## Abstract

I model the nonlinear fiber laser using an expanded Ginzburg-Landau equation (GLE) which includes the self-steepening (SS) and intrapulse Raman scattering (IRS) effects. I show that above a minimum value of the Raman effect, it is possible to find two chirped solitary pulses for the laser system. The smaller chirped solitary wave corresponds to the dispersion-managed (DM) regime whereas the larger chirped solitary wave corresponds to the so-called *similariton* regime.

© 2006 Optical Society of America

## 1. Introduction

A nonlinear laser system is a complex device that requires extensive and careful numerical simulations to optimize its operation. Distributing the linear and nonlinear elements along the axis of the laser results in a nonlinear differential equation [1]. The solitary pulse solution of this approximate distributed model gives an average behaviour of the pulse profile of the system which can be helpful when carrying out numerical simulations. In a previous paper [2], I have shown that the spectrum of the chirped solitary pulse of the distributed GL model yields a realistic temporal profile of the unchirped pulse generated by the laser. As a matter of fact, in a DM fiber laser operating in the negative average dispersion regime, the predicted pulse width (see Eq. (6) in Ref. [2]) was confirmed by several experimental results. This particular operating point (Γ*L*=2*π*,*β*=1) also implies that all these systems will generate pulses with the same peak power for any average negative dispersion. For the positive average dispersion regime, no specific operating point emerges from this model. Hence, in this paper, after extending the GL distributed model by including higher-order nonlinear terms such as the SS and IRS effects, I shall show that the resulting solution of this equation leads to two specific values for the chirp parameter *β*. The solitary pulse having the largest chirp parameter *β* has a spectral profile typical of the profile observed experimentally and numerically in the so-called *similariton* regime [3–5].

## 2. The distributed laser model

The typical fiber laser cavity consists of two concatenated fiber segments with normal dispersion, gain and anomalous dispersion followed by a mode-locking mechanism. In order to study numerically this complex nonlinear system, a master differential equation [6] has been derived and is commonly known as the extended nonlinear Schroedinger equation (ENLS) where the different physical effects are uniformly distributed along the propagation axis x. Here, in order to study the propagation of very short pulses, I have included the third-order dispersion (TOD) term (*β*
_{3}), SS (
$\frac{\gamma}{{\omega}_{0}}$
) and the IRS (*T*_{R}
) effects. The second order disperion (SOD) term *β*
_{2} is complex (*β*
_{2}=$\overline{\beta}-2{\mathit{\text{igT}}}_{0}^{2}$), where $\overline{\beta}$ is the group velocity dispersion (GVD), *T*
_{0} is the inverse gain bandwidth, g stands for the gain and *l* for the loss. The nonlinear parameter *γ* is complex (*γ*=*γ*
_{0} (1+*iε*
_{0})) where *γ*
_{0} is the Kerr nonlinearity and where the small saturable absorber parameter *ε*
_{0} stands for an approximation of the mode-locking mechanism. Following the presence of gain in this ENLS, this master equation can be called an extended Ginzburg-Landau equation (EGLE) and is given by:

As it is the case for the GL differential equation, the EGLE supports a chirped solitary wave solution [7] and is given by:

The main difference between the solitary wave given by Eq. (2) and the usual solitary wave of the GLE comes from the phase shift term (a) relative to the central frequency *ω*
_{0}. In appendix A, it is shown how the six charateristic parameters (*V*
_{0},*β*,*α*,*b*,Γ,*a*) of the solitary pulse are related to the several internal parameters of the laser system. I also show, in appendix A, that eight real equations can be formally derived for the six parameters. However two complex compatibility relations impose certain restrictions on the internal parameters. I suppose, at this stage, that those restrictions can be supported by the dynamic operation of the laser system.

In a previous analysis [2] of such a system where the SS and the IRS effects were not included in the model, I was able to find out that the solitary pulse had a chirp parameter *β*≅1 to explain most of the experimental observations when the average dispersion is negative (*$\overline{\beta}$
L*<0). For positive average dispersion (*$\overline{\beta}$
L*>0), no fixed operation point for the chirp parameter *β* could be found neither from the model nor from the experiments. Here this extended model yields a direct access to a unique chirp parameter for the different operating regimes. This result comes from the analysis of Eq. (A3) after seperating the real and imaginary parts and assuming that *β*
_{3} is real. When this is carried out, the following characteristic equation can be deduced for the chirp parameter *β*:

According to Eq. (3), three chirp parameters *β* can be calculated from this third-order algebric relation. Here, in the framework of the laser system under study, the IRS term *T*_{R}*ω*
_{0} must be positive and the chirp parameter must also be positive [see Eq. (A8)]. Hence, as shown in Fig. (1), when *T*_{R}*ω*
_{0}>1.65, two positive chirp parameters *β* are found for a given value of the IRS effect. Assuming that the saturable absorber parameter *ε*
_{0} is very small, we can deduce from Eq. (A11) that for *β*<√2, the average dispersion must be negative. Then, according to Eq. (3), DM operation in the negative average dispersion regime is always possible if th IRS term is larger than 4 (*T*_{R}*w*
_{0}≥4). Assuming that the Raman term *T*_{R}
=3*fs* and 5*fs* respectively, the Raman parameter *T*_{R}*ω*
_{0} are epproximately equal to 3.64 and 6 for a fiber laser operating at 1550*nm* whereas for a laser operating at a wavelength of 1030*nm*, the Raman parameter is 5 and 9 respectively. When *T*_{R}*ω*
_{0}<4, two operating modes of the laser can be achieved in the same positive average dispersion regine ( $\overline{\beta}$ *L*>0):one with a low chirp parameter *β* and the other corresponding to a large value of the chirp parameter *β*. For the case corresponding to *T*_{R}*ω*
_{0}>4, the operation of the laser is in the DM negative average dispersion regime or in a highly-chirped positive dispersion regime. The case corresponding to the situation where *β*=0 is simply the pure solitonic regime, which occurs in the anolamous dispersion region. Figure (1) summarises well what can be deduced from Eq. (3) and more specifically, we show the operating regimes corresponding to the case where *T*_{R}*ω*
_{0}=5.

## 3. The highly-chirped (similariton) regime

For a realistic IRS parameter (*T*_{R}*ω*
_{0}>5), Eq. (3) always yields one large possible value of the chirp parameter *β*. The chirped solitary pulse will be obviously highly-chirped for large values of *β*. As from now, the discussion will be restricted to this situation and I will show that this solitary wave solution has most of the features of the so-called *similariton* operating regime which occurs for large positive average dispersion in a fiber laser. Figure (2) shows the amplitude of the spectrum [(a) and (c)] of the solitary pulse (Eq. (2) of Ref. [2]) for different values of the chirp parameter *β*. It is to be noted that the amplitude of the various depicted spectra, while being arbitrary, ensures that all the shown spectra have the *same* energy. The spectrum tends to a nearly rectangular shape as the value of chirp parameter *β* increases. This profile appears to be close to the shape of the *similariton* previously measured and calculated [9] in a fiber laser where a clear and steepest decay is observed near
$\nu =\frac{{\nu}_{f}}{2}$
. We also depict in Fig. (2) the phase profile [(b) and (d)] for each value of *β*.

The spectral full-width-half-maximum (FWHM) of the pulse is given by [2]:

For large values of the chirp parameter *β*, Eq. (4) can be approximated by:

In a real fiber system [9], the pulse will propagate through the mode-locking device before being ejected from the resonator. In the present work, I have modelled the saturable absorber with a small nonlinear parameter *ε*
_{0}. In appendix B, it is shown that the passage of a rectangular spectral profile *V̂*
_{0} (*ν*) through the absorber will be transformed to:

The spectral amplitude is shown in Fig. 3(a) and is typical of the one calculated and measured in the laser system of Refs. [5, 9]. Now, assuming that the pulse is chirp-free outside the resonator, the temporal profile is depicted in Fig. 3(b).

For Eqs. (6a) and (6b), the calculated time-bandwidth product (TBP) is:

The so-called *similariton* regime was first introduced by *Ilday et al.* [3] and was inspired at first from the idea of using the parabolic pulse asymptotic solution of the nonlinear Schroedinger equation (NLS) which was first reported by *Anderson et al.* [8]. However, they were well aware that this asymptotic solution cannot be compatible with the periodic boundary condition of a laser resonator. According to the present model, the average pulse profile inside the laser is given by:

where the small frequency shift effect a has been neglected. In order to show that this profile is more realistic that a parabolic one in such a highly-chirped regime, I shall compare it with the results reported recently by *Ruehl et al.* [9]. In this paper, the authors model the laser system using the usual split-step Fourier algorithm and they use a two-level-model to calculate the parameters of the gain fiber. The calculated temporal phase and amplitude profile corresponds to Fig. 1(a) of their article. The pulse width of 6.5*ps* corresponds to a root-mean-square (RMS) width of 2.76*ps* for a Gaussian profile and to 3.34*ps* for the chirped pulse distribution given by Eq. (8) which is to be compared to their calculated RMS width of 3.1*ps*. For large values of *t*, their phase profile is linear with a slope close to *αβ*=19 whereas for the pulse distribution given by Eq. (8), the predicted slope will be given by *αβ* and when combined with the result of Eq. (5), for large *β*, gives:

where I have used their value of *ν*_{f}
=7 for their calculated spectral width. The measured spectrum appears to be shifted as inferred by the present model through the presence of the TOD. From the result of Eq. (9) and from the knowledge of the pulse width, the value of *β* can be estimated to be around 80 which implies an IRS term of *T*_{R}*ω*
_{0}≈20. It is important here to understand that the distributed model gives only an *averaged* information of the pulse that propagates in the laser and no accurate information on the local temporal and phase profiles is available. However, as pointed out and discussed in a previous paper [2], the spectrum is generally less affected by the nonlinear effect during the propagation and therefore its profile can be very realistic of the physical situation. Finally, for very large values of the chirp parameter *β* and assuming that the saturable absorber term *ε*
_{0} and the frequency chirp to be very small, the main characteristics of the chirped averaged similariton pulse can be obtained approximately from the equations of appendix A:

where the total phase shift for a laser of length *L* is given by:

assuming that for a parabolic gain profile, the gain parameter (2${\mathit{\text{gT}}}_{0}^{2}$) can be approximated by:

where *ν*_{g}
is the frequency gain bandwidth. Using Eqs. (A11) and (11), it can be shown that:

Refering again to the laser system of Ref. [9], with *$\overline{\beta}$
L*=0.013, Eq. (13) yields a phase shift (Γ*L*=*π*). With the estimated chirp parameter *β*=80, we can predict via the use of Eq. (13) that the gain frequency width should be 10*THz* which is close to the gain baindwidth of 12.7*THz* used in Ref. [9]. In a recent paper by *Zhao et al.* [12] on the propagation of gain-guided solitons in a DM laser operating with a large positive dispersion, the authors approximately report the experimental observation of the same typical spectrum. Finally, following the numerical integration of the GL equation, they have obtained a similar type of spectrum [13] as the one described analytically in the present paper.

## 4. Conclusion

After including the SS and IRS terms into the GL differential equation, I have shown how to derive a characteristic equation for the chirp parameter *β*. This main result defines clearly three different operating regimes for a fiber laser system. The first regime is the solitonic regime which appears as a unique point in Fig. (1). Two other values for the chirp parameter *β* can also be found to satisfy the phase of the chirped pulse solution. The smaller one defines the DM regime laser operation either in anomalous or normal average dispersion regime. The anomalous dispersion regime is always possible for a realistic Raman parameter and the chirp parameter will be close to 1. The normal average dispersion regime seems to be possible if the Raman parameter is very small and if the chirp parameter is larger than √2. However, for a laser operating in a positive dispersion regime, the chirp parameter is always large and the corresponding highly-chirped pulse appears to have the characteristics features of the so-called *similariton* regime recently introduced as a modification of what is commonly known as the temporal parabolic pulse regime [3, 4, 5, 9]. The parabolic pulse is an asymptotic solution of a pulse propagating into an amplifier and is not compatible with the periodic conditions imposed by a laser resonator. The parabolic pulse propagating into an amplifier of gain (*g*-*l*) will have a RMS chirp parameter *C* given by:

according to Ref. [10]. For the chirped hyperbolic-secant distribution, the RMS chirp parameter, for large *β*, is given by:

This result shows that the laser resonator *doubles* the optimal chirp that the amplifier could have achieved. The present model has thus demonstrated that it is *not* necessary to introduce the self-similar temporal parabolic pulse in order to explain the *similariton* regime. However, the chirped solitary pulse used here is without doubt a self-similar pulse and hence the term *similariton* [11] should continue to be used to define this high energy laser regime.

## A. Appendix A

It has already been observed [7] that the chirped hyperbolic secant ansatz given by Eq. (2) is a solitary wave solution to the EGLE given by Eq. (1) under specific conditions. After direct substitution of ansatz (2) in Eq. (1), the resulting equation can be seperated into a time-symmetrical part and a time-antisymmetrical part. Ansatz (2) is a solution of the symmetrical differential equation if:

and

In order to force the antisymmetrical part to satisfy ansatz (2), the two following additional complex conditions need to be satisfied:

These four complex relations result into eight real equations for the six characteristic parameters (*V*
_{0},*a*,*b*,Γ,*α*,*β*) of ansatz (2). However, Eqs. (A1) and (A3) as well as Eqs. (A2) and (A4) impose certain relations among some of the internal and external parameters of the laser system. Here, I will assume that *β*
_{3} is real (it is to be pointed out that this is *not* the case in Ref. [7]) and as a consequence of this assumption, Eq. (A3) can be solved and allows one to fix the chirp parameter *β* relative to the Raman term namely *T*_{R}
such as:

Next, after imposing the compatibility relation between Eqs. (A1) and (A2), one finds that the phase shift parameter *a* must satisfy the following relation:

The phase shift parameter *a* is of course assumed to be much smaller than the central frequency *ω*
_{0} and from here I shall neglect all the contribution in
$\frac{{a}^{2}}{{\omega}_{0}^{2}}$
. Therefore, within this approximation, the rest of the external parameters are given by:

Finally, the two residual compatibility relations read as:

Notice that if the the self-steepening is not included into the model (*ω*
_{0}→∞), Eq. [A3] fixes the chirp parameter to *β*≡1, and as discussed in Ref. [2], this chirp parameter (*β*≡1) is typical of the negative DM regime.

## B. Appendix B

The propagation of a pulse *V* (*τ*,*x*) propagating through a saturable absorber is modelled by the differential equation:

Here the propagation is assumed to be on a short virtual distance *x*
_{0} and it can be assumed that the pulse profile will not change significantly during this short distance. It can be shown [14] that the Fourier transform of |*V*|^{2}
*V* where *V*=*V*
_{0} [*sech*(*ατ*)]^{1-iβ} is related to the Fourier transform of *V* (*V̂*) through the following relation:

Taking the Fourier transform of the differential equation (B1), the output spectrum *V̂* can be related to the incoming spectrum *V̂*_{i}
via the following relation:

For very large values of *β* this relation reads as

where the spectral width for large values of *β* has been used accordingly with Eq. (5). When *β* is large, the incoming spectrum *V̂*_{i}
is rectangular and has been normalised to 1. Assuming that *ν*_{f}
is still the FWHM of the output spectrum, it is straightforward to show that:

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