LEGENDRE FUNCTIONS AND THE THEORY OF HOLE-BURNING

Based on the momentum method the Fokker-Planck equation is derived for a velocity distribution perturbed by monomode laser excitation (hole burning). If the used collision kernel applies to marked forward scattering and if the collision partner is assumed to be infinitely heavy, the FPE degenerates into the Legendre equation. While a proportional term accounts for the decay of atoms, an inhomogeneity accounts for their creation. The considered species is metastable argon ions. These ions are probably created by electronic ionization, while both electronic deexcitation and metastability exchange interfere in their destruction. The laser keeps the solution of the FPE at zero level for any velocity whose component along the laser beam upsilon(z,0) follows from the laser frequency by the Doppler condition. We treat this constraint as a boundary condition for the solution. The latter is put up as a superposition of the Legendre function which solves the homogeneous part, with a special solution of the inhomogeneous equation. Free parameters are adjusted to the boundary condition. The thus obtained solution is finally integrated over all the velocity groups of atoms which under our assumptions are not coupled collisionally, to yield the expected shape of the signal. The velocity groups were assumed to be distributed according to Maxwell. We devised two types of experiments to measure the shape of the hole. Both are suited to measure holes on the wings as well as on the centre of the line. Remarkable similarity of the measured and the calculated signals is found.