Slowly varying envelope

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... 

Consider the case when the external field is a monochromatic circularly polarized pulse, E(f) = Aexp(-iojf), where A is a slowly varying envelope function. For this pulse the phase angle of dE/df is rotated by r/2 from the direction of E. From Eqs. (5.17) and (5.18) we then find... 

The models we have discussed so far correspond to continuous (CW) lasers with a fixed sharp frequency and constant intensity. They can be easily adapted to the case of pulsed lasers that have slowly varying envelopes. They can furthermore have a chirped frequency—that is, a frequency that changes slowly with time. For periodic (or quasi-periodic) semiclassical Hamiltonians, the Floquet states are the stationary states of the problem. Processes controlled by chirped laser pulses include additional time-dependent parameters (the pulse... 

The analytic form of the pump and dump pulses in the optimal phase-unlocked pump-dump control is />( >)( )= p( >)(f) exp (—/cOe f)- -exp megt), where Tipp) is a slowly varying envelope of the helds, and (s>eg is the energy difference between the minima of the excited and the ground states. The objective in the ground state is represented in the Wigner formulation by an operator A = A(r) g)(g. A(r) is the Wigner transform of the objective in the phase space T = of coordinates and momenta, and... 

Fig. 7.24 (a) Optical nutation in CH3p observed with CO2 laser excitation at A = 9.7 pm. The Rabi oscillations appear because the Stark pulse lower trace) is longer than in Fig. 7.23. (b) Optical free-induction decay in I2 vapor following resonant excitation with a cw dye laser at = 589.6 nm. At the time = 0 the laser is frequency-shifted with the arrangement depicted in Fig. 7.22 by Au = 54 MHz out of resonance with the I2 transition. The slowly varying envelope is caused by a superposition with the optical nutation of molecules in the velocity group Vz = o) — (oo)/k, which are now in resonance with the laser frequency oj. Note the difference in time scales of (a) and (b) [705]... 

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

This optical induction free decay can be measured with a beat technique at time r = 0 the frequency cu of a cw laser is switched from co = con to aJ o) 2 out of resonance with the molecules. The superposition of the damped wave at o) 2 emitted by the coherently prepared molecules with the wave at co gives a beat signal at the difference frequency Aco = co 2— o), which is detected [12.70]. If Aco is smaller than the Doppler width, the laser at cu interacts with another velocity subgroup of molecules and produces optical nutation, which superimposes the free-induction decay and which is responsible for the slowly varying envelope in Fig. 12.21b. 

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... 

In the following, we briefly discuss a general approach to extract the individual contributions and spectroscopic signals from the overall polarization,and discuss the simplifications that arise when the usual assumptions (i.e. nonoverlapping and weak laser fields, slowly-varying envelope assumption, and RWA) are invoked. 

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... 

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... 

Consider a molecular system described by the Hamiltonian operator Ho (r, R) in interaction with a pulsed laser field with slowly varying envelope and frequency. Below, for ease of notation, the dependence of the various operators and wavefunctions on the electronic and nuclear coordinates r and R is dropped. In the semiclassical dipolar approximation, the total Hamiltonian operator of the system reads... 

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... 

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