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We report on, for the first time to our knowledge, the existence of (1+1)-dimensional dipole solitons. We numerically find that such solitons can be supported by one-dimensional optical lattice in Kerr-type nonlinear media. Two dipoles of solitons are completely located at a broad optical lattice whose transverse modulation period is larger than ~1.5 times of the soliton width. The inherent repulsive interaction between dipoles can be effectively overcome by the lattice. The stable swing motion of dipole solitons in such lattice is also demonstrated.
©2006 Optical Society of America
1. Introduction
Spatial solitons have been a subject of many studies since their first theoretical prediction [1]. Recent researches focused on more complex structures than nodeless ground models. Many composite multimode solitons are associated with dipole solitons (DSs). The first theoretical prediction and experimental observation of DSs have been demonstrated by Refs. [2] and [3], respectively. Multipole and multicomponent dipole solitons have been also exploited [4–11]. However, the phase difference between their dipoles, in general, is out of phase, which leads to instability of DSs due to the repulsion between the dipoles. To achieve stabilization of DSs, most authors let the DSs couple with a nodeless component in a vectorial (two-component) system both theoretically and experimentally [2, 3, 7, 10, 12, 13] due to the trapping of the dipole mode by the soliton-induced waveguide. Another method of stabilizing the DSs is to use the optical lattices that can trap the dipoles against repulsion, and therefore, obtain the stationary DSs in experiment [9, 14, 15] and get rotary multipole solitons in theory [8, 10].
The propagation of light in media with periodically transverse refractive index modulation is of great importance to steer photonics through all-optical control and exhibits many interesting phenomena [16]. The experimental observations of discrete solitons were first demonstrated in one-dimensional waveguide arrays [17–19]. Such solitons were extensively investigated both theoretically and experimentally in one- and two- dimensional optical lattices [19–22]. Optical lattice with modulation period larger than soliton width can also stabilize solitons and provide unique propagation properties of solitons in nonlinear optics [23–25] and in Bose-Einstein condensates [26, 27].
To date, all theoretical and experimental researches for DSs are only based on the (2+1)-dimensional (two transverse plus one longitudinal dimensions) ones; however, the studies of (1+1)-dimensional (one transverse plus one longitudinal dimensions) DSs have not been exploited yet, to our knowledge.
In this paper, by use of one-dimensional optical lattice with large modulation period, we find that (1+1)-dimensional DSs can be supported due to the balance between the repulsive force of dipoles and the centripetal force of dipoles; the latter arises from the effect of lattice potential on dipoles. The lattice period is required to be larger than ~1.5 times of the soliton width. The trapped DSs can also display stable swing motion around the center of the lattice.
2. The model
The evolution equation of soliton propagation along the longitudinal direction z in one-dimensional waveguide with periodic refractive index modulation in the transverse direction x and Kerr-type self-focusing nonlinear media is
where p is the modulation depth parameter, V(x)=cos(x/T) describes the profile of transverse periodic refractive index distribution, and T describes the modulation period. Notice that Eq. (1) has several conserved quantities, including the energy flow U and the Hamiltonian H:
Next we search the stationary soliton solution. For p=0, the solution form of Eq. (1) is u(z,x)=Asech[A(x-x 0)]×exp[iα(x-x 0)+iβz+iϕ], here A is the soliton amplitude, x 0 is initial center position of the soliton, α is the incident angle (or slop of the phase front), β is the propagation constant along z direction, and ϕ is the initial phase. By modulating the soliton solution through multiplying it with sine function, analogous to the modulation of cosine function as the two-dimensional DSs in Ref. [12], the stationary solution of DSs with ϕ=0 is
For Eq. (4), soliton have two symmetrical humps that are out of phases, and the light intensity in the middle of the solition is zero. Such soliton possesses the characters of DS, and the two humps form the two dipoles of DS.
In this paper, two dipoles are located at a lattice, which is in contrast to the two-dimensional DSs [14], where two dipoles of solitons are set in two adjacent lattices. Substituting the DS solution Eq. (4) to the Eq. (3), one gets the transverse motion equation of center positions of DS
where the effective potential Veff (x 0) is given by [23, 28]
The center of DS trapped corresponds to the position of the effective potential minimum [Eq. (6)]. From Eqs. (5) and (6), the center position of DS remains at the lattice center in propagation with x 0=0 and α=0, i.e., DS linearly propagates along z. When the initial center position of DS is off the lattice center x 0≠0 or DS has an incident angle α≠0, the center position of DS produces swing motion around the lattice center as a result of the effect of lattice potential. The swing motion has been investigated for nodeless solitons [23–25].
3. Numerical results
We investigate the first- and second-order DSs corresponding to A=1 and A=2 in Eq. (4), respectively. The profiles of the DSs are shown in Figs. 1(a) with A=1 and 1(b) with A=2. The second-order DS has larger intensity and smaller width (or space between dipoles) than the first-order DS, which implies the second-order DS possesses stronger repulsion than the first-order DS. These DSs can be trapped by optical lattice with appropriate lattice period T and modulation depth p [Fig. 1(c)]. To further study the DSs’ robustness, in all numerical simulations below, we add a noise to the input DSs by multiplying them with [1+ρ 1,2(x)],
Fig. 1. Profiles of dipoles solitons (DSs) for (a) the first-order DS A=1 and (b) the second-order DS A=2. (c) Lattice modulation depth p versus modulation period T for A=1 and A=2, to achieve the stabilization of DSs.
where ρ 1,2(x) is a Gaussian random function with <>=0 and <>= (we choose that σ1,2 is equal to 10% the input soliton amplitude). Our direct simulations of Eq. (1) are based on slit-step Fourier method [29], the propagation step size is Δz=0.1, the number of discrete points of soliton profiles is 4096 and the space between points is Δx=0.01, most evolutions of DSs are plotted by extracting soliton profile every 1/4 of a diffraction length in propagation.
Fig. 2. (Color online) Evolutions of unstable DSs along direction z. (a) T=3, p=0 and A=1. (b) T=3, p=0 and A=2. (c) T=6, p=5 and A=1. (d) T=6, p=10 and A=2.
When the lattice is absent, DSs cannot be stably formed in propagation due to the strong repulsive force between dipoles [Figs. 2(a) and 2(b)], leading to the rapid breakaway of two dipoles. When the value of p is too small to support DSs, DSs undergo a drastic oscillation, where two dipoles expand and shrink accompanying with large radiant loss [Figs. 2(c) and 2(d)]. The oscillation is because that in initial stage, the repulsive force of dipoles exceeds the centripetal force of the dipoles (arising from the trapping effect of lattice potential on dipoles), two dipoles are detached quickly and the repulsive force is weaken so that the centripetal force becomes dominant, two dipoles of DSs begin to shrink simultaneously toward the center of the lattice. But two dipoles cannot mutually touch all the time due to the inherent repulsion. When two dipoles are very close, the shrinkage of the dipoles will stop and the dipoles begins repulse. Such oscillation badly damages the stabilization of DSs.
Figures 3(a)–3(f) show that DSs are well trapped in lattice under proper parameters that satisfy a relation between p and T as shown in Fig. 1(c), where the instability of DSs is greatly overcome because the balance between the repulsive force and the centripetal force of the dipoles is reached. In all cases, the modulation period 2πT of the lattice is larger than ~1.5 times of the soliton width, and the two dipoles are completely located at a lattice. The modulation depth p must be enhanced with the increase of lattice period T [Fig. 1(c)]. In addition, under the same value of T, the p for the second-order DSs A=2 are larger than those for the first-order ones [Fig. 1(c)]. This is because that the repulsive force between dipoles of A=2 is stronger than that of A=1 as described above, therefore, it needs larger lattice modulation depth p for A=2 to support DS than that for A=1.
Fig. 3. (Color online) Evolutions of DSs trapped by lattice for z=200. The first-order DSs A=1 for (a) T=3, p=10, (c) T=6, p=40 and (e) T=8, p=70. The second-order DSs A=2 for (b) T=3, p=50, (d) T=6, p=190 and (f) T=8, p=340.
When the initial center position of DSs is off the center of lattice or DSs are input with an incident angle, DSs produce transverse swing motion. To maintain the stability of DSs, the initial center position and incident angle cannot exceed the critical value shown in Figs. 4(a) and 4(b), respectively. The critical values of both incident angle and center position for the first-order DS are larger than those for the second-order one. The reason is that the second-order DS, has larger repulsion and stronger lattice than the first-order DS as described above, can excite larger energy losses in swing than the first-order DS. These critical values depend on the soliton amplitude A, lattice period T and lattice depth p. Figures 4(c)–(f) show when the center position of DSs x 0< or the incident angle α<αcr , the trapped DSs exhibit stable swing motion in propagation for z=200. The transverse trajectory of center positions of DSs can be determined by the Eqs. (5) and (6), which is similar to the cases of nodeless soliton swing [23–25]. This stable swing of DSs displays synchronous motion of single pendulum around the lattice center almost without disorder. Through changing x 0 and α, the output positions of DSs can be controlled. If x 0 and α are not zero, DSs’ swing will result from the simultaneous effects of both x 0 and α. In this case, only if the maximum of DSs’ center position in swing is not larger than the critical center position, DSs can exhibit stable swing motion [Figs. 4(g) and 4(i)]. But when the center position x 0≥ or the incident angle α>αcr as in Figs. 4(h) and 4(j), respectively, DSs exhibit unstable swing motion due to the very large radiate losses.
The experiment observation is feasible because that the model of Eq. (1) can be implemented by using the Kerr-type self-focusing media imprinted the optical lattice. The optical lattice can be produced by using coherent beam interference [20–22] or via amplitude modulation of a partially coherent beam [30, 31]. The dipoles structure can be obtained by inputting light across a phase mask (or glass slide) [5]. In addition, the model of Eq. (1) extensively appears in other nonlinear systems, e.g., attractive Bose-Einstein condensates [26, 27]. So our results are useful to stimulate further study in all relevant nonlinear systems, especially, including Bose-Einstein condensates.
Fig. 4. (Color online) Critical incident angle (a) and critical center position (b) versus period T for both A=1 and A=2. Evolutions of DSs exhibit stable swing motion around the center of lattice: (c) T=3, p=10, x 0=0.2 and A=1, (d) T=6, p=40, ∫=0.5 and A=1, (e) T=6, p=190, x 0=0.3 and A=2, (f) T=6, p=190, ∫=0.5 and A=2, (g) T=3, p=10, x 0=0.2, ∫=0.1 and A=1, and (i) T=6, p=190, x 0=0.2, ∫=0.1 and A=2. Evolutions of DSs exhibit unstable swing motion: (h) T=3, p=10, x 0=0.8, ∫=0.2 and A=1, and (j) T=6, p=190, x 0=0.3, ∫=1.5 and A=2.
4. Summary
To summarize, (1+1)-dimensional DSs whose two dipoles lie in a lattice are found to be supported in one-dimensional optical lattice with large modulation period (>~1.5 times of the soliton width). This is the first report on the existence of stable (1+1)-dimensional DSs. In the absence of optical lattice, soliton cannot exist because of the repulsive force between the dipoles. However, in the presence of the lattice, the lattice can trap the two dipoles against the repulsion, leading to the formation of DSs. The stable swing motion for such solitons is also investigated. Such complex structure of soliton might be of importance for potential applications in future digital-imaging processing devices based on soliton light spots. In addition, they have also the potential application in all-optical switch by the wide variety of the interactions between the two dipoles. Our results will be useful to boost the research for (1+1)-dimensional vector DSs as the (2+1)-dimensional vector ones.
Acknowledgments
This work is supported by National 973 (2004CB719804) Project of China, the National Natural Science Foundation of China (10674183).
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