## Abstract

We observe experimentally, the first time to our knowledge, two types of composite gap solitons in optically induced one-dimensional nonlinear lattice in LiNbO_{3} crystal. We observe the staggered bright composite gap soliton when a single Gauss probe beam is incident at Bragg angle as well as a dipole probe beam is incident at normal incidence. When a single Gauss beam is at normal incidence, the in-phase bright composite gap solitons are observed.

© 2006 Optical Society of America

## 1. Introduction

Waveguide arrays are one or two dimensional photonic crystal structures .The wave dynamics in waveguide arrays is equivalent to the transport dynamics of electrons in semiconductors [1]. The wave propagation in periodic lattices differs fundamentally from that of their homogeneous counterparts. In nonlinear periodic lattices, the interplay between linear coupling effect among adjacent waveguides and nonlinearity plays a dominant role. The balance between these two effects can result in a self-localized state, which is often called as a “discrete” soliton or “lattice” soliton. The novel phenomena of light behavior in nonlinear lattice, such as discrete diffraction, lattice solitons, have attracted a strong interest these days. Of particular interest are the solitonic phenomena, which are universal in nature, from biological molecules [2], to solid-state system [3] and Bose-Einstein condensates [4].

The first successful experimental observation of optical lattice solitons in fabricated AlGaAs waveguide arrays was reported in 1998 [5]. In 2002, it was predicted that lattice solitons were possible in photorefractive crystals [6]. Then the experimental observation of discrete solitons was demonstrated in the optically induced real time waveguide arrays in SBN crystals by employing its screening nonlinearity and strong electro-optic anisotropy [7]. Recently, one-dimensional waveguide arrays fabricated by titanium in-diffusion in LiNbO_{3} crystal were used to form the spatial gap solitons [8]. The method of optical induction allows real time control of lattice spacing and potential well depth, so it opens up an exciting possibility for creating reconfigurable photonic structures. In the most of reported experiments, to observe lattice solitons in optically induced waveguide arrays requires the lattice-forming waves experience the lowest nonlinearity, at the same time the probe wave should experience the highest possible nonlinearity. Under this condition, the lattice is operated in linear regime; the nonlinear self-action of the lattice-forming beams as well as any cross action from a probe beam can be neglected. Recently, the lattice that is operated in nonlinear regime attracted strong interest because it is expanded beyond the limits of weak material nonlinearity. It was predicted that the stable nonlinear photonic lattices could be established in the photorefractive crystals even if the lattice-forming waves experience strong nonlinearity [9, 10]. Such a lattice interacts with the incoherent probe beam strongly, through the nonlinear XPM (cross-phase-modulation) effect, facilitates the formation of a novel type of a composite gap soliton, where one of the components be trapped and localized by a nonlinear periodic structure created by the other component [11]. The experimental observation of a composite gap soliton in two-dimensional nonlinear lattice optically induced in the SBN crystal was reported in 2004 [12].

In this letter, we report on an experimental observation of two kinds of composite gap solitons in the optically induced one-dimensional nonlinear lattice in LiNbO_{3}: Fe crystal. The lattice is created by interfering a pair of extraordinary-polarized coherent plane waves in the sample, which possess the self-defocusing nonlinearity arising from the bulk photovoltaic effect. The nonlinear refractive index change is given by

where *r* is the electro-optic coefficient, *E _{sc}* is the space charge field,

*E*is the amplitude of the photovoltaic field,

_{p}*I*is the light intensity, and

*I*is the dark irradiance. The maximum nonlinear index change optically induced by the photovotic nonlinearity can be very high (about 0.003[13]). In our experiment, the Δ

_{d}*n*is approximate to 10

^{-4}. With sufficiently high nonlinearity, the staggered composite gap solitons are observed when a single Gauss probe beam is incident at Bragg angle (k

_{x}=π/

*D*,

*D*is the lattice period) as well as a dipole probe beam is at normal incident. The in-phase composite gap solitons are observed when a single Gauss probe beam is launched into the lattice at normal incidence (k

_{x}=0), with sufficiently high nonlinearity.

## 2. Experimental setup and methods

A He-Ne laser beam (*λ*=632.8 nm) is split into two parts called lattice-forming beams which interfere in the sample to create the waveguide arrays. The lattice-forming beams which have the same power of 0.8 mw are extraordinarily polarized and create a grating with period *D* = 14*μ*m. The intensity of each lattice-forming beams is 0.84 mW/mm^{2}. Extraordinarily polarized beam from another He-Ne laser is also split into two parts: one of which is focused onto the input face of the sample by a spherical lens L_{1} (f=3.5 cm) to act as the probe beam; the other is used to interfere with the signal output beam to reveal the relative phase of solitonic components. The power of the probe beam is 1μw and the light intensity ratio of lattice-forming beam to probe beam is 1.1/1. The light distributions at the input and output faces of crystal are imaged onto a CCD camera by the lens L_{2} (12× magnification) , respectively. The LiNbO_{3}:Fe crystal employed in the experiment is doped with iron of 0.05wt% concentration and its dimensions are 9×9.5×10 mm^{3}(x×y×z).

First, we create a one-dimensional nonlinear lattice by illuminating sample with the lattice-forming beams for 20 minutes. Then we block the lattice-forming beams and monitor the intensity distribution of probe beam at the sample’s output face immediately after the probe beam is switched on (t=0). The discrete diffraction is observed as the result of probe beam’s linear propagation through the lattice. To observe the lattice solitons, we launch the probe beam and lattice-forming beams simultaneously and monitor the intensity distribution of the probe beam at the output face of the sample as it evolves in time, eventually forming the composite gap solitons. We use three kinds of probe beam’s incident conditions to observe the composite gap solitons: one of which is a single Gauss probe beam be incident at Bragg angle; another is a dipole probe beam be incident at normal incidence; the last one is a single Gauss probe beam be incident at normal incidence. After the formation of composite gap solitons, we switch on the interfering plane beam to reveal the relative phase of solitonic components. At last, we use the probe beam as a reading plane beam with the focusing lens removed to observe the lattice.

## 3. Experimental results and discussion

#### 3.1 A single Gauss beam be incident at Bragg angle

When the probe beam is incident at the Bragg angle ( *α*=0.59°), the staggered composite gap solitons are observed. Figure 2 shows the forming process of an even symmetry composite gap soliton. At the input face of sample, the probe beam is centered between two adjacent intensity maxima of lattice-forming beams’ interfering pattern [Fig. 2(a)]. At the time t=0, the intensity profile of probe beam at output face of sample has roughly 11 channels’ width [Fig. 2(b)]. As illumination time increasing, the pattern begins to narrow: at t=90 minutes it shrinks to 6 channels’ width [Fig. 2(c)], at t=130 minutes it concentrates at two channels accompanied by two slightly excited neighbor channels [Fig. 2(d)]. The two intensity peaks of the formed gap soliton reside on the input two adjacent intensity maxima of lattice-forming beams’ interfering pattern. Figure 3 shows the forming process of an odd symmetry composite gap soliton when the probe beam is centered on the intensity maximum of lattice-forming beams’ interfering pattern. At t=150 minutes, the odd symmetry composite gap soliton forms whose intensity peak resides on the input intensity maximum of the lattice-forming beams’ interfering pattern [Fig. 3(d)]. The patterns of lattice after the formation of the composite gap solitons are shown in Fig. 2(e) and Fig. 3(e).

Because the nonlinear effect in LiNbO_{3}: Fe crystal develops rather slowly, there is no nonlinearity caused by the probe beam at the time t=0. So Fig. 2(b) and Fig. 3(b) display the discrete diffraction, which is resulted from coupling between adjacent channels and multiple interference effects, fundamentally different from the linear diffraction in homogeneous media. The existence of gap solitons is closely linked to the structure of the linear wave spectrum. In linear regime, the wave propagating in a waveguide array is subjected to a periodic potential, so the dispersion relation (propagating constant*β* to Bloch wave number*K*) is divided into transmission bands, separated by gaps in which propagating mode is forbidden. In our experiment, the optically induced lattice (Δ*n* ≈ 10^{-4}) belongs to the low index-step lattice, so the first band is primarily excited by the probe beam [14].When the probe beam is incident at the Bragg angle, it is at the edge of first band and experiences the anomalous diffraction. As illumination time increasing, the probe beam induces a negative defect in the lattice employing the sample’s self-defocusing nonlinearity as well as the strong interaction between the probe beam and the lattice-forming beams through nonlinear XPM effect. With sufficiently high nonlinearity, the negative defect causes the propagating constant βof corresponding Bloch wave down into the Bragg reflection gap between the first and second transmission bands. Then the probe beam with the lattice-forming beams forms a composite band-gap soliton, where the nonlinear periodic lattice created by the lattice-forming beams traps the probe beam. It is clearly shown that a negative defect exists in the lattice and some waveguide channels are deformed after the formation of a composite gap soliton [Fig. 2(e), Fig. 3(e)]. The fact that the intensity peaks of the composite gap solitons are located on the input intensity maxima of lattice-forming beams’ interfering pattern indicates that the formed composite gap solitons are staggered type. To prove this, we use a plane beam to interfere with the output soliton to reveal the relative phase of solitonic components. The interferograms show that each channel of the formed composite gap solitons is out of phase with its neighbors [Fig. 4(b), 4(d)]. The interferograms confirm that the odd and even symmetry composite gap solitons observed in our experiments are indeed of the staggered type.

#### 3.2 A dipole probe beam be incident at normal incidence

The twisted modes are generated by introducing a tilted glass plate into half of the probe beam and adjusting the tilted glass to make both parts of the beam have a relative phase of *π*. The dipole probe beam is launched into the lattice at normal incidence with its two bright parts aligned at two adjacent intensity minima of lattice-forming beams’ interfering pattern at the input face of sample [Fig. 5(a)]. At the time t=0, the intensity profile of probe beam at the output face of sample has roughly 10 waveguide channels’ width [Fig. 5(b)]. As illumination time increasing, the pattern begins to narrow: At t=40 minutes, it has 8 waveguide channels’ width [Fig. 5(c)]; at t=90 minutes, it shrinks to 6 waveguide channels’ width [Fig. 5(d)]. Eventually, the pattern concentrate at two channels accompanied by two slightly excited neighbor channels at t=140 minutes [Fig. 5(e)]. In periodic lattice, gap solitons have the profile closing resembling modulated Bloch waves near the corresponding band edges [15]. So, if we shape the probe beam to match the profile of the modulated Bloch-wave properly, we can realize the spatial gap solitons’ formation. The gap solitons that arise from the edge of first band have the “phase signature” of the Bloch mode associated with the same transverse momentum [7]. When the dipole beam is launched at normal incidence, it can efficiently excite spatial gap solitons arising from the edge of first band because such an input beam has the resembling profile with the gap soliton just as Ref. [8] mentioned. At the edge of first band, the diffraction is anomalous, only bright gap solitons can form, because of the self-defocusing nonlinearity possessed of the LiNbO_{3} sample. At the time t=0, we observed the discrete diffraction. With sufficiently high nonlinearity, the nonlinear self-action of probe beam as well as the strong interaction between the probe beam and the lattice-forming beams make the probe beam be localized in the lattice. So the dipole probe beam with the lattice-forming beams creates the composite gap solitons, which is in good agreement with the case of a single Gauss beam being incident at Bragg angle. But the excitation method of a single Gauss beam be incident at Bragg angle may lead to a nonpure excitation of gap solitons in the lattice whose index step is not very low [14, 16]. The gap solitons formed in the lattice, which is operated in the linear regime, are the result of only probe beam’s nonlinear self-action; while in our experiment, the probe beam’s nonlinear self-action as well as the interaction with the lattice-forming beams cause the formation of composite gap solitons.

#### 3.3 A single Gauss beam be at normal incidence

When the probe beam is launched into the lattice at normal incidence, the in-phase composite gap solitons are observed. Figure 6 shows the forming process of an odd symmetry composite gap soliton. At the input face of sample, the probe beam is centered on the intensity minimum of lattice-forming beams’ interfering pattern [Fig. 6(a)]. At the time t=0, the intensity profile of probe beam at output face of sample has roughly 11 channels’ width [Fig. 6(b)]. As illumination time increasing, the pattern broadens further and reaches maximum width at t=30minutes as Fig. 6(c) shows. After that time, the pattern begins to narrow: at t=120 minutes it shrinks to 5 channels’ width [Fig. 6(d)], at t=220 minutes it concentrates at the input channel with two slightly excited neighbor channels [Fig. 6(e)]. The intensity peak of an odd symmetry composite gap soliton resides on the input channel. Figure 7 shows the forming process of an even symmetry composite gap soliton when the probe beam is centered on the intensity maximum of the lattice-forming beams’ interfering pattern. At t=200 minutes, the even symmetry composite gap soliton is observed, with most of energy concentrated in the input two channels [Fig. 7(e)]. The patterns of lattices after the formation of the composite gap solitons are shown in Fig. 6(f) and Fig. 7(f).

When the probe beam is at normal incidence, it is at the base of the first band and experiences normal diffraction. At the time t=0, the discrete diffraction is observed. As illumination time increasing, the probe beam induces a negative defect in the lattice employing the sample’s self-defocusing nonlinearity as well as the strong interaction between the probe beam and the lattice-forming beams through nonlinear XPM effect. The intensity profile of probe beam broadens further in 30 minutes is because the induced negative defect is still shallow and can’t trap the probe beam. With higher nonlinearity, the negative defect becomes deep and gradually traps the light due to the repeated Bragg reflect if the defect mode resembling the profile of input beam [17, 18]. The fact that a deep negative defect can trap the light must be contributed to the repeated Bragg reflection of optical waves in the lattice, just like what happens to electronic waves in semiconductors. In Such a negative defect where probe beam tend to escape from it, if a localized mode can be found, then this mode must reside inside the bandgap of the periodic medium. In Ref. [18], the defect modes are studied theoretically. When a negative defect is weak, the defect mode is rather weakly confined; as the negative defect becomes strong, the mode gets more confined; however, the mode becomes less confined again when the defect is too strong. Such a situation is evaluated: a Gauss beam is launched at zero angle into a negative defect site whose lattice intensity is half of the nearby sites and the probe Gauss beam takes the profile of the defect mode. The result shows that the probe beam will propagate stationarily and not diffract at all. In our experiment, the defect is induced by the probe beam, so they are automatically phase matched. With high enough nonlinearity, the defect mode is pushed down into the Bragg reflection gap and localized. Our experimental results are in good agreement with the simulation demonstrated in Ref. [18]. The fact that the intensity peaks of formed composite gap solitons reside on the input intensity minima of lattice-forming beams’ interfering pattern indicates the formed composite gap solitons are in-phase type. The corresponding interferograms show that each channel of the composite gap solitons is in-phase with its neighbors [Figs. 8(b), 8(d)]. The in-phase bright gap solitons will not be formed in the lattice which is operated in linear regime and possessing self-defocusing nonlinearity. Because such lattices are not “flexible” and will not form large negative defects, which can trap probe beams due to repeated Bragg reflection. The formation of in-phase composite gap solitons needs almost 50% longer time than the formation of staggered composite gap solitons is because the deeper defect is needed for the probe beam to be localized. The larger negative defects can be observed in the lattice after the formation of the in-phase composite gap solitons comparing with the staggered composite gap solitons case. The even symmetry in-phase composite gap soliton is unstable and tends to transform into a nonsymmetrical structure as a result of small perturbation of the probe beam’s position, because the power will swaps between the input two unequal parallel channels [19,20].

## 4. Conclusion

We have demonstrated the experimental observation of in-phase and staggered composite gap solitons in optically induced nonlinear one-dimensional lattice possessing a saturable self-defocusing nonlinearity in LiNbO_{3}: Fe crystal. The staggered composite gap solitons are observed when a single Gauss beam is incident at Bragg angle as well as a dipole beam is incident at normal incidence. Being incident at normal incidence, the single Gauss beam forms the in-phase composite gap solitons with the lattice-forming beams. It is the first time to our knowledge to observe two types of composite gap solitons in optically induced one-dimensional lattice.

## Acknowledgments

This research was supported by the National Natured Science Foundation of China (grants: 60278006, 60378013 and 10474047).

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