We demonstrate the cnoidal wave formation in a two-laser system with a saturable absorber in the cavity of one of the lasers. Another laser is used to activate the saturable absorber in order to control the pulse shape, width, intensity and frequency. Using the three-level laser model based on the Statz - De Mars equations, we show that for any value of the saturable absorber parameter there exists a certain modulation frequency for which the pulse shape is very close to a soliton shape with less than 5% error at the pulse base. Such a device may be prominent for optical communication and laser engineering applications.
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A cnoidal wave is also known as a soliton pulse train. Even though the first solitary wave to be reported was by Russell in 1834 , it was not until the late 19th century when Lord Rayleigh attempted for the first time to give explanation of this phenomenon , the subject was left to rest until the 1960’s . The first phenomenological definition of a soliton that comes to mind is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed in a nonlinear dispersive medium. A balancing mechanism between nonlinearity and dispersion is responsible for this phenomenon .
Optical solitons can be either spatial  or temporal [6,7]. Hasegawa and Tappert  suggested that the balance between self-phase modulation and anomalous dispersion could explain soliton formation in optical fibers. Many exactly solvable models have soliton solutions, including the Korteweg - de Vries, the nonlinear Schrödinger, the coupled nonlinear Schrödinger, the Sine-Gordon, and the Manakov equations; actually this represents a very active field of mathematical and physical research. The main soliton practical feature, that makes it a good carrier for reliable optical communications, is its capacity to maintain its energy and shape (frequency) while propagating along a fiber. However, until now, the soliton production in a laser system has been quite expensive [8–10].
In this paper we propose a hybrid system (Fig. 1 ) composed by two lasers and a saturable absorber (SA) placed inside the cavity of one of the lasers, which acts as a passive Q-switching element for this laser. The radiation from another laser modulated by an electro-optical modulator (EOM) is injected directly into the SA to activate it and by such a way to control the output of the first laser. We demonstrate, for the first time to our knowledge that such a system can act as a cnoidal wave generator in a wide frequency range at quite a reasonable cost.
2. Theoretical model
The dynamics of the laser system presented in Fig. 1 can be described by the model based on the Statz - De Mars equations , which originally were deduced to describe oscillations in a maser, then this model has undergone many modifications to be adopted for laser systems. The Statz - De Mars equations for a three-level laser with a SA  without modulation are written as follows:
We rewrite Eqs. (1) in an adimensional form [12,13] by introducing new parameters and variables defined as t´ = t/τ, G = τ/T, δ = τ/τa, ρ = 2σa/βσ, α = ΓνσTN, αa = -ΓνTk 0 a(la/l), n(t´) = ΓνσTN(t´), na(t´) = -Γν(la/l)Tka(t´), and m(t´) = 2πβστS(t´)/hw, and including the normalized control harmonic modulation (1 + cos(ωt))/2 to αa in the third equation which describes the SA:
While an external signal injected into an optical cavity has often been used to modify the shape of the output signal [14,15], to our knowledge, this is the first time that it has been injected directly into the SA in order to modify the system dynamics and generate cnoidal waves. In this context, the SA acts as an active device since the laser output is regulated by it.
To evaluate necessary conditions for lasing, we find fixed points of Eqs. (2) and perform their linear stability analysis. In order to do so, we transform the non-autonomous system of Eqs. (2) into an autonomous one by making a change of variables, so that the new system is written as:
The characteristic equation for a perturbed stationary solution (i.e. ms = 0, ns = α, nas = αa, and xs = 0) is:Fig. 2 . Thus, to obtain oscillations in the laser with SA , α and αa should be chosen inside the dashed region shown in Fig. 2.
In the numerical simulations, we choose the parameters typical for a dye laser : G = 200, α = 4, δ = 1, ρ = 0.001, and the initial conditions near one of the critical stable points of Eqs. (2), i.e. m 0 = 0.25, n 0 = 0, and na 0 = 0.152. Since the parameter αa depends on geometrical values and on the absorbent centers density in the SA (the dye concentration in a dye SA cell), we use it as the SA defining parameter and call it absorption ratio.
Figure 3 shows the temporal dynamics of the laser output for fixed αa = 15 and different modulation frequencies ω. For small ω (lower than the laser relaxation oscillation frequency), the laser generates pulse trains with localized undulation windows, which are the damped relaxation oscillations (Figs. 3(a)–3(c)). For higher ω, only one frequency remains, i.e. the laser oscillates with the modulation frequency (Figs. 3(d)–3(f)), and the pulse shape strongly depends on ω. One important aspect is that as ω is increased; the peak amplitude first increases, reaches a maximum, and then decreases, thus going from a sech 2 (when the amplitude is maximum, Fig. 3(d)) to almost harmonic oscillations (Fig. 3(f)). While the peak amplitude is decreasing, the laser intensity never falls down to zero again; the continuous background appears because the frequency applied to the SA is so high that it has not enough time, neither to relax to its ground state nor to saturate. As ω further increases, the signal behavior becomes more and more sinusoidal with relatively small amplitude. We repeat the simulations for different αa with a step of 5 and find that the results shown in Fig. 3 for αa = 15 follow exactly the same qualitative pattern for any other αa ∈ [5, 60].
As discussed in  and following our results, the cnoidal wave behavior is bounded by two values of a control parameter; in our case, at high modulation frequencies the cnoidal waves are transformed to sinusoidal waves and at low frequencies to sech 2. Note, that the generated cnoidal waves are asymptotically stable due to quadratic nonlinearities of the SA [16,17].
Another interesting feature of the observed dynamics is that when the pulse train amplitude reaches its maximum, a sech 2 (soliton-like) shape approximates the pulse shape with a very good precision as demonstrated in Fig. 4 . This is confirmed by overlapping one pulse with a sech 2 waveform; the difference that appears on the base right hand side is very small (in the order of 2%, and always less than 5%). We find that for every saturable absorber coefficient αa there is an optimal modulation frequency ωs for which this soliton-shape approximation has better precision than for other frequencies. As seen from Fig. 5 , ωs increases approximately linearly with αa with two jumps at αa = 5 and αa = 30. We should note that several theoretical and experimental works report the existence of solitons [18–22] meaning that the pulses shape obtained at the output presents the soliton characteristic functions (sech and sech 2). While in the cited works, the difference between the reported pulses and the soliton shape is larger than 5%; our system allows the soliton generation with a higher precision, which makes it prominent for optical communication purposes.
We have numerically demonstrated with a modified Statz – De Mars model that a laser with an active saturable absorber under the influence of a control radiation from another laser can generate cnoidal waves within a certain range of control parameters, which bound soliton-like and sinusoidal regimes. No matter the physical saturable absorber characteristics, there is always a certain value of the modulation frequency that would result in the cnoidal waves generation. When we compared the resulting pulses with a typical soliton shape (sech 2), we obtained less than 5% error at the pulse base. The proposed system can be a base for building a reliable and cheap device to generate cnoidal waves as efficient information carriers for optical communication; the proposed system is economic due to the elements involved, at using general purpose laser elements and not ultra-fast optics elements, the experimental implementation of the presented scheme results less expensive.
We acknowledge CONACYT (Mexico) for the financial support through ANUIES-ECOS project M08-P02 and project No. 100429.
References and links
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