Self-consistent field equations, trial

However, if the correction terms are introduced as demonstrated above, it is just a way of approaching the limit in which the trial function is expressed as ff(r, t)X(R, t) with no restriction on the form of X(R, f), i.e., it is not for instance a GWP at all times We shall denote this limit the self-consistent field limit (SCF) or rather the time-dependent SCF limit (TDSCF). The reason for this designation is that the equations for the two wavefunctions are solved self-consistently, i.e., the TDSE for one mode involves the average interaction with the other and vice versa. Thus, if we insert a product type wavefunction r r, t)X R, t) in the TDSE, equation (7), using the expansion (1), multiply fi om the left with X(R, t) and integrate over R we get... 

In order to proceed, we will accept that the transverse components of the electromagnetic field are the only ones that are relevant in the problem on the basis of the exact calculation that we have performed for the fundamental Gaussian beam. Instead, we will use trial functions for u that will lead to self-consistent expressions for the transverse components of Gaussian beams of arbitrary order when substituted into the vector Helmholtz equation. The derivation is clearest for the fundamental. We will redrive the transverse field components of the fundamental Gaussian beam here. The deviation of higher order modes is outlined in the Appendix. 

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