New precise results on dynamics of weak-pulse excitation of mirrorless slab of two-level atoms

Abstract

We consider by precise calculations the result of passing a very short coherent electromagnetic pulse through a gaseous slab of resonant (“two-level”) atoms bounded by vacuum on both sides. By “very short” we mean that at time t=0 (just after the passage) the system is left in an initial condition in which (except in a negligibly small region next to the exit face) the atoms have not yet felt reflection from the exit boundary; we assume only weak excitation from the ground state. A large rôle is played in our calculations by what we call ENR (exact numerical results) which are obtained by summing over a very great number of eigenmodes in the modal expansion of our formulas. Among other things, we show that, for thicknesses exceeding 5 resonant wavelengths λ, the forward flux of radiation is almost precisely inverse to the thickness, contrary to Beer’s law. We also introduce a generalized definition of frequency shift by regarding the forward flux spectrum as a mass distribution and finding the shift in its center of mass. With this definition we are able to exhibit oscillations of extremely small amplitude but extremely well defined period in the shift as a function of thickness, starting around 5λ and losing no quality even at 80λ. We have given a new discussion of the landmark 1969 paper of Burnham and Chiao in which the slowly varying envelope approximation (SVEA) was used to solve the atomic excitation problem exactly subject to neglect of the backward wave. We have elucidated the distinction between the thick-slab approximation (decay of excitation faster than return from back face), used there, and our medium-thickness problem (return from back face faster than local decay). We have shown how the method of Burnham and Chiao can be adapted to the medium-thickness problem, yielding an SVEA result almost identical to the one they presented. Other new results are summarized in the Introduction.