The behavior of the photon echo produced by an inhomogeneously broadened spectral line is analyzed, with particular emphasis on the effect of spatial degeneracy of the molecular energy levels. The theoretical results are used to aid in the identification of the states involved in the echoes produced from gaseous S
F6 excited by pulses of C
O2 laser radiation near 10.6 \ensuremath{\mu}. The analysis demonstrates that for the $|{\ensuremath{\Delta}}_{j}|=1$ transitions (
P and
R branches) the echo induced by two linearly polarized pulses propagating along the same direction, with an angle $\ensuremath{\psi}$ between their electric vectors, is polarized at an angle greater than $\ensuremath{\psi}$ with respect to the first pulse, and the echo intensity decreases to a minimum, but not zero, as $\ensuremath{\psi}$ approaches $\frac{\ensuremath{\pi}}{2}$. An exception is the $j=1\ensuremath{\leftrightarrow}j=0$ transition, for which the echo is always polarized along the second pulse and the intensity varies as ${cos}^{2}\ensuremath{\psi}$. For ${\ensuremath{\Delta}}_{j}=0$ transitions (
Q branch) the echo polarization generally lies at an angle smaller than $\ensuremath{\psi}$, i.e., between the first two pulses, and its intensity also decreases as $|\ensuremath{\psi}|$ approaches $\frac{\ensuremath{\pi}}{2}$. The exceptions for the ${\ensuremath{\Delta}}_{j}=0$ transitions are the cases $j=\frac{1}{2}\ensuremath{\leftrightarrow}j=\frac{1}{2}$ and $j=1\ensuremath{\leftrightarrow}j=1$. In the former case, the echo is polarized at an angle $2\ensuremath{\psi}$ with respect to the electric field vector of the first pulse, and the echo intensity is independent of $\ensuremath{\psi}$; whereas in the latter case, the echo behaves like that for $j=1\ensuremath{\leftrightarrow}j=0$. The dependence of the echo behavior on the angular momentum of states involved in the transition can serve to identify the transitions when measurements of the echo polarization and intensity are possible.
The theoretical analysis on photon echoes is reported in a resonant multi-level system with level degeneracy, by a general consideration. With respect to the system possessing more than three-fold degeneracy, the intensity of photon echoes depends not only on the areas of the exciting optical pulses, but also on their polarization characteristics. In particular, it was found that some anomalous echoes change their intensities with the polarization directions of exciting optical pulses which have different frequencies from those of the echoes because of the coherent coupling of resonant transitions with common energy level and the interference effect of transitions through the degenerate sublevels. In the first half of this paper the general formulation on the multi-level system with arbitrary degeneracy is performed, and in the second half, the application of the general theory to some typical examples is also given.
The polarization properties of the photon echo generated by two linearly polarized pulses of resonant radiation at the (6s6p)3 P 1 ↔ (6s 2)1 S 0 transition of 174Yb are investigated. A complicated polarization behavior of the photon echo versus an angle between the polarization vectors of the excitation pulses is revealed in a mixture of ytterbium vapor with inert gas. For the angles ranging from 0° to 75°, a conventional echo with its linear polarization coinciding with the second excitation pulse dominates and the echo amplitude decreases with an increasing angle. For the angles ranging from 75° to 89°, the photon echo is elliptically polarized. Finally, for an angle of 90°, the conventional echo disappears and the collision echo becomes linearly polarized along the first excitation pulse.
We measured the polarization of the photon echo which was stimulated in a gas of S
F6 molecules by either linear-linear, circular-linear, or linear-circular pulse sequences from the
P(14),
P(16),
P(18),
P(20),
P(22), and
P(24) C
O2 laser lines near 10.4 \ensuremath{\mu}m. For
P(16) a linear-linear pulse sequence with angle $\ensuremath{\beta}$ between the linearly polarized pulses produced a linearly polarized echo at angle $\ensuremath{\phi}$ with the second pulse, and $\ensuremath{\phi}$ rotated in the same sense as $\ensuremath{\beta}$; a left-circular-linear sequence produced an echo with right elliptic polarization, and a linear-left-circular sequence produced an echo with left-circular polarization. For all other lines a linear-linear sequence produced an echo polarized along the second pulse, and results were similar to the measurement of Gordon et al. on
P(20). For
P(14),
P(18), and
P(20) a circular-linear sequence produced an echo linearly polarized along the second pulse. For
P(14) through
P(20) a linear-left-circular sequence produced a left-circular echo. A theory is developed for these pulse sequences for arbitrary values of angular momentum
Ja, and a simplification found for large
Ja. For laser radiation in direction
Y and the linear polarization of the second pulse along
Z, the polarization vector for a linear-linear sequence is approximately $\ensuremath{\mp}(\frac{1}{2})\stackrel{^}{X}sin\ensuremath{\beta}+\stackrel{^}{Z}cos\ensuremath{\beta}$, and for a left-circular-linear sequence $\ifmmode\pm\else\textpm\fi{}(\frac{i}{2})\stackrel{^}{X}+\stackrel{^}{Z}$, where the upper sign refers to ${J}_{a}={J}_{b}\ifmmode\pm\else\textpm\fi{}1$ or the
P and
R branches and the lower sign to
Ja=Jb or
Q branch. For a linear-left-circular sequence the echo is left circular for
Ja=Jb, ${J}_{b}\ifmmode\pm\else\textpm\fi{}1$ or the
P,
Q, and
R branches. The general features of the linear-linear sequence are similar to those suggested by Gordon et al. when a sign ambiguity is considered. Only
P(16) was in good agreement with theory. The $\ensuremath{-}(\frac{1}{2})\stackrel{^}{X}sin\ensuremath{\beta}$ or the $+(\frac{i}{2})\stackrel{^}{X}$ terms were not observed for the other lines.
In the limit of small exciting light-pulse areas the amplitude and polarization of the tri-level photon echo in gases formed by the exciting pulses with arbitrary shapes are obtained. It is shown that the dependence of the tri-level photon echo signal on the mutual orientation of the exciting pulse polarization vectors may be used to determine the optical coherence-matrix relaxation characteristics for an optically forbidden transition. The conditions under which a tri-level photon echo pulse repeats the shape of each of the exciting pulses are defined. It is proved that the tri-level photon-echo polarization properties do not depend on the shapes of the exciting pulses, either for small light-pulse areas or for the tri-level echo formation on narrow spectral lines.