1. Introduction

X-waves, first introduced as diffraction- and dispersion-free localized solutions of linear wave equations [1, 2, 3], are emerging as a general paradigm that explain several nonlinear phenomena in settings described by a linear Schrödinger operator of the hyperbolic type [4, 5]. This is indeed the case for laser beams subject to diffraction and normal group-velocity dispersion (GVD) in bulk, where envelope X-waves where first observed in second-harmonic generation [6], and later extended to explain dynamic filamentation [7] as well as parametric generation in water [8]. X-waves also play a relevant role in periodic media such as photonic crystals or Bose condensed gases (where a negative effective mass due to a periodic potential mimics normal GVD) [9, 10], and have been observed in discrete systems [11, 12, 13] such as waveguide arrays (WGA) [14, 15]. In parallel with these cases described by conservative models, X-wave (hyperbolic) patterns have also been envisaged in dissipative systems [16, 17]. In this letter, we address a different aspect related to the physics of strongly driven-damped systems (lasers), namely the role that X-waves can have in passive mode-locking (ML). Specifically, we theoretically demonstrate the first X-wave which is (i) mode-locked in a cavity and (ii) a global attractor of the system.

Introduced before the advent of lasers [18], ML is a general phenomenon [19, 20, 21, 22, 23, 24] and still a topic of active research in optics [25, 26, 27]. In particular, passive ML can be induced by exploiting spatial nonlinear effects (Kerr lens ML) in conjunction with temporal soliton pulses [28, 29]. However, ML mediated by the formation of more complex spatiotemporal structures such as nonlinear mode-locked X-waves, has not been reported to date.

 

Fig. 1. Schematic of a ring cavity configuration with passive (WGA for ML operation and a dispersive fiber section) and active (Erbium-doped fiber amplifier) elements. The input and output of the WGA are labeled A and B, respectively. The dominant physical effect is indicated for each component. The right graph demonstrates the ideal saturable absorption curve of the percentage of transmitted energy in the center wave-guide (E) versus intensity (I) generated by the WGA.

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Specifically, we consider the case where the role of intracavity saturable absorption (or intensity discrimination) is played by the nonlinear spatial effects which occur during propagation in a WGA [30, 31], the physics of which is now well understood [14, 15]. We show that, when the average cavity GVD is normal, the nonlinear mode-coupling in the WGA allows the electromagnetic field in the cavity to self-organize into a ML X-wave. The spontaneously generated X-wave is the global long-term attractor of the nonlinear evolution (in contrast with conservative settings [4, 5, 6, 7, 8, 11, 12, 13]) and very robust against large pertubations. Indeed, the nonlinear cavity appears to generate an infinite basin of attraction for the manifestation of X-wave phenomena. Also, at variance with previous studies on X-waves, here the ML X-wave are periodically generated at each round-trip despite the fact that GVD acts separately from the diffraction and the dominant nonlinear self-phase modulation (SPM) in the cavity. In this respect, the ML pulses achieved are actually periodic solutions over one round-trip, similarly to stretched-pulse or dispersion-managedML lasers [28, 29], or so-called exploding solitons [25]. As far as the stability and robustness of ML pulses over a round trip is concerned, we are effectively considering the Floquet stability of a Poincare section in an infinite-dimensional dynamical system [32]. Our numerical simulations imply that this Poincare section admits a global attractor which is the X-wave. Crucial to this behavior is the GVD accumulated over the round-trip, which makes the physics very different from other spatio-temporal localized structures (cavity bullets) emerging in externally driven, purely passive, nonlinear resonators [33].

2. Governing equations

We consider, without loss of generality, the output of the WGA (B in Fig. 1) in the realistic cavity shown in Fig. 1, where an optical amplifier is followed by a dispersive and weakly nonlinear element (typically a passive fiber section) and a strongly nonlinear diffractive WGA element. The evolution of the normalized electric field envelope Q along the fiber section, subject to normal GVD, SPM, linear attenuation, and bandwidth limited gain (fixed by τ), is ruled by the equation

where the saturable gain is

 

Fig. 2. Temporal (top) and spectral (bottom) pulse shapes emerging at point B (see Fig. 1) in the central waveguide, showing formation of a ML pulse from white-noise initial data. Here L_f=1 m and L_a=3 mm, the average cavity GVD is normal (k″=15 ps²/km), and gain parameters are τ=0.1,e₀=1, and g₀=3.5.

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Here, we adopt the standard scaling for nonlinear Schrödinger systems [30]: the retarded (in the pulse frame) time T, the distance Z, and the envelope Q are given in units of an arbitrary full-width at half-maximum duration T₀ associated dispersion length Z₀=(T ₀/1.76)²/k″, and onesoliton reference amplitude Q ₀ (|Q ₀|²=(γZ ₀)^-1, where γ=2π n₂/(λ₀A_eff) is the fiber nonlinear coefficient), respectively. Since extensive simulations performed with different values of the parameters do not show qualitative changes, we discuss below specific results obtained for a free-space wavelength λ ₀=1.55 µm, an average cavity dispersion k″=15 ps²/km, a nonlinear coefficient γ=1.7 km^-1W^-1 (Kerr index n ₂=2.6×10^-16 cm²/W and effective cross-sectional area A_eff=60 µm²), and lossα=ΓZ₀(Γ=0.2 dB/km). Furthermore we take T ₀=200 fs, which gives for a typical band-limited spectral gain of the order of Δλ=(λ²₀/2πc)Ω=20–40 nm, a bandwidth normalized parameter τ=(1/Ω²)(1.76/T ₀)²≈0.08–0.32.

In addition to the cavity model (1) and (2), evolution in the WGA must be considered. The leading-order equations governing the nearest-neighbor coupling (discrete diffraction [14, 15]) of electromagnetic energy in the WGA are given by [14, 15]

where A_n represents the normalized envelope (again in units of Q ₀) in the n-th of the 2N+1 waveguides (n=-N, …,-1,0,1, …,N). Here, Ẑ is scaled in units of the WGA physical length L_a, and hence C=κL_a and β=(γ_aL_a/γZ₀), γ_a being the WGA nonlinear coefficient. To make connection with experiments on physically realizable WGAs [11, 12, 13, 14], we take L_a=3 mm, a linear coupling coefficient κ=0.82 mm^-1 (C=2.46), γ_a=3.6 m^-1W^-1 (β=7.3), and N=20. Note that for a typical length of the fiber section L_f=1-2 m, the ratio of the nonlinear figures of merit between the fiber and the WGA, γ_aL_a/γL_f~6–12, is such that nonlinear SPM effects are dominated by the WGA. In contrast, dispersive effects are dominated by the fiber section since they are negligible in the WGA (L_a≪L_f) and hence are not included in Eq. (3). Here, the WGA acts as an ideal saturable absorber (intensity discrimination) element for which an idealized saturable absorption curve applies (See Fig. 1).

 

Fig. 3. Input (A) and output (B) temporal power distribution in the WGA. At the input, energy is only launched in the center waveguide (A₀), while at the output the energy has spontaneously formed into the X-wave configuration involving about eleven guides. Only energy in the A₀ mode is preserved upon re-entry into the fiber section of the cavity.

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3. Laser dynamics

To illustrate the spontaneous formation of the X-wave structure in the normal GVD regime, we integrate numerically the proposed infinite-dimensional map by alternating Eqs. (1) and (2) for a length L_f and Eqs. (3) for a length L_a according to Fig. 1. Thus Q of Eq. (1) becomes A₀ in Eq. (3) when entering or leaving the waveguide array. Importantly, upon exiting the WGA, the system is strongly perturbed since the energy from all the neighboring channels (A _i where i=±1,2,3, …) are expelled from the laser cavity. Nevertheless, we observe the formation of a stable ML pulse, as illustrated in Fig. 2, which shows the field A₀ at the output (B in Fig. 1). The white-noise is quickly reshaped (over 10 round trips) into the ML pulse of interest. Thus the ML pulse acts as a global attractor to the laser cavity system. The simulation further implies that the ML behavior is stable in the sense of Floquet [32] since it is a periodic solution in the cavity. The spectral shape clearly indicates that the ML pulse is highly chirped, in analogy to what is found for 1D (no spatial dynamics) solutions of the master ML equations in the normal GVD regime [28, 29].

While Fig. 2 depicts the output behavior in the central wave-guide of the WGA, the overall electromagnetic field actually experiences a strong spatio-temporal reshaping per cavity round trip that involves stable coupling of a significant portion of the incoming WGA power to neighboring waveguides with nontrivial timing. The input and output (A and B of Fig. 1, respectively) time-domain intensities in all the waveguides, once nonlinearML has been achieved are displayed in Fig. 3. As shown, the interplay of accumulated GVD, discrete diffraction, and nonlinearity drives the field into a self-organized nonlinear X-waves, whose main signature is a central peak accompanied by pulse splitting occurring in the external channels. To show more clearly the X-shape of the ML wave-packet generated at the ouput (B) of the waveguide array, Fig. 4 depicts a topographical plot of the time-domain (top) of all the waveguides. The distinctive X-wave structure is clearly evident. To lend further evidence to the existence of the X-wave structure, we plot the 2D Fourier transform of the time-domain. The bottom panel of Fig. 4 demonstrates that the spectrum is also X-shaped, as expected for X-waves [4, 5, 6]. It should be noted that other non-trivial spectral splitting phenomenon have been observed in all- normal fiber lasers [34]. However, the specific X-shaped time- and spectral-responses observed here are specific to X-wave systems.

 

Fig. 4. Time-domain profiles and its two-dimensional Fourier transform at the output (B) in the WGA after steady-state ML has been achieved. The X-wave structure is clearly seen in the topographical plot (top) of the output time-domain profiles of Fig. 3. Further, the expected wavenumber versus frequency dependence in the X-wave is shown in the Fourier domain (bottom).

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Fig. 5. Evolution to the steady-state output (B) in the neighboring waveguides A ₁, A ₂, and A ₃. The bottom right graph is a bar graph of the steady-state distribution of energy (∫∞ -∞ |A_j|² dT) in the waveguides. The symmetry about the center waveguide results from the initial condition being applied only in this waveguide. Note the significant redistribution of energy in the waveguides.

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To further characterize the ML X-wave dynamics, Fig. 5 illustrates the ML to the global attractor in the neighboring waveguides A ₁, A ₂ and A ₃. Once again, generic white-noise initial data quickly self-organize into the steady-state ML pattern. Note the characteristic pulse splitting (dip in the power) in the neighboring waveguides. This shows, in part, the generated X-wave structure. The final panel in Fig. 5 gives the energy (∫∞ -∞ |A_j |² dT) in each of the waveguides and shows that a significant portion (more than 50%) of the electromagnetic energy has coupled to the neighboring waveguides. This is in sharp contrast to ML with anomalous GVD for which less than 6% is lost to the neighboring waveguides [31] and no stable X-waves are formed. The significant loss of energy in the cavity to the neighboring waveguides is compensated by the gain section and shows that the laser cavity is a strongly damped-driven system.

4. Conclusions

To conclude, we emphasize that nonlinear X-waves have been so far considered as normal GVD light-bullets generated by Gaussian wave-packets. However, the current manuscript shows significant new aspects to the X-wave dynamics. Specifically, dissipative ML systems display multi-dimensional X-shaped global periodic attractors. From the perspective of laser physics, for normal dispersion and in the presence of diffractive effects, laser ML is fostered by this new form of spatio-temporal coherence. Our results are not limited to the discrete geometry considered here and have the potential to contribute to the field of Bessel-beam/axicon lasers [35], where the need of a conical mirror in the cavity can be circumvented by non-linear morphogenesis. Our analysis, which is based on a quantitative, fully comprehensive approach, predicts that the mode-locked X-wave laser can be constructed with currently available, state-of-the-art technology; and our results can lead to a novel class of X-wave-driven active optical devices and better understanding of multi-dimensional self-organized localized structures.

From a physical perspective, we point out that the normal dispersion of the fiber provides the required chirp for the X-wave pulse splitting in the lateral waveguides of the WGA. Such an out of axis splitting is the basic ingredient for X-wave generation. Theoretically, the only analytically tractable approach to characterize this is through a mean field model that reduces the map composed by fiber and WGA to a single governing equation. Only in the normal dispersion regime does this equation admit the X-wave solution; showing the need for normal dispersion. However such an asymptotic analysis is not strictly valid in the regime considered in our manuscript and future theoretical investigations are required. Further evidence for the requirement of normal dispersion is confirmed by the fact that in the case of anomalous dispersion, such splitting is not obtained [29, 30, 31, 36]. Indeed, numerical computations of the governing system with the sign of the dispersion flipped in the fiber shows that no X-wave is formed.