Oscillatory behavior of spatial soliton in a gradient refractive index ...

Appl. Phys. B 84, 465–469 (2006)

Applied Physics B

DOI: 10.1007/s00340-006-2289-8

Lasers and Optics

Oscillatory behavior of spatial soliton in a gradient refractive index waveguide with nonlocal nonlinearity

l.w. dongu h. wang

Institute of Information Optics, Zhejiang Normal University, Jinhua 321004, P.R. China

Received: 30 January 2006/Final version: 30 April 2006 Published online: 23 June 2006 • © Springer-Verlag 2006 ABSTRACT Oscillatory behavior of spatial solitons in a transverse parabolic gradient refractive index distribution (GRIN) waveguide with both local and nonlocal nonlinearity is investigated. Dynamics of such solitons are analyzed by the effective-particle approach method. For weak nonlocal nonlinearity, solitons oscillate in transverse direction periodically during propagation. The normalized width and maximum of refractive index variation of the waveguide play a key role in determining the oscillating period while the position of soliton oscillatory center is slightly influenced by the nonlocal nonlinearity. Stronger nonlocal nonlinearity leads to instability of the oscillatory solitons. Furthermore, the dynamics of the solitons are simulated numerically and good agreements are obtained between the analysis and numerical results. This behavior may be used in all-optical routers, switches etc. PACS 42.65.Tg;

1

42.65.Jx; 42.65.Wi

Introduction

Optical spatial solitons are self-trapped optical beams of finite spatial cross section that travel without the divergence associated with freely diffracting beams [1]. Due to their novel physics as well as potential applications, spatial solitons are under intensive study these days and much properties have been studied and found out in the last decade [1–6]. Some possible applications of spatial solitons such as optical pulse compression [7], all optical switching [8], logic devices [9] etc. have been proposed. To explore all possible applications, it is extremely important to extract and understand the generic properties of spatial solitons. In nonlinear materials, the presence of light modifies their properties (refractive index, absorption, or conversion to other frequencies). The universal principle unifying all solitons is that the wave-packet (beam or pulse) creates, by virtue of the nonlinearity, a potential well and captures itself in it. It becomes a bound state of its own induced potential well. As is well known, one of the most important properties of spatial solitons is that their dynamics are alike a particle moving in a light-induced potential well [1, 10–12]. This particlelike property allow us to treat the spatial solitons as a particle. u Fax: +86 579 2298831, E-mail:[email protected]

Another situation one often encounters is that the nonlinear medium with a transverse inhomogeneous refractive index distribution, that is, medium with both GRIN distribution and material nonlinearity. Aceves et al. [10, 11] developed an effective-particle theory and implemented this methods to the study of dynamics of spatial solitons due to interface separating nonlinear media. Scheuer and Orenstein derived two-particle theory to study the interactions of two solitons in the vicinity of the interface of two nonlinear media [8]. Another interesting property of solitons in a waveguide with transverse Gaussian linear refractive index profile is reported in [13]. Later, a simple model with triangular linear refractive index profile is studied [14]. More recently, a series of theoretical studies on the dynamics of the spatial soliton in a transverse periodical inhomogeneous medium have been reported by Y.V. Kartashov by the same method [5, 6, 15–17]. However, most of related studies above concerned only the localized nonlinearity of the materials except for [5] that deals with the problem of harmonic modulation refractive index with nonlocality. Other papers [18–20] discussing the nonlocal nonlinearity are not involved in the transverse inhomogeneous refractive index distribution. Under appropriate conditions, the nonlinearity of materials might be highly nonlocal, a phenomenon that importantly affects the properties of solitons supported by such media [21]. For example, nonlocal diffusion nonlinearity becomes significant for narrow light beams in photorefractive crystals [22]. Understanding the influence of nonlocality on soliton formation and propagation is therefore an important issue. In this paper, we consider the trapped and oscillatory behavior of spatial solitons in waveguide constituted by media with both local and nonlocal components of Kerr-type nonlinearity and transverse GRIN distribution. In present scheme, the GRIN medium is chosen as parabolic distribution while nonlocal nonlinearity depends on the derivative of the intensity of beam. The study begin by starting with the perturbed nonlinear Schrödinger equation (NLSE). To elucidate the problem completely and clearly, we derive the modulated NLSE and present the model that will be discussed in Sect. 2. Dynamics of solitons are derived analytically by effectiveparticle approach method in Sect. 3. Simulations of the propagating dynamics of solitons are given in Sect. 4 and Sect. 5 gives the conclusions.

466

2

Applied Physics B – Lasers and Optics

Model

The theoretical model describing the dynamics of a laser beam propagating in the nonlinear waveguide can be derived from the Maxwell equations. Taking the paraxial approximation and neglecting the anisotropic properties of the crystal, the model in the X – Z plane is simplified as following iβ

∂E 1 ∂ 2 E n2 + + β 2 |E|2 E = 0 , 2 ∂Z 2 ∂X n0

(1)

where E is the slowly varying amplitude envelope of the beam propagating along the z direction, β the wavevector of the guided mode, n 0 , n 2 the linear and nonlinear refractive index of the medium respectively. Equation (1) can be rewritten in dimensionless form as, ∂A 1 ∂ A + + |A|2 A = 0 , ∂z 2 ∂x 2 2

i

(2)

√ where x = βX , z = βZ , A = n 2 /n 0 E . Since we discuss the case of weak GRIN and nonlocal nonlinear medium, the refractive index modulation can be treat as a slight perturbation of the NLSE.

i

∂A 1 ∂ 2 A + + |A|2 A = −VA , ∂z 2 ∂x 2

∂|A|2 , ∂X

a(x) =

d2x dv = = − p−1 dz 2 dz

∞

−∞

dV A A∗ dx . dx

(6)

According to the effective-particle approach theory, one can assume that |A A∗| is only a function of x − x(z). Since power p is a conservational quantity, (6) may be written as following, d2x ∂U(x) =− . dz 2 ∂x

a(x) =

(4)

Obviously, (7) depicts the movement of a particle in a Newton potential U . Considering the fact that the medium is nearly homogeneous to the propagation of soliton profile but inhomogeneous to the motion of soliton in transverse direction when the refractive index perturbation due to GRIN and nonlocal response is small, we applied quasi-homogenous approximation and the perturbed solution becomes

where b is the normalized width of the waveguide, ∆n 0 the maximum index variation and µ describes the magnitude of the nonlocal component of nonlinearity. Note that in the present model, the refractive index distribution depends not only on the light intensity and the parabolic GRIN but also on derivatives of the light intensity. This nonlinearity occurs in photorefractive crystals (PRC) with diffusion component of nonlinear response. The use of the additional incoherent background illumination of PRC enables one to realize nonlinearity close to that in Kerr materials. The influence of the spatially nonlocal diffusion component of the nonlinear response gives an additional refractive index contribution, which is proportional to the derivative of the light intensity on the transverse coordinate [23]. For physical reasons, we assume that the depth of the refractive index modulation ∆n 0 is small compared to the unperturbed index and is of the order of the nonlinear correction to refractive index due to Kerr effect. The parameter µ is also assumed to be small, consistent with the fact that in practice the nonlocal contribution to the nonlinearity is small compared to the local one. 3

“acceleration”

(3)

where V = ∆n(x) + n non (x) and ⎧ 0 x < −b ⎪ ⎪ ⎨ x2 ∆n(x) = ∆n 0 (1 − ) −b ≤ x < b ⎪ b2 ⎪ ⎩ 0 x≥b n non (x) = µ

approach method [10] on such solitons as long as the potential induced by the GRIN and nonlocal contribution is relative small. In such an approach, the light beam can be considered as a particle whose position is given by x(z), where z is treated as the “time” variable. To find an equation of motion for the particle (beam), we begin by defining the center of ∞ the beam as [10, 11] x(z) = −∞ x A A∗ dx/ p(z), where p(z) ∞ is the power defined as p(z) = −∞ A A∗ dx . “velocity” ∞ dx 1 −∞ (A∂x A∗ − A∗∂x A) dx v(z) = = i (5) dz 2 p(z)

Effective particle approach – analysis

A qualitative picture of the soliton properties in the GRIN waveguide with both local and nonlocal nonlinearity can be grasped by implementing the effective-particle

A(x, z) = ηsech [η(x − x(z))] exp [i(v(z)x + σ(z))] .

(7)

(8)

Note that the velocity is now a function of propagation distance z in that v(z) = dx(z)/ dz and dσ(z)/ dz = [η2 − v(z)2 ]/2 compared to the case of uniform medium. The explicit formulae of the acceleration can be found by substituting (4) and (8) into (6), i.e., a(x) = a1 (x) + a2(x) ,    d2x 2∆n 0 CC a1 (x) = = ηb(TT − T) − ln , dz 2 b2 η p C  2µη5 SS(9 + 2CC 2 + 4CC 4 ) a2 (x) = 15 p CC 5  S(9 + 2C 2 + 4C 4 ) + , C5

(9) (10)

(11)

where C = cosh(η(b + x)), S = sinh(η(b + x)), T = tanh(η(b + x)), CC = cosh(η(b − x)), SS = sinh(η(b − x)), TT = tanh(η(b − x)), a1 and a2 corresponding to the acceleration induced by the GRIN and nonlocality, respectively. Acceleration depends on parabolic GRIN and nonlocality synchronously. One may infer from (9) that a1 is proportion to the parameter ∆n 0 and approximately inverse proportion to the parameter b while a2 depends critically on η, the amplitude of soliton and µ. While η, i.e., the power of soliton is large, a2

DONG et al.

Oscillatory behavior of spatial soliton in a gradient refractive index waveguide with nonlocal nonlinearity

467

FIGURE 1 Acceleration profile for different soliton and waveguide parameters b, η and µ, ∆n 0 = 0.3

will dominate the dynamics of soliton. Since the dependence of the acceleration on the parameters of soliton, as written in (9), is rather difficult to understand, we show in Fig. 1 the acceleration profiles of the “effective particle” with different parameters η, b and µ. Solitons are placed at the center of potential well induced by the parabolic GRIN. It is clear that a1 is an antisymmetric function of x while a2 is symmetric. The acceleration of the “effective particle” has a maximum at certain x for each η which is determined by da(x)/ d(x) = 0. The antisymmetry decreases with b or µ increasing or ∆n 0 decreasing. We can also predict the movement of soliton from the acceleration profile according to Newton mechanics if a2 is relatively small. If a input beam is launched at a position deviating to the center of waveguide with a zero initial velocity (defined by (5)), it will be subject to an “effective centripetal force” and moves back and forth within the scope of −x → x in the waveguide. Assuming that the beam is launched at −x 0 , it suffers immediately an accelerative process. The velocity achieves a maximum when it reaches the center of the waveguide. Thereafter, the “particle” decelerates because of the reverse sign of the acceleration. The velocity becomes zero as the “particle” reaches x 0 . It will be forced back to the center by the opposite acceleration compared to the initial one. For a fixed propagation distance, the frequency of the oscillation is proportion to the acceleration. Thus, the oscillatory frequency increases with the parameter ∆n 0 increasing or b decreasing. We should point out, the above conclusions are approximately true when the nonlocal response is weak. When the nonlocality is stronger and the soliton remains stable, the maximum of soliton velocity will slightly deviate from the center of waveguide and the motion of soliton is no longer symmetric due to the asymmetric acceleration profile. Further, the acceleration profile becomes highly asymmetric, which will destroy the periodical oscillation and stability of soliton while a2 is the dominant. The rule of nonlocality can be interpreted as that it imposes a variational “force” on the soliton in a fixed direction, thus forces it to escape from the potential well induced by GRIN and Kerr nonlinearity. 4

Simulation of the dynamics of the spatial solitons

To prove the analytical results in the previous section, we simulate the propagation of solitons by means of split-step Fourier methods. An input beam is launched at a position deviating from the center of waveguide with a zero initial velocity and propagates along the z axis. The input condition with a initial velocity is rather complicate and discussion of it is beyond the scope of present paper. Radiation occurs immediately as the soliton begin propagating, then

a stable soliton bound state forms. Figure 2 gives out the evolution of solitons with different parameter b and ∆n 0 . Clearly, the solitons propagate stably and oscillate periodically. The frequency of oscillation of the smaller b value is higher than that of larger one, while the case becomes opposite for parameter ∆n 0 . The center of oscillation slightly deviates from the center of waveguide induced by parabolic GRIN due to the weak nonlocality. This is in accordance with the analysis prediction in Sect. 3. Evolution of solitons with different η and µ are displayed in Fig. 3. The solitons will be unstable if η is either too big or too small. On one hand, stability of solitons with lower power may be destroyed by the perturbation induced by parabolic GRIN (Fig. 3a) and remains almost unchanged for relatively higher power solitons (Fig. 3b) by the same perturbation. The reason is that the index perturbation due to GRIN is strong to the lower power soliton (Fig. 3a) and weak to the higher power soliton (Fig. 3b and c). On the other hand, the stability of soliton with higher power is very sensitive to the nonlocality as in Fig. 3c and d. This phenomenon can also be drawn by (9). Dynamics of soliton is now chiefly determined by nonlocality of the medium. The fact that a2 is proportion to the η5 lead to a much bigger disturbance to soliton and destroy its stability. One can see clearly the influence of nonlocality on the dynamics of solitons from Fig. 4. The input condition is A = ηsech(η(x − x)). Soliton with η = 1.5, µ = 0.3 becomes quasi-stable and the oscillation is distorted. For even larger µ, the soliton may be completely unstable with the other parameters fixed. Figure 4c shows the maxima of amplitude of the oscillation same as Fig. 4a with the propagation distance. This amplitude oscillation may be interpreted by the internal mode theory [24, 25]. The oscillation of the amplitude is robust and survives for very long distance and its frequency is different from that of oscillation induced by parabolic GRIN. The intensity profile of the soliton in half a oscillatory period is shown in Fig. 4d. The profiles at z = 3.95 and z = 4.26 is used to examine that the propagation distance is actually half a period. Several features of dynamics of solitons are listed below. The first one is that the oscillatory center deviates from the position of x = 0 which illustrates the role of nonlocal nonlinearity and testifies the prediction in Sect. 3. The second one is that the velocity is not symmetric about the oscillatory center. This is in accordance with (9) and Fig. 1. The third feature is that higher power soliton is more sensitive to nonlocality (µ = 0.05) than that of lower power soliton (µ = 0.25, Fig. 2b). As has been discussed in [5, 6], the soliton will bend when we only apply the weak nonlocality (Fig. 4e). We also perform the simulation for a larger z and the main features remain the same.

468

Applied Physics B – Lasers and Optics FIGURE 2 Propagation of the solitons with different parameters b, ∆n 0 , η = 1, x = 5, µ = 0.25

FIGURE 3 Propagation of the solitons with different η and µ. ∆n 0 = 0.3, b = 5 in (a), (b); ∆n 0 = 0.1, b = 0.5 in (c), (d)

5

Conclusions

In conclusion, we investigated the dynamics of spatial soliton in GRIN waveguides with both local and nonlocal

kerr nonlinearity. By means of an effective-particle approach method, we derive the explicit acceleration formulae of the soliton. In such a scheme, solitons oscillate periodically with a variational acceleration in transverse direction during propa-

DONG et al.

Oscillatory behavior of spatial soliton in a gradient refractive index waveguide with nonlocal nonlinearity

469

FIGURE 4 Influence of nonlocality on dynamics of solitons with x = 5, ∆n 0 = 0.3, b = 5, η = 1.5; (a) µ = 0.05; (b) µ = 0.3; (c) maxima of amplitude with initial input as in (a); (d) intensity profile with propagation distances in half an oscillatory period of a soliton as in (c); (e) evolution of soliton only disturbed by nonlocal nonlinearity

gation when the nonlocal nonlinearity is relative small. The oscillatory periods of solitons decrease with the increasing b or decreasing ∆n 0 for a fixed propagation distance. The nonlocal nonlinearity slightly influences the center of oscillation and distort the shape of oscillation when the power of soliton is lower. However, when the nonlocal response is stronger, it affects the stability and dynamics of soliton remarkably by shifting the soliton away from the center of waveguide or lead to the instability. The soliton oscillates in the scope determined by the position of the input beam for weak nonlocal response. Simulation results of the propagation dynamics of the solitons are detailed discussed and in good agreement with the analysis predictions. ACKNOWLEDGEMENTS The authors are indebted to Dr. Fangwei Ye for useful discussions. The work is supported partly by the National Natural Science Foundation of China (Grant No. 60477039 and 10575087).

REFERENCES 1 G.I. Stegeman, M. Segev, Science 286, 1518 (1999) 2 C.K. William, B.A. Malomed, P. Chu, Phys. Rev. E 57, 1092 (1998) 3 T.J. Alexander, Y.S. Kivshar, A.V. Buryak, R. Sammut, Phys. Rev. E 61, 2042 (2001) 4 A. Trombettoni, A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001)

5 Y.V. Kartashov, V.A. Vysloukh, L. Torner, Phys. Rev. Lett. 93, 153 903 (2004) 6 Y. Kartashov, L.-C. Crasovan, A. Zelenina, V.A. Vysloukh, A. Sanpera, M. Lewenstein, L. Torner, Phys. Rev. Lett. 93, 143 902 (2004) 7 A.D. Reitze, H. Weiner, D.E. Leaird, Opt. Lett. 16, 1409 (1991) 8 J. Scheuer, M. Orenstein, Opt. Lett. 24, 1735 (1999) 9 Y.H. Pramono, M. Geshiro, T Kitamura, IEICE Trans. Electron. E 11, 1755 (2000) 10 A.B. Aceves, J.V. Moloney, A.C. Newell, Phys. Rev. A 39, 1809 (1989) 11 A.B. Aceves, J.V. Moloney, A.C. Newell, Phys. Rev. A 39, 1828 (1989) 12 R. Scharf, A.R. Bishop, Phys. Rev. E 47, 1375 (1993) 13 F. Garzia, C. Sibilia, M. Bertolotti, Opt. Commun. 139, 193 (1997) 14 A. Suryanto, E.V. Groesen, J. Nonlinear Opt. Phys. Mater. 10, 143 (2001) 15 Y.V. Kartashov, V.A. Vysloukh, Phys. Rev. E 70, 026 606 (2004) 16 Y.V. Kartashov, V.A. Vysloukh, L. Torner, Opt. Lett. 29, 1399 (2004) 17 Y.V. Kartashov, A.A. Egorov, A.S. Zelenina, V.A. Vysloukh, L. Torner, Phys. Rev. Lett. 92, 033 901 (2004) 18 N.I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J.J. Rasmussen, P.L. Christiansen, Opt. Lett. 29, 286 (2004) 19 A.A. Sukhorukov, Y.S. Kivshar, Phys. Rev. Lett. 91, 113 902 (2003) 20 A.W. Snyder, D.J. Mitchell, Science 276, 1538 (1997) 21 W. Krolikowski, O. Bang, Phys. Rev. E 63, 016 610 (2000) 22 V. Aleshkevich, Y. Kartashov, V. Vysloukh, Phys. Rev. E 63, 016 603 (2001) 23 V. Aleshkevich, V. Vysloukh, Y. Kartashov, Opt. Commun. 174, 277 (2000) 24 J. Yang, Phys. Rev. E 66, 026 601 (2002) 25 L. Dong, F. Ye, J. Wang, T. Cai, Y. Li, Physica D 194, 219 (2004)