Physical Review

Photon Echoes in Gases

J. P. GORDON, C. H. WANG, C. K. N. Patel, R. E. Slusher, W. J. TOMLINSON

Published Mar 10, 1969• 150 citations

Abstract

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${\mathrm{F}}_{6}$ excited by pulses of C${\mathrm{O}}_{2}$ 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.

DOI: 10.1103/physrev.179.294

Related papers

Polarization of photon echoes from S F6 molecules

C. V. Heer, Robert J. Nordstrom

Physical Review AFeb 1, 197520 citations

Photon echo measurements in S F6 and Si F4

W. M. Gutman, C. V. Heer

Physical Review AAug 1, 197713 citations

Conventional and collision photon echo in ytterbium vapor

Н. Н. Рубцова, Sergei A. Kochubei, В. Н. Ищенко +2 more

Laser PhysicsFeb 1, 20099 citations

Transmission of Coherent Optical Pulses in Gaseous S F6

C. K. Rhodes, A. Szöke

Physical ReviewAug 5, 196979 citations

Superradiant Damping Effects in Photon-Echo Theory

Antonio M. Ponte Goncalves, Andree A. Tallet, R. Lefèbvre

Physical Review AMay 1, 197013 citations

Theory of modulated photon echoes

Axel Schenzle, S. B. Grossman, Richard G. Brewer

Physical Review AMay 1, 197623 citations

Physical Review

Photon Echoes in Gases

J. P. GORDON, C. H. WANG, C. K. N. Patel, R. E. Slusher, W. J. TOMLINSON

Published Mar 10, 1969• 150 citations

Publisher

Abstract

{\mathrm{F}}_{6}

excited by pulses of C

{\mathrm{O}}_{2}

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.

DOI: 10.1103/physrev.179.294

Related papers

Polarization of photon echoes from S F6 molecules

C. V. Heer, Robert J. Nordstrom

We measured the polarization of the photon echo which was stimulated in a gas of S ${\mathrm{F}}_{6}$ 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 ${\mathrm{O}}_{2}$ 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 ${J}_{a}$ , and a simplification found for large ${J}_{a}$ . 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 ${J}_{a}={J}_{b}$ or $Q$ branch. For a linear-left-circular sequence the echo is left circular for ${J}_{a}={J}_{b}$ , ${J}_{b}\ifmmode\pm\else\textpm\fi{}1$ 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.We measured the polarization of the photon echo which was stimulated in a gas of S${\mathrm{F}}_{6}$ 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${\mathrm{O}}_{2}$ 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 ${J}_{a}$, and a simplification found for large ${J}_{a}$. 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 ${J}_{a}={J}_{b}$ or $Q$ branch. For a linear-left-circular sequence the echo is left circular for ${J}_{a}={J}_{b}$, ${J}_{b}\ifmmode\pm\else\textpm\fi{}1$ 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.

Physical Review AFeb 1, 197520 citations

Photon echo measurements in S F6 and Si F4

W. M. Gutman, C. V. Heer

The polarization of photon echoes stimulated by linear-linear, circular-linear, and linear-circular excitation sequences has been studied in S ${\mathrm{F}}_{6}$ with the $P(12)$ through $P(26)$ laser lines of C ${\mathrm{O}}_{2}$ near 10.6 \ensuremath{\mu}m and in Si ${\mathrm{F}}_{4}$ with the $P(28)$ through $P(34)$ lines near 9.6 \ensuremath{\mu}m. A linear-linear pulse sequence with angle $\ensuremath{\beta}$ between the linear polarizations produces an echo at angle $\ensuremath{\phi}$ with respect to the second pulse. $\ensuremath{\phi}$ rotates in the same sense as $\ensuremath{\beta}$ with $P(16)$ in S ${\mathrm{F}}_{6}$ and in the opposite sense as $\ensuremath{\beta}$ with the other lines in S ${\mathrm{F}}_{6}$ , and with the lines in Si ${\mathrm{F}}_{4}$ . A left-circular-linear sequence produces a right-elliptic echo with $P(16)$ in S ${\mathrm{F}}_{6}$ , and a left-elliptic echo in the other cases. A linear-left-circular sequence produces a left-circular echo in all cases. These results are in accord with the large- $J$ polarization theory of Heer and Nordstrom. For a linear-linear excitation sequence, this theory predicts an unnormalized echo electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\cong}\ensuremath{\mp}\stackrel{^}{X}(\frac{1}{2}sin\ensuremath{\beta})+\stackrel{^}{Z}cos\ensuremath{\beta}$, where the negative sign applies to a $P$ - or $R$ -branch transition, and the plus sign to a $Q$ -branch transition. For a left-circular-linear sequence, the theory predicts an unnormalized echo electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\cong}\ifmmode\pm\else\textpm\fi{}\frac{1}{2}i\stackrel{^}{X}+\stackrel{^}{Z}$, where the plus sign refers to a $P$ - or $R$ -branch transition, and the negative sign to a $Q$ -branch transition. Echo decay rates determined from relaxation measurements are used to calculate molecular cross sections, and these cross sections are compared with calculations based on the refractive indices and the Slater-Kirkwood equation. Experimental cross sections are in general smaller than those found from the Slater-Kirkwood equation.The polarization of photon echoes stimulated by linear-linear, circular-linear, and linear-circular excitation sequences has been studied in S${\mathrm{F}}_{6}$ with the $P(12)$ through $P(26)$ laser lines of C${\mathrm{O}}_{2}$ near 10.6 \ensuremath{\mu}m and in Si${\mathrm{F}}_{4}$ with the $P(28)$ through $P(34)$ lines near 9.6 \ensuremath{\mu}m. A linear-linear pulse sequence with angle $\ensuremath{\beta}$ between the linear polarizations produces an echo at angle $\ensuremath{\phi}$ with respect to the second pulse. $\ensuremath{\phi}$ rotates in the same sense as $\ensuremath{\beta}$ with $P(16)$ in S${\mathrm{F}}_{6}$ and in the opposite sense as $\ensuremath{\beta}$ with the other lines in S${\mathrm{F}}_{6}$, and with the lines in Si${\mathrm{F}}_{4}$. A left-circular-linear sequence produces a right-elliptic echo with $P(16)$ in S${\mathrm{F}}_{6}$, and a left-elliptic echo in the other cases. A linear-left-circular sequence produces a left-circular echo in all cases. These results are in accord with the large-$J$ polarization theory of Heer and Nordstrom. For a linear-linear excitation sequence, this theory predicts an unnormalized echo electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\cong}\ensuremath{\mp}\stackrel{^}{X}(\frac{1}{2}sin\ensuremath{\beta})+\stackrel{^}{Z}cos\ensuremath{\beta}$, where the negative sign applies to a $P$- or $R$-branch transition, and the plus sign to a $Q$-branch transition. For a left-circular-linear sequence, the theory predicts an unnormalized echo electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\cong}\ifmmode\pm\else\textpm\fi{}\frac{1}{2}i\stackrel{^}{X}+\stackrel{^}{Z}$, where the plus sign refers to a $P$- or $R$-branch transition, and the negative sign to a $Q$-branch transition. Echo decay rates determined from relaxation measurements are used to calculate molecular cross sections, and these cross sections are compared with calculations based on the refractive indices and the Slater-Kirkwood equation. Experimental cross sections are in general smaller than those found from the Slater-Kirkwood equation.

Physical Review AAug 1, 197713 citations

Conventional and collision photon echo in ytterbium vapor

Н. Н. Рубцова, Sergei A. Kochubei, В. Н. Ищенко +2 more

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.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.

Laser PhysicsFeb 1, 20099 citations

Transmission of Coherent Optical Pulses in Gaseous S F6

C. K. Rhodes, A. Szöke

Strong and short coherent pulses of electromagnetic radiation propagate through extended absorbing media with anomalously low absorption, they are delayed and reshaped. This article presents experimental results and calculations for propagation of 10.6-\ensuremath{\mu} wavelength pulses through gaseous S ${\mathrm{F}}_{6}$ . The degeneracy of overlapping levels of S ${\mathrm{F}}_{6}$ in that spectral region is important, and enables us to use a simple model to describe observed phenomena. We assume a continuous distribution of dipole moments. Results are particularly interesting in two regimes of excitation. One corresponds to very strong excitation and \ensuremath{\sim}20 absorption lengths in the medium: Pulses sharpen at their leading edge, while their front part gets absorbed. The other pertains to much lower excitation and only 2 absorption lengths: Reradiation can be distinguished and long delays result which qualitatively resemble the behavior of a two-level system in self-induced transparency. Fairly conclusive agreement is obtained between a previous theoretical treatment of a homogeneously broadened multilevel medium and experiment. The agreement is substantiated by computer calculations. In three Appendixes we present the observation of focusing effects, the theory of an experiment related to transient nutation, and some remarks on photon-echo polarization experiments in optically thick media.Strong and short coherent pulses of electromagnetic radiation propagate through extended absorbing media with anomalously low absorption, they are delayed and reshaped. This article presents experimental results and calculations for propagation of 10.6-\ensuremath{\mu} wavelength pulses through gaseous S${\mathrm{F}}_{6}$. The degeneracy of overlapping levels of S${\mathrm{F}}_{6}$ in that spectral region is important, and enables us to use a simple model to describe observed phenomena. We assume a continuous distribution of dipole moments. Results are particularly interesting in two regimes of excitation. One corresponds to very strong excitation and \ensuremath{\sim}20 absorption lengths in the medium: Pulses sharpen at their leading edge, while their front part gets absorbed. The other pertains to much lower excitation and only 2 absorption lengths: Reradiation can be distinguished and long delays result which qualitatively resemble the behavior of a two-level system in self-induced transparency. Fairly conclusive agreement is obtained between a previous theoretical treatment of a homogeneously broadened multilevel medium and experiment. The agreement is substantiated by computer calculations. In three Appendixes we present the observation of focusing effects, the theory of an experiment related to transient nutation, and some remarks on photon-echo polarization experiments in optically thick media.

Physical ReviewAug 5, 196979 citations

Superradiant Damping Effects in Photon-Echo Theory

Antonio M. Ponte Goncalves, Andree A. Tallet, R. Lefèbvre

A theoretical model for superradiant damping effect in inhomogeneously broadened media and its application to photon-echo problems are discussed. Only optically thin systems are considered, and all nonradiative relaxation mechanisms are neglected. Numerical calculations are performed for a very wide square inhomogeneous line shape and square applied pulses. It is shown that excitation of the system becomes increasingly difficult as the superradiant lifetime ${\ensuremath{\tau}}_{s}$ becomes comparable with ${T}_{2}^{*}$ . The energy reradiated after the end of the exciting pulse is considerable for ${\ensuremath{\tau}}_{s}\ensuremath{\approx}{T}_{2}^{*}$ and for some applied pulse areas. Two-pulse sequences produce multiple echoes as the microscopic polarizations spontaneously rephase. The first-echo amplitude maximum is found to occur for first-applied-pulse areas increasingly above $\frac{1}{2}\ensuremath{\pi}$ as ${\ensuremath{\tau}}_{s}$ decreases. For a given first-pulse area, the echo intensities increase initially as ${\ensuremath{\tau}}_{s}$ decreases (negligible damping) and finally decrease and vanish (important damping). The ratio of the second- to first-echo amplitudes is found to be proportional to $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$ for small superradiance levels and to decrease when ${\ensuremath{\tau}}_{s}$ approaches ${T}_{2}^{*}$ . An approximate estimate of this ratio is derived for the region of linearity in $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$.A theoretical model for superradiant damping effect in inhomogeneously broadened media and its application to photon-echo problems are discussed. Only optically thin systems are considered, and all nonradiative relaxation mechanisms are neglected. Numerical calculations are performed for a very wide square inhomogeneous line shape and square applied pulses. It is shown that excitation of the system becomes increasingly difficult as the superradiant lifetime ${\ensuremath{\tau}}_{s}$ becomes comparable with ${T}_{2}^{*}$. The energy reradiated after the end of the exciting pulse is considerable for ${\ensuremath{\tau}}_{s}\ensuremath{\approx}{T}_{2}^{*}$ and for some applied pulse areas. Two-pulse sequences produce multiple echoes as the microscopic polarizations spontaneously rephase. The first-echo amplitude maximum is found to occur for first-applied-pulse areas increasingly above $\frac{1}{2}\ensuremath{\pi}$ as ${\ensuremath{\tau}}_{s}$ decreases. For a given first-pulse area, the echo intensities increase initially as ${\ensuremath{\tau}}_{s}$ decreases (negligible damping) and finally decrease and vanish (important damping). The ratio of the second- to first-echo amplitudes is found to be proportional to $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$ for small superradiance levels and to decrease when ${\ensuremath{\tau}}_{s}$ approaches ${T}_{2}^{*}$. An approximate estimate of this ratio is derived for the region of linearity in $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$.

Physical Review AMay 1, 197013 citations

Theory of modulated photon echoes

Axel Schenzle, S. B. Grossman, Richard G. Brewer

Photon echoes, observed recently in ruby and in ${\mathrm{NH}}_{2}$ D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.Photon echoes, observed recently in ruby and in ${\mathrm{NH}}_{2}$D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.

Physical Review AMay 1, 197623 citations

{\mathrm{F}}_{6}

excited by pulses of C

{\mathrm{O}}_{2}

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

We measured the polarization of the photon echo which was stimulated in a gas of S

{\mathrm{F}}_{6}

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)

{\mathrm{O}}_{2}

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

{J}_{a}

, and a simplification found for large

{J}_{a}

. 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

{J}_{a}={J}_{b}

Q

branch. For a linear-left-circular sequence the echo is left circular for

{J}_{a}={J}_{b}

, ${J}_{b}\ifmmode\pm\else\textpm\fi{}1$ 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.

The polarization of photon echoes stimulated by linear-linear, circular-linear, and linear-circular excitation sequences has been studied in S

{\mathrm{F}}_{6}

with the

P(12)

through

P(26)

laser lines of C

{\mathrm{O}}_{2}

near 10.6 \ensuremath{\mu}m and in Si

{\mathrm{F}}_{4}

with the

P(28)

through

P(34)

lines near 9.6 \ensuremath{\mu}m. A linear-linear pulse sequence with angle $\ensuremath{\beta}$ between the linear polarizations produces an echo at angle $\ensuremath{\phi}$ with respect to the second pulse. $\ensuremath{\phi}$ rotates in the same sense as $\ensuremath{\beta}$ with

P(16)

in S

{\mathrm{F}}_{6}

and in the opposite sense as $\ensuremath{\beta}$ with the other lines in S

{\mathrm{F}}_{6}

, and with the lines in Si

{\mathrm{F}}_{4}

. A left-circular-linear sequence produces a right-elliptic echo with

P(16)

in S

{\mathrm{F}}_{6}

, and a left-elliptic echo in the other cases. A linear-left-circular sequence produces a left-circular echo in all cases. These results are in accord with the large-

J

polarization theory of Heer and Nordstrom. For a linear-linear excitation sequence, this theory predicts an unnormalized echo electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\cong}\ensuremath{\mp}\stackrel{^}{X}(\frac{1}{2}sin\ensuremath{\beta})+\stackrel{^}{Z}cos\ensuremath{\beta}$, where the negative sign applies to a

P

- or

R

-branch transition, and the plus sign to a

Q

-branch transition. For a left-circular-linear sequence, the theory predicts an unnormalized echo electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ensuremath{\cong}\ifmmode\pm\else\textpm\fi{}\frac{1}{2}i\stackrel{^}{X}+\stackrel{^}{Z}$, where the plus sign refers to a

P

- or

R

-branch transition, and the negative sign to a

Q

-branch transition. Echo decay rates determined from relaxation measurements are used to calculate molecular cross sections, and these cross sections are compared with calculations based on the refractive indices and the Slater-Kirkwood equation. Experimental cross sections are in general smaller than those found from the Slater-Kirkwood equation.

{\mathrm{F}}_{6}

. The degeneracy of overlapping levels of S

{\mathrm{F}}_{6}

in that spectral region is important, and enables us to use a simple model to describe observed phenomena. We assume a continuous distribution of dipole moments. Results are particularly interesting in two regimes of excitation. One corresponds to very strong excitation and \ensuremath{\sim}20 absorption lengths in the medium: Pulses sharpen at their leading edge, while their front part gets absorbed. The other pertains to much lower excitation and only 2 absorption lengths: Reradiation can be distinguished and long delays result which qualitatively resemble the behavior of a two-level system in self-induced transparency. Fairly conclusive agreement is obtained between a previous theoretical treatment of a homogeneously broadened multilevel medium and experiment. The agreement is substantiated by computer calculations. In three Appendixes we present the observation of focusing effects, the theory of an experiment related to transient nutation, and some remarks on photon-echo polarization experiments in optically thick media.

{T}_{2}^{*}

. The energy reradiated after the end of the exciting pulse is considerable for ${\ensuremath{\tau}}_{s}\ensuremath{\approx}{T}_{2}^{*}$ and for some applied pulse areas. Two-pulse sequences produce multiple echoes as the microscopic polarizations spontaneously rephase. The first-echo amplitude maximum is found to occur for first-applied-pulse areas increasingly above $\frac{1}{2}\ensuremath{\pi}$ as ${\ensuremath{\tau}}_{s}$ decreases. For a given first-pulse area, the echo intensities increase initially as ${\ensuremath{\tau}}_{s}$ decreases (negligible damping) and finally decrease and vanish (important damping). The ratio of the second- to first-echo amplitudes is found to be proportional to $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$ for small superradiance levels and to decrease when ${\ensuremath{\tau}}_{s}$ approaches

{T}_{2}^{*}

. An approximate estimate of this ratio is derived for the region of linearity in $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$.

Photon echoes, observed recently in ruby and in

{\mathrm{NH}}_{2}

D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.