Nonlinear coupled mode, equations

In this study we suppose nonlinear organic material shows optical Kerr effect as n = n0+n2lEl2 and n2 = X<3)/(2n0). Moreover for simplification, we suppose the waveguides allow single mode propagations and TE polarization. After appropriate handling we get the following nonlinear coupled mode equations [ 12] ... 

Evaporated PDA(12-8) film was used as a nonlinear optical medium in a layered guided wave directional coupler. The directional coupling phenomenon happens in two adjacent waveguide by periodical energy transfer. The theory of linear directional coupler was exactly established [11]. It can be reduced to coupled mode equations ... 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

The existence of the nonlinear polarization field does not ensure the generation of significant signal fields. With the exception of phenomena based on an intensity-dependent refractive index, the generation of the nonlinearly produced signal waves at frequency cos can be treated in the slowly varying amplitude approximation with well-known guided wave coupled mode theory (1). As already explicitly assumed in Equation 1, the amplitudes of the waves are allowed to vary slowly with... 

It is also assumed that efficient nonlinear coupling occurs only between two guided modes propagating in the +z direction, the fundamental mode a and the SHG mode 2- In the slowly varying envelope approximation (SVEA), the governing equations become... 

All the terms in this equation are analytical functions of the parameter z except for A(z), which is the numerical solution of Equation (6.181). Equation (7.27) is a system of coupled nonlinear equations with removable singularity at z = -1 that may cause instability of the numerical solution. Note, however, that the nonlinearity of this equation is totally due to the factor 9 z) [see Equation (6.193)], which is related to the normalization condition for the vector U. Thus, for a given mode y the solution can be written in the form... 

The coefficients of this mode-coupling functional are the basic control parameters of this idealized version of MCT. One sees that Eqs. [46] and [47] amount to a set of nonlinear equations for the correlators S(q,t) that must be solved self-consistently. 

Our findings lead us in a number of useful directions. One of these directions is a generalization of our basic instantaneous approach to dynamics. Our original linear INM formalism assumed the potential energy was instantaneously harmonic, but that the coupling was instantaneously linear [Equation (15)]. We still need to retain the harmonic character of the potential to justify the existence of independent normal modes (at least inside the band), but we are free to represent the coupling by any instantaneously nonlinear function we wish. A rather accurate choice for vibrational relaxation, for example, is the instantaneous exponential form ... 

In the case of nonlinear polarizability coupling, the ratio of the cascaded to direct fifth-order response can be expressed in terms of the ratio of the first and second derivative of the polarizability [Equation (32)]. Using instantaneous normal mode simulations, Murry et al. (40) have calculated the relative ratios of a 11 and at2> for 5000 intermolecular modes in CS2. Although their results show this ratio to be somewhat randomly... 

A more reahstic and more general treatment would presumably lead to a set of equations like Eqs. (52), with the potential V(x) fluctuating as a consequence of couplings with nonreactive modes (see Section III). For the sake of simplicity, we study separately the two different aspects. While Section III was devoted to pointing out the role of multiphcative fluctuations (derived from nonlinear microscopic Liouvillians) in the presence of additive white noise, this subsection is focused on the effects of a non-Markovian fluctuation-dissipation process (with a time convolution term provided by a rigorous derivation from a hypothetical microscopic Liouvillian) in the presence of a time-independent external potential. 

We perform a nonlinear analysis that will allow us to obtain amplitude equations to characterize the evolution of the unstable modes. Gross and Volpert (4) have studied the 1-D case for loss of neutral stability at the wavenumber s = 0. This corresponds to sufficiently small values of the tube circumference L. The analysis in this instance results in a single Landau-Stuart equation which governs the weakly unstable modes. If, however, the tube circumference L is large, loss of stability will occur for some s > 0. Our nonlinear analysis yields a coupled set of Landau amplitude equations. 

Fig. 1(b)). The growth of A reflects the nonlinear nature of the time-evolution equation of the pattern and hence the mode-mode coupling effect (6). The... 

In Kawasaki s approach to mode-mode coupling, one does not use the complete set of Mori s equations for the complete set of variables. Rather the procedure is to keep only the equation for discard the equations for the nonlinear variables, and employ various alternative techniques to solve the equation for A. This approach is the one we shall discuss, as otherwise simplification to a very few nonlinear variables is impossible. 

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