Focusing of Carr-Purcell photon echoes, and collisional effects

Since for echo formation the off-diagonal density matrix element σab evolves from σba=σab* of the previous pulse, the eikonal φ-ωt reverses sign with each pulse. If the radius of curvature of the first π2 pulse is R₁, the radius of curvature of the subsequent π pulses R₂, R₃, etc., the radius of curvature of the photon echo which follows the second pulse is 1RE1=2R₂-1R₁, the radius of curvature of the Carr-Purcell echo following the third pulse is given by 1RE2=2R₃-2R₂+1R₁, and for subsequent Carr-Purcell echoes by 1RE3=2R₄-2R₃+2R₂-1R₁, etc. The influence of motion, adiabatic collisions, and velocity-changing collisions on the decay of echo amplitude is considered for curved wave fronts. Since the change in position from the constant-velocity position is important in the decay, a statistical theory for small displacement and its relationship to the forward scattering cross section is given. These results with r^-3 and r^-6 interactions between colliding molecules are compared with the experimental data for CH₃F and with an r^-6 potential with the data for SF₆ and SiF₄.