility, funding, market, etc.) have been particularly analyzed. Since the creation of this tool in December 2014, members of SPV posted more than 39,000 and archived over than 448shared archives was analyzed by the authors (inventory obtained in December of 2015).

We are interested in the social aspects of the community, content and form of sharing experiences, information, and knowledge. The use of the netnography allows researchers to be less intrusive in the process of research and thorough understanding relationships between members. It is a complementary qualitative method allowing us to enrich the results of the series of interviews and non-participating observations.

To analyses this material, we used the triangulation of data and coding in a dictionary of themes emerging (Strauss & Corbin, 2004). Three objectives were present: I) identify the functioning and evolution of SPV particularly in relation to the interactions between these members in the management of events and practice and the dynamics of innovation; II) capture the logic of coopetition, III) detail the set of transformations that occurred in the network and in relation to innovation practices.

RESULTS

The results of this study firstly focus on the cooperation relationship existed in the CoP to then present the results based coopetition relationship developed by the time. The effects of these relationships appear on three levels: at CoP, at the local innovation ecosystem, and at the city.

The community studied

The movement of creation of the Brazilian startups has emerged in 1999/2000 after the techbubble burst and was resurrected only in the past 18-24 months. As a result, stakeholders had to grow stronger and persevere through the good and the bad times by which passed the Internet industry. The advantage is that the Brazilian startup ecosystem seems willing to work hardthere is potential, the talent of entrepreneurs, capital, and entrepreneurial passion.

Belo Horizonte is one of the biggest entrepreneurial centers in the Brazil. A Strong dependence of the large and standardized worldwide characterize the organizational dynamic of Belo Horizonte ICT's cluster. However, it is also remarkable and representative a growing niche space occupied by a group of young and heterogeneous startups, which take part in distinct market structures that lead to diversified behavior and strategies of software development according to the market area. For this reason, the sector lacks more appropriate analyzes because of its particular dynamism and segmentation.

About more than 200 startups host in the city, which is more than triple from two years ago. In 2005, the sector was particularly marked by the purchase of the startup founded in 2000, Akwan Information Technology by Google Inc.

In this context directed to innovation, it has emerged the CoP San Pedro Valley (SPV) in 2011.

The adoption of this nickname appears during the conversation in a brewery of a group of startups entrepreneurs located in the district of São Pedro. The idea was to allude to the Silicon Valley American.

After this, others entrepreneurs have begun to take part in this ritual of meetings in that brewery, to relax in the afternoon, but also to discuss their current business. SPV has become a selfgoverning community of entrepreneurs sharing common goals. In fact, the members of this community believe that this kind of informal meeting represents the best way to achieve their goals thanks to the sharing of information, knowledge and experiences. Now, San Pedro Valley is already one of the greatest startup community in Brazil. It is interesting to note that the SPV community there is neither office nor an official representative.

The theoretical principles of the CoP identified by [START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF] are present in the network relationship of SPV. Table 2 highlights the features of this community. According to the members, become a membership helps to save learning costs on the management of startups and implementation of theirs strategies. The progressive establishment of a common cognitive capital constituted by the circulation and comparison of individual practices of each member at SPV community is the specific investment based the concrete of the community. In addition, this can be considered the basis of "social standards" that guide the behavior of agents. Furthermore, the recognition of this community helps attract investment and increase the visibility of local startups. The startups of SPV chose to exchange information by learning to interaction as the main one for the development of incremental improvement and innovation processes. The startups use the external sources of information to improve their technical and managerial capabilities. As soon, the quality of information and learning exchanged tend to be better and more optimized.

There are various source of information used by the startups. These sources are provided by a formal meeting organized by the members to informal conversation in leisure times. According to the interviews, the members believe that these informal events are important sources of exchange and reinforce of partnership, but also to detect interests, experiences, knowledge and necessity that strengthen the group's community spirit.

Among the sources of information, it was possible to identify:

 Participation in practicing communities: for members that were interviewed this is the main source of information. In the practice, the members trough interacting and cooperating in the creation of innovation, learning and advantage the cluster system.  Fairs and events: they are not so frequently organized in the CoP, however, the participation of SPV members in external events in Brazil and abroad is regarded by entrepreneurs as another important information sources according to the interviews;  Costumer information of the startups: it is a part of the database to capture, organize, and disseminate information;  Courses and training: organized by the institutions of the cluster and by the SPV is another efficient source of information;

For knowledge sharing the presence of a greater intensiveness in the capacity building and learning, processes are latent. This frequent exchange of information by interaction between SPV members, institutions and other actors of the TIC environment has been essential to place this SPV CoP in a development level higher than the others CoP. It is a specific individual investment, at the collective level, underlying the community cement and the base 'social standards', which guide the members' relationship (Brousseau, 2000).

« The first information, which I believe is the most important of all from SPV, is that we are a community. In addition, it changes everything. The entrepreneurs grow together. The main advantage that exists within a system as the SPV is that information interchange remains fluid. In addition, it is very rare to produce mainly on the part of large firms. Normally, the software factories are in competition". ( ET2)

"In general, the media want to spread the story, but if you want to understand the community, the functioning of the ecosystem, you have to go into the ends of the chain." (ET5)

"We try to stay in touch with everyone, meet face to face because the relationship of trust feeds the fluidity of our relationships [...] This kind of event where nothing is requested, it will be just be there and to be who you are, is very important. "(ET5)

However, of course, the logic of cooperation is not stable over time, especially as the number of members continues to grow. This recall that strategy is enacted in the performance of everyday activity (Rouleau, 2005), as people appropriate the subject positions organizationally constructed for them. Then, the agents in the CoP need to question how to manage the massive growing without losing the control? Entrepreneurs must continually exchange the signs on their desire to continue to adopt a cooperative behavior in this non-opportunistic logic.

Gradually, as they develop a cooperative process, the growth of cognitive capital contributes to making the sharing of increasingly less costly signals, since the potential opportunistic behavior is detected. An example of this practice is that community members tend to share information with those members that are closest to the meetings and interacting more frequently. We realize that not all members of the community are in direct interaction, but those who are mostly active tend to be benefited with the cognitive capital gains from the SPV community.

The remoteness and a low degree of formal relationship between local entrepreneurs and the entities in the ICT cluster, according to our collected data, is due initially to divergence of interests between startup entrepreneurs and the others entrepreneurs from organizations and government agencies Cluster. But, ultimately, SPV entrepreneurs will have to prove their worth by way of execution. That implies design strategies together or not with the cluster, in order that fit the local market, and then implementing with excellence, efficiency, and an orientation to cash-flow. This is a commitment to working closely with the portfolio of the CoP SPV to build-up an entrepreneurial culture to go along.

These events allow us to identify three distinct phases of evolution from SPV connected to the participation and cooperation of entrepreneurs: This evolution allows observing that the practice is moving whereas the entrepreneur's network evolves without however weakening the embeddedness of practice and organizational structure of the startups.

Coopetition and reification in the CoP

Although the cooperative aspect of the relationship, the startups of SPV also competes with each other for market share, investors, technology, customer considering that many of these firms were represented in the same markets. Furthermore, some members compete for representation of SPV at structuring meetings. Indeed, some dates suggest that it is always the same entrepreneurs who give interviews to the press and that SPV is a closed group of firms who want to make themselves known. To manage these contradictions, members must exchange permanently on the signals that show commitment and adoption of a cooperative intention in detriment of opportunistic logic. This is manifested by control of participation.

"Joining SPV is easy, it is just sign up, and there you will have information on the different event [...]. However, being part of the WhatsApp Group is harder... So they do a search frequently to see who participates or not to the group. And people that who have published nothing and who do not exchanges are removed from the group to give to leave place for others entrepreneurs more motivated to exchange." (ET4) "There are many new people who join us now with the Slack. However, the WhatsApp Group is more restrictive. Therefore, we organized a SPVCerva [title of the event of the community, which aims meetings over a glass of beer] to meet those who want to drink a beer and discuss... » (E4-sur Slack)

However, the literature on CoP tends to not explore the phenomena of exclusion [START_REF] Castro Gonçalves | La face cachée d'une «communauté de pratique technologique[END_REF] and assume generally that the risk of opportunistic remains very measured, for two major reasons:

-because the frequency of interactions tends to intensify and reinforces the establishment of social norms and shared routines -and helps to reduce the risk of hold-up and the problems of moral hazard (Wenger, 2008). Therefore, these theoretical principles do not exclude the emergence of coopetition relationships in the CoP.

Taking into account the level of competition to the local ecosystem, the startups seek to establish different alliances in accord to their level of evolution (entrepreneurial traction phase -ETentrepreneurial maturation phase -EM, entrepreneurial validation phase -EV) Analyze also shows the presence of competition between CoP members and others stakeholders in the ecosystem of local innovation. All of them looking for the access to the same market, the use of similar technologies, and the access to the government grants or to leading position in the ecosystem (face the players, the representative institutions of the sector at national level, the government or the investors).

DISCUSSION: THE EFFECTS OF THE COOPETITION AND THE DIFFERENT LEVELS OF ANALYSIS

This research presented that the logic of cooperation and coopetition are taking place simultaneously inside the CoP but go beyond its borders. The nature of the partnership with the community changed over time. Cooperation, competition, and coopetition are thus expressed at different levels:

At the level of the CoP and the NoP: the common platform of the interaction of SPV raises the cooperation among its members nevertheless the study finds little evidence of "control actions" even if there is not a "formal" sense of commitment to structure and identity this in their relationships. Within SPV, the "startup domain" guides the questions, so stimulating members to present their ideas for introducing or contributing to a discussion of problems and solutions. The group of entrepreneurs grows and reinforces actions to meet regularly, but not in a very structured way. The mutual engagements, built by members of the community of SPV leads to the creation of it shared meaning on issues in startups sector. The community becomes a locus that enables startup entrepreneurs to learn by engaging in simple tasks, participation and involvement in the community live. According to the community of practice model Wenger's (1998), the three dimensions were observed: joint enterprise, mutual engagement and shared repertoire. In spite of the organizational culture conducive to shared governance, some forms of leaderships emerge. Some members are more involved than others in events; in media representation; in meetings with others actors (public power and private power) and in contact with institutional organizations of the ICT ecosystem, (being part of this small group appears as a form of power and "dispute" between the others startups). The entrepreneurs "leaders" are part of the group of entrepreneurs in the traction phase (the startups that are already well stabilized in the market) but also to, entrepreneurs in other phases, have strong legitimacy in the network. This group of entrepreneurs has a high degree of official recognition in the NoP and in the ICT ecosystem, even though they deny this power. They exert a control over the content of the networks. The majority of these companies are considered "success case" in their sector. The embeddedness of the practice (Agterberg et al., 2010) appears to play a decisive role on the actions of the NoP to keep common social identity despite the geographical dispersion and the number of members increased. The growing of SPV inside the ICT ecosystem is also supposed to favor the co-creation of value with other communities and networks outside [START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF].

At the level of local innovation ecosystem:

The inter-organizational collaborations between the NoP become an important part of local innovation ecosystem strategy to deal with business dynamics and higher uncertainties of ICT cluster. The series of phases of cooperation and competition does not exclude cooperation in the competition phase. Indeed, the rival firms continue to work together, combining these two activities in an inherent paradox. Dagnino, 2007). The NoP simultaneously stimulates cooperation through the promotion of the ecosystem of ICT of the city (investment in the necessary technological infrastructure for the sector, the workforce skills, support for the creation of a support for startups and incentives for the internationalization of enterprises of the network). These actions show a positive relationship between coopetition and market performance in the city [START_REF] Ritala | Coopetition strategy: when is it successful? empirical evidence on innovation and market performance[END_REF].

Coopetition transversal relationships have also been introduced within the ecosystem. For example, SPV has exceeded the boundaries of the region and aims to represent all the startups of the ICT cluster of the city and therefore receive more visibility by attracting the interest of private investors, Government and Society. Social responsibility is another example of transversal knowledge domain, representing an opening of practice as a vector to maintain a mutual engagement of its members, the embeddedness of practice and avoid that competition effect negatively the cooperative dynamic that exist between members. It is a source of collective learning encouraging the interaction of the CoP with its ecosystem of innovation and the city where it evolves. This configuration, evolution and organization of SPV occur along a continuum between CoP and NoP generated by a progressive opening movement to the participation of new members within the network. This movement gave him more legitimacy in the ecosystem of ICT.

The recognition of the legitimacy of SPV implies the need to manage competing and opportunistic logics to relationships of coopetition. Cooperation within competing firms can lead to win-win potential conditions. The effects of this dynamic of cooperation on the network appear as a true source of value creation through the establishment of trust relationships [START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF] between the startups of SPV. Relations of competition encourage startups to improve in order to maintain their competitiveness, while cooperation provides access to new resources [START_REF] Bengtsson | Cooperation and Competition in Relationships Between Competitors in Business Networks[END_REF], 2000).

Conclusion

Bengtsson et al. (2010, p. 210) stress, "There is a lack of knowledge about the effects of coopetition and different types of coopetitive interactions" and underline the need to follow a "systematic empirical research" based on their conceptual development. This study seeks to in with a focus on the context of innovative startup community.

Organizational analysis of the coopetition within an inter-organization community of practice by the dynamics of participation/reification [START_REF] Wenger | Communities of Practice -Learning, Meaning and Identity[END_REF] [START_REF] Wenger-Trayner | Learning in Landscapes of Practice: Boundaries, identity, and knowledgeability in practice-based learning[END_REF]. This is future research directions that would provide a better understanding of the dynamics of innovation in a specific geographical area.

Domaine de savoir partagé

Partage de connaissances et échange d'expériences sur la création et le fonctionnement de startups de haute technologie Fonctionnement Rencontres informelles en coprésence. La participation aux rencontres n'est pas obligatoire mais les entrepreneurs s'impliquent spontanément et régulièrement. Fondée sur un principe de coopétition puisque certains membres coopèrent malgré le fait qu'ils soient des concurrents Offre spontanée d'aide lorsqu'un membre de la communauté est en difficulté par la proposition de solutions et le partage de connaissances (par ex. échange sur de bonnes pratiques de gestions, le choix de serveurs, informations et actualisations et nouveautés des langages de programmation, la programmation en général) (engagement mutuel). SVP a un jargon propre et des histoires partagées (par ex. lors de discussions pour le développement de leur plateforme d'échanges, l'un des membres fait une blague en faisant allusion à un langage spécifique informatique spécifique utilisée par les entrepreneurs et un autre récemment arrivé ne l'a pas comprise. L'histoire est devenue une dessinée qui raconte une partie de l'histoire de la communauté). Implication de certains membres de la communauté à l'écosystème d'innovation en TIC de la ville de BH (ces membres de SPV participent aux réunions du organisées par le gouvernement de l'Etat de Minas Gerais et d'autres dans le cadre de programmes pour le développement de startups et/ou d'innovations. Disponibilité d'un répertoire de ressources partagé par les membres (sur les chaînes thématiques de Slack e sur le site web de SPV).

Capacité de la pratique

Reconnaissance partagée de l'amélioration de la pratique des membres de par leur participation à la communauté (par ex. : le développement de nouvelles technologies dans les startups du réseau, l'identification de nouveaux marchés par une startup en utilisant les services d'outre startup, l'établissement de partenariats entre startups de marchés complémentaires, la création d'un programme d'aide financière et technique d'un startup déjà consolidée dans le marché et en phase de traction pour soutenir une startup en phase de validation d'idée). Alimentation du répertoire partagé avec de nouvelles histoires (une commission a été nommée pour la création d'une rubrique sur le site web de SPV avec l'objectif de mieux exposer les startups membre de la CoP), d'outils technologiques (site web, fan page sur Facebook, un groupe dans le WhatsApp et un groupe sur Slack) et de symboles (la marque SPV avec dépôt de droit et un slogan « We ? SanPedroValley »).

Appartenance Institutionnelles de personnes interviewées

Figure 1

 1 Figure 1 illustrates how to investigate the coopetition under this framework.

Figure 1 :

 1 Figure 1: Analytical framework of coopetition in CoP and NoP.

"

  There is always a startup on the front and one behind [making allusion to the competitive position of the startups]. It is what makes that information either flue and hence reduce the learning curve in the community" (ET4)

  The cooperative relations are expressed according to the integrated mode(Pellegrin-Boucher, 2010). These strategies (cooperation and competition) incorporate the SPV practices and allow the startups to differentiate other competitors as a dialectic 'differentiation-integration'[START_REF] Pellegrin-Boucher | Coopetitive Strategies in the ICT Sector: Typology and Stability[END_REF]. For example:-Startup (ET2) exchanges knowledge and more tangible resources as the formation of an employee competitor that already holds the expertise (ET1); -Startup (ET4) gets a market study by a competing startup (EV2) and which in this case becomes business partner; -Resolution of several technical issues of information by competitors members of the CoP. This type of exchange takes place daily in the Slack group and Meetups organized by members; -An already well-developed startup (ET6) that hosted free competitors that did not have the means to pay the rental following the abrupt closure of a public incubation program (Seed MG in the month of April 2015).

  by exchanges between entrepreneurs of innovative startups effect of different natures in the ICT sector in terms of open innovation (Chesbrough, 2003) : (I) Sharing of expertise of firms contribute to reduce the learning curve (very relevant for startups who must innovate faster) ; (II) Reduction of the dependency and the need of external resources through the sharing and exchange of resources within the inter-organizational network; (III) Recognition of the community power can attract investors and new partners of the local innovation ecosystem. The network of startups can be considered as an institutional innovation as it reduces the limitations of companies and promotes innovation in the territory. CoP appears as a broker (manager facilitators) in the local innovation ecosystem in that it allows, and a boundary objects (forum at Slack, meeting) through the relations of coopetition of its members to disseminate widely knowledge (Castro Gonçalves, 2015). The issues highlight by this dynamic, based on social relationships, evolves with respect to the development of the territory and under a movement of "host" of new members within the network. Another issue was the enhancement of the relationship with other players at the ecosystem. These results are in line with the research on the development of knowledge in the extension of organizational boundaries (Castro Gonçalves, 2012). It is also observed by analysis that spontaneous coopetition (Pellegrin-Boucher, 2010) is advantageous to the startups by the possibility to overcome knowledge asymmetries by the establishment of internal interconnections (between firms) and external interconnections (between customers, suppliers and with the companies of the same sector through the formation of networks) in the city of Belo Horizonte. These interconnections change the competition practices in the field of ICT and promote an intensification of the activities of intermediation and construction of the NoP. Strategies of coopetition may also generate defensive strategies against external or internal threats of the CoP. The community becomes an actor to the development of innovation in the ecosystem as a collective response to the lack of resources and power. The originality of this research could be strengthened by analyzes of other inter-organizational communities in other areas of innovation. The results of this research are indeed very attached to the local context. From the theoretical framework, a more pronounced analysis on the interaction between different CoPs or other innovation networks would further highlight the coopetition wider effects on the body of knowledge

Table 2 -The three key elements of a CoP (Wenger, 1998) Cooperation and participation in the CoP

 2

The community of San Pedro Valley (SPV) has emerged from the conversations and encounters a group of entrepreneurs of startups that «suffered» for the lack of resources. It appears as a reaction to the lack of support and offset in terms of operation compared to the other players in the local innovation ecosystem.

"I see two distinct industries [SPV and cluster of Belo Horizonte], as two distinct sectors. '' We do

not need to go to the MGTI because we have different logics. The logic of a startup is a different logic of a large factory of software.

" (ET1)

At the level of the city: the

  The dynamics of interaction of SPV integrates the local ecosystem. Faced to the institutional pressures of the Government of the State of Minas Gerais in an attempt to control and homogenize the startups, SPV expresses defensive cooperation practices. A group of SPV entrepreneurs met to write public letters against the «attempt of creating a control manifested by attempt to incorporate SPV firms to the public programs of the city of Belo Horizonte» in defense of the "independence and economic and political autonomy of SPV" and against the closure of Seed, the accelerator program created by de Government of Minas Gerais. role played by CoP and boundary relations in the development of NoP consolidates intra-organizational learning in the city. The NoP seeks to know its impact on job creation and the market share in the city in order to be better organized, creating a reliable database. This will allow SPV assert itself as an institutional actor in the city. The tools of technological NoPs (Slack, E-mail, WhatsApp, Facebook page) enable to innovate entrepreneurs to identify the technologies and practice on similar projects and to find who knows what and where they are located (Social Embeddedness of the NoP). This is detected by the growth in the number of new members in the practice network (passage of 3 members in 2011 to almost 300 members in 2015) and especially by the mechanisms of integrating knowledge scattered among entrepreneurs of startups. SPV has an impact of changing and unstable environmental conditions on firms' coopetitive strategic behavior (Padula and

	opinions have been expressed in several meetings recorded and documented. These records and
	documents are available to interested parties on request' "(Public Letter written by members of SPV
	collectively in Slack)
	"The program was internationally known and considered one of the most important
	entrepreneurship development initiatives in Brazil. […] Many entrepreneurs have manifested their
	reactions to SEED's HQ closing. The consensus is that we will do everything we can to keep the
	program legacy alive.
	San Pedro Valley existed long before SEED, and will keep existing after its gone -welcoming new
	friends made through this wonderful program.
	Our community will remain strong and promoting entrepreneurship -apart from politics. Our
	beliefs are solid, and we will keep developing entrepreneurs for the future of Brazil" (Entrepreneurs
	of San Pedro Valley in medium.com)
	"San Pedro Valley (SPV), a community with over 300 startups of the metropolitan region of Belo
	Horizonte, has just publicly clarify the present relevant points of the speech of the Governor [...] and
	his Secretaries about alleged integration of SPV or rigging with a public entity. 'SPV does not
	respond or do not belong to any political party. The community will remain independent and will not
	be controlled by an organization, a party, a coalition or political interest of right, left, center, present
	in power or the opposition.' 'SPV will be always open to innovation and entrepreneurship and is
	open to speaking and to help you. About the subject of the SEED, our suggestions, critics and

Echantillonnage de la recherche et codes

		Nombre d'interviews	Codes
		(moyenne de 1h30	utilisés pour
		par entretien)	les verbatim
	Associations professionnelles et syndicales locales et		ASPRO1,
	nationales présentes dans l'écosystème d'innovation	3	APRO2,
			ASPRO 3
	Entrepreneurs membres de SPV	18	ET1, ET2,
			ET3...
	Institutions mixtes partenaires (Université, entreprises et
	état local), chargées d'accélérer le développement d'idées	3	INST1,
	d'innovation (programme SEED) et de développer la		INST2,
	recherche de partenaires (programme MIC)		INST3
	TOTAL	24 entretiens / 37 heures d'entretiens
	Tableau 3 :

Objectifs

Répondre à des besoins ponctuels S'adapter à une situation donnée

Tester la coopétition Répondre à des besoins stratégiques concernant l'innovation, la productivité, l'internationalisation. Limiter la coopétition à certains domaines de l'entreprise.

Répondre à des besoins stratégiques concernant l'innovation, l'internationalisation, la productivité. Généraliser la coopétition à tous les domaines d'activité de l'entreprise.

Instruments privilégies

Mise en oeuvre de partenariats ponctuels par la direction de l'entreprise ou certains managers.