Bounded multiple pulses

As it was found (see, for example [4,7,8,9]), the multipulse generation can be caused by the formation of the bounded soliton-like complexes with strong correlation between components. The stability of the spectrum modulation in the experiment suggests the nearly constant inter-pulse distance and phase difference. The presence of the different mechanisms of the ultrashort pulse destabilization somewhat complicates the picture. The number of the pulses in the bounded soliton-like complex does not depend on the initial field only in the regions, where there is no multistability. Outside these regions the number of pulses, their intensity and durations depend on the initial conditions. The inter-pulse distance is more sensitive to the initial conditions. The multiple pulses formed by the continuum amplification have, as a rule, random distances because the pulse can rise at the arbitrary moment within the cavity period. However, the inter-pulse distance approaches the constant for the regular initial signal due to an interaction of the pulses arising from the common seed. In the case of the multiple pulse generation caused by the bounded perturbation growth, the pulses evolve through the dissociation of the initial single pulse. Because the pulses stay close-coupled all the time, interaction between the pulses plays a major role and the inter-pulse distance becomes less dependent on the initial conditions. In practice the presence of the mode locking start-up assistance (for example, due to acousto-optical modulator [26]) will regularize the initial field and produce the repeatability of the inter-pulse distances. However, the dissociation of the bounded multiple pulse complex always has the nonzero probability if only a fast saturable absorber is involved. The interaction between the pulses in the bounded soliton-like complex can result from the soliton interaction mechanism and from the binding potential formed by dissipative factors. The mechanism of the Schrödinger solitons interaction was analyzed in the framework of the inverse-scattering formalism [46]. It was found that the motion of the soliton pair can be described as to be influenced by the effective force, which depend on their relative phase $\varphi$, amplitude and distance $\delta$. In the specific case of $\varphi$=0 and the equal amplitude of the interacting solitons, such interaction causes the periodic collapse of the bounded multiple pulse complex with the period [47], where $\delta_0$ is the initial inter-soliton distance normalized to the soliton duration and $z$ is normalized to the dispersion length. If we take the parameters of Fig. , this formula gives time of 2$\times$10$^6$ cavity round-trips as the double pulses collapse period. It is obvious, that it is too small to explain the observed steady-state behavior of the multiple pulse complexes, which are interferometrically stable over seconds in the experiment. Hence, the soliton mechanism can not be considered as the single and main source of the pulse interaction. First, this mechanism does not describe the situation of the positive GDD and second, the contribution of the spectral dissipation reduces the interaction of the Schrödinger solitons [48,49,9]. As it was found in [7], the perturbation of the nonlinear Schrödinger equation by the linear and nonlinear loss terms and by the spectral filtering produces the oscillating soliton wings. The overlapping of these wings forms the soliton interaction potential, which has a minimum. In fact, the interaction through the oscillating pulse wings is suppressed in our case due to the damping of the side-band generation (see Fig. and [50]). Nevertheless we suppose, that the spectral loss [4] and the fast absorber saturation may contribute to the interaction of the pulses. Let us consider Eq. () from this point of view. There is the method to analyze the pulse interaction, which is based on the study of the conserved momenta of Eq. () [8]: and . Multiplying Eq. () by $a^{*}$, adding the complex-conjugate, and integrating over $t$ results in:

which is the energy balance equation for the laser field. Let us consider as the simplest example a superposition of two pulses:

($\delta$ is the half-distance, $t$ and $\delta$ are normalized to the pulse duration). It is obvious, that the interference between the pulses alters the energy "transmission" increment (right-hand side of Eq. ()). Fig. , a shows the energy loss due to the spectral filtering (solid curve), the absorber saturation (dashed curve) and their common action (dotted curve) in the dependence on $\delta$ ($\varphi$=0). One can see, that there exists the minimum of spectral loss for $\varphi$=0 and $\delta \approx 1$. The appearance of this minimum is demonstrated in Fig. . The merged pulses suffer the larger spectral loss as a result of the widest spectrum (solid curve). The $\delta$ increase splits the spectrum and concentrates the energy in its central part (dashed curve). The spectral loss is minimal in this case. The further distance increase forms the more uniform modulation with the rise of the high-frequency part (dotted curve). As a result, the spectral loss approaches that of the single pulse. The spectral loss decrease produced by this mechanism is larger for the shortest pulses. Hence, region of its action extends for the distances $\sim$ 5 $\div$ 8 $t_p$. The chirped pulses interact even stronger.

Figure: Energy loss for the spectral filter (solid), the saturable absorber (dashed) and their common action (dotted). (b): Functions $F_1$ (solid), $F_2$ (dashed) and $F_3$ (dotted). $a_0$=0.1, $\sigma$=1, $\varphi$=0, $t_p$=12 $t_f$, $\gamma$=0.02. Inter-soliton half-distance $\delta$ is normalized to the pulse duration $t_p$.

Figure: Spectrum of the double-pulse complex with $t_f$=$t_p$/1.76, $\delta$=0 (solid line), 0.6 (dashed line), 1.8 (dotted line). $\delta$ is normalized to $t_p$.

The next obvious mechanism is the fast absorber saturation, which favors the pulses merging due to the stronger loss saturation for the overlapping pulses. In the calculation we took into account only the first term in the expansion on $\sigma \vert a\vert^2$ in the last term of Eq. () (dashed curve in Fig. , a; this treatment is similar to that for the explanation of the colliding-pulses regime in [13], where the double-pulse generation resulted from the stronger Kerr-lensing for the colliding pulses). The combination with the spectral filtering can lead to the "absorbtion" of the pulses into the "potential well" for the comparatively small $a_0$ and $\sigma$ (dotted curve in Fig. , a). But the increase of the field amplitude and the saturation parameter lead to the pulse merging in the simplified model of Eq. () because the last term approaches the maximal value of 2$\gamma$ as a result of the pulse amplitude or $\sigma$ growth. At this moment, it is not quite clear what mechanism prevents the pulses from collapse. It is obvious, however, that the distance between the pulses is a result of the balance between the pulling force (the fast absorber saturation) and the yet unclear repulsion mechanism. The pulse intensity growth causes stronger saturation of the absorber and therefore increases the pulling force. This is illustrated by Fig. $a$ and , where inter-pulse distance decreases with intensity growth caused by $\sigma$ increase and $D\rightarrow$0, respectively. Some additional aspects of the pulse interaction can be revealed by the consideration of the second momentum of Eq. (), which describes the force acting on the pulse along the $t$ axis:

where $F_1(\delta)$ and $F_2(\delta)$ are the functions of $\delta$ and describe the linear loss and gain action and the spectral filtering, respectively, and is the function of $\delta$, $\varphi$, peak intensity $a_0^2$ and $\sigma$ and describe the fast absorber action. $F_1$, $F_2$ and $F_3$ have analytical form, but the expressions are overcomplicated and not instructive. A typical case is illustrated by Fig. , b. Besides the trivial stationary points $\varphi$=0, $\pm \pi$, there exists some inter-pulse distance causing zero interaction due to spectral filtering (intersection of $F_2(\delta)$ with zero line). As $F_3$ depends also on $\varphi$ and $\sigma a_0^2$, the right-hand side of Eq. () can vanish at values of $\varphi$ which differ from 0, $\pm \pi$. Thus, the stable pulses can have various phase differences. Our calculations demonstrate, that the phase difference changes in the process of the pulse evolution, in agreement with [4,9]. The simulation also reveals the existence of the "attracting" set of $\varphi$ values. The vicinity of these points is the most probable place for the pulse to stay, as demonstrated by the histogram in Fig. . This histogram was accumulated from 750 multiple pulse regimes corresponding to the various system parameters as well as the various initial conditions. We suppose, that the existence of this attracting set explains the spectrum regularity, which is observed experimentally.

Figure: Histogram of the phase differences accumulated for the different double and triple pulses regimes.

It should be noted, that the unbounded multiple pulse complexes can be stable over the whole simulation time. As the difference between the pulse intensities in the soliton-like complex, as a rule, is small enough, this causes the constant phase difference during the large time period. The last results in the experimentally observed regular spectrum for the pulses with large and nonuniform $\delta$.

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