Focusing of Carr-Purcell photon echoes, and collisional effects

Since for echo formation the off-diagonal density matrix element ${\sigma }_{\mathrm{ab}}$ evolves from ${\sigma }_{\mathrm{ba}}={\sigma }_{\mathrm{ab}}^{*}$ of the previous pulse, the eikonal $\varphi -\omega t$ reverses sign with each pulse. If the radius of curvature of the first $\frac{\pi }{2}$ pulse is $R_1$, the radius of curvature of the subsequent $\pi$ pulses $R_2$, $R_3$, etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{R_{E1\mathrm{}}}=\frac{2}{R_2}-\frac{1}{R_1}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{R_{E2\mathrm{}}}=\frac{2}{R_3}-\frac{2}{R_2}+\frac{1}{R_1}$, and for subsequent Carr-Purcell echoes by $\frac{1}{R_{E3\mathrm{}}}=\frac{2}{R_4}-\frac{2}{R_3}+\frac{2}{R_2}-\frac{1}{R_1}$, 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 $r^{-3\mathrm{}}$ and $r^{-6\mathrm{}}$ interactions between colliding molecules are compared with the experimental data for ${\mathrm{CH}}_3$F and with an $r^{-6\mathrm{}}$ potential with the data for ${\mathrm{SF}}_6$ and ${\mathrm{SiF}}_4$.