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56 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Rodas-Verde et al.

Stable propagation of pulsed beams in Kerrfocusing media with modulated dispersion

María I. Rodas-Verde

Área de Óptica, Facultade de Ciencias de Ourense, Universidade de Vigo, As Lagoas s/n,Ourense, ES-32005 Spain

Gaspar D. Montesinos

Departamento de Matemáticas, Escuela Técnica Superior de Ingenieros Industriales,Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

Humberto Michinel

Área de Óptica, Facultade de Ciencias de Ourense, Universidade de Vigo, As Lagoas s/n,Ourense, ES-32005 Spain

Víctor M. Pérez-García

Departamento de Matemáticas, Escuela Técnica Superior de Ingenieros Industriales,Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

Received March 1, 2005; revised June 3, 2005; accepted June 10, 2005

We propose the modulation of dispersion to prevent collapse of planar pulsed beams that propagate in Kerr-type self-focusing optical media. As a result, we find a new type of two-dimensional spatiotemporal soliton sta-bilized by dispersion management. We have studied the existence and properties of these solitary waves bothanalytically and numerically. We show that the adequate choice of the modulation parameters optimizes thestabilization of the pulse. © 2006 Optical Society of America

OCIS codes: 190.3270, 190.5530.

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. INTRODUCTIONhe analysis of the propagation of high-power pulsed la-er beams is among the most active fields of study in non-inear optics, in which the main dynamical phenomenons the dependence of the refractive index of the materialsith the amplitude of light fields. For propagation in ma-

erials showing a linear dependence of the refractive in-ex with the laser intensity (optical Kerr effect), theathematical formulation of the beam dynamics is ad-

quately described by the cubic nonlinear Schrödingerquation1 (NLSE). In this case, the excitation of opticalolitons is one of the most significant phenomena.2–5

Despite the success of the concept of the soliton, thesetructures mostly arise in one-plus-one-dimensional con-gurations. An important exception are quadratic materi-ls, which display a linear dependence of the refractivendex with the amplitude of the field. In this kind of me-ia, multidimensional soltions were predictedheoretically6,7 and verified experimentally.8 However, inerr media this is not possible, mainly owing to the well-nown collapse property of the cubic NLSE in multidi-ensional scenarios. This implies that a two-dimensional

aser beam that propagates in a Kerr-type nonlinear me-ium will be strongly self-focused to a singularity if the

0740-3224/06/010056-6/$15.00 © 2

ower exceeds a threshold critical value, whereas forower powers it will spread as it propagates.

In that collapse prevents the stability of multidimen-ional soliton bullets in systems ruled by the cubic NLSE,great effort has been devoted to search for systems with

table solitary waves in multidimensional configu-ations.9–11 It has been recently shown that a modulationf the nonlinearity along the propagation direction in theptical material can be used to prevent the collapse ofwo-dimensional laser beams.12,13 The concept has beenxtended to the case of several incoherent optical beams14

nd to the case of matter waves.15,16

In this paper we use a similar idea to stabilize againstollapse of pulsed laser beams propagating in planaraveguides. This configuration is of great importance, as

t is the common experimental procedure used for excitingpatial optical solitons. Instead of making a modulation ofhe nonlinearity, our idea is to act on the chromatic dis-ersion term of the NLSE. Thus, our procedure resembleshat used to obtain dispersion-managed temporal solitonsn optical fibers. Preliminary results on this kind of sys-em were described by Bergé et al.17 and more recently byatuszewski et al.18 In this paper we present a explora-

ion in a wide range of parameters showing that an ad-

006 Optical Society of America

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Rodas-Verde et al. Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 57

quate choice provides an optimum stabilization for longropagation distances.

. PHYSICAL MODELe consider the paraxial propagation along z of a pulsed

eam of finite size in time t and spatially confined by aaveguide along the y axis. Thus, diffraction acts only in

he x direction and is balanced by the self-focusing non-inearity given by a refractive index of the form n=n0n2�E�2. The dynamics of the slowly varying amplitude of

he pulse E is described by a one-plus-two-dimensionalLSE of the form

2i� �E

�z+ k0�

�E

�t � + k0��2E

�t2 +1

k0

�2E

�x2 + 2k0

n2

n0�E�2E = 0,

�1�

here k0, k0�, and k0� are, respectively, the wavenumber inacuum, the inverse of the group velocity, and the group-elocity dispersion coefficient.

As we mentioned above, for continuous beams an ad-quate modulation of n2 along z prevents collapse of theight distribution and yields to stabilized two-dimensionalolitons. From a mathematical point of view, one canchieve an equivalent effect by suitably modifying theecond-order derivatives of the propagation equation.19–21

owever, it is physically impossible to change the sign ofiffraction of a continuous beam during its evolution. Forhe case of pulses, only the temporal derivative in Eq. (1)an be modulated by one’s changing the dispersion pa-ameter k0�, which can take positive as well as negativealues. In this paper we will show that it is possible totabilize a pulsed beam against collapse by acting only onhe dispersion during beam propagation, without alteringiffraction.For a linearly polarized Gaussian pulsed beam with

ower P=��E�2dzdx, an initial beam waist w0,x and tem-oral width w0,t, it is useful to write the above Eq. (1) indimensional form by means of the Fresnel length Fk0w0,x

2 in the reduced-time frame. Thus, we make theescaling: �= �t−k0�z� /w0,t, �=z /F, �=x / �F /k0�1/2, and u�w0,xw0,t /k0��1/2E /P1/2. Then, if we consider that disper-ion is modulated by a periodic function along propaga-ion, we will finally yield to a generalized NLSE of the fol-owing form:

i�u

��+

1

2d���

�2u

��2 +1

2

�2u

��2 + g�u�2u = 0. �2�

ere, d���=Fk0� /w0,t2 and g=k0n2PFk0� /n0w0,xw0,t. It is not

bvious what kind of periodic function d��� will be moredequate to stabilize pulsed beams. A simple choice (andatural for applications) is to take d��� as a piecewise-onstant function of the form d���=da for ���a and���=db for �a����b (see top of Fig. 1) where da, db, �a,nd �b are free parameters that can be varied to fit ex-erimental requirements and to optimize the stabilizationf the pulsed beam.

For usual glasses with second-order dispersion k0�10 fs2 mm−1, the adimensional values of our analysis

orrespond to an experiment performed with a mode-

ocked laser with wavelength �0=785 nm, beam waist0,x=0.2 mm, and pulse length w0,t=20 fs. This yields toalues of the Fresnel length F�320 mm, and thus �a0.02 corresponds to 6.4 mm length layers. All theseuantities are common in usual experiments with com-ercial mode-locked lasers. If glasses with high nonlin-

arity are used,22 such as Ga:La:S �n2�210−4 cm2/GW�, and the previous data are taken into ac-

ount, the peak power of the pulse for our simulations isn the range of 5 KW, which can be easily achieved experi-

entally with available femtosecond mode-locked lasers.

. VARIATIONAL ANALYSISo get some insight on the dynamics given by Eq. (2), weave performed an analytic study by means of the time-ependent variational approach.23 Notice that Eq. (2) cane obtained from the Lagrangian density:

2L = i�uu* − u*u� + d���� �u

���2

+ � �u

���2

− g�u�4, �3�

here the overdot denotes the derivative with respect to. We choose a Gaussian ansatz of the form

u = A exp− � �2

2w�2 +

�2

2w�2� + i����2 + ���

2� . �4�

he �-dependent parameters in the above equation havehe following meaning: A is the amplitude; w�, w� are thepatial and temporal widths; and ��, �� are the initial cur-ature and chirp. Although Gaussians are not exact solu-ions of Eq. (2), our choice simplifies the calculations ands a usual light distribution in experiments. The standardariational calculations23 lead to the equations describinghe dynamics of the pulsed beam spatial and temporalidths:

w� =1

w�3 −

1

2

g

w�2w�

, �5a�

w� −d���

d���w� =

d2���

w�3 −

1

2

gd���

w�2w�

. �5b�

umerical simulations of Eqs. (5) show that the beam cane stabilized for many choices of the model parameters.hese equations also predict two types of oscillation foridth w�: a fast one due to the modulation of dispersionnd a slow one that almost coincides with the variation of�. This low-frequency oscillation is generated by the in-

ernal nonlinear dynamics of the system.

. LAYERS OF EQUAL SIZEariational models usually provide only simple and intui-ive predictions that must be complemented with the in-egration of Eq. (2). In the first place, we will present aetailed analysis of the simplest case: two layers of equalize and different dispersions. All the results to be pre-ented in this paper are based on direct simulations of Eq.

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58 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Rodas-Verde et al.

2) by using a split-step Fourier method on a 520520 grid and absorbing boundary conditions to get rid of

adiation.In the first place, we must notice that, for constant

nomalous dispersion and a given value of the beam flux, there is a critical threshold for the nonlinearity gcr forhich a stationary solution of Eq. (2) exists. This is theell-known unstable Townes soliton, u�� ,� ,����� ,��exp�i���, where � is a free scaling parameter.wing to the scaling invariance of the cubic NLSE, a fam-

ly of Townes solitons can be generated by making��r , t�=���� ,�� ,�2��. Thus, the stretching symmetry

f the cubic NLSE allows a continuum of Townes solitonsith different sizes and different peak amplitudes. The

ondition for collapse is fixed by gcr for a given value of P.The size of the layers and the amount of dispersion can

ary in a wide range. Taking into account previous stud-es on soliton stabilization by modulation of theonlinearity,14–16 we have centered our simulationsround values of �a and �b in the range of 0.02, which cor-esponds to a layer of 6.4 mm thickness for a Fresnelength F=320 mm. The amount of dispersion variessymmetrically around zero in the form of da=1+8, db1−8. This choice minimizes radiation losses duringropagation. It is important that the distribution of dis-ersion is asymmetric to keep the pulse below the col-apse threshold. If the average dispersion is positive or

ig. 1. (Color online) (top) Sketch of the propagation of a pulsedeam through a system with periodic modulation of dispersionade by our joining two different planar Kerr waveguides. The

orward–backward trajectory can be obtained by reflection atirrors M1 and M2. (bottom) Oscillations of the peak amplitude

f stabilized pulsed beams propagating in the above system. Thepper curve corresponds to a value g=1.074gc (surface plots inhe top figure). For the lower curve, g=1.034gc. The modulations of the form da=1+8, db=1−8, �a=�b=0.02. In (a) and (b) de-ails of the fast oscillations are shown.

ero, the beam will collapse. We must notice, however,hat the range of the parameters for stabilization is wide.elf-confined propagation can be obtained at least withinn order of magnitude of variation around the previousalues. In addition, we will show below that stabilizations robust against addition of noise.

In Fig. 1 we plot the evolution of the peak amplitude ofwo different pulsed beams in a system sketched in Fig. 1top). In Fig. 1 (bottom) the upper curve corresponds to aalue g=1.074gc, where gc�0 is the critical value of theonlinearity for collapse. For the lower curve, g=1.034gc.he modulation is of the form da=1+8, db=1−8, �a=�b0.02. In Figs. 1(a) and 1(b), details of the fast oscillationsre shown.In Fig. 2 we plot the width oscillations of the stabilized

oliton from Fig. 1 with g=1.074gc. As the variationalquations predict, the spatial width displays only a low-requency oscillation, whereas the temporal width pre-ents small-amplitude fast variations over the same low-requency main oscillation. This is clearly seen in therequency spectra displayed in Figs. 2(c) and 2(d).

To check the validity of the variational approach devel-ped in Section 3, we compare the predictions of Eqs. (5)ith the results of the numerical integration of Eq. (2)

orresponding to the case shown in Fig. 2. That is, theodulation is of the form da=1+8, db=1−8, 2�a=2�b0.04. As can be seen in Fig. 3 there is a minor quantita-

ive deviation in both the period and the amplitude of thescillations. Nevertheless, the variational equations givegood qualitative behavior of the oscillations and, there-

ore, the range of parameters to be used in direct numeri-al simulation of the full NLSE.

In Fig. 4 we plot the results of the simulation of Eq. (2)or the same input pulsed beams of Fig. 1. In this case weave chosen waveguides of half size, resulting in a fre-uency of modulation that is twice that of Fig. 1. Stabili-ation of both beams is achieved, showing that there is aide range of the values of the parameters involved. How-

ig. 2. Oscillations of the spatial (lower) and temporal (upper)idths of the stabilized soliton with g=1.074gc from Fig. 1. (a)nd (b) Insets display details of the curves. (c) and (d) Insetshow the frequency spectra of the spatial and temporal width os-illations, respectively.

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Rodas-Verde et al. Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 59

ver, some changes can be appreciated when a compari-on is made with the low-frequency oscillations of Fig. 1.n the first place we must stress that the amplitude of theeak oscillations is higher for both stabilized solitons; thisan be understood by considering an effective stabilizedoliton of higher width, compared with the layers’ thick-ess. Thus, the effect of the nonlinearity will be a higherendency to collapse. Concerning the frequency of thelow oscillations, we have not observed any regularity.or the soliton with g=1.074gc the frequency is multipliedpproximately by �2. For the soliton with g=1.034gc, theactor is approximately 3. This means that the dynamicsf the slow oscillations is also dependent on the input con-ition.

. ANALYSIS OF THE ROBUSTNESSn this section we will explore the robustness of the sta-ilization method. To this aim, we will analyze two varia-ions on the case of the previous section. In first place weill analyze the effect of changes in the modulation func-

ion, studying both changes in its shape by changing theype of the modulation function. On the other hand, weill investigate the stability of the solitons against theddition of noise.

ig. 3. Comparison between the evolutions of the spatial andemporal widths of the stabilized soliton with g=1.074gc ob-ained from the variational equations (5) and from the NLSEEq. (2)]. The modulation is of the form da=1+8, db=1−8, �a�b=0.02. The plots correspond to (a) � (variational), (b) �

variational), (c) � (numerical), and (d) � (numerical).

ig. 4. Oscillations of the (a) peak amplitudes and (b) widths ofhe same input solitons from Fig. 1. In this case the modulations given by da=1+8, db=1−8, �a=�b=0.01. In (b) the oscillationsf the spatial and temporal widths are overlapped (insets).

. Stabilization with Layers of Different Sizeset us begin with the case of two layers of different sizes.t is intuitive that the total balance of dispersion shoulde kept constant in order to achieve stabilization; thus ifne of the layers is reduced to half its value, the value ofhe dispersion in this region should be doubled. Takinghis into account, we show the results obtained in Fig. 5,here we have taken layers of size 2�a=0.01, 2�b=0.03.herefore, to keep dispersion constant, we need values ofa=1+17, db=1−17/3. As can be seen in the graphs, theffect in the low-frequency oscillations is quite similar toig. 4. Our explanation of this is that the average thick-ess of the layers is the same in both cases; this adds ex-ra support to the previous qualitative explanation inerms of an effective wider soliton for thinner layers.

. Stabilization with Harmonic Modulationnother interesting situation corresponds to a waveguide

hat has been prepared to display a periodic modulationlong propagation different from the step case studiedreviously. Although the detailed shape that could be ob-ained depends on the fabrication process, by one’s chang-ng the dispersion relation along the waveguide, a har-

onic function is a simple choice to study the effect ofifferent modulation functions in the stabilization pro-ess.

In Fig. 6 we show the results of numerical simulationshen the dispersion is taken as d�t�=d0+d1 cos��t� with0=1, d1=8, and �=80, which correspond to the values ofection 4. It can be appreciated that the change from atep to a sinusoidal function does not much affect the sta-ilization. This adds extra support to the robustness ofhe procedure. Compared with Fig. 1, the main differ-nces appear in the frequency and amplitude of the low-requency oscillations. The soliton with g=1.034gc is moreffected by the change in the modulation function.

. Effect of Noiseinally, we will briefly study the stability of the solitonsbtained against the addition of noise. This is shown inig. 7 where we have perturbed the initial beams with aandom noise of 1% in amplitude. We have taken sinu-oidal modulation with d0=1, d1=8, and �=157 (which

ig. 5. Oscillations of the (a) peak amplitudes and (b) widths ofhe same input solitons from Fig. 1. In this case the modulations given by da=1+17, db=1−17/3, �a=0.005, �b=0.015. In (b) thescillations of the spatial and temporal widths are overlapped.

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60 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Rodas-Verde et al.

orresponds, in the step dispersion case, to layers of sizea=0.01, �b=0.01). The main effect of noise is a little dis-urbance in the low-frequency oscillations, which appearess regular than in the previous figures. We have checkedhat the stabilization is robust below a noise threshold ofpproximately 5% of the peak amplitude of the input sig-al.

. CONCLUSIONSn conclusion, we can say that in this paper we have de-cribed two-dimensional spatiotemporal solitons that aretabilized against collapse by means of dispersion man-gement. We have studied their stability and propertiesoth analytically and numerically. We have shown thathe adequate choice of the modulation parameters will op-imize the stabilization of the pulse. We have also checkedhe robustness of the method by strong changes in theodulation function and the addition of noise. Our results

re of importance in the field of high-power pulse propa-ation in nonlinear optical materials.

ig. 6. Oscillations of the (a) peak amplitudes and (b) widths ofhe same input solitons from Fig. 1. In this case the modulations given by a sinusoidal function with d0=1, d1=8, �=80. In (b)he oscillations of the spatial and temporal widths areverlapped.

ig. 7. Oscillations of the (a) peak amplitudes and (b) widths ofhe same input solitons from Fig. 1 when a random noise of 1% inmplitude is added to the initial data. In this case the modula-ion is sinusoidal with d0=1, d1=8, �=157. In (b) the oscillationsf the spatial and temporal widths are overlapped.

A remaining open question concerns the stability ofigher-dimensional beams. In fact, at the present timehe mechanism of stabilization of fully three-dimensionalolitons, which have been predicted to collapse in the casef modulation of nonlinearity, is not clear.15 Thus, our re-ults on dispersion management would be an importanttep forward in stabilization of light bullets in combina-ion with modulation of nonlinearity.

G. D. Montesinos and V. M. Pérez-García are partiallyupported by Ministerio de Educación y Ciencia underrant BFM2003-02832 and Consejería de Educación yiencia of the Junta de Comunidades de Castilla-La Man-ha under grant PAC-02-002. G. D. Montesinos acknowl-dges support from grant AP2001-0535 from MECD. H.ichinel and M. I. Rodas-Verde acknowlege support from

he Spanish MEC project (FIS2004-02466) and Xunta dealicia DXID project (PGIDIT04TIC383001PR).M. I. Rodas-Verde, the corresponding author, can be

eached by e-mail at mrodas@uvigo.es.

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