Hamiltonian chaos in nonlinear optical polarization dynamics

HAMILTONIAN CHAOS IN NONLINEAR OPTICAL POLARIZATION DYNAMICS

D. DAVID, D.D. HOLM and M.V. TRATNIK Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS-B258, Los Alamos, NM 87545, USA

NORTH-HOLLAND

PHYSICS REPORTS (Review Section of Physics Letters) 187, No. 6 (1990) 281—367. North-Holland

HAMILTONIAN CHAOS IN NONLINEAR OPTICAL POLARIZATION DYNAMICS D. DAVID, D.D. HOLM and M.V. TRATNIK Theoretical Division and Center for Nonlinear Studies, Los Ala,nos National Laboratory. MS-B258, Los Alarnos, NM 87545. USA Received August 1988

Contents: 1. Introduction 2. Field amplitude equations 3. Lie—Poisson geometry and the phase space reduction to spheres 4. Fixed point analysis and bifurcations in the two-beam problem 4.1. Case 1. u = 0, = K 4.2. Case 2. u = 0, ~K 43. Case 3. o- ~ 0. i~ = K, r 4.4. Case 4. a-, K arbitrary (regular cases) 4.5. Singular subcases. a- = ±(~— K) 5. Kinks and solitary waves 5.1. Case 1. a- = 0, K~= K 5.2. Case 2. a- = 0, K~ K 5.3. Case 3. (r~0, Kj=K. ~=r

,~ ,~

283 285 291 299 301 304 307 31)) 311 314 316 32)) 325

5.4. Case 4. r = — K) 6. Horseshoe chaos and Arnold diffusion 6.1. Physical interpretation of horseshoe chaos in the twobeam problem 2 xR 6.2. Horseshoe chaos on S 6.3. Horseshoe chaos on 5 x S2 6.4. Arnold diffusion on S2 X S X R~ 7. Conclusion Appendices A. The analogy with the gyrostat and horseshoe chaos for the single Stokes pulse B. (‘ritical points, hifureations and phase portraits of the one-beam problem References

328 331 333 334 337 339 342

342 346 365

Abstract: This paper applies Hamiltonian methods to the Stokes representation of the one-beam and the two-beam problems of polarized optical pulses propagating as travelling waves in nonlinear media. We treat these two dynamical systems as follows. First, we use the reduction method of Marsden and Weinstein to map each of the systems to the two-dimensional sphere, S~The resulting reduced systems are then analyzed from the viewpoints of their stability properties and of bifurcations with symmetry; in particular, several degenerate bifurcations are found and described. We also establish the presence of chaotic dynamics in these systems by demonstrating the existence of Smale horseshoe maps in the three- and four-dimensional cases, as well as Arnold diffusion in the higher-dimensional cases. The method we use to establish such complex dynamics is the Mel’nikov technique, as extended by Holmes and Marsden, and Wiggins for the higher-dimensional cases. These results apply to perturbations of homoclinic and heteroclinic orbits of the reduced integrable problems for static, as well as travelling-wave, solutions describing either a single optical beam, or two such beams counterpropagating. Thus, we show that these optics problems exhibit complex dynamics and predict the experimental consequences of this dynamics.

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D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

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1. Introduction In this paper we treat polarization dynamics in the Stokes description for a single radiative beam, or two such counterpropagating beams interacting within a nonlinear optical medium, e.g., laser pulses in an optical fiber. We show that laser polarization dynamics has the potential to produce chaos in an experimentally reproducible, high-resolution, context. Maker et a!. [1964] first demonstrated that the polarization of a light beam propagating in a nonlinear medium can exhibit novel and interesting behavior. They considered the self-interaction of a plane electromagnetic wave in a nonlinear, isotropic medium, and both predicted and observed that for elliptically polarized light the major axis of the ellipse rotates as a function of propagation length, assuming steady state in time. More recently Lytel [1984],Kaplan [1983], and Kaplan and Law [1985] have discussed the multiple spatially stable configurations of the nonlinear interaction of two counterpropagating plane waves in an isotropic medium. Yumoto and Otsuka [1985]and Otsuka et al. [1985] considered such an interaction in a cubic crystal and found polarization bistability as well as numerical evidence for spatial chaos. Gaeta et al. [1987a]also included time dependence with a finite relaxation time for the nonlinearity and in a numerical investigation found indications of temporal chaotic behavior in the polarization and intensity. These researchers have worked with complex electric field amplitudes, which for the two-beam problem has the phase space C2 x C2. From the viewpoint of dynamical systems analysis, it proves to be of considerable advantage to work with the so-called Stokes description, which reduces the phase space of the two-beam problem to the direct product of two Poincaré spheres, S2 x S2. For each coherent electromagnetic field there are three independent Stokes parameters, two angles and a conserved radius, which are bilinear products of the vector components of the complex electric field amplitude. The Stokes parameters are real and contain the same information as the electric field amplitude except for the absolute phase, which is usually not of interest, but which can be calculated by returning to the phase space C2 x C2 via explicit reconstruction procedures once the solution on S2 x 52 is known: these can be written down in terms of quadratures (see, for instance, Marsden et al. [1989]).The polarization of the field is easily and elegantly visualized on the surface of the Poincaré sphere (shown in fig. 2.1). This approach was first introduced in the context of polarization dynamics by Kubo and Nagata [1980,1981], who studied the effects of stress fields on the propagation of light in a linear anisotropic medium. Subsequently it has been extended to the study of nonlinear polarization dynamics by Sala [1984],Wabnitz and Gregori [1986],Wabnitz [1987],Tratnik and Sipe [1985, 1986a, b, 1987a, b, c, d], and Zakharov and Mikhailov [1987]. Sala [1984]considers the spatial polarization dependence of a single plane wave propagating through an isotropic medium in the presence of a dc field-induced birefringence, while Gregori and Wabnitz [1986]consider a cubic crystal with the induced birefringence lying along one of the symmetry axes. Wabnitz and Gregori [1986]also consider spatially periodic birefringence, which can lead to polarization chaos as they have shown numerically, and discuss fixed points of the two-beam problem in a parity-invariant isotropic medium. Tratnik and Sipe [1987a,b, c] systematically treat the one- and two-beam problems for all material symmetries. They identify the integrable cases, find and physically interpret the invariants and fixed points, indicate several types of periodic solutions and also construct a rigid body analogy for the one-beam problem. Upon including time dependence (but with zero relaxation time of the nonlinearity) Tratnik and Sipe [1987d]find an interesting family of what appear to be polarization solitons for the special case of an electrostrictive medium. In another special case, Zakharov and Mikhailov [1987]show that, when self-interaction of each wave is neglected, the resulting

284

D. David et al., I-i’amiltonian chaos in nonlinear optical polarization dynamics

partial differential equations are those of the SO(3)-valued sigma model, which they had previously shown to be completely integrable, having soliton, multisoliton, and kink solutions. The moving kink solutions separate different domains of constant polarization; the kink width and velocity depend upon the two intensities and other parameters in the system. The present paper treats polarization dynamics of a quasi-monochromatic optical beam propagating, or two such beams counterpropagating, in a nonlinearly polarizable medium, by using methods that have recently been developed for Hamiltonian dynamical systems [Holmes and Marsden 1982a, b, 1983; Wiggins 19881. The paper uses Hamiltonian methods in the following steps: (1) formulate the equations of motion; (2) determine conservation laws and reduce the number of phase space dimensions; (3) identify equilibrium solutions as critical points of conserved quantities; (4) establish stability criteria for the equilibrium solutions; (5) investigate bifurcations and changes of stability on the reduced phase space as parameters are varied in the Hamiltonian; (6) break symmetries in the vicinity of the bifurcation points and use analytical methods to establish Poincaré—Birkhoff—Smale chaos (due to Smale horseshoe mappings). Section 2 sets notation and presents the physical motivation and approximations leading to the nonlinear partial differential equations for optical polarization dynamics. We discuss the case of an infinite homogeneous medium; but the same equations also describe nonlinear polarization dynamics in an optical fiber under standardly used approximations. In particular, the pulse length is assumed to he long with respect to the space and time scales defined by optical frequencies and wavelengths, as is certainly the case for a typical laser. (Ultrashort pulses violating this approximation are outside the scope of the present paper.) The electromagnetic field is described by a “fast” running wave modulated by a slowly varying envelope amplitude which takes into account the small effects of anisotropy and nonlinearity. It is assumed that the frequency of the field is sufficiently far from any resonances in the medium so that dissipation and group velocity dispersion can be neglected. Section 3 provides the Hamiltonian formulation of the problem of determining travelling-wave solutions for two counterpropagating nonlinearly interacting optical beams, thereby reducing the dynamics to a system of ordinary differential equations. (One may also seek static, spatially varying solutions as a special case of the travelling waves.) In terms of the initial physical variables (the components of the complex electric field amplitudes), the phase space is the eight-real-dimensional product space C2 x C2 possessing a Poisson bracket arising from a canonical symplectic form. In terms of the Stokes formalism, the phase space is a six-dimensional Poisson manifold defined on the dual space of the Lie algebra so(3) ~ so(3) and possessing a Lie—Poisson structure reminiscent of the problem of two coupled rigid bodies [Krishnaprasad and Marsden 1987]. This Poisson manifold naturally reduces to the direct product of two Poincaré spheres, S2 X S2, each of which is symplectic. In the presence of an additional Si symmetry (in particular, for parity-invariant materials isotropic about the propagation direction) we further reduce the four-dimensional system on 52 x S2 to a twodimensional one, defined on the sphere S2 (modulo either one, or two singularities, depending upon the values of the beam intensities), where it becomes completely integrable. The term reduce is used here in the sense of Marsden and Weinstein 119741 and does not imply that some physical information is lost; we present a Reconstruction Lemma in section 3, which shows how to reconstruct from the dynamical information on the reduced space ~2 all of the properties of the physical solutions defined on the original phase space C2 >< C2. The genera! reconstruction method can be found in Abraham and Marsden [1980];an improvement of it in terms of connections is in Marsden et al. [1989].

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

285

We perform a fixed-point analysis for static, or travelling-wave solutions of the two-beam problem without birefringence in section 4 and determine the stability properties of the equilibrium solutions. We also illustrate the types of possible orbits by showing phase portraits that exemplify certain degenerate types of bifurcations taking place on S2. In relation to this last point, and as pointed out in section 3, the reduction procedure does not always map the initial manifold onto the two sphere, ~2; it so happens that, when certain functional identities are satisfied between two of the parameters and one of the constants of motion, the reduction map is not to S2 rather it is either to a cylinder, or to one of two types of paraboloids. Alternatively, one may regard the reduction map as being to S2 with singularities. It is known that singularities occur only at phase points with symmetry (see Arms et al. [1981])and our results are consistent with this. Finally, in section 4 we locate the various saddle points on the reduced S2 and determine the conditions on the parameters of the problem for which these points remain saddles. Saddle points are associated with heteroclinic and homoclinic orbits, i.e. separatrices on the reduced ~2, which correspond to particular solutions when lifted back to the Stokes polarization variables. In section 5, we present explicit coordinate representations of some of the heteroclinic and homoclinic orbits on the reduced phase space S2 for the two-beam problem. Then, by virtue of the reconstruction lemma presented in section 3, we reconstruct solitary wave and kink solutions in the Stokes polarization variables corresponding to these homoclinic and heteroclinic orbits. We also construct particular solutions that appear for special parameter values when the reduction is forced to be onto a paraboloid or a cylinder, instead of onto the sphere. In section 6, we attack the issue of chaos for our system, by deforming the Hamiltonian function of the system to introduce small space- or time-periodic perturbations; some of these perturbations break the isotropy of the system so that the phase space can no longer be reduced to a sphere: the motion rather takes place in S2 X S2. Then using the Mel’nikov method in conjunction with the Poincaré— Birkhoff—Smale theorem, we prove that such perturbations cause homoclinic (or heteroclinic) tangles and, hence, produce horseshoe chaos. We also discuss in section 6 how to measure this horseshoe chaos in an experimental situation. The appendices are devoted to the one-beam problem with birefringence. In appendix A we show that the one-beam problem contains as a special case an invariant subsystem of the dynamics of a gyrostat (a rigid body with flywheel attached), which is shown to possess horseshoe chaos in Koiller [1984,1985]. The perturbations of a gyrostat treated by Koiller [1984,1985] correspond to modulations of birefringence in the one-beam problem. In appendix A we show that modulations of the nonlinear susceptibility also lead to horseshoe chaos for the one-beam problem with birefringence, by showing analytically that the Mel’nikov function for this problem has simple zeros. In appendix B we determine Lyapunov stability criteria for equilibrium solutions of the one-beam problem, using the energy— Casimir stability algorithm of Holm et a!. [1985].For each parameter regime of physical relevance, we classify the fixed points (and invariant sets whenever degeneracies occur) of the system and their stability type. Moreover, we present sequences of figures showing how the phase portrait on S2 changes as material parameters are varied. These figures illustrate the types of bifurcations that take place and set the stage for understanding the rich variety of dynamical behavior exhibited in this problem.

2. Field amplitude equations To set the notation and state our assumptions, we begin by outlining the derivation of the electric field equations in the slowly varying amplitude approximation. The macroscopic polarization P(r, t) is

286

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

given in terms of the macroscopic electric field E(r, t) up to third-order nonlinearity by (see Shen [1984] and Bloembergen [19651) P,(r,

t)

=

f +

dt’

~~

1(t

—

t’)E1(r, t’)

J f J dt’

dt”

dt”

~k~(t

—

t’, t

—

t”, I

—

t”)E1(r, I’ )E~(r,t”)E1(r. t”),

where the subscripts denote Cartesian components and are to be summed over i, repeated. The Fourier transforms of the susceptibilities, x~1(w)=

f

(2.1)

j, k,

I

=

1,2, if

dt ~~(t) exp(iwt),

x~(w’,w”, w”)

=

f f J dt’

dt”

(2.2) dt”

t”, t”) exp[i(w’t’

~~t’,

+

w”t” +

which respectively characterize the linear and nonlinear response of the medium, are required to satisfy the involution identities *

x13(w) = x~1(—w),

(3)

,

,

X~Jk1(w w w ,

,,,

)

(3) =

XiJkI

*

,

(—w —w —w ,

,

),

(2.3)

for the fields E(r, t) and P(r, t) to be real. We further assume a lossless medium, and therefore also have [Butcher 1985, Shen 1968]

x~1(w)=x~(w),

X~k~(w,~

_w)x~*(w,w,

(2.4)

—w).

It is then convenient to write the linear susceptibility x13(w) as the sum of an isotropic and anisotropic part,

8~ x~1(w) x(w) 1+ x~(w),

(2.5)

where x(w) is one third the trace of ~~1(w)and is real by the first of eqs. (2.4). The traceless matrix x,~t)(w),which is hermitian by the same equation, describes the linear anisotropy of the medium and includes both birefringence and magnetic effects (optical activity). It is assumed that the linear anisotropy as well as the nonlinearities are small in magnitude, i.e.,

1~I, Ix~E2I 1

Ix~1

~

(2.6)

,

where E is a typical electric field amplitude. We consider an electromagnetic field which consists of counterpropagating quasi-monochromatic waves, perhaps with different frequencies, E(r, t) = e~(z,t) exp[i(k÷z

—

w~t)]+ e(z, t) exp[—i(kz + wt)] + c.c.

,

(2.7)

where e÷(z,t) and e_(z, t) are supposed to vary slowly on space and time scales defined by optical frequencies and wavelengths,

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

kI1~ôe_Iôz~,wi k1~ôe~!s9z(, 4i~8e~Iat~, wI’~ôe_Iôt~ ~

ej, e_~ .

287

(2.8)

The wavenumbers k÷,k and frequencies w~,w_ are related through the linear isotropic susceptibility, _

22

2

22

2

k~c=w+e(w÷), k_c w_e(w_), (2.9) 4irx(w). The running waves thus describe where as usual the dielectric constant defined as 1 + whereas the slowly varying amplitudes the linear isotropic component of the isresponse of s(w) the medium, e~ (z, t) and e (z, t) take into account the perturbation introduced by anisotropy and nonlinearity. It is assumed that our system is sufficiently far from any resonances so that the frequency dependence of and x~is very weak, or alternatively that the frequencies w~.and w_ are equal, so that we can set -

x~1~(w+)~~,1~(w_) =

~

(2.10)

,

(2.11) where the susceptibility coefficients x ~ and x are constants. Furthermore, we expand the linear isotropic component of the susceptibility about the carrier frequencies w~and w_, keeping only the first-order terms, x(w)~x(w+)+(w—w+)x’(w÷),

x(w)~x(w_)+(w—w_)x’(w_),

(2.12)

which, in effect, amounts to neglecting group velocity dispersion. Substituting (2.7) and (2.1) into Maxwell’s equations and keeping only first-order derivatives of the slowly varying amplitudes e+(z, I) and e_(z, t) now produces 1 ôe. t9e. —n +

—

2lTik =

+

[x~1 ~e~

3

1+ 3X~(e+Je+ke)+ 2e+Je_ke~l)],

1 de

—

. —

ôe.

=

2lTik ~

3) [x~~e_1+ 3X~kl(e_Je_ke_l + *

2e_Ie÷ke+l)1, *

(2.13)

where i, j, k and I label the two transverse components of e~and e_, the longitudinal components being negligible by assumptions (2.6). In obtaining (2.13) we have used the fact that x~is symmetric under exchange of j and k (intrinsic permutation symmetry, see Butcher [1985]and Shen [1968])and under exchange of i and 1 by virtue of (2.4). The group velocities are defined to be v.~~dw(k)IdkIk,

v_ ~dw(k)Idk~k,

(2.14)

and ~ e(w~),e e(cu_). If we change the independent variables to the following dimensionless local time coordinates: 2irk —

2irk

u +

~

~

(z + v_t),

~

u_ +

~_

(z

—

v+t),

(2.15)

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

288

the equations of motion become =

3X)Ikl(e+Ie+ke+/ * —i[x~(1) 1 e÷1+ (3)

=

—i[x~

(1)

3

+

2e÷Jekel)1 * (2.16)

(3) * * 3Xiikt(eie_kel + 2e_Ie+ke+l)].

e1 +

In what follows we take v~= v, k~= k, and e~= E’ in the light of (2.10) and (2.11). Under a change of polarization basis the amplitudes e~,and e, transform as spin-i representations of SU(2). It proves convenient to work with bilinear products of these field amplitudes, which transform as ~ = 1 ~ 0, that is, as a SO(3) vector with an invariant magnitude. We define such a Stokes vector for each of the two counterpropagating fields using the standard convention [Born and Wolf 1959] u~= e÷~J(~)Jke+k, t~= e~

(2.17)

1(~)1~e~ ,

where o-~,with i = 1, 2, 3, are the Pauli spin matrices. One can easily show that the magnitudes of the real, three-component vectors u and U are equal to their respective intensities, =

e1e~3,

i~ =

(2.18)

e~1e1.

These Stokes vectors describe the intensity and polarization of each field, but not their absolute phases. However, the phases can be deduced from eqs. (2.16) by quadrature, once the Stokes parameters are known. The correspondence between the directions of the Stokes vector u and the polarization is elegantly depicted on the Poincaré sphere in Stokes vector space (see fig. 2.1). Each point on the sphere defines a direction which corresponds to a particular polarization. All vectors u lying in the 1—3 plane represent linearly polarized light; rotating through an angle 0 in the 1—3 plane corresponds to rotating the plane of polarization through an angle 0/2 in real space. The directions 2 and —2 correspond to left- and 2

___

I

)

t—.~

I

L/

/ /

If!

I

I

.,..‘-‘

\ ~ /~ i 7 //j,i t / U \‘~/ r~ iiA /F \ \V\ \‘ ~“t—~-4~-____J__-—----I~ 7’ / /

‘V

3

Fig. 2.1. The Poincaré sphere.

I

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

289

right-handed circular polarizations, respectively. The remaining directions on the upper and lower hemispheres to left- and right-handed elliptical polarizations; moreover, opposing directions correspond to orthogonal polarizations. From (2.16) and (2.17), we deduce equations of motion for the Stokes vectors u and i~,

~

~

±u

(2.19)

where the vectors b, f1,~and fl,~are defined as

b~~=a+(Iu~+2Ii~I)c,~

1J,~=Wu,

fl~=Wu,

(2.20)

and the constant vectors a and c and the second-rank tensor W are obtained by contracting the susceptibilities with Pauli spin matrices, ak

=

,

,

(°k)J~x~J1~ C~= ~(°k)/1x~j~ii

W,,~,,= ~

.

(2.21)

These material parameters are real and W is symmetric; so one can always transform to a polarization basis in which Wis diagonal, i.e., W= diag(A

1, A2, A3), which we will assume in most of what follows.

The first terms in each of eqs. (2.19) involving the vector a represent linear anisotropy, as is clear from the first of eqs. (2.21). They lead to precession of the Stokes vectors about the direction of a. The second terms involving the vector c lead to precession at a frequency that depends on the two intensities ul and The vectors fl~,and Il~in the remaining terms can be thought of as nonlinearly induced ~

anisotropies that depend on the polarizations. The self-interaction terms [the middle terms on the right-hand side of (2.19)1 are those of a rigid body while the cross interaction terms [the last terms in (2.19)] comprise the SO(3) sigma model studied by Zakharov and Mikhailov [1987].

The definition (2.17) of the Stokes vectors is assumed to be in a linear polarization basis. This particular choice has no physical significance; but, with the basis fixed, we can identify the physical effects associated with the components of a, c, and W. These are a1

—*

birefringence,

c1

—~

induced birefringence,

a2

—~

optical activity,

c2

—~

induced optical activity,

a3

—~

birefringence,

c3

—~

induced birefringence,

W11, W~3,W33 W12, W23 W22

—~

—*

—~

(2.22)

linear—linear polarization interaction,

linear—circular polarization interaction,

circular—circular polarization interaction.

Material symmetries place certain constraints on these parameters, which we classify according to parity invariance and rotation symmetry C~,n = 1, 2,. ~ about the propagation axis, see table 2.1. Of course a given material can have more than one rotation symmetry depending on the propagation .

.

,

direction. The “2” components describe magnetic effects so those parameters with a single “2” index must vanish in parity-invariant media. See Tratnik and Sipe [1987a]for further discussion of the rotation symmetries C3, C~,n

5.

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

290

Table 2.1 Propagation axis symmetries

b

W ,WII

C. C,

nonparity invariant

(6, 6,, 6%)

W1, ~Wli

,

parity invariant

(6 0. h, )

)) WI,

W

12 W12 W, W22 W2, W11 (I B’2 ))

W12

W11~

nonparity invariant

(0. 6,, 0)

() W1, 0 W,, )) B1, (I B’,,,

parity invariant

(0, 0, 0)

B’11 0 14’,

nonparity invariant

(0, b,. 0)

W11

C4

~WII

C,. C~,n ~ 5

parity invariant

0 0 14’ 0 0

(0,0,0)

(I W2 (1

B1,

0

I) ~22

(I I) W,, 0

14 0 0

B1

In this paper, we shall be concerned with a special case of the general system (2.19). The case we consider is the coupled set of equations for two optical beams counterpropagating in an isotropic, parity-invariant medium, for which b is identically zero and W has two degenerate eigenvalues, W=diag(A1, A2, A1), 9u/~=fl~xu+2fl~Xu,

~üh~’=fl~xu+2fl~xu.

(2.23)

Furthermore, we consider travelling waves, u(ij,

~)

=

f)),

u(T(i~,

ü(1),

~)

=

ü(r(ij,

i)),

(2.24)

for which one can easily show that the magnitudes u~and üj are constant. The travelling-wave variable T is defined as (2.25)

T~r~/K + rrj!K,

where K and are constants and we have set r I ul, ~ jul. The full partial differential case is not treated here, but even in the travelling-wave case we find a rich variety of behavior. Equations (2.23) then become i~

=

(,~/i)(f2~ X u + 2fl~X u),

=

=

ü = (KIr)(fl~x ü + 2fl~x

where the dot denotes differentiation with respect to

T.

ii),

(2.26)

Equations (2.26) have a noncanonical

Hamiltonian structure with a Lie—Poisson bracket defined on the dual space of the direct-sum Lie

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

291

algebra so(3) c~ so(3), as discussed in the next section. Furthermore, due to the degenerate eigenvalues of W for isotropic materials (A1 = A3), this system has an additional ~1 symmetry. This 51 symmetry

2 The homoclinic and leads, via reduction, to areduced two-dimensional Hamiltonian system S heteroclinic orbits of this system (discussed in section 4) on are the kinksphere and solitary wave solutions of the original equations (2.23). We also demonstrate analytically in section 6 that under various perturbations which lift the degeneracy of the eigenvalues of W the integrability breaks down, giving horseshoe chaos.

Equations (2.23) are invariant under the parity and time reversal transformation, u 1(T)i—~u1(—r)

,

U2(T)~3u2(T),

t~1(T)I—*fi1(—T)

(2.27)

t~2(r)E—~—i,~2(—T),

u3(r)i—*u3(—r),

They are also invariant under the simultaneous transformations K’~K,

~

u’——u,

(2.28)

i~v-~—ii,

so without loss of generality we may assume K >0. Furthermore, since the equations are symmetric with respect to barred and unbarred quantities (and interchanging and ~)we may also assume I I K without loss of generality. Remark. Until now, we have been considering propagation in an infinite homogeneous medium; but the same equations describe polarization dynamics in an optical fiber, upon averaging over the transverse mode profiles [Crosignaniand Di Porto 1985]. These equations also occur in other contexts such as nonlinear couplers [Dainoet al. 1986] and phase conjugation [Trilloand Wabnitz 1988]. In appendices A and B, we treat the birefringent single-pulse equations (I i~I = 0), ,~

(2.29)

Ü=buXU+fluXU~

A stability analysis of the equilibrium solutions of the integrable system (2.29) is given there using the energy—Casimir method, and the one-beam problem with spatial modulation of the susceptibilities is found to be related to the problem of a rigid body with an attached flywheel, which has been shown to possess horseshoe chaos in Koiller [1984].

3. Lie—Poisson geometry and the phase space reduction to spheres

The equations of motion for u and ti, and in fact all of the dynamics for optical beam interaction, can be set into a convenient and natural geometric framework. In this section, we show that Lie_Poisson*) Hamiltonian formulations exist for both the one- and two-beam problems. ~ Let g and g* be, respectively, an arbitrary Lie algebra and its dual. The Lie—Poisson bracket oftwo real-valued functions on g, F and G is defined as (F, G} = (pS, [éFIôjs, ~9G/dtel), where ~eEg* and éFlép., c1Gh3~zEg. The bracket [ ] is the Lie product on g, and ( ) is a nondegenerate real pairing between g* and g- This definition is due to Lie [1893,vol. 3, ch. 25, sec. 115, fonnula (75)]. See also various articles mentioned in Marsden [1984]for further discussion of Lie—Poisson brackets. Such a bracket satisfies the defining relations of a Poisson bracket: it is bilinear, skew-symmetric, and satisfies the Jacobi identity.

,

,

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

292

As discussed in the previous section, a system of two counterpropagating optical beams has travelling-wave solutions satisfying the following dynamical equations for the Stokes parameters u and u: ti = (,~7F)(b~~ +

+

2fl~)x u,

i~= (KIr)(b~~ + fl~+

2fl,~)X ü,

(3.1)

where the terms within parentheses are defined in eqs. (2.20). These equations are Hamiltonian, with a Lie—Poisson bracket defined on the dual space of the direct-sum Lie algebra so(3) ~ so(3), with dual coordinates u and ü~.Explicitly, for F, G: so(3)* ~ so(3)* —* R, the Lie—Poisson bracket is given by*) -

i~

~G\

/c2F

K

-

foF

3G~

(3.2)

where K and are constants, and r = juj and F= al are constants of motion [cf. eq. (2.25)]. With respect to the Lie—Poisson bracket (3.2), eqs. (3.1) may be expressed in Hamiltonian form as ,~

F = {F, H)

(3.3)

,

by choosing F to be u and ü, in turn, and setting H equal to the optical energy, H°= ~

W~u

+

~

W~u

+

2ü~W u

+

u ~

+ i~.b~,

(3.4)

whose derivatives are defined through dH°= (buü +r

+

—1

W u + 2W~i~) du + (b~~ + W~ ii .

—

—--1

—

+

2W u) dü .

—

—

(c.u+2c’u)u~du+r (c’u+2c~u)udu.

(3.5)

The last two terms in (3.5) contribute nothing to the equations of motion since they produce vectors collinear with u and ü in the cross products in (3.2). Indeed, any function C of r and F is a Casimir (or distinguished) function for the Lie—Poisson bracket (3.2), i.e., {C(r, F), H)

VH(u, ii),

=

0.

(3.6)

That is, C(r, F) Poisson-commutes with every Hamiltonian expressible in the dynamical variables u and u. In particular, the factors of r and Fin the Lie—Poisson bracket (3.2) can be regarded as constant scale factors. The manifold

S~(direct product of two spheres in R3 x R3) obtained by setting r and F both equal to constants is determined by the Lie—Poisson bracket (3.2), since a point on .~rFremainson this manifold under the action of the Poisson bracket for any Hamiltonian. On ~rj’ as we shall see explicitly in a moment, symplectic coordinates can be introduced (locally) and the space R3 x R3 can be foliated by the symplectic manifolds, or “leaves” .I~,i.e. by the direct product S~x S~of spheres of radii r and F centered at the origin of each copy of R3. Mathematically, the manifold ~r,F is a coadjoint ~

=

X

*) In the Lie—Poisson bracket (3.2), u and ii are elements of R3 identified with the dual of the Lie algebra so(3), the cross product is the Lie product on R3 (R3 is its own dual), and the dot product is the pairing.

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

293

orbit of the group SO(3) x SO(3) acting on the dual of its Lie algebra. A theorem of Kostant, Kirillov,

and Souriau states that the coadjoint orbits are symplectic manifolds. The dimension of a symplectic leaf 1~rFis even, and is generally equal to four, except on the sets {(u, 0)) and {(O, i~)},where it equals two, and at the point (0, 0), whose dimension is zero. The property that a Lie—Poisson manifold may be foliated by symplectic leaves is also discussed in Marsden and Weinstein [1974]and in Holmes and

Marsden [1983]. Remark on the one-beam problem. The dynamics of a single Stokes pulse is also describable as a Hamiltonian dynamical system. (It is the subcase of the two-beam Hamiltonian system with ii absent.) To be more specific, the one-beam system is defined on the Lie—Poisson manifold so(3)* with the

following bracket: (3.7)

and the following Hamiltonian function: HO:so(3)*~~÷R, u~b~u+~uW~u,

(3.8)

where W is the diagonal matrix diag( A1, A2, A3), and the vector b = a + rc defined in terms of constant vectors a and c depends linearly on the intensity r = ul. In components, the equations of motion for the one-beam problem are

U2 =

(A2 — A3)u2u3 + b2u3 (A3 — A1)u3u1 + b3u1

U3

(A1 — A2)u1u2 + b1u2

=

=

—

b3u2, b1u3,

—

b2u1.

—

(3.9)

As with any Hamiltonian function of u, the one-beam system (3.9) restricts to a system on the symplectic sphere .Z,. (the Poincaré sphere). We can coordinatize this by transforming to spherical coordinates, namely u1 = r sin 0 sin u2 = r cos 0, u3 = r sin 0 cos 4. The induced Hamiltonian function and Hamiltonian vector field in these coordinates are ~,

2(A

20 sin24

1sin

H°= ~r

+

A

20 + A 2 cos

2O cos2~)+ r sin 0 (b

1 sin ~

3 sin

+

b3 cos

4’) + b2r cos 0, (3.10)

XH0(rsln0)

—1

0

0

(H~t90—H~ô~),

and the equations of motion on

o

=

=

b1 cos b2

—

4’— b3 sin 4’ + (A1

(b1 sin

(3.11)

are —

A3)r sin 0 cos

4’ sin 4’, 24’ + A

4’ + b3 cos 4’)cot 0— r(A1 sin

24’ 3 cos

—

(3.12)

A

2) cos 0.

Note that the system (3.12) is completely integrable, since it is a Hamiltonian system defined on a

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

294

two-dimensional phase space, hence is formally a one degree of freedom system, possessing a conserved energy. The one-beam problem is discussed further in appendices A and B. Further reduction of the two-beam problem due to isotropy (A3 = A1). In (3.4) W is the diagonal matrix A1

0

W= 0

A2

0 0

0

0

A3

(3.13)

.

11 induced by the Lie—Poisson structure is given via The Hamiltonian formula (3.2) as vector field corresponding to H XHO

= {.,

H°)= (i~/F)(VH° X u)

.‘~7

(K/r)(VH°x

+

(3.14)

i~).‘~7

and the associated equations of motion are expressed in Hamiltonian form by a = {u, H°)= XHO(u)

ü~{i~, H°}= X~s(i~)

For an isotropic parity-invariant medium (a

0

c, A 1

Li2

=

(,~IF)[(A7 A1)u2u3

=

2(i~IF)A1(u1t~3 u1i.~1)

—

—

A5)u1u2

(i~/F)[(A1

=

(,/r)[(A.

t12

=

2(K/r)A1(I~1u3 i~3u1)

U3

=

(,c/r)[(A1

—

—

(3.16)

2A

A~)ü~ü32A1ü2u3 —

—

A1) these equations read, in component form,

,

2A1u1t~2+

=

—

=

2A1u2i.~3+ 2A2u1ü~2J,

—

—

(3.15)

.

,

+

1u2L~~I 2A2i~3u5], (3.17)

,

A2)ü~ü~2A2ü~u2+ 2A1ü~u1J. —

The reduction theorem of Marsden and2 xWeinstein implies that~ themay dynamical system (3.1) S2. These [1974] symplectic leaves be assigned the usual restricts the symplectic leaves = S spherical tocoordinates: (0, 4’; 0, 4’)~r,F E ~~rF’ using the axes u, and u

2 as the polar axes,

=

r sin 0 sin

4’,

ü1

=

F sin 0 sin 4’,

u2=rcosO.

ü2=FcosO,

u3=rsinOcosçb,

ü~=Fsin0cos4’.

(3.18)

The Poisson structure induced on the symplectic leaves may be calculated directly by transformation of variables starting from (3.14) and using (3.18) in the chain rule to yield the following (local) expression for the Hamiltonian vector field X110: ~

(3.19)

D. David et aL, Hamiltonian chaos in nonlinear optical polarization dynamics

295

The Lie—Poisson bracket induced on ‘ri’ is symplectic, with symplectic pairs (4’, K cos 0) and (4’, i~cos 0). For an isotropic parity-invariant material, a = 0 = c, A1 = A3, and the Hamiltonian function H°in (3.4) may be expressed in terms of these new coordinates as H°(0,4’; 0,

2(A 20 + A 2O) + ~F2(A 20 + A 20) ~r 1sin 2 cos 1sin 2 cos + 2rF[A 1 sin 0 sin 0 cos(4’ 4’) + A2 cos 0 cos 0].

4’) =

(3.20)

—

From this expression, it is now a simple task to derive the equations of motion on Note that only four equations are needed, since r and F are invariant quantities. These equations are ~

o

= =

4) =

XH0(0)

=

—2A1,~sinOsin(4)

X~o(4)) =

—(A1

—

Xj~o(4))= —(A1

—

—4)),

0= X~o(0)= 2A1K sin 0 sin(4’

A2)(i~rIF)cos0 —2i~[A1cot Osin Ocos(4)

—

4,)

—

—

4)),

A2 cos 0],

(3.21)

A2)(KF/r) cos 0— 2K[A1 cot 0 sin 0 cos(4) —4’)— A2 cos 0].

The two-beam system of equations (3.21), defined on ~ admits yet a further reduction for the isotropic material we are considering, with W= diag(A1, A2, A1). To see this, notice that the quantity o-=Kcos0+KcosO

(3.22)

is a constant of the motion for the system (3.21). The vector field associated with this quantity, (3.23)

X~= (Ki~IrF)(9~1,+ ~)‘

represents a simultaneous translation of the azimuthal angles 4’ and 4’ by the same amount. Under this transformation, the Hamiltonian H°in (3.20) is invariant, since it depends on the two variables 4’ and 4’ only in the invariant combination 4) 4). Using2,the the dynamical system conjugate (3.21) reduces to a (w, set a)ofand equations on the plusS’ansymmetry additional (3.23) quadrature. The canonically quantities (o-, f3) defined sphere S by —

w=Kcos0—KcosO,

o-=KcosO+Kcos0,

a=4)—4’,

/3q5+4),

(3.24)

provide coordinates for reduction of the system (3.21) to a cylinder. In order to reduce to a sphere, we replace w by another angular variable 4’; see fig. 3.1. The definition of this angle depends on the relative magnitude of o- with respect to those of —

(1)

~

K

.

—

K.

=

cos4’

K

and

i~

as follows:

I,~I(1—cosO)—IKI(1—coso)= Ialsin2(012)—IKIsin2(012) al(1—cosO)+IKI(1—coso)

Ialsin2(012)+IKIsin2(0/2)’

(3.25a)

(2)IKI—I,~Io’I,~I—IKI:cos4i=cos0,

(3.25b)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

296

Cs)

‘P

R

Rcos’41

2R

Fig. 3.1. The geometrical relationship between the cylindrical variable w and the spherical coordinate ~i.

(3) ~jKl—IKI: cos4’= IKI(1+coso)—IKI(1+coso) IKl(1+cOsO)+IKk1+cosO) -

=

2(O/2)—j,~jcos2(0/2) jKIcos KIcos2(0/2)+lKjcos2(0!2) (3.25c)

There are other definitions for the angle 4’, which could imply that the dynamics would still take place on a sphere, but these alternatives would in general restrict the dynamics to a polar subinterval of [0,ir]. We choose as our new variables a-, 4’, a, and /3 as defined in (3.24) and (3.25), and reduce to a sphere coordinatized by the angles 4’ and a. In terms of these new variables, the Hamiltonian function may be rewritten as follows. First, to condense the notation, we introduce the quantity w w(4’), given as (1) a-I,~j—IKI: w=~Kj—i,~j+(iKl+i,~j—a-)cosçb, (2)

IKI—lKj

jKl—IKI:

w=—u+2jKjcos4l,

(3.26a) (3.26b)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

(3) o~IKI—l,~I:w1KI—IKI+(IKI+IKI+o)cos4J, where cos

4’ is given

297

(3.26c)

by formulae (3.25); thus, in all cases, we may write co in the form

ww0+Rcos4i,

(3.27)

where ~ and R are given in (3.26). We also introduce the quantities 2 (a0(4’) = f(4,)f(4’), (3.28) f(41) = \f4,c~ (a- + f(tp) = ~4 where w = rt(4’), as prescribed by formula (3.27). By substituting relations (3.24) and (3.26) into the —

—

—

)2

)2

Hamiltonian function (3.20) and using (3.28) we obtain the following reduced Hamiltonian: H°= ~A 2+ F2) + k(A 2(u+ w)2 + (FIi~)2(a-— )2] 1(r

2

+ ~(rFIKi~)[A

—

—

A1)[(rIK)

w2) 2 +A

2(u

1 0(4’) cos a].

(3.29)

The vector field associated with this reduced Hamiltonian is (with H~= 0) XHO

=

~

4’)(H°ad~ H~a + R sin 4, H~ô~). —

(3.30)

The equations of motion on the reduced sphere are then

4, = =

—A1Q(4’) sin aI(R sin 4’), —2A2w + ~(A2 A1)[(rI~IFK)(a- + w) —

—

f3

=

(3.31)

A1[(flf)(u

+ w) —

(flf)(a-

—

—

(FKIr,~)(a- w)] —

w)] cos a,

(3.32)

2A2u + ~(A2 A1)[(rI~IFK)(a- + w) + (FKIr,~)(a- w)] —

—

A1[(flf)(u + w) + (flf)(u

—

—

w)] cos

a.

(3.33)

Note that the right-hand sides of these equations do not depend on the variable /3. Thus, the system (3.21) has been reduced to dynamics on a sphere, plus an additional quadrature for the variable /3, to be performed once a and t/i are known; in other words, we have separated the variables and are now dealing with a two-dimensional subsystem. This is usual when reducing a Hamiltonian system: any (continuous) symmetry of the Hamiltonian yields a reduction by one degree of freedom, i.e. by two variables. Equations (3.31) to (3.33) for the two-beam problem will be used in the next two sections, where we study special orbits on the reduced phase space, i.e. on the sphere ~2 with azimuthal and polar angles a and 4,, respectively. Later, in section 6, the equations for the full system will be shown to have solutions that exhibit horseshoe chaos and, more generally, show higher-dimensional chaotic behavior. We conclude this section with the following remark. The reduction scheme discussed above can be

depicted by the following sequence of mappings:

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

298

2

c

x C2

~ H~ Stokes restriction S3 x S3 ~HH x s2

Hopf reduction

P reduction via S’ symmetry

The initial configuration space is the eight-dimensional manifold C2 X C2, whose coordinate functions are the components of the complex electric field amplitudes e~and e. The Stokes map H~projects the initial manifold to the six-dimensional manifold S3 x S3 this projection suppresses the absolute phases of the electric field amplitudes and uses conservation of their magnitudes, I e j and j e j. (The projection or restriction [I~ could also be naturally viewed as a mapping ~l~’ X S3 X S3 C ~ x where is the algebraic field of quaternions.) Then, the Hopf map HH brings the system down to the fourdimensional symplectic manifold S~x S~.The combined map HH TI~can be regarded as reduction of C2 x C2 with respect to S1 x ~ (the conserved quantities are the radii r and F). Finally, reduction by the S1 symmetry associated with the conserved quantity a- takes the symplectic system on x S2 to the symplectic reduced sphere S2, plus an additional quadrature for the 5’ angle variable /3, conjugate to a-. One interesting caveat remains: the map P given in (3.24) to (3.26): x is not everywhere differentiable. As we discuss in section 4, P is a mapping x S2 52r where is the manifold obtained from S2 by removing one or both of its poles. Alternatively, we could view P as a map of x S2 onto considered as a manifold with singularities. This viewpoint does not yield any particular difficulties with the dynamical analysis. The singular set in fact consists of one or two single points, namely the poles, in the cases treated in section 4. Inconnection with this matter, it is worthwhile, however, to jump ahead for a moment and examine the geometrical origin of these exceptional cases. As will be seen in section 4, these situations will be characterized, at the level of phase space portraits on by bifurcations of an unusual kind involving cusped closed curves. Analysis shows that these cusped curves consist of two components. In two of the cases in section 4, we have a single homoclinic ioop connected to a pseudo saddle point; the third case is a limit of the first two, for which we have two heteroclinic orbits connected to two pseudo saddle points, forming a homoclinic 2-cycle. It turns out that these pseudo saddle points behave like normal saddle points, except for having two branches instead of four. The disappearance of two of the branches occurs via the mapping P and is related to the magnitude of the conserved quantity a-. Other situations exist in which these sorts of cusped bifurcations are associated to singularities; see, e.g., Patrick [1989]. We end this section by showing that in reducing our system from C2 x C2 to no loss of information has occurred, even though we have eliminated six out of eight dimensions. (For a discussion of the general theory of Poisson reduction, see Marsden and Ratiu [1986].) -

+

~‘—*

~,

~‘

°

~2

~2

~2

~2

~2

~2s

~2

~2,

~2

Reconstruction Lemma: The solutions of (3.31) and (3.32) on travelling-wave solutions of the system (2.16) on C2 x C2.

~2

determine, up to quadratures, the

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

299

Proof. The converse statement is the reduction procedure explained earlier in this section. To prove the direct statement, we show how to reconstruct e~and e_. Assume that (3.31) and (3.32) are solved for 1/i and a. Then eq. (3.33) can be integrated to give /3. Recall that cris a given constant of the motion. 2. The passage to Then, using (3.25) and (3.26), eqs. (3.24) can be inverted to give 0, 0, 4’ 4~~ X S S3 x S3 consists of a simple embedding determined by specifying r and F (which are given constants of the motion as well): we thus invert eqs. (3.12) to obtain the Stokes vectors u and ii. If we write the electric field amplitudes as ~2

e~= =

[Ije+~Iexp(iz.t~)+ 9Ie+~Iexp(—i4÷)]exp(i~1’÷), (3.34)

[Ije_~Iexp(izl_) + Me_NI exp(—iz.1_)] exp(i~I’_),

where and ~I’~are the relative and absolute phases, respectively, then knowledge of the Stokes vectors determines the moduli of the amplitudes and relative phases through the relations ~±

=

~f(juj + u 3)12,

Ie+~j= \J(jul

—

u3)12,

e~l= V(Ial + u3)12, (3.35)

___________

je_~I=V(Ial—i,~3)/2,

~ =—~arctan(u2Iu1),

& =—~arctan(ü2/ü1),

and it remains to integrate the travelling-wave form of (2.16) to determine the absolute phases ‘I’÷and ‘p. As mentioned before, the reconstruction procedure is explained in Abraham and Marsden [1980], and Marsden et al. [1989]. Remark. Mathematically, the system (3.16) and (3.17) represents Hamiltonian motion on the group SO(4), for which general integrability conditions (i.e. conditions expressed in terms of the matrix W) are given in Veselov [1983] (see also Reyman and Semenov-Tian-Shanskii [1986]). These general conditions include the present case of isotropic materials, of course, and provide extensions of this case which remain to be studied.

4. Fixed point analysis and bifurcations in the two-beam problem In this section we describe the dynamics of the two-beam problem on the reduced phase space Specifically, we locate the fixed points of the system (3.31), (3.32) and determine their stability as a function of material parameters. Of course, eq. (3.33) for the additional variable /3 must be integrated after determining the motion on the reduced 52 in order to lift the dynamics to the full phase2 will spacenot P given by the product of the two Poincaré spheres. Consequently, fixed points in the space S necessarily be fixed in the space P. In the case where the critical points are saddle points, the separatrices connecting them (the heteroclinic and homoclinic orbits) correspond to special types of solutions, which behave like solitary waves and kinks. Such solutions are discussed in detail in the next section. To determine the fixed points and the qualitative behavior on S2, it proves advantageous to split the analysis into several cases. In effect, we will consider situations when either a- = 0 or a- ~ 0 on one hand,

300

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

‘~I

and either Kj = lid or KI ~ on the other hand. The reason for this splitting is to identify special cases. For instance, setting a- = 0 defines a subspace of the parameter space for which the dynamics is richer than in the more generic situations, and is also qualitatively different, leading to what we call a butterfly bifurcation, in which a half great circle of fixed points appears. In what follows we will always assume that K >0 and I K unless we state otherwise. In view of the symmetry (2.28) and the symmetric nature of the system (2.26) with respect to barred and unbarred quantities, these assumptions cause no loss of generality. We also introduce the quantity L A2/A1, which will be the bifurcation parameter. To find the stability properties of a given fixed point, we linearize the equations in its neighborhood. In the neighborhood of a critical point (4,, d), define

çb=4’+sg,

a”a+eh,

(4.1)

and write the equations of motion (3.31) and (3.32) in the form

4’

=

—A1 12(4,) sin aI(R sin

a

=

—2A2w

—

~(A, +

—

4’)

P(4’i, a),

A2)[(rl~IFK)(a- + w)

w)

—

(f/f)(a-

—

(FKIr1~)(a- w)]

—

(4.2)

—

w)] cos a

Q(4i, a),

where ~u2 and the quantities f and fare defined in formulae (3.28). Then the following system governs the linearized dynamics of g and h:

(~ (g~(8P/o~4i 9P/ôa~(g )M~h)~Q/o4, ôQ/ôa)~h

43 ‘

where the stability matrix M is evaluated at (~‘,i) and has entries ~PId4i= —A, sin a [(a- + w)(f/f)

—

(a-

—

w)(f!f)] + A1 ff sin a cot t/i/(R sin 4,),

~9P/t9a= —A1ffcos a/(R sin 4,), cQId4’ = A1R sin 4, {—2F +

(4.4b)

2(f/f3)

[f/f + f/f+ (a- + w)

+(a-—w) (fIf~)+2(a-—w)/ff]cosa}, 2

c9Q/da

=

3

~‘

A 1[(u + o4(f/f)

—

(4.4a)

(a-

—

2

—

w)(flf)] sin a,

(4.4c) (4.4d)

and to, f, f are defined as before through (3.26) and (3.28). The quantity T in (4.4c) is given by f= ~(L

—

l)p

—

L,

p = (rl~IFK)+ (FKIrl~).

(4.5)

The nature of a given critical point is then found by examining the spectrum of the stability matrix M in (4.3). This stability analysis could also be performed using the energy—Casimir method as we do in appendix B for the one-beam problem.

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

301

4.1. Case1.cr0,I,~IK Using formulas (3.25b) and (3 .26b), the equations of motion (4.2) become

4’ = —2A~sin a sin 4,,

a

=

—4A1K(cos a

—

F) cos

4,.

(4.6)

As usual, the fixed points are determined by requiring that the right-hand sides of (4.6) vanish. This gives us the following list of critical points:

4’ =

‘rrI2,

a = 0;

(4.7a)

4J~r1T/2,

a7T;

(4.7b)

çfr=0,

cosa=F;

(4.7c)

4i=ir,

cosa=F.

(4.7d)

I

Naturally, the last two points exist only if IF ~ 1, since a must be real. Special subcases of the first two cases arise when F = 1, —1: then 4’ is allowed to be arbitrary in (4.7a) and (4.7b), respectively. These situations correspond to an interesting bifurcation, examined below, which we call the “butterfly bifurcation”. The matrix corresponding to the critical point (4.7a) is

( 0 —2A1K\ M_~4AK(1F) 0 ~

(

4

.8)

with eigenvalues R~= ±2A1K\f~J(T— 1).

(4.9)

There are three possibilities to take into account. If F < 1, then the eigenvalues are purely imaginary and thus (4,, a) = (irI2, 0) is a center. If F> 1, then the roots are real, so that the fixed point is of saddle type. Finally, if F = 1, then we encounter a degenerate case: the matrix M becomes nilpotent. This corresponds to a continuum of fixed points distributed on the half great circle defined by a =0. Indeed, setting F = 1 in the equations of motion (4.6) makes it clear that a = 0 is a sufficient condition for the right-hand sides of (4.6) to vanish, independently of the value of the second variable 4’. For the next fixed point condition, we see that the matrix corresponding to (4.7b) is

( 0 M_~4AK(1+F)

2A1K\ 0 )‘

(4.10)

with eigenvalues R~= ±2A1K1/—2(F+ 1). We again have three cases. When F> —1, this point

(4.11)

(4’, a) =

(‘n~I2,ir) is a center. It is a saddle point

302

[).

David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

whenever F < —1. The case F = —1 is a degenerate one and is associated with a half great circle of fixed points given by a = iT. Let us now turn to the two last fixed points, which are the poles of the sphere. For the north pole (4.7c), we have the diagonal matrix /_2A1KV~F2

M=~

o

o 4AKV1_12

(4.12)

.

Thus the eigenvalues are the elements on the diagonal; note that they are multiples of each other. We also have three cases here. First, if F <1, then the eigenvalues are real valued and the north pole is a saddle point. Here, though, we must be careful. This first case is one for which our reduction is not to S~ but, as explained at the end of section 3, to the singular manifold obtained from it by transforming its poles into cusp points. In this situation, the poles do act as saddle points, although only two heteroclinic orbits connect them, thus forming a homoclinic 2-cycle. Secondly, if F> 1, then the poles are centers. Thirdly, we again have a degenerate situation when F takes the values ±1. This degeneracy is associated with the curves of fixed points that we described above for the first two critical points, in the sense that the north pole is one of the extremities of that curve of fixed points. Finally, for the south pole, the matrix M is just minus that for the north pole. The south pole is therefore characterized exactly as the north pole is, i.e., both of the poles are saddle points when IF <1, are centers when F> 1, and are degenerate (to all orders!) when F= ±1. The butterfly bifurcation. As an illustration of the bifurcation sequence corresponding to this case, we choose F= rand i~= i; then p =2 and F= —~(L+ 1). Note that the values F= ±1correspond to L = —3 and 1, respectively. The sequence of pictures in fig. 4.1 shows the evolution of the level surfaces of the Hamiltonian on the reduced phase space as the parameter L is varied. When L—* (F—* +x), the poles are centers and the2,equator is a circle of fixed points (see fig. 4.la). For L ~ —3 (F 1) and 71-) is a center; so are the poles, which are enclosed by two homoclinic finite, connected the point (4,, = (IT! point at (4,, a) = (ir/2, 0) (see fig. 4.lb). As L approaches the value —3 orbits to a) a saddle from below, the homoclinic orbits and the periodic orbits near them become more oblong and approach a half great circle (shown in figs. 4.lc, d, e). The first butterfly bifurcation occurs at L = —3 (F 1) when the homoclinic orbits collapse into a half great circle of fixed points given by a = 0 (see fig. 4.le). As L increases above —3, the point (4,, a) = (ir/2, 0) transforms into a center and the poles become saddle points, now connected by two heteroclinic orbits, these last being half great circles at some azimuthal coordinates a = ±~, depending on the value of L; the point (ir/2, ~) remains a center (see fIg. 4.lf). As L is further increased the two heteroclinic orbits rotate azimuthally towards a = IT (shown in figs. 4.lg, h, i). This angle is, in fact, reached when L = 1 = —F, at which point a second butterfly bifurcation, identical to the first one but with the order of events reversed, takes place. Thus, the heteroclinic orbits collapse together to form a half great circle of fixed points at a = IT (see fig. 4.lj). When L >1 (F< —1), the poles become centers again and the point (IT!2, IT) becomes a saddle point to which are connected two homoclinic orbits encircling the poles. The point (IT!2, 0), for its part, remains a center (see fig. 4.1k). Thus the topology of the portrait for F < —1 is identical to that for F> 1 but for a rotation of a by IT. As L ~ (F—~—Do), the two homoclinic orbits collapse together onto the equator to give the same portrait as that obtained when F—* ~, but with velocities reversed on all orbits. Diagram 4.1 depicts the bifurcations just described for this case. —~

—p

D. David el a!., Hamiltonian chaos in nonlinear optical polarization dynamics

Orientation

a. L =

c.L=-3.4

d.L=-3.04

-9

/

_________

f. L = 2.96

g.

i.L=0.6

b. L =

303

-5

e.L=-3

i

\,

L = -2.7

h. L = -1

j.L=1

Fig. 4.1. Evolution of the reduced phase portrait as the parameter L

=

A

2/A1 is varied, for case 1 (the butterfly bifurcation); r = F,

K = i~, U =

0.

N

p

____

L=-=

L=-3

L=l

L==

Diagram 4.1. Bifurcation diagram for case 1. The thick plain and dotted lines represent centers and saddle points, respectively, and the thin plain line represents curves of fixed points. N = (0, r, 0), S = (0, —r, 0), F = (0,0, r), B = (0,0, —r).

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

304

4.2. Case 2. a-0, Il~HK For this case, the equations of motion take the form

4’ =

—2A1 KV(K!K)2

24, sin a, —

cos

(4.13)

á=2A 1K{2F—[F(4i)+1!F(4i)]cosa} cos4i, where 2 cos24i]!sin24i.

F(4i)

~f!f= V[(i~!K)

(4.14)

—

Clearly, the right-hand side of the 4, equation in (4.13) can only be allowed to vanish for sin a = 0; requiring that the square root vanish would make the a equation singular. The a equation can only be stationary, if either 4, = 71-12, or the expression within the curly brackets in (4.13) vanishes, F2 + 1

=

±2FF.

(4.15)

The solutions to (4.15), for a = 0 and F+=F±VF2—1,

IT,

are, respectively,

F~—1,

(4.16a)

F=—F±VF2—1, F1.

(4.16b)

Substituting these into (4.14) we obtain an expression for the polar coordinate of the fixed point, cos

4’ =

±\/~2

—

(,~!K)2]!(F2—1),

sin 4, =

~[(~!K)2

—

1]/(F2 —1).

(4.17)

Note that these solutions are well defined so long as

Fl

F2> (~IK)2.

1,

(4.18a, b)

In view of condition (4.18b), let us reexamine (4.16). For a = 0, F.,. are monotonically decreasing and increasing functions, respectively, on the interval F ~ —1. Thus, in order to be compatible with the requirement that J ,~l> K, we must reject one branch of F. Specifically, we must keep only F = F (F2 1)1/2. Similarly, for a = IT, it turns out that the only acceptable choice is F = —F + (F2 1)1/2. Furthermore, (4.18b) implies, using (4.15) and the above considerations, that Fmust satisfy the inequality —

—

—

Ft

(K2 +

I~2)I2KI,~I y,

(4.19)

i.e., these fixed points exist only when I F~exceeds the ratio of the arithmetic mean to the geometric mean of K2 and ,~2 We therefore have the following list of fixed points: a0,

çli=irI2;

(4.20a)

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

alT,

4i=irI2;

a

4,=arcsin \/(~II~_1 =arccos

a

0,

= IT,

4’ = arcsin

305

(4.20b)

\/(~!Ic)~—

1

=

F=F—\/F2 —1; ~j’~2’~2,

arccos ~jF2_(I~IK)2

F= —F + \1F2 —1.

(4.20c)

(4.20d)

Note that (4.20c) and (4.20d) each represent pairs of fixed points (since, e.g., arccos is a bivalued function), and that these pairs cannot exist simultaneously, i.e., if the critical points given by (4.20c) exist then those specified by (4.20d) do not exist, and vice versa. Let us now examine the nature of the critical points in (4.20). For that given by (4.20a), the matrix Mis

(

—2A

0

1Ii~I\

M= ~,—2A1K[2F—IKIlK KIIIdI]

0

—

(4.21)

,,~

and has eigenvalues R

=

±2A1\1~j~j V21— 11(1/K

—

KIIKI.

(4.22)

Hence, the point (i/i, a) = (irI2, 0) is a saddle point as long as F> y, and a center otherwise. For the fixed point given by (4.20b), the matrix M takes the form

(

M= —2A1K[2F+

II~+ KIIKI]

2AJKI)

(4.23)

1K11K + KIIKI).

(4.24) y, and a center otherwise. Now, for

with eigenvalues

R

=

±2A1\1~j~j\/—(2F+ 2,

Thus, (4.20c),thewepoint have (4,, a) =

/

2

IT)

is a saddle point when F <

0

—

—

(IT!

1)[(kIK)2

M = ~ —2A1 K(F

—

F2] IF3 sin 4’

—2A

)

0 1KFsin4,\

‘

(4.25)

with eigenvalues R~= ±(2A

2 1)[(,~IK)2 F2]. (4.26) 1KIF)\I(F These values are purely imaginary, since F2 > (i~IK)2> 1. Therefore, the pair of fixed points determined by (4.20c) are centers. Finally, for the other pair given by (4.20d), we find that M is just minus what we had for the preceding pair. It then follows that we find exactly the same eigenvalues and therefore the very same conclusion can be drawn, namely that this pair of fixed points is, whenever it exists, a pair of centers. —

—

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

306

>~\

(~

~

-1 ______

L_~~

Orientation

a. L

= -32

b. L =

-8

(~, ~

/

c. L = -4.4

e. L

d. L = -3.62

I

=

3.5

~

~H1

/

~ ,,~

f.L=-2

\~N

g.L=1

h.L=4

~Th /

~

L~:T~ .-

i.L=t()

j.1.=28

Fig. 4.2. Evolution of the reduced phase portrait as the parameter L is varied, for case 2; r = 2,

,—

4.

K

1, s~= 2.

=

U =

0.

As an illustration (see fig. 4.2), let us choose i~= 2, K = 1, F= 4, and r = 2. We then have p = 2, y = + ~ and F = ~(L + 1). When L—3 the poles are centers and the2,equator constitutes 0) is a saddle point an to invariant of fixed points. When L < orbits ~ (Fenclosing < the a)type = (IT! which arecurve connected a pair of homoclinic thepoint centers of (4.20c), and (IT!2, IT) is a center (see fig. 4.2a). This triplet (saddle + two centers with two homoclinic orbits) then proceeds to collapse into a single center as the value of the parameter L is increased towards ~(shown in figs. 4.2b, c, d). At this value of L, a bifurcation occurs which transforms this triplet into a single center at (IT!2, 0); the fixed point at (IT!2, IT) remains a center by itself (see fig. 4.2e). This situation, i.e. two centers at (IT!2, 0) and (IT!2, IT), persists until L reaches the value L = (F= when a second bifurcation takes place. This bifurcation transforms the center (IT!2, IT) into a new triplet, in all respects similar to the first one, except that the saddle point is now located at (IT!2, IT) (see figs. 4.2f, g, h). As L becomes large and positive, the two centers of type (4.20d) inside the homoclinic orbits connected to that saddle point are seen to slowly migrate towards the poles (see figs. 4.2i, Asymptotically, when L—* (F—~—rDo), we again have two centers at the poles as the equator becomes a degenerate curve of fixed points; note, however, that all velocities are now opposite to those for F—i’ These bifurcations are summarized in diagram 4.2. —

—Do, —

— ~),

(4’,

—

~

~),

j).

Do~

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

307

N

F

B

L=-=

L=-7/2

L=3/2

L==

Diagram 4.2. Bifurcation diagram for case 2. The thick plain and dotted lines represent centers and saddle points, respectively, and the thin plain line represents curves of fixed points. N = (0, r, 0), S = (0, —r, 0), F= (0,0, r), B = (0,0, —r).

4.3. Case 3. a-0, I,(I=K,r=r The equations of motion for this case are A sin a

4’(2K—Ia-I)sin4’~

I

—

a__2A

(a-+w)

K

(a-w)

14K

\

(4.27)

(4K2+u2—w2)cosa

1

2 V4K2 ( w)2~ ito, V4K2 (a- + to) where to (2K Ia-I) cos 4’. Clearly, we can only allow the 4’ equation to vanish when sin a = 0. From the a equation, we see that there are two ways to make this equation stationary. The first is to set to = 0, i.e., 4, = ITI2 [if b] 2K, then, in view of eq. (3.26a), to is forced to be zero when I~I = K]. The second alternative is to require that the square bracket be zero,

1[_F+

—

—

—

—

—

(a- + to)2 V4K2

where ± refer to a = 0 and a for to, whose solution is to

=

±(a-2+ 4K2

—

=

(a-

IT,

— to)2

F ± (4K2 + a-2

—

to2)

=

0,

(4.28)

respectively. Squaring this expression leads to a biquadratic equation

±4a-KF/~/F2— 1)1/2,

(4.29)

where the two ± signs are independent. Now, the term 4K2 + a-2 to2 in (4.28) is positive since ItoI S 2K. Hence, for a = 0 or a = IT, F is forced to be positive or negative, respectively, in order for (4.28) to have solutions. Let us then consider expression (4.29) for to and use the fact that I I is bounded by zero from below. Requiring that the radicand vanish in (4.29) and recalling that K >0 then forces the choice of + or from the ± sign in the square root for each of a =0 and a = IT. Omitting details, we finally obtain to in the form —

—

D. David et al., Hamihonian chaos in nonlinear optical polarization dynamics

308 to =

±(a-2+

4~2 ~

4Ia-IKF!VF2

—

(4.30)

1)l2,

where the ~ refer to a = 0 and a = IT, respectively. Recalling that the 4, coordinates of these fixed points as /

~,a-+4i t/i=arccos1,~± 2

/ 2 ±4a-KF!vF 2K—Ia-I

2—

to = (2K

—

1)\t12 \

).

a-I) cos 4,, we can write

(4.31)

Let us now determine the nature of the fixed points which we have obtained for this case. As all the fixed points are characterized by sin a = 0, it follows that the stability matrix of the system will, in all cases, adopt an antidiagonal form. First, for (4,, a) = (IT!2, 0), the matrix M is 0 M— ~—2A1(2K

—

Ia-I)[F

(42

-A

2)] +

a-2)!(4K2

—

0 1(2K+uI)~

a-

(4.32)

The eigenvalues of this matrix are R

±A

=

2a-2) \/F

—

—

(4K2 +

a-2) !(4~2 a-2).

(4.33)

—

1~2(4K Note that the first square root is always real valued since, for this case, a- = K cos 6 + cos U immediately implies that Ia-H 2K. Hence, (IT!2,O) is a saddle point whenever F>(4K2 + a-2)!(4K2 a-2) (hence F must be positive); otherwise it is a center. For the next critical point, (IT!2, IT), we have the matrix ,~

—

/

0

M= ~—2A1(2K The eigenvalues are

—

R

Ia-I)[F+

A 2)!(4K2

—

1(2K+Ia-I)\ 0

a-2)]

(4.34)

(4K2 + a-

2) \/—[F + (4K2 + a-2)!(4K2 a-2)]. (4.35) aThus, this fixed point is a saddle point whenever F< —(4K2 + a-2) !(4i2 a-2) (hence F must be negative); otherwise it is a center. Consequently, the two points (IT /2, 0) and (IT / 2, IT) always form a saddle—center pair. Let us now examine the other critical points which occur when a = 0, IT, and 4’ is given by (4.31). Using the quantities f, f, for a = 0, the stability matrix M takes the form =

—

±A

1\/~~jK2

—

—

/

2to2 sin

M = ~64A1Ka-

0

4’ (a-2 + to2 + 2K2)!(ff)3

—A

0 ), 1ff!2Ksin4,\

(4.36)

and therefore has the eigenvalues R

+2 to2 + 2i~~2I(ff). (4.37) 1a-toVaThese being imaginary numbers, it follows that these fixed points, when they exist, are always centers. For a = IT, we obtain the additive inverse of that for a = 0; hence we obtain the same eigenvalues and therefore draw the same conclusions, namely that these fixed points are always centers (whenever they exist). =

±4iV’~A

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

309

We observe that the phase space portrait on S2 for this case evolves in a manner completely similar to that of the preceding case, so that we need not describe it any further; refer to figs. 4.2. It is worthwhile pointing out that case 1 is recovered as a limit of the present case, when u—i’ 0. Moreover, the butterfly bifurcation sequence is seen to be a singular phenomenon. This is of interest in view of our future analysis about the chaotic behavior of the system. Indeed it means that the half-circles of fixed points a = 0, IT (see figs. 4.le, j) are not structurally stable. For instance, consider a perturbation of the type A 2 —i’ A2 + e cos[ v(z z0)]. Deriving the equations for the perturbed motion and performing a fixed-point analysis shows that these half-circles do not exist for all “times” but blink periodically as cos[ v(z z0)] equals some constant fixed by r, K, and F. This fact will imply that we should not expect to see the usual type of chaos. The nature of the phase portrait and the limit to the butterfly sequence is illustrated in fig.2 4.3. The + 4K2)! two curved lines are obtained from formula (4.30) by requiring to to vanish, F±(a-) = (a(a-2 4K2), and represent those values of Fat which saddle + two centers triplets bifurcate into centers (or vice versa). For nonvanishing values of a- (note that the diagram is symmetric under u—i’ a-), the bifurcations proceed as follows. For F < F~,we have a triplet T 1 in the front of the sphere and a center C1 in the back. At F = F~,we see that T1 bifurcates into a single new center C2 while C1 remains unchanged. When F = F_, C1 splits into ~new triplet T2 in the back of the sphere while C2 remains unchanged. The butterfly sequence is obtained by restricting to the F-axis. In view of what we said above, let us consider what happens when we follow, for instance, the path F~ (a-). For a- ~ 0, T1 bifurcates into the half-circle —

—

—

—

F~(a)

~/7///~,1/I

~

F_(a)

Fig. 4.3. Bifurcation portrait for cases 1 and 2. Case 1 is recovered by restricting to the I’-axis.

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

310

= 0. The nature of this phenomenon is best seen by following a path near and parallel (leftwards) to F~(a-).On such a path, let T1 consist of the saddle point S and the two centers C.,.; the saddle point S 2± ç1i~. If a-s~0.then 4i~—*0 as lies on the equator, and the centers C+ have a =0 and 4’+ = IT! F—i’ F(o-). However, as a-—i’ 0, then çli 2. Thus, when a- = 0, perturbing the system by periodically 0—i’ IT! modulating the parameter A 2 will amount to defining a back and forth motion on the F-axis, about F —1; this means that the phase portraits will periodically bifurcate, in a discontinuous manner, between two configurations:~~~ C.,. L—i’ C—i’ L—i’ C.. L—i’ C~

a

—i’

4.4. Case 4. a-,

i~, K

—i’

—i’

arbitrary (regular cases)

This last case is more complicated to analyze than the earlier cases. The equations are the full ones given by (4.2), and finding the fixed points explicitly is unfeasible. From the requirement that the right-hand side of the 4, equation must vanish, it follows, again, that sin a = 0. Hence, the fixed points are distributed along that great circle. To find the 4, coordinates, we need to solve for to in the second of eqs. (4.2). However, in this case to obeys an algebraic equation of the sixth order, which is not solvable in general. Consequently, for qualitative analysis of the dynamics, we resort to numerical study of the 2. level of the Hamiltonian on the reduced sphere As surfaces an illustration, let us consider = 2, K = 1, F= 4, r =S 3, and a- = 2. The pictures in fig. 4.4 show the corresponding dynamics. Numerical investigations tend to suggest that for a generic choice of the above parameters (i.e., such that we do not recover any of the previous cases) the phase space portraits consist only of two fixed points, which are centers for all values of L. As shown in fig. 4.4, when we vary L from —Do to these centers, which are located on the great circle sin a = 0 as mentioned earlier, essentially migrate from one pole to the other (see also the bifurcation diagram 4.4); they are in fact identified with the poles asymptotically in the limit IL i.e., I Fl—i’ Do ,~

Do,

—i’

I

Orient.ili,,ri

I

‘I

:——~‘

—

Do,

1)

I

,

I ——I

—

I .

I / I’

/ I 1=.’

,

—4

-

I, I

- Ill

Fig. 4.4. Evolution of the reduced phase portrait as the parameter L is varied, for casc 4; r = 3. F= 4,

K

1. ,/ = 2,

rr

=

2.

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

311

N

L=-=

L==

Diagram 4.4. Bifurcation diagram for case 4. The thick plain lines represent centers. N = (0, r, 0), S = (0, —r, 0).

4.5. Singular subcases. a-

=

±(Ii~I K) —

There are two special situations of interest corresponding to specific values of the constant of the motion a-, and of the parameters K and for which the final reduction of our dynamical system, namely the projection P: x 52~52 from the Poincaré spheres, has singularities. We recall from case 1 that singularities occur for a- = 0 and = ~ in addition, an unusual bifurcation takes place in the limit IFI—i’°, with both poles being singular. This circumstance indicates that for this choice of parameters the projection to the reduced phase space P should be x S2—i’ 52\{4, = 0, 4, = IT). Removing the poles of S2 this way ensures that P is a homeomorphism, and in fact reduces our dynamical system onto a manifold diffeomorphic to a cylinder. It turns out that exactly two other choices of the invariant a- yield analogous situations; namely, the values a- = K, and a- = K For each of these choices, one of the poles becomes singular. For a- = K it is the south pole; implying that in order to preserve its smoothness we must view the map P as 52 x 4, = IT) so that the reduced phase space, as a differentiable manifold, is diffeomorphic to a paraboloid. Dropping this smoothness requirement would mean that P would project to a topological set obtained from by pinching it at the south pole, i.e., this pole would become a cuspidal point. Similarly, for a- = K the singular point is the north pole so we can redefine P as x S2 52\{ 4’ = 0), once again obtaining a reduced phase space diffeomorphic to a paraboloid. Note, finally, that these two special situations merge together when = i: then a- = 0 and we recover case 1 where both poles become singular. Let us now examine each of the other two singular subcases in more detail. We begin with a- = K. The generic fixed points are determined by requiring that sin a = 0, so that they are located on the great circle of arc a = 0 and IT, and their 4, coordinates are found by applying the formula 4, = arccos[(w IR] for each real solution of the above mentioned sixth-order algebraic equation. In addition to these fixed points, another one can be derived as follows. We_require that 11(4,) = 0 be satisfied in the limit 4, IT. Now, consider the quantity f/f. Both f and f go to zero in this limit. However, their ratio does not vanish; indeed, using l’Hôpital’s rule, we obtain ~2

I ‘~I, I ‘~I

~2

I’~I I ~I

—

—

—

~2

52\{

~2

—

~2

I

—

—

to

0)

—i’

I ‘~I,

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

312

1L~f1fVi~P~ Naturally, P(4i, a) Q

(4.38)

0 identically. As for the a equation of motion in (4.2), we must have

=

2A~{Fw+ ~a- ~[(a-+ w)\i~~ —(a-

=

~(L

—

—

w)VK!I~I]cos a}

=

0, (4.39)

—

1)[(rI~7~K)— (iK!rl~)].

Solving for cos a, we obtain cos a

_

((ii + =

_

(4.40)

.

(a-+to)\/~f7~-(a--to)\/K!IKt

Using the fact that a- =

a

____

2(Fto_+_~1a-) =

arccosl,~

I~1—

F)K

—

K

(i.1

and that —

to

=

—a- +

2K

cos 4,

—(K +

‘~I)~ we can

rewrite this as

F)Ii~I\

2v~i~~i )

(4.41)

.

As usual, the a coordinate is real if, and only if, Icos al ~ 1. Expressing F and zl in terms of the parameter L, this condition reads

—4\1~j~J~ [(rh),~ + (~I~I!rI~)K](L —1)— 2(K + ,~I)L~ 4\1~J~[. We now determine the nature of this fixed point. Using the fact that l’Hôpital’s rule we find that the matrix M is / A (Fto + z.1u) tan a M—~ 0 2A —

(4.42)

f= f = 0

in (4.4) and using

0 1(Fto+th)tana)’

(4.43)

and therefore has real eigenvalues. It then follows that this fixed point at the south pole is a saddle point, whenever it exists. In view of our previous comments and observations, it certainly acts as a saddle point. The qualitative description of the phase portrait is essentially the same as what we will describe for the next case. Let us now consider the opposite situation, where a- = K I ‘~I.Besides the fixed points determined by sin a = 0 and the real roots to of the sixth-order algebraic equation discussed earlier, there is also the north pole. Indeed, in the limit 4,—i’ 0, we observe that f = f = 0, so that the 4, equation identically vanishes: P(çfi, a) = 0. Moreover, analogously to (4.38), we have —

~ f!f=VIi~I/i.

(4.44)

The a equation yields eq. (4.39) once more, so that cos a is again given by (4.40), and condition (4.42) for a still holds. The stability matrix for the fixed point at the north pole is given by the negative of M in (4.43) for the fixed point at the south pole. Consequently, the qualitative behavior of the phase space portraits in the two cases will be the same.

D. David et a!., Hamihonian chaos in nonlinear optical polarization dynamics

313

The teardrop bifurcation. As an illustration of the behavior of the phase space portrait for this singular case when the above special critical points may appear, let us choose r = K = 1, i = k = 4; hence we set a- = —3. These values also imply that p = 2, to = —a- + 2K = 5, and Zi = 0. Formula (4.40) then reads cos a = ~Fand condition (4.42) reads IFI s4!5, or, equivalently, L E [—2.6,0.6]; these are the values for which the north pole is defined as a (pseudo) saddle point. The evolution of the portrait on S2 is depicted in fig. 4.5. When the value of L is large and negative, the portrait consists of two centers C 12 located quite near the north and south poles with a coordinates 0 and IT, respectively (C1 is visible in fig. 4.5b); as mentioned before, the a coordinates of these centers will remain constant through all these pictures. As L the centers C12 asymptotically approach the poles and the portrait consists only of, circular orbits parallel to the equatorial plane. For finite negative L, C1 moves away from the north pole along a = 0, while C2 moves away from the south pole along a = ~ moreover, the trajectories deform to become oblong near the north pole (see figs. 4.5c—f). A first teardrop bifurcation takes place when L reaches a value near —2.6. A critical orbit through the north pole then develops a cusp at the north pole and thereby becomes a single homoclinic orbit connected to the pole and enclosing the center C1 lying along a = 0 (see fig. 4.5g). As we continue to increase the value of L, this homoclinic orbit takes the shape of a teardrop and proceeds farther south (see fig. 4.5). Eventually, the orbit passes through the south pole and moves back again towards the —+

Orientation

a. L = .10

—~,

b. L = -4

~

/

c. L = .3

)

~

~

j

e. L = -2.61

f. L = -2.59

h.L=-1.6

i.L=0

j.L=0.4

k.L=O.59

I.L=0.62

m.L=1

n.L=4

o.L=1O

d. L = -2.7

/

g. L = -2.4

Fig. 4.5. Evolution of the reduced phase portrait as the parameter L is varied, for a singular subcase of case 4 (the teardrop bifurcation); r ?=4, K=l, ~=4, o’—3.

1,

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

314

north pole, but from the a =

side of the sphere (see figs. 4.5i—k). Meanwhile C2 has been moving northwards along a = IT and the homoclinic loop now surrounds C2 and the angular aperture at the north pole of the two branches of the ioop consequently increases from 0 to a value greater than IT. We also observe that C1 moves towards the south pole. When L reaches the critical value 0.6, a second bifurcation occurs as the cusp at the north pole smoothens and the orbit becomes periodic with finite period. The center then tends toward the pole as L becomes large. Also, the orbits tend toward circular (see figs. 4.5~—o).As L—*Do, the phase portrait returns to that for L—i’ with velocities reversed. Diagram 4.5 illustrates these bifurcations. IT

—Do,

N

L=-2.6

1=0.6

1=-’

Diagram 4.5. Bifurcation diagram for a singular subcase of case 4. The thick plain and dotted lines represent centers and saddle points, respectively. N = (0, r, 0), S = (0, —r, 0).

This ends the present section describing the various types of bifurcations taking place on the reduced phase space, as the material parameter L = A2!A1 is varied. A natural question arising from this analysis concerns the corresponding physical solutions. That is, the question concerns how the orbits on the reduced appear when lifted back on 52 x the product of two Poincaré spheres for the two polarization vectors u and i~. We can give a partial answer to this question. As we have seen, the pictures show the existence of distinguished orbits on the reduced phase2 space, the heteroclinic X Innamely the following section, and homoclinic orbits. These orbits will lift to particular solutions on S we will examine a few of these particular solutions associated with distinguished orbits and show that some of these solutions behave like kinks and solitary waves. ~2

~2

~2•

5. Kinks and solitary waves In this section, we associate the homoclinic and heteroclinic orbits on the reduced sphere to particular travelling-wave solutions on the product S2 X S2 of Poincaré spheres. These particular solutions turn out to be either kinks (domain walls separating two domains of different polarization states), or solitary waves (localized excitations). ~2

315

D. David et a!., Hamihonian chaos in nonlinear optical polarization dynamics

Recall that we are working in a coordinate system in which W is diagonal, A1

W= 0 0

0 0 A2 0 , 0 A3

(5.1)

and where the first and third eigenvalues are equal for an isotropic medium, A1 = A3. If the eigenvalue A1 is negative we make the transformation u’an—u,

W’an—W,

,i’an—ü,

(5.2)

which preserves the form of the equations of motion (2.26), but in terms of the primed variables with A >0. So, without loss of generality, we can assume A1 >0. Also recall that eqs. (2.26) are invariant under the simultaneous transformations K’-4K,

u~—i’—u,

~

ü~—*—ü’,

(5.3)

so without loss of generality we may assume K >0, and considering the symmetric nature of the equations with respect to barred and unbarred quantities we may also assume K without loss of generality. 2 lead to infinite solitary wave Periodic solutions the Hamiltonian the reduced sphere trains on the productof space S2 x S2 of system Poincaréonspheres. Much moreS interesting from the physical viewpoint are the homoclinic and heteroclinic orbits of the reduced system, which give kink and solitary wave solutions of the original system (2.23). First, we recall some of the relevant formulas. The Hamiltonian (3.29) is

I I

H= ~A

2+

~2)

+

~A

2+ ~a-2+ Ea-to + V4K2 —(u+

1~KK [Fto

1(r

to)2 ~

—

(a-

—

to)2

cos a],

(5.4)

where we have defined Fan ~(1—L)(~ +

—

L,

4an ~(1

—

L)(~ +

~) +

L,

Ean ~(1

—

L)(~

—

(5.5) and the equations of motion are

4, =

—A

2

—

(a- + to)2 ~42

—

(a-

—

to)2

sin a!(R sin

(5.6)

I/i),

1\[4K =

A

2 ~a4K —(u—to) 1(2Fto + Ea-) + Al(~4K2 (a- + to) A 1(24a- + Eto) A1(~~~:(a- + W): (a4K —(a-—to)

—

to) +

=

—

—

—

(a- to)2 ~a-+ w)) cos a, —(a-+to)

(5.7)

(a- to) ~a-+ —(a-+w)

(5.8)

—

4K

—

to)+

—

4K

where the variables are defined in (3.18), (3.24), (3.25), and (3.26).

—

—

to))

cos a,

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

316

5.1. Case 1. a-0, kI=K

to

First consider the case a- = 0 and I = K, which falls into the category of case 1 in section 4. Then 2i. cos 4, and the Hamiltonian and equations of motion become

=

+ 2A 24t + sin24i cos a), H = ~ A1(r2 + 1(riK!I~)(Fcos 4, = —2A 1 K sin 4, sin a, a = —4A1 K cos 4, (cos a F),

(5.9a)

.2)

/3 = 2A1 KE cos 4,.

—

(5.9b)

The hyperbolic fixed points of this system are (A) (B) (C)

2, 41IT! 4, = IT!2,

4,=Oand4,=IT,

a0; F>1, a = IT; F < —1, a=±arccosF; [‘1<1.

(5.10)

5.1.1. Case 1A For the first fixed point sin 4, = 1, cos 4, = 0, sin a = 0, cos a = 1 and the Hamiltonian evaluated at this point is H(IT!2, 0) = 2A 1 Krr!K. Equating this to the value of the Hamiltonian on the homoclinic orbit H(IT!2, 0) = H(4i, a) gives the following relation between 4, and a: sina=V7~—1 cos4’y

24, 1—Fcos 1—cos4,

cosa

62024,

sin4i

(5.11)

Substituting into (5.9b) gives 21

—

=

~

4’

—

cos24i,

(5.12)

—2A~~VF

which upon integration yields sin 4,(~)=~1 52sech2~, —

~=\/2!(F+

1),

~~2A

2 —1 1K6VF

(T— r 0).

(5.13)

Then substituting into (5.11) leads to

2~ 1 flY sech cosa(~) 1—ô2sech2~ The /3 equation becomes

.

52VF2

—

1 sech ~ tanh ~

—

‘

/3 = 2A

1KE8 sech ~

1—52sech2~

.

(5.14)

(5.15)

which upon integration yields =

+

\/~

arcsin(tanh ~).

(5.16)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

Now, by using eqs. (3.26), (3.24), (3.18) and recalling that asolution in terms of the variables on S2 x u~(~) = r\/i

—

~2

sech2~sin[~J3(~)+ ~a(~)],

~

=

=

0 and

F~i

—

~2

~

=

K

317

we may reconstruct the

sech24 sin[~l3(~)

—

u 2(~)= re5 sech ~,

2sech2~cos[~f3(~)+ ~a(fl],

u2(4) = —sgn(i~)Fsech~, (5.17) u3(5~)= Fyi— ~2sech2~cos[~f3(~)— ~a(~)],

u3(~)=r\I1 6 where is given by (5.16), a(~)by (5.14), with ~ and ~ defined in (5.13). This solution is valid, for a given material and intensities, over the range of K which satisfies the inequality in (5.bA). In the limit this solution approaches a fixed point of eqs. (2.26), —

/3(e)

~ —*

u 1

=

rsin(~/30

—

~

,

—~

ii~= Fsin(~/30 ~P) —

u2=0,

ü~2=0,

(5.18)

u3=rcos(~/30—~P),

zi3=Fcos(~/30—~~),

which represents equal linear polarizations. In the limit namely u1=rsin(~130+~),

~—* +co

it approaches another fixed point,

a1=Fsin(~/30+~), (5.19)

u3

=

r cos( ~/3~+ ~CI’),

u3

=

F cos( ~/3~+ ~

which also represents equal linear polarizations but rotated through an angle ‘P with respect to (5.18), where ~ is given by 2 —1. (5.20) = ~ITE/VF So this is a kink solution in which the asymptotic linear polarizations simply differ by a rotation through an angle 4~.If cP is an integer multiple of 2IT then the asymptotic polarizations at ±~are identical and we obtain a localized polarization wave, or in other words a solitary wave solution. For example if = —1, K = 1, and the ratio of the intensities satisfies

r/F=~~~ ~

n=1,2,...,

(5.21)

then CP = 2irn and the solution is a solitary wave. 5.1.2. Case lB For the second fixed point, (5.1OB), sin 4, = 1, cos 4, = 0, sin a = 0, cos a = —1 and the Hamiltonian evaluated at this point is H(IT/2, IT) = —2A 1 KTT/K. Equating this with the Hamiltonian on the homoclinic orbit gives the following relation between 4, and a:

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

318

24, cosa=— 1+Fcos 1—cos 4, where now 8 ~\/2!(1 cos a( ~) =

—

sina=VF2_1~~4’V62_cos24,,

sin 4’

(5.22)

F). Then (5.12) and (5.13) follow as before, but instead of (5.14) we have sin a( ~) = 82VF2

—

—

1 sech~t:nh~

(5.23)

The /3 equation (5.15) and solution (5.16) are still valid, as is (5.17), the solution on S2 x S2. Now, however, as the solution asymptotically approaches ~—i’

=

—Do

rsin(~/3 0+

~IT —

~

u2=0,

i~i=

Fsin(~/30

—

~IT —

ü~2=0,

u3=rcos(’~f3~+ ~IT—

~

(5.24)

~3=Fcos(~$0—SIT— UI),

which represents orthogonal linear polarizations, in contrast to case 1A, where the solution asymptotically approached equal linear polarizations. In the limit f—i’ +~ it again approaches orthogonal linear polarizations, =

rsin(~I30+

~IT +

HI),

i~ =

Fsin(~/30

—

1~2=0, u1—rcos(2f30+2IT+2~),

~IT +

pb), (5.25)

~

but rotated through an angle tP compared to (5.24). So as in athe previous 2IT, becomes solitary wave.case this is a kink-like solution, which, when cP is an integer multiple of 5.1.3. Case 1C At the two fixed points (5. 1OC) the Hamiltonian is found to be 2A 1 KrrF!K, which implies that on the heteroclinic orbit connecting these two points ±arccosF.

(5.26)

Substituting this in (5.9b) gives 2 sin 4,, 4’ = ~2A1,(V1 F which upon integration yields

(5.27)

a

=

—

4,(~)=2arctan(e~),~an2AlKV1_F2(T_TO). The

/3 equation of motion /3

=

(5.9d)

(5.28)

then becomes

±2A 1KEtanh~,

(5.29)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

319

from which we deduce /3=

/3~±

p2 ln(cosh ~).

(5.30)

The solution on S2 x ~2 can now be reconstructed, ~

~

u

2(~)=±rtanh4,

u2(~)=rsgn(i~)Ftanh~,

~

(5.31)

~

where J3(~)is given by (5.30), a(e) by (5.26), and ~ is defined (5.28). In the limit f—i’ for ,~>0, this solution approaches the constant value —~,

u1=0, u2=~r,

(5.32)

ü2=±F,

u3=0, which represents opposite circularly polarized waves corresponding to a fixed point of eqs. (2.26). In the opposite limit ~—i’ + it again approaches a fixed point,

u2=±r,

a2=rF,

u30,

u3—0,

(5.33)

also representing opposite circularly polarized waves, but with the rotations reversed as compared to (5.32). This is also a kink-like solution since it asymptotically approaches different fixed point polarizations as i—i’ ±~.The polarization profile as ~varies from to +~ is depicted in fig. 5.1 on the ui and Iui spheres. For the + sign in (5.26) the u trajectory asymptotically emanates from the south pole, crosses the equator at a longitude of ~(130 + arccos F), and then asymptotically approaches the north pole. Furthermore, we see from (5.30) that as —~

~~—‘

(5.34)

so the trajectory spirals out of the south pole and then spirals towards the north pole with the opposite sense of rotation (not to be confused with the opposite senses of rotation of the circular polarizations

represented by the north and south poles). The ii trajectory on the other hand spirals out of the north pole, crosses the equator at a longitude of (l3~ arccos F), then spirals towards the south pole with ~

—

the opposite sense of rotation. These solutions are depicted in fig. 5.1.

For the minus sign in (5.26) the interpretation is similar. In this case the u trajectory begins at the t~trajectory.

north pole and approaches the south pole, and vice versa for the

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

320

7/

Orienlation

__ ~-

—=

/

n~

Orientation

Fig. 5.1. Polarization profile on the uI (top) and

On the other hand, if

‘~I(bottom)

Poincaré spheres for case IC.

<0 then as a—i’ —Do the solution (5.31) approaches the fixed point

i~

u1=0, u2=÷r,

(5.35)

t~2=~F,

u3—0, which represents circularly polarized waves with the same sense of rotation, in contrast to (5.32), which in the same limit describes opposite circularly polarized waves. In the limit +cc one obtains ~

u2=±r,

u2=±r,

u3=0,

u3—0,

(5.36)

which are again circularly polarized waves with the same sense of rotation, but opposite to that of (5.35). So this is also a kink-like solution, approaching different fixed points at ± The polarization profile as ~ varies from to +~ is essentially that in fig. 5.1 except that now both the u and ü trajectories begin at the south pole (north pole) and end at the north pole (south pole).

~.

—~

cr0,I,~Is~K

5.2. Case 2.

Now we4’.consider I ‘~I~ K butand stillequations with a- = of 0. motion This corresponds 2K cos The Hamiltonian become to case 2 in section 4, for which to =

H = ~A

2 + F2) + (2A 1(r

4’

24i + sin 4’ V(,~/K)2 cos2t/i cos a],

—2A

1 Krr!K)[F cos 2 K2 cos2tfr sin a, —

1\/,~

—

(5.37) (5.38)

’

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

a

2

4A1KFcos ~p—2A( y~2K2cos24’+ ~ sin 1/1

=

K2 Sin

4’

~ —

sin4’ K

cos24

—

~

K2

321

)cosacos4’,

(5.39)

cos24’

P=2A1KEcos4’+2A1(y~ 2_K2cos24’

sin4’

(5.40)

)cosacos4’,

while the hyperbolic fixed points are 2, a0; F>y

(A)

4’IT!

(B)

4rIT!2,

where ‘yan(K2 +

(5.41) F<—y,

aIT;

,~2)/2KiI~i.

5.2.1. Case2A Evaluating the Hamiltonian at the fixed point and equating it to that on the homoclinic orbit H(4’, a) = H(IT!2, 0) = 2A cos a =

I’~I sin 4,

—

1rFsgn(,~)gives a relation between 2l/J KFcos —

K2

4’ and a,

(5.42)

cos2l/J

from which we deduce an expression for sin a, sin a =

cos4,VF2bV82cOS24’ ________ sin 4, V(~!K2 cos24’

8_Vi(FI~I/)2/(F21)

(5.43)

—

Substituting this into the =

4’ equation of motion gives

2 —1. cos 4’ sin 4’

—2A

—

cos24i,

(5.44)

1K\[F

which upon integration yields cos4’(~)=ôsech~,

sin4’(~)=\i-82sech2~’, ~—2A

2—bô(T—r 1K\/F

and then from (5.42) and (5.43) we obtain 2sech2~ cos a(~)= i~I/K F6 ~2 sech2~\I(i~IK)2 ~2 sech2~‘ __________

0),

(5.45)

_____________

—

—

—

___ —

sina(~) —

1

sech ~ tanh ~

82 sech2~\/?~7K)2 32 sech2 —

Using these results in the /3 equation we find

(5.46)

322

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

~ =2A1~3Esech~ +2A1~(~!~ F82sech2~)(~ —

2sech2~ 1— 62sech2~)6sech ~, —

)2

8

(5.47)

which after a tedious integration yields E =

/3~+

~2

(F~k!!K) ~ arcsin(tanh ~ + ~/p2 1 .

—

—

~

—

32

/ sinh~ arctank\~ —

52

(5.48)

(1—FI,~I!K)

+

—1 ~(~7K)2

(

arctan~

~~Isinh~

~2

—

K282

and the square roots are all well defined since I’~I K, F> I and 8 <1. Then by using eqs. (3.24) and (3.18) and recalling that a- = 0 we may reconstruct this solution in terms of the variables u and i~, =

r~i—

~2

sech2~sin[~/3(~)+ ~a(~)],

2~sin[~f3(4)

u 1(~) =

F\/1

—

~2

—

sech

u 2(~)=rS sech ~,

u2(~)——sgn(,~)i8sech 4~

~

(5.49)

~

where a(~)is given by (5.46), /3(4) by (5.48), 8 is defined in (5.43), and ~ in (5.45). In the limit ~—i’ —Do, this solution approaches a fixed point of eqs. (2.26), u1=rsin(~/30—~P),

ü1=Fsin(~J30—~cI3),

u2=0,

t~2=0,

u3=rcos(2/302cP),

i~3=Fcos(~/30—~~),

(5.50)

which represents equal linear polarizations. In the limit ~ —i’ + ~ it approaches another fixed point, namely u1=rsin(~/30+~k), ü~1=Fsin(~f30+~), (5.51) u3—rcos(2/30+2~),

~3=Fcos(~/30+~I),

which also represents equal linear polarizations but rotated through an angle cP with respect to (5.50),

where cP is given by ~IT(

E

+

(F—Il~I!K) 2—iVi—s2

+

(K—FII~I) VF2_1V~2_K282

.

( 552

2 \yF2_1 Vr So this is again a kink-like solution in which the asymptotic linear polarizations simply differ by a

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynwnics

323

rotation angle ‘~I.If this angle is an integer multiple of 2 IT then the asymptotic polarizations at ±cc are identical and we obtain a solitary wave solution. The polarization profile as 4~varies from —cc to cc for different values of r is depicted in figs. 5.2. The u and ü trajectories are confined to the northern and southern hemispheres respectively, so they are

drawn on the same sphere. Notice that the trajectories can cross themselves (but not simultaneously) since each sphere by itself is not the phase space of a dynamical system. Each of the trajectories emanates from the equator, circles the respective pole any number of times, depending on the value of ~, and then asymptotically approaches the equator again at an angle c1 with respect to the starting point. The number of times the trajectory encircles the pole is the winding number of the solution. a

Remark. Both solitary wave and kink solutions emerge in S2 X S2 from the homoclinic orbits through 0, IT and 4’ = IT!2, since these points lift to products of equatorial circles in S2 x S2. This reflects the

=

S1 symmetry of the Hamiltonian. 5.2.2. Case 2B In this case, instead of (5.42), we find that on the homoclinic orbit, cos a =

—I’~I—2KFcos2çf/ 2 2 —

sinçfr~K

K

i/s

and a are related by (5.53)

cos4’

and sin a is still given by (5.39) but with 6 redefined as

Orientation

a.r=48

b.r= 12

c.r=6

d.r=4

e.r=3.8

g.r=3.74

h.r=3.734

f.r=3.75

Fig. 5.2. Polarization profiles on the ui Poincaré sphere for case 2A for various values or solitons when the initial and final points of the trajectory are identified.

of the intensity r. These solutions are reminiscent of kinks,

324

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

\Ii

S The

i/s

—

(F + ~I!K)2!(F2—1).

(5.54)

equation of motion (5.44) and solution (5.45) are still valid but instead of (5.46) we now have

—kI

cosa(~)=

2 —

S sech

KFS2 sech2~ K 8 sech ~

— 2

—2

~

2

2

2

—

(5.55)

______

82VF2_1

sina(~)

—

sech ~tanh~

sech24

~2

\/(,~!K)2

—

~2

sech2~

and the /3 equation is slightly altered to the following:

=

2A

2 sech2~)( 1~8Esech ~

—

~2

sech2~+ 1

—

2A1K(I~I!K + F8

82sech2~)8sech ~, (5.56)

whose solution is =

/3~+

E

(F+I,~I!K)

.

v~2 1 arcsin(tanh 4:) + Vi2 —

1+FIi~I!K

+

VT~~ V(,~7K)2 ~

— ~2

—

Vi

~

—

/ sinh~ arctan~\~ —

~2

/ I,~Isinh4:\ arctan(\y 2 K282h1 —

.

(5.57)

The reconstructed solution on x is given by (5.49) with /3(4:) given by (5.57), a(4:) by (5.55), and the redefinition (5.54) of 8. In the limit 4:—* —cc this solution approaches the following fixed point of eqs. (2.26): ~2

u1 = rsin(~/30+

~IT

—

~),

u2=0, u3

=

rcos(~f30+

~2

t~i=

Fsin(~f30

—

~IT

—

ü2=0, ~IT

—

~P),

U3

=

(5.58)

Fcos(~f30

—

~IT

—

which represents orthogonal linear polarizations in contrast to case 2A, where we found equal linear polarizations. In the limit 4:—* +cc it approaches another fixed point, namely =

rsin(~/30+

~IT

+ ~I3),

ü~= Fsin(~/30

—

~IT

+

~P),

u2=0,

u2—0,

u3=rcos(~/30+~IT+~),

I~3=Fcos(~/30—~IT+~),

(5.59)

which also represents orthogonal linear polarizations but rotated through an angle ~Pcompared to (5.58),

where

325

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

E

~1T( —

2

(F+I,~iIK)

+

+

VF2—lVb—62

~

(K+FII~I)

(

Vr2—iV~2—K262

560 .

)

So these are also kink-like solutions whose asymptotic polarizations differ by a rotation angle. The polarization profiles as 4: varies from —cc to +cc are similar to those of figs. 5.2 except that the trajectories initially begin and end on opposite sides of the equator (representing orthogonal polarizations). As in the previous cases, if the winding number ‘P is an integer multiple of 2 IT then the solution becomes a solitary wave. 5.3. Case 3. a-~0,IKIK,rr Next consider the parameter values oO, I’~i= K, and F= r. Then ia-i) cos i/i and the Hamiltonian and equations of motion are H= A 2 + ~A 2I~K)[Fto2 + 4a-2 + ~4~2 1r 1(r =

(2~

—

—

kri) sin 4’ V4K2 ,I 2 I 14K

á=2A

2A1th

—

Al(\[4K: 4K

—

+ to)2 V4K2

(a-

—

I 2 14K

2

2(a-—to)—V4

—

K

(

(a- + w)~\/4K2

—(a-+to)

1Fto+A1I~\/4 2 =

—

—

(a-

to)

(a- +

~): ~a-

—(u—to)

—

to)

4K

—

to)2

to(4r)

(2K

—

cos a],

sin a,

—(a-—to)

2

2(a-+w) cosa,

2_

+ /4K2

(a-

becomes

(5.61)

K —

to)2

—

to(i/i)

(u+to)

(a- ~): ~a-+ to)) cos a, —(a-+to) —

—

which have the following hyperbolic fixed points:

(A)

i/i= IT!2,

a

0;

(B)

çfi=ir!2,

aIT;

F>(4K2+u2)I(4K2—u2), (5.62) F<—(4K2+u2)I(4K2—a-2).

5.3.1. Case3A Evaluating the Hamiltonian at the first fixed point and equating this to the value on the homoclinic orbit gives the following relation between 4’ and a: 2

cosa= ~f4K2_(a-+

an

2

—a- —Fto

to)2 \/4K2_(a-_

\18K2!(F + 1)— 2o-21(F

Substituting into the

2

4’ equation

—

/2

, to)2

1).

.

sina=

/~‘2

vF —ltovS ~4K2—(a-+

to)2

2

—to

V4K2_(u_

to)2

563 (5.64)

of motion (5.61),

~_AlVF2_1V~2_(2K_iuI)2cos24’~,

(5.65)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

326

and integrating yields cos

4’(4:) =

8 sech

4:,

8

~!(2~

—

Ia-l)~

A1VF2 —1(2K

4:

—

Ia-I)(T

—

re).

(5.66)

Then substituting back into (5.63) gives (2K

cosa(4:)= V4K2

—

—

[a+ (2K

Ia-I)[(2K + Ia-I) —

Ia-I)8 sech

—

(2K

~42

4:]2

a-I)F82 sech24:]

—

—

[

(2~ Ia-I) 8 sech 4:12

—

—

______

(2K

sin a(4:)= V4K2

[a-+ (2K

—

—

a-I)2VF2

—

Ia-I)8 sech

—

4:]2

1

~2

sech

~42

—

(5.67)

4: tanh 4:

[a- (2K —

—

Ia-I)8 sech

4:]2

Using these solutions to integrate the /3 equation in (5.61), gives after lengthy manipulations

/3(4:)

s2VF2i(~ Ia-I)

+

=

+

—

8

[_

tanh

—

Vi2

F

1(2K+a-)

4:

—

—

VF2 1

arcsin(tanh

4:)

+

/ 2K+a-+~cosh4: \ / 2K+a-—~cosh4: ~arcsin~ ) + ~arcsin~ \8+(2K+a-)cosh.4:’ \—5+(2K+a-)cosh 4:

—

/—(2K—a-)+~cosh4:\ 1 / 2K—u+~cosh4:\1I ~arcsin( + ~arcsin~ 8 (2, a-) cosh 4: / ~8 + (2K a-) cosh 4:’ ~1

.

.

—

.,

—

.

(5.68)

—

The solution on S2 x S2 then takes the form u~(4:)=

(r!2K)\/4K2

[a+ (2K

-

u 2(4:)

=

(r!2K)[a- + (2K

Ia-I)~sech 4:],

—

=

(r!2K) sgn(i~)[a-

=

-

-

-

-

-

—

—

[a-

Ia-I)~sech 4:], Ia-I) Ssech

(2K — (2K —

—

4:,

r~i

—

(a-!2K)2

sin((14:

—

cos[~/3(4:)

—

~‘P),

‘~~(4:) = r~i

~‘P),

u2(4:)= ra-sgn(k)!2K, u 2 cos(Q4: 3(4:) = r\[1 (a-!2K)

u2(4:)= ra-!2K, 2 =

4:]2

-

this solution approaches

u =

4:]2

(5.69)

(r!2K)~4K

For large negative

u3(4:)

sin[~/3(4:) + ~a(4:)],

=

(r!2K)~K2

2

1(4:)

4:]2

[a-+ (2K Ia-I)~sech 4:12 cos[~ /3(4:) + ____________________________ (r!2K)\/~K2 [a- (2K Ia-I)8 sech sin[~/3(4:)

u~(4:)=

U3(4:)

Ia-I)~sech

-

r~i— (a-!2K)

cos(fI4:

—

—

—

(a-!2K)2

sin(~l4:

—

(5.70) —

327

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

(82z1—1)a-

Q —

F

‘P=—

II)’

52VF2 1(2

Vr2—i

(2K—Ia-i~i \ 2K+a- ~ ~

F

Vr’2—i

(571)

2~

This describes a periodic solution where the vectors u and ii point in the same direction (,~ >0) making a constant angle arccos(a-!2K) to the polar axis and precessing about this axis at an angular frequency of (1 (with respect to the variable 4:). For large positive 4: it again approaches a periodic solution, u

2 sin(Q4: +

1(4:) = r~i (a-!2K) u 2(4:)= ra-!2K,

~‘P),

fi

1(4:) = r~i

—

2 sin(Q4: + ~‘P),

—

(a-!2K)

u2(4:)— ra-sgn(I~)!2K, (5.72) u 2 cos(114: + ~‘P), = r\/1 (a-12K) 3(4:) = rVi (a-!2K) which is the same as (5.70) except for a phase shift of ‘P. Thus, the effect of this solitary wave is simply 2 cos(fl4: + ~‘P),

—

—

to introduce a net phase shift in the periodic precession of the Stokes vectors about the polar axis. 5.3.2. Case 3B Evaluating the Hamiltonian at the fixed point (5.62B) and equating this to the value on the homoclinic orbit gives the following relation between 4’ and a: cosa=

4K / 2 V4K

2

+a- 2 —Fto 2

—(a-+to) 2/V4K —(u—to) 2

,

(5.73)

2

and sin a(i/s) takes the same functional form as before with ~ redefined as an ~

The

82

4’ equation

(5.74)

+ 2~2

(5.65) and solution (5.66) still hold but now (2K

cosa(4:)= ~4K2

—

and upon proceeding as

[a-+

—

(2K

a-I)[—(2K + —

before we

2a-(82L1 + 1) 1(2

82~2 +

11 —

I

1.

—

Ia-I)

—

(2K

Ia-I)~sech 4:]2 ~42

—

—

IuI)F82 sech24:] [a- (2K Ia-I)~sech 4:]2 —

H)

SF tanh VF2—1(2K+a-)

4:

F

Vr2—i arcsin(tanh 4:)

—

/ 2K+u+~cosh4:\ arcsinl\8+(2K+a-)cosh J +

1

1

—

(5.75)

find /3(4:) is given by the slightly modified expression [cf. (5.68)],

2K+a-—~cosh4: \ ~arcsin(/\—5+(2K+a-)cosh ., 4:!j . / 2K—a-+c~cosh4: ~arcsiffl/—(2K—a-)+~cosh4:\ .~ + ~ arcsm(

+ ~2

,

—

\8—(2K—a-)cosh4:’

\8+(2K—a-)cosh4:

(5.76)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

328

The solution on S2 x S2 is again given by (5.69) with the above mentioned redefinitions. For large negative 4: the solution approaches u

2 sin(Q4: +

~P),

~IT —

ii

1(4:) = r\/1 (a-!2K) u 2(4:)= ra-!2K, u3(4:) =

r\[1

=

+

(52~

2 cos(Q4: + —

2 sin(Q4: 1(4:)

—

(a-!2K)

1)a-!82VF2 —1 (2K

—

—

—

—

~IT —

(a-!2K)

u2(4:)— ra-sgn(l~)!2K, ü~~(4:) = r\/1 (a-!2K)2 cos(Q4:

~‘P), Ia-I)~

~ir

r\/1

—

(5.77) —

~IT —

(5.78)

and ‘P is given by (5.71). This describes another periodic solution, where, for i~> 0, the two vectors make the same angle arccos(a-!2K) to, but on opposite sides of, the polar axis and precess about this axis. For large positive 4: it again approaches a similar periodic solution, u

2 sin(114: +

~IT +

~P),

ü~(4:)= r\/1

+

~P),

ü~(4:)= r\/1

1(4:) = r\/1 (a-!2K) u 2(4:)=ra-!2K,

(a-!2K)2 sin(f14:

—

t~2(4:)=ra-sgn(l~)!2K, (a-!2K)2 cos(O4:

—

—

r + ~P),

—

2 cos(Q4: +

~IT

—

(5.79) ~IT +

~I),

u3(4:) = r~i (a-!2K) which compared with (5.77) is shifted by a phase of ‘P. This is analogous to case 3A, where the effect of the solitary wave is a net phase shift in the periodic precession. —

a-±(Is~I—K)

5.4. Case 4.

Now consider the value a- = (3.26). In this case to(i/i) = K

—

H= ~A

I’~I

—

which lies on the boundary of regions (1) and (2) as defined in

2+ F2) + ~A 1(r

2 +

1(rF!Ki~)[(F + ~1 E)(~~~ K) —

+ 4K2Fcos24J + 4K Sin

4I=—2A

a

K,

I ~I+ 2K cos i/i, and the Hamiltonian and equations of motion become 4, ~

—

(kI

—

K)(E

—

2F) cos

4’

—

K

—

K C05

4,)2

cos a],

2—(I,~I—K—Kcos4I)2sina, 1Vl~

=

A 1(K

—

E) + 4A1KFcos 4,

kI)(2F—

/K(kI—K—Kcos4’)sin4l +2A1~12 \/K (IKHKKc0S4’) -

/3

2K(I,~I

—

=

2A1~(I~~I K) + A1E(K —

(5.80) cos4s

2

—

‘-2

-

~-2

—

VK—(IKI—K—Kcos4’))cosa~

situ/i

I’~I + 2K cos 4’)

(K(I,~I—K—Kcos4i)sin4’ -

VK —(IKI—K—Kcos4’)

cos4’

2+~

VK—(IKHKKcos4’))cosa.

sin/s

The 4’ and a equations have a fixed point in the limit as i/i IT. Clearly the 4’ equation vanishes for 4i = IT, whereas the ~ equation involves the ratio of two quantities which both go to zero as 4’ IT. This limit is evaluated by first squaring to eliminate the root and then using L’Hôpital’s rule. One finds —~

—~

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

sin/i hm ~~rV,~2(i,~IKKcos/i)2

=

1

329

.

(5.81)

—c— .V’~

Consequently, the fixed point at the south pole is described by

I~k1,

a±arccos~,

I/IIT,

(5.82)

where

panIkI!K,

~

and is hyperbolic provided

I I <1. The two angles a

(5.83)

F are the direction of approach to the south pole of the homoclinic orbit. Evaluating the Hamiltonian at the fixed point and equating it to the value on .the homoclinic orbit, H(/i, a) = H(IT, ±arccos~), implies that 4, and a are related by =

±arccos

cosa=+c05~[_c05~F_hl~h’2] sin/iVp~2_(~L._1_cos/i)2

(5.84)

____

sina=

V1+cos4,V(1_cos/i)(2~L._1_cos/i)_[(is_cos/i)F_(,L_1)EI2]2 sin /i V2~s 1 cos 4’ —

—

Substituting the expression (5.84) for sin a into the equation of motion for 4, and formally integrating leads to

J

sin/id/i

=_2A

2

1KJdT, (1 + cos

/i)V’~1

—

~2

cos 4, + 83 cos

where

4’ ~2 an2~(1—

2,

8~an~—1— [~tF—(.t

—

(5.85)

1)E!2]

F2) + (~s 1)FE, —

83 an(1

—

F2). (5.86)

The change of variable q an (1 + cos ~ 1 simplifies this to dq

=

—2A 1K

dr,

(5.87)

~ V(6i+82+&3)q2_(82+283)q+ou

which is found in standard integration tables. Then from (5.87) cos/i(4:)=

(~2

+ 28k)

(82 +

—

263) +

—

~

—

~

cosh cosh

4: , 4:

(5.88)

where we have defined

4: an 4A1KV~ ~i

—

~2

~

—

~

T~an

8A1v~~1 —

~2

ln(8~ 46~33). —

(5.89)

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

330

An analytical expression for a(4:) can now be obtained by substituting (5.88) into (5.84). Next, the equation of motion for the variable /3 after substituting (5.84) for cos a(4’) becomes (1+cos/i)[(~—cos4’)F—~(~—1)EI /3=A1K(~t—1)(2L1—E)—2A1KEcos4i—2A1K(~L—1) (1—cos4’)(2~i—1—cosi/i) (5.90) Substituting

(5.88) for cos 4’(4:) and integrating produces the following analytical expression for /3(4:): (~—1)(2~—E)+2E—2(/L—1)F4:ET1 w156 540 m322 540 lSBT

—

2 4v~=~f~ (N+M)E 1 (exp(4:)+N—VN2—1 ____

—

2VN2_1~y1_~2

~(~2

(ji—1)(F—~E)(N+M) 4~~1—~2

—

1 (2exp(4:)+N—M—\/(N—M)2—4

1 ~(N—M)2—4

(~—1)(F+~E)(N+M)

\()y()2

1

~[(2~ —1)N—M]2—4~2 x ln(~ exp(4:) + (2~ 1)N M ~[(2~ 1)N M]2 4~~1—

—

~2

—

—

—

2~exp(4:) + (2~i I)N— M + ~[(2~ —

—

—

—

1)N— MI2

—

4~L2)

(5.91)

— 4~2

where we have defined two more constants N~(8 2+281)!\/8~—48183,

M_=(85+2&~)I\/5~—4S183.

Tracing back through eqs. (3.24), (3.18) and recalling that a- = terms of the variables u and ü,

I’~I

—

K,

(5.92) we reconstruct this solution in

ui(4:)~2r[8i(82+S3)_Si(Si+S2)+(8t+S2+S3)\/8~_48i83cosh4:]12 <

u2(4:)=r u3(4:)

=

=

82 +281 —~8~—4S163cosh4: 82+283+\/~—481S3cosh4:

2r[83(82 + 83)— ><

u1(4:)

sin[~/3(4:) +

S2+25~+~5~—4818~cosh4:

~

+

11 of t\ ~ cos~2p~j

+ 283 + 4FVI —

~2

[~2

~2)

+

(5~+

12a

+

83)~S~48~83cosh —

,

45~S3cosh + 283—4(1 —

—

4: +

~2)

sin[~/3(4:)—_~a(4:)J + 283 + cosh — ~

~2

4:

\/8~ 45~8~ cosh —

4:]u2

4:]t~2

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

8(1_4:2) + 283 + U3(4:)

=

4FV1

—

~

[8~+ 283

—

VS2

4(1

—

~

cosh

4~)+

2

—

—

331

)4: 48~83cosh

cos[~/3(4:)— ~a(4:)] 2

4:]1/2

+

263

+

Vs2

—

~

cosh

4:

where a(4:) is given by (5.84) upon substituting for cos 4,(4:), and /3(4:) is given by (5.91). In either limit 4: ±cc this solution approaches the same constant polarization, —~

u1=0, u2=r,

t~2=sgn(i~)F,

(5.94)

u30, which represents opposite (i~ >0) or equal (,~ <0) circularly polarized waves. So this solution represents a solitary wave. The u trajectory spirals out of the south pole, approaches the equator but does not cross it and then asymptotically spirals back toward the south pole again. The ,~trajectory spirals out of the north pole (i~> 0), approaches the equator, then spirals back to the north pole again. That the trajectories spiral in and out of the poles is deduced from (5.90) which as I 4:I—~cc and cos /i(4:)—~—1 approaches /3

—*

2A14(I,~I

—

K)

—

AiE(II~I+ K).

(5.95)

We emphasize that each trajectory asymptotically approaches the same pole, which is in contrast to the previous kink-like solution (5.31), where each trajectory asymptotically approaches opposite poles. This concludes our discussion of the case a- = ‘~I K. Another interesting parameter value is a- = K which lies on the boundary of regions (2) and (3) as defined in (3.26). The corresponding solitary wave solution can be obtained from (5.93) by transforming u2 u2 and U2 ~ In this case as 4:—÷±ccthe u and t~trajectories asymptotically approach —

I

I’~I~

—

—*

u1=0,

~

u2=r,

u2—sgn(K)T,

u3=0,

u3—0,

—

—~ —

(5.96)

which complements (5.94). This solution is also a solitary wave since the asymptotic polarizations at + cc and —cc are equal. 6. Horseshoe chaos and Arnold diffusion In this final section, we use Mel’nikov’s technique (as described in Guckenheimer and Holmes [1983], and Wiggins [1988]) to investigate the behavior of orbits when material parameters in the two-beam problem are perturbed. In particular, we prove the existence of horseshoe maps and, thus, the existence of chaotic orbits under small, spatially periodic perturbations of the W eigenvalues. The Mel’nikov function, in the case of a one degree of freedom Hamiltonian system, is defined to be

D. David et a!.. Hamiltonian chaos in nonlinear optical polarization dynamics

332

M(t0)

=J

{H°,H’}(q~(t+t0), p~(t+t0))dt,

(6.1)

where { , } is the Poisson bracket, H = H°+ rH’ is the Hamiltonian, and the integral is taken along a homoclinic orbit for the unperturbed dynamics. At first order in the perturbation parameter, r ~ 1, the Mel’nikov function M(t0) is proportional to the (signed) distance between stable and unstable manifolds of the homoclinic point (q0, p0) with unperturbed homoclinic trajectory given by (q~(t),p~(t))and parametrized by the phase variable t0. Melnikov’s theorem [Mel’nikov 19631 states that simple zeros of M(t0) (i.e., M(t0) = 0, M’(t~)~ 0) imply transverse intersections of the stable and unstable manifolds of the homoclinic point. The Mel’nikov technique allows us to prove that the Poincaré map for the flow starting near (q0, p~~) has Smale horseshoes by using the Poincaré—Birkhoff—Smale theorem. (See Guckenheimer and Holmes [1983] or Wiggins [1988] for details.) When the Melnikov function has simple zeros, following the dynamical evolution of a rectangular region of initial conditions near the homoclinic point under iteration of the Poincaré map shows that the region is folded, stretched, contracted, and eventually mapped back over itself in the shape of a horseshoe. This horseshoe map is the underlying mechanism for chaos. Thus, the presence of simple zeros of the Mel’nikov function allows us to conclude that transverse intersections of homoclinic orbits occur during the perturbed evolution on phase space, while also giving an analytic criterion for the existence of Smale horseshoes. In this situation, the region in the vicinity of each transverse intersection develops an extremely complicated Cantor set structure, whose iterated Poincaré maps can be shown to contain countably many unstable periodic motions of all possible periods, and uncountably many unstable nonperiodic motions. (See Moser [1973]and Wiggins [1988] for the methods of proof of these statements and further descriptions of homoclinic tangles.) The application of the Mel’nikov technique consists of four steps [Wiggins1988]: (1) Parametrize the unperturbed homoclinic orbit either continuously on phase space, or using a Poincaré section. (2) Define the (signed) distance between perturbed stable and unstable manifolds in a moving coordinate system along the unperturbed homoclinic orbits. This signed distance is proportional to the Mel’nikov function. (3) Ascertain whether the Mel’nikov function has simple zeros; if so, conclude to horseshoe chaos. (4) Demonstrate the horseshoe map (usually this is done numerically) and determine the type of physical behavior implied by the horseshoe chaos. Often this physical behavior is evidenced by a type of random, or intermittent switching between states that is identifiable with a symbolic shift, e.g., a binary sequence for a Bernoulli shift on two symbols. Many times (and, in particular, for the nonlinear optics cases we study here) the Mel’nikov function is an oscillating trigonometric function of the orbital phase parameter t0 (the initial time, for example) with —Do ~ ~ Do~ This situation provides evidence that the perturbed homoclinic orbit has broken into a countable infinity of heteroclinic points. For example, the horseshoe chaos in the case of a periodically perturbed single Stokes pulse (discussed in appendix A) corresponds physically to intermittent switching from one elliptical polarization state to another one whose semimajor axis is approximately orthogonal to that of the first state, with a passage close to unstable circular polarization during each switch. This intermittency is realized on the Poincaré sphere by an orbit which spends most of its time near an unperturbed “figure eight’ shape with a (homoclinic) crossing at the north pole (circular polarization). Under periodic perturba-

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

333

tions of the W eigenvalues, this orbit switches randomly from one lobe of the figure eight to the other each time it returns to the crossing region where the homoclinic tangle is located. Thus, for the

one-beam problem we predict random switching under spatially periodic perturbations of the material parameters, as the optical polarization state passes through a homoclinic tangle near circular polarization. This is how “spatial chaos” may be measured for either a static, or a travelling-wave single Stokes pulse. For more details concerning the one-beam problem, see appendix A. In the two-beam problem, where the perturbed motion may take place in more than two dimensions, we use a generalization of Mel’nikov’s technique originally due to Holmes and Marsden [1982a,b, 1983] and developed further in Wiggins [1988]. In these cases, the Mel’nikov vector function given in formulae (4.1.101) and (4.1.102) of Wiggins [1988]and reexpressed here as eq. (6.38) is proportional at order e to the vector distance between stable and unstable manifolds of a hyberbolic set (e.g. a hyperbolic torus). Zeros of the Mel’nikov vector imply transverse intersections of the stable and unstable manifolds of the perturbed hyperbolic set in higher dimensions, leading thereby to higherdimensional generalizations of horseshoe maps called transition chains and, thus, to Arnold diffusion (provided certain nonresonance conditions given in Arnold [1978, appendix 8] are satisfied). The main physical consequence of Arnold diffusion for optical pulses is that (given sufficient time) polarization can be transferred back and forth among various nonlinear modes of the system in a very irregular manner, in contrast to the regular quasi-periodic energy transfer occurring between modes in linear, or integrable systems. Three cases of perturbations of W = diag( A1, A2, A3) are treated below for the two-beam problem. These are labelled to the space on whichpolarization the perturbed motion takes place.are 2 xcases R1 case. Periodicaccording modulations of phase the circular—circular interaction coefficient S introduced as perturbations, A 1. In this case, the perturbed 2—~ A~52 = A2 cos[v(z z0)], s ~ 1, of z ER motion develops horseshoe chaos on X + R’sdue to the appearance homoclinic tangles in the vicinity of the unperturbed saddle points. S2 x S2 case. Constant perturbations are introduced that break material isotropy in the circular— linear polarization interaction coefficient, A 3 = A1 + e ~ 1. In this case, according to the KAM theorem, sufficiently irrational tori (preserved for small perturbations) serve as boundaries to regions of homoclinic motions on the three-dimensional surface. 2 x S2 Xchaotic R1 case. Spatially periodic perturbationsenergy are introduced to break material isotropy, S A 3 = A1 + e cos[ v(z z0)]. In this case, the solution can diffuse from torus to torus along transition chains. Arnold diffusion occurring via transition chains is described in more detail in Holmes and Marsden [1982b]. In the present situation, the main conclusion is that periodic motions of arbitrarily high period can be found close to these transition chains, just as in the standard two-dimensional horseshoe example. —

—

6.1. Physical interpretation of horseshoe chaos in the two-beam problem For the case of two counterpropagating beams, the physical interpretation of the horseshoe chaos can be described terms ofThe the motion difference of azimuthal angles offrom the polarization the phase 2 x ~2 as infollows. switches intermittently the vicinityvectors of theintwo stable space s configurations at 4, 4, = ±IT (mod 2ir), by passing through a heteroclinic tangle in the region of 4, 4, =0, 2 IT. Upon identifying IT, + IT with the symbols L = 0, R = 1 (for left and right), this motion can be characterized by a random binary sequence, whose choice depends sensitively on the initial conditions in the region of the homoclinic tangle. Thus, a given orbit on S2 x S2 starting near the —

—

—

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

334

unperturbed homoclinic orbit meanders through the tangle and emerges moving randomly leftward (0) or rightward (1). This random switching of the difference in azimuthal polarization angles under either constant or spatially periodic perturbations of the circular—linear interaction coefficient and under spatially periodic modulations of the circular—circular interaction coefficient implies that spatial chaos may be measured for two counterpropagating Stokes pulses, either as static solutions with spatial variation, or as travelling waves. The remainder of this section provides the proofs of the presence of Mel’nikov zeros under perturbations of homoclinic orbits in each of the three cases discussed above, using for the unperturbed homoclinic orbits the particular solutions found in section 5. 6.2. Horseshoe chaos on S2 X R’ Recall that for an isotropic medium the first and third eigenvalues of the tensor Ware equal, which results in an ~1 symmetry of the Hamiltonian for the two-beam polarization equations. First, we consider a periodic perturbation which preserves this symmetry, A,

0

0

/0 0 0

W= 0 A 0

2 0 +scos[v(z—z0)](0 0 A, \o

1 0 0 0

(6.2)

where e ~ 1 and i’ is the spatial frequency of the perturbation. Physically, this perturbation models an isotropic medium whose response to circular polarization is periodically modulated as a function of space. The unperturbed and perturbed Hamiltonians calculated from (6.2) and (3.4) are given by H°=

~u1W11u1 + ~u~W,1ii1+ 2u~W~1ü1,

H’

r cos[P(z

=

—

(6.3a)

z0)] (~u~m~1u1 + ~u~m,1u1 + 2u,m~1i~1),

(6.3b)

where m is defined as /0 0 0\ m=(0 1 0). \o 0 oJ

(6.4)

In terms of spherical polar coordinates r, F, 0, 0, 4,, 4, these Hamiltonians become 2(A 20 + A 20) + ~F2(A,sin20 + A 20) H°= ~r 1sin 2 cos 2 cos + 2rF[A, sin 0 sin 0 cos(4, ~) + A 2 cos 0 cos 0], 2cos20 + ~F2cos20 + 2rFcos 0 cos 0), H’ = e cos[v(z z0)](~r while in terms of the variables a-, to(/i), a and /3, they are written —

-

(6.5)

—

H°= ~A,(r2 + F2)

~(A

—

2(a- +

1 + ~(rF!KI~)[A

2 2(u

—

—

A2)[(r!K)

to2)

+ Ajl(/i)

to)2

+

cos a],

(Fh~)2(a-

—

to)2]

(6.6)

335

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

where 11(4’) is given in eq. (3.28), and H1

~.ecos[v(z

=

z

—

2(a-+ to)2 + (FI~)2(a-

—

0)] [(rIK)

4(rF!KI~)(a-2

w)2 +

—

(6.7)

to2)].

In order to render the calculations manageable, we consider a relatively simple case, namely,

a-0,

K—l,

F=r,

Kl,

(6.8)

for which the Hamiltonians become H°= A 2 1r

—

~(A,

—

3A 2to2 2)r

—

~A 2(4 1r

—

to2)

cos a ,

H’

=

~er2cos[v(z

—

z

2

0)] (—a-

+

3w2)

(6.9)

One can easily verify that this perturbation preserves a-, as a consequence of preserving the 1 symmetry. Furthermore, for the special case (6.8) the variable /3 is also constant, /3 = 4, + 4, = 0, so one can view this perturbation as taking place on the reduced space (to, a). Thus the usual, onedimensional, Mel’nikov method applies. The partial derivatives with respect to to and a of the unperturbed and perturbation Hamiltonians are, in the special case (6.8), =

~r2(3A

2(4— to2) sin a,

2 A, +2A, cos a)w, 9H°Iaa= ~A,r t9H’/dw = ~sr2 cos[v(z z 1Iaa = 0, 0)]w, dH from which we calculate the Poisson bracket of H°with H1

(6.10)

—

—

{H°,H1} = ~F’ ~9JJ

~9H

—

=

~eA,r4cos[v(z

—

z

2)to

0)} (4— to

sin a.

(6.11)

The first of the kink solutions (5.15) in the special case of (6.8) turns out to be only a function of

time, whereas we are considering a perturbation periodic in space. The second kink (5.24) is only a function of space, which, in terms of to and a becomes to(z)

=

2tanh[(2A,r sin a 0)z],

a(z)

a0anarccos{—[(A, + A2)/2A1]}

=

(6.12)

.

Then substituting (6.12) into (6.11) and formally integrating over the heteroclinic orbit implies the

following Mel’nikov function, from (6.1): 4 sin a M(z0)

=

e6A1r

0

f

2(2A dz sech

1r sin a0z) tanh(2A,r sin a0z) cos[v(z

Upon expanding the cosine in (6.13), cos[v(z term

—

z0)]

—

z0)].

(6.13)

cos(vz) cos(vz0) + sin(vz) sin(vz0), the first

=

produces an odd integrand whose integral vanishes, leaving 4 sin a M(z0)

=

e6A1r

0 sin(i’z0)

J

2(2A,r sin a dz sech

0z) tanh(2A,r sin a0z) sin(vz).

(6.14)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

336

Integrating by parts in (6.14) gives

f

M(z0) = ~er3v sin(vz0)

2(2A,r sin a dz sech

0z) cos(vz).

(6.15)

This integral is found in standard tables, yielding

3rvrr

/ ,‘

.

2

~

/

‘~

PIT

1 csch~ \4A,r sin a sin(vz0) , M(z0) = e~ \8A, sin a0 0 2

(6.16)

.

2

which clearly has simple zeros as a function of z0. Mel’nikov’s theorem then implies horseshoe chaos for spatially periodic perturbations of this kink solution. For the other of the kink solutions, (5.15), we consider a perturbation which is periodic in time, as opposed to space, but otherwise is of the same form as (6.3b), H’

=

e cos[v(t

—

t0)] (~u~m~1u1 + ~ii~m~1tii1 + 2u,m,1ü1) ,

(6.17)

where rn,1 is again given by (6.4). Then (6.5) and (6.7) follow as before with cos[v(z z0)] replaced by cos[ v(t t0)]. We again consider a relatively simple case, to make the calculations manageable, —

—

a-=0,

K=l,

rr,

K=l,

which differs from (6.8) by the sign of 2

—

H°= A1r

i~.

(6.18) For this case, the Hamiltonians become

2to2 + ~A,r2(4 w2)cos a ,

~(A, + A

H’

—

=

~er2cos[~(t t

2

—

2)r

0)] (3a-

—

to2)

(6.19)

The variables a- and /3 are again both constant, so we can calculate the Mel’nikov function on the reduced space (to, a). The partial derivatives of the Hamiltonians with respect to to and a are aH°Ic9w=

—

~r2(A, + A

ÔH’Ie9w =

2 + 2A, cos a)to, —~rr2cos[v(t— t

2(4

0)]to,

9H°h9a=

—

—

Sin a,

to2)

i,A,r

(6.20)

9H’h9a =0,

and the Poisson bracket of H°with H’ is found to be 4cos[v(t t 2)to sin a {H°,H’) = ~eA,r 0)] (4— to The first kink solution (5.15) for the parameter values (6.18) is, in terms of —

to(t)

=

.

—2 tanh[(2A,r sin a 0)t],

a(t) = a0 an arccos[( A,

—

3A2) !2A1]

,

(6.21) to

and a, (6.22)

which is only a function of time. Substituting (6.22) into (6.21), integrating over all time, and

proceeding as we did in the previous case yields the Mel’nikov integral,

D. David et al., Hamiltonian chaos in nonlinear /

i

2

~

rvIT

optical polarization

dynamics

337

/

I

PIT

M(t0)=—s~ . 2 1;cschi\4A,rsina . 0 sln(vt0), \8A,sm a0

(6.23)

2

which clearly has simple zeros as a function of t0. Hence, horseshoe chaos again arises. Now we consider the third kink solution, (5.43), which asymptotically approaches linear polarizations. For the parameter values (6.18) this solution becomes

w(t) = 2\/~7~ sech(2A,r\/~t),

sin a(t)

=

—S2V~sech(2A,r~/~t)tanh(2A,r\I~ t) (6.24)

which when substituted into (6.21) gives for the Poisson bracket in the Mel’nikov function, 4\f~(S 2(2A,r\/~t) tanh(2A,r\/~t) cos[v(t t {H°,H’) = —2eA1r 2/S3)sech 0)]. —

(6.25)

Though the functional forms of w(t) and a(t) differ from the previous cases, the Poisson bracket of H° with H’ on this orbit is essentially the same as before. Integrating over all time along the unperturbed orbit and proceeding as we did earlier now yields the Mel’nikov function, M(t0)

=

—~e(f)

csch(4A~’,~,,~~_) sin(vt0),

(6.26)

which again has simple zeros as a function of t0, implying horseshoe chaos for the third kink solution as well. 2 x S2 6.3. Horseshoe chaos on S

Next, we consider a perturbation that lifts the degeneracy of the eigenvalues A, and A

3, thereby breaking the S’ symmetry of the equations of motion. Physically, this corresponds to breaking the

isotropy of the the propagation direction. this case, the perturbed definedTon 2 X S2 and wemedium will needabout the multidimensional Mel’nikovIntechniques developed in system Wigginsis[1988]. he S“inverse moment of inertia tensor” is now A, 0 W= 0 A

0

0

),

2

0

0

A,+eJ

(6.27)

where e 4 1 represents a small constant perturbation in the third eigenvalue. The perturbation Hamiltonian is given by 1 = s( ~u,d H 11u1 + ~u,d11u1 + 2u,d11ü1),

where d.1 is defined by

(6.28)

338

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

/0 0 0

d=(0 0

o

(6.29)

.

0 1

\o

In terms of spherical polar coordinates, this perturbation becomes H’

=

e(~r2sin20 cos24, + ~F2sin20cos2/ + 2rF sin 0 sin 0 cos 4, cos 4,),

while in terms of the variables to, H’

(rr)[1

=

(~r

Kr 4K 4y~-[

—2

+

+

~4K2

—

[42

—

a,

(

a-, and /3 it is written —

a-)2]

!~![4K2

+

2

2

—(to—a-)]——.[4K

(to +

a-)2

(6.30)

V4~2 (to —

—

—

(to +

u)21)(1 + cos a cos /3)

2

—(to+a-)])sinasin/3

a-)2 (cos

/3 + cos a)].

(6.31)

This situation falls into the category of Wiggins’ system III [Wiggins1988], which in the present case has the following relevant Mel’nikov function: M(/3o)=—fdr~--(to,a,a-,$+/3o),

(6.32)

where simple zeros as a function of ~ imply horseshoe chaos. As before, we consider the relatively simple case (6.8) for which the partial derivative of II’ with respect to /3 is given by

~H’Io13= ~.er2(1

—

cos a)(4

—

to2)

sin /3

(6.33)

.

Upon substituting the analytical expressions for the heteroclinic orbit (6.12) (the second kink solution) and recalling that in this case the unperturbed a and /3 are constant, we obtain

c9H’Io/3

=

rr2(1

—

cos a

2[(A, sin a 0)sech

0)r] sin f3~ .

(6.34)

Substituting (6.34) into (6.32) and integrating gives the Mel’nikov function

M(130)=—er 22cosa0 . sin f3~, sin a0

(6.35)

which clearly has simple zeros as a function of /3~,implying horseshoe chaos. The second relatively simple case (6.18) also gives (6.33) but for an unimportant minus sign change in the first bracket. Consequently, for the heteroclinic orbit (6.22) (the first kink solution) we again obtain (6.35) for the Mel’nikov function, apart from this minus sign and a redefinition of a0 according to (6.22). Thus, we have horseshoe chaos in this case, as well.

339

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

6.4. Arnold diffusion on S2 x S2 x R’ Now we consider a third type of perturbation: one that both breaks the 5’ symmetry according to (6.27), and is spatially periodic. The “inverse moment of inertia” tensor in this case is A

1 0 W= 0 A2

0

0

0

/0 0 0\ 0 +ecos[v(z—z0)])0 0 0J, A,

\o

(6.36)

0 11

while the perturbation Hamiltonian is given by H’

=

ecos[v(z

z0)] (~u,d11u1+ ~u~d~1a1 +2u~d~1u1),

—

(6.37)

where d.1 is defined by (6.29). In terms of to, a, a-, and /3, the perturbation Hamiltonian is given by (6.31), multiplied by the periodic factor cos[ v(z z0)]. This case also falls into the category of Wiggins’ system III, but the relevant Melnikov function is a two-component vector given by —

I

M,(z0, f3~) j dz

M2(z0, /3~)—

J

/~H° 9H’ !~-~-~--~-

oH°oH’ —

-~-

-~-

OH°OH’\ —

-~-

OH°I

-i-)

+

-~-j dz

OH’

dz~—.

(6.38)

If these components both vanish simultaneously, the implication is Arnold diffusion [Arnold 1964]. We consider again the choice of parameters (6.8) for which OH°IOa-vanishes and OH’Id/3 is given by (6.33) multiplied by cos[ v(z z0)], while the partial derivatives of H°and H’ with respect to to and a —

are found to be 2(3A 2(4 to2) sin a, OH°h3to= ~r 2 A, + 2A, cos a)to, OH°/c9a= A,r OH’/Oto=—~ecos[v(z—z 2(1—2cosa—2cos/3+cosacosJ3)to, 0)]r ~9H’IOa= ~jecos[P(z z 2(1 cos /3)(4 to2) sin a —

—

~

—

—

(6.39)

—

0)] r

Substitution of these formulae into (6.38) gives for the Mel’nikov vector

M,(z

4 0,130)= ker

J

M

dzcos[v(z—z

2)wsina, 0)][3A2—

J

~(A,+A2)cos/3](4—to

(6.40)

2 dz cos[v(z—z 2(z0,/30)=—~sr

2)sin/3. 0)] (1— ~cosa)(4—w

Evaluating (6.40) on the heteroclinic orbit (6.12) and noting that the unperturbed /3 is

a constant

gives

340

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

M,(z0, /3~)= er4 sin a0 [3A2 ~(A, + A2) cos f3~] —

J

X

2[(2A,r sin a dz sech

2(1

—

0)z] tanh[(2A,r sin a0)z] cos[P(z

—

cos a

(6.41)

0) sin ~

M2(z0, /3~)= —er

2[(2A,r sin a xJ dz sech

0)z] cos[v(z

—

z0)].

Expanding the cosine as cos[v(z z0)] = cos(vz) cos(vz0) + sin(vz) sin(vz0) and dropping terms that lead to odd integrands leaves 4 sin a M,(z0, /3~)= er 0 [3A2 ~(A, + A2) cos f3~]sin(i~’z0) —

—

J

x

M2(z0,

/3~)=

2(1 —

2[(2A,r sin a

dz sech —

0)z] tanh[(2A,r sin a0)z] sin(vz),

cos a

0) sin /3~cos(Pz0)

er

f

(6.42) 2[(2A,r sin a dz sech

0)z] cos(Pz)

Finally, integrating by parts in the first integral in (6.42) gives essentially the second one, which is found in standard tables, thereby yielding M,(z0,

/3~)=

~(

16A~sin2ao) csch(4A~

ao)[3A2

—

2(A, + A2) cos

$~]sin(~z0), (6.43)

M2(z0,

/3~)=

_E(

4A

.2)(1

—

cos a0) csch( 4A rn a0) sin /30

This Mel’nikov vector has two sets of simple zeros. The first set is z0nIT/v,

f30—mIT,

n,m0,±1,±2,...,

(6.44)

while the second set is z0

(n

cos f30=2A21(A,

+ ~)ITIv,

+

A2),

n =0, ±1,±2,...,

(6.45)

provided A, and A2 are such that Icos 130I 1. Thus, this system for the choice of parameters (6.8) exhibits Arnold diffusion in the vicinity of the homoclinic orbit (6.12) under a periodic perturbation that breaks the 5’ symmetry. For the other heteroclinic orbit (6.22) in the special case of parameter values (6.18), we consider a periodic perturbation in time, as opposed to space, but otherwise of the same form as (6.37), 1 = e cos[v(t t H 0)] (~u~d,1u1 + ~ü~d~1ii1 + 2u~d~1i~1), (6.46) —

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

341

where d~1is again given by (6.29). Then instead of (6.39) the partial derivatives of H°and H’ become 2(A, + A 2(4 to2) sin a, (6.47a) OH°/Oto= ~r 2 + 2A1 cos a)to, OH°/e9a = ~A,r —

—

—

OH’IOw

2(1 +cos /3 0)] r 0cos a + 2cos a +2cos f30)to, (6.47b) 2(1 + ~ cos /3~)(4 to2) sin a = ~s cos[~(t t0)] r and in this case also OH°10f3vanishes while oH’/O/’3 is given by (6.33) apart from an unimportant minus in the first bracket and a redefinition of a 0 according to (6.22). Proceeding as we did in the previous case results in the following Mel’nikov vector: =

—~

cos[v(t— t

—

—

M,(t0, /3~)=

—

~(

16A~sin2ao) csch(4A~ ao)~3A1 A2) cos —

—

A2] sin(~t0), (6.48)

M2(t0, /3~)= s(4A

2

)(l

+

~cos a0) csch( 4A ,~na0) sin f3~

where a0 is now defined in (6.22), which has the following simple zeros: t0=nITIv,

f30=rnir,

and also the second set (if cos

t0=(n+~)ITIP,

n,m=0,±1,±2,...,

(6.49)

/30I ~1)

cos/30=2A21(3A1—A2), n=0,±1,±2,...,

(6.50)

implying Arnold diffusion in this case also. Remarks

(1) Throughout this section we have been assuming that both spatially and temporally periodic perturbations of the material parameters preserve the class of travelling waves, rather than exciting solutions of the full partial differential equations (2.19). Whether this assumption is valid is an open question. (2) Other types of perturbations exist besides periodic modulations of material parameters. For

example, one could include in the Hamiltonian additional interaction terms depending nonlinearly on the field amplitudes to higher order. Whether such perturbations would induce chaos is itself a subject

for a separate investigation. (3) The Mel’nikov technique applies, provided the perturbations do not destroy the hyperbolic nature of the critical point near which the Smale horseshoe develops. In the parameter regime for which the butterfly bifurcation of section 4 takes place, perturbations on the homoclinic side of the bifurcation

could radically change the location of the hyperbolic points, perhaps resulting in a new type of chaos which would produce something even more complex than horseshoes in two dimensions. Perturbing the half great circle of fixed points at the butterfly bifurcation could also lead to exponentially slow chaotic dynamics near the singular points at the north and south poles, which are neutrally stable to all orders (see Holmes et al. [19891).

342

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

(4) Exponentially slow chaos may also appear upon perturbing the teardrop bifurcation in section 4, in the vicinity of the cusp at the north pole of the reduced phase space. Again, this calls for a separate investigation.

7. Conclusion In this paper, we have applied Hamiltonian methods in the Stokes description of the one-beam and two-beam problems of polarized optical pulses in nonlinear media to determine integrability conditions, study bifurcations on the reduced spaces, and establish the existence of chaotic dynamics in the form of horseshoe chaos in the two- and three-dimensional cases, and Arnold diffusion in the higherdimensional cases. The method we have used is Marsden—Weinstein reduction [Marsden and Weinstein 1974] in combination with the method of Mel’nikov [1963], as developed for the higher-dimensional cases by Holmes and Marsden [1982a,b, 1983] and by Wiggins [1988].The results have been shown to apply to periodic perturbations of homoclinic orbits of the reduced integrable problems, for static and travelling-wave solutions describing either a single optical beam, or two such beams counterpropagating. We have shown that these problems possess complex dynamics and have predicted the experimental consequences of this dynamics. We have also identified a number of interesting (degenerate) bifurcations that take place in the integrable reduced problems. In particular, the “butterfly” and “teardrop” bifurcations in section 4 (associated with singularities on the reduced manifold) are excellent candidates for the generation of new types of chaotic behavior in the presence of material parameter perturbations of the same three types as those considered here for the homoclinic orbits.

Acknowledgements We are grateful to J. Ackerhalt, H. Brandt, C. Doering, P. Krishnaprasad, P. Milonni, Y. Oh, G. Patrick, C. Scovel, and A. Weinstein for their interest and helpful comments. We would also like to thank Jerry Marsden, Harvey Segur, and Steve Wiggins for several detailed discussions of the ideas in this work. We are particularly indebted to Steve Wiggins for an advance copy of his book [Wiggins 1988] and for instructive explanations of the geometry of the Mel’nikov technique in higher dimensions. Appendix A. The analogy with the gyrostat and horseshoe chaos for the single Stokes pulse Tratnik and Sipe [1987a] show that the equations of motion (2.29) for the Stokes polarization parameters of a single optical beam propagating as a travelling wave in a nonlinear medium (the single Stokes pulse) may be interpreted as the equations of a gyrostat, a free rigid body with a rigid flywheel attached. As we shall discuss, the system of coupled equations (2.29) for the single Stokes pulse is completely integrable as a Hamiltonian system. Indeed, for the symmetric case when the medium is isotropic about the direction of propagation [so, e.g., b = (0, b2, 0), W= diag(A,, A2, A,)], the solutions of (2.29) are expressible in terms of elliptic functions. This symmetric situation in optics corresponds for the gyrostat to the case when the flywheel attachment axis coincides with the 2 principal axis of the rigid body and both flywheel and rigid body are symmetric about this axis.

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

343

Koiller [1984]has examined the bifurcations and separatrices on the angular momentum sphere for the latter, symmetric, case of the gyrostat equations. Analytical formulas can be given for the separatrices. Moreover, a small imperfection in the 5’ symmetry of the flywheel causes the separatrices to split with transverse intersection, producing Smale horseshoes. Koiller [1984, 1985] investigates the separatrix splitting using the technique of Mel’nikov [1963] as developed by Holmes and Marsden [1982a,b, 1983]. In this appendix, we complete the analogy between the sets of equations for the single Stokes pulse and the gyrostat, by showing that the two problems have nonzero intersection: a special case of the optics problem is an invariant subsystem of the gyrostat dynamics; so the Hamiltonians, Lie—Poisson brackets and equations of motion for the two problems can be made to be effectively identical for this

invariant subsystem. We then consider perturbations that are physically realistic from the viewpoint of nonlinear optics.

Namely, we consider spatially periodic modulations of the circular—circular polarization self-interaction coefficient A2 in W. In the case that the unperturbed medium satisfies the additional condition A2 = A3, the Mel’nikov technique leads to an analytically manageable integral for the Mel’nikov function, which is shown to have simple zeros. Thus, horseshoe chaos is predicted for this case in the dynamics of the single Stokes pulse. We also discuss the physical implications for measuring this horseshoe chaos in an

experimental situation. The Hamiltonian for the single Stokes pulse is recalled from (3.20),

(A.1)

H=b~u+~u’Wu,

where W= diag(A,, A2, A3) describes self-induced ellipse rotation and b = a + IuIc, with a and c constant vectors, describes linear and induced anisotropies. We assume an isotropic medium for which b = (0, b2, 0) to make contact with the gyrostat equations in Koiller [1984,1985], in which case the Hamiltonian (A.1) becomes 2 H= ~

+

~A3u~+ ~A2(u2+ b2IA2)

—

~b~IA 2.

(A.2)

The Lie—Poisson bracket given in Koiller [1984,1985], OG

OH

OGOH

OGOH

~

(A.3)

[defined on the dual space of the Lie algebra so(3) ~ so(2)] produces the following equations of motion, upon substituting {u,, u2, u3, b2, 1i} into the Hamiltonian form G = {G, H}, ü,=(A2—A3)u2u3+b2u3, /320,

~

ü2=(A3—A,)u3u,,

ü3=(A,—A2)u,u2—b2u,,

(A.4)

The first three equations in (A.4), deriving from the u piece of the Lie—Poisson bracket (A.3), are the correct equations of motion for the single Stokes pulse. Note that the equations in (A.4) decouple: the

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

344

first three equations describe integrable motion on the Poincaré sphere, u~+ u~+ u~=

1u12

=

const.;

(A.5)

and, using this motion, the last two equations are integrable by quadratures. In the optics problem, there is the additional physical relation b 2

=

a2 + IuIc2

=

const.,

(A.6)

which is preserved by the dynamical equations (A.4) and, thus, may be imposed as an initial condition, which is thereafter maintained by these dynamical equations. The Hamiltonian (A.2) may be brought into exactly the same form as that for the gyrostat in Koiller [1984, 1985], by relabelling variables after moving into a frame in the configuration space SO(3) x (5’ x R’) in which the phase angle ‘1’ ES’ advances in time with constant angular frequency b218. This time-independent phase change may be accomplished by adding a term proportional to b~to the Hamiltonian (A.2) and does not affect the equations for the optical polarization u, obtained from the u piece of the Lie—Poisson bracket (A. 3). (If one were talking about a simple rigid body, transforming to a rotating reference frame with angular velocity vector ul~would be accomplished by adding a term u to the Hamiltonian. This, however, would not affect the angle cP. Likewise, the time-dependent phase change for ~Pdoes not affect the polarization variable, u.) Thus, the analogy between the single Stokes pulse in an isotropic medium and the gyrostat can be stated as follows. The Hamiltonians can be made identical for the two cases and a special case of the Stokes pulse dynamics is an invariant subsystem of the gyrostat dynamics, with initial condition (A.6) preserved by the dynamics in both cases. A.1. Periodic perturbations of optical material properties We now calculate the Mel’nikov integral for another type of perturbation, which is not considered in Koiller [1984, 1985] but is relevant for Stokes pulse propagation. Consider a non-parity-invariant material with C4 rotation symmetry about the axis of propagation, for which the material tensors take the form b = (0, b2, 0) and W= diag(A,, A2, A3). The Hamiltonian in this case is expressible in terms of the conjugate variables u2 = r cos 0 and 4, on the Poincaré sphere, 24, + A H = ~(A, sin

24,)(r2 3 cos

—

u~)+ ~A 2u~+ b2u2.

The corresponding equations of motion are given by 2 u~), U2 = —OHIO4, = —(A, A3) sin 4, cos 4, (r 4=OHIOu 24,+A 24,—A 2=b2—(A,sin 3cos 2)u2.

(A.7)

—

—

One may easily verify that two fixed points are

(A.8)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

345

¶~+r(A-A3) =

±r,

sin 4,

=

~

(A.9)

r(A, —A3)

and a linear stability analysis shows that, whenever the intensity and material parameters are such that 4, is real, these fixed points are saddles. We concentrate on the north pole u2 = 1, 4, = 4, and evaluate the conserved Hamiltonian at this point to find a relation between u2 and 4, on the homoclinic orbit, U2=

2b2 2

2

A, sin 4, + A3 cos 4,

—

A2

—r,

(A.10)

which, when substituted into the equation of motion for 4,, gives 2~ cos24,). r(A, A3)(cos Upon integrating (A.11) we obtain (with ~ = z +

(A.11)

—

=

—

tan 4, =tan(~)/tanh(~q) , from which we deduce 24, = sin24 sin 1 cos24, sech2( ~~j)

~~sin ~cos

Vt,

the travelling-wave variable)

r(A

~

3 cos24,

A,),

(A.12)

cos2~tanh2( ~

=

1

—

—

(A.13)

cos24, sech2( ~

—

Substituting these formulae into (A.10) gives an analytical expression for u 2 on the homoclinic orbit, 2~sech2( ~i~)] 2b2[1 cos 24, + A 24,.~ A 24, sech2( ~) —r. 2= A, sin 3 cos 2 + (A2 A3) cos We consider a periodic perturbation of the eigenvalue A 2, that is, u

— —

(A.14)

—

A~= A2 + e cos[ii(~

—

~)] ,

(A.15)

where e 4 1 and v is the modulation frequency. Then from (A. 1) the perturbation Hamiltonian is H’

=

~su~cos[v(~

(A.16)

— ~j,,)],

and we easily calculate the canonical Poisson bracket of this perturbation with the unperturbed

Hamiltonian 2 {H°,H’) = r(A,

—

—

u~)u

A3) sin 4, cos 4, (r

2cos[P(~ flu)],

(A.17)

—

which when formally integrated becomes the Mel’nikov function M(~0)= e(A,

—

A3)

J

5(~)[r2 u~(7))]u —

sin çb(~)cos q

2(?))cos[v(~

—

i~)] d~.

(A.18)

D. David et a!.. Hamiltonian chaos in nonlinear optical polarization dynamics

346

In the particular case A.

A3, this integral is manageable. However, this does not reduce our system to that considered by Koiller [1984, 1985] (A, = A3); in addition, the perturbation considered here is of a different nature, not physically realizable in the gyrostat. Substituting (A.13) and (A.14) into (A.18) gives =

~ f

M(~,,)= r

2( ~)

tanh( ~) sech

~

x [r(A,

A

—

2~sech2( ~i~)] cos[ 2) —2 cos

v(~ ~j —

0)jd~.

(A. 19)

Expanding the cosine in (A.21) as cos[v(~ ~j~)] = cos(v~~) cos(vm,) + sin(v71) sin(~ij0)and noting that the first term leads to an odd integrand, whose integral vanishes, then leads to —

3~ / 1 2 4sin~cos A sin(vTh))~r(Al A,) j tanh(~ij)sech (~‘i~)sin(P7J) dsj .

M(i~ 0)=

E

—

)2

—

f

2~ tanh(~)sech4(~)sin(P~)d31). —2cos

(A.20)

These integrals are found in standard tables. Hence, M(~o)=e(2IT)( (A

.

1

—

~)2csc~( ~)[r(A,

A2) sin 4,

sin 24,

~

(A.21) 6s,n 4,

which clearly has simple zeros as a function of ij0 implying horseshoe chaos. Physically, the horseshoe chaos in the case of a periodically perturbed single Stokes pulse corresponds to intermittent switching from one elliptical polarization state to another one, whose semimajor axis is approximately orthogonal to that of the first state, with a passage close to unstable circular polarization during each switch. This intermittency is realized on the Poincaré sphere by an orbit which spends most of the time near an unperturbed “figure eight” shape as in fig. B.4c with a (homoclinic) crossing at the north pole (circular polarization). Under periodic perturbations of the W eigenvalues, this orbit switches randomly from one lobe of the figure eight to the other each time it returns to the crossing region where the homoclinic tangle is located. Thus, for the one-beam problem we predict random switching under spatially periodic perturbations of the material parameters, as the optical polarization state passes through a homoclinic tangle near circular polarization. This is how “spatial chaos” may be measured for either a static or a travelling-wave single Stokes pulse.

Appendix B. Critical points, bifurcations and phase portraits of the one-beam problem In this appendix, we reduce the single Stokes pulse equations onto the Poincaré sphere (guI = constant), classify the critical points of the reduced Hamiltonian for the six inequivalent classes of optical media reviewed in Tratnik and Sipe [1987a], and sketch the phase portraits of the reduced systems on the Poincaré sphere, using the ratios of the three eigenvalues of W as bifurcation parameters. Complications due to nonzero birefringence and intensity dependence are also discussed.

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

347

The algorithm that we use to classify the critical points is known as the energy—Casimir method and can be described as follows (see Holm et al. [1985]for a more extensive discussion). Suppose we are given a finite-dimensional dynamical system defined on a manifold M with Poisson bracket { , }, enabling the equations of motion to be expressed in Hamiltonian form, ii

XH(u)

=

=

{u, H),

(B.1)

where u E M, ü is its time derivative and H is the Hamiltonian function. Suppose a is an equilibrium solution of (B.1) and C is a constant of the motion such that the sum H + C has a critical point at a. The constants C are typically Casimir functions (also called distinguished functions) of the Poisson bracket for system (B. 1); these are functions C, which Poisson-commute with all functions G: M A, i.e., VG {C,, G) = 0. The equilibrium solution a lies in a class of equilibria that are critical points of the sum = H + 4,({C,}). If there were several Casimirs, one would choose 4, to be a function of all of them to maximize effectiveness by treating the stability of the entire class of equilibria at once. For the equilibrium a, we calculate D2~C= .,tl,~du1 du1, where .4~is the Hessian matrix of ~‘ evaluated at a. The —~

finite-dimensional system (B.1) will be Lyapunov stable at u = a, provided the matrix .At(â) is of definite sign; since, in that case, the level surfaces of ~ upon which the perturbed motion takes place, will be “nested” in generalized ellipsoids around the critical point and, therefore, the perturbed motion will stay near the original equilibrium solution. For further details see Holm et al. [1985],section 2, and references cited therein. This appendix determines the conditions for stability of the single Stokes pulse by using the energy—Casimir method. This method provides a complete classification of all stationary solutions for the system, thereby verifying and completing partial results appearing elsewhere in the literature [Gregori and Wabnitz 1986; Daino et al. 1986]. Moreover, homoclinic and heteroclinic orbits are identified, by plotting level surfaces of the Hamiltonian on the reduced phase space consisting of the sphere Bifurcations of these orbits appear as changes of stability when material parameters are varied. Thus, the energy—Casimir method provides the basis for a rather complete description of the dynamical features of the single Stokes pulse. As explained in section 3 and appendix A, the2one-beam problem be angle reduced fromanSO(3) with coordinatized by acan polar 0 and azimuthal coordinates (u,, u2, u3) to the Poincaré sphere S angle 4,. We recall that the Hamiltonian function and the equations of motion on SO(3) read, respectively, as [see (3.20) and (3.21)] ~2

H°(u,,u 5,u3)= ~(A,u~+A2u~+A3u~)+(b,u,+b,u2+b3u3),

U2

while on

=

(A2

—

A3)u5u3 + b2u3

—

b3u5,

=

(A3

—

A1)u3u, + b3u,

—

b,u3,

=

(A,

—

A2)u1u, + b,u2

—

b5u,,

(B.2)

(B.3)

we have 2[(A,sin24, + A 24,) sin2O + A 20] + r[(b, sin 4, + b H°(0,4,) = ~r 3 cos 2 cos 3 cos 4,) sin 0 + b2 cos 0], ~2

(B.4)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

348

0=b,cos4,—b3sin4,+(A, —A3)rsinOsin4,cos4,, 24, + A =

b2 —(b, sin 4,

b3 cos 4,)cot 0— r(A~sin

+

24, 3 cos

—

(B.5)

A 2)cos 0.

Tratnik and Sipe [1987a]show that the vector b with components (b,, b1, b3) and the matrix Wwith eigenvalues (A,, A2, A3) in the above equations can be transformed, in view of the different types of optical media, into one of the following six physically inequivalent forms: Case 1.

b= (0,0,0),

W=diag(A,, A2, A,);

Case 2.

b=(0,0,0),

W=diag(A,,A2,A3);

Case 3.

b=(O, b7,0),

W=diag(A,, A2, A,);

Case 4.

b=(0,b2,0),

W=diag(A1,A2,A3);

CaseS.

b=(b,,0,b3),

W=diag(A,,A.,,A3);

Case 6.

b=(b,,b2,b3),

W=diag(A,,A2,A3).

(B.6)

(Parity-invariant media are characterized by b2 = 0. Also note the degeneracies in some of the A’s.) In this appendix, for each of these cases, we use the energy—Casimir method to determine stability conditions for the equilibrium solutions of (B.3) on SO(3). In addition, we use standard techniques (see, e.g., Guckenheimer and Holmes [1983]and La Salle complementary [1976]) to characterize bifurcations anda 2. These methodsthe permit us to give homoclinicpicture orbits of of the the reduced S structure for the single Stokes pulse. complete behaviorsystem of the (B.5) phaseonspace Since there is only one Casimir for the Lie—Poisson bracket given in (A.3), namely r = u~,we may write ~ as (recall that b, = a. + rc,) ~‘{=

H + 1i(~)= ~A~u~’ + (a, + rc 1)u~+ D(fl,

(B.7) 2 and is to be determined for each where t~J is solution. an a priori arbitrary of its argument ~ = the ~r unreduced system (B.3) when using equilibrium It turns out tofunction be convenient to work with the energy—Casimir method to determine the equilibria and their stability criteria, and to work with the reduced system (B.5) when examining bifurcations and orbit structures. The requirement that ~Chave a critical point when evaluated at an equilibrium solution a of the system (B.3) reads D~°(ã) = 0. Explicitly, an equilibrium solution a is a critical point of X when the following three conditions hold (i, k, / = 1,2,3): +

Ak + c~ü ,Ir] ~ + (ak + rCk) = 0.

(B.8)

The critical point condition (B.8) determines all of the equilibrium solutions of the system (B.3). The components of the Hessian matrix .Al of ~ evaluated at ii are given in [D2~r](a) =

~kl(a)

duk du 1

=

{[cP’(E)

+

Ak

+ c~a~/r]8k, +

[cP”(~)l2kal

+ (Clak + CkU,)/r

—

~

duk du1. (B.9)

349

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

When the quadratic form (B.9) associated to the Hessian matrix .iU is definite, then the corresponding equilibrium solution ~ is stable. Sylvester’s theorem [Gantmacher 1959] determines necessary and sufficient conditions for such a quadratic form to be of definite sign. Let A be a n x n matrix, and consider its principal subdeterminants D, (i = 1, . . . , n). Then the quadratic form associated to A is positive definite iff D, >0 for all i, and negative definite iff D, <0 and sgn(D,~,)= sgn(D,) for i > 1. For each of the above cases, after determining the equilibrium solutions a on SO(3) and their corresponding stability conditions, we pass to the reduced space S2, in order to characterize each —

equilibrium as either a center or a saddle in this two-dimensional space. This characterization is obtained by linearizing the system (B.5) in a neighborhood of the examined fixed point (0, 4,) and using the standard spectral method. (See also, e.g., Arnold [1973]and Arnold [1982].)This linearization may be obtained by substituting a perturbed solution of the form

0=~+ef,

(B.10)

4,=~+eg,

into the system (B.5) on 52 written as é=P(0,4,),

4,=Q(0,4,).

(B.11)

Then, to order e the linearized equations become (f’~~(f~(oPIo0OP/e9cb\(f ~g) ~g)\oQIo0 OQIOçb)\g

B12

(

The eigenvalues of the matrix X are given by R±

2ifoPoQ~J(oPoQ~2(oPoQoPoQ~T 1. 00 04, \ 00 04,1 \ 00 04, 04, 001 J —

(B13)

It follows that (& ~) is a center whenever Re(R+) <0, and a saddle point otherwise (there cannot be any source or sink points, since we are dealing with a Hamiltonian system).

B.1. Case 1. b=(0,0,0), W=diag(A,,A 2,A,)

From the equations of motion (B .3), or (B .5), we find that the set of fixed points consists of the two poles of the sphere, given by 0 = 0, IT, or a = (0, ±r, 0), together with2.allLettheuspoints on the great circle first consider the poles. at the equator, given by 0 = ir!2, i.e., a = (u1, 0, u3) with u~ + u~ = r Using (B .8) and (B .9) allows us to write the Hessian matrix as (here and below we exchange the indices 1 and 2 in order to obtain a simpler sequence of principal subdeterminants) r2P”(~) 0 ft~= 0 A,—A

0 2

0

0

0

.

A,—A2

The conditions for definiteness of .Af~ are

(B.14)

D. David et a!., [-lamiltonianchaos in nonlinear optical polarization dynamics

350

either A,

A2

and

cP”( ~)>0,

or A,
and

cP”(~)<0.

>

(B. 15)

Clearly, it is always possible to choose a satisfactory function cI(~)for each of these conditions separately and, hence, we deduce that the poles are always stable fixed points, i.e. centers. (We ignore the trivial degenerate case, A1 = A2.) Consider now the great circle of fixed points at the equator; the Hessian matrix is 0 =

0

0

,I3”(~)

0

U3U)(~)

(B.16)

143(1~~)

â~I”(~)

Since the second and third lines of ~it are linearly dependent, the determinant vanishes identically, 1 symmetry of the fixed points at 0, for any choice of the function ~ This occurs because of the S the equator when A 3 = A,. The phase space portrait consists of two centers at the poles of the sphere and the degenerate set of fixed points at the equator. The orbits are all closed and parallel to the equatorial plane, thus they are circles (see fig. B.1). The sense of rotation changes for orbits below or above the equator, and the azimuthal angular velocity increases as one goes away from the equator; this’ can be seen from the equation of motion for the variable 4,: 4, =(A2— A1)rcosO.

(B.17)

B.2. Case 2. b=(0,0,0), W=diag(A,,A2,A3) For this case, the equations of motion have six fixed points: â=(0,±r,0),

i.e.,sin0=0;

a

i.e.,

Ô = ir/2, sin ~ =

i.e.,

~=

=

(±r, 0,0),

â=(0,0,

±r),

0;

(B.18)

~I2, ~ =

/

/ Orientation

/~i

H Fig. B.1. Phase space portrait of the reduced system for case 1.

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

351

fl~P~ a. M

Orientation

~

= -6

c.M=0

d.M=0.5

f.M=1.5

=

2I”(~) 0 r 0 A 1 A2 0

e.M=1

g.M=2

Fig. B.2. Evoiution of the reduced phase portrait for

±r,

-0.5

~, ~

~

We consider first a = (0,

b. M

h.M10

ease

2 as the parameter M

=

)t

3IA1 is varied.

0), i.e. the poles, for which the Hessian matrix is 0

—

0

0

A3—A2

.

(B.19)

This matrix is definite, when either of the following sets of conditions are satisfied: 2~”(~)>0, either (A3 A2) >0, (A, A2) >0, r (B.20) or (A 2I”(~)<0. 3—A2)<0, (A,—A2)<0, r Adequate functions 1( ~) exist for each of these cases. Looking at the first two inequalities of each of the two above conditions, we see that the poles are stable fixed points (centers) if, and only if, A 2 is the smallest or the greatest among the three quantities A1, A2, and A3 if this is not satisfied, then the poles are saddle points. The results for the four other fixed points are similar. For ~ = (±r, 0, 0), the stability condition is that A, be the smallest or the greatest among A,, A2, and A3. Finally, a = (0,0, ±r)are stable fixed points whenever A3 is the smallest or the greatest among A,, A2, and B .2system shows 2 forA3.theFigure reduced the typical orbits which are to be expected in the phase space portrait on S (B .5) (the figures are labelled by M A 3/A,). The sequence of pictures shows how the phase portrait evolves as the parameter A3 is varied while A, and A2 are kept fixed. The bifurcations which occur are —

—

the same as those for the rigid body, see Arnold [1978].The bifurcation sequence for orbits depicted in fig. B .2 is shown schematically in diagram B .2.

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

352

Diagram B.2. Bifurcations for case 2. The thick plain and dotted lines represent centers and saddle poinls, respectiveiy, and the thin plain line represents curves of fixed points. N = (0, r, 0), S = (0, —r, 0), F= (0,0, r), B = (0,0, —r), L (—r, 0,0), R = (r, 0,0).

B.3. Case 3. b = (0, b2, 0), W= diag(A,, A2, A,)

In this case, we have fixed points at the two poles and a line of fixed points on a circle parallel to the equatorial plane. Explicitly, these critical points are â=(0,±r,0),

i.e., ~=0and

~=~r; (B.21)

a=(u,,b2/(A,—A2),u3)ja~=r, i.e.,~arbitrarywithcosè=b2r’(A1—A7)’. Let us begin with the fixed points at the poles, diagonal,

a

=

(0, ±r, 0). We find that the Hessian matrix is

2~”(~)±a r

2Ir

0 0

=

A,

—

0 A2 ~ b2/r 0

0 0

.

(B.22)

The conditions for ~4tto be definite then read as either of the two following pairs of conditions: 2ii”(~)> ±a either A, A2 ~ b2Ir>0, r 2Ir, (B .23) 2’Ii”(~)<±a or A,—A2+b2Ir<0, r 2Ir. —

Here again, there always exist choices of functions 1( ~) for which the above conditions will be satisfied, irrespective of the magnitude r of the beam intensity. Hence, the poles are always stable, i.e.

353

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

centers. In the special case when A, A2 = ±b2/r, two of the eigenvalues of 4~vanish. This, however, does not affect the above conclusions, namely that the poles are stable centers, as can be seen by integrating the equations of motion (B.5). Indeed, it is easily found that the orbits are given by 0 = 0~= constant and 4, = b2(1 ±cos 00)t, i.e., they are circular orbits at constant latitude 0~: —

—

hence the poles are centers.

Let us now examine the stability issue for the circle of fixed points in (B .21). The Hessian matrix is +

2c2â21r + A2

—

O,02K+c20,Ir + c2u3Ir

A,

I2,02K +

c20,!r ~203K + c2031r

i2~K

O,â3K

,

(B.24)

O,03K

3. The last two rows (or columns) of this matrix are linearly dependent, so where K 0c1”( ~) c2ü21rthe circle of fixed points is S1 degenerate for isotropic materials. The phase that identically; space portraits for this case are similar to those for case 1, except that the circle of fixed points is no longer on the equator (see fig. B.3, where we have set b —1), butbyat setting a latitude given by 2, is2 =recovered b 0 = arccos[b2Ir(A1 A2)]; the latitude for case 1, 0 = IT! 2 = 0 in the —

—

preceding formula. According to (B .5), the angular velocities on the circular closed orbits are given by ~=b2—(A,—A2)rcos0.

(B.25)

B.4. Case 4. b = (0, b2, 0), W= diag(A,, A2, A3) For this case, we may have up to six distinct fixed points. Two of them are the poles ~ = 0, IT or a = (0, ±r, 0); these are always present. The four other fixed points, however, exist only under certain

conditions and their nature, as we will see below, is related to the nature of the fixed points at the poles, i.e., whether the poles are centers or saddle points. These four additional fixed points are given by

n N

—

Orientation

Fig. B.3. Evolution of the reduced phase portrait for case 3 as the parameter M is varied; b, =

—1.

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

354

a

=

a

=

(±\/r2 b~/(A1 A2)2, b2/(A, —

—

(0, b21(A3

—

A2), ~

—

A2), 0),

i.e., sin ~

=

1, cos 0 = b2Ir(A,

i.e., sin ~ =0, cos

=

b2Ir( A.,

—

A2),

(B.26)

—

A2).

(B.27)

Let us first consider the poles. The Hessian matrix is diagonal,

.111=

0 0 A,—A2~b2Ir 0 0 A3—A2÷b2/r

0 0

(B.28)

.

The conditions for stability are thus 2t1”(~)>±a A1—A2~b1Ir>0, A3—A2~b2/r>0, r 2Ir, 2cI”(~)<±a or A,—A2÷b2Ir<0, A3—A2~b2Ir<0, r 2Ir.

either

(B.29)

Notice that (unlike the rigid body) stability in this case depends upon the intensity r of the beam. Since the quantities A,, A2, A3 are fixed parameters for any given optical medium, it follows that some beam intensities r might be too strong, or too weak, to enforce the two stability conditions given in terms of A1, A2, and A3, thereby resulting in instability. Now, whenever the first two inequalities in (B.29) are satisfied, it is possible to find functions c1~(~) such that the third inequality is satisfied as well. Hence, the stability conditions (B.29) for the north and south poles become, respectively,

(north pole),

(A1

—

A2

b2Ir)(A3

—

A2

—

b2Ir)>0

(A,

—

A2 + b2Ir)(A3

—

A2

+

b2/r)>0 (south pole).

—

(B.30)

All other situations yield instabilities of either, or both, of the poles. 2, notably its property of being a We point out that the topology of the reduced phase space, S manifold without boundaries, permits various configurations of critical points which do not occur on manifolds with boundaries; for instance, a pair of centers is allowed on S2 but not on R2 (i.e., the noncompleted real plane). The stability conditions on the poles have a direct impact on the nature of the four other fixed points defined through (B.26) and (B.27). Consider first those fixed points given by (B.26) for which the Hessian matrix is [as in case 3, we use the quantity K = 1”( ~) c 3] —

2i221r

~

~

0 A3—A1)

Ka~+2c 2a2!r+A2—A, Ki2,t22+c2â,Ir 0 K~,a2+c2a,Ir Kü~ 0 0 0 A3—A,

The matrix ill is therefore positive (negative) definite if, and only if, A3 above is positive definite. Noting that can be reduced to /A —A .~j=ü~det~ c2/r

cIr~ C)

—

.

(B.31)

A, >(<) 0 and the matrix ~

(B.32)

D. David rt al., Hamiltonian chaos in nonlinear optical polarization dynamics

355

implies that the fixed points given in (B.26) are stable, provided either of the following sets of inequalities are satisfied: 2/K, A,)> (c2/r) (B.33) or A 2!K. 3—A,<0, K<0, (A2—A,)<(c2!r) For the first set of inequalities, K > 0 implies A 2 A, > 0 for the third inequality in this set to be satisfied. Similarly, for the second set, we need A2 A, <0. Thus, we may rewrite conditions (B.33) as 21(A either A2>A,, A3>A,, K>(c2/r) 2—A,), (B.34) 2/(A or A2
either A3

—

A, >0,

K >0,

(A2

—

—

—

Now, for either case, it is always possible to construct a convenient function ~P(4:) such that the third inequalities in (B.33) and (B.34) are satisfied. Hence the fixed points given by (B.26) are stable, provided only that (A2 A,)(A3 A,) >0, i.e., that A, is not the intermediate eigenvalue. Notice, however that these critical points only exist when the square root in (B.26) is real valued. The stability analysis for the other pair of fixed points, determined by eq. (B.27), is the same as the previous analysis, up to permutation of the indices 1 and 3. Thus, if (A2 A3)( A, A3) >0, then these fixed points are stable, provided u3 is real in (B .27). The dynamics taking place on the reduced phase space for this case is illustrated by the sequence of pictures given in fig. B.4 (figs. B.4A and B.4B give views of the northern and southern hemispheres, respectively). We have chosen r = 3, A2 = 0, A, = 1, and b2 = —1. Thus A, A2 = 1 >0 and, therefore, the fixed points given by formula (B.26) exist for all values of A3 and are saddle points [provided the square root in (B.26) is real] when A3 > 1. The fixed points given by (B.27) exist provided IA3~ 1/3, and they are saddle points only if 1/3
—

—

—

~2

—

A3<—~

—~
~
A3>1

north pole

saddle

center

center

center

south pole (0,, 02,0) (0, 03)

saddle center center

saddle center

center center saddle

center saddle center

02,

—

According to this table, three bifurcations occur as A3 (or M) passes through the values —1/3, 1/3, and 1. Asymptotically, as A3 —oo, the great circle defined by cos 4, = 0 becomes a line of fixed points and all other (regular) orbits are circles parallel to that great circle; in addition, there are two centers lying on the equator at 4, = 0, ir. For (large) finite negative values of A3, each of the poles is the asymptotic connecting point for two homoclinic loops (see figs. B.4a); the asymptotic phase portrait for A3 results as these homoclinic loops ultimately extend from one pole to the other, thus merging together to create the above mentioned pair of curves of fixed points. As A3 is increased, each of the two figure-eight patterns (generated by a polar saddle point and its two associated homoclinic loops) gradually gets smaller (see figs. B.4b—d). This shrinking happens faster for the northern hemisphere pattern, which actually vanishes when A3 reaches the value —1/3, at which point a first bifurcation —~

—~

356

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

)rtCflLjtloIl

H

.~‘ .7

a NI = -10

h NI

-

M

-~

t

=

-06

//_

-

o M=tJ

-

..,

-

\I=(lln

d \I

~

(1

-

g.\I=1)4

Ii

.~

-

=

NI=0S

/‘

I~. .--~:-.r

I

\I =

ft.

\I = I

k

\I =4

NI =

0

3, A,

=

(A) Fig. B.4. Evolution of the reduced phase portrait for case 4 as the parameter Mis varied~b 2 (B) southern hemisphere.

=

—1, r

=

1,

~

=

0. (A) Northern hemisphere.

occurs as the north pole transforms from a saddle point to a center. The topology of the phase portrait then remains unchanged (see figs. B.4e) until A3 = 1/3. As A3 approaches this value, the angular aperture at the south pole for each of the two remaining homoclinic loops (the difference between the incidence angles of each branch of the loop, or, equivalently, of the stable and unstable manifolds of the saddle point) increases. At the bifurcation value A3 = 1/3, this aperture is equal to ~ thus the two ioops become tangent to each other. The south pole then undergoes a transition from saddle point to center, while two new saddle points appear near the south pole and start to move along the great circle defined by sin 4, = 0 (see figs. B.4f—i). These new saddle points reach a circle C0 parallelling (and below) the equatorial plane when A3 = 1. At this point, the phase portrait is identical to that of case 3 (see figs. B.3 and B.4j); namely, C0 is a curve of fixed points and all orbits are circular and parallel to C0. This curve of fixed points is unstable (as shown in case 3) and immediately breaks 2, down while into two four new heteroclinic orbits connecting two saddle points, which emerge on C0 at 4, = ±IT! centers appear, also on C 0, at 4, = 0, ir in addition to those at the poles. As we continue to increase the value of A3, these heteroclinic orbits close down upon the poles as a pair (see figs. B.4k—~). Asymptotically, as A3—s~~, they merge together to form a pattern similar to that when A3—~ i.e., the poles eventually lie on the great circle of fixed points given by cos 4, = 0. This bifurcation sequence is shown schematically in diagram B.4. —~,

357

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

Orientation

____

~

a. M = -~0

~&

b.M

~\

e.M=0

c. M =

= -2

-0.36

~f=~H-~

f.M=0.36

~J

h.M=0.5

g.M=0.4

~

~\

~

~

~

~

i.M=0.8

d. M =

-0.5

~

k.M=4

j.M=1

l.M=10

(B) Fig. B.4. Cont’d.

F F

N

S

Q

--

B

K M=-~’

M=-t/3

M=1/3

M=t

M=~

Diagram B .4. Bifurcations for case 4. The thick plain and dotted lines represent centers and saddle points, respectively, and the thin plain line represents curves of fixed points. N= (0, r, 0), 5 = (0, —r,0), F= (0,0, r), B = (0,0, —r), L= (—r, 0,0), R = (r, 0,0), J = (V~,—1,0), K= (—VL —1,0), P = (0, —1, V~),0 = (0, —1, —Vs).

358

D. Davtd et a!.. Hamiltonian chaos in nonlinear optical polarization dynamics

B.5. Case 5. b

(bk, 0, b3), W= diag(A1, A2, A3)

=

The first set of two fixed points for this case are 2 b~/(A, A,)2 b~/(A 2, b a = (b,/(A2 A1), ±\/r 2 A3) 3/(A2 —

—

—

—

—

—

A3)).

(B.35)

The other fixed points are of the form (B.36) with constraints 2, +

=

(A

r

3

—

A,)0,03 + b301

—

b~03= 0.

(B.37a, b)

These constraints are satisfied by up to four additional critical points, the number of which depends upon the values of the various parameters present in the problem. From (B.37b), we deduce that 03

=

b3i211[b,

+

(A,

—

A3)0,I

(B.38)

.

Substituting this into (B.37a) we obtain a quartic equation for ü~,whose solutions are the real roots of the polynomial 2— + w)v + ~(w±~w2 + 4~r), (B.39) + (~1 ±~r

where

f3~=

b,/(A,

—

A

3). and w is any real solution of the cubic equation 3—(+—r2)w2—4f3~j3~0.

(B.40) w Let us begin by finding the stability conditions for the fixed points given by formula (B.35). The Hessian matrix takes the form —2 Ku 2

—

-

—

—

Ku,u2+c,u2/r

—

—

Ku3u2+c3u2Ir

~ll= KO,O2+c~0,/r (A1—A2)+K0~+2c~0,/rK0,t23+(c,~3+c30,)/r K0302 + c302/r

K0103 + (c,03 + c3t2,)/r

(A3

—

,

(B.41)

A2) + K0~+2c303/r

3. The stability conditions are, as usual, that .ill be definite. Positive where K = 1”(definiteness, ~) (c, 0, +respectively, c303) /r and negative are prescribed by the following sets of inequalities: —

K>0, (A,—A

2>0,

(A

2)K—(c1/r)

2]—(A,—A 3—A2)[(A,—A2)K—(c,/r)

2>0; 2)(c3/r) (B .42a)

K<0, (A

2>0, 1—A2)K—(c,/r)

(A

2]—(A,—A 3—A2)[(A,—A2)K—(c,/r)

2<0. 2)(c3/r)

(B.42b)

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

359

We can rewrite these in a simpler form. In fact, using the first inequalities with the second ones and the first two with the third ones, it is seen that the above conditions are respectively equivalent to A,>A2 ,

A3>A2 ,

A,
A3
2/(A, —A 2/(A K>(c,/r) 2)+(c3!r) 3—A2) ; 2/(A, —A 2!(A K<(c,/r) 2)+(c3Ir) 3—A2).

(B.43a) (B.43b)

Furthermore, one can always choose functions 14:) such that the third inequalities are satisfied and hence the stability conditions reduce to (A,

A2)(A3

—

—

(B.44)

A2)>0.

Thus, the fixed points under consideration are centers if, and only if, A, is either the smallest or the greatest among the quantities A,, A2, A3. Let us now examine the fixed points given by formula (B.36) with constraints (B.37). The Hessian matrix is

~~A2—A,)—b,IO, =

(

KO~+ 2c,i2,!r

—

~3), b,/0,

(B.45) KO,03 + (c,03 + c3t2,)!r ,

+ (c103 + c30,)/r

K0~+ 2c3ui3!r

—

(B.46)

b3/03

and K is defined as above. The conditions for .At to be positive or negative definite read, respectively, 2>0; (B.47a) (A2—A1)—b,h2,>0, ~[‘,>0, ~13j=’I’,~P3—[KO,O3+(c,O3+c3i2,)/r] (A 2 >0, (B.47b) 2 A,) b,IO, <0, <0, ~ = ~iç [KO,03 (c103 + c30,)!r] where —

=

)[‘,

—

KO~+ 2c,O,!r

—

—

—

b,/0 1 ,

~

=

Kt2~+ 2c3t23!r

—

(B.48)

b3!03.

We can reexpress these conditions as follows. For conditions (B.47a), the second and third inequalities imply ~ >0; similarly, for conditions (B .47b), we deduce that 1P3 <0. Thus we can rewrite (B .47a, b) as 2!1I.’ (A2

—

A,)> b,/O, ,

V.’, >0,

‘W,

>

[KO~ii3+ (c,i23

+

c30,) /r]

1 >0; 2iW~<0.

(B.49a) (B.49b)

(A2— A,)
D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

360

of the equations of motion in their neighborhood on the reduced space and using formula (B.13). In this case, the right-hand sides P and Q of the equations of motion for 0 and 4,, respectively, are ~2

P = b, cos 4,—b3 sin 4, + (A,

Q=

A3)r sin 0 sin 4, cos 4,, 24, + A 24, —(b, sin 4, + b3 cos 4,) cot 0 r(A, sin 3 cos —

—

A 2) cos 0.

—

One can verify that OP!00 and OQ!04, vanish identically at the fixed points since these quantities have an overall multiplicative factor cos 0 R

=

(02

=

0~cos0

=

0), so that (B.13) reduces to

±y(0PI04,)(~9Q/80),

(B.50)

where 24, =

—(b, sin 4,

b3 cos 4,) + (A,

+

—

sin24,),

A3)r(cos

—

oQ/00 = (b, sin 4, + b

(B.51) 24,

24, + A

3 cos 4,) + r(A, sin

—

A

3 cos

2).

Orientation

K H~ ~

I~

~H/ ~

~

M=-036

‘.:

CM~2

~

~

g

K

h M~-03

I

M—0

Fig. B.5. Evolution of the reduced phase portrait for case 5 as the parameter M is varied; b,

=

=

—1, r

=

3,

A,

=

1,

A2

=

0.

D. David et al., Hamiltonian chaos in nonlinear optical polarization dynamics

361

These fixed points are stable (Le. centers) whenever aPIO4, and dQIoO have opposite signs, and unstable (i.e. saddle points) when they have the same sign. Hence, via (B.51), instability follows from either of the following conditions:

24, + A 24, A 3 cos 4,> —r(A, sin 3 cos 2), (B.52) 24, sin24,) < b, sin 4, + b 24, + A 2cb A or (A, A3)r(cos 3 cos 4, < —r(A, sin 3 cos 2). 2 illustrate graphically in fig. of level surfaces of the Hamiltonian (B .4) onthat S are likely to occur for this B.5Numerical the types calculations of distributions of fixed points, orbits, and bifurcations case. We choose b 1 = b3 = —1, A, = 1, A2 = 0, r = 3, and vary the parameter A3. Four bifurcations occur. When A3 is large and negative, we observe two saddle points near the poles connected by four heteroclinic orbits, and four centers lying on the equator (see fig. B.5b). All orbits are closed; those around the opposing centers C,,2 rotate in one direction and those around the other pair C34 rotate in the opposite direction. As A3 the above mentioned saddle points S,2 are identified with the north and south poles, respectively. Moreover, the heteroclinic orbits collapse pairwise to two half great either (A,

—

A3)r(cos24,

—

sin24,)> b, sin 4, + b

—

—

—

—

—~ —~,

k.M=O.058

j.M=O.04

n.M=t

m.M=0.5

~H

~Ji1~\\

p.M=2

~

o.M= 1.8

4~HI~

~

\~\‘~\

q.M=4

S.

M= 20

Fig. B.5. Cont’d.

r.M=10

D. David et al, Hamiltonian chaos in nonlinear optical polarization dynamics

362

circles of fixed points given by 4,

As for the four centers, C34 vanish and C,2 have their 4, respectively. In this limit, there remain only two families of closed circular =

±ir/2.

coordinate tend to 0, IT, orbits around C1,2. As the value of A3 is increased to finite negative values less than 1/3, S1 .2 gradually move towards 2. the getting nearer to eachthe other, while their 4, coordinate increases from IT! (Seeequator, figs. B.5c—h. Note especially “ladybug” shapecommon in fig. B.5e.) At the same time, away the two regions of closed orbits around the centers C 23 shrink until A3 reaches a critical value near —1/3. The first bifurcation occurs there as S12 and C2 collide and disappear to leave a single saddle point S3. During this bifurcation, two of the heteroclinic orbits vanish and the two remaining separatrices transform into a pair of homoclinic loops connected at S3 (see fig. B.5i). As we continue to increase the value of A3, the homoclinic loop encircling the center C3 diminishes in size (see figs. B.5j, k) until A3 gets to a critical value near 0.06, whereupon a second bifurcation occurs as C3 collides with S3. At the bifurcation value, the homoclinic loop which did not shrink forms a cusped closed orbit, which immediately smoothens as A3 continues to increase (see figs. B.5~—n).At this point, the phase portrait thus consists of two centers (C14) and closed orbits. While this phenomenon is taking place, a third bifurcation takes place, when A3 reaches a critical value near 0.1, on the equator near 4, = 0, as C, transforms into a saddle point S4 with two homoclinic orbits enclosing two new centers C56 lying above and below the equator, respectively (this pattern begins to be visible on fig. B .5o). A final bifurcation (invisible on figs. B.5), identical to the second but in reverse order, occurs when A3 passes by a critical value near 2, at which point one of the closed orbits becomes singular by developing a cusp on the equator near 4, = irI2. This cuspidal point then gives rise to a saddle point S, and a center C7 lying 2, alsorespectively, on the equator. As A3 while C becomes large and positive (see figs. B.5p—t), S45 move towards 4, = ±IT! 56 migrate towards the north and south poles, respectively; as for C37, they move towards 4, = ir and 0, respectively. These six critical points tend toward these limits as A3—~ci~. The bifurcations that we have just described are summarized in diagram B.5. —

—

N

N

F p

P

V

V

.

L

....

K

w

.

B

.

Q

, ,.

w

.

,,.,,,‘

K S

K

B

,,..‘

S

~

M=~=

\~—~/..S

~l.~S/3

‘M—I/,F

\I=I+’.S/.’

Diagram B.5. Bifurcations for case 5. The thick plain and dotted line represent centers and saddle points, respectively, and the thin plain line represents curves of fixed points. N = (0, r, 0), 5 = (0, —r, 0), F= (0,0, r), B = (0~0,—r), L = (—r,0, 0), R (r, 0,0). 3 = (1,0, V~),K = (1,0, -Vs), P=(V~/2.0,vi~I2), Q=(V~I2,0.-\/~!2),V=(-V~/2.0,V~/2),W=(-v’~/2,0,-\/~I2).

363

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

B.6. Case 6. b=(b,,b2,b3), W=diag(A,,A2, A3) For this last case, the fixed points are given by 02

=

b20,![b, + (A,

—

A2)O,],

03

=

b30,![b1 + (A,

—

A3)O,],

(B.53)

where 0~is a solution of the following sixth-order equation: .6

+

‘5 2(y2 + y3)b,u, +

2 [(72

+ 2[(y, + y,)(y273b, 2

—

+ [7273(b, + b2 + b3) 222

2

r

+ 47,73 + 2

2 2 y3)b, + 72b2 + 73b3 r 22

22

2=4

—

) + y273(72b2 + 73b3)]b1u,

2

22

2 — (72

22

+ ~Y2Y3 +

.~3

y3)r ]b,u1 22

2=2

—

27,73(72 +

y3)r b,u, 23=

7273b1 224

—

=

0, (B.54)

with

72

1!(A1

=

—

A2) and y~= 1/(A,

—

A3). The Hessian is

Ki2~+2c2t2,/r—b,!O, Kz2102+(c,02+c201)/r KO,03+(c,i23+c30,)/r =

K01L22 + (c102 + c20,)/r KO,i23 + (c,03 + c30,)!r

KI2~+

2c202/r

—

b,h22

K0203 + (c2t23 + c302)!r

K0203 + (c203 + c302)!r KI2~+2c303/r

—

,

(B.55)

b3!113

3. The stability conditions for the Hessian to be definite in sign are that either where = ‘1”( ~) of the K following setsc~ ofâ!rinequalities are satisfied: —

either .A(~~>0, or

~

~11~~22

1.1~22~i2>0, .~~i2>0,

Ijtt~>0, I=4t~<0.

(B.56)

As in case 5, we have been unable to write these stability conditions in a manageable analytical form; it is best to perform a numerical calculation of how the various fixed points change stability as the values of Wand b are varied, by examining the level surfaces of the Hamiltonian (B.4) on the reduced space As an illustration of the dynamical features associated with this last case, we consider the choice of parameters r = 3, A, = 1, A 2 = 0, b, = —1, b2 = —2, b3 = —1, and vary the parameter A3. The evolution of the phase space portrait is depicted on figs. B .6. When A3 (or M) is large and negative, the portrait consists of a saddle point 5, near the north pole, a center C, near the south pole, and two additional centers C23 slightly below and above the equator, having a coordinates around 0 and IT, respectively. Moreover, C23 are enclosed within homoclinic orbits connected 2, with (see fig. 0),5,(IT!2, IT), B.6b). As A3—~ —cc, S1 tends toward the north pole while C23 move to (i/i, a) = (IT! respectively; in addition, the two homoclinic orbits asymptotically approach the great circle defined by cos a = ±~‘2 (C, tends to the south pole and then vanishes as the homoclinic orbits merge together). This great circle is a curve of fixed points and all of the orbits are closed and circular, parallel to that curve. As A 3 approaches a ciritical value near —1.4, the homoclinic orbit enclosing C3 shrinks (see figs. B .6d—f). When this critical value is reached, 5, and C3 collide together and vanish as the homoclinic

D. David et a!., Hamiltonian chaos in nonlinear optical polarization dynamics

364

/

//

/

~

/

Orientation //(,

-

-____

~\

,\

\~ 7’’

HK

H

~

/

a.M=-II)

~H

/

I

b M=-4

c

M

~

____ d

M=

16

H

__~~ e

M=-145

/ f.M-l

~Hi~ ~I~>

~ ~ g.M=0 ~I’ h.M=2 ~ t MIO Fig. B.6. Evolution of the reduced phase portrait for case 6 as the parameter M is varied; b = b

3 = —1, b2

—2, r = 3,

A~=

1. A~= 0.

loop around C, becomes smooth. At this point the phase portrait consists of closed orbits and two centers C, and C2 (see fig. B.6g). This configuration remains unchanged (except for movements of C,2) until A3 reaches a second critical value near 2, when one orbit develops a cusp, thereby giving rise to a new saddle point a new center C4, and a pair of homoclinic orbits surrounding C, and C4 this bifurcation is similar to the first one, but reversed (not visible on figs. B.6). As A3 becomes large and positive, S2 moves towards the south pole; so do C,4 while C2 approaches the north pole (see figs. B.6h—j). Asymptotically, as A3 cc, the phase portrait is the same as for A3 cc, except that the orbits have reversed .directions. Diagram B .6 summarizes these bifurcations. This concludes our qualitative analysis of the one-beam problem. The above work is complete, as far as dynamical behavior is concerned, only for the first four cases. For cases 5 and 6, and especially for the last one, the problem becomes very complex and needs to be further investigated. This complexity in qualitative behavior arises because there are six parameters in the problem (two ratios of the As’s, three of the b1’s, and the beam intensity r). The situation is more tractable in the study of the two-beam problem in the text when the vector b is set equal to zero. However, all of the present complexity and more will arise in the study of the two-beam problem with birefringence (b 0). ~2’

—~

—~ —

D. David et a!., Hamiltonian chaos in nonlinear optical polarization

dynamics

365

N

B

F

S

—

M=-=

M=-O.25

----

M=2

-

M==

Diagram B.6. Bifurcations for case 6. The thick plain and dotted lines represent centers and saddle points, respectively, and the thin plain line represents curves of fixed points. N =

(0, r, 0), S = (0, —r, 0), F = (0,0, r), B = (0,0, —r).

The bifurcation analysis presented here sets the stage for studying homoclinic and heteroclinic chaos in these systems under periodic perturbations. Examples of homoclinic chaos are given for the

two-beam problem in section 6 and for the one-beam problem in appendix A. References R. Abraham and J.E. Marsden, 1980, Foundations of Mechanics, 2nd Ed. (Benjamin/Cummings, Reading, MA). J.M. Arms, J.E. Marsden and V. Moncrief, 1981, Symmetry and bifurcations of momentum mappings, Commun. Math. Phys. 78, 455—478. V.1. Arnold, 1964, Instability of dynamical systems with several degrees of freedom, Soy. Math. — Dokl. 5, 581—585. V.1. Arnold, 1973, Ordinary Differential Equations (MIT Press, Cambridge, MA). V.1. Arnold, 1978, Mathematical Methods of Classical Mechanics (Springer, New York). V.1. Arnold, 1982, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York). N. Bloembergen, 1965, Nonlinear Optics (Benjamin, New York). M. Born and E. Wolf, 1959, Principles of Optics (Pergamon, Oxford). P.N. Butcher, 1985, Nonlinear Optical Phenomena Bulletin 200 (Ohio State Univ., Columbus, OH). E. Caglioti, S. Trillo and S. Wabnitz, 1987, Stochastic polarization instability: limitation to optical switching using fibers with modulated birefringence, Opt. Lett. 12, 1044—1046. B. Crosignani and P. Di Porto, 1985, Intensity-induced rotation of the polarization ellipse in low-birefringence, single-mode optical fibers, Opt. Acta 32, 1251—1258. B. Daino, G. Gregori and 5. Wabnitz, 1986, New all-optical devices based on third-order nonlinearity of birefringent fibers, Opt. Lett. 11, 42—44. AL. Gaeta, R.W. Boyd, JR. Ackerhalt and P. Milonni, 1987a, Instabilities and chaos in the polarizations of counterpropagating light fields, Phys. Rev. Lett. 58, 2432—2435. AL. Gaeta, R.W. Boyd, P.W. Milonni and J.R. Ackerhalt, 1987b, Instabilities in the propagation of arbitrarily polarized counterpropagating waves in a nonlinear Kerr medium, in: Optical Bistability III, eds. H.M. Gibbs, P. Mandell, N. Peyghambarian and S.D. Smith (Springer, Berlin). FR. Gantmacher, 1959, The Theory of Matrices (Chelsea, New York) Vol. 1, Ch. X. G. Gregori and S. Wabnitz, 1986, New exact solutions and bifurcations in the spatial distribution of polarization in third-order nonlinear optical interactions, Phys. Rev. Lett. 56, 600—603. J. Guckenheimer and P. Holmes, 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematics Sciences, Vol. 42 (Springer, New York).

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