Slowly varying amplitude approximation

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... 

Even though the laser pulses are approximated as delta functions, the slowly varying amplitude approximation can still be applied to pulses as wide as tens of femtoseconds, where the time scales for the nuclear degrees of freedom remain much slower than the pulse width (1). 

Within the slowly varying amplitude approximation, the generated electric field amplitude grows linearly with respect to the distance from the front boundary, z, of the optical sample. Direct integration over z gives the... 

Before we start with quantum description, let us recollect the classical solutions which will be used later in the method of classical trajectories to study some quantum properties of the fields. Equations (56) are valid also for classical fields after replacing the field operators a and b by the c-number field amplitudes a and p, which are generally complex numbers. They can be derived from the Maxwell equations in the slowly varying amplitude approximation [1] and have the form. 

E (k2, w)E(k3, co). E (k2, 0)) denotes the complex conjugate of E(k2, boundary conditions the solution of Eq. I yields... 

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." ... 

Solving Maxwell s equations using Equations (6.1) and (6.2) in the slowly varying envelope approximation, one can calculate the amplitude of the electric field of the TH to be (Ward and New 1969 Bjorklund 1975) ... 

Slowly varying envelope approximation Approximation in which it is assumed that the amplitude of an... 

An approximation that is generally valid is the so-called slowly varying envelope approximation. In effect, a particular frequency component Ej of the total electric field can be represented by a product of a fast (temporally and spatially) oscillatory term exp[r( y f-ci)yO] and an amplitude function y(f, t), whose temporal and spatial variations are on a much slower scale. Therefore, we have... 

The paraxial approximation is essentially a Taylor series expansion of an exact solution of the wave equation in powers of p/w, terminated at ( p/tv), that allows us to exploit the rapid decay of a Gaussian beam away from the optical axis. We will develop a more precise criterion in the sequel. We will also show that the phase and amplitude modulation of the underlying plane wave structure of the electromagnetic field is a slowly varying function of distance from the point where the beam is launched. 

To solve Eq. (18), we use the usual slowly vary-ing-amplitude approximation. Let... 

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

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