## Abstract

We show analytically and numerically that parabolic pulses and similaritons are not always synonyms and that a self-phase modulation amplification regime can precede the self-similar evolution. The properties of the recompressed pulses after SPM amplification are investigated. We also demonstrate that negatively chirped parabolic pulses can exhibit a spectral recompression during amplification leading to high-power chirp-free parabolic pulses at the amplifier output.

©2006 Optical Society of America

## 1. Introduction

Self-similar propagation in high-power fiber amplifiers has generated considerable interest since the first theoretical and experimental demonstration of the phenomena in 2000 [1]. It can be shown that any initial waveform propagating under conditions of normal dispersion, non-linearity and gain will evolve asymptotically into a similariton, a pulse characterized by a parabolic intensity profile with a linear positive chirp. This pulse will then undergo self-similar (SS) dynamics with its peak power and temporal and spectral widths increasing exponentially with the propagation length. The only issue is one of the time/length scales required to achieve this. The generation of parabolic pulses has been demonstrated in various fiber-based amplifiers, including both rare-earth doped amplifiers (Ytterbium [1–3] and Erbium [4]), and Raman-amplifiers [5]. Typical characteristics of asymptotic behaviour, such as the property that the generated parabolic pulse is independent of the initial pulse shape, have already been experimentally verified [6].

However, to date, no clear distinction has been made in the literature between the asymptotic self-similar evolution and the amplification of pulses that simply have a parabolic input pulse shape. However, with the rapid advances in similariton lasers [7, 8], and the use of linear pulse shaping techniques, for example those based on superstructured fiber Bragg gratings [9], it has become possible to reliably generate pulses with a parabolic pulse shape (either with or without a linear chirp), and to use these pulses as an input to a fiber amplifier system. The generalised expression for a parabolic pulse of energy *U*_{P}
= *4 P*_{P}* T*_{P}
/ *3* √2 is given by :

where *P*_{P}
is the peak power of the pulse, *T*_{P}
is the temporal full-width at half-maximum (FWHM) and *C*_{P}
the linear chirp coefficient.

We demonstrate herein that the dynamics involved in the amplification of parabolic pulses can be distinctly different from the asymptotic self-similar evolution usually referred to when speaking about parabolic amplification. We show that the dynamics in the initial stages of pulse evolution is not self-similar at all, but dominated by self-phase modulation (SPM). Only after a sufficient propagation length do the pulses become self-similar. We also investigate the quality of pulse compression that can be achieved using simple linear chirp compensation and show that a parabolic pulse shape represents the optimal initial pulse shape in terms of compressed pulse quality within the SPM amplification regime. We also study the effect of an initial linear chirp on pulse evolution and show that a negative initial chirp can lead to a spectral compression, resulting in chirp-free high-power parabolic pulses at the amplifier output.

## 2. Self-similar and self-phase modulation amplification regimes

#### 2.1 Influence of the temporal initial width

We consider in this section the amplification of chirp-free Gaussian pulses with an FWHM temporal width *T*_{0}
and an initial energy *U*_{0}
. The evolution of the complex electric field *ψ* can then be modelled using the non-linear Schrödinger equation (NLSE) including a longitudinally and spectrally constant gain coefficient g, a second order dispersion *β*_{2}
and a non-linear coefficient *γ*. Note that we have ignored the effect of higher order dispersion, finite gain bandwidth of the amplifier, and stimulated Raman scattering throughout the analysis that follows;

To illustrate our analysis, we have considered in this paper the amplification of pulses with the same initial energy *U*_{0}
= 50 pJ but with different initial temporal widths *T*_{0}
ranging between 0.8 and 16 ps, in an amplifier based on a erbium doped fiber with parameters *β*_{2}
= 40.10^{-3} ps^{2}.m^{-1}, *γ* = 6.10^{-3} W^{-1}.m^{-1} and g = 3 dB.m^{-1} at telecom wavelengths (1550 nm). We have plotted in Fig. 1(a) the longitudinal evolution of the temporal width of the pulses during amplification. We can see that shortest pulses quickly converge to the asymptotic SS solution *(red circles)* which corresponds to a parabolic pulse with parameters *P*_{P_SS}*, T*_{P_SS}
and *C*_{P_SS}
as given in [1], regardless of the initial shape :

The longer the initial pulses, the greater the distance required to converge to the asymptotic SS behaviour. Indeed, there is a vast region within which longer pulses do not exhibit any change in their temporal width. In practice, limiting effects not included in Eq. (2), such as the finite gain bandwidth [10] or Raman scattering [11] may prevent the pulses reaching the asymptotic SS regime.

We can calculate the ratio between the linear length *L*_{D}
and the non-linear length *L*_{NL}
[12]:

SS propagation requires a balance between the linear and nonlinear effects. However, in the case of long initial pulses (for a given initial pulse energy) the non-linear effects dominate in the early stages of propagation. In this instance the propagation of longer pulses (ps pulses in our case) can to a first approximation be modelled using Eq. 2 but setting *β*_{2}
= 0. In this case, the temporal pulse shape is unaffected by the amplification process [12], as shown in Fig. 1(b) where we have plotted the evolution of the misfit parameter *M* between the pulse intensity profile |*ψ(t)*|^{2} and a parabolic fit |*ψ*_{P_FIT}*(t)*|^{2}:

For short pulses, *M* quickly decreases to zero, confirming that the initial Gaussian pulse converges to the parabolic intensity profile. For longer pulses, the beginning of this evolution comes later. We can also see that the decrease of *M* is not monotonic, which suggests that the dynamics towards the SS regime for longer pulses may be more complex than just a decrease in the energy contained in the exponentially decreasing similariton wings [4, 13, 14].

#### 2.2 Comparison between different initial pulse shapes

We now compare the evolution of pulses with the same energy *U*_{0}
= 50 pJ and temporal width but with different initial shape, (Gaussian, sech, and parabolic). We have plotted Fig. 2(a) the evolution of *M* for pulses with 0.8 and 8 ps initial pulse durations.

Let us first consider the use of 0.8 ps initial pulses. We see in this case that the initial pulse shape which converges the fastest to the parabolic shape is the Gaussian pulse. This behaviour is qualitatively consistent with the conclusions of Ozeki *et al*. [15] who predicted that the optimum initial pulse shape was a Gaussian chirp-free pulse associated with an initial *L*_{D}
/*L*_{NL}
ratio of 2.1 *β*_{2}
/ g (which leads to a value < 1 for usual amplifier parameters). We note that the initial 800 fs parabolic pulse does not retain its parabolic shape during propagation, leading to a significant increase in the misfit function. To explain this behaviour, let us note that the ratio *L*_{D}
/*L*_{NL}
is only 9 for 800 fs, 50pJ pulses. One has to keep in mind that a parabolic pulse remains parabolic only in the high intensity regime [16], i.e. when the effects of nonlinearity are dominant, which is not the case for 800 fs pulses. To outline the dramatic effects of dispersion on the evolution of a parabolic pulse, we have plotted in Fig. 2(b) the evolution of a 800 fs parabolic pulse in a purely dispersive medium (non-linearity is neglected here).

We now consider the evolution of 8 ps pulses. As already mentioned in part 2.1, for such an initial temporal width, the non-linearity becomes the predominant effect (the ratio *L*_{D}
/*L*_{NL}
has increased by an order of magnitude compared to the 0.8 fs initial pulse case). In these conditions, we see that an initial parabolic pulse remains parabolic, with only a small increase in the *M* factor. Regarding Gaussian and sech pulses, both shapes have not yet completely entered the asymptotic regime after 14 meters of propagation (see Fig. 2(a)). Let us note that here, we have launched a transform-limited parabolic pulse, and not a similariton, which would have initially a linear chirp with a value *C*_{P0_SS}
= g/3 *β*_{2}
and for which the energy *U*_{P0_SS}
and temporal width *T*_{P0_SS}
would follow the relation:

This would lead to a value of the energy of the initial pulse of 26 nJ. The value used here for the 8 ps pulse is very far from this optimum value required to directly obtain asymptotic self-similar behaviour [14].

## 3. Parabolic pulses in pure self-phase modulation regime

We now concentrate on the evolution of initial parabolic pulses *ψ*_{P0}
(*t*) in the SPM amplification regime (where *U*_{P0}*, P*_{P0}*, T*_{P0}
and *C*_{P0}
represent the initial properties of the pulse). We introduce the normalised field *$\widehat{\psi}$*(*t*) and parameter *$\widehat{\gamma}$* defined by :

so that Eq.(2) can be rewritten under the NLSE-like from:

In the first instance, we completely neglect the effects of chromatic dispersion, so that the normalised parabolic temporal intensity profile will undergo no changes. Under theses conditions, the resulting field can be expressed as :

The effect of the propagation is then a modification of the linear chirp coefficient which can be expressed as:

The other parameters of the pulse evolve as:

A comparison of Eq. (10) and (11) with expressions obtained in the asymptotic regime (Eqs. (3)) highlights several physical differences, such as the fact that expressions (10) and (11) are dependant on the initial temporal pulse width and chirp whereas the asymptotic solution is completely independent of these parameters. In contrast to the SS regime, where *C*_{P_SS}
is only determined by *β*_{2}
and g, the linear chirp coefficient *C*_{P_SPM}
is γ-dependent and evolves along z in the SPM amplification regime.

The difference between the SPM and SS amplification regimes can also be clearly seen in Fig. 3(a) in which we present the longitudinal evolution of the pulse chirp coefficient during amplification. The figure emphasizes the fact that the amplification of longer parabolic pulses is dominated by SPM at the beginning of the propagation (blue diamonds, Eq. (10)) and only after then by SS evolution (red circles, Eq. (3)). As a comparison, we have also plotted the evolution of an initial 800 fs Gaussian pulse which converges more quickly to the asymptotic regime.

We also consider the evolution of the spectral properties of the parabolic pulses within Fig. 3(b). The spectrum of a parabolic pulse, in the limit of a highly linearly chirped pulse [14, 16], is expressed by a parabolic spectral profile with a linear spectral chirp. The FWHM spectral width *f*_{P}
can then be evaluated by :

Figure 3(b) illustrates the distinctive difference between the SPM and SS regimes. We also note that the amplification of parabolic shaped input pulses leads to spectrally broader pulses than those obtained using Gaussian or sech pulse shapes. This is illustrated in the example of Fig. 3(c), where it is also shown that parabolic pulses lead to nearly parabolic spectra, whereas the use of other pulse shapes lead to highly modulated spectra, a typical signature of the effects of SPM [12, 17].

## 4. Recompression of pulses in the self-phase modulation regime

We study in this section the quality of compression that can be achieved in the SPM regime with a simple linear chirp compensation. The initial parabolic pulse shape leads to the best quality pulses, exhibiting only small pedestals. Indeed, the expression *ψ*_{CP}
of a perfect parabolic pulse after recompression [18] is given by :

where *J*_{1}
is the first order Bessel function of the first kind. The intensity profile is then proportional to the (*J*_{1}*(x)/x)*^{2}
function, a function which is also involved in the well-known Fraunhoffer diffraction pattern of a circular aperture. The FWHM temporal width *T*_{CP_SPM}
and peak power *P*_{CP_SPM}
of the recompressed parabolic pulses can be expressed by :

where Γ is the gamma function.

As can be seen Fig. 4, using an initial parabolic shape, less energy remains in the wings (Fig. 4(a)) and the temporal width of the recompressed pulse is shorter (Fig. 4(b)) compared to the results obtained with either Gaussian or sech pulses. As a result, the peak-power of the pulses after recompression is significantly enhanced by using an initial parabolic shape (Fig. 4(c)).

## 5. Influence of the initial pulse chirp

In this final section, we examine the influence of an initial chirp on the pulse evolution. To date, studies have been restricted to only parabolic pulses with a positive linear chirp. But with the advances of Fiber Bragg gratings pulse shaping methods, it has also become possible to generate chirp-free initial pulses, or even parabolic with a negative chirp profile.

#### 5.1 Description taking into account the dispersion

To have a more complete description of the behaviour of the parabolic pulse we extend our analysis to incorporate the effects of dispersion thereby allowing any potential effects due to temporal intensity profile changes to be taken into account. Note that the analysis presented here is an extension of that presented by Anderson *et al*. for the case of a constant *γ* coefficient [16]. We first express the normalised field *$\widehat{\psi}$*(*t*) in terms of amplitude *Â*(*t*) and phase *φ*(*t*) so that Eq. (8) leads to a system of two coupled differential equations :

We consider the situation of a high intensity pulse (for which the term 1/*Â* ∂^{2}
*Â*/∂*t*
^{2} can be neglected) and we introduce the chirp *δω* = -∂*φ*/∂*t* and power *P̂*(*t*) = |*Â*(*t*)|^{2}:

Using the general expression of a parabolic pulse (Eq. (1)) with the normalised peak power *P̂*_{P}
, we obtain a system of three differential equations :

that can be rewritten in the following form :

We now have a system of equations which completely describes the evolution of the parameters of the parabolic pulse within the amplifier (similar expressions have already been derived by [13] and [19] in the context of the asymptotic propagation of positively chirped pulses). The first second-order differential equation can be solved numerically with the initial condition *T*_{P0}
and (∂*T*_{p}
/∂*z*)_{z=0} = *β*_{2}* C*
_{P0}
*T*
_{P0} . The initial chirp will then determine the initial evolution of the pulse, leading to an initial phase of temporal broadening for positive *C*_{P0}
values and a compression for negative values (for an initially chirp free pulse, *T*_{P}
will remain unchanged in the first stages of propagation, consistent with the assumption made in Section 3).

In Fig. 5(a) we plot the longitudinal evolution of the chirp coefficient for initial pulses with different initial chirp values (0, 0.064 and -0.064 THz/ps, blue, red and green curves respectively). We can see very good agreement between the numerical integration of (2) and the solution obtained based on equations (18), the general modelling proposed by Eqs (18) being valid both in the SPM and SS amplification regimes, for any sign of initial chirp.

We have also plotted the evolution of the spectral width *f*_{P}
. By using Eqs. (12) and (18), the spectral width of the parabolic pulse can be evaluated by :

We can see Fig. 5(b) that the agreement between Eq. (19) and numerical integration of (2) is not perfect. This discrepancy is mainly due to the fact that Eq. (12) is strictly valid only for highly chirped pulses, which is not the case for chirp-free initial pulses below the distance of 7 metres. The same approximation affects the initially negatively chirped pulse at a distance of between 5 and 7 meters along the amplifier. Special attention will be devoted to the behaviour of the parabolic pulses in this region below.

#### 5.2. Spectral recompression of parabolic pulse

We study more carefully the propagation of parabolic pulses with an initial negative linear chirp. In this case, we can see Fig. 5(b) that the pulse undergoes an SPM spectral compression [20–22]. Indeed, during the amplification, the linear chirp coefficient progressively increases and at a distance *z*_{c}
becomes null. At this distance, the pulse is a chirp-free parabolic pulse. Such an evolution can also be qualitatively understood from Eq. (10) which shows that after a distance *z*_{c}
= *ln[(1 - C*_{P0}${T}_{P\mathit{0}}^{\mathit{3}}$* g)* /*U*_{P0}
3√2 *γ]* / g, the chirp *C*_{P}
is expected to be zero. We can draw here a parallel between the action of the dispersion which can compensate a positive linear spectral chirp by introducing an opposite linear spectral chirp, and the action of non-linearity which can compensate a temporal negative chirp by introducing a linear temporal chirp for the case of a parabolic pulse. As a consequence, dispersion provides a compression in the temporal domain, whereas non-linearity provides a compression of parabolic pulse in the spectral domain.

The detailed evolution of this spectral compression is illustrated in Fig. 6(a). A large spectral compression is achieved with the resulting spectrum after a distance of *z*_{c}
= 6.6 m becoming characteristic of a transformed-limited parabolic pulse. We have compared the spectral recompression obtained with an initial parabolic pulse and the results obtained both with sech and Gaussian pulses (the different results are compared at optimal spectral compression distance). We can see in Fig. 6(a) that use of parabolic pulses leads to the most efficient spectral compression, with only weak spectral substructure compared to the large wings that appear for other input pulse shapes. The quality of the spectrally recompressed parabolic pulses is confirmed in Fig. 6(b) where the temporal intensity and phase profiles of spectrally recompressed parabolic, sech and Gaussian pulses are presented. Each of the different pulse shapes lead to a flat phase at the center of the pulse however both the Gaussian and sech exhibit large deviations of the phase in their wings, resulting in non transform-limited pulses [21].The only pulse which can be considered as transform-limited is the one arising from the use of a parabolic input pulse shape.

We have thus demonstrated a very promising means to generate high-power parabolic transformed-limited pulses, through SPM spectral recompression of parabolic pulses with an initial negative chirp. Such pulses can be easily generated either directly by using linear pulse shaping of short laser pulses [9], or by combining the output of similariton lasers or amplifiers with a pair of dispersion gratings which will overcompensate the positive linear chirp of the similaritons.

## 6. Conclusion

We have studied the amplification of parabolic pulses within fiber amplifier systems, highlighting the fact that for longer pulse widths an SPM dominated regime precedes the SS regime. Analytical and numerical results show that the evolution and the properties of the parabolic pulses in this regime are distinctly different from those of similaritons. Compared to Gaussian or sech pulses, parabolic input pulses amplified under the SPM regime can lead to a significant improvement in the quality of the compressed pulses. The behaviour of parabolic pulses with differing degrees and signs of chirp have also been investigated. A negative linear chirp coefficient can lead to a spectral recompression process due to the effects of SPM, generating transform-limited parabolic pulses. We believe that this detailed analysis of the behaviour of parabolic pulses in the non asymptotic regime may have major implications in the development of high-power parabolic amplifier chains.

## Acknowledgments

Christophe FINOT gratefully acknowledges financial support for this work through the European Union Marie-Curie Fellowship scheme.

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