Physics Letters A 372 (2008) 4395–4398

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Physics Letters A

www.elsevier.com/locate/pla

Soliton control in optical lattices with periodic modulationof nonlinearity coefficient

Jun Zhou a,b,∗, Chunhua Xue b, Yihong Qi b, Senyue Lou a

a Department of Physics, Faculty of Science, Ningbo University, Ningbo 315211, Zhejiang, Chinab Department of Optical Engineering, School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, Jiangsu, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 November 2007Received in revised form 27 March 2008Accepted 3 April 2008Available online 8 April 2008Communicated by A.R. Bishop

PACS:42.65.Pc42.65.Tg42.65.Wi

Keywords:Nonlinear opticsPulse propagation and solitonsSpatial soliton

We address the dynamics of solitons in the optical lattices with periodic modulation of the nonlin-earity coefficient. Based on the quasi-particle approach, the properties of fundamental soliton localizedin optical lattices are theoretically analyzed and shown its potential application for controllable solitonswitching. Moreover, the phenomena of multi-soliton splitting and the single-soliton constituent trappingin the optical lattices are illustrated and discussed.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

In recent decades, the properties of nonlinear propagation oflight in periodic photonic structures have been an attractive topicof investigation for its unique possibilities for control of light [1].The effects of discrete self-localized modes in periodicity nonlinearphotonic structures such as weakly coupled waveguide arrays, hadbeen theoretically predicted [2] and experimentally observed [3,4],and exhibit a wealth of potential application, for example, multi-port soliton switching [5–9] and discrete soliton control [10–14].And, the dynamical features of collisions between solitons havebeen investigated analytically and numerically in detail by the dis-crete nonlinear Schrödinger (DNLS) model [15] for optical latticeand Gross–Pitaevskii (GP) equation [16] for Bose–Einstein conden-sate (BEC) loaded in the optical lattice, respectively. Furthermore,the quasi-continuous nonlinear photonic structures such as nonlin-ear optical arrays, which can be formed by shadow refractive-indexmodulation, offer other possibilities for soliton control [17–21].

Another available technique for light control is nonlinearitymanagement, which assumes the nonlinearity coefficient periodi-cally changes along the propagation direction of light. Nonlinearity

* Corresponding author at: Department of Physics, Faculty of Science, NingboUniversity, Ningbo 315211, Zhejiang, China. Tel.: +86 574 87609980; fax: +86 57487600744.

E-mail address: ejzhou@yahoo.com.cn (J. Zhou).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2008.04.003

management can control the soliton energy flow at fixed width forimproving light transmission [22–24]. However, few studies so farare reported for light control in photonic structures with periodicmodulation of the nonlinearity coefficient. In fact, such photonicstructures can offer more opportunities for all-optical manipula-tion. In this Letter, the properties of light propagation in a mediumwith periodic modulation of nonlinearity coefficient are theoreti-cally investigated, and the single-soliton mobility and the multi-soliton splitting in the lattices are also simulated numerically.

2. Theory expression

We consider that the optical lattices is alternately composedof two kinds of self-focusing Kerr-type nonlinear medium, whichhave same linear refractive index n0 but different cubic nonlin-earity coefficient, n21 and n22 (n21 > n22). The evolution of beampropagation in such optical lattices can be described by the non-linear Schrödinger equation (NLSE)

i∂q

∂ξ+ 1

2

∂2q

∂η2+ [

1 + σ(η)]q|q|2 = 0, (1)

where q(η, ξ) = (Ldif/Lnl)1/2 A(η, ξ)I−1/2

0 , A(η, ξ) is the slowlyvarying envelope, I0 is the input peak intensity, Ldif = n0ωx2

0/c isthe diffraction length, Lnl = 2c/ωn2 I0 is the nonlinear length, ω isthe carrying frequency, n2 = (n21 + n22)/2 is the mean values ofthe nonlinearity coefficient and n2 > 0, η = x/r0 is the transverse

4396 J. Zhou et al. / Physics Letters A 372 (2008) 4395–4398

coordinates normalized to the beam width r0, ξ = z/Ldif is thelongitudinal coordinates normalized to the diffraction length Ldif,the lattice function σ(η) = σ0sgn[cos(2πη/T )] describes the trans-verse nonlinearity coefficient profile, σ0 = (n21 − n22)/(n21 + n22)

is the lattice depth and T is the transverse lattice period. For theconvenience of investigation, we set T = π/2. In the form of a per-turbed NLSE, Eq. (1) can be rewritten as

i∂q

∂ξ+ 1

2

∂2q

∂η2+ q|q|2 + V (q)q = 0. (2)

where V (q) = σ(η)|q|2 is the perturbed potential function, whichdepicts the effect of periodic modulation of nonlinearity coef-ficient on the soliton beam. Here we consider that the funda-mental soliton launched into the medium satisfies q(η, ξ = 0) =χ sec h(χη)exp(iα0η), where χ is the form factor or inversebeam width and α0 is the incident angle. For σ0 � 1 the spa-tial soliton propagating in the medium may be represented as aquasi-particle and can be approximated by q(η, ξ) = q0 sec h[χ(η−〈η〉)] exp[iα(η − 〈η〉) + iφ], where q0 is the amplitude, α = α(ξ) isthe propagation angle of the soliton, which agrees with the in-cident angle α0 at ξ = 0, φ is the phase, U = ∫ ∞

−∞ |q|2 dη is the

energy flow and 〈η〉 = (1/U )∫ ∞−∞ |q|2ηdη describes the evolution

of the integral beam center. With the aid of the quasi-particle ap-proach [21], propagation of the soliton can be given by

d2

dξ2

⟨(η − 〈η〉)2⟩

= 2

U

∞∫−∞

[∣∣∣∣ ∂q

∂η

∣∣∣∣2

+ |q|2(η − 〈η〉)dV

dη− 1

2|q|4

]dη

+ 2

U 2

( ∞∫−∞

∂q

∂ηq∗ dη

)2

, (3)

d2

dξ2〈η〉 = 1

U

∞∫−∞

|q|2 dV

dηdη, (4)

where 〈(η − 〈η〉)2〉 = (1/U )∫ ∞−∞(η − 〈η〉)2|q|2 dη depicts the evo-

lution of the mean-square width of the beam. First-order expan-sion of σ(η) in Fourier series yields σ(η) ≈ σ1 cos(2πη/T ), whereσ1 = 4σ0/π . Assumed that the form factor of soliton does notchange upon propagation, substitution of q(η, ξ) and σ(η) intoEqs. (3) and (4) obtains:

d2

dξ2

⟨(η − 〈η〉)2⟩

= 2

3

(χ2 − q2

0

) − 2

3σ1q2

0π2/Tχ

sinh(π2/Tχ)

×[(

1 − π2

T 2χ2

)π2/Tχ

tanh(π2/Tχ)+ 2π2

T 2χ2

]cos

(2π〈η〉

T

), (5)

d2

dξ2〈η〉 = −2π

T

σ1q20

3

(1 + π2

T 2χ2

)π2/Tχ

sinh(π2/Tχ)sin

(2π〈η〉

T

). (6)

Considering the stationary soliton propagated in central waveguidecorresponding to the conditions 〈η〉 = 0, d〈(η − 〈η〉)2〉/dξ = 0 andd2〈(η−〈η〉)2〉/dξ2 = 0, the stationary soliton amplitude is got fromEq. (5)

q20 = χ2 ×

{1 + σ1

π2/Tχ

sinh(π2/Tχ)

×[(

1 − π2

T 2χ2

)π2/Tχ

tanh(π2/Tχ)+ 2π2

T 2χ2

]}. (7)

Fig. 1. Soliton propagation in the optical lattices with σ0 = 0.05 at (a) α0 = 0.101and (b) α0 = 0.103, for χ = 1 and αcr ≈ 0.102. Dependence of the critical angle on(c) the lattice depth and (d) the form factor, respectively.

And, it also be deducted that q0 ≈ χ for σ0 � 1. Eq. (6) describestrajectories of the integral beam center in the optical lattice. Thebeam is launched into the lattice at the point η = 0, consequently,〈η〉ξ=0 = 0 and (d〈η〉/dξ)ξ=0 = α0. Therefore, solutions of Eq. (6)must satisfy

〈η〉 ={

Tπ sin−1[ α0

αcr× sn(

παcrξT ,

α0αcr

)], at α0 < αcr,

Tπ sin−1[sn(

πα0ξT , αcr

α0)], at α0 > αcr,

(8)

where

αcr = 2χ

[4σ0

3π

(1 + π2

T 2χ2

)π2/Tχ

sinh(π2/Tχ)

]1/2

. (9)

The significance of Eq. (8) is that, the beam is located in thecentral waveguide and oscillates periodically at the point η = 0when α0 < αcr, but crosses central waveguide when α0 > αcr. Con-sequently, there is a critical value of the incident angle separatingthe two regimes, which is given by Eq. (9).

3. Results and discussions

The theoretical analysis above can be verified numerically bythe split-step Fourier method. The propagation of soliton in themedium is shown in Fig. 1(a) and (b) for an incident angle nearbythe critical angle. When the incident angle is smaller than thecritical angle, the soliton is located around the central waveguideand travels along the center waveguide at an oscillation period(Fig. 1(a)). However, the soliton strides over the central waveguideand travels obliquely when the incident angle is larger than thecritical angle (Fig. 1(b)). The relation of critical angle and latticedepth is shown in Fig. 1(c), and the theoretical curve is rather con-sistent with the numerical one at small lattice depth but separatesslightly at large lattice depth. In Fig. 1(d), similarly, the theoreticalresults of critical angle versus the form factor agree very well withthe numerical one at a small form factor but deviates lightly at alarge form factor.

The energy loss of the quasi-particle is not involved in theanalysis above. However, suitable energy loss of the soliton uponpropagation may result in soliton trapping. Obviously, large lat-tice depth can give rise to energy loss of the soliton. For large

J. Zhou et al. / Physics Letters A 372 (2008) 4395–4398 4397

Fig. 2. (a) Soliton switching to the eighteenth channel in the optical lattices at χ = 1and α0 = 0.6. Dependence of the number of output channel on (b) the incidentangle at χ = 1 and (c) the form factor at α0 = 0.5, respectively. In all cases σ0 = 0.2.

lattice depth, as is shown in Fig. 2(a), the inclined soliton radi-ates slowly in the process of propagation, its propagation angledecreases asymptotically and then the soliton is trapped in one ofthe waveguide channels. Here the nonlinear system with energyloss can be regarded as a damped system, and the motion of thesoliton can be seen as damped motion. The number of the outputchannel trapped soliton can be controlled quantitatively by varyingthe incident angle and the form factor of the soliton. The soliton istrapped in the nth channel when (n − 1/2)T � 〈η〉 � (n + 1/2)T atξ → ∞. Dependence of the number of output channel on controlparameters are illustrated in Fig. 2(b) and (c). The number of out-put channel increases monotonically with the incident angle butdecreases monotonically with the form factor. It is useful for theapplication of soliton switching by manipulating of control param-eters, i.e., the incident angle and the form factor.

Moreover, numerical results show the phenomena of solitontrapping also appear in two other fascinating cases. One is σ0 = 1,i.e., the optical lattices are composed of a linear medium and aself-focusing nonlinear medium. In such optical lattices, the sta-tionary soliton exists due to balance between strong nonlinearself-focusing effect and linear diffraction (Fig. 3(a)). The other oneis σ0 > 1, i.e., the optical lattices are made of a strong self-focusingnonlinear medium and a weak self-defocussing nonlinear medium.In such optical lattices the stationary soliton exists because of sup-pression of strong self-focusing effect from weak self-defocussingeffect. However, in these two cases the width and the ampli-tude of the soliton exhibit strong breathing phenomena at smallnumber of the output channel (Fig. 3(b)), and the breathing phe-nomena fade away with increase of the number of the outputchannel (Fig. 3(c)). To explore the mechanism of the phenom-ena, we search for stationary solutions of Eq. (1) in the form ofq(η, ξ = 0) = w(η)exp(ibξ), where w(η) is a real function and bis a real propagation constant. Stationary solutions of Eq. (1) canbe obtained with Newton-iterative method. For the case of σ0 = 1,at the same propagation constant, the stationary solution is al-most equal for the amplitude to the fundamental soliton, but ismore narrow obviously than the fundamental one (Fig. 3(a)). It in-dicates that, when the fundamental soliton is launched into the

Fig. 3. (a) Comparison between the fundamental soliton (solid line) and the station-ary solution (dashed line) at b = 0.32. (b) Breath phenomena of soliton trappedin the second channel at χ = 0.8, σ0 = 1 and α0 = 0.6. (c) Disappearance ofbreath phenomena of soliton trapped in the eighth channel at χ = 0.8, σ0 = 1 andα0 = 0.85. Dependence of the number of output channel on (d) the incident angleat χ = 0.8 and (e) the form factor at α0 = 0.5, respectively.

lattices, the width and amplitude of the stationary solution ex-hibit strong breathing phenomena because of total self-focusingeffect (Fig. 3(b)). Lastly, due to the energy loss during the solitoncrossing the waveguide, the soliton shape comes to be stabilizedand the breathing phenomena fade away. Thus the breathing phe-nomena are not observed for the soliton which is trapped in theoutput channel with large number (Fig. 3(c)). Similar phenomenaalso exist in the lattices with σ0 > 1. The parametric dependenceof soliton switching is demonstrated in Fig. 3(d) and Fig. 3(e) forthese two lattices.

In addition, we find out from Eq. (9) that the critical angle isin proportion to the form factor of the input soliton. This meansthat a soliton with large form factor can be trapped in a latticeeven with small lattice depth. The phenomenon exhibits particu-lar application for controlling the multi-soliton bound states split-ting. The multi-soliton states are made up of set of several singlesolitons with the different amplitudes corresponding to Zakharov–Shabat eigenvalues and with zero binding energy [25]. Becausethere is no binding energy between the single-soliton constituents,suitable perturbations can be used to split them into their individ-uals [26–28]. We consider the initial conditions of the N-solitonbound states q(η, ξ = 0) = N sec h(η)exp(iα0η). Here N = 3 corre-sponds to three-soliton bound states, the amplitude and incidentangle of the kth constituent soliton are given by χk = 2k − 1 and

4398 J. Zhou et al. / Physics Letters A 372 (2008) 4395–4398

Fig. 4. Splitting of a three-soliton bound states and trapping of their single-solitonconstituent at (a) σ0 = 0.01, α0 = 1; (b) σ0 = 0.01, α0 = 1.2; (c) σ0 = 0.1, α0 = 0.1;(d) σ0 = 0.1, α0 = 1.2; (e) σ0 = 0.2, α0 = 0.3; and (f) σ0 = 0.2, α0 = 1.2.

αk = α0 (k = 1, . . . , N), respectively. The representative examplesof N-soliton splitting are illustrated in Fig. 4. Firstly, for the caseof shallow lattices (σ0 = 0.01), the χ5 constituent soliton is al-ways localized in the central waveguide unless the incident angleis large enough (Fig. 4(a) and (b)). However, the χ1 constituentsoliton and the χ3 constituent soliton can travel in the mediumfreely no matter how the incident angle is large. Secondly, for thecase of deeper lattices (σ0 = 0.10), the χ3 constituent soliton islocalized in one of the output waveguides at small incident an-gle (Fig. 4(c)), but travels in the medium freely at larger incidentangle (Fig. 4(d)). Also, the χ5 constituent soliton is localized incentral waveguide and the χ1 constituent soliton can motion inthe medium freely for arbitrary large incident angle. Lastly, withfurther increase of the lattice depth (σ0 = 0.20), the χ5 constituentsoliton and the χ3 constituent soliton are always localized in cen-tral waveguide and the adjacent waveguide, respectively (Fig. 4(e)and (f)).

4. Conclusions

In conclusion, soliton mobility in a photonic structure withperiodic modulation of nonlinearity coefficient is investigated. Atshadow optical lattices, propagation of the soliton is simple andregular because its energy loss can be neglected. However, forlarge lattice depth, the results shown the advantage of controllablesoliton switching, which is anticipated theoretically and simulatednumerically for the control parameters, i.e., the incident angle andthe form factor of soliton. Furthermore, the interesting behaviorof controllable soliton switching is presented for two special op-tical lattices, one of which is composed of a linear medium and aself-focusing nonlinear medium, and another of which is made of astrong self-focusing nonlinear medium and a weak self-defocussingnonlinear medium. Finally, splitting of a three-soliton bound stateand trapping of their single-soliton constituent in the lattices issimulated numerically. These results are pertinent nearly to thesoliton control experiment. They are also significant for the all-optical manipulation of light signals.

Acknowledgements

This work was supported by the Fund of National Natural Sci-ence of China (Grant No. 10574058) and the Fund of Jiangsu Uni-versity (Grant No. 04JDG041).

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