The Lineshape of Saturation-Broadened Transitions in the Gas Phase

Abstract The equations of motion for the density matrix are manipulated in spherical tensor form to derive formulas for the lineshapes of saturation-broadened spectroscopic transitions for gas-phase molecules. The formulas, which include consideration of the spatial degeneracy, are given for both plane-polarized and circularly polarized radiation and are in the form of weighted sums of the reciprocals of the eigenvalues of a matrix. In most practical cases, the eigenvalues will be in the form of Lorentz denominators, which expresses the lineshape as a sum of saturation-broadened Lorentz lineshapes. The approximations required to make the lineshapes equal to a sum of Karplus–Schwinger saturation-broadened lineshapes are described. The lineshape formula for plane-polarized radiation is equivalent to the continued-fraction form previously reported (R. H. Schwendeman, J. Chem. Phys. 73, 4838–4847 (1980)).