We theoretically demonstrate that optical discrete X-waves are possible in normally dispersive nonlinear waveguide arrays. We show that such X-waves can be effectively excited for a wide range of initial conditions and in certain occasions can be generated in cascade. The possibility of observing this family of waves in AlGaAs array systems is investigated in terms of pertinent examples.
© 2005 Optical Society of America
Discrete optical systems exhibit several intriguing properties that have no analog whatsoever in the bulk . This is primarily due to the fact that the linear diffraction properties of such discrete configurations are significantly altered as a result of structural periodicity. This itself has several interesting consequences in both the linear and nonlinear regime [1, 2, 3]. For example, anomalous diffraction  is possible in the linear domain whereas discrete optical solitons can form under nonlinear conditions . Thus far, discrete solitons have been observed in several physical systems such as for example in Kerr nonlinear AlGaAs waveguide arrays [2, 4, 5], in two-dimensional biased photorefractive lattices [6, 7, 8], as well as in quadratic (χ (2)) arrays . In addition, other processes like discrete modulational instability , staggered or gap solitons [11, 12], and discrete vortex solitons [13, 14] have also been reported in these settings.
Quite recently another class of waves, the so-called X-waves has attracted considerable attention. In general X-waves represent stationary or invariant field configurations that can propagate free of diffraction and or dispersion effects. These waves involve infinite energy and are possible in bidispersive systems, e.g. physical arrangements that exhibit both normal and anomalous dispersive/diffracting behavior in different spatial or spatio-temporal coordinates. 3-D X-wave solutions were first obtained within the context of ultrasonics  whereas their 2-D analogues and 3-D generalizations have only recently been found . The generation of nonlinearly induced X-waves has been experimentally and theoretically considered in both quadratic and Kerr normally dispersive media [17, 18]. In particular, spontaneously generated X-light bullets were experimentally observed in lithium triborate crystals .
An important aspect associated with AlGaAs is that this material system, besides from being highly nonlinear, exhibits appreciable normal dispersion around 1.5µm. In essence in AlGaAs arrays two basic linear processes are in play: discrete diffraction and normal dispersion. As a result, for in phase waveguide excitation, the array behaves in a bidispersive fashion, e.g., the sign of the dispersion term in the evolution equation is opposite to that of diffraction. Following the previous discussion, it is natural for one to ask whether X-resembling wave configurations are possible in these arrays. In this study, we show that such arrays can support a novel class of waves: discrete X-waves. Even though these waves are closely related to the X-family encountered in the bulk, as we will see their behavior is very particular to discrete systems. Even more importantly, in space-time they exhibit features that can not be observed in either bulk or single-waveguide configurations.
2. System analysis
To analyze discrete X-wave excitation, consider an array of uniformly spaced weakly-coupled single-mode waveguides. From coupled-mode theory, the modal field amplitude Un in the n-th waveguide evolves according to [3, 19]:
where β 2 is the group velocity dispersion, C the linear coupling strength, n 2=(n̂2 n)/(2η 0), n̂2=2.47×10-13 cm 2/W is the AlGaAs self-focusing Kerr coefficient and k 0=2π/λ 0. n is the linear refractive index of AlGaAs and η 0=(µ 0/ε 0)½. We consider a 1cm long array consisting of 41 waveguide elements with an effective cross-sectional area of 5µm2. The distance between waveguides is taken to be 10µm. This array is used at λ0=1.55 µm where the AlGaAs normal dispersion is β2=+1.3 ps2/m. The coupling constant is assumed to be C=728 m-1 which is close to that encountered in previous experiments [2, 4, 5]. In all the examples, Eq. (1) is solved numerically by means of a symmetrized split-step Fourier scheme, with the nonlinear part treated on the basis of a fourth order Runge-Kutta method. The accuracy of the simulations is checked against the invariants of this equation during propagation. These correspond to the total power P and the Hamiltonian H of the system, i.e.,
The array is excited in time with hyberbolic secant-pulses. In the transverse plane (waveguide sites), we use either single or multiple waveguide excitation as initial conditions.
3. Discrete X-wave formation
X-wave formation in such a system is investigated over a wide range of input conditions. This is done in terms of the initial peak-power and temporal width. In this section, we varied the initial peak-power from 0.5kW to 1.5kW, where the 2- and 3-photon absorption is negligible compared to the Kerr-effect in AlGaAs waveguides. The initial pulse duration was taken to be either 1ps or 200fs. In all the figures, |Un|2=1 represents the peak-power initially used.
3.1 Single waveguide excitation
We will first examine X-wave formation by keeping the temporal width of the initial pulse at 1ps and by varying only the input peak-power. For a low input peak-power (0.5kW) the coupling among waveguides dominates the process and the initial pulse spreads in the transverse direction, as shown in Fig. 1. It is evident that the process of X-wave formation has already begun even at this power level. Clearly the nonlinearity is not strong enough to prevent the initial pulse from spreading in space and as a result it discretely diffracts.
At an intermediate input peak-power (0.7kW), discrete X-waves are more effectively excited. The interplay between diffraction and nonlinearity causes this pattern to arise in the following way. During propagation the initial pulse reaches a stage of maximum time-compression and edge-steepening occurs, leading to an X-like (or H-like) pattern. This pattern owes its shape to the confinement of the highly nonlinear part of the pulse in the central waveguide and to the linear diffraction of the low-power pulse edges. After this compression, both the linear edges (that diffract far from the central waveguide) as well as the main part of the pulse continue to broaden due to normal dispersion. The broadening of the main body of the pulse reduces the level of nonlinearity and it is therefore more amenable to diffraction. As a result, the central part of the remaining pulse diffracts, giving birth to two ultrashort spikes that keep moving apart. The X-like pattern that is generated by this procedure is depicted in Fig. 2.
At a high input peak-power (1.5kW) nonlinearity dominates over coupling, thus confining the major part of the initial pulse within the central waveguide, as shown in Fig. 3. The side lobes again diffract in a manner similar to that described previously, but they are now weaker. In a way, the high nonlinearity in the central waveguide decouples it from the rest of the array and the evolution of the initial pulse resembles that of pulse broadening in normal-dispersive nonlinear fibers. In the array however, this pulse profile is considerably steeper at the edges, due to the discrete diffraction of the low power lobes in the transverse direction. This process shows promise in terms of generating square-like pulses.
By keeping the input peak-power at 1.5kW and by using a shorter 200fs pulse at the input, the effect of group velocity dispersion becomes even more pronounced. As depicted in Fig. 4, appreciable pulse broadening occurs and this fact leads to a totally different scenario. In particular, this leads to an X-formation and after that high-intensity spikes are generated sequentially. These spikes in turn produce other new X-waves in cascade.
3.2 Multiple waveguide excitation
We next investigate the behavior of this array when a Gaussian beam is used to excite the waveguides at the input. In this case, the diffraction properties of the system depend on the excitation angle, that is on the initial phase difference Δφ among adjacent waveguides . For in-phase excitation (Δφ=0), the X-pattern is again observed as shown in Fig. 5 when the peak power in the central waveguide is 0.5kW and the initial temporal width of the pulse is set to be 1ps.
When the initial phase difference is set to values other than zero the system response changes. For the same initial power and temporal conditions but for Δφ=π, where the diffraction in the system is anomalous, the beam discretely diffracts (Fig. 6(a)), being unable to support any kind of X-like formation. For Δφ=π/2, where the diffraction in the system is zero, a more complicated V-like pattern is observed (Fig. 6(b)) due to higher-order diffraction effects. Again, the non-bidispersive nature of this configuration inhibits any X-like pattern.
In conclusion, we have demonstrated that X- wave formation is possible in nonlinear normally dispersive waveguide arrays. We have shown that the interplay between diffraction and nonlinearity leads to features that can not be otherwise observed in either bulk or single-waveguide configurations. The possibility to excite such X-waves in AlGaAs arrays was also considered in terms of relevant examples. The derivation of the dispersion relations of spatial-temporal waves and the study of stationary solutions are of great importance and will be considered in our future work.
This work was supported in part by the Greek Ministry of Education through the Heracletos grants, by NSF, and by an ARO MURI.
References and links
4. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999). [CrossRef]
5. J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003). [CrossRef]
6. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef]
7. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002). [CrossRef]
8. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92, 123902 (2004). [CrossRef]
10. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902 (2004). [CrossRef]
11. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003). [CrossRef]
13. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004). [CrossRef]
14. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004). [CrossRef]
15. J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992). [CrossRef]
16. D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three- dimensional bidispersive optical systems,” Opt. Lett. 29, 1446–1448 (2004). [CrossRef]
17. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously Generated X-Shaped Light Bullets,” Phys. Rev. Lett. 91, 093904 (2003). [CrossRef]
18. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett. 90, 093904 (2003). [CrossRef]
19. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 11, 2637–2644 (2002). [CrossRef]