Spatial soliton switching in strongly nonlocal mediawith longitudinally increasing optical lattices

Zhiping Dai and Qi Guo*

Laboratory of Photonic Information Technology, South China Normal University, Guangzhou 510006, China*Corresponding author: guoq@scnu.eud.cn

Received September 28, 2010; revised November 5, 2010; accepted November 11, 2010;posted November 15, 2010 (Doc. ID 135820); published December 21, 2010

We present the impact of imprinted longitudinally gradually increasing optical lattices on the soliton mobility instrongly nonlocal nonlinear media. Based on the Ehrenfest theorem in quantum mechanics, we show that theintroduction of a longitudinal modulation offers an opportunity for soliton steering. The position of the outputchannel can be varied by small changes of the growth rate, a property that might find applications in opticalswitching schemes. Numerical simulations confirm the theoretical result. © 2010 Optical Society of America

OCIS codes: 190.0190, 190.6135, 190.4360.

1. INTRODUCTIONThe propagation of light in local nonlinear media is profoundlyaffected by the imprinted optical lattice, which exhibits awealth of opportunities for all-optical control, such as all-optical soliton steering and switching (for a comprehensivereview, see [1]). Lattice modulation in the longitudinal direc-tion provides a powerful way for the control of beam propa-gation, which has been intensively studied both theoreticallyand experimentally recently. Especially, dynamic localizationof light [2], inhibition of light tunneling [3], and higher-orderand mixed dynamic localization resonances [4] have been ob-served in curved optical lattices. Rabi oscillations are knownto occur in longitudinally properly modulated lattices [5,6].Much richer phenomena are expected in three-dimensional(3D) optical lattices [7–9], but their experimental demonstra-tion remains a challenge owing to the difficulties in fabricatingthe desired 3D lattice structures. Recently, the difficultieshave been overcome, and 3D optical lattices have been cre-ated with the optical induction technique [10,11]. Zhang et al.demonstrated experimentally coherent destruction of tunnel-ing, anomalous diffraction, and negative refraction in opticallyinduced 3D optical lattices [12].

However, under appropriate conditions, the nonlinear re-sponse of media might be nonlocal, a phenomenon that dras-tically affects the propagation of light (see [13] for a review). Ithas been shown that the nonlocal nonlinear response allowssuppression of the modulation instability of the plane waves,collapse arrest of multidimensional beams, and soliton stabi-lization in nonlocal nonlinear media [14–16]. The impact ofimprinted optical lattices on the mobility of solitons in a med-ium with weak nonlocality [17,18], arbitrary degree of nonlo-cality [19], and strong nonlocality [20] has been studied, and away to control the mobility of solitons has been put forward.In particular, [19] predicts the possibility of almost radiation-less propagation of solitons across the nonlocal opticallattices, which is addressed in [20] in the strongly nonlocalcase. Moreover, recent progress in creation of reconfigurableoptical lattices in photorefractive crystals [21–23] and nematic

liquid crystals [24–26] opened a direct way to explore theproperties of solitons by varying the lattice parameters.

As the nonlocality reduces drastically the value of thePeierls–Nabarro potential barrier [19,27,28], radiative lossesof solitons across the lattice can be very small in strongly non-local media. This means that, unlike the local case [29], it ishard to trap a strongly nonlocal lattice soliton into a specificchannel. In the present paper, we investigate the soliton mo-bility in the strongly nonlocal nonlinear medium with a long-itudinally gradually increasing optical lattice. We show that adifferent growth rate of the longitudinal modulation can beused to position the soliton at a prescribed location in the lat-tices, which provides an opportunity for optical control.

We organize the article as follows. In Section 2, we study thesoliton mobility with the aid of the Ehrenfest theorem. We dis-cover that the longitudinal modulation introduces new effectsinto the soliton transversemobility. Specifically, the soliton tra-veling across the lattices may be trapped in one of the guidingchannels, and the position of the output channel can be variedby small changes of the growth rate. In Section 3, we perform anumerical simulation to show the soliton trapping. We also ob-tain a switching curve depicting the dependence of the outputchannel on the growth rate. In Section 4, we put forward thepossibility of applications based on our findings. The conclu-sion is given in Section 5.

2. THEORETICAL MODEL ANDSOLITON MOBILITYWe consider the following dimensionless nonlocal nonlinearSchrödinger equation[14–16,30,31] with an increasing opticallattice [32,33]:

i∂ ~U∂z

þ 12∂2 ~U∂2x

þ ~UZ

Rðx − ξÞj ~Uðξ; zÞj2dξþ ~UpQðx; zÞ ¼ 0;

ð1Þ

where Uðx; zÞ is the complex amplitude envelope of the lightbeam, x and z are transverse and longitude coordinates,respectively, RðxÞ is the real symmetric nonlocal response

134 J. Opt. Soc. Am. B / Vol. 28, No. 1 / January 2011 Z. Dai and Q. Guo

0740-3224/11/010134-05$15.00/0 © 2011 Optical Society of America

function, and Qðx; zÞ ¼ cos2ðΩxÞð1 − expð−δzÞÞ describes theoptical lattices (Ω is the lattice frequency and δ is its growthrate) (Fig. 1). In practice, such a kind of lattice can be inducedoptically. Specifically, the refractive index modulation in thetransverse direction can be induced with two interfering planewaves [33–35]. Longitudinal modulation can be created byvarying intensities [12], intersection angles, or carrying wave-length [36] of the lattice-forming plane waves.

As has been previously indicated in [31], the width of theresponse function RðxÞ is much broader than the beam widthin the strongly nonlocal case [14–16,31]. Equation (1) can bereduced to the following equation:

i∂ ~U∂z

þ 12∂2 ~U∂2x

þ ~UZ �

R0 þ12R000ðx − ξÞ2

�j ~Uðξ; zÞj2dξ

þ ~UpQðx; zÞ ¼ 0; ð2Þ

where R0 ¼ Rð0Þ and R000 ¼ R00ð0Þ. By defining the beam power

P ¼Z

j ~U j2dx; ð3Þ

and the integral beam center [30]

qðzÞ ¼Rξj ~Uðξ; zÞj2dξR j ~Uðξ; zÞj2dξ ; ð4Þ

Equation (2) turns into

i∂ ~U∂z

þ 12∂2 ~U

∂2xþ ~UR0P þ 1

2~UR00

0Pðx − qÞ2 þ 12~UR00

0

Zðξ − qÞ2dξ

þ ~UpQðx; zÞ ¼ 0: ð5Þ

By the transformation [30]

U ¼ ~U expf−i½R0P þ R000

2

Zz

0dz0

Zðξ − qÞ2j~Uðξ; zÞj2dξ�zg; ð6Þ

Equation (5) is reduced to

i∂U∂z

þ 12∂2U

∂x2−U2γPðx − qÞ2 þ UpQðx; zÞ ¼ 0; ð7Þ

where γ ¼ −R000 > 0. Note that the phase difference between ~U

and U does not affect the mobility of the beam.Comparing with the one-dimensional Schrödinger equation

[37], we obtain the equivalent potential energy of Eq. (7) as

V ¼ 12γPðx − qÞ2 − pQðx; zÞ: ð8Þ

Based on the Ehrenfest theorem [20,37,38], one can readilyget a motion equation for the integral beam center as follows:

d2q

dz2¼

R ð−∂V=∂xÞjU j2dxR jU j2dx : ð9Þ

Since we focus on the strongly nonlocal case, we considerthe following Gaussian-type beam:

Uðx; zÞ ¼ A exp�−ðx − qÞ22w2

�exp½iβðx − qÞ þ iϕ�; ð10Þ

where A is the beam amplitude, which is large enough for thebeam to propagate in a soliton state, w is the beam width, β isthe incident angle with respect to the propagation axis, and ϕis the phase.

Substitution of Eqs. (8) and (10) into Eq. (9) yields thefollowing equation:

d2q

dz2þ Ω2

0

2Ω sinð2ΩqÞ ¼ 0; ð11Þ

where Ω0 ¼ ½2pΩ2ð1 − expð−δzÞÞ expð−Ω2w2Þ�1=2 defines thefrequency of small-amplitude oscillations of the soliton center.The notion of a particle for the soliton is valid when the so-liton’s width w is substantially smaller than the period of theoptical lattice T ¼ π=Ω [20,37,38]. For the sake of clarity, forall the results reported here, we keep this criterion by choos-ing Ωw ¼ 1=2, although our investigations show that we cankeep the particle notion even when the two length scales arecomparable in magnitude, i.e., w ∼ 1=Ω.

Alternatively, we assume that, without loss of generality, asoliton is launched into the medium at the point x ¼ 0 with anangle β0, so that qjz¼0 ¼ 0 and dq=dzjz¼0 ¼ β0. If the lattice isnot modulated longitudinally, i.e., Qðx; zÞ ¼ cos2ðΩxÞ, [20] hasshown that the soliton’s mobility is separated by a critical an-gle, which is given by βc ¼ ½2p expð−Ω2w2Þ�1=2. When the inci-dent angle is smaller than the critical angle, the soliton istrapped by the first waveguide formed by the lattices. Abovethe critical angle, the soliton travels across the lattices.

In the presence of a longitudinal modulation, i.e., Qðx; zÞ ¼cos2ðΩxÞð1 − expð−δzÞÞ, the critical angle βc ∼ Ω0=Ω ¼ ½2pð1−expð−δzÞÞ expð−Ω2w2Þ�1=2. Thus, βcðz¼0Þ¼0, which meansthat initially the system corresponds to a continuous medium,where the soliton will propagate without any difficulty in anydirection. Then, βc increases with the propagation distance.As a result, the soliton may be trapped by one of the guidingchannels at a certain distance where the condition βc > βðzÞ issatisfied.

3. NUMERICAL SIMULATIONTo confirm the analytical result, we numerically simulate thepropagation of a Gaussian-type beam in a strongly nonlocal

Fig. 1. (Color online) Profile of a longitudinally increasing latticewith δ ¼ 0:03 and p ¼ 1.

Z. Dai and Q. Guo Vol. 28, No. 1 / January 2011 / J. Opt. Soc. Am. B 135

nonlinear medium with an increasing optical lattice. We takeEq. (1) as the evolution equation and Uðx; zÞjz¼0 in Eq. (10) asthe initial condition. The numerical arithmetic we use hereis the split-step Fourier method [39]. The material responseis supposed to be a Gaussian function, i.e., RðxÞ ¼ expð−x2=2w2

mÞ=ðffiffiffiffiffi2π

pwmÞ, and we introduce a nonlocal parameter α to

define the degree of nonlocality, where α is the ratio of thebeam width w to the characteristic nonlocal response lengthwm of the media [14–16,30,31]. The smaller the nonlocal para-meter, the stronger the degree of nonlocality.

Figures 2(a) and 2(b) show the propagation trajectories ofsolitons in such lattices with different lattice parameters whenα ¼ 0:1 (the strongly nonlocal condition). Although bothfigures illustrate the phenomenon of soliton trapping, the lo-cations where trapping occurs are different, which impliesthat we can steer a soliton from one channel to another byvarying the lattice parameter.

Figures 3(a) and 3(b) demonstrate the dependencies ofthe output channel on the growth rate for different latticedepth when α ¼ 0:1. We assume the soliton is trapped inthe nth channel if nπ=Ω − π=ð2ΩÞ ≤ qðzÞ ≤ nπ=Ωþ π=ð2ΩÞwhen z → ∞. The switching curves suggest the possibilityof steering a soliton into a prescribed channel by changingthe growth rate. At small growth rates, a small variation ofthe growth rate leads to a large change of the number of the

output channel. By carefully checking Figs. 3(a) and 3(b),we can see the switching curve moves right when the latticedepth decreases. This means that, to trap a soliton in the samechannel, a larger growth rate is needed for the lattice with asmaller depth. Moreover, we calculated the dependencies ofthe output channel on the growth rate for different degreesof nonlocality at a fixed lattice depth and plotted the corre-sponding switching curves in Figs. 3(c) (α ¼ 0:5) and 3(d)(α ¼ 0:05). The curve in Fig. 3(c) moves left compared to theone shown in Fig. 3(a). The reason is that the radiative lossesincrease with the decrease of the degree of nonlocality, whichhelps to trap the soliton. However, once the strongly nonlocalcondition is satisfied, the radiative losses are negligible, andthe switching curve is almost the same as that shown inFig. 3(a). Figure 3(d) confirms this point.

4. POTENTIAL APPLICATION BASED ONTHE FINDINGSWe have found that the number of the output channel dependson the lattice growth rate and the lattice depth in the stronglynonlocal medium. Here we propose a possible scheme to steera soliton from one channel into another based on the upperresults.

Suppose that a soliton with an incident angle β0 is launchedinto a strongly nonlocal nonlinear medium (SNNM) with anincreasing optical lattice (here we neglect the reflection in themedium surface), as shown in Fig. 4. We assume that, withoutloss of generality, the soliton is trapped in the nth channel atfirst. The dependence of the number of the output channel onthe lattice growth rate suggests that we can steer the solitonfrom the initial channel into others. Slowly changing thegrowth rate, one can steer the soliton into a desired channel.

Fig. 2. (Color online) Propagation dynamics of solitons in longitud-inally increasing optical lattices with different lattice parameters. (a)δ ¼ 0:06, p ¼ 1, Ω ¼ 0:5; (b) δ ¼ 0:04, p ¼ 0:9, Ω ¼ 0:5. For both cases,β0 ¼ 0:7, α ¼ 0:1, and w ¼ 1. The red lines show the evolution of thesoliton center.

Fig. 4. (Color online) Contour plots showing control of the solitonswitching by varying (b) the lattice growth rate and (c) the latticedepth. Parameters: (a) p ¼ 0:9, δ ¼ 0:025, Ω ¼ 0:5; (b) p ¼ 0:9,δ ¼ 0:04, Ω ¼ 0:5; (c) p ¼ 1:2, δ ¼ 0:025, Ω ¼ 0:5. For all the cases,β0 ¼ 0:7, α ¼ 0:1, and w ¼ 1. The dashed lines show the sketch ofthe optical lattices.

Fig. 3. Dependence of the number of the output channel on thegrowth rate at Ω ¼ 0:5, β0 ¼ 0:7. (a) p ¼ 1:2, α ¼ 0:1; (b) p ¼ 0:9,α ¼ 0:1; (c) p ¼ 1:2, α ¼ 0:5; (d) p ¼ 1:2, α ¼ 0:05.

136 J. Opt. Soc. Am. B / Vol. 28, No. 1 / January 2011 Z. Dai and Q. Guo

For example, to steer the soliton into the mth channel, weonly need to change the lattice growth rate to an appropriatevalue according to their dependence. On the other hand, thedependence of the output channel on the lattice depth indi-cates that we can also steer a soliton into a prescribed channelby changing the lattice depth.

A concrete example for this control process is given in Fig. 4.The soliton is launched into the SNNM with an increasing op-tical lattice at an incident angle β0 ¼ 0:7. At first, when the lat-tice growth rate is 0.025 and the lattice depth is 0.9, the solitonis trapped in the eighth channel (Fig. 4(a)). When we vary δ to0.04 and keep the other parameters unchanged, the soliton isswitched into the fifth channel (Fig. 4(b)). One can also steerthe soliton into the fifth channel by changing p rather than δ.Figure 4(c) depicts the latter case.

5. CONCLUSIONTo summarize, we have studied the impact of the imprintedlongitudinally gradually increasing optical lattice on the soli-ton mobility in the strongly nonlocal nonlinear medium. Withthe aid of the Ehrenfest theorem, we investigate the propaga-tion dynamics of solitons and discover that the introduction ofa longitudinal modulation introduces new effects into thesoliton transverse mobility. The soliton traveling across thelattices may be trapped in one of the guiding channels, andthe position of the output channel can be varied by smallchanges of the growth rate. Moreover, the presence of thegrowth rate also leads to a dependence of the output channelon the lattice depth. The finding enriches the possibilities ofsoliton control in the lattices and might find applications inoptical switching schemes.

It is worth stressing that the soliton trapping uncoveredhere is considerably different from the radiative trapping ad-dressed in [29], where radiation losses result in a reduction ofthe transverse mobility of solitons and causes soliton trap-ping. In contrast, the physical feature we presented here isthe tunability of the lattice growth rate, and the radiationlosses of the output solitons are very small.

ACKNOWLEDGMENTSThis research was supported by the National Natural ScienceFoundation of China (NSFC) (grants 11074080 and 10904041),the Specialized Research Fund for the Doctoral Program ofHigher Education (grant 20094407110008), and the NaturalScience Foundation of Guangdong Province of China (grant10151063101000017).

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