1. Introduction

Fiber lasers can generate ultra short and high power pulses, and they are today replacing solid-state lasers in several applications. Polarization additive pulse mode-locking technique allows fiber lasers to self start the generation of pulses at high repetition rates. Dispersion management (DM) of the fiber loop allows different regimes of stable laser operation. The modeling of this type of lasers was best made by Haus et al [1] who showed how the mode-locking mechanism was effective. The nonlinear pulse shaping, which happens as a result of the propagation through the fiber under the combined influence of the gain, dispersive and dissipative effects, is usually made through careful numerical simulations depending on several local parameters that are not directly measurable. However it is possible to develop a simple master equation model [1] after properly distributing the gain, loss, non linearity and dispersion along the fiber loop. The steady state solution of the master equation (known also as the GL equation (GLE)) is a chirped solitary wave. Unfortunately, this mathematical solution gives limited information such as the average width and chirp of the pulses circulating in the loop. Therefore, in this short communication, I will first suggest a different interpretation to this solitary wave solution that yields a direct access to the chirp-free pulse profile. Then, in the next section, the pulse width and bandwidth of the chirp-free solution is related, through simple formulas, to the total accumulated dispersion of the laser loop.

2. Profile of the chirp-free pulse.

I now consider the propagation of a pulse in a ring cavity configuration, as technically described by several authors [2,3]. The cavity consists of two fiber segments with anomalous dispersion and normal active dispersion respectively. The pulse evolution is governed by the master Eq. (A1) given explicitly in appendix A where the gain, loss, nonlinearity and dispersion effects have been distributed along the propagation distance x [1,4]. In this analysis, the effect of the mode-locking mechanism is not included but can easily be taken into account by adding a complex nonlinearity term in the master Eq. (A1). This is done in order to limit the number of parameters and does not change the form of the following mathematical result. The steady state solution for a pulse propagating in such a medium is given by the chirped solitary wave :

The phase constant Γ, the chirp β, the width α and the amplitude V ₀ are related to the gain, loss, self-phase-modulation, dispersion and gain bandwidth as shown in appendix A.

In the laser cavity, the pulse evolves from a positive to a negative chirp resulting in some case to a very large stretching from the middle of one of the fiber sections to the junction between the normal and anomalous dispersion fiber segments. The solitary wave given by Eq. (1) is an exact solution of the distributed model that reflects an averaged behavior of the pulse along the fiber loop. To obtain a chirp-free pulse in practical laser system, an additional fiber or dispersive element [5] is added at the exit port. At first, one may be tempted to mathematically dechirp the solution (1) in order to calculate the minimum width predicted by the distributed model. However, this will result in an averaged width of the pulse which will be far from the exact minimum width observed somewhere along the fiber loop.

Hence, in an effort to tackle this problem, I instead postulate that the amplitude spectrum of the solitary wave solution , of the distributed model, contains all the information about the spectrum of the chirp-free pulse with minimum width generated in the loop. The temporal profile of the envelope of this minimum chirp-free pulse can be obtained by calculating the inverse Fourier Transform of the modulus of the spectrum of the solitary wave. The modulus of the spectrum of the solitary wave has been calculated in closed form in Ref. [6] by Paré et al. and reads as:

The pulse profile at its minimum width location will thus be given by an inverse Fourier Transform of the square root of Eq. (2):

I depict, in Fig. 1, the resulting time profile as well as the modulus of the spectrum for some characteristic values of the chirp parameter β. The spectrum has a hyperbolic secant shape for the NLS soliton (β = 0) and tends to a top-hat shape for large chirp parameter (see Ref. [6]) as can be seen from Fig. 1(d) which corresponds to a chirp value of β = 2. We also notice that this particular behavior of the spectrum generates oscillations in the temporal profile which are typical of the radiation less solitary waves that propagate in DM fiber communication link. Therefore, this feature in itself is a good indication that this approach is right.

 

Fig. 1. Temporal and spectral profiles corresponding to Eqs. (2) and (3) for various values of β.

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Figure 2 shows the time bandwidth product (TBP) (τ_fν_f ) (dotted line) and the time bandwidth variance (TBV) (σ ² σ̂²) (full line) versus the chirp parameter β. A minimum value of the TBV that is very close to the typical value of a pure Gaussian pulse (0.25) is obtained for β between 0.8 and 0.9. We also point out that for this particular range of values of β, the TBP of the corresponding distribution is around 0.44 which is also reminiscent of a pure Gaussian pulse. However, from Eq. (2), we clearly see that the spectrum of this pulse has an exponential asymptotic decay rather than Gaussian.

 

Fig. 2. TBP (dotted line) and TBV (full line) of the pulse given by Eq. (2) as a function of the chirp parameter β

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The analysis of a fiber laser, as far as the propagation part is concerned, is very close to that of a DM fiber communication link. For example, in a DM link in the zero-average dispersion regime, a TBV σ ² σ̂² = 0.2789 has been calculated [7] for a lossless system. From Fig. 2, we find that this value is close to the point corresponding to a chirp parameter of β = √2 which, remarkably, coincides exactly with the chirp parameter of the zero average dispersion regime. The spectral and temporal pulse profiles calculated in Ref. [7], when compared to this chirp-free pulse, are very similar. The main discrepancy is that the spectrum of the lossless system is slightly narrower at high frequencies due to the absence of the parabolic gain medium. For a lossless DM communication system, it has been demonstrated [8] that the pulse having the minimum width is chirp-free and is always located in the middle of the anomalous fiber segment. For a loss system, several numerical simulations have shown that the location of the pulse with minimum width is slightly displaced from the middle of the fiber segment [9]. In a fiber laser, the output port is generally located at the junction of the anomalous and normal fiber segments where the chirp is at its maximum. The linear decompression of the chirped pulse will be very close to the minimum chirp-free pulse proposed here because the local nonlinear effects are very small for one transit in the cavity.

3. Minimum pulse width and total dispersion

The full-width-half-maximum (FWHM) of the pulse spectrum can be calculated from its distribution given by Eq. (2) and is given by:

The parameter α can be replaced in this relation using the Eqs. (A3) and (A5) and the following relation for the pulse width relative to the time bandwidth gain parameter T ₀ is derived:

Figure 3 depicts this last relation for a phase shift parameter ΓL=2π. A clear minimum is reached for the chirp parameter β close to one. Moreover as mentioned in appendix, a minimum of energy of the pulse is also observed close to this operation point. Assuming that a negative dispersion fiber laser system always yields the minimum achievable pulse within the framework of this model, it should operate with a chirp parameter β close to 1 and their time bandwidth product should be close to 0.47. Replacing the gain parameter in equation (4) by the relation (A6) the pulse width for an average negative dispersion fiber laser reads:

The width is now related to a measurable physical parameter (β ₂ L) the total dispersion, and to one intrinsic parameter to total phase shift (ΓL). From the results of several experimental systems Dennis and Duling [17] have shown that the minimum pulse width can be approximated by:

Their finding is essentially the same as this analytical derivation if one fixes the phase shift ΓL=2π. There was no reason ,a priori, to choose this value for the phase shift (thanks to a Reviewer) but here it appears that this choice ensures a total minimum of the pulse width. Several fiber lasers, having negative total dispersion, have been experimentally realized and characterized [2,3,5,10,11,12].From their reported measurements of the minimum temporal and/or spectral width, I have found that the characteristic parameter β is comprised between 0.9 and 1.1. This is somehow surprising to find nearly the same characteristic parameter for a fiber laser loop ranging from several meters (85m) [2] to a few meters.

 

Fig. 3. The minimum pulse width normalized to the gain and bandwidth plotted as a function of β.

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It is interesting to observe that for this characteristic parameter (β = 1), the time bandwidth product is 0.47 which is close to the limited bandwidth product achieved by a Gaussian pulse. But, as shown in Fig. 1, the pulse has some small oscillations in the time domain.

Figure 4 depicts the general relation between the total dispersion and minimum pulse width for any characteristic parameter β and when the phase shift parameter ΓL=2π.

 

Fig. 4. Relation between the minimum pulse, dispersion and the characteristic parameter β.

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The fiber lasers having positive total dispersion are known to produce higher energy and shorter pulses than the negative ones. From the experimental measurements published in Refs. [3,5,9,11,12,13,14], I was not able to find a coherent characteristic parameter β. As for the case of DM communication links in the positive average dispersion regime, proper operation is still possible but only for an average dispersion close to zero. The published measurements have average dispersion so close to zero that the inherent calculated dispersion error is too large to assess the present model. The main problem with the evaluation of the total dispersion is the lack of an accurate value for the dispersion parameter of the doped and Flexcor fiber. For example, Haus and al. [11] have made a laser system with a total average positive dispersion of +0.004 ± 0.006 ps² and the width of the spectrum was 6.125ps ^-1. Within the margin of error, it can be assumed here that the dispersion of their fiber laser could be zero. For zero average dispersion, the present model gives a minimum width of 90 fs which is exactly the width they have estimated assuming a Gaussian pulse. The TBP of the published results are all estimated to be around 0.6 to 0.68. According to the model derived in this paper, the characteristic operation point should be 1.8≤β≤2.2. An iterative procedure could be derived with this first estimate of the TBP for finding the final characteristic parameter . However, we must be aware that the model of the parabolic gain profile starts to achieve its limit of validity in this region of very short pulses.

4. Conclusion

In this communication, I have claimed that the spectrum of the chirped solitary wave of the GLE contains information about the minimum pulse profile generated in fiber laser. For negative dispersion fiber laser a natural operation point was found, and published results seem to confirm it. For positive dispersion fiber no such operation point was found, however, the measured time bandwidth products are consistent with the present model. Positive dispersion fiber laser generates also pulse pairs at higher power [13,14,15]. This regime of operation has been recently successfully modeled by a cubic-quintic Ginzburg-landau equation (CQGLE) [16]. This work confirms that a distributed model, for such piece-wise propagation systems, can be productive and should be more widely used for describing complex nonlinear devices.