1. Introduction

The dynamics of many lasers can no longer be described satisfactorily by the master equation of modelocking [1], which is based on reasonably small changes in the lasers pulse shape within one roundtrip. This is especially so for fiber lasers operating in the stretched pulse mode [2] or similariton mode [3, 4]. The variational method is largely used for the analysis of dispersion managed transmission systems [5]. It was recently applied to the analysis of fiber and solid state mode-locked lasers [6, 7, 8, 9, 10]. To gain insight into dynamics and pulse stability it seems a natural alternative to time consuming full numerical simulations. In this paper we show that this method provides a general framework for a semi-analytical characterization of the intra-cavity pulse dynamics of passively mode-locked lasers with large pulse shaping per roundtrip and their linear stability, and apply it to the two relevant regimes of stretched pulse [2] and similariton laser operation [4].

2. Theory and system modeling

Pulse propagation is described by the extended nonlinear Schrodinger equation (NLSE):

where u = u(z,t) is the complex pulse envelope and t is the time in the co-moving frame. We denote by $\beta ″$ (z) the group velocity dispersion, γ(z) is the self phase modulation coefficient and g_s(z) is the saturated gain coefficient given by g_s(z) = g ₀(z)/[1+E(z)/E _sat], where g ₀ (z) is the small signal gain coefficient, E(z) is the pulse energy and E _sat is the gain saturation energy. We assume a parabolic spectral gain profile, g̃(z, ω) ≃ g_s(z) [1 - (ω/Ω_g)²]¹, although a similar analysis could be performed for a generic gain profile [8]. Due to the heterogeneity of the resonator, each parameter of the NLSE is periodically dependent on z in a step-wise fashion. The conservative part of the NLSE can be derived from the Lagrangian [5] ∫^+∞ _-∞ = ℒ(u,u_z;u ^*,u ^* _z)dt, with the lagrangian density ℒ(u,u_z;u ^*,u ^* _z) defined as follows:

where subscript z denotes differentiation with respect to z. The experimental evidence of a linear chirp [2, 4] in both stretched and similariton lasers suggests adopting the following general ansatz in order to evaluate the evolution of its parameters in the fiber ring:

where f(s) is the (real) pulse amplitude normalized to have unit energy, ∫^∞ _-∞ f ²(s)ds = 1, τ = τ(z) is the pulse width, ρ = ρ(z) is related to the chirp coefficient β = β(z) by β = ρτ², t ₀ = t ₀(z) is the time displacement and Ω = Ω(z) is the deviation from the optical center frequency. We denote by I_k and J_k the kth moments of f(s) and its Fourier transform f̃(λ) respectively:

Defining also P ₄ = ∫^+∞ _-∞ f ⁴ (s)ds, it is straightforward algebra to calculate the Lagrangian for the ansatz (3):

which expectedly depends only on the moments I ₂, J ₂ and P ₄. The equations of motion for the pulse parameters can be derived by applying the Euler-Lagrange equations, appropriately modified to include the non-conservative terms [9]:

where x is one of the six pulse parameters and x_z is its derivative with respect to z. Introducing the mean square pulse bandwidth, ℬ² = ∫(ω - Ω)²∣ũ(ω)∣²dω/ ∫ ∣ũ(ω)∣²dω which is related to the other pulse parameters by ℬ² = J2/T2+l2p2T2, it is possible to arrange the equations of motion as follows:

where we have made use of the additional definition $S={\int }_{-\infty }^{+\infty }{s}^2{f}^{\prime 2}ds={\int }_{-\infty }^{+\infty }{\lambda }^2{\mid \tilde{f}\prime \mid }^2\frac{\mathrm{d}\lambda }{2\pi }.$.

The equations of motion have useful insights. From Eq. (8) one can see that the energy is increased by the saturated gain and decreased by the gain filtering in proportion to the mean square bandwidth and to the square frequency deviation. Note that this equation could be directly derived from the NLSE multiplying both sides by u ^* and integrating the real part [11]. Equation (9) shows that the gain filtering contributes to the change in pulse width depending on the sign of the quantity 1 - S + I ₂ J ₂. In fact, for 1 - S + I ₂ J ₂ < 0 the pulse width is reduced by gain filtering, whereas for 1 - S + I ₂ J ₂ > 0 there is a critical value of the chirp, β_C = √(1 - S + I ₂ J ₂) / (I ₄ -I ² ₂) such that for ∣β∣ < β_C the gain filtering broadens the pulse. This value discriminates the two different ways the gain filtering influence the pulse width, namely increasing the pulse width when filtering for β < β_C and reducing the pulse width by reducing the chirp for β > β_C (for the Gaussian ansatz, for instance, β_C = 1). From Eq. (11) one can see immediately that the frequency offset is forced to vanish by a permanent restoring force; in consequence of this the derivative of the time shift Eq. (12) also vanishes and t ₀ tends to a constant value that can be set to zero without loss of generality. In what follows we can safely set Ω = 0 and t ₀ = 0.²

We model the effect of the saturable absorber simply by a lumped transmission function, such that the transmitted pulse, which we denote by u ₊(t), is given by u ₊(t) = Γ^1/2 _SA [1 - l ₀/(1 + ∣u∣²/P _sat)] u(t), where l ₀ is the modulation depth, Psat is the saturation power and Γ_SA is a coupling coefficient accounting for linear losses that affect the pulse in the cavity and that, without loss of generality, can be lumped together after the saturable absorber. By comparison of the second order Taylor expansion of the pulse (around t = 0) at the input and output of the saturable absorber, one can derive expressions for the pulse amplitude A (which is related to the energy by E = A ²τ) and for the pulse width:

 

Fig. 1. (a)Schematic diagram for the stretched pulse laser. (b) Intracavity pulse dynamics: (i) Energy, (ii) rms pulsewitdth and (iii) bandwidth. Solid lines are the solution to Eqs. (8)-(10); dashed lines refer to full numerical simulations.

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where f ₀ = f(0) and we have denoted by the subscript “+” the parameters of the transmitted pulse u ₊(t). The energy of the transmitted pulse is obtained by using E ₊ = A ² ₊τ₊ along with Eqs. (13), (14), whereas the parameter ρ is not affected by the saturable absorber.

3. Stretched pulse laser and similariton laser: intracavity pulse dynamics

In order to apply our analysis to the stretched pulse laser and to the similariton laser we need to choose an appropriate ansatz. While the choice seems to be mandatorily gaussian for the first case, self-similar pulses are of slightly different shape [3, 12]. Nevertheless we use also for the similariton laser a gaussian pulse shape, having found no significant changes occurring by different choices. The design of the cavity for the stretched pulse laser is based on ref. [13], whereas for the similariton laser is based on ref. [4]. The steady state values of the pulse parameters at any location in the resonator are found by using the shooting method [14] and then are assumed as the initial condition to integrate the equations for one roundtrip. The results are shown in Figs. 1 and 2, where we have plotted the evolution of the energy, the root mean square pulse width and bandwidth versus the normalized position in the ring. We have compared the results of the variational analysis to the results of full numerical simulations based on the split step method, which are reported by thick dashed lines. The evident agreement with the numerical simulations confirms the validity of this approach. The small deviation in terms of pulse width and bandwidth in Fig. 2 seems to be due to the fact that the shape of the similariton is also evolving in the cavity, but this feature is not included in the present analysis . An estimate of the maximum energy in the cavity, E _max, can also be obtained by neglecting the term accounting for gain filtering in the integration of Eq. (8). This is allowed by the fact that the energy dynamics is dominated by gain saturation [11]. Neglecting also the change in pulse width at the saturable absorber, we obtain E _max ≃ E_L [2g ₀ L_g + log(Γ_SA(1 -l ₀)²)] /[1 - Γ_SA(1 - l ₀)²]. For the systems of Figs. 1 and 2 this gives E _max ≃ 0.36nJ and E _max ≃ 34nJ respectively, which are very close to the real values.

 

Fig. 2. Same as fig. 1 for the similariton laser oscillator.

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4. Linear stability analysis

As an application of the theory, we study the stability of the two systems under scrutiny against perturbations and against the growth of cw radiation. In the first regard, solving the equations of motion over many map periods and for different initial conditions (which requires a few seconds on a personal computer) insures this kind of stability. We may also quantify the stability by computing the Floquet coefficients [15] of the system. To do so we linearize the equations of motion (8-10) around the steady state solutions and obtain a set of linear equations with periodically varying coefficients:

where Δx is the perturbation column vector, Δx = (ΔE, Δτ, Δρ)^T, A(z) is a 3 × 3 periodic matrix with period L _t and the matrix B _L accounts for the lumped contribution of the saturable absorber and also the grating pair in the similariton laser. The matrix Q(z) of the independent solutions of (15) can be written as Q(z) = U(z)exp(λz), being U(z) a periodic matrix with period L _t. The diagonal matrix λ contains the Floquet coefficients λ_i, i= 1,2,3: their real parts determine the decay rate of the perturbations, whereas their imaginary parts may account for relaxation oscillation type of behavior. They can be evaluated by using that Q(L_t) = exp(λL_t) for Q(0) = 1. The Floquet coefficient with the smallest absolute real part determines the stability of the perturbations. The Floquet coefficient for the frequency shift, which plays a key role in the analysis of the timing jitter, can be evaluated explicitely linearizing Eq. (11): λ_Ω = - 4 ∫₀ ^L_t g_s(z)ℬ²(z)/Ω_g ²dz/L _t. With regard to the stability against cw radiation, we assume as a stability criterion that the net (power) gain for the cw radiation must be less than unit for the laser to be stable (this criterion is equivalent to the one assumed in ref. [6]):

 

Fig. 3. Solid lines are the boundary of the stable region, whereas dashed lines correspond to the stable cw gain G _cw = 0.9 for (a) stretched pulse laser and (b) similariton laser. Corresponding, insets show the negative real part of the Floquet coefficients λ_i (solid lines) and λΩ (dot-dashed lines) normalized to the total fiber length L _t.

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where g_s(z) is determined by the steady state pulse and L_g is the length of the gain fiber. The gain for cw radiation is sensitive to the modulation depth of the saturable absorber and to the gain bandwidth. The results of the analysis of such dependence are reported in Fig. 3, where we have plotted the boundary of the stable region in the plane (l ₀,Ω_g/2π) for (a) the stretched pulse laser and for (b) the similariton laser. In the insets, for the values of l ₀ (and Ω_g) corresponding to G _cw = 0.9, we have plotted the negative real part of the Floquet coefficients λ_i (solid lines, only two because two coefficients are complex conjugate) and λ _Ω (dot-dashed lines), normalized to the ring length L _t. The plots show that, for the range of parameters under scrutiny, the two systems manifest the same robustness against perturbations (which is set by the negative real part of the Floquet coefficient closest to zero), despite the significantly higher energy of the similariton laser, which, interestingly, appears to be even more robust against frequency fluctuations.

5. Conclusion

In conclusion, we have developed a variational theory to study the dynamics and stability of passively mode-locked lasers that show large temporal and spectral changes in the pulse within one roudtrip of the cavity. The theory is used to study the dynamics of the perturbations and the stability of these systems against the growth of cw radiation.