Among various photon echo phenomena which have been predicted and observed in vapours [1], stimulated photon echo (three pulse echo) is a very attractive tool to investigate angular dependence of collisional processes. Already applied to the study of small angle elastic scattering [2] this method may also prove adequate for studying quasi-resonant inelastic processes involving excited species such as depolarizing collisions, collisional transitions between hyperfine, fine or rotational sublevels, energy pooling collisions. For all these processes total cross sections have already been obtained in the past by different experimental ways. However they deserve to be reexamined in the perspective of their angular dependence.
It has been shown recently that molecular gas samples excited with coherent light can display a variety of transient phenomena, similar to those found in nuclear magnetic resonance. This article elucidates how these coherence effects can be used to isolate or unfold molecular collision mechanisms that normally remain hidden within the optical line shape. Elastic collisions, for example, are easily detected here in two-pulse photon echo experiments for a
13CH3F vibration-rotation transition. The echo-decay function which provides a signature for the velocity-changing collision diffusion mechanism, is not just a simple exponential in time but exhibits an $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ contribution for short times and an $\mathrm{exp}(\ensuremath{-}\ensuremath{\Gamma}t)$ decay for long times. This behavior, which is unknown heretofore, contrasts with spin echoes in molecular liquids where Brownian motion leads only to the cubic decay law. An $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ behavior can be understood in terms of a solution of the Fokker-Planck equation which describes the effect of Brownian motion on echo decay. Such a treatment is valid for small phase excursions; in the case of a gas, this implies the Doppler phase factor $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\ll}1$, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ is the propagation vector of light, $\ensuremath{\Delta}u$ is a characteristic velocity jump for a binary collision, and $\ensuremath{\tau}$ is the echo-pulse delay time. When $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\gg}1$ as in the long-time regime, the Fokker-Planck solution fails. We, therefore, present a new solution to the Boltzmann transport equation using a weak collision model and find agreement with the entire echo time dependence observed. The echo measurements indicate very small changes in longitudinal velocity per
13CH3F-
13CH3F collision, i.e., $\ensuremath{\Delta}u=85$ cm/sec, thereby justifying the weak-collision model. The total elastic collision cross section is 430
A˚2. It follows that elastic collisions lead to velocity thermalization in a time of \ensuremath{\sim}5 sec when the
13CH3F pressure is 1 mTorr. A comparison is also made of the
13CH3F dephasing time ${\ensuremath{\tau}}_{2}$ in a coherent Raman beat decay, which is independent of velocity-changing collisions, with the longitudinal decay time
T1. Here,
T1 represents the molecule-optical interaction time, due largely to jumps in molecular rotation (
J) and orientation (
M) state, and is obtained from a delayed nutation measurement. The fact that the pressure dependent part of ${\ensuremath{\tau}}_{2}={T}_{1}$ shows that
T1 is also independent of velocity diffusion. Futhermore, when a hole is burned in the Doppler distribution, population recovery must be due to inelastic rather than elastic collisions. Optical Carr-Purcell echoes, multiple pulse echoes, provide a direct measure of the
13CH3F transverse dephasing time
T2 without the effect of elastic collisions while being sensitive to "phase interrupting collisions." Again, we find that
T2=T1 so that phase interruptions are negligible. Had such a process dominated the two-pulse echo, an $\mathrm{exp}(\ensuremath{-}\frac{t}{{T}_{2}})$ damping would have been noticed with no $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ contribution. Thus, the present study covers several new aspects of molecular collisions. It represents the first detailed examination of velocity-changing collisions by coherence methods and without the complication of Doppler broadening.
Since for echo formation the off-diagonal density matrix element ${\ensuremath{\sigma}}_{\mathrm{ab}}$ evolves from ${\ensuremath{\sigma}}_{\mathrm{ba}}={\ensuremath{\sigma}}_{\mathrm{ab}}^{*}$ of the previous pulse, the eikonal $\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t$ reverses sign with each pulse. If the radius of curvature of the first $\frac{\ensuremath{\pi}}{2}$ pulse is
R1, the radius of curvature of the subsequent $\ensuremath{\pi}$ pulses
R2,
R3, etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{{R}_{E1}}=\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{{R}_{E2}}=\frac{2}{{R}_{3}}\ensuremath{-}\frac{2}{{R}_{2}}+\frac{1}{{R}_{1}}$, and for subsequent Carr-Purcell echoes by $\frac{1}{{R}_{E3}}=\frac{2}{{R}_{4}}\ensuremath{-}\frac{2}{{R}_{3}}+\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, 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 ${r}^{\ensuremath{-}3}$ and ${r}^{\ensuremath{-}6}$ interactions between colliding molecules are compared with the experimental data for
CH3F and with an ${r}^{\ensuremath{-}6}$ potential with the data for
SF6 and
SiF4.
Effects of electric dipole reorienting collisions on photon echoes are investigated. In an extension of the author's earlier work, a general expression is derived, including terms from binary collisions, for the photonecho dipole moment amplitude. The explicit results are worked out for simple cases such as ${j}_{1}=\frac{1}{2}\ensuremath{\leftrightarrow}{j}_{2}=\frac{1}{2}$, ${j}_{1}=1\ensuremath{\leftrightarrow}{j}_{2}=0$, ${j}_{1}=1\ensuremath{\leftrightarrow}{j}_{2}=1$, and ${j}_{1}=\frac{3}{2}\ensuremath{\leftrightarrow}{j}_{2}=\frac{1}{2}$ transitions. While it is shown that echoes resulting from the above simple transitions will not be depolarized by atomic collisions, photon echoes induced by any other transitions are necessarily depolarized by collisions. In addition, it is possible to show that the general expression for the echo dipole moment can be written as a sum of multipole moments. Only the odd moments contribute to the echoes. The importance of this picture is that collision-induced relaxation of each moment is described by a single time constant---as in the model commonly used for Hanle-effect experiments. As a result, for an echo arising from a ${j}_{1}\ensuremath{\leftrightarrow}{j}_{2}$ transition (${j}_{1}\ensuremath{\ge}{j}_{2}$), there must be
21j1(j1+1) time constants to describe the collision damping of the echo intensity,
j1 being an integer. The corresponding formula for the half-integer case is
21(j2+21)(j2+23).
Elastic and depolarizing collisions between excited ytterbium atoms (6s6p
3P1 level) and noble gases (He, Ne, and Xe) are investigated using a stimulated-photon-echo technique. The method is sensitive to collisional velocity changes concerning both the total population ${\ensuremath{\rho}}^{0}$ and the alignment ${\ensuremath{\rho}}^{2}$ of the level. More specifically, it enables us to compare precisely the velocity changes associated with elastic scattering and with alignment-preserving scattering. This represents an improvement with regard to previous works. A detailed theoretical discussion of depolarizing collisions in laser spectroscopy is presented. A simplified density-matrix equation is derived under appropriate approximations. Stimulated photon-echo data for elastic scattering are interpreted within the limits of small-angle scattering in a long-range van der Waals potential; total elastic scattering cross sections are derived from comparison between experiment and calculation: ${\mathit{\ensuremath{\sigma}}}_{\mathrm{He}}^{\mathrm{el}}$=220(20) A
˚2, ${\mathit{\ensuremath{\sigma}}}_{\mathrm{Ne}}^{\mathrm{el}}$=320(20) A
˚2, ${\mathit{\ensuremath{\sigma}}}_{\mathrm{Xe}}^{\mathrm{el}}$=1240(60) A
˚2. Total rates for destruction of alignment are measured in an auxiliary standard photon-echo experiment; the corresponding cross sections are ${\mathit{\ensuremath{\sigma}}}_{\mathrm{He}}^{(2)}$=94(3) A
˚2, ${\mathit{\ensuremath{\sigma}}}_{\mathrm{Ne}}^{(2)}$=123(7) A
˚2, ${\mathit{\ensuremath{\sigma}}}_{\mathrm{Xe}}^{(2)}$=300(20) A
˚2. Stimulated photon-echo data for depolarizing collisions are successfully compared with a calculation that we develop starting from a long-range anisotropic model potential.