Negative GDD

First, let us consider the case, when $D$ $<$ 0, which assures the quasi-Schrödinger soliton formation. As it was shown in [17] on the basis of the aberrationless approximation, the soliton-like pulse, i. e. the pulse with the $sech$-time profile and the negligible chirp, exists in the parameter range, which is wider than it follows from the soliton model, and becomes unstable in the vicinity of $D$ = 0 due to the automodulational instability. The behavior of the pulse width in our case is presented in Fig. in dependence on the GDD variation. Approaching zero GDD results in the generation of the multiple pulses (up to 24 for given parameters, the example of the triple pulses is shown in Fig. , the intensity of filling corresponds to value of $\log\vert a(z,t)\vert^2$). In Fig. we depict only parameters of the double and the triple pulses. The single pulse shortening caused by decreasing $\vert D\vert$ transforms into the double pulse generation, then the triple pulse generation etc. Each bifurcation is accompanied by the drastic pulse width increase. The characteristic feature is the multistable lasing in the region of the multiple pulse generation: there exists a distinct hysteresis in the pulse parameters behavior (see Figure ). The pulse multiplication results in the strong instability in the vicinity of $D$ = 0. Such behavior is not the noise or the continuous-wave operation, but the multiple pulse generation with strong coupling between the pulses and nonregular changes of their parameters.

Figure: The dependence of the pulse width on the GDD coefficient. Only single, double and triple steady-state pulse characteristics are shown. Captions at points denote the inter-pulse distances. P= $8\times 10^{-5}$, $\sigma$=10, $\epsilon$=1.3 $\times 10^{-3}$, $\gamma$=0.02, $\rho$=0.01. Other parameters are mentioned in the text.

Figure: The contour-plot of the $\vert a(z,t)\vert^2$ logarithm for the stable triple pulse operation. P= $1.6\times 10^{-4}$, $\sigma$=10, $D$=-2188 $fs^{2}$ (other parameters correspond to Fig. ). Darker regions correspond to higher intensity. $z$ is the number of round-trips.

To analyze the nature of the ultrashort pulse destabilization we shall consider the influence of the basic lasing factors: self-phase modulation, GDD, self-amplitude modulation due to the fast saturable absorber and the gain saturation, and the spectral filtering. First, let us examine the nonlinear phase shift contribution. The solid curves in Fig. show the phase-retardation $\phi$ at the steady-state pulse peak after the full cavity round-trip. One can see, that the phase shift is close to that for the Schrödinger soliton (dashed curves) and is small in comparison to $\pi$ (especially for the lower intensity multipulse regimes). Hence the self-phase modulation can not produce the pulse spectrum fragmentation. Moreover, approaching the stability boundary does not cause any significant phase shift. As a result, there is no the spectrum fragmentation in our case.

Figure: The dependence of the single-transit phase retardation on the GDD coefficient. The negative GDD branches correspond to parameters of Fig. 1, the positive GDD branches correspond to P= $1.6\times 10^{-4}$, $\sigma$=20. The dashed curves show $\phi$ corresponding to Schrödinger soliton.

As an additional destabilizing factor, the dispersive perturbations in the laser system with self-phase modulation and negative net-GDD give rise to the spectral side-bands [37,1,38]. Their position in the absence of the higher-order dispersion can be found from the condition = $k_{s}\pm k_{p}$, where = , $k_{s}$ = $2D/t_{s}^2$, and $k_{p}$ = $2\pi j$ are the normalized to inverse cavity length wave numbers of the dispersive wave, solitary wave, and periodic perturbation, respectively; $\omega_{sb}$ is the side-band frequency, $t_s$ is the width of the soliton-like pulse with the shape $sech(t/t_s)$, $j$ is an integer. When $t_s^2 \gg \vert D\vert$, we have . Our simulations show that the spectral loss for the side-bands is too large in our case. Thus, the side-band generation does not contribute to the ultrashort pulse destabilization for the given parameters. We thus see that in our case neither the self-phase modulation induced spectral fragmentation nor the side-bands generation caused by the dispersion perturbations produce the multiple pulse operation. However, the former mechanism can cause the chaotical behavior for $\vert D\vert\rightarrow 0$, where the abrupt nonlinear transformation due to self-phase modulation is not compensated by GDD. The latter mechanism, i. e. the side-band generation, can significantly influence the lasing due to the dispersion wave amplification when the pulse width approaches $\sqrt { \vert D\vert}$. The transition to the multiple pulse generation can not be comprehended without taking into account the dissipative laser factors, such as the spectral filtering, the saturable, linear and spectral losses. Fig. shows the dependencies of the absorber loss saturated by the pulse peak intensity and the spectral loss for the soliton-like pulse on GDD. One can see, that the absorber is saturated (solid curves) and almost does not contribute to lasing. At the same time, approaching zero GDD leads to the shortening of the pulse width and broadening of the spectrum. This produces the pronounced spectral loss growth (dashed curves). Alternatively, for the comparatively large pulse durations caused, for example, by the large $\vert D\vert$, the essential spectral broadening results from the chirp growth in the vicinity of the stability boundary (but without spectral fragmentation). In both cases the spectral loss decreases the pulse energy. Consequently, the gain saturation is reduced and the net-gain $\alpha-\rho-\gamma$ increases and becomes positive (Fig. ). This causes the background amplification on the pulse wings resulting in the multiple pulse generation (the analysis of the stability loss in the case of $\sigma \gamma \ll$ 1 is given in [39]). The rise of the background with the subsequent multiple pulses appearance is clearly visible in Fig. .

Figure: The dependence of the saturated absorber loss $\gamma/(1+\sigma \vert a_{max}\vert^2)$ (solid) and spectral loss (dashed) on the GDD coefficient for the parameters of Fig. . $\vert a_{max}\vert^2$ is the steady-state pulse peak intensity.

Figure: The dependence of the net-gain coefficient $\alpha-\rho-\gamma$ on $D$ for the parameters of Fig. .

Hosted by www.Geocities.ws