Accessible spatiotemporal parabolic-cylinder solitons

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Accessible spatiotemporal parabolic-cylinder solitons

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2013 J. Phys. B: At. Mol. Opt. Phys. 46 075401 (http://iopscience.iop.org/0953-4075/46/7/075401) View the table of contents for this issue, or go to the journal homepage for more

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JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

doi:10.1088/0953-4075/46/7/075401

J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401 (7pp)

Accessible spatiotemporal parabolic-cylinder solitons Wei-Ping Zhong 1 , Milivoj R Beli´c 2,3 , Boris A Malomed 4 , TingWen Huang 2 and Goong Chen 5 1

Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province, Shunde 528300, People’s Republic of China 2 Texas A&M University, PO Box 23874, Doha, Qatar 3 Institute of Physics, University of Belgrade, PO Box 68, Belgrade, Serbia 4 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv IL-69978, Israel 5 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Received 1 January 2013, in final form 25 February 2013 Published 22 March 2013 Online at stacks.iop.org/JPhysB/46/075401 Abstract We study analytically and numerically ‘accessible’ spatiotemporal solitons in a three-dimensional strongly nonlocal nonlinear medium. A general localized soliton solution of the ‘acceptable’ type is obtained in the Cartesian coordinates, using even and odd parabolic-cylinder functions. Characteristics of these accessible spatiotemporal solitons are discussed. The validity of the analytical solutions and their stability is verified by means of direct numerical simulations. (Some figures may appear in colour only in the online journal)

1. Introduction

the characteristic nonlocality length being much larger than the beam’s width. The recent surge in the studies of selftrapped optical beams in strongly nonlocal nonlinear media is motivated by the experimental observation of nonlocal spatial optical solitons in nematic liquid crystals (NLCs) [9] and in lead glasses [10], as well as by a number of theoretical predictions [11–13]. Many of the predicted and demonstrated properties of nonlocal nonlinear models suggest that, in such optical media, one should expect stabilization of diverse types of nonlinear wave structures, such as, for example, necklaces [2, 14, 15] and soliton clusters [16, 17] in the two-dimensional (2D) transverse space. The nonlocality of the nonlinearity prevents the beam from collapse in media with the cubic nonlinearity in all physical dimensions, resulting in the prediction of stable multidimensional solitary waves [18, 19]. Stable three-dimensional (3D) spatiotemporal solitons in cubic nonlocal nonlinear media were reported in [20, 21]. Light bullets formed via the synergy of reorientational and electronic nonlinearities in liquid crystals were proposed and discussed in [22]. As said above, a highly nonlocal situation arises in a nonlinear medium in which the characteristic size of the material response is much wider than the size of the

Hermite–Gaussian [1] and Laguerre–Gaussian solitons [2] are well-known exact solutions of the strongly nonlocal nonlinear Schr¨odinger (NLS) equation in Cartesian and polar coordinates, respectively. Their theoretical and practical importance was established mainly because they form two complete bases of orthogonal modes, and because their profiles and widths remain unchanged with the propagation distance. Another natural ansatz, which may be used for finding exact analytical solutions in the form of spatiotemporal solitons (‘light bullets’) in strongly nonlocal media, may be provided by the parabolic-cylinder functions, which is the objective of this work. The nonlocal nonlinearity, which occurs in many physical systems such as plasmas [3], Bose–Einstein condensates [4] and different optical materials [5–7], has attracted considerable interest in recent years. According to the degree of nonlocality—as determined by the relative width of the response function with respect to the size of the light beam— four categories of the nonlocality may be identified [8]: local, weakly nonlocal, generically nonlocal and strongly nonlocal. In particular, the strongly nonlocal case is the one with 0953-4075/13/075401+07$33.00

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W-P Zhong et al

2. The model and its parabolic-cylindrical soliton solutions

excitation itself. This situation naturally arises in NLCs, where both experiments [5, 9] and theoretical calculations [23] demonstrated that the nonlinearity is highly nonlocal. However, even a high degree of the nonlocality does not guarantee the existence of stable soliton structures. In particular, the orientational nonlinearity in NLCs is highly nonlocal, but the nonlinear response is not perfectly quadratic, implying that, if one launches a Gaussian beam into the cell, an oscillating soliton will be observed. Still, the beam propagation in NLCs neatly aligns with the ‘accessible soliton’ model of Snyder and Mitchell [24]. Although this model seems to be locally linear, it still describes the behaviour of strongly nonlocal solitons quite well. Typical experimental parameters corresponding to a NLC can be summarized as follows. The cell size is a few mm in each spatial dimension; the diffraction length is of the order of 100 micrometers; the beam’s widths extends over a few micrometers; the beam’s wavelength is in the green; the ordinary refractive index is about n = 1.5; the bias voltage across the cell is of the order of 1 V; and optical powers are in the mW range. The typical experimental lengths over which the strongly nonlocal solitons—nematicons—are observed in NLCs amount to about 2 mm. To date, only fundamental nematicons were reported; NLCs are natural candidates for media in which higher-order parabolic-cylinder solitons reported below might be observed. In this work, we demonstrate that a class of 3D accessible spatiotemporal parabolic-cylinder solitons can be supported in the model of the strongly nonlocal nonlinear medium. Such solutions, constructed by means of the method of the separation of variables as products of complex modulation functions and Gaussian beams in the Cartesian coordinates, form stable soliton patterns in the propagation. It is well known that stationary solitons in 3D media with the local cubic nonlinearity are always unstable against collapse. The strong nonlocality helps to stabilize the new solutions, because in that limit they approach the Snyder–Mitchell model of accessible solitons [24]; as mentioned above, that model is locally linear. Hence, it cannot feature instability. There are not too many areas of nonlinear spatiotemporal dynamics where such nonlocal nonlinear limits naturally occur, but the generation of nematicons in liquid crystals is one of the fields to which our method naturally applies [22, 23]. The paper is organized as follows. In section 2, we introduce the general 3D strongly nonlocal nonlinear model and obtain exact accessible soliton solutions. Using the separation of variables and products of a modulation function and the Gaussian beam in the Cartesian coordinates, we construct a new class of 3D spatiotemporal parabolic-cylinder solitons. In section 3, we present some solutions as relevant examples for some specific parameters. We find that the new class of 3D strongly nonlocal spatiotemporal solitons may display various forms. In section 4, the validity of the analytical solutions and its stability is verified by means of direct numerical simulations. Finally, we summarize our results in section 5.

To start, we assume that the spatiotemporal beam propagates along coordinate z. In the case of the strongly nonlocal medium, the optical beam with the complex amplitude u (z, x, y, τ ) is governed by the following 3D paraxial wave equation [21]: ∂u 1 i + ∇ 2 u − sρ 2 u = 0, (1) ∂z 2 where s > 0 is the parameter proportional to the beam’s power, which is a conserved quantity. The proportionality of s to the total power is how the implicit global (extremely nonlocal) nonlinearity is introduced in the Snyder–Mitchell model. In ∂2 ∂2 ∂2 + ∂y equation (1) ∇ 2 = ∂x 2 + ∂τ 2 is the spatiotemporal 2 Laplacian and ρ = x2 + y2 + τ 2 is the ‘spatiotemporal radius’, τ standing for the local (retarded) time in the frame of reference moving with the pulse. Equation (1) is a linear partial differential equation with the quadratic potential, formally equivalent to the Schr¨odinger equation for the quantummechanical harmonic oscillator. We search for a spatiotemporal soliton solution of equation (1) by writing the solution as a product of a complex modulation function, uF (z, x, y, τ ), and the Gaussian beam, uG (z, ρ): u (z, ρ) = uF (z, x, y, τ ) uG (z, ρ ) ,

(2)

where uG (z, ρ) satisfies the following equation: ∂uG 1 i (3) + ∇ 2 uG − sρ 2 uG = 0, ∂z 2 which is formally equivalent to equation (1). This equation has an exact self-similar localized solution [25] 2

− ρ 2 +iθ (z) k uG (z, ρ) =  e 2w0 , (4) 3 w02  where θ (z) = a0 − 2w3 2 z, s = 1 2w04 , k is the normalization 0 constant, w0 is the initial width of the Gaussian soliton beam and a0 is the initial phase. Substituting equation (2) into equation (1), we obtain an equation for uF:   ∂uF 1 ∂uF ∂uF 1 ∂uF + ∇ 2 uF − 2 x +y +τ = 0. (5) i ∂z 2 ∂x ∂y ∂τ w0

To find complex solutions of equation (5), we use the method of the separation of variables, and split equation (5) into three independent equations [26]. We thus obtain (1 + 1)D partial differential equations in each of the transverse dimensions. For example, the equation in the x direction is ∂UF 1 ∂UF 1 ∂ 2UF i − 2x + = 0. (6) 2 ∂z 2 ∂x w0 ∂x We solve these equations by the self-similar method. We assume that UF (z, x) = A (z) F (), where A (z) is x is the selfthe amplitude of the beam and  (z, x) = μ(z) similar variable, where μ (z) is a z-dependent scaling factor to be determined. Substituting UF (z, x) into equation (6) and 2

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separating the variables again leads to the following equations for F, μ and :   1 ∂ 2F ∂F −i ν+ F = 0, (7) − i ∂2 ∂ 2 2μ

∂μ 2iμ2 − 2 = 1, ∂z w0

  1 2μ2 ∂A =− ν+ , A ∂z 2 where ν is the separation constant. Letting λ = equations (8) one obtains      w02  2i 2 C exp z − 1 , λ =  2 2 w0

Using the same process, we obtain the other two components of the full solution of equation (5), along the y- and τ - directions: 1     2i −im− 4 (1−i) (e,o) im+ 14 (1−i)  UF (z, y) = |C − 1| C − exp − w 2 z  0 y2

×e− 4λ2 Gm(e,o) (z, y),

(8a)

UF(e,o)

(8b) √ iμ, from

   1 1  2i − 2 ν− 4 A (z) = A0 C − exp z , w02 

where

x2

(13c)

 m 1 iy2 , (z, y) = e 1 F1 − , , 2 2 2λ2 √   m − 1 3 iy2 iy −i y22 o 4λ Gm (z, y) = e , , 2 , 1 F1 − λ 2 2 2λ   2 τ2 l 1 iτ Gel (z, τ ) = e−i 4λ2 1 F1 − , , 2 , 2 2 2λ −i

Gem

(9a)

(9b)

y2 4λ2



and

√   l − 1 3 iτ 2 iτ −i τ 22 = e 4λ 1 F1 − , , 2 . λ 2 2 2λ Here m (m = 0, 1, 2 . . .) and l(l = 0, 1, 2 . . .) are the mode numbers of the beam along the y- and τ -axes, respectively. Now, the 3D solution of equation (1) in the Cartesian coordinates can be readily constructed as a product of the 1D solutions of the form (13), namely k u (z, ρ) =  UF(e,o) (z, x) UF(e,o) (z, y) UF(e,o) (z, τ ) 3 w02 Gol (z, τ )

−

×e

ρ2 2w 2 0

+i(a0 −

3 2w 2 0

z)

.

(14a)

Thus, in equation (14a) any combination of parities is possible. Equation (14a) is the exact solution of equation (1). It is evident that the shape of the spatiotemporal even and odd parabolic-cylindrical solitons is described by the three mode numbers, (n, m, l). They play the role of the standard mode indices, such as those in the Hermite–Gaussian modes of lasers. Recall that w0 is the initial width of the Gaussian beam, and a0 is the initial phase.

(12b)

3. Discussion

where 1 F1 is the confluent hypergeometric function, and n is the mode number of the beam in the x-axis direction. For Gen , n is a non-negative integer (n = 0, 1, 2, . . .) and for Gon , to ensure < 0, n should be a positive integer (n = 1, 2, 3 . . .). − n−1 2 Note that the subscript ν is changed into n in equations (12). By collecting the above results and rearranging the terms, we obtain a solution of equation (6): 1     2i −in− 4 (1−i) 1 UF(e,o) (z, x) = |C − 1|in+ 4 (1−i) C − exp − 2 z  w0 × e− 4λ2 Gn(e,o) (z, x),

(z, τ ) = |C − 1|

  −il− 14 (1−i)   C − exp − 2i z   2 w0 

τ2

There exist two independent even and odd paraboliccylinder functions Geν (z, x) and Goν (z, x) that are solutions to equation (10) [27]:   2 1 i 1 ix2 −i x 2 e 4λ , (11a) + ν, , Gν (z, x) = e 1 F1 4 2 2 2λ2 √   3 i 3 ix2 ix −i x22 o . (11b) Gν (z, x) = e 4λ 1 F1 + ν, , λ 4 2 2 2λ2   If we pick ν = i n + 1 2 , these solutions are rewritten as   2 n 1 ix2 −i x 2 e 4λ , (12a) Gn (z, x) = e 1 F1 − , , 2 2 2λ2   √ x −i x2 n − 1 3 ix2 2 4λ , , 2 , (z, x) = i e 1 F1 − λ 2 2 2λ

il+ 14 (1−i)

×e− 4λ2 Gl(e,o) (z, τ ),

where C is a real integration constant; as a parameter, C is called a ‘distribution factor’, as it controls the physical size 1 1 of the beam. The choice of A0 = |C − 1| 2 ν+ 4 (with C = 1) normalizes the amplitude, A (z = 0) = 1. Assuming F () = i 2 G () e 4  transforms equation (7) into the well-known differential equation for the parabolic-cylinder functions:   ∂ 2G 1 2  − iν G = 0. + (10) ∂2 4

Gon

(13b)

In this section, we display and discuss solutions given by equation (14a) for different possible combinations of mode numbers. It should be pointed out that, according to the choice of the parity, there exist eight types of spatiotemporal solitons in the form of equation (14a) (see the appendix). Thus, different classes of spatiotemporal even and odd parabolic-cylindrical solitons can be constructed using different choices for (n,m, l).Furthermore, we find that when C = 0 is chosen, λ2 = w02 2 becomes a constant, and the beam diffraction and dispersion are exactly balanced by the potential term originating from the nonlinearity. The beam becomes a shape-invariant accessible soliton [27]. Thus, the accessible spatiotemporal parabolic-cylindrical soliton

(13a)

where λ and Gn(e,o) are determined by equations (9a) and (12). 3

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(a)

(b)

(c)

Figure 1. Isosurface plots of the parabolic-cylinder soliton structures with the even combination Gen (z, x) Gem (z, y) Gel (z, τ ). The parameters are: (a) n = m = l = 0; (b) n = m = l = 2; (c) n = m = l = 4.

solution of equation (14a) can be written down in the following form:

 −i(n+m+l)− 34 (1−i) 2i k exp − 2 z Gn(e,o) (z, x) u (z, ρ) =  w 3 2 0 w0  ρ2 −

×Gm(e,o) (z, y) Gl(e,o) (z, τ ) e

w2 0

+i a0 −

3 2w 2 0

z

.

For the single combination of the odd parabolic-cylinder functions Go, Gon (z, x) Gom (z, y) Gol (z, τ ), one obtains another class of the parabolic-cylinder solitons. Figure 2(a) shows the intensity distribution of the soliton mentioned above, with parameters n = m = l = 2; the shape of this soliton extends along each of the axes. For n = m = 4 but different l, these solitons form more complex structures. Two examples, for l = 2 and l = 5, are depicted in figures 2(b) and (c). Next, we investigate intensity distributions of spatiotemporal solitons with mixed-parity combinations of paraboliccylinder functions Ge and Go. As a typical example, we here pick Gon (z, x) Gem (z, y) Gel (z, τ ). We fix n = 1 and change the other two parameters. The solitons are plotted in figure 3. When m = l = 2, the soliton is composed of two disjoint vertical structures, presented in figure 3(a). When m is set equal to n, m = 1, and l is increased to 3, the soliton acquires the form as shown in figure 3(b). Increasing l to l = 4, the soliton forms a completely different structure, see figure 3(c). In all the cases, still, the intensity at the central position remains zero. Another typical mixed example is provided by the choice of Gon (z, x) Gom (z, y) Gel (z, τ ). We here take larger values for n and l, such as n = l = 8, but pick different m. Figure 4 presents a collection of intensity distributions for these solutions. For m = 0 the soliton forms four interconnected spheres; see figure 4(a). For m = 2 in figure 4(b), the soliton elongates into four ellipsoids along the y direction. When m increases to 4, four ellipsoids become eight spheres joined together; see figure 4(c).

(14b)

It is straightforward to see that |u (z, ρ )| vanishes at ρ → ∞, i.e. equation (14b) represents a localized solution. It should be emphasized that although both equations (14a) and (14b) are solutions to the strongly nonlocal NLS equation, their forms are very different. The solution presented by equation (14a) in general oscillates along the z direction, while the solution in (14b) is shape invariant. In [1], the coordinate arguments of Hermite functions are real, while in this paper the arguments of the confluent hypergeometric functions are imaginary; of course, the intensity is still real. Thus, some of their features are also very different. In this paper, we fix λ = 1/2 and the initial beam width w0 = 1. We only consider the analytical soliton solution (14b); in the following, different combinations of even and odd invariant parabolic-cylinder functions are investigated. Their intensities do not depend on z, and the Gn(e,o) factors do not depend on z either. In figure 1, we pick the combination of even paraboliccylinder functions Ge , i.e. Gen (z, x) Gem (z, y) Gel (z, τ ). Obviously, when all of the three parameters are zero, the soliton forms a sphere which we call a fundamental Gaussian soliton; see figure 1(a). Note that the third axis is temporal; therefore, in all figures we see the structures in the x–y plane evolving as pulses in the local time, all the structures being invariant along the propagation axis z. In figures 1(b) and (c) we display two special cases when the three parameters are the same positive even numbers; there exists a similar structure in which eight adjacent spheres are connected to each other. The soliton forms a hollow cubic structure. As the three parameters get larger, additional spherical structures appear inside. The maximum optical intensity is reached at the farthermost spheres, the intensity being zero at the central point.

4. Numerical simulations The spatiotemporal parabolic-cylinder mode (14b) is the exact solution to the linear equation (1). As such, it should be stable, and should not show any tendency to spread or collapse. Still, to confirm the validity of solution (14b) and test its actual robustness, we have performed a numerical study of its propagation; this was accomplished using the split-step 4

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(a)

(b)

(c)

Figure 2. Intensity distributions of the parabolic-cylinder solitons, with combination G0n (z, x) G0m (z, y) G0l (z, τ ). The parameters are: (a) n = m = l = 2; (b) n = m = 4, l = 2; (c) n = m = 4, l = 5.

(a)

(b)

(c)

Figure 3. Typical examples of mixed even and odd parabolic-cylinder functions Ge and Go; the combination is Gon (z, x) Gem (z, y) Gel (z, τ ). The parameters are: (a) n = 1, m = l = 2; (b) n = m = 1, l = 3; (c) n = m = 1, l = 4.

(a)

(b)

(c)

Figure 4. Another typical example of the mixed even and odd parabolic-cylinder functions Ge and Go, with combination Gon (z, x) Gom (z, y) Gel (z, τ ). The parameters are: (a) n = l = 8, m = 0; (b) n = l = 8, m = 2; (c) n = l = 8, m = 4. 5

J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401

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(a)

(b)

Figure 5. Comparison of the analytical solution with the numerical simulations, for the beam Gon (z, x) Gom (z, y) Gol (z, τ ). Parameters: n = 2, m = l = 4 (left) and n = 5, m = l = 4 (right). (a) The intensity distribution as predicted by the analytical expression (14b). (b) The corresponding numerical solution of equation (1), after passing propagation distance z = 100.

beam propagation method [28–30]. To perform numerical simulation, we select the beam as Gon (z, x) Gom (z, y) Gol (z, τ ) and the initial width w0 = 1. The exact solution was deliberately inserted into the numerical scheme with the singleprecision accuracy, which, in a sense, represents a perturbation around the exact solution. This did not produce any effect on the stability of the numerical solution. Figure 5 compares the exact solution of equation (14b) with the simulations of equation (1) for different parameters (n, m, l). As expected, no collapse or spreading is seen, and a very good agreement with the analytical solution is obtained. Similar behaviour is seen for an even initial condition, as well as for different odd– even combinations.

Analytical soliton solutions to the (3 + 1)D strongly nonlocal spatiotemporal NLS equation are obtained. The 3D accessible solitons are constructed with the aid of the well-known even and odd parabolic-cylinder functions in Cartesian coordinates, and their properties are discussed. Comparison with the numerical simulations is carried out. Stability of such solitons is demonstrated. The spatiotemporal parabolic cylindrical solitons may appear in different forms, depending on the values of different parameters, in particular the mode numbers.

5. Conclusions

This work was supported by the National Natural Science Foundation of China under grant no 61275001. The work at the Texas A&M University at Qatar was supported by the NPRP 09-462-1-074 project with the Qatar National Research Fund.

Acknowledgments

We have demonstrated the existence of localized 3D accessible spatiotemporal parabolic-cylinder solitons in the strongly nonlocal medium, both analytically and numerically. 6

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Appendix Different possible solutions (14).

combinations

of

exact

Type

Solution combination

1 2 3 4 5 6 7 8

Gen (z, x) Gem (z, y) Gel (z, τ ) Gon (z, x) Gom (z, y) Gol (z, τ ) Gon (z, x) Gem (z, y) Gel (z, τ ) Gon (z, x) Gom (z, y) Gel (z, τ ) Gen (z, x) Gom (z, y) Gol (z, τ ) Gen (z, x) Gem (z, y) Gol (z, τ ) Gen (z, x) Gom (z, y) Gel (z, τ ) Gon (z, x) Gem (z, y) Gol (z, τ )

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