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Accessible spatiotemporalparabolic-cylinder solitonsWei-Ping Zhong1, Milivoj R Belic2,3, Boris A Malomed4,TingWen Huang2 and Goong Chen5
1 Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province,Shunde 528300, People’s Republic of China2 Texas A&M University, PO Box 23874, Doha, Qatar3 Institute of Physics, University of Belgrade, PO Box 68, Belgrade, Serbia4 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,Tel Aviv University, Tel Aviv IL-69978, Israel5 Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
Received 1 January 2013, in final form 25 February 2013Published 22 March 2013Online at stacks.iop.org/JPhysB/46/075401
AbstractWe study analytically and numerically ‘accessible’ spatiotemporal solitons in athree-dimensional strongly nonlocal nonlinear medium. A general localized soliton solution ofthe ‘acceptable’ type is obtained in the Cartesian coordinates, using even and oddparabolic-cylinder functions. Characteristics of these accessible spatiotemporal solitons arediscussed. The validity of the analytical solutions and their stability is verified by means ofdirect numerical simulations.
(Some figures may appear in colour only in the online journal)
1. Introduction
Hermite–Gaussian [1] and Laguerre–Gaussian solitons [2]are well-known exact solutions of the strongly nonlocalnonlinear Schrodinger (NLS) equation in Cartesian andpolar coordinates, respectively. Their theoretical and practicalimportance was established mainly because they form twocomplete bases of orthogonal modes, and because their profilesand widths remain unchanged with the propagation distance.Another natural ansatz, which may be used for finding exactanalytical solutions in the form of spatiotemporal solitons(‘light bullets’) in strongly nonlocal media, may be providedby the parabolic-cylinder functions, which is the objective ofthis work.
The nonlocal nonlinearity, which occurs in many physicalsystems such as plasmas [3], Bose–Einstein condensates[4] and different optical materials [5–7], has attractedconsiderable interest in recent years. According to the degreeof nonlocality—as determined by the relative width of theresponse 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
the characteristic nonlocality length being much larger thanthe beam’s width. The recent surge in the studies of self-trapped optical beams in strongly nonlocal nonlinear mediais motivated by the experimental observation of nonlocalspatial optical solitons in nematic liquid crystals (NLCs)[9] and in lead glasses [10], as well as by a number oftheoretical predictions [11–13]. Many of the predicted anddemonstrated properties of nonlocal nonlinear models suggestthat, in such optical media, one should expect stabilizationof diverse types of nonlinear wave structures, such as, forexample, necklaces [2, 14, 15] and soliton clusters [16, 17] inthe two-dimensional (2D) transverse space. The nonlocality ofthe nonlinearity prevents the beam from collapse in media withthe cubic nonlinearity in all physical dimensions, resulting inthe prediction of stable multidimensional solitary waves [18,19]. Stable three-dimensional (3D) spatiotemporal solitons incubic nonlocal nonlinear media were reported in [20, 21]. Lightbullets formed via the synergy of reorientational and electronicnonlinearities in liquid crystals were proposed and discussedin [22].
As said above, a highly nonlocal situation arises ina nonlinear medium in which the characteristic size ofthe material response is much wider than the size of the
J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401 W-P Zhong et al
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 notguarantee the existence of stable soliton structures. Inparticular, the orientational nonlinearity in NLCs is highlynonlocal, but the nonlinear response is not perfectly quadratic,implying that, if one launches a Gaussian beam into the cell, anoscillating soliton will be observed. Still, the beam propagationin NLCs neatly aligns with the ‘accessible soliton’ modelof Snyder and Mitchell [24]. Although this model seems tobe locally linear, it still describes the behaviour of stronglynonlocal solitons quite well.
Typical experimental parameters corresponding to a NLCcan be summarized as follows. The cell size is a few mmin each spatial dimension; the diffraction length is of theorder of 100 micrometers; the beam’s widths extends overa few micrometers; the beam’s wavelength is in the green;the ordinary refractive index is about n = 1.5; the biasvoltage across the cell is of the order of 1 V; and opticalpowers are in the mW range. The typical experimental lengthsover which the strongly nonlocal solitons—nematicons—areobserved in NLCs amount to about 2 mm. To date, onlyfundamental nematicons were reported; NLCs are naturalcandidates for media in which higher-order parabolic-cylindersolitons reported below might be observed.
In this work, we demonstrate that a class of 3D accessiblespatiotemporal parabolic-cylinder solitons can be supportedin the model of the strongly nonlocal nonlinear medium.Such solutions, constructed by means of the method of theseparation of variables as products of complex modulationfunctions and Gaussian beams in the Cartesian coordinates,form stable soliton patterns in the propagation. It is wellknown that stationary solitons in 3D media with the localcubic nonlinearity are always unstable against collapse. Thestrong nonlocality helps to stabilize the new solutions, becausein that limit they approach the Snyder–Mitchell model ofaccessible solitons [24]; as mentioned above, that model islocally linear. Hence, it cannot feature instability. There arenot too many areas of nonlinear spatiotemporal dynamicswhere such nonlocal nonlinear limits naturally occur, but thegeneration of nematicons in liquid crystals is one of the fieldsto which our method naturally applies [22, 23].
The paper is organized as follows. In section 2, weintroduce the general 3D strongly nonlocal nonlinear modeland obtain exact accessible soliton solutions. Using theseparation of variables and products of a modulation functionand the Gaussian beam in the Cartesian coordinates, weconstruct a new class of 3D spatiotemporal parabolic-cylindersolitons. In section 3, we present some solutions as relevantexamples for some specific parameters. We find that the newclass of 3D strongly nonlocal spatiotemporal solitons maydisplay various forms. In section 4, the validity of the analyticalsolutions and its stability is verified by means of directnumerical simulations. Finally, we summarize our results insection 5.
2. The model and its parabolic-cylindrical solitonsolutions
To start, we assume that the spatiotemporal beam propagatesalong coordinate z. In the case of the strongly nonlocalmedium, the optical beam with the complex amplitudeu (z, x, y, τ ) is governed by the following 3D paraxial waveequation [21]:
i∂u
∂z+ 1
2∇2u − sρ2u = 0, (1)
where s > 0 is the parameter proportional to the beam’s power,which is a conserved quantity. The proportionality of s to thetotal power is how the implicit global (extremely nonlocal)nonlinearity is introduced in the Snyder–Mitchell model. Inequation (1) ∇2 = ∂2
∂x2 + ∂2
∂y2 + ∂2
∂τ 2 is the spatiotemporal
Laplacian and ρ =√
x2 + y2 + τ 2 is the ‘spatiotemporalradius’, τ standing for the local (retarded) time in the frame ofreference moving with the pulse. Equation (1) is a linear partialdifferential equation with the quadratic potential, formallyequivalent to the Schrodinger equation for the quantum-mechanical harmonic oscillator.
We search for a spatiotemporal soliton solution ofequation (1) by writing the solution as a product of a complexmodulation 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:
i∂uG
∂z+ 1
2∇2uG − sρ2uG = 0, (3)
which is formally equivalent to equation (1). This equation hasan exact self-similar localized solution [25]
uG(z, ρ) = k
3
√w2
0
e− ρ2
2w20+iθ (z)
, (4)
where θ (z) = a0 − 32w2
0z, s = 1
/2w4
0, k is the normalizationconstant, w0 is the initial width of the Gaussian soliton beamand a0 is the initial phase.
Substituting equation (2) into equation (1), we obtain anequation for uF:
i∂uF
∂z+ 1
2∇2uF − 1
w20
(x∂uF
∂x+ y
∂uF
∂y+ τ
∂uF
∂τ
)= 0. (5)
To find complex solutions of equation (5), we use themethod 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 thetransverse dimensions. For example, the equation in the xdirection is
i∂UF
∂z+ 1
2
∂2UF
∂x2− 1
w20
x∂UF
∂x= 0. (6)
We solve these equations by the self-similar method.We assume that UF (z, x) = A (z) F (�), where A (z) isthe amplitude of the beam and � (z, x) = x
μ(z) is the self-similar variable, where μ (z) is a z-dependent scaling factorto be determined. Substituting UF (z, x) into equation (6) and
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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401 W-P Zhong et al
separating the variables again leads to the following equationsfor F, μ and �:
∂2F
∂�2− i�
∂F
∂�− i
(ν + 1
2
)F = 0, (7)
2μ∂μ
∂z− 2iμ2
w20
= 1, (8a)
2μ2
A
∂A
∂z= −
(ν + 1
2
), (8b)
where ν is the separation constant. Letting λ = √iμ, from
equations (8) one obtains
λ2 = w20
2
∣∣∣∣C exp
(2i
w20
z
)− 1
∣∣∣∣ , (9a)
A (z) = A0
∣∣∣∣C − exp
(2i
w20
z
)∣∣∣∣− 1
2 ν− 14
, (9b)
where C is a real integration constant; as a parameter, C iscalled a ‘distribution factor’, as it controls the physical sizeof the beam. The choice of A0 = |C − 1| 1
2 ν+ 14 (with C �= 1)
normalizes the amplitude, A (z = 0) = 1. Assuming F (�) =G (�) e
i4 �2
transforms equation (7) into the well-knowndifferential equation for the parabolic-cylinder functions:
∂2G
∂�2+
(1
4�2 − iν
)G = 0. (10)
There exist two independent even and odd parabolic-cylinder functions Ge
ν (z, x) and Goν (z, x) that are solutions
to equation (10) [27]:
Geν (z, x) = e−i x2
4λ21F1
(1
4+ i
2ν,
1
2,
ix2
2λ2
), (11a)
Goν (z, x) =
√ix
λe−i x2
4λ21F1
(3
4+ i
2ν,
3
2,
ix2
2λ2
). (11b)
If we pick ν = i(n + 1
/2), these solutions are rewritten as
Gen (z, x) = e−i x2
4λ21F1
(−n
2,
1
2,
ix2
2λ2
), (12a)
Gon (z, x) =
√ix
λe−i x2
4λ21F1
(−n − 1
2,
3
2,
ix2
2λ2
), (12b)
where 1F1 is the confluent hypergeometric function, and n is themode number of the beam in the x-axis direction. For Ge
n, n isa non-negative integer (n = 0, 1, 2, . . .) and for Go
n, to ensure− n−1
2 < 0, n should be a positive integer (n = 1, 2, 3 . . .).Note that the subscript ν is changed into n in equations (12).By collecting the above results and rearranging the terms, weobtain a solution of equation (6):
U (e,o)F (z, x) = |C − 1|in+ 1
4 (1−i)
∣∣∣∣C − exp
(− 2i
w20
z
)∣∣∣∣−in− 1
4 (1−i)
× e− x2
4λ2 G(e,o)n (z, x), (13a)
where λ and G(e,o)n are determined by equations (9a) and (12).
Using the same process, we obtain the other twocomponents of the full solution of equation (5), along they- and τ - directions:
U (e,o)F (z, y) = |C − 1|im+ 1
4 (1−i)
∣∣∣∣C − exp
(− 2i
w20
z
)∣∣∣∣−im− 1
4 (1−i)
×e− y2
4λ2 G(e,o)m (z, y), (13b)
U (e,o)F (z, τ ) = |C − 1|il+ 1
4 (1−i)
∣∣∣∣C − exp
(− 2i
w20
z
)∣∣∣∣−il− 1
4 (1−i)
×e− τ2
4λ2 G(e,o)
l (z, τ ), (13c)
where
Gem (z, y) = e−i y2
4λ21F1
(−m
2,
1
2,
iy2
2λ2
),
Gom (z, y) =
√iy
λe−i y2
4λ21F1
(−m − 1
2,
3
2,
iy2
2λ2
),
Gel (z, τ ) = e−i τ2
4λ21F1
(− l
2,
1
2,
iτ 2
2λ2
),
and
Gol (z, τ ) =
√iτ
λe−i τ2
4λ21F1
(− l − 1
2,
3
2,
iτ 2
2λ2
).
Here m (m = 0, 1, 2 . . .) and l(l = 0, 1, 2 . . .) are the modenumbers of the beam along the y- and τ -axes, respectively.Now, the 3D solution of equation (1) in the Cartesiancoordinates can be readily constructed as a product of the1D solutions of the form (13), namely
u (z, ρ) = k
3
√w2
0
U (e,o)F (z, x)U (e,o)
F (z, y)U (e,o)F (z, τ )
×e− ρ2
2w20+i(a0− 3
2w20
z). (14a)
Thus, in equation (14a) any combination of parities ispossible. Equation (14a) is the exact solution of equation (1).It is evident that the shape of the spatiotemporal even andodd parabolic-cylindrical solitons is described by the threemode numbers, (n, m, l). They play the role of the standardmode indices, such as those in the Hermite–Gaussian modesof lasers. Recall that w0 is the initial width of the Gaussianbeam, and a0 is the initial phase.
3. Discussion
In this section, we display and discuss solutions givenby equation (14a) for different possible combinations ofmode numbers. It should be pointed out that, accordingto the choice of the parity, there exist eight types ofspatiotemporal solitons in the form of equation (14a) (seethe appendix). Thus, different classes of spatiotemporal evenand odd parabolic-cylindrical solitons can be constructedusing different choices for (n, m, l). Furthermore, we find thatwhen C = 0 is chosen, λ2
(= w20
/2)
becomes a constant,and the beam diffraction and dispersion are exactly balancedby the potential term originating from the nonlinearity. Thebeam becomes a shape-invariant accessible soliton [27]. Thus,the accessible spatiotemporal parabolic-cylindrical soliton
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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401 W-P Zhong et al
(a) (b) (c)
Figure 1. Isosurface plots of the parabolic-cylinder soliton structures with the even combination Gen (z, x) Ge
m (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 followingform:
u (z, ρ) = k
3
√w2
0
[exp
(− 2i
w20
z
)]−i(n+m+l)− 34 (1−i)
G(e,o)n (z, x)
×G(e,o)m (z, y) G(e,o)
l (z, τ ) e− ρ2
w20+i
(a0− 3
2w20
z). (14b)
It is straightforward to see that |u (z, ρ)| vanishes at ρ →∞, i.e. equation (14b) represents a localized solution. It shouldbe emphasized that although both equations (14a) and (14b)are solutions to the strongly nonlocal NLS equation,their forms are very different. The solution presented byequation (14a) in general oscillates along the z direction, whilethe solution in (14b) is shape invariant. In [1], the coordinatearguments of Hermite functions are real, while in this paperthe arguments of the confluent hypergeometric functions areimaginary; of course, the intensity is still real. Thus, someof their features are also very different. In this paper, we fixλ = 1/2 and the initial beam width w0 = 1. We only considerthe analytical soliton solution (14b); in the following, differentcombinations of even and odd invariant parabolic-cylinderfunctions are investigated. Their intensities do not depend onz, and the Gn
(e,o) factors do not depend on z either.In figure 1, we pick the combination of even parabolic-
cylinder functions Ge, i.e. Gen (z, x) Ge
m (z, y) Gel (z, τ ).
Obviously, when all of the three parameters are zero, the solitonforms 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 aspulses in the local time, all the structures being invariant alongthe propagation axis z. In figures 1(b) and (c) we display twospecial cases when the three parameters are the same positiveeven numbers; there exists a similar structure in which eightadjacent spheres are connected to each other. The soliton formsa hollow cubic structure. As the three parameters get larger,additional spherical structures appear inside. The maximumoptical intensity is reached at the farthermost spheres, theintensity being zero at the central point.
For the single combination of the odd parabolic-cylinderfunctions Go, Go
n (z, x) Gom (z, y) Go
l (z, τ ), one obtains anotherclass of the parabolic-cylinder solitons. Figure 2(a) shows theintensity distribution of the soliton mentioned above, withparameters n = m = l = 2; the shape of this soliton extendsalong each of the axes. For n = m = 4 but different l, thesesolitons form more complex structures. Two examples, forl = 2 and l = 5, are depicted in figures 2(b) and (c).
Next, we investigate intensity distributions of spatiotem-poral solitons with mixed-parity combinations of parabolic-cylinder functions Ge and Go. As a typical example, we herepick Go
n (z, x) Gem (z, y) Ge
l (z, τ ). We fix n = 1 and changethe other two parameters. The solitons are plotted in figure 3.When m = l = 2, the soliton is composed of two disjoint ver-tical structures, presented in figure 3(a). When m is set equalto n, m = 1, and l is increased to 3, the soliton acquires theform as shown in figure 3(b). Increasing l to l = 4, the solitonforms a completely different structure, see figure 3(c). In allthe cases, still, the intensity at the central position remainszero.
Another typical mixed example is provided by the choiceof Go
n (z, x) Gom (z, y) Ge
l (z, τ ). We here take larger values for nand l, such as n = l = 8, but pick different m. Figure 4 presentsa collection of intensity distributions for these solutions. Form = 0 the soliton forms four interconnected spheres; seefigure 4(a). For m = 2 in figure 4(b), the soliton elongatesinto four ellipsoids along the y direction. When m increasesto 4, four ellipsoids become eight spheres joined together; seefigure 4(c).
4. Numerical simulations
The spatiotemporal parabolic-cylinder mode (14b) is the exactsolution 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 itsactual robustness, we have performed a numerical study ofits propagation; this was accomplished using the split-step
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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401 W-P Zhong et al
(a) (b) (c)
Figure 2. Intensity distributions of the parabolic-cylinder solitons, with combination G0n (z, x) G0
m (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) Ge
m (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 combinationGo
n (z, x) Gom (z, y) Ge
l (z, τ ). The parameters are: (a) n = l = 8, m = 0; (b) n = l = 8, m = 2; (c) n = l = 8, m = 4.
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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401 W-P Zhong et al
(a)
(b)
Figure 5. Comparison of the analytical solution with the numerical simulations, for the beam Gon (z, x) Go
m (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) Thecorresponding numerical solution of equation (1), after passing propagation distance z = 100.
beam propagation method [28–30]. To perform numericalsimulation, we select the beam as Go
n (z, x) Gom (z, y) Go
l (z, τ )
and the initial width w0 = 1. The exact solution wasdeliberately inserted into the numerical scheme with the single-precision accuracy, which, in a sense, represents a perturbationaround the exact solution. This did not produce any effecton the stability of the numerical solution. Figure 5 comparesthe exact solution of equation (14b) with the simulations ofequation (1) for different parameters (n, m, l). As expected,no collapse or spreading is seen, and a very good agreementwith the analytical solution is obtained. Similar behaviour isseen for an even initial condition, as well as for different odd–even combinations.
5. Conclusions
We have demonstrated the existence of localized 3Daccessible spatiotemporal parabolic-cylinder solitons in thestrongly nonlocal medium, both analytically and numerically.
Analytical soliton solutions to the (3 + 1)D strongly nonlocalspatiotemporal NLS equation are obtained. The 3D accessiblesolitons are constructed with the aid of the well-known evenand odd parabolic-cylinder functions in Cartesian coordinates,and their properties are discussed. Comparison with thenumerical simulations is carried out. Stability of such solitonsis demonstrated. The spatiotemporal parabolic cylindricalsolitons may appear in different forms, depending on the valuesof different parameters, in particular the mode numbers.
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China under grant no 61275001. The work atthe Texas A&M University at Qatar was supported by theNPRP 09-462-1-074 project with the Qatar National ResearchFund.
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J. Phys. B: At. Mol. Opt. Phys. 46 (2013) 075401 W-P Zhong et al
Appendix
Different possible combinations of exact analyticalsolutions (14).
Type Solution combination
1 Gen (z, x) Ge
m (z, y) Gel (z, τ )
2 Gon (z, x) Go
m (z, y) Gol (z, τ )
3 Gon (z, x) Ge
m (z, y) Gel (z, τ )
4 Gon (z, x) Go
m (z, y) Gel (z, τ )
5 Gen (z, x) Go
m (z, y) Gol (z, τ )
6 Gen (z, x) Ge
m (z, y) Gol (z, τ )
7 Gen (z, x) Go
m (z, y) Gel (z, τ )
8 Gon (z, x) Ge
m (z, y) Gol (z, τ )
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