Nonlinear envelope equation

Several types of unidirectional propagation equations are widely used in the nonlinear optics literature. The most important examples are Non-Linear Schrodinger (NLS) equation [32], Nonlinear Envelope Equation [33] (NEE), the First-Order Propagation equation [31] (FOP), Forward Maxwell s equation [34] (FME), and several other equations that are closely related to these. The derivations found in the literature differ from equation to equation, and in some cases the physical meaning of the required approximations may not be readily evident due to a number of neglected terms. 

The Nonlinear Envelope Equation [33] is a paraxial equation with some additional approximations related to chromatic dispersion. This equation appears to be extremely close to the paraxial version of UPPE. 

The theoretical approach is based on the solution to the mixed type linear/nonlinear generalized Schrodinger equation for spatiotemporal envelope of electrical field with account of transverse spatial derivatives and the transverse profile of refractive index. In the quasi-static approximation, this equation is reduced to the linear/nonlinear Schrodinger equation for spatiotemporal pulse envelope with temporal coordinate given as a parameter. Then the excitation problem can be formulated for a set of stationary light beams with initial amplitude distribution corresponding to temporal envelope of the initial pulse. 

The -propagated approach is much more common in nonlinear optics simulations based on envelope equations, often related to NLS. The time-propagated approach is on the other hand common for solvers based on direct integration of Maxwell s equations. 

The Nonlinear Schrodinger Equation (see [32] for applications in optics) (NLS) is a prototype propagation equation in nonlinear optics. This is also the simplest case suitable to illustrate the general procedure outlined above. One characteristic feature of NLS and of other envelope type equations is the presence of a reference frequency. Usually, one chooses the reference angular frequency Wr as the central frequency of the initial pulse, but this is not necessary. Actually it is useful to keep in mind that wr is to a certain extent a free parameter, and that the obtained results must be almost independent of its concrete choice. If a numerical simulation turns out to be sensitive to the choice of ur, it means that an envelope equation is being used outside of its region of validity. 

One prominent application of large XPM in our structure would be the creation of an all-optical switching[73] based on semiconductor material. For two matched Gaussian pulses, the nonlinear phase shift can be determined from the slowly varying envelope equation[74]. With the... 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

Noncompetitive inhibition 476,477 Nonheme iron proteins. See Iron-sulfur and diiron proteins Nonlinear equations 460 Nonmetallic ions, ionic radii, table 310 Nonproductive complexes 475 Norepinephrine (noradrenaline) 553,553s in receptor 555s Nuclear envelope 11... 

Before we proceed in solving Equation (27.15), we give some explanations for Equation (27.17). If there were not the last term in Equation (27.15), the one with which comes from the nonlinear term in Equation (27.13), we would expect the solution in the form Eie " + cc instead of Equation (27.17). This would be a modulated wave with a carrier component e " and an envelope F. We will see later that the modulation factor F will be treated in a continuum limit whereas the carrier wave will not. In other words, the carrier component of the modulated wave includes the discreteness and the procedure is called semi-discrete approximation. 

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... 

But for a weakly nonlinear case, if we want to derive the associated Landau -Stuart (LS envelope) evolution equation, it is suffisant to assume that the amplitude of the wave packet envelope is a function of 2 = 0 only. 

The integration in Equation [4] considers the existence of several frequency combinations matching the condition = Ey cOy, within the bandwidth of the applied field. A single laser whose bandwidth is large enough to include frequency components that match 0) -0) = O) can drive the nonlinear response. However, more frequently the laser bandwidth is smaller than o) and different beams with distinct centre frequencies at needed to match the Raman resonance. Let us now consider a field composed of a superposition of quasi-monochromatic beams, i.e. the amplitude at a carrier frequency is modulated by a complex envelope that defines its spectral shape. The central frequencies are chosen to match the Raman resonant four-wave mixing scheme (see Table 1), which can be later adapted to other colour choices. Three laser beams are present of frequencies o)q, cOp, cOg, with 0) -0) = 0) as the only resonant frequency combination. 

Proceeding along similar lines as before one obtains the envelope amplitude equations that are now PDEs that describe the nonlinear interactive behaviour of the wavepackets accounting for the dynamics of all the modes included in the annulus (2D) or spherical shell (3D) of width in... 

This is a curious result, as it indicates that a nonlinear property can be calculated from linear data, but it has been found to describe accurately the response of polymeric liquids at sufficiently low shear rates. The low-shear-rate limiting behavior indicated by the above equations, which involves monotone increasing functions, is always shown in plots of nonlinear data the nonlinear responses involve overshoots in the material functions, but should always start out at short times, when the strain is still small, by following the low-shear-rate, LVE curve. Then, as the shear rate increases, the nonlinear data fall below the linear envelopes at shorter and shorter times [44,45]. These features can be seen in Fig. 10.9, which shows the data of Menezes and Graessley [46] for shear and first normal stress difference in start-up of simple shear. The dashed lines are calculated from the linear spectrum using Eqs.4.8 and 10.49. As expected, the... 

An argument to resolve the discrepancy between the failure envelopes obtained for different modes of straining is indicated by the work of Blatz, Sharda, and Tschoegl [42]. These authors have proposed a generalized strain energy function as constitutive equation of multiaxial deformation. They incorporated more of the nonlinear behavior in the constitutive relation between the strain energy density and the strain. They were then able to describe simultaneously by four material constants the stress-strain curves of natural rubber and of styrene-butadiene rubber in simple tension, simple compression or equibiaxial tension, pure shear, and simple shear. 

---