Slowly-varying amplitude and phase

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... 

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. 

In the case of coherent laser light, the pulses are characterized by well-defined phase relationships and slowly varying amplitudes (Haken, 1970). Such quasi-classical light pulses have spectral and temporal distributions that are also strictly related by a Fourier transformation, and are hence usually refered to as Fourier-transform-limited. They are required in the typical experiments of coherent optical spectroscopy, such as optical nutation, free induction decay, or photon echoes (Brewer, 1977). Here, the theoretical treatments generally adopt a semiclassical procedure, using a density matrix or Bloch formalism to describe the molecular system subject to a pulsed or continuous classical optical field, which generates a macroscopic sample polarization. In principle, a fully quantal description is possible if one represents the state of the field by the coherent or quasi-classical state vectors (Glauber, 1965 Freed and Villaeys, 1978). For our purpose, however. 

Thus the original differential equation (6-90) of the second order has been replaced by the system (6-96) of two first order differential equations in terms of the amplitude a and the phase 9.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. 

Total effect of these two components shows up as a combination of two factors- one on a slow scale given by the first quantity inside the square bracket and the second quantity that varies on the original scale given by the second cosine function. As the two components are infinitesimally apart in the spectral plane, the slowly var 4ng part can be viewed as the amplitude of the total effect. As energy of a wave system is proportional to the square of the amplitude of the wave, one can view the energy of the system to vary following the phase variation of the amplitude. Therefore the speed... 

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. 

A simplex telemetry system is employed for the transmission using straightforward modulation schemes such as amplitude modulation, on-off keying, amplitude-shift keying, and phase-shift keying [101-103]. Because the physiological signals vary slowly, such modulation schemes are preferable for miniaturized and low-power wireless biosensors. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

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