Slowly varying envelope approximation

These results can be used for comparative analysis in the elaboration of nonlinear implementations of more rigorous computational methods, free of the slowly varying envelope approximation. 

The parabolic equations derived in a slowly varying envelope approximation that describe the second harmonic generation (SHG) of ultrashort pulses in media with locally inhomogeneous wave-vector mismatch, have the form ... 

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) ... 

Our interest is in long smooth pulses and so we assume the second-derivative terms on the left side are much smaller than the first derivatives (the slowly varying envelope approximation, SVEA). We define the relationship between k and to be... 

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

Note that the first two approximations employed (classical external field and dipole approximation) are usually well justified in molecular physics, whereas the slowly varying envelope approximation may approach the hmits of validity in the case of femtosecond pulses with optical frequencies. The validity of the assumption of an optically thin sample, i.e. the complete neglect of pulse propagation effects, depends on the specific experiment under consideration. To simplify the notation, we henceforth suppress the x-dependence of polarizations and fields, and also drop the prefactor of the polarization in Eq. (6). 

In order to derive the TFG SE signal from this dehnition, it is standard practice in the literature to neglect the retardation effects (r = 0) and to invoke the slowly-varying-envelope approximation, i.e. 5 P(t) —uPP t), where a is the carrier frequency. This is tantamount to the assumption... 

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... 

The transient transmittance PP signal is defined through the difference polarization Ppp (0 = Ppp(t) —Ppf(f) [1]. Here Ppp ( ) is the pump-offpolarization induced solely by the probe pulse, which is obtained from Eqs. 9.29 and 9.30 with = 0. Within the RWA and the slowly varying envelope approximation, the integral (int) and dispersed... 

With the slowly varying envelope approximation (SVEA) which is valid for almost all presently realistic cases one derives the equation... 

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 orientation angle follows the interference grating between the input e wave ) and its scattered noise component o wave and may be expressed as 0(z,f) = + c.c. Using the slowly varying envelop approximation, that is, d Q z,t)ldz and [q dQ z,f)lbz] are negligible compared with (z, f), the equations of... 

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