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Bimodal counterpropagating spatial solitary-waves
Optics Communications North-Holland
103 (1993)
OPTICS COMMUNICATIONS
145-152
Full length article
Bimodal M. Haeltennan,
counterpropagating A.P. Sheppard
spatial solitary-waves
and A.W. Snyder
Optical Sciences Centre. Australian National University, Canberra. Australia Received
14 April 1993; revised manuscript
received
A simple analysis of the standard coupled nonlinear of a new class of spatial solitary waves in Kerr media. constitute the fundamental and second modes of the two distinct, parallel self-guided beams. The existence simple physical arguments.
1.
16 June I993
Schrijdinger equations for counterpropagating beams reveals the existence These solitary waves are bound states of two counterpropagating fields that waveguiding structure they induce. For certain parameters they represent and the features of these particular solutions are explained on the basis of
Introduction
Owing to both their theoretical interest and their fundamental role in many practical nonlinear systems, the processes of self-interaction of counterpropagating light beams in nonlinear materials have attracted growing interest in recent years. Many aspects of the problem have been investigated, revealing a very rich spectrum of complex behaviors. Instabilities, self-oscillations and temporal chaos have been reported in simple plane-wave models of counterpropagating beams in the cases of materials with finite response time [ 11, chromatic dispersion [ 21 and with a tensorial nonlinearity when accounting for the polarization of the field [ 3 1. It was shown that selfinteraction between counterpropagating polarized waves leads to new types of nonlinear polarization eigenstates [4] and may induce periodic, multistable, and chaotic spatial distributions [ 5 1. More recently, much attention has been paid to diffractive transverse effects in Kerr-type nonlinear media. Linear stability analyses of the coupled nonlinear Schrodinger (NLS) equations that govern counterpropagating beams in Kerr media show that plane waves are unstable against transverse modulation independently of the sign of the nonlinearity [ 6,7 1. Numerical studies in two-dimensional geometry revealed that this instability may lead to periodic pattern formation, hysteresis, symmetry-breaking, oscillations and chaos [ 6 1. Recent three-dimensional simula0030-4018/93/$06.00
0 1993 Elsevier Science Publishers
tions of diffractive transverse effect predicted the formation of transverse patterns of hexagonal symmetry [ 8 1, a result which is in agreement with the theoretical and experimental studies of mirrorless degenerate four-wave-mixing oscillations in atomic vapor [9]. In this paper we show that diffractive transverse effects due to self-interaction of two counterpropagating beams in a Kerr medium lead to a hitherto unrecognized class of spatial solitary wave. These solitary waves are peculiar in that they constitute a bound state of the fundamental and the second modes of the waveguiding structure they induce together. For certain values of the wave parameters they represent a bound state of two identical solitons propagating in parallel. The nature of these solitary waves can be foreseen by means of simple physical arguments.
2. Theoretical
model
We consider counterpropagation of two beams in one transverse dimension as shown schematically in fig. 1. In practice this situation can be obtained by considering monomode propagation in a planar slab waveguide [lo] or, alternatively, by using a twobeam interference technique [ 111. In this preliminary study we consider only the stationary states of the field. The questions of build-up transient dynam-
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Fig. 1. Schematic of the counterpropagating beam configuration considered in the text. F and B are the forward and backward beam envelopes. x and z are the transverse and longitudinal coordinates, respectively.
its and of stability of the solitary waves require elaborate numerical investigations beyond the scope of this communication. In the scalar approximation of the field, the envelope amplitudes of the two stationary counterpropagating beams, say F(x, z) and B( x, z), obey the following set of coupled NLS equations [6] -i~=~~+[IFl’+(l+h)IBI’]F,
(la)
i~=~~+[lei2+(l+h)lF12]u,
(lb)
where x and z represent, in dimensionless units, the transverse and longitudinal coordinates, respectively. The parameter h describes the effect of the index grating due to the standing-wave interferences. Its value, which lies between zero and unity, parametrizes the diffusion process inherent to the type of Kerr nonlinearity considered, e.g., h is zero for thermally induced nonlinearity and tends to unity for local nonlinearity. Note that we consider only the selffocusing Kerr nonlinearity. In the presence of a single beam, i.e., when either For B is zero, eq. ( 1) reduces to the NLS equation whose soliton solutions have been extensively studied in the literature. Much attention has been paid to the problem of interaction between copropagating solitons and in particular, two identical solitons have been shown to interact through a force that depends sinusoidally on their relative phase [ 121. As a consequence, a pair of 180” out-of-phase solitons and a pair of in-phase solitons exhibit interaction forces of equal amplitudes and opposite signs. This result suggests that the superposition of in-phase and out-ofphase soliton pairs would lead to a perfect balance 146
1 November 1993
between attraction and repulsion forces resulting in the formation of two distinct solitons propagating in parallel. We shall see in the following that the superposition of two counterpropagating soliton beams pairs allows for such an ideal balance between opposite forces. Moreover, though this reasoning holds for well separated soliton beams for which the interaction force depends sinusoidally on their relative phase [ 121, we may anticipate that for arbitrarily closer beams a field configuration still exists that preserves the balance between attraction and repulsion.
3. Solitary-wave solutions The counterpropagating field configurations F(x, z) and B(x, z) that correspond to solitary waves, i.e., waves that are uniform along the z-axis, have the form F(x, z) =f(x)
exp(@)
,
B(x, z) =b(x)
exp( -ij3z)
(2a) ,
(2b)
where the beam envelopes f and b, and the corresponding propagation constants ~1 and /I are real quantities. Substituting these expressions into eq. ( 1) leads to a set of coupled second order nonlinear differential equations y”+f3+(1+h)b2f-+O,
(3a)
+b”+b3+ (1 +h)f2b-/lb=O ,
(3b)
where primes denote x-derivatives. Sincefand b are independent of z, these equations can be considered as linear wave equations for the propagation modes of two distinct waveguides. The fields f and b would correspond to the mode of the waveguide whose index profiles are proportional to f2+ ( 1-t h)b2 and respectively. The fact that the forb2+(l+h)f2, ward and backward waves see different effective index profiles comes from the well known grating induced nonreciprocity inherent to counterpropagation of beams in nonlinear materials (see e.g. ref. [ 13 ] 1. Note that when h=O both profiles are identical. Equations (3) are the equations of motion for a unit mass in a conservative potential of the two-dimensional space V; b). The corresponding hamiltonian is
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1.5
(4) For a diffusive nonlinearity, that is, when h = 0, this hamiltonian exhibits an additional conservation law and is considered integrable [ 141. In that particular case it is identical to that derived in ref. [ 15 ] for the study of temporal vector solitons in birefringent nonlinear dispersive media. The analytical form of the solitary-wave solutions proposed in ref. [ 15 ] will be discussed below. For h#O the hamiltonian H is not integrable and the solutions f and b of eq. (3) must be calculated numerically. Since we are interested in solitary-wave solutions we must look for the separatrix trajectories of H, i.e., the solutions that obeyf; b, f' and b’+O as x+ *co. Naturally the hamiltonian H exhibits symmetries that reveal the existence of simple separatrix trajectories that correspond to analytical expressions of the solitary-waves. For example, if b (or f) is zero the separatrix follows a straight trajectory along the f-axis (b-axis), which, according to eq. (2) corresponds to the solution f(x)= 2psech(flpx), b(x)=0 [or b(x) =$$sech( 2px), f(x) =O]. These soluK tions correspond naturally to the well known spatial solitons obtained by unidirectional beam propagation in Kerr materials. When p= p other separatrices are found that follow the bissectrice lines in the plane CfTb), i.e., f= k b. In this case eqs. (2a) and (2b) become identical NLS equations and yield
f(x)=
fb(x)=dmsech(,&x)
.
(5)
These solitary-wave solutions represent the superposition of two identical counterpropagating NLS solitons whose intensity is adjusted so as to compensate for the presence of the other. In the following we use the notation CP solitons to denote these particular counterpropagating spatial soliton states given by eq. ( 5 ). Note that, due to cross-phase modulation, the intensity [ 299/ (2 + h) ] of the counterpropagating solitons is in general (h > 0) less than half the intensity (247) of a single unidirectional soliton of the same width. The separatrices that correspond to these analytical solutions are depicted in fig. 2. We shall see in the following that they represent asymptotic states of a family of peculiar bound solitary-wave solutions.
b
cb) i 1.01
(4 0.5
-
(4
/ OL
0.5
1.0
1.5
f -0.5
-
\
(4
Fig. 2. Separatrix trajectories in the plane Cr;b) corresponding to the unidirectional (a) forward and (b) backward NLS spatial solitons. Curves (c ) and (d ) correspond to the counterpropagating (CP) soliton solutions given in eq. (5) with q= 1. Before proceeding to the analysis of more general solitary-wave solutions of eq. (2 ) it is convenient to discuss the physical meaning of the CP soliton SOlutions eq. (5). In the case of a single self-guided beam, the NLS soliton solution represents a field configuration that constitutes the fundamental mode of the monomode waveguide it induces itself through the nonlinearity [ 161. In the case of counterpropagating soliton beams this self-consistency relation can be easily generalized when the nonlinearity is diffusive (h=O). Indeed, in that situation the grating induced by the standing-wave interferences is completely washed-out and the induced waveguide is uniform. It is the same waveguide than that induced by a unidirectional NLS soliton beam and naturally, the field distributions f and b of eq. (5) in this case constitute the fundamental mode of the waveguide they induce together. According to eq. (5) each counterpropagating soliton contributes for a half of the waveguide index profile. Note however that, for h=O, eq. (5) only yields a particular case of the CP soliton solutions as the ratio between the soliton intensities is not necessarily 50% but can have any value. On the other hand, when h # 0 it is easy to verify that eq. (5) yields the only solution (in which the solitons have the same intensity). In that case the index of refraction is modulated by the standing-wave
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interferences. In the limit of local Kerr nonlinearity (h= 1) the modulation depth reaches 100% and it is conceptually difficult to consider the resulting refractive index structure as a waveguide, especially because the induced grating is perfectly resonant with the counterpropagating waves. Nevertheless, as is well known [ 13,171, due to the particular phase conditions between the standing wave and the induced grating, the index modulation acts only on the phase of the waves and does not induce any amplitude coupling. The field of each counterpropagating wave of eq. (5) therefore behaves as if it was in a perfectly uniform waveguide whose effective refractive index is increased by a factor of 1 + h/2 (according to the grating induced phase changes). This result allows us to consider the refractive index structures induced by counterpropagating solitary waves as uniform waveguides. In particular, we may state the selfconsistency relation by saying that the CP soliton solution eq. (5) represents the two counterpropagating fundamental modes of the monomode waveguide they induce together. We shall see in the following that this self-consistency principle can be generalized to bimodal waveguides. Let us now consider general solitary-wave solutions to eq. (2). Let us notice that, by means of a simple change of variables (x+x/p”‘, f+f~“~, b+brpL12, and /3-+/3/q), eq. (3 ) can be normalized in such a way that the phase parameter ~1is equal to unity. We can therefore analyze the solutions with respect to the single wave-parameter /I. Let us first consider the case where the forward beam is a soliton that induces a waveguide for a very low amplitude backward beam. In that situation b
.
(6)
This second order linear differential equation has solutions which decay at infinity. They correspond to the modes of the waveguide as seen by the backward beam. The fundamental mode is [ 18 ] b(x)=sech”(@x)
(7)
where s is dependent on h through the relation s(s+ 1)=2(h+ 1). Its propagation constant is&=s2. The higher order modes are given in terms of the Legendre functions of the first kind [ 181 148
b(x)=P:-“[tanh(
1 November 1993
(8)
with the restriction n ds (n being the order of the modes). Their propagation constants are /Zn= (n -5) 2. The maximum value of s corresponds to h = 1 (local nonlinearity) and is s = 1.56. This shows that the effective waveguide supports a maximum of two modes. In the limiting case of a highly diffusive nonlinearity, h = 0, s is equal to unity and the second mode is at cut-off (p,=O), as is the case for the waveguide seen by the forward soliton beam itself [ 161. In the following we only consider the case of a local Kerr nonlinearity h = 1. This restriction does not affect the generality of our conclusions. The propagation constants of the two modes are therefore PO= 2.43 and pi = 0.3 14. This linear perturbation analysis of eqs. (3) can be summarized saying that, in the linear regime of the backward wave, a soliton beam can guide two backward modes, a fundamental mode (zero-node beam envelope) and a first order mode (one-node antisymmetric envelope). By extending this result up to the nonlinear regime of the backward wave b, we can foresee the existence of two types of bound spatial solitary-waves. One type would consist of a bound state of the two counterpropagating fundamental modes of the nonreciprocal waveguide they induce themselves, while the other would involve the fundamental mode and the first order mode of the “forward” and “backward” induced waveguides, respectively. We have calculated numerically (by means of a standard shooting technique) the separatrix trajectories of H in the general situation where p# 1. From the linear perturbation analysis we know that when /3 is close to PO or pi the composite solitary wave approaches the forward soliton solution, i.e. the corresponding separatrix trajectory lies on the f-axis (b= 0). We first study the separatrix trajectories of H in the vicinity of PO. We verified that the separatrices only exist for values of /? smaller than PO. Examples of trajectories are shown in fig. 3 for several values of p. We observe that, as /3 decreases from 2.43, the trajectory progressively bends and leaves the f-axis. When /3 tends to unity the separatrix approaches the bissecting line f= b which corresponds to the CP soliton solution given in eq. (5) (with @= 1). In fig. 4 we plotted an example of field envelopes f and b of this type of solitary waves together with the cor-
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f
Fig. 3. Separatrix trajectories obtained in the case of a local Kerr nonlinearity (h= l), with (o= 1, and (a) P=2.3, (b) 8=2.0, (c) /I= 1.5, (d) /I= 1.1. Fig. 5. Separatrix trajectories obtained in the case of a local Kerr nonlinearity (h= I), with qr= 1, and (a) p=O.36, (b) p=O.78, (c) /?=0.95. k b corresponding to a pair of CP solitons (separated by an infinite distance). In fig. 6 (a, b and c) we have plotted the solitary wave envelopes f and b corresponding to the three separatrices of fig. 5 together with the respective effective index profiles If’+ (1 +h)b2] and [b2+ (1 +h)f2]. For p~O.36 (fig. 5a) the forward solitary wave has a symmetric bell shape close to the sech profile of the forward soliton obtained with /3=p,, whereas the backward solitary wave has an antisymmetric profile with a single node at the origin x=0. Naturally, since both these waves represent field distributions that are uniform along the z-axis, they constitute modes of the waveguide they induce. The graph of fig. 6a show that the waveguide index profiles are bell-shaped and symmetric. Clearly, the forward and backward waves correspond to the fundamental mode and the second mode of this induced nonreciprocal waveguide, respectively. For larger /3 (see fig. 6b) the effective waveguide index profile is broader and exhibits two maxima. Correspondingly the forward wave envelope (fundamental mode) has two maxima and the amplitude of the backward wave (second mode) is much larger. As illustrated in fig. 6c, when /I approaches unity (i.e., when p-q) the intensity maxima go further apart and the intensity at the origin decreases dramatically. As a result a two-core planar waveguide is induced. The index profile of each core tends to the ideal sech2-shape which would be obtained in the
f=
-2
0
2
x
-8
0
2
x
Fig. 4. Solitary wave envelopesj; b and the corresponding effective index profiles Lf’+ (1 +h)b2] and [b*+ (1 +h)f*], respectively, for the parameters of curve (c) of fig. 3.
index responding effective profiles, i.e. lf2+(l+h)b2] and [b2+(1+h)f2] for the forward and backward waves, respectively. We see that these solutions do not lead to any new particular features with respect to the CP soliton solutions. For this reason we do not study them in more detail. We now consider the second type of solitary waves predicted by the linear perturbation analysis of eqs. (3). The corresponding separatrix trajectories have been calculated by imposing the condition f ’ =O when b=O which corresponds to the symmetry of the solitary-wave solution in the linear regime eq. (7). Figure 5 illustrates the influence of p on the separatrix trajectories. We see that, when /I is increased from pi, the separatrix which is initially along the faxis forms a loop that broadens progressively while the amplitude on the f-axis decreases (curve a). As a result, under further increase of p the loop divides into two symmetric lobes that narrow progressively (curve b). Finally, as p approaches unity (curve c), the lobes tend asymptotically to the bissecting lines
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bound pair of CP spatial NLS solitons obtained when /I approaches q can be physically interpreted in terms of a balance between attraction and repulsion forces: the pair of CP solitons propagating in parallel is constituted by the superposition of the forward even mode that consists of attracting in-phase solitons and the backward odd mode that consists of repulsing outof-phase solitons.
4. Asymmetric solitary-waves
-5
0
5
x
Fig. 6. Solitary wave envelopesf; b and the effective index profilesf’+ (l+h)b’ (solid line) and b2+ (1 +h)f* (dotted line) for whichfand b are the fundamental and the second mode, respectively. The parameters are those of curves (a), (b) and (c) of fig. 5.
limit p= 1 corresponding to the CP soliton solutions eq. (5 ). In this limit the bound solitary-wave splits into a pair of independent parallel CP spatial solitons infinitely separated. Since the propagation constants of the fundamental and the second modes tend to the same value, the intensity profiles of these two mode become identical, although they keep their even and odd symmetry, respectively. In summary, as /3 increases from pi to unity, the bimodal solitary wave structure passes from the zero amplitude conditions for the first order mode (the single forward soliton) to the degeneracy between the fundamental and the first order modes (pair of independent CP solitons). It is worth noting that the features of these apparently complicated counterpropagating bound solitary-waves can be described by means of elementary concepts of linear optical waveguide theory. We recall that the existence of the 1.50
In the above developments we restricted our analysis to the family of the bimodal solitary wave that are symmetric with respect to the_6axis and that exhibit a single intersection with this axis. Naturally, the search for other separatrix families could reveal the existence of other types of bound counterpropagating solitary waves (that could not be predicted by the above linear perturbation analysis). Although this matter is beyond the scope of the present communication, we briefly consider it here for the simple case of an idealized diffusive nonlinearity (h = 0). In this situation the hamiltonian H shows an additional conservation law and can in principle be integrated analytically [ 141. For the sake of brevity we shall not consider here the analytical forms of the solitarywave solutions. An example of such solution is given in ref. [ 15 1. Figure 7 shows an example of separatrices of two different families for /?=0.9. One (dashed line) belongs to the family of symmetric separatrices considered above. The other (solid line) is not symmetric with respect to thef-axis and thus clearly shows the existence of another family. Naturally, in accordance with the symmetries of H, to this asymmetric trajectory corresponds a third one (not represented on fig. 7) that constitutes its mirror image with respect to thef-axis. Note that it is this family of asymmetric separatrices of H that is considered in ref. [ 15 1. In fig. 8 we plotted the asymmetric solitary wave envelopesfand b as well as the corresponding index profilef*+ b*. This result shows that the self-consistency principle for counterpropagating beams, according to which the two bound solitary waves constitutes the fundamental and second modes of the waveguide they induce, can be generalized to asymmetric waveguides. As indicated by this example which reveals the richness of the so-
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Fig. 7. Example of symmetric and asymmetric separatrices obtained for q= 1, andpz0.9 in the case of a diffusive nonlinearity (h=O).
I
:,.
-5
0
5x
Fig. 8. Solitary wave envelopesf; band the index of intensity profilef’+ b2, corresponding to asymmetric separatrix depicted by the solid line in fig. 7.
lutions of the nonlinear counterpropagating wave equation, one may expect that additional separatrix families of H could be found that would lead to other types of solitary waves.
1 November
1993
sistency principle of self-guided beams from singlemode to two-mode waveguides. We have shown that in the limit of identical propagation constants, the refractive index structure becomes that of a two-core planar waveguide which corresponds to the propagation of parallel solitons without any mutual interaction force. The suppression of the interaction force is the result of the superposition of in-phase soliton and out of phase soliton pairs that is made possible by the use of counterpropagating beams. Our purpose in this communication was to demonstrate the existence of previously unrecognized bimodal spatial solitary waves in counterpropagating beam configuration. As illustrated above with a simple example, the hamiltonian H is expected to possess additional separatrix families that would lead to other types of bound solitary waves. The fundamental problems of transient build-up dynamics and stability of the solitary waves also remain open. These counter-propagating bound solitary waves were foreseen on the basis of the sinusoidal phase dependence of the mutual interaction force between solitons. This feature not only holds for NLS solitons, but more generally for self-guided beams ( 1D or 2D) of media whose refractive index depends arbitrarily on intensity [ 19 1. Therefore one may anticipate the existence of counterpropagating spatial solitary waves in more general physical situations. Finally, note that the waveguide induced by the parallel bound CP solitons constitutes a uniform two-core waveguide, or in other words a directional coupler, that could lead to versatile applications to all optical switching. All these aspects of the problem are now under investigation and will be the subject of future reports.
Acknowledgements 5. Conclusions In conclusion, a simple analysis of the standard coupled NLS equation that describes counterpropagation of light beams in self-focusing Kerr-type nonlinear media has allowed us to show the existence of a novel type of spatial solitary waves. These solitary waves are bound states of two counterpropagating waves that constitute the fundamental and second modes of the waveguiding structure they induce. In this sense, we have generalized the self-con-
We acknowledge fruitful discussions with N.N. Akhmediev and D.J. Mitchell. We are indebted to the anonymous Referee for several suggestions which have been very helpful to us in improving the presentation of this work.
References [ 1] Y. Silberberg and I. Bar Joseph, Phys. Rev. Lctt. 48 ( 1982) 1541.
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[2] CT. Law and A.E. Kaplan, Optics Lett. 14 ( 1989) 734. [3] A.L. Gaeta, R.W. Boyd, J.R. Ackerhalt and P.W. Milonni, Phys. Rev. Lett. 58 ( 1987) 2432. [4] A.E. Kaplan, Optics Lett. 8 ( 1983) 560. [5] S. Wabnitz, Phys. Rev. Lett. 58 (1987) 1415; S. Watnitz and G. Gregory, Optics Comm. 59 ( 1986) 72. [6] W.J. Firth and C. Pare, Optics Lett. 13 (1988) 1096; W.j. Firth, A. Fitzgerald and C. Pare, J. Opt. Sot. Am. B 7 (1990) 1087. [7] G.G. Luther and C.J. McKinstrie, J. Opt. Am. B 7 (1990) 1125. [8]R. Chang, W.J. Firth, R. Indik, J.V. Moloney and E.M. Wright, Optics Comm. 88 (1992) 167. [9] G. Grynberg, Optics Comm. 67 (1988) 363.
[IO] S. Maneuf, R. Desailly and C. Froehly, Optics Comm. 65 (1988)
152
193.
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[ 111 A. Barthelemy, C. Froehly, S. Maneuf and F. Reynaud, Optics Lett. 17 ( 1992) 844. [ 121 J.P. Gordon, Optics Lett. 8 (1983) 596. [ 131 A.E. Kaplan and P. Meystre, Optics Comm. 40 ( 1982) 229. [ 141 J. Hietarinta, Phys. Rev. A 96 (1983) 273. [ 151 D.N. Christodoulides and RI Joseph, Optics Lett. 13 ( 1988) 53. [ 161 A.W. Snyder, D.J. Mitchell and L. Poladian, J. Opt. Sot. Am.B8(1991) 1618. [ 171 J.H. Marburger and F.S. Felber, Phys. Rev. Lett. A 17 (1978) 335. [ 181 A.W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). [ 191 D.J. Mitchell, A.W. Snyder and L. Poladian, Electron. Lett. 27 (1991) 848.
103 (1993)
OPTICS COMMUNICATIONS
145-152
Full length article
Bimodal M. Haeltennan,
counterpropagating A.P. Sheppard
spatial solitary-waves
and A.W. Snyder
Optical Sciences Centre. Australian National University, Canberra. Australia Received
14 April 1993; revised manuscript
received
A simple analysis of the standard coupled nonlinear of a new class of spatial solitary waves in Kerr media. constitute the fundamental and second modes of the two distinct, parallel self-guided beams. The existence simple physical arguments.
1.
16 June I993
Schrijdinger equations for counterpropagating beams reveals the existence These solitary waves are bound states of two counterpropagating fields that waveguiding structure they induce. For certain parameters they represent and the features of these particular solutions are explained on the basis of
Introduction
Owing to both their theoretical interest and their fundamental role in many practical nonlinear systems, the processes of self-interaction of counterpropagating light beams in nonlinear materials have attracted growing interest in recent years. Many aspects of the problem have been investigated, revealing a very rich spectrum of complex behaviors. Instabilities, self-oscillations and temporal chaos have been reported in simple plane-wave models of counterpropagating beams in the cases of materials with finite response time [ 11, chromatic dispersion [ 21 and with a tensorial nonlinearity when accounting for the polarization of the field [ 3 1. It was shown that selfinteraction between counterpropagating polarized waves leads to new types of nonlinear polarization eigenstates [4] and may induce periodic, multistable, and chaotic spatial distributions [ 5 1. More recently, much attention has been paid to diffractive transverse effects in Kerr-type nonlinear media. Linear stability analyses of the coupled nonlinear Schrodinger (NLS) equations that govern counterpropagating beams in Kerr media show that plane waves are unstable against transverse modulation independently of the sign of the nonlinearity [ 6,7 1. Numerical studies in two-dimensional geometry revealed that this instability may lead to periodic pattern formation, hysteresis, symmetry-breaking, oscillations and chaos [ 6 1. Recent three-dimensional simula0030-4018/93/$06.00
0 1993 Elsevier Science Publishers
tions of diffractive transverse effect predicted the formation of transverse patterns of hexagonal symmetry [ 8 1, a result which is in agreement with the theoretical and experimental studies of mirrorless degenerate four-wave-mixing oscillations in atomic vapor [9]. In this paper we show that diffractive transverse effects due to self-interaction of two counterpropagating beams in a Kerr medium lead to a hitherto unrecognized class of spatial solitary wave. These solitary waves are peculiar in that they constitute a bound state of the fundamental and the second modes of the waveguiding structure they induce together. For certain values of the wave parameters they represent a bound state of two identical solitons propagating in parallel. The nature of these solitary waves can be foreseen by means of simple physical arguments.
2. Theoretical
model
We consider counterpropagation of two beams in one transverse dimension as shown schematically in fig. 1. In practice this situation can be obtained by considering monomode propagation in a planar slab waveguide [lo] or, alternatively, by using a twobeam interference technique [ 111. In this preliminary study we consider only the stationary states of the field. The questions of build-up transient dynam-
B.V. All rights reserved.
145
Volume 103, number I ,2
OPTICS COMMUNICATIONS
Fig. 1. Schematic of the counterpropagating beam configuration considered in the text. F and B are the forward and backward beam envelopes. x and z are the transverse and longitudinal coordinates, respectively.
its and of stability of the solitary waves require elaborate numerical investigations beyond the scope of this communication. In the scalar approximation of the field, the envelope amplitudes of the two stationary counterpropagating beams, say F(x, z) and B( x, z), obey the following set of coupled NLS equations [6] -i~=~~+[IFl’+(l+h)IBI’]F,
(la)
i~=~~+[lei2+(l+h)lF12]u,
(lb)
where x and z represent, in dimensionless units, the transverse and longitudinal coordinates, respectively. The parameter h describes the effect of the index grating due to the standing-wave interferences. Its value, which lies between zero and unity, parametrizes the diffusion process inherent to the type of Kerr nonlinearity considered, e.g., h is zero for thermally induced nonlinearity and tends to unity for local nonlinearity. Note that we consider only the selffocusing Kerr nonlinearity. In the presence of a single beam, i.e., when either For B is zero, eq. ( 1) reduces to the NLS equation whose soliton solutions have been extensively studied in the literature. Much attention has been paid to the problem of interaction between copropagating solitons and in particular, two identical solitons have been shown to interact through a force that depends sinusoidally on their relative phase [ 121. As a consequence, a pair of 180” out-of-phase solitons and a pair of in-phase solitons exhibit interaction forces of equal amplitudes and opposite signs. This result suggests that the superposition of in-phase and out-ofphase soliton pairs would lead to a perfect balance 146
1 November 1993
between attraction and repulsion forces resulting in the formation of two distinct solitons propagating in parallel. We shall see in the following that the superposition of two counterpropagating soliton beams pairs allows for such an ideal balance between opposite forces. Moreover, though this reasoning holds for well separated soliton beams for which the interaction force depends sinusoidally on their relative phase [ 121, we may anticipate that for arbitrarily closer beams a field configuration still exists that preserves the balance between attraction and repulsion.
3. Solitary-wave solutions The counterpropagating field configurations F(x, z) and B(x, z) that correspond to solitary waves, i.e., waves that are uniform along the z-axis, have the form F(x, z) =f(x)
exp(@)
,
B(x, z) =b(x)
exp( -ij3z)
(2a) ,
(2b)
where the beam envelopes f and b, and the corresponding propagation constants ~1 and /I are real quantities. Substituting these expressions into eq. ( 1) leads to a set of coupled second order nonlinear differential equations y”+f3+(1+h)b2f-+O,
(3a)
+b”+b3+ (1 +h)f2b-/lb=O ,
(3b)
where primes denote x-derivatives. Sincefand b are independent of z, these equations can be considered as linear wave equations for the propagation modes of two distinct waveguides. The fields f and b would correspond to the mode of the waveguide whose index profiles are proportional to f2+ ( 1-t h)b2 and respectively. The fact that the forb2+(l+h)f2, ward and backward waves see different effective index profiles comes from the well known grating induced nonreciprocity inherent to counterpropagation of beams in nonlinear materials (see e.g. ref. [ 13 ] 1. Note that when h=O both profiles are identical. Equations (3) are the equations of motion for a unit mass in a conservative potential of the two-dimensional space V; b). The corresponding hamiltonian is
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(4) For a diffusive nonlinearity, that is, when h = 0, this hamiltonian exhibits an additional conservation law and is considered integrable [ 141. In that particular case it is identical to that derived in ref. [ 15 ] for the study of temporal vector solitons in birefringent nonlinear dispersive media. The analytical form of the solitary-wave solutions proposed in ref. [ 15 ] will be discussed below. For h#O the hamiltonian H is not integrable and the solutions f and b of eq. (3) must be calculated numerically. Since we are interested in solitary-wave solutions we must look for the separatrix trajectories of H, i.e., the solutions that obeyf; b, f' and b’+O as x+ *co. Naturally the hamiltonian H exhibits symmetries that reveal the existence of simple separatrix trajectories that correspond to analytical expressions of the solitary-waves. For example, if b (or f) is zero the separatrix follows a straight trajectory along the f-axis (b-axis), which, according to eq. (2) corresponds to the solution f(x)= 2psech(flpx), b(x)=0 [or b(x) =$$sech( 2px), f(x) =O]. These soluK tions correspond naturally to the well known spatial solitons obtained by unidirectional beam propagation in Kerr materials. When p= p other separatrices are found that follow the bissectrice lines in the plane CfTb), i.e., f= k b. In this case eqs. (2a) and (2b) become identical NLS equations and yield
f(x)=
fb(x)=dmsech(,&x)
.
(5)
These solitary-wave solutions represent the superposition of two identical counterpropagating NLS solitons whose intensity is adjusted so as to compensate for the presence of the other. In the following we use the notation CP solitons to denote these particular counterpropagating spatial soliton states given by eq. ( 5 ). Note that, due to cross-phase modulation, the intensity [ 299/ (2 + h) ] of the counterpropagating solitons is in general (h > 0) less than half the intensity (247) of a single unidirectional soliton of the same width. The separatrices that correspond to these analytical solutions are depicted in fig. 2. We shall see in the following that they represent asymptotic states of a family of peculiar bound solitary-wave solutions.
b
cb) i 1.01
(4 0.5
-
(4
/ OL
0.5
1.0
1.5
f -0.5
-
\
(4
Fig. 2. Separatrix trajectories in the plane Cr;b) corresponding to the unidirectional (a) forward and (b) backward NLS spatial solitons. Curves (c ) and (d ) correspond to the counterpropagating (CP) soliton solutions given in eq. (5) with q= 1. Before proceeding to the analysis of more general solitary-wave solutions of eq. (2 ) it is convenient to discuss the physical meaning of the CP soliton SOlutions eq. (5). In the case of a single self-guided beam, the NLS soliton solution represents a field configuration that constitutes the fundamental mode of the monomode waveguide it induces itself through the nonlinearity [ 161. In the case of counterpropagating soliton beams this self-consistency relation can be easily generalized when the nonlinearity is diffusive (h=O). Indeed, in that situation the grating induced by the standing-wave interferences is completely washed-out and the induced waveguide is uniform. It is the same waveguide than that induced by a unidirectional NLS soliton beam and naturally, the field distributions f and b of eq. (5) in this case constitute the fundamental mode of the waveguide they induce together. According to eq. (5) each counterpropagating soliton contributes for a half of the waveguide index profile. Note however that, for h=O, eq. (5) only yields a particular case of the CP soliton solutions as the ratio between the soliton intensities is not necessarily 50% but can have any value. On the other hand, when h # 0 it is easy to verify that eq. (5) yields the only solution (in which the solitons have the same intensity). In that case the index of refraction is modulated by the standing-wave
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interferences. In the limit of local Kerr nonlinearity (h= 1) the modulation depth reaches 100% and it is conceptually difficult to consider the resulting refractive index structure as a waveguide, especially because the induced grating is perfectly resonant with the counterpropagating waves. Nevertheless, as is well known [ 13,171, due to the particular phase conditions between the standing wave and the induced grating, the index modulation acts only on the phase of the waves and does not induce any amplitude coupling. The field of each counterpropagating wave of eq. (5) therefore behaves as if it was in a perfectly uniform waveguide whose effective refractive index is increased by a factor of 1 + h/2 (according to the grating induced phase changes). This result allows us to consider the refractive index structures induced by counterpropagating solitary waves as uniform waveguides. In particular, we may state the selfconsistency relation by saying that the CP soliton solution eq. (5) represents the two counterpropagating fundamental modes of the monomode waveguide they induce together. We shall see in the following that this self-consistency principle can be generalized to bimodal waveguides. Let us now consider general solitary-wave solutions to eq. (2). Let us notice that, by means of a simple change of variables (x+x/p”‘, f+f~“~, b+brpL12, and /3-+/3/q), eq. (3 ) can be normalized in such a way that the phase parameter ~1is equal to unity. We can therefore analyze the solutions with respect to the single wave-parameter /I. Let us first consider the case where the forward beam is a soliton that induces a waveguide for a very low amplitude backward beam. In that situation b
.
(6)
This second order linear differential equation has solutions which decay at infinity. They correspond to the modes of the waveguide as seen by the backward beam. The fundamental mode is [ 18 ] b(x)=sech”(@x)
(7)
where s is dependent on h through the relation s(s+ 1)=2(h+ 1). Its propagation constant is&=s2. The higher order modes are given in terms of the Legendre functions of the first kind [ 181 148
b(x)=P:-“[tanh(
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(8)
with the restriction n ds (n being the order of the modes). Their propagation constants are /Zn= (n -5) 2. The maximum value of s corresponds to h = 1 (local nonlinearity) and is s = 1.56. This shows that the effective waveguide supports a maximum of two modes. In the limiting case of a highly diffusive nonlinearity, h = 0, s is equal to unity and the second mode is at cut-off (p,=O), as is the case for the waveguide seen by the forward soliton beam itself [ 161. In the following we only consider the case of a local Kerr nonlinearity h = 1. This restriction does not affect the generality of our conclusions. The propagation constants of the two modes are therefore PO= 2.43 and pi = 0.3 14. This linear perturbation analysis of eqs. (3) can be summarized saying that, in the linear regime of the backward wave, a soliton beam can guide two backward modes, a fundamental mode (zero-node beam envelope) and a first order mode (one-node antisymmetric envelope). By extending this result up to the nonlinear regime of the backward wave b, we can foresee the existence of two types of bound spatial solitary-waves. One type would consist of a bound state of the two counterpropagating fundamental modes of the nonreciprocal waveguide they induce themselves, while the other would involve the fundamental mode and the first order mode of the “forward” and “backward” induced waveguides, respectively. We have calculated numerically (by means of a standard shooting technique) the separatrix trajectories of H in the general situation where p# 1. From the linear perturbation analysis we know that when /3 is close to PO or pi the composite solitary wave approaches the forward soliton solution, i.e. the corresponding separatrix trajectory lies on the f-axis (b= 0). We first study the separatrix trajectories of H in the vicinity of PO. We verified that the separatrices only exist for values of /? smaller than PO. Examples of trajectories are shown in fig. 3 for several values of p. We observe that, as /3 decreases from 2.43, the trajectory progressively bends and leaves the f-axis. When /3 tends to unity the separatrix approaches the bissecting line f= b which corresponds to the CP soliton solution given in eq. (5) (with @= 1). In fig. 4 we plotted an example of field envelopes f and b of this type of solitary waves together with the cor-
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f
Fig. 3. Separatrix trajectories obtained in the case of a local Kerr nonlinearity (h= l), with (o= 1, and (a) P=2.3, (b) 8=2.0, (c) /I= 1.5, (d) /I= 1.1. Fig. 5. Separatrix trajectories obtained in the case of a local Kerr nonlinearity (h= I), with qr= 1, and (a) p=O.36, (b) p=O.78, (c) /?=0.95. k b corresponding to a pair of CP solitons (separated by an infinite distance). In fig. 6 (a, b and c) we have plotted the solitary wave envelopes f and b corresponding to the three separatrices of fig. 5 together with the respective effective index profiles If’+ (1 +h)b2] and [b2+ (1 +h)f2]. For p~O.36 (fig. 5a) the forward solitary wave has a symmetric bell shape close to the sech profile of the forward soliton obtained with /3=p,, whereas the backward solitary wave has an antisymmetric profile with a single node at the origin x=0. Naturally, since both these waves represent field distributions that are uniform along the z-axis, they constitute modes of the waveguide they induce. The graph of fig. 6a show that the waveguide index profiles are bell-shaped and symmetric. Clearly, the forward and backward waves correspond to the fundamental mode and the second mode of this induced nonreciprocal waveguide, respectively. For larger /3 (see fig. 6b) the effective waveguide index profile is broader and exhibits two maxima. Correspondingly the forward wave envelope (fundamental mode) has two maxima and the amplitude of the backward wave (second mode) is much larger. As illustrated in fig. 6c, when /I approaches unity (i.e., when p-q) the intensity maxima go further apart and the intensity at the origin decreases dramatically. As a result a two-core planar waveguide is induced. The index profile of each core tends to the ideal sech2-shape which would be obtained in the
f=
-2
0
2
x
-8
0
2
x
Fig. 4. Solitary wave envelopesj; b and the corresponding effective index profiles Lf’+ (1 +h)b2] and [b*+ (1 +h)f*], respectively, for the parameters of curve (c) of fig. 3.
index responding effective profiles, i.e. lf2+(l+h)b2] and [b2+(1+h)f2] for the forward and backward waves, respectively. We see that these solutions do not lead to any new particular features with respect to the CP soliton solutions. For this reason we do not study them in more detail. We now consider the second type of solitary waves predicted by the linear perturbation analysis of eqs. (3). The corresponding separatrix trajectories have been calculated by imposing the condition f ’ =O when b=O which corresponds to the symmetry of the solitary-wave solution in the linear regime eq. (7). Figure 5 illustrates the influence of p on the separatrix trajectories. We see that, when /I is increased from pi, the separatrix which is initially along the faxis forms a loop that broadens progressively while the amplitude on the f-axis decreases (curve a). As a result, under further increase of p the loop divides into two symmetric lobes that narrow progressively (curve b). Finally, as p approaches unity (curve c), the lobes tend asymptotically to the bissecting lines
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bound pair of CP spatial NLS solitons obtained when /I approaches q can be physically interpreted in terms of a balance between attraction and repulsion forces: the pair of CP solitons propagating in parallel is constituted by the superposition of the forward even mode that consists of attracting in-phase solitons and the backward odd mode that consists of repulsing outof-phase solitons.
4. Asymmetric solitary-waves
-5
0
5
x
Fig. 6. Solitary wave envelopesf; b and the effective index profilesf’+ (l+h)b’ (solid line) and b2+ (1 +h)f* (dotted line) for whichfand b are the fundamental and the second mode, respectively. The parameters are those of curves (a), (b) and (c) of fig. 5.
limit p= 1 corresponding to the CP soliton solutions eq. (5 ). In this limit the bound solitary-wave splits into a pair of independent parallel CP spatial solitons infinitely separated. Since the propagation constants of the fundamental and the second modes tend to the same value, the intensity profiles of these two mode become identical, although they keep their even and odd symmetry, respectively. In summary, as /3 increases from pi to unity, the bimodal solitary wave structure passes from the zero amplitude conditions for the first order mode (the single forward soliton) to the degeneracy between the fundamental and the first order modes (pair of independent CP solitons). It is worth noting that the features of these apparently complicated counterpropagating bound solitary-waves can be described by means of elementary concepts of linear optical waveguide theory. We recall that the existence of the 1.50
In the above developments we restricted our analysis to the family of the bimodal solitary wave that are symmetric with respect to the_6axis and that exhibit a single intersection with this axis. Naturally, the search for other separatrix families could reveal the existence of other types of bound counterpropagating solitary waves (that could not be predicted by the above linear perturbation analysis). Although this matter is beyond the scope of the present communication, we briefly consider it here for the simple case of an idealized diffusive nonlinearity (h = 0). In this situation the hamiltonian H shows an additional conservation law and can in principle be integrated analytically [ 141. For the sake of brevity we shall not consider here the analytical forms of the solitarywave solutions. An example of such solution is given in ref. [ 15 1. Figure 7 shows an example of separatrices of two different families for /?=0.9. One (dashed line) belongs to the family of symmetric separatrices considered above. The other (solid line) is not symmetric with respect to thef-axis and thus clearly shows the existence of another family. Naturally, in accordance with the symmetries of H, to this asymmetric trajectory corresponds a third one (not represented on fig. 7) that constitutes its mirror image with respect to thef-axis. Note that it is this family of asymmetric separatrices of H that is considered in ref. [ 15 1. In fig. 8 we plotted the asymmetric solitary wave envelopesfand b as well as the corresponding index profilef*+ b*. This result shows that the self-consistency principle for counterpropagating beams, according to which the two bound solitary waves constitutes the fundamental and second modes of the waveguide they induce, can be generalized to asymmetric waveguides. As indicated by this example which reveals the richness of the so-
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Fig. 7. Example of symmetric and asymmetric separatrices obtained for q= 1, andpz0.9 in the case of a diffusive nonlinearity (h=O).
I
:,.
-5
0
5x
Fig. 8. Solitary wave envelopesf; band the index of intensity profilef’+ b2, corresponding to asymmetric separatrix depicted by the solid line in fig. 7.
lutions of the nonlinear counterpropagating wave equation, one may expect that additional separatrix families of H could be found that would lead to other types of solitary waves.
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sistency principle of self-guided beams from singlemode to two-mode waveguides. We have shown that in the limit of identical propagation constants, the refractive index structure becomes that of a two-core planar waveguide which corresponds to the propagation of parallel solitons without any mutual interaction force. The suppression of the interaction force is the result of the superposition of in-phase soliton and out of phase soliton pairs that is made possible by the use of counterpropagating beams. Our purpose in this communication was to demonstrate the existence of previously unrecognized bimodal spatial solitary waves in counterpropagating beam configuration. As illustrated above with a simple example, the hamiltonian H is expected to possess additional separatrix families that would lead to other types of bound solitary waves. The fundamental problems of transient build-up dynamics and stability of the solitary waves also remain open. These counter-propagating bound solitary waves were foreseen on the basis of the sinusoidal phase dependence of the mutual interaction force between solitons. This feature not only holds for NLS solitons, but more generally for self-guided beams ( 1D or 2D) of media whose refractive index depends arbitrarily on intensity [ 19 1. Therefore one may anticipate the existence of counterpropagating spatial solitary waves in more general physical situations. Finally, note that the waveguide induced by the parallel bound CP solitons constitutes a uniform two-core waveguide, or in other words a directional coupler, that could lead to versatile applications to all optical switching. All these aspects of the problem are now under investigation and will be the subject of future reports.
Acknowledgements 5. Conclusions In conclusion, a simple analysis of the standard coupled NLS equation that describes counterpropagation of light beams in self-focusing Kerr-type nonlinear media has allowed us to show the existence of a novel type of spatial solitary waves. These solitary waves are bound states of two counterpropagating waves that constitute the fundamental and second modes of the waveguiding structure they induce. In this sense, we have generalized the self-con-
We acknowledge fruitful discussions with N.N. Akhmediev and D.J. Mitchell. We are indebted to the anonymous Referee for several suggestions which have been very helpful to us in improving the presentation of this work.
References [ 1] Y. Silberberg and I. Bar Joseph, Phys. Rev. Lctt. 48 ( 1982) 1541.
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[2] CT. Law and A.E. Kaplan, Optics Lett. 14 ( 1989) 734. [3] A.L. Gaeta, R.W. Boyd, J.R. Ackerhalt and P.W. Milonni, Phys. Rev. Lett. 58 ( 1987) 2432. [4] A.E. Kaplan, Optics Lett. 8 ( 1983) 560. [5] S. Wabnitz, Phys. Rev. Lett. 58 (1987) 1415; S. Watnitz and G. Gregory, Optics Comm. 59 ( 1986) 72. [6] W.J. Firth and C. Pare, Optics Lett. 13 (1988) 1096; W.j. Firth, A. Fitzgerald and C. Pare, J. Opt. Sot. Am. B 7 (1990) 1087. [7] G.G. Luther and C.J. McKinstrie, J. Opt. Am. B 7 (1990) 1125. [8]R. Chang, W.J. Firth, R. Indik, J.V. Moloney and E.M. Wright, Optics Comm. 88 (1992) 167. [9] G. Grynberg, Optics Comm. 67 (1988) 363.
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[ 111 A. Barthelemy, C. Froehly, S. Maneuf and F. Reynaud, Optics Lett. 17 ( 1992) 844. [ 121 J.P. Gordon, Optics Lett. 8 (1983) 596. [ 131 A.E. Kaplan and P. Meystre, Optics Comm. 40 ( 1982) 229. [ 141 J. Hietarinta, Phys. Rev. A 96 (1983) 273. [ 151 D.N. Christodoulides and RI Joseph, Optics Lett. 13 ( 1988) 53. [ 161 A.W. Snyder, D.J. Mitchell and L. Poladian, J. Opt. Sot. Am.B8(1991) 1618. [ 171 J.H. Marburger and F.S. Felber, Phys. Rev. Lett. A 17 (1978) 335. [ 181 A.W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983). [ 191 D.J. Mitchell, A.W. Snyder and L. Poladian, Electron. Lett. 27 (1991) 848.