Spatial solitons pairing by cross phase modulation

Optics Communications 88 (1992) 419-423 North-Holland

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Spatial solitons pairing by cross phase modulation Ra/fl D e L a F u e n t e Laboratorio de Optica, Facultad de Fisicas, Universitad de Santiago de Compostella, Spain

and Alain B a r t h e l e m y Institut de Recherches en Communications Microondes et Optiques, u.a. au CNRS n ° 356, Equipe Optique, Facultb des Sciences, 123, rue A. Thomas, 87060 Limoges cedex, France

Received 9 September 1991

We show that two beams of different color can simultaneouslypropagate undistorted as solitary waves in a homogeneousKerr medium of both focusing or defocusing type thanks to cross phase modulation. These beams are obtained by pairing bright or/ and dark spatial solitons of identical width but different intensities.

1. Introduction

Spatial solitons have been been extensively investigated recently, in large part because of the potential use of these phenomena for optical switching and processing applications [ 1 ]. Spatial optical solitons are self-trapped optical beams which propagate in nonlinear Kerr material without changing their transverse profile because of the competing effects of diffraction and self-focusing in a two-dimensional frame. The evolution of spatial soliton is governed by the nonlinear Schr6dinger equation as is the case for soliton pulses, their temporal counterpart that propagate in single mode optical fibers. There exist bright and dark spatial solitons (a lack of light on a constant background) depending on the sign of the intensity dependent contribution to the refractive index [ 2 ]. The first have already been observed in liquid CS2 [ 3 ] and nonlinear glasses [ 4 ] while the second have been observed more recently in Na vapor and in ZnSe crystal [ 5 ]. The coherent superposition of two fundamental bright soliton beams of identical size and wavelength leads to the propagation of a soliton of second order [ 6 ]. When two fundamental spatial solitons of the same wavelength do not overlap but are launched close to each other their inter-

action gives rise to their attraction or repulsion owing to their phase difference [ 1 ]. The purpose of this letter is to show that two spatial solitons ofidentical size but different wavelength can be superimposed and continue their self-guided propagation if their intensity is changed to satisfy the conditions presented in the following of this letter. This possibility results from the combined effects of diffraction, self phase modulation (SPM) and crossphase modulation (XPM). The nonlinear contribution to the refractive index, essential for the observation of solitons, also gives rise to cross phase modulation between two overlapping waves of different frequencies during their common propagation. In the temporal domain XPM introduces important new features to nonlinear pulse propagation in fibers. For example it has been shown that an optical pulse can propagate undistorted as a bright pulse in the normal dispersion regime when it couples through XPM to a dark pulse of different wavelength in the anomalous dispersion regime [ 6 ]. In the spatial domain, XPM has also great consequences on the nonlinear propagation of simultaneously launched intense beams. For example modulational instability of the transverse profiles may result from the interaction of two waves in a Kerr

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defocusing medium, while in the single beam case the occurence of this instability is only expected with focusing nonlinearity [ 7 ]. Trillo et al. [6] and Afanasjev et al. [8] have demonstrated that, in single mode fibers, XPM allows the propagation of a new kind of solitary pulses that corresponds to particular solutions of the coupled nonlinear wave equations. They consist in bound states obtained by the pairing of "bright" a n d / o r "dark" fundamental solitons of the SchiSdinger nonlinear equation. We have expanded their work to the particular case of spatial soliton phenomena. Although the starting equations are very similar, conditions for the existence of stationary pairs of soliton are established and discussed, that are specific to the spatial domain. Our approach is then compared to earlier works on nonlinear pairing of soliton pulses.

2. Propagation equations In order to consider the transverse distortions supported by two beams when they propagate simultaneously at high intensity in an optical Kerr medium we write the electric fields as the general superposition of two monochromatic waves of frequency to1 and co2:

+E2(r, to2) exp( -it02 t) +c.c.

( 1)

We assume further that both waves are linearly polarized and that their respective orientations are either parallel or perpendicular. Starting with the Maxwell equations for the electric field and introducing in the nonlinear polarization vector the contributions of SPM and XPM we obtain t h e well known propagation equation V× V X E i - (ki)2E~-2(k2n2/no,) X (IE~ 12+~clE3_~ 12)G = 0 ,

2iki (OEi/0z) -t- (02/~//Ox 2 ) "]-2 ( k 2n2/noi ) X(IEiI2+IclE3_iI2)Ei.=O,

i=1,2,

i=1,2.

(3)

In the nonlinear contributions to this equation the first term in parenthesis accounts for SPM while the second term is responsible for XPM. The coupling coefficient x is a constant that depends on the experimental conditions. For two waves of parallel polarization x = 2 . For cross polarization x = ] if the nonlinearity is of electronic origin as is the case for glass, and x = - 1 if the nonlinearity is due to molecular reorientation as is the case for certain liquids (CS2 for example). Diffraction enters via the second derivative of the fields. As long as we are concerned by linearly polarized waves eq. (3) can be written with scalar fields. Furthermore this equation can be simplified making use of the following normalization

Ui=aokl(ln21/nol)l/2Ei,

i=1,2,

~=(1/kla~)z,

(4)

where ao is a parameter related to the beam width. So Ui, U2 satisfy "OU1 02U1 +sign(n2) (IUII2+tclU212)UI=O 1-'-~- "l"/ 00"-"--~

. 0u2

02u~

1-~-- + ½~

+sign(n2)

x a ( x l U112+ I U212)U2 = 0 ,

(5)

with (2)

where k,= toinodc is the wave vector, noi is the linear refractive index at frequency 09, n2 the nonlinear coefficient of the refractive index and x a constant that depends on the experimental conditions. We assume also that the frequency difference does permit neither the growing of stimulated scattering nor energy transfer through four-wave mixing. 420

Further simplifications can be made assuming first that free propagation takes place in a two-dimensional frame, the (x, z) plane for instance, the waves being infinite or guided along the perpendicular dimension (y), and secondly that the study is restricted to paraxial diffraction. These assumptions lead to the following form of the wave equations

a=X/ao,

E(r, t) =El (r, 091) exp( --ito, t)

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fl=k2/kl =~-i no2/,~.2nol , ot=fl;tl/22 = (,~ l /2 2) 2no2/ noi . Before analyzing the system of equations (5) we consider the case where x can be set equal to zero. This case corresponds to the situation where the two waves are well separated in space and do not overlap. In such a case, depending on the sign of n2, two

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kinds o f stable solutions exist that are the well known bright and dark solitons (2)

Ui=Aitanh(a) exp(i#i(),

i = 1, 2 ,

A2 =sign(n2) ( x - o t ) / o t ( 1 - 1 c 2 ) ,

U1 = sech ( a ) exp ( i ( / 2 ) , UI = t a n h ( a ) e x p ( - i O

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,

(6)

for n2 > and n2 < 0 respectively. Similar solutions exist for U2.

3. Spatial soliton pairs

It is straighforward to look for solutions o f the coupled equations corresponding to 1¢# 0, by pairing solutions of the above types. This leads to three kinds o f soliton couples, each o f which depends on the sign o f the nonlinearity but also on the values o f the parameters x and or. Direct substitution in eq. (5) o f two bright solitons gives the stationary solutions Ui=Aisech(tr) exp(ifli~) ,

(8)

A22= sign (n2) ( x a - 1 ) / a ( 1 - x z) with

#,=-1,

#2=-1/#.

Conditions on a for the existence o f solution (8) are the same as that listed in table 1 for two bright solitons, except that the sign o f n2 must be negative. The last kind o f solution corresponds to the superposition o f a bright soliton beam with a dark one. By looking for solutions o f the form Ul = A l sech(a) e x p ( i f l () , U2 = A z t a n h ( a ) exp(i#2~) ,

(9)

we obtain the following intensities and propagation constants for the trapped waves

i = 1, 2

with A~=sign(n2) (ot-x)/a(1-x2)

A2 =sign(n2) ( a - x ) / o t ( l - x 2 ) , A~ =sign(n2) ( l - x o t ) / o t ( 1 - x 2 ) ,

(7)

,

A~ = sign (n2) (xc~- 1 ) / a ( 1 - x 2) ,

#t=½, [31=1/2#.

with

This result depicts the fact that two bright spatial solitons o f different color but identical size can be superimposed and continue to propagate as solitary waves if there respective intensity is changed to satisfy the conditions o f eq. (7). For the propagation o f those soliton pairs a self-focusing Kerr m e d i u m is needed and the wavelengths ratio ot must range between the values given in table 1. There is no solution for r = - 1. Also, for tangent hyperbolic shaped beams we obtain

fll = ( o t x 2 + a - - 2 t c ) / 2 o t ( 1 - - x 2) ,

Table 1 Conditions to be fullfilled by the wavelength ratio a for the propagation of a bright spatial soliton pair in a focusing nonlinear medium and for a dark soliton pair in a defocusing nonlinear medium as well. Experimental situation

a

Polarizations tl", r = 2

1/2<

<2

Polarizations ~--,, x= 2/3

2/3 <

< 3/2

(10)

#2 = (rot-- 1 ) / # ( 1 - - x 2 ) .

In table 2 we express the conditions required for these solitary waves to be self-trapped. For x > 0 the coupled wave equations exhibit soliton solutions o f the above type for both negative and positive refractive index nonlinearity, depending on the value o f a. We see also that the allowed values o f a are just Table 2 Conditions to be fullfilled by the wavelength ratio a for the propagation of a bright spatial soliton coupled to a dark soliton, owing to the sign of the intensity dependent contribution to the refractive index+ Experimental situation

n2

a

Polarizations 1"?,x= 2

>0 <0

< 1/2 >2

Polarizations t-,, x= 2/3

>0 <0

>3/2 <2/3 421

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the missing values in table 1. There is no solution for x<0.

4. Discussion In this paragraph we first wish to emphasize the differences between the temporal and spatial situation. In fibers, for pulses to have significant interactions, the two waves considered must travel at the same group velocity. This condition is only met in practice in the two following situations: (i) in single mode fibers for two specific wavelengths situated on either side of the group velocity dispersion minimum (thus one wave lies in the positive dispersion domain, the other one in the negative dispersion domain), and (ii) in multimode fibers for two waves travelling on a different mode and for a suitable choice of the wavelength difference (thus the mode dispersions may be different, and the XPM is altered by the overlap integral between the mode transverse patterns). However, in the spatial domain it is very easy to send two superimposed parallel beams that can interact on long propagation distances. Except for the medium transparency there is no requirement for the choice of wavelength at this step. Furthermore, the diffraction effect, spatial analog of the group velocity dispersion, is directly proportional to the wavelength, which reduces the parameters involved in the computation. Let's emphasize now the differences between our approach of the spatial soliton case and the earlier works of Trillo et al. and Afanasjev et al. In the Trillo paper the pairing of bright and dark pubes is only considered. The transposition of their solutions to the spatial domain requires a medium whose nonlinearity is of opposite sign for the two wavelengths. This is only encountered in resonant media exhibiting at the same time high absorption. Afanasjev et al., in their work, look only to the existence of the nonlinear pairing of soliton pulses of identical am. plitude, in the special situation where the dispersion is of identical modulus for the two wavelengths. There is no spatial analog of this later situation since the diffraction strength is always different for two distinct wavelengths. They then derived the conditions to be satisfied by the nonlinear coupling coefficients for the observation of those soliton pairs (this im422

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plies the use of multimode fibers). In our spatial case, for a given material, knowing its nonlinear coefficient and the soliton beams wavelengths we can find the good choice of soliton profiles and intensity to obtain their simultaneous trapping with the help of tables 1 and 2. The energies of the two solitary waves are separately conserved during their propagation along z. These energies differ from the energies of a fundamental soliton of the SNE, and their summation as well, in contrast to the temporal situation.

$. Intuitive interpretation of soliton pairing For each beam to be trapped, the linear phase shift introduced by diffraction must be cancelled by the one resulting from the induced refractive index change. Equalizing the amplitude of these phase shifts of opposite sign (for the waves considered) leads to the fact that the induced refractive index must scale like (22/no) for each beam. Therefore the nonlinear phase of beams i and j must only differ by the ratio (,~,j/,~i)2(noi/noj) (or parameter ot). Let us first deal with two beams of identical shapes. For each wave, the intensity of the other wave contributes x times more than their own intensity to their "own refractive index gradient". Thus the two beams help each other to cancel the diffraction effect. Solving the equation system for the two nonlinear refractive indices gives directly the intensities required for the pairing. These intensities scale respectively as ( 1 - x~ ot) for beam j and ( 1 - rot ) for beam i. This shows effectively that ot must range between x and 1Ix. Out of this range several situations may occur; for example the two beams can self-focuse or one can spread and the other one can self-focus, and so on. Note that for the first two families of soliton pairs described above, the intensity of each beam is always less than needed for a single fundamental spatial soliton. If at least the intensity of one beam approaches this reference (for a value of ot close to x or 1Ix) the corresponding intensity for the other wave fails to zero. So it can be expected to trap a beam of soliton shape but very weak intensity by a superimposed spatial soliton each time the wavelength ratio between the two beams (21/22) (no2/nol)l/2 reaches x 1/2 or x-1/2. When the two beams are of opposite

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shape (sech2(x) and tanh2(x) ) X P M diminishes the amplitude o f the induced refractive index, in contrast to the previous case. This allows a hyperbolic secant beam carrying an intensity higher than the fundamental soliton intensity, not to focus and remain trapped. This also offers the opportunity for a hyperbolic tangent profile to be guided in a focusing nonlinear m e d i u m and for a bright spatial soliton to be guided in a defocusing Kerr medium. Finally, note that for given values o f n2, x and ix, there exists only one solution for the two waves transverse profiles and intensities corresponding to a stable self-guided propagation. U p to now we have only considered exact solutions o f eqs. (5). If the intensity differs from that given in eqs. (7), ( 8 ) and ( 10 ) the beams will spread or focus during their propagation eventually in an oscillatory fashion. Nevertheless if one o f the input beams is launched with much more intensity than the other one, the weak field does not affect significantly the strong field. I f this latter one corresponds to a spatial soliton (single beam case) the weak signal, even if its shape differs from that o f a soliton, and even if its width is different, can be guided by the stable refractive index gradient induced through XPM, as it has been demonstrated in recent experiments (ref. [ 9 ] ).

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Acknowledgements This work was supported by the Centre National d'Etude des Telecommunications (grant n ° 90 8B 098 LAB) and R. De La Fuente thanks Caixa Galicia for the payment o f a scholarship during his stay at IRCOM.

References [ 1] F. Reynaud and A. Barthelemy, Europhys. Lett. 12 (1990) 401. [ 2 ] V.E. Zakharov and A.B. Shabat, Soy. Phys. JETP 34 (1972) 62. [ 3 ] A. Barthelemy, S. Maneuf and C. Froehly, Optics Comm. 55 (1985) 201. [4] J.S. Aitchinson,A.M. Weiner, Y. Silberberg, M.K. Oliver, J.L. Jaekel, D.E. Leaird, E.M. Vogel and P.W.E. Smith, Optics Lett. 15 (1990) 471. [ 5 ] S.R. Skinner, G.R. Allan, D.R. Andersen and A.L. Smirl, OSA Annual Meeting, Boston, Technical Digest (1990) p. 167. [ 6 ] S. Trillo, S. Wabnitz, E.M. Wright and G.I. Stegeman, Optics Lett. 13 (1988) 871. [7] G.P. Agrawal, J. Opt. Soc. Am. B 7 (1990) 1072. [8] V.V. Afanasjev, E.M. Dianov and V.N. Serkin, IEEE J. Quantum Electron. 25 (1989) 2656. [9] R. De La Fuente and A. Barthelemy, Optics Lett., to be published.

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