Forward propagating field

This is an exact system of equations that describes the evolution of modal amplitudes along the z-axis for the forward propagating field. A similar equation holds for the backward propagating component, of course. 

In other words, to obtain a closed system to solve numerically, we must require that the nonlinear polarization is well approximated by the nonlinear polarization calculated only from the forward propagating field. This means that the equation is only applicable when the back-reflected portion of the field is so small that its contribution to the nonlinearity can be neglected. 

When the propagation constant p is positive, the fields of the waveguide propagate in the positive z-direction in Fig. 11-1 (a). Under the transformation p- —p, the backward-propagating fields are simply related to the forward-propagating fields. If we denote forward and backward by -I- and — superscripts, then we deduce from Eq, (30-5) that there are two possibilities. Either... 

It can be seen from the algorithm model stated above that in the Krotov method the electric field obtained in the feth iteration is used immediately to propagate /(f), which has a direct contribution to the new electric field in the next time step. In one iteration, the Krotov method involves three wave packet propagations, that is, the forward propagations of and x (t) in Steps 8.3... 

The total transverse fields of the fiber at its endface, ( and can be represented by the transverse portion of the eigenfunction expansion of Eq. (11-2). Clearly only the forward-propagating modes are required. We use the representation of Eq. (11-6) for the bound-mode fields, and denote the transverse portions of the radiation fields at z = 0 by Et and H,j. Continuity of the transverse fields at the endface requires that... 

On a circular fiber which propagates only the fundamental modes, scattered power that is not radiated can only be directed into the forward- and backward-propagating, even and odd fundamental modes. If the forward-propagating, even HE,i mode is incident on the nonuniformities in Fig. 22-i(a), then the bound portion of the total electric field is given by... 

The total fields of an optical waveguide are expressible as a summation over discrete bound and leaky modes, together with the space-wave fields. This representation is formally identical to the expansion of Eq. (11-2). If we consider only forward-propagating modes, then... 

We relate these fields to the fields of the forward-propagating mode through Eq. (11-7), and from Eqs. (31-1) and (31-5) deduce... 

The coupled mode equations of the previous section can be derived intuitively. This also provides insight into the physical mechanism of the coupling process. Consider a differential section of the perturbed waveguide of length dz, as shown in Fig. 31-2, and its effect on the k th forward-propagating bound mode. The z dependence of the fields, hi(z) of Eq. (31-45), is expressible as... 

The coupling coefficients of Eq. (3 l-65c) are expressible in a more compact form, as we show here. Combining the field components of Eq. (31-58) and substituting the fields for the yth forward-propagating local mode of Eq. (19-2), we have... 

The first term on the right accounts for the change in phase, and the second term for the change in amplitude. To determine the contribution to da -, we examine the reflected and transmitted fields when the k th forward-propagating local mode is incident on the section. 

The total transverse electric field is continuous across the section, i.e. E, = E, where E, is composed of the fields of the kth forward-propagating and all backward-propagating local modes at z, while E is composed of the fields of all forward-propagating local modes at z -f dz. Hence... 

In addition to the beating pattern of the forward propagating SPP field, FDTD simulations in Figure 8.6(e) predict a backward propagating wave... 

Even within the small numbers of studies conducted to date, we are already seeing potentially dramatic effects. Free radical polymerization proceeds at a much faster rate and there is already evidence that both the rate of propagation and the rate of termination are effected. Whole polymerization types - such as ring-opening polymerization to esters and amides, and condensation polymerization of any type (polyamides, polyesters, for example) - have yet to be attempted in ionic liquids. This field is in its infancy and we look forward to the coming years with great anticipation. 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

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