The nonlinear dynamics of doubly-degenerate vector four-wave mixing (FWM) are studied analytical and numerically, in phase space and in Stokes space. Depending on the initial conditions, vector FWM can evolve aperiodically or periodically, but not chaotically. The dynamics of vector FWM are similar to, but richer than, the dynamics of scalar FWM.
©2005 Optical Society of America
The nonlinear dynamics of scalar three- and four-wave interactions have been studied since the 1960s [1, 2, 3]. Three-wave interactions occur in χ (2) media and satisfy the frequency-matching condition ω 3=ω 1+ω 2, where wave 3 is the (higher-frequency) pump, and waves 1 and 2 are the (lower-frequency) products. If the associated wavenumber-matching conditions are also satisfied, the products grow until the pump is depleted. If the pump power is high, the effects of self-phase modulation (SPM) and cross-phase modulation (CPM), which depend on χ (3), are also important. By modifying the pump and product wavenumbers, SPM and CPM can detune the interaction before complete pump depletion occurs. Four-wave interactions occur in χ (3) media. Degenerate interactions satisfy the frequency-matching condition 2ω 2=ω 3+ω 1, where wave 2 is the pump and waves 3 and 1 are the products. Nondegenerate interactions satisfy the frequency-matching condition ω 2+ω 3=ω 4+ω 1, where waves 2 and 3 are the pumps and waves 4 and 1 are the products. Since four-wave mixing (FWM), SPM and CPM all depend on χ (3), their effects are always of comparable importance. The effects of pump depletion, SPM and CPM on scalar three-wave interactions, and degenerate and nondegenerate scalar four-wave interactions, have been described in detail [4, 5, 6].
FWM occurs repeatedly in optical communication systems. For example, signal-signal FWM in multi-channel systems produces coherent perturbations that degrade the signal quality. These perturbations are minimized by the use of polarization interleaving, in which configuration each channel is perpendicular to its neighbors . Other examples are parametric devices, such as amplifiers [8, 9], frequency-convertors [10, 11] and regenerators [12, 13], which are based on FWM. The output powers and polarizations of the processed signals and generated idlers depend on the input powers and polarizations of the pump(s) and signals. Since transmission fibers are not polarization-maintaining, these devices are required to work for all input signal polarizations.
The initial (linearized) evolution of vector FWM was discussed in a recent review . In this paper a preliminary study is made of the nonlinear evolution of vector FWM. Each wave is characterized by its Jones vector |A〉, which contains two complex amplitudes (one for each polarization component). Thus, the nondegenerate-FWM equations involve sixteen real variables and the degenerate-FWM equations involve twelve . Such large sets of equations are daunting. Recall that in degenerate FWM, idler generation is seeded by a weak input signal, and the signal and idler powers become comparable after a few gain lengths. In this paper a reduced set of FWM equations is considered, in which the Jones vectors of the signal and idler are set equal a priori. Doubly-degenerate vector FWM satisfies the frequency-matching condition 2ω 2=2ω 1. In a fiber with random birefringence, it is governed by the equations
where D denotes the spatial derivative d=dz, the hermitian operators
and γ is the nonlinearity coefficient of the fiber . In Eq. (3) βj is a wavenumber, 〈Aj |jAj 〉 is the inner product of |Aj 〉 with itself, k is the Stokes vector of wave k≠j and is the Pauli spin vector. The Jones-vector and Stokes-vector (Poincaré-sphere) formalisms, and many mathematical identities that relate them, were described in detail by Gordon and Kogelnik . These notations and results will be used throughout this paper, without further comment. We were motivated to study doubly-degenerate FWM to test mathematical methods and develop physical insights, which could be applied to subsequent studies of degenerate and nondegenerate FWM. Nonetheless, the reduced FWM equations (1) and (2) describe a physical interaction between two waves with the same frequency, but different spatial eigenmodes, in a multi-mode fiber or other three-dimensional medium. For example, filamentation involves a transfer of power from the lowest-order mode of a beam to a set of higher-order modes .
Like their counterparts for degenerate and nondegenerate FWM, Eqs. (1)–(3) describe the effects of linear and nonlinear (SPM- and CPM-induced) wavenumber mismatches, and (CPM-induced) polarization rotation, on the power transfer between the waves. Terms of the form k · cause the Stokes vectors of the waves to rotate rigidly about a common axis in Stokes space . This rotation has no effect on the power transfer between the waves. The SPM and CPM terms are similar to the corresponding terms in the scalar-FWM equations. The nonlinear detuning of scalar FWM has been discussed in detail [5, 6]. For the purposes of this paper, it is sufficient to consider the equations
This paper is organized as follows: In Section 2 the lagrangian and hamiltonian formalisms associated with the vector FWM equations (4) and (5) are described. These formalisms are used to determine the conserved quantities associated with the FWM equations and a reduced set of FWM equations called the inner-product equations. In Section 3 the inner-product are solved analytically, and the phase space associated with them is illustrated. In Section 4 the FWM equations are rewritten in terms of the action (photon) fluxes and phases of the polarization components of the waves. These flux-phase equations confirm the predictions of the inner-product equations and reveal aspects of the dynamics that are not described by the inner-product equations. In Section 5 numerical solutions of the FWM equations are displayed in Stokes space, to illustrate the polarization dynamics of FWM. The effects of fiber loss and polarization-mode dispersion (PMD) are discussed briefly in Section 6. Finally, in Section 7 the main results of this paper are summarized.
2. Dynamical formalism and conserved quantities
in which the canonical displacements are |Aj 〉 and its hermitian conjugate 〈Aj |, together with the Euler-Lagrange equations
and their hermitian conjugates. In the lagrangian formulation 〈Aj |=2i is the canonical momentum associated with the canonical displacement |Aj 〉 and -|Aj 〉/2i is the momentum associated with the displacement 〈Aj |.
It follows from Eq. (6) and the definition
that the Hamiltonian
The FWM equations are equivalent to the Hamiltonian (9), together with the Hamilton equations
and their hermitian conjugates. In the hamiltonian formulation |Aj 〉 and 〈Aj | are canonically-conjugate variables. (There is no factor of 2i in the definition of the canonical momentum.)
Since the Hamiltonian (9) does not depend explicitly on position, it is a constant of motion. Other constants of motion can be deduced from physical considerations. Since the inner products 〈Aj |Aj 〉=Fj are photon fluxes , it is natural to derive equations for all the inner products 〈Aj |Ak 〉. By combining the Hamilton equations (10), which are identical to Eqs. (4) and (5), one finds that
One can often facilitate the study of polarization effects by relating evolution in Jones space to evolution in Stokes space. Each Jones vector jAji is associated with the Stokes vector j =〈Aj | |Aj 〉. For reference, | j |=〈Aj |Aj 〉. By combining Eqs. (4) and (5), one finds that
Even in the presence of FWM, the total Stokes vector = 1+ 2 is constant. By replacing the operator with any other hermitian operator O, one can derive an infinite family of conserved quantities. Equation (14) corresponds to the choice O=I, where I is the identity operator. Since every linear operator can be expressed as a linear combination of the identity and spin operators, the aforementioned conserved quantities are linear combinations of the fluxes and Stokes-vector components in Eqs. (14) and (18).
Although the (complex) Jones-vector inner-product 〈A 1|A 2〉 does not have a simple interpretation in Stokes space, its magnitude does. By using the identity |Aj 〉〈Aj |=Fj (1+ j · )/2, where j = j /Fj , one finds that
The invariant associated with Eq. (21) is denoted by J.
Just as Eqs. (15)–(17) are generalizations of Eqs. (11)–(13), so also does Eq. (22) appear to be a generalization of Eq. (20). However, by using the identity stated before Eq. (19), one finds that Eq. (22) is the negative of (20).
3. Nonlinear evolution of vector four-wave mixing
Equations (4) and (5) describe the evolution of two complex Jones vectors. Such a system is characterized by four complex variables. The inner-product equations (11)–(13), which arose in the search for conserved quantities, describe the evolution of only two real variables and one complex variable. Nonetheless, these equations form a closed set and are suitable for further study.
It is convenient to define 〈A 1|A 1〉=u, 〈A 2|A 2〉=v and 〈A 2|A 1〉=x+iy, and to measure the individual fluxes in units of the total initial flux u 0+v 0, which requires distance to be measured in units of the interaction length 1/γ(u 0+v 0). By using this notation, one can rewrite Eqs. (11)–(13) as the real equations
which reflects the fact that the total flux is constant [Eq. (14)]. Let u=1=2+w, in which case v=1/2-w. Then
Equation (32) implies that the phase space associated with the model equations (28)–(30) is a sphere. Since the model equations apply to both scalar and vector FWM, the topology of the scalar and vector phase-spaces are identical. The initial value u 0 is bounded by 0 and 1, so -1/2≤w 0≤1/2. Recall that x 2+y 2=|〈A 1|A 2〉|2. It follows from the triangle inequality that +≤u 0 v 0=1/4- and, hence, that ++≤1/4: The radius of the phase sphere cannot exceed 1/2. At this point one must distinguish between scalar and vector FWM.
First, consider the scalar process, for which x=(uv)1/2 cosϕ and y=(uv)1/2 sinϕ, where ϕ is the phase difference between A 1x and A 2x. It follows from these relations that +=u 0 v 0, so the phase sphere has radius 1/2. The phase sphere for scalar FWM is illustrated in Fig. 1. The six points where the phase sphere intersects the (w, x and y) coordinate axes are equilibrium points: (0,-1/2,0), (0,0,-1/2), (0,1/2,0) and (0,0,1/2) are centers, whereas (-1/2,0,0) and (1/2,0,0) are saddle points. The centers correspond to the flux condition u 0=v 0=1/2 and the phase conditions ϕ=-π, -π/2, 0 and π=2, respectively, whereas the saddle points correspond to the flux conditions u 0=0 and v 0=0, respectively. If |x 0|=|y 0| the motion is aperiodic. In contrast, if |x 0|≠|y 0| and both values are nonzero, the motion is periodic.
Second, consider the vector process, for which +≤1=4: The values of x 0 and y 0 are independent. This independence has interesting consequences. For example, suppose that w 0=0:25 and x 0=0.00. In the scalar process y 0=±0.43. The initial point (0.25,0.00,-0.43), which is denoted by a circle in Fig. 1, lies on a periodic orbit. However, in the vector process y 0 can have any value between -0.43 and 0.43. If y 0=0 the radius of the phase sphere is 0.25 (rather than 0.50) and the initial point (0.25,0.00,0.00) is an equilibrium point, as illustrated in Fig. 2. Now suppose that w 0=0.00 and x 0=0.25. In the scalar process y 0=±0.43. The initial point (0.00,0.25,-0.43), which is also denoted by a circle in Fig. 1, lies on a periodic orbit. In the vector process y 0 can have any value between -0.43 and 0.43. If y 0=-0.20, the radius of the phase sphere is 0.32 and the initial point (0.00,0.25,-0.20) lies on a different periodic orbit, as illustrated in Fig. 3. There is a simple way to relate the scalar and vector interactions. Suppose that w 0, x 0 and y 0 were chosen for the vector case, let r 0=2(++)1/2 and let w̄0=w 0/r 0, x̄0=x 0/r 0, ȳ0=y 0/r 0. Then ++ȳ20=1/4: The evolution of the vector interaction with initial condition (w̄0,x 0,y 0) is qualitatively similar to the evolution of the scalar interaction with initial condition (w 0,x 0,y 0). It follows from Eqs. (28)–(30) that the length scales of the scalar and vector interactions differ by a factor of r 0.
Having discussed briefly the general topology of the phase sphere associated with Eqs. (28)–(30), we now discuss in detail some specific solutions of these equations. First, consider aperiodic motion, for which |x|=|y|. By combining Eq. (28) with Eq. (29), or Eq. (32), one finds that
where the critical value wc =(+2)1=2. For reference, this critical value equals . Equation (33) has the solution
where the reference point z 0=∓tanh-1(w 0/wc )=2wc . [The inverse function tanh-1(w) is tabulated.] It follows from Eqs. (32) that
Formulas for u and v=1-u follow from solution (34) and the definition w=u-1/2. The solution associated with the initial point (-0.40,-0.21,-0.21) is displayed in Fig. 4. For reference, Eq. (33) can be rewritten in the form
where the critical values w±=(++±|-|)1/2. For reference, these critical values equal (J 0±|H 0|)1/2. Equation (37) is equivalent to the equation for a particle moving in the potential well V(w)=-4(w 2--w 2)(w 2+-w 2), as illustrated in Fig. 5(b). Equation (37) has the solution
where sn is the elliptic sine function of modulus m=w-/w+, the wavenumber k=2w + and the reference point z 0=-sn-1(w 0/w-|m)=k. [The inverse function sn-1(w|m)=F(sin-1 w|m), where F is the elliptic integral of the first kind, is tabulated.] It follows from Eqs. (31) and (32) that
Formulas for u and v=1-u follow from solution (38) and the definition w=u-1/2. The recurrence length
where K is the complete elliptic integral of the first kind. The solution associated with the initial point (-0.4,0.0,-0.3) is displayed in Fig. 6. For the special cases in which |x 0|=|y 0|, Eqs. (37) and (38) reduce to Eqs. (36) and (34), respectively.
4. Flux-phase formulation
The Jones-vector formalism developed in Section 2 facilitated the search for conserved quantities and led to the inner-product equations, the consequences of which were studied in Section 3. Although the inner-product equations capture the essence of vector FWM, they do not provide a complete description of this process (which involves eight real variables rather than four). One can rectify this omission by making a complimentary analysis of vector FWM, which is based on the flux and phase equations. For reference, the scalar flux-phase equations are solved in Appendix A.
where the relative phases θ 12=θ 1-θ 2 and ϕ 12=ϕ 1-ϕ 2, together with the Hamilton equations
and similar equations for ϕj and Gj . Stated explicitly, the first four dynamical equations are
The Hamiltonian (42) depends on the four fluxes F 1, F 2, G 1 and G 2, and the two relative phases θ 12 and ϕ 12. It does not depend on the average phases (θ 1+θ 2)=2 and (ϕ 1+ϕ 2)/2, which are ignorable. Since the Hamiltonian does not depend explicitly on position, it is a constant of motion, as stated in Section 2. It follows from Eqs. (46) and (48), and their counterparts for G 1 and G 2, that
Equations (49) and (50) imply that the total parallel flux T‖ and the total perpendicular flux T⊥ are (independent) constants: There is no way to transfer flux from one polarization component to the other.
Let |E‖〉 and E⊥〉 denote any pair of orthogonal (perpendicular) Jones vectors of unit length and let |A𢌪=A‖|E‖〉+A⊥|E⊥〉 denote any Jones vector. Then the associated Stokes vector is defined by its components
(Greek letters are used to label components because the numbers 1 and 2 are used to label waves.) In terms of the fluxes and phases, the components of the total Stokes vector are
By adding and subtracting Eqs. (49) and (50), one finds that the total flux T and the first component of the total Stokes vector sα are constant. Thus, Eqs. (49) and (50) are equivalent to Eq. (14) and the first component of Eq. (18). Although the constancy of sb and sg is not obvious from formulas (55) and (56), it is required by the second and third components of Eq. (18).
In the notation of Section 3 the differential flux 〈A 1|A 1〉-〈A 2|A 2〉=2w and the inner product 〈A 2|A 1〉=x+iy. In terms of the fluxes and phases, the (squared) differential flux
and the (squared) inner product
The preceding analysis shows that the evolution of vector FWM can be described completely by six real variables (four fluxes and two relative phases), and is constrained by five constants of motion (H, T and the three components of ).
It is convenient to use the initial polarization vector of the pump wave (2) to define the first (parallel) axis in Jones space and any perpendicular vector to define the second. In this framework the initial fluxes F 2≠0 and G 2=0 by construction (and without loss of generality). In general, the product wave has components that are parallel and perpendicular to the pump wave.
The evolution of vector FWM is illustrated in Fig. 7, for the case in which the initial fluxes F 1=0.2, G 1=0.2, F 2=0.6 and G 2=0.0, and the initial phase differences θ 12=π/4 and ϕ 12=0. (Since G 2=0, ϕ 2 is undefined. The results are insensitive to the value of ϕ 12.) For the stated initial conditions w 0=-0.10, x 0=0.25 and y 0=0.25. Since |x 0|=|y 0| the motion is aperiodic [Eqs. (34) and (35)]. It is clear from the figure that F 1 increases as F 2 decreases and G 1 decreases as G 2 increases: The total parallel and perpendicular fluxes T ‖ and T ⊥ are constant. In Section 3 it was shown that the system approaches the equilibrium point (,0,0) asymptotically as z→∞. For the stated parameters =0.36, so the asymptotic values of the product and pump fluxes u=F 1+G 1 and v=F 2+G 2 are 0.86 and 0.14, respectively. In the scalar process (G 1=0) the motion is aperiodic: Steady state is reached when the pump flux has been transferred to the product (F 2=0). The polarization vectors of the pump and product are perpendicular, but only in a trivial sense. In the vector process the motion is also aperiodic. However, both waves have finite flux in steady state: The polarization vectors of the pump and product are perpendicular in a nontrivial sense.
The evolution of vector FWM is also illustrated in Fig. 8, for the case in which the initial fluxes F 1=0.2, G 1=0.2, F 2=0.6 and G 2=0.0, and the initial phase differences θ 12=π/8 and ϕ 12=0. For these initial conditions w 0=-0.10, x 0=0.32 and y 0=0.13. Since |x 0|≠|y 0| the motion is periodic [Eqs. (38)–(40)]. Once again, F 1 increases as F 2 decreases and G 1 decreases as G 2 increases: The total parallel and perpendicular fluxes T‖ and T⊥ are constant. As the parallel fluxes F 1 and F 2 oscillate they attain several different maxima. The perpendicular fluxes G 1 and G 2 also attain several different maxima, at locations that have no obvious correlations with the locations of the parallel maxima. Despite this disorder, the oscillations of the total pump and product fluxes are perfectly periodic. In Section 3 it was shown that the extreme values of w are ±(J 0-|H 0|)1/2. For the stated parameters (J 0-|H 0|)1/2=0.21, so the extreme values of the pump and product fluxes are 0.71 and 0.29.
5. Stokes-space visualization
It is often useful to visualize polarization dynamics in Stokes space. (For example, CPM causes the individual Stokes vectors to rotate rigidly about a common axis.) In this section the figures were obtained by solving Eqs. (45)–(48) numerically, and using parts of Eqs. (54)–(56) to relate the individual fluxes and phases to the individual Stokes vectors. For reference, FWM is described directly in Stokes space in Appendix B.
The Stokes vectors are plotted parametrically in Fig. 9, for the same initial conditions as Fig. 7. In the figure the α, β and γ axes are labeled by A, B and C, respectively. The common viewpoint is (2,5,2), in arbitrary units. As distance increases, the phase point (tip of the Stokes vector) for wave 1 moves upwards, whereas the phase point for wave 2 moves downwards: The total Stokes vector is conserved. Since the wave evolution is aperiodic, the Stokes-space trajectories (of the phase points) are simple.
The Stokes vectors are also plotted parametrically in Fig. 10, for the same initial conditions as Fig. 8. The common viewpoint is (2,5,2). Once again, the total Stokes vector is conserved. Although the fluxes evolve periodically in Fig. 8, the flux components evolve in more-complicated ways. in Fig. 10 this complexity is manifested as bounded, but not-quite-periodic motion. As an aid to visualization, the Stokes-space trajectory of wave 1 is also illustrated in Fig. 11, from the viewpoints (2,2,5) and (2,5,-2). These figures illustrate polarization dynamics for moderate distances. The long-distance trajectory of wave 1 is illustrated in Fig. 12, for the same initial conditions as Fig. 10 and the same viewpoints as Fig. 11. The figure suggests that the trajectory of wave 1 is restricted to a well-defined surface in Stokes space. Further work is required to determine the precise nature of this surface, and to determine whether the trajectory is closed.
In the presence of loss, the right sides of Eqs. (4) and (5) include the terms -α|A 1〉/2 and -α|A 2〉/2, respectively. By making the substitutions |Aj 〉=|Bj 〉e-αz/2 and ζ=(1-e-αz)/α in Eqs. (4) and (5), one finds that the equations for |Bj (ζ)〉 are the same as the lossless equations for |Aj (z)〉: As the waves propagate, their powers decrease and the rate at which they exchange power decreases concomitantly.
Doubly-degenerate FWM was introduced as a simplified model of degenerate and nondegenerate FWM in a fiber. It is also true of the latter processes that loss reduces the powers of the constituent waves and the rates at which they exchange power. Several parametric-amplification experiments were made recently, with highly-nonlinear fibers and high-power pumps [8, 9]. In these experiments the loss coefficients a were about 0.4 dB/Km, the nonlinearity coefficients γ were about 10/Km-W and the pump powers were of order 1 W. For parameters such as these, the interaction length is of order 0.10 Km, whereas the loss length is of order 10 Km: Loss is less important than nonlinearity.
In highly-nonlinear fibers PMD causes the polarization vectors of waves with different frequencies to become misaligned . The aforementioned experiments involved waves with (real) frequency differences of order 5 THz (wavelength differences of order 40 nm), for which the decorrelation lengths were of order 1 Km. Waves with such large frequency differences were used to demonstrate the suitability of parametric amplifiers for multiple-channel communication systems. For moderate frequency differences, of order 1 THz, the decorrelation length is of order 30 Km: PMD is less important than nonlinearity. Thus, it is possible, with current technology, to observe the nonlinear polarization dynamics of degenerate and nondegenerate FWM in fibers.
Doubly-degenerate four-wave mixing (FWM) involves two waves, whose frequencies satisfy the matching condition 2ω 1=2ω 2. It is a simple model of degenerate and nondegenerate FWM, for which the matching conditions are 2ω 2=ω 3+ω 1 and ω 2+ω 3=ω 4+ω 1, respectively, and filamentation, in which power is transferred between the spatial eigenmodes of a beam.
In this paper the nonlinear polarization dynamics of doubly-degenerate vector FWM were studied. As the constituent waves evolve, the total photon flux and the total Stokes vector remain constant. Both of these quantities are measurable. The vector-FWM equations [(4) and (5)], which describe a system with eight degrees of freedom, were used to derive a self-contained set of inner-product equations [(11)–(13)], which describe a system with four degrees of freedom. The inner-product equations were solved analytically. Depending on the initial conditions, the photon-fluxes of the waves can evolve aperiodically or periodically, but not chaotically. Although the fluxes usually evolve periodically (Fig. 8), the flux-components evolve in more-complicated ways: The parallel and perpendicular components of wave 1 exchange flux with the corresponding components of wave 2, but the locations of the parallel maxima have no obvious correlations with the locations of the perpendicular maxima. The dynamics of vector FWM were illustrated in phase space (Section 3) and Stokes space (Section 5). Although the total Stokes vector is constant, the tips of the individual Stokes vectors move along complicated trajectories (Figs. 10–12): In Stokes space the motion is bounded, but not-quite-periodic.
Finally, the mathematical methods and physical insights developed in this paper can be used to study degenerate and nondegenerate vector FWM, which occur in a variety of optical systems.
Appendix A: Nonlinear evolution of scalar four-wave mixing
In this appendix the nonlinear evolution of scalar FWM is reviewed briefly. Scalar FWM is governed by the complex amplitude equations
where the relative phase θ=θ 1-θ 2, and the flux equations
together with the Hamilton equations
Each phase is a canonical displacement and each flux is the associated canonical momentum. Since the Hamiltonian (66) does not depend on distance, it is a constant of the motion, which is denoted by H 0. Since the Hamiltonian depends on the relative phase, rather than the individual phases, the total flux F 1+F 2 is also constant.
It is convenient to measure the fluxes in units of the total flux, which requires distance to be measured in units of the interaction length 1/γ(F 1+F 2). Let F=F 1 and ′ denote a normalized distance derivative. Then one can replace Eqs. (66)–(68) by the reduced Hamiltonian
and the reduced Hamilton equations
The phase plane associated with Eqs. (69)–(71) is illustrated in Fig. 13. The equilibrium points (0,1=2) and (π/2,1/2) are centers: Both fluxes are nonzero (F 1=1/2=F 2) and the relative phase is a multiple of π/2 [cos(2θ)=±1 and sin(2θ)=0]. These conditions allow each wave to modify the phase of the other, but prevent a flux transfer between the waves [Eqs. (62)–(65)]. In contrast, the equilibrium points (π/4,0), (π/4,1), (3p/4,0) and (3π/4,1) are saddle points.
where =[F 0(1-F 0)cos(2θ 0)]2, and F 0 and q0 are abbreviations for F 1(0) and q (0), respectively. Equation (72) is mathematically equivalent to the equation for a particle with displacement F moving in the potential well V(F)=-4F 2(1-F)2 with total energy E=-4, as illustrated in Fig. 14. (These definitions omit the usual factors of 1/2.) Points A and B correspond to saddle points in Fig. 13, whereas point C corresponds to centers.
Suppose that 0<F 0<1 and cos θ 0=0. For these initial conditions, which correspond to a point on the dot-dashed line in Fig. A2, between the equilibrium points, Eq. (72) has the aperiodic solution
which describes an irreversible flux transfer from wave 2 to wave 1. (F 1 increases monotonically from F 0 to 1, whereas F 2 decreases monotonically from 1-F 0 to 0.) Solution (73) can be rewritten as
where the reference point z0 =tanh-1(1-2F 0). Solution (74) corresponds to the condition sin(2θ 0)=1. If the condition sin(2θ 0)=-1 is imposed, there is an irreversible flux transfer from wave 1 to wave 2. One obtains the solution for this case from the stated solution by replacing z with -z. Although the methods of analysis used in Section 3 and this appendix are different, solution (74) is equivalent to solution (34).
Now suppose that 0<F 0<1 and cosθ 0≠0, and let
denote the roots of the polynomial function E-V(F), listed in ascending order. For these initial conditions, which correspond to a point on the dashed line in Fig. A1, between the turning points, Eq. (72) has the periodic solution
where sn denotes the elliptic sine function of modulus m=[(c-b)(d-a)/(d-b)(c-a)]1/2, the wavenumber k=[(d-b)(c-a)]1/2 and the reference point z0 is determined by the definite integral
Solution (79) describes a reversible flux exchange between waves 1 and 2. Solution (79) corresponds to the condition sin(2θ 0)>0, which causes the particle to move from its initial position toward point c, before moving back toward point b. If the condition sin(2θ 0)<0 is imposed, the particle starts to move toward point b. One obtains the solution for this case from the stated solution by replacing z with -z.
Figure A2 shows that the potential function V(F) is symmetric about the value F=1/2. Let w=F-1/2. Then one can rewrite Eq. (72) as
where the critical values w±=(1±4|H 0|)1/2/2. Equation (81) is identical to Eq. (37), so solution (79) must be equivalent to solution (38). In principle, one can reconcile solutions (79) and (38) directly, by using a Gauss transformation to convert an elliptic function with modulus 2(w+w-)1/2/(w++w-) and wavenumber w++w- to one with modulus w-/w+ and wavenumber w+, and using double-argument and shift formulas to convert the wavenumber from w+ to 2w+ and shift the argument by a quarter period . We established the consistency of the two solutions by evaluating them both numerically.
Appendix B: Stokes-space formulation
It is often convenient to use the polar Stokes coordinates
If the evolution equations for ρ, θ and ϕ can be solved, the cartesian Stokes coordinates follow from the inverse relations
The Jones vector of each wave can be written in the form
By substituting formula (84) in Eq. (6), one finds that the polar Lagrangian
where cj =cos(θj /2), sj =sin(θj /2), and the phase differences ψ 12=ψ 1-ψ 2 and ϕ 12=ϕ 1-ϕ 2. The initial conditions are often defined in terms of fluxes and phases. In such cases the polar initial conditions follow from the relations
where the subscripts f denote angles that appear in the flux-phase formulation. If the Euler-Lagrange equations associated with the Lagrangian (85) can be solved, the fluxes and phases follow from the inverse relations
It follows from the Lagrangian (85) that p=ρ and q=ρs 2 are the canonical momenta associated with the canonical displacements ψ and ϕ, respectively. Consequently, the polar Hamiltonian
The polar initial conditions follow from the relations
If the Hamilton equations associated with the Hamiltonian (88) can be solved, the fluxes and phases follow from the inverse relations
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