Amplitude slowly varying

Thus the original differential equation (6-90) of the second order has been replaced by the system (6-96) of two first order differential equations in terms of the amplitude a and the phase 9. Moreover, as Eqs. (6-96) contain the small factor (i on the right-hand side, the quantities, a and 9 are small, that is, both a and 9 are slowly varying functions of time and one can assume that during one period T = 2nfca, the trigonometric functions vary but slightly. 

The following cases are somewhat more subtle. Figure 2c depicts a slowly varying amplitude Pnw with a very slowly varying phase-... 

If the gauge fields have small amplitude and are slowly varying relative to scale Pf, the fast modes are decoupled from low energy physics. The low energy... 

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." ... 

A common way to treat the problem of a picosecond pulse propagation in regular dielectric waveguide with Kerr nonlinearity was to solve the nonlinear Schrodinger equation (NLSE) for the slowly varying temporal amplitude of electrical field ... 

Here complex amplitude of the pulse envelope E x, z, t) is also a slowly varying function of z and t. Spatiotemporal distribution of the electric field is described by E x,z,t) = E x,z,t)ex-p[iujQt— i/Sz), f3 being the longitudinal wavenumber of the waveguide mode at the pulse peak. 

Here E x,z) is slowly varying amplitude of the total electrical field E x, z) = E x, z) exp if3z — iujt), /3 is a parameter responsible for fast oseillations of the total field in longitudinal direction. 

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) ... 

It is now seen that assumption (a) is, indeed, valid provided / Aay, and T are slowly varying functions of the energy. That is, by integration over the intermediate states in eq. (2-34) a function is obtained having the same energy dependence as the original (assumed) transition amplitude. Equation (2-39) may be solved for Aga. The result is ... 

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... 

A similar analysis can be made for quasi-periodic signals which consist of a sum of sine waves with slowly-varying amplitude and instantaneous frequency each of which is assumed to pass through a single filter. 

In addition to FM signals with nested modulators, Justice also considered a class of signals modeled by a harmonic sum of sine waves with a slowly varying amplitude, typical of many speech and music sounds... 

The deterministic component of the model consists of sinusoids with slowly-varying amplitude and frequency, or in musical terms the partials of the sound24. The... 

FIGURE 2.15 Comparison among different views of a snapshot of a subsystem satisfying the Swift-Hohenberg equation—a simplified model of convection in the absence of mean flow. Panel (a) shows the detailed flow directions, whereas panels (b) and (c) exhibit the amplitude and phase, respectively. The latter are slowly varying and allow for easier identification of the grain boundaries of the flow. 

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... 

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. 

We write u = ipip, 2)e , where we assume that ipi p, z) is a slowly varying amplitude and phase function as discussed in preceding text. We will model the (complex) phase modulation by a function The... 

---