The interaction dynamics of X-waves in an AlGaAs waveguide array is theoretically considered. The nonlinear discrete diffraction dynamics of a waveguide array mediates the generation of spatio-temporal X-waves from pulsed initial conditions. The interactions between co-propagating and counter-propagating X-waves are studied. For the co-propagating case, the initial phase relation between the X-waves determine the attractive or repulsive behavior of the X-wave interaction. For the counter-propagating case, the collisions between X-waves generate a nonlinear phase-shift. These dynamics show that X-waves interact in a manner similar to solitons.
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The study of X-waves was first introduced in the context of linear phenomenon. The so-called X-waves were localized solutions of linear wave equations in the diffraction- and dispersion-free limit [1, 2]. Despite its genesis in linear theory, X-waves have emerged as a general model capable of describing nonlinear phenomena in settings whose underlying physics is described by a linear (hyperbolic) Schrödinger operator . Thus the application of X-waves to optical systems is not surprising given the central role of the linear Schrödinger operator in characterizing laser beams subject to diffraction and normal group-velocity dispersion (GVD) in bulk. Indeed, envelope X-waves were first observed in second-harmonic generation  and later extended to explain dynamic filamentation  as well as parametric generation in water . 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) [7, 8] and have been observed in discrete systems  such as waveguide arrays (WGA) [10, 11]. The ubiquitous nature of X-waves also suggests that they have applications outside of conservative models. Indeed, such X-wave patterns have also been predicted in dissipative systems [11, 12].
Given the number of nonlinear systems in which X-waves play a role, whether X-waves are linear or nonlinear objects is still a topic of current debate and interest. A recurring issue in this debate is that X-waves are spatio-temporal structures, but they are not stationary or periodic in terms of the propagation distance. Thus, as the X-wave evolves forward in time (or propagation distance) the nonlinear strength of the X-wave decreases. After sufficiently long times, this decrease reduces the impact of the nonlinearity and the X-wave becomes a linear object. However, before the amplitude of the X-wave is sufficiently reduced, nonlinearity plays an important role in its dynamics.
In this manuscript, we study the case where the nonlinearity plays a dominant role and show that there is a certain symmetry between the understanding of solitons and X-waves by comparing the interactions of X-waves to the interactions of solitons. Solitons are fully nonlinear solutions that interact linearly upon collision except for the inclusion of nonlinear phase-shifts and time-lags . Further, whether solitons attract or repel is dependent upon the relative phase-difference between the solitons . To date, the interactions between pairs of X-waves have not been considered. In this manuscript, a variety of X-wave interactions are characterized in the ideal optical setting of a waveguide array (WGA) . Much like solitons, the theoretical study shows that the nonlinear X-wave interactions produce what appear to be linear collision dynamics aside from nonlinear phase changes that are intensity dependent. In the context of co-propagating X-waves, the initial phase difference determines an effective attraction or repulsion dynamics. Thus, the spatial-temporal X-waves behave in much the same manner as solitons.
The paper is outlined as follows: In Sec. 2, the interaction dynamics of both co-propagating X-waves and counter-propagating X-waves are considered. Each of these systems has a different governing set of equations that are formulated in the corresponding subsections. Section 2 represents the critical findings of the paper. Section 3 provides concluding remarks following the numerical findings.
2. Nonlinear X-wave Interactions
The interactions between co-propagating X-waves and counter-propagating X-waves are considered in the context of an experimentally realizable AlGaAs WGAs . The parameters used in simulations are physically relavent for a 3 mm long AlGaAs WGA with input pulses generated from a mode-locked laser producing FWHM pulsewidths of 200 fs . The linear coupling coefficient is taken to be c = 0.82 mm−1 and the nonlinear self-phase modulation parameter is taken to be γ = 3.6 m−1W−1. In waveguides, chromatic dispersion also is present. However in keeping with previous works, dispersion has been neglected in this system . In any experimental fiber-waveguide system, the dispersive effects would be dominated by the fiber and so may be neglected in the waveguide. Hence, we take D = 0 in all that follows. Additionally, simulations with the inclusion of realistic dispersion values as measured experimentally  show the same qualitative dynamics as are demonstrated in this manuscript.
Typical peak powers in the WGA are on the order of kilowatts, and the total number of waveguides is 41 [10, 14]. Because of the parameters and initial conditions used, trivial modifications to recent experiments [9, 10, 14] should suffice to confirm the numerical simulations presented here.
2.1. Co-propagating X-waves
The first type of X-wave interaction is between co-propagating X-waves. In this interaction, pulses are injected into different but nearby waveguides in the WGA, and the leading-order equations governing the nearest-neighbor coupling (discrete diffraction ) of electromagnetic energy in the WGA are given by
where An represents the electric field envelope in the n−th of the 2N + 1 waveguides (n= −N, ···,−1,0,1,···, N) where N = 20 for 41 waveguides. This set of governing equations has been shown to accurately reproduce experimental findings for pulses with kilowatt peak powers and pulsewidths of hundreds of femtoseconds .
To begin, two pulses are launched into two adjacent waveguides with
and An(0, T) = 0 for n ≠ 0,1 where η 0, η 1 = 2.0 and ΔT = 1. At this value, fully nonlinear X-waves are formed during the propagation in the WGA. The dynamics of the X-waves depend on the relative phase difference between the injected pulses. Figure 1 demonstrates a time-history of propagation of (1) with an initial phase difference of Δθ = 0. From the pulse-initial condition, the characteristic X-wave structure forms. Although the majority of the energy in the X-wave remains confined in the initial waveguide, the low amplitude sections interact and perturb the X-waves. Ultimately, a pair of X-waves results with a slightly larger separation than the original pair of pulses, i.e. they repel. In contrast, consider the initial conditions in (2) with identical amplitudes but with Δθ = π. A time-history of (1) with these initial conditions is shown in Fig. 2. The WGA still generates an X-wave from each of the initial pulses as shown in the second panel in Fig. 2. However, the separation in T between the resulting X-waves is negligible, i.e. they attract. Therefore, for X-waves with identical amplitudes, the resulting dynamics depends on the initial phase-difference between the pulses.
Furthermore, the dynamics do not qualitatively depend on the initial separation between the two X-waves. Figure 3 shows the final separation between the X-waves as a function of phase difference for three different initial separations, ΔT = 0.5, 1.0, and 1.5. In all cases, for Δθ near zero the X-waves repel, and for Δθ near π the X-waves attract. The strength of interaction does depend upon the initial separation. As shown in Figure 3, X-waves that have a small initial separation will both repel more strongly and also attract at smaller values of Δθ. Likewise, X-waves with larger initial separations will repel and attract less strongly. Indeed, this agrees with the limiting case of X-waves that are so far separated that they are non-interacting and will neither attract nor repel.
Similarly, the differences in the maximum pulse amplitude determine the amount of attraction or repulsion but, for sufficiently large initial amplitudes, do not change the qualitative behavior of the X-waves. Figure 4 shows the final separation of two co-propagating X-waves that are initially in-phase and out-of-phase for a variety of different initial pulse amplitudes. Although the final position depends on the pulse amplitude, the in-phase pulses, shown in black, still repel while the out-of-phase pulses, shown in blue, attract. These behaviors persist for a large range of pulse amplitudes, though the amount of attraction or repulsion appears to decrease as the intensity increases. The results in Figure 4 were obtained using ΔT = 1, but the same qualitative results occur even if other separations are used.
Figure 3 and Figure 4 taken together show that the phase-dependence of the pulse-dynamics is a generic phenomenon and does not depend on either the initial separation or the pulse amplitude provided the initial pulse amplitude is sufficiently large. With low-amplitude initial conditions the interaction is dominated by linear effects and therefore this qualitative understanding of the X-wave interaction will not hold. As stated previously, the linear case is not the focus of this manuscript and therefore the results, though qualitatively different from nonlinear case, will not be presented here. Because the governing equations are a system of nonlinear and Hamiltonian PDEs, the exact initial condition is important in determining the exact solution at the end of the waveguide. There are no attractors that make one configuration more favorable than another. However, the initial phase-difference can be used to qualitatively describe the dynamics for a range of physically relevant initial conditions controlled by two physically relevant parameters, amplitude and phase.
In the previous case, the two initial pulses were of equal amplitude. However, if pulses of different amplitude are launched, the pulse with the largest peak power dominates the interaction. Figure 5 shows the interaction dynamics of two X-waves with initial conditions given by Eq. (2) where η 0 = 2.0 and η 1 = 1.0 and ΔT = 1 and Δθ = π. Unlike the equally sized pulses, the larger pulse dominates the dynamics and incorporates the smaller pulse into a single X-wave structure regardless of the phase difference. Therefore, it is the amplitude difference and not the phase-difference that determines the resulting dynamics.
To summarize: the interactions between high amplitude X-waves are nonlinear in nature and exhibit soliton-like dynamics . The dynamics of two co-propagating X-waves is dependent upon both the relative phase-difference and the absolute difference in amplitude of the two injected pulses. For sufficiently large initial conditions, the qualitative dynamics of the X-waves are independent of both the pulse size and initial separation. Therefore, the X-waves in this situation are not linear objects and share many of the qualities of solitons when they are placed in close proximity.
2.2. Counter-propagating X-waves
Another interaction of interest is the interaction between identical but counter-propagating X-waves. This situation could be generated experimentally by butt-coupling input fibers to opposite ends of a waveguide. The governing equations for counter-propagating waves must include both the forward- and backward-propagating fields. The governing equations, (1), must be modified to account for the second field yielding
where An is the forward-propagating field of the nth waveguide and Bn is the backward-propagating field of the nth waveguide. In these equations, the nearest-neighbor coupling and self-phase modulation terms are retained from the co-propagating case, but an additional cross-phase modulation term appears along with a group-velocity term determining the forward (+ σ) and backward (−σ) directions of propagation.
The collision of counter-propagating X-waves can be accomplished by launching initial pulses on both sides of the waveguide simultaneously. Thus the initial conditions take the form:
where 2ΔZ measures the initial spatial separation between the right moving (forward-propagting) and left moving (backward propagating) X-waves. It must be stressed that unlike the co-propagating interaction the governing equations of this system occur in a stationary lab frame and the initial condition is defined for all Z at T = 0 and not in the usual optical coordinate system. In this case, the waves collide at Z = 0. Figure 6 demonstrates the basic collision dynamics. The two X-waves pass through each other without visible deformation. Thus the collision appears to be linear. However, there is an induced nonlinear phase shift due to the collision. The resulting phase shift depends upon the initial launch intensity of the counter-propagating pulses.
Figure 7 demonstrates the phase shift incurred by the X-waves as a function of initial launch intensity. To calculate the phase-shift and time-lag, two simulations were performed. The first simulation includes both the forward- and backward-propagating X-waves. The second simulation only included the forward-propagating X-wave. Both simulations used the same physical and computational parameters as well as the same initial condition for the forward-propagating X-wave. Indeed, the only difference is the presence of the backward-propagating X-wave. After the interaction occurs, the cross-correlation of the forward-propagating X-wave in the 0th waveguide was taken. The maximum of the cross-correlation determines both the time-lag of the X-wave and the relative phase difference between them. It is clear from Figure 7 that X-waves act as linear waves for sufficiently low initial intensities. The nonlinear phase shift approaches zero as the initial pulse heights decrease. As the injected pulses are made more intense, the nonlinearities begin to exhibit themselves and the phase-shift increases in a monotonic fashion. As exhibited in Figure 6 however, the interaction still appears to be linear to the eye. Indeed, regardless of the pulse height there is no observable spatial lag generated by this interaction.
The dynamics of the X-wave formation may be studied by considering the two initial pulses in Eq. (4) for various values of ΔZ. Figure 8 shows the phase-shift due to X-wave interaction for five different initial values of η +, η −. In all cases, there is no measurable delay due to the interaction, but the amount of phase-shift is dependent upon both the initial separation and the initial pulse height. The results of Figure 8 may be broken down into two separate regimes, the low-amplitude initial conditions and the high-amplitude initial conditions.
The low amplitude cases are shown in the red, teal, and purple lines in Figure 8 and correspond to η +, η − = 1.00, 0.75, 0.50, and are characteristic of lower amplitude solutions as well. In these case, the amount of phase-shift decreases as the distance between the initial pulses increase. At relatively short initial separations, the low-amplitude X-wave has little time to distribute energy to neighboring waveguides through discrete diffraction. Because the phase shift is generated solely by nonlinear effects, the short distances allow the X-waves to be as nonlinear as possible given their initial conditions. Indeed, as the separation is increased the phase-shift decreases because the X-wave distributes energy to the outer waveguides and acts in a linear fashion. With an initial separation of 1.5, there is essentially no noticeable phase-shift.
On the other hand, with sufficiently intense initial conditions the X-waves can interact non-linearly even after propagating for a significant distance because the nonlinear mode-coupling traps energy in the original waveguide. At these higher intensities, the structure of the X-wave begins to play a role in the dynamics. At small separations, the amount of phase-shift is relatively large. At intermediate distances, the system is transitioning between the initial pulse-shape and the X-wave structure. Hence, the phase-shift decreases for a short period of time. After the X-wave structure has formed, the phase-shit increases again as shown in Figure 8. The separation for which this occurs depends upon the initial amplitude of the input pulse, and there is an ideal distance for which the interaction between X-waves causes a maximum amount of phase-shift. Because X-waves are not permanent structures, the phase-shift will eventually decrease as propagation distance increases. Regardless of the initial condition, given enough time energy will be coupled out to the outer waveguides and the interaction will become linear.
In all the cases presented in this manuscript, the initial phase-difference has not been mentioned. Due to the form of the governing equations in Eq (3), the interaction between the forward and backward propagating X-waves depends solely on the intensity of the X-wave. Therefore, it should be stressed that the results are the same regardless of the initial phase-difference between the X-waves. In this way, the counter-propagating case is qualitatively different from the co-propagating case where such phase-differences are critical.
The nonlinear interaction between counter-propagating X-waves is difficult to detect because it is not evident in the intensity of X-wave. As demonstrated, counter-propagating X-waves interact in a fashion that does not produce visible changes in the X-wave structure or a measurable delay in the X-wave itself. Nonetheless, the presence of a nonlinear phase-shift that depends on the initial pulse amplitude demonstrates that these X-waves indeed interact in a nonlinear fashion and can be likened to the nonlinear phase shifts experienced by solitons upon collision .
To conclude, we emphasize that nonlinear X-waves have been typically considered as normal GVD light-bullets generated by initial Gaussian wave-packets. However, the current manuscript shows new aspects to the X-wave dynamics. Specifically, we have provided the first numerical studies of X-waves dynamics under collision and interaction in a waveguide-array model that can be easily implemented experimentally . The results show that the X-waves, whose genesis is in linear theory, behave much like solitons, whose genesis is in a fully nonlinear theory. For co-propagating waves, the X-wave interactions can be attractive or repulsive when the two pulses are out-of-phase or in-phase respectively, similar to solitons . Attractive waves coalesce in time and space while repulsive X-waves separate further than their initial starting value. Counter-propagating X-waves collide leaving little evidence of nonlinear interaction. Indeed, only an intensity dependent nonlinear phase shift is generated. This nonlinear phase shift is much like soliton collisions. However, unlike soliton collisions, no timing shift is measured during the collision process.
The analysis of X-wave interaction, which is based on the quantitative model of waveguide arrays, predicts that the X-wave interactions can be verified with currently available, state-of-the-art technology. The results lead to a better understanding of multi-dimensional self-organized localized structures and to a more fundamental understanding of X-waves as fully nonlinear structures that behave more like solitons than linear structures.
J. N. Kutz acknowledges support from the National Science Foundation ( DMS-1007621) and the US Air Force Office of Scientific Research (AFOSR) ( FA9550-09-0174).
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