Time dependence of the fields

Consider a one-dimensional optical structure that consists of nonlinear layers. The structure is illustrated in Fig. 1. Assuming no field variation in the X and z directions and the convention exp(icot) for the time dependence of the fields, Maxwell s equations take the form... 

We choose the time dependence of the field strengths of the pump and Stokes fields to be... 

A more widely used approach for organic molecules is based on second-order perturbation theory. Here the dipolar contribution to the field induced charge displacement is calculated by inclusion of the optical field as a perturbation to the Hamiltonian. Since the time dependence of the field is included here, dispersion effects can be accounted for. In this approach the effect of the external field is to mix excited state character into the ground state leading to charge displacement and polarization. The accuracy of this method depends on the parameterization of the Hamiltonian in the semi-empirical case, the extent to which contributions from various excited states are incorporated into the calculation, and the accuracy with which those excited states are described. This in turn depends on the nature of the basis set and the extent to which configuration interaction is employed. This method is generally referred to as the sum over states (SOS) method. 

The operators P and obey the usual equal time anticommutation relations. The time-dependence of the field operators appearing here is due to the Heisenberg representation in the L-space. In view of the foregoing development which parallels the traditional Schrodinger quantum theory we may recast the above Green function in terms of the interaction representation in L-space. This leads to the appearance of the S-matrix defined only for real times. We will now indicate the connection of the above to the closed-time path formulation of Schwinger [27] and Keldysh [28] in H-space. Equation (82) can be explicitly... 

This completes our brief description of the computational methods used in these studies. In the following sections some recent results will be presented and discussed. We will cover the calculation of ionization rates, the photoelectron energy distributions, the determination of the residual excited state populations remaining after excitation by a short pulse and finally show some photoemission spectra. The shape of the pulse envelope clearly can affect all these observable quantities. For example, the final state populations are found to be very sensitive to the pulse width and the peak intensity. Such results emphasize the point that in a strong, short pulsed field, the time dependence of the field envelope is reflected in the time evolution of the excitation dynamics. During the pulse. 

There exists a different pathway for a selective population of state 2). In a first step, a selectively transfer from the ground to the first excited state is performed. We already showed that this is possible with a 100% yield. The second step then involves a change of the target state from 1) to 2). A numerical example for this successive excitation process is shown in Fig. 24. Until a time of 2ps, the same features as already discussed above (i.e., Fig. 22) are found, namely, the stepwise increase of the population B (t) and a pulse-train structure of the field. Afterwards, a more complex time-dependence of the field is encountered. This is because now the vibrational dynamics in the intermediate as well as the target electronic state enters into the construction scheme for the field. The control in this two-step process is more effective if compared to the direct transfer (Fig. 23). Here, we achieve an almost complete transfer of population into the target state 2). 

Upon applying a field modulation technique, it was possible to record directly field-induced changes in the AT/T spectra. Therefore, the kinetic traces in Fig. 2.2 reflect the time dependence of the field-induced differential transmission (AT/T)fm. which is the difference between AT/T recorded in the presence and absence of the electric field (AT/T)fm = (AT/T)f-(AT/T)f o. 

Generally, the interference of two fields results in an elliptic polarization. It can be shown if we consider time dependencies of the fields. After the liquid crystal layer, the output fields are ... 

This result is particularly useful in calculating the linear response of a system which is under the influence of an external field (often, a time-dependent electromagnetic field). We assume that the interaction of such a field with the system can be represented by adding terms to the Hamiltonian of the form A(F)F(t), where F(t) represents the explicit time-dependence of the field. The Liouville equation for such a system can be written as... 

The vector E = Eg + iE is called phasor. The full time dependence of the field can be written as ... 

The time dependence of the field correlation function is therefore determined by the time correlation function of a spatial Fourier component of the concentration, namely... 

Dynamic light scattering examines the time dependence of the field correlation function. There is an enormous literature, much contradictory, on direct calculation of g q, t) from the forces between the diffusing particles. This section treats the direct calculation, but only for the simplest of model systems, namely a suspension of colloidal spheres. There are corresponding calculations for nondilute polymer molecules, but these calculations are even more complicated than what follows, in part because neighboring beads on the same chain are required to stay attached to each other. The presentation here shows the tone of the approach, based on papers by Carter and Phillies(25) and Phillies(26,27). Several excellent alternative treatments are available, e.g., Beenakker and Mazur(28,29) and Tokuyama and Oppenheim(30,31). 

To obtain the group velocity, we need a generalized reciprocity theorem which allows for variations in the frequency w of the implicit time dependence of the fields. We let E and H be the fields of the jth mode at wavelength L These fields satisfy the source-free Maxwell equations, and if we allow for variations in the permeability p, they have the form... 

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