Physical Review Letters

Photon Echoes in Gases

C. K. N. Patel, R. E. Slusher

Published May 13, 1968• 201 citations

Abstract

Photon echoes following two pulses of C${\mathrm{O}}_{2}$ laser radiation incident on S${\mathrm{F}}_{6}$ gas have been observed. Homogeneous relaxation times for S${\mathrm{F}}_{6}$ colliding with S${\mathrm{F}}_{6}$, He, Ne, and ${\mathrm{H}}_{2}$ are shown to be easily measurable from the decay of photon-echo amplitude.

DOI: 10.1103/physrevlett.20.1087

Related papers

Photon Echoes in Gases

J. P. GORDON, C. H. WANG, C. K. N. Patel +2 more

Physical ReviewMar 10, 1969150 citations

Polarization of photon echoes from S F6 molecules

C. V. Heer, Robert J. Nordstrom

Physical Review AFeb 1, 197520 citations

Propagation of a Cavity-Dumped C O2 Laser Pulse through S F6

Peter K. Cheo, C. H. Wang

Physical Review AFeb 1, 197021 citations

Coherent optical transient study of molecular collisions: Theory and observations

P. R. Berman, Joel M. Levy, Richard G. Brewer

Physical Review AMay 1, 1975137 citations

Photo Echo and Optical Nutation in Molecules

Richard G. Brewer, R. L. Shoemaker

Physical Review LettersSep 6, 1971338 citations

Photon echo measurements in S F6 and Si F4

W. M. Gutman, C. V. Heer

Physical Review AAug 1, 197713 citations

Physical Review Letters

Photon Echoes in Gases

C. K. N. Patel, R. E. Slusher

Published May 13, 1968• 201 citations

Publisher

Abstract

Photon echoes following two pulses of C

{\mathrm{O}}_{2}

laser radiation incident on S

{\mathrm{F}}_{6}

gas have been observed. Homogeneous relaxation times for S

{\mathrm{F}}_{6}

colliding with S

{\mathrm{F}}_{6}

, He, Ne, and

{\mathrm{H}}_{2}

are shown to be easily measurable from the decay of photon-echo amplitude.

DOI: 10.1103/physrevlett.20.1087

Related papers

Photon Echoes in Gases

J. P. GORDON, C. H. WANG, C. K. N. Patel +2 more

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.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.

Physical ReviewMar 10, 1969150 citations

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

Propagation of a Cavity-Dumped C O2 Laser Pulse through S F6

Peter K. Cheo, C. H. Wang

Coherent interactions between short C ${\mathrm{O}}_{2}$ laser pulses and a resonant absorbing S ${\mathrm{F}}_{6}$ gas medium have been examined carefully, using a laser cavity-dumped rectangular pulse with a width of 20 nsec and input intensities from 10 to ${10}^{4}$ W/ ${\mathrm{cm}}^{2}$ . Parameters studied included the width, shape, and delay time of the transmitted pulse after propagation through several absorption lengths of S ${\mathrm{F}}_{6}$ cell. In addition to these measurements, we have observed two new phenomena, namely, optical free induction decay and edge echo, which are induced by this short rectangular C ${\mathrm{O}}_{2}$ laser input pulse. Results indicate that although level degeneracy in S ${\mathrm{F}}_{6}$ may be involved, the simple two-level model can account qualitatively for most observed phenomena, with the exception that the observed shape of a $2\ensuremath{\pi}$ pulse deviates significantly from the symmetric hyperbolic-secant function predicted by the simple two-level theory. In the $\ensuremath{\pi}$ pulse region, we have demonstrated that the free-induction-decay tail can be valuable for studies of atomic and molecular collision processes.Coherent interactions between short C${\mathrm{O}}_{2}$ laser pulses and a resonant absorbing S${\mathrm{F}}_{6}$ gas medium have been examined carefully, using a laser cavity-dumped rectangular pulse with a width of 20 nsec and input intensities from 10 to ${10}^{4}$ W/${\mathrm{cm}}^{2}$. Parameters studied included the width, shape, and delay time of the transmitted pulse after propagation through several absorption lengths of S${\mathrm{F}}_{6}$ cell. In addition to these measurements, we have observed two new phenomena, namely, optical free induction decay and edge echo, which are induced by this short rectangular C${\mathrm{O}}_{2}$ laser input pulse. Results indicate that although level degeneracy in S${\mathrm{F}}_{6}$ may be involved, the simple two-level model can account qualitatively for most observed phenomena, with the exception that the observed shape of a $2\ensuremath{\pi}$ pulse deviates significantly from the symmetric hyperbolic-secant function predicted by the simple two-level theory. In the $\ensuremath{\pi}$ pulse region, we have demonstrated that the free-induction-decay tail can be valuable for studies of atomic and molecular collision processes.

Physical Review AFeb 1, 197021 citations

Coherent optical transient study of molecular collisions: Theory and observations

P. R. Berman, Joel M. Levy, Richard G. Brewer

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 $^{13}\mathrm{C}$ ${\mathrm{H}}_{3}$ F 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 $^{13}\mathrm{C}$ ${\mathrm{H}}_{3}$ F- $^{13}\mathrm{C}$ ${\mathrm{H}}_{3}$ F collision, i.e., $\ensuremath{\Delta}u=85$ cm/sec, thereby justifying the weak-collision model. The total elastic collision cross section is 430 ${\mathrm{\AA{}}}^{2}$ . It follows that elastic collisions lead to velocity thermalization in a time of \ensuremath{\sim}5 sec when the $^{13}\mathrm{C}$ ${\mathrm{H}}_{3}$ F pressure is 1 mTorr. A comparison is also made of the $^{13}\mathrm{C}$ ${\mathrm{H}}_{3}$ F dephasing time ${\ensuremath{\tau}}_{2}$ in a coherent Raman beat decay, which is independent of velocity-changing collisions, with the longitudinal decay time ${T}_{1}$ . Here, ${T}_{1}$ 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 ${T}_{1}$ 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 $^{13}\mathrm{C}$ ${\mathrm{H}}_{3}$ F transverse dephasing time ${T}_{2}$ without the effect of elastic collisions while being sensitive to "phase interrupting collisions." Again, we find that ${T}_{2}={T}_{1}$ 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.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 $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F vibration-rotation transition. The echo-decay function which provides a signature for the velocity-changing collision diffusion mechanism, is not just a simple exponential in time but exhibits an $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ contribution for short times and an $\mathrm{exp}(\ensuremath{-}\ensuremath{\Gamma}t)$ decay for long times. This behavior, which is unknown heretofore, contrasts with spin echoes in molecular liquids where Brownian motion leads only to the cubic decay law. An $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ behavior can be understood in terms of a solution of the Fokker-Planck equation which describes the effect of Brownian motion on echo decay. Such a treatment is valid for small phase excursions; in the case of a gas, this implies the Doppler phase factor $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\ll}1$, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ is the propagation vector of light, $\ensuremath{\Delta}u$ is a characteristic velocity jump for a binary collision, and $\ensuremath{\tau}$ is the echo-pulse delay time. When $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\gg}1$ as in the long-time regime, the Fokker-Planck solution fails. We, therefore, present a new solution to the Boltzmann transport equation using a weak collision model and find agreement with the entire echo time dependence observed. The echo measurements indicate very small changes in longitudinal velocity per $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F-$^{13}\mathrm{C}$${\mathrm{H}}_{3}$F collision, i.e., $\ensuremath{\Delta}u=85$ cm/sec, thereby justifying the weak-collision model. The total elastic collision cross section is 430 ${\mathrm{\AA{}}}^{2}$. It follows that elastic collisions lead to velocity thermalization in a time of \ensuremath{\sim}5 sec when the $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F pressure is 1 mTorr. A comparison is also made of the $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F dephasing time ${\ensuremath{\tau}}_{2}$ in a coherent Raman beat decay, which is independent of velocity-changing collisions, with the longitudinal decay time ${T}_{1}$. Here, ${T}_{1}$ represents the molecule-optical interaction time, due largely to jumps in molecular rotation ($J$) and orientation ($M$) state, and is obtained from a delayed nutation measurement. The fact that the pressure dependent part of ${\ensuremath{\tau}}_{2}={T}_{1}$ shows that ${T}_{1}$ is also independent of velocity diffusion. Futhermore, when a hole is burned in the Doppler distribution, population recovery must be due to inelastic rather than elastic collisions. Optical Carr-Purcell echoes, multiple pulse echoes, provide a direct measure of the $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F transverse dephasing time ${T}_{2}$ without the effect of elastic collisions while being sensitive to "phase interrupting collisions." Again, we find that ${T}_{2}={T}_{1}$ 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.

Physical Review AMay 1, 1975137 citations

Photo Echo and Optical Nutation in Molecules

Richard G. Brewer, R. L. Shoemaker

Photon echo and optical nutation have been easily observed in ${\mathrm{C}}^{13}$ ${\mathrm{H}}_{3}$ F and N ${\mathrm{H}}_{2}$ D by applying Stark pulses which shift the molecular levels into resonance with cw laser radiation. Numerical computations of the nutation effect agree with observation, and echo characteristics closely follow predictions of existing theories. The $T_{2}^{}{}_{}{}^{\ensuremath{'}}$ pressure dependence, from infrared-echo measurements, indicates that ${\mathrm{C}}^{13}$ ${\mathrm{H}}_{3}$ F relaxes primarily by rotational energy transfer.Photon echo and optical nutation have been easily observed in ${\mathrm{C}}^{13}$${\mathrm{H}}_{3}$F and N${\mathrm{H}}_{2}$D by applying Stark pulses which shift the molecular levels into resonance with cw laser radiation. Numerical computations of the nutation effect agree with observation, and echo characteristics closely follow predictions of existing theories. The $T_{2}^{}{}_{}{}^{\ensuremath{'}}$ pressure dependence, from infrared-echo measurements, indicates that ${\mathrm{C}}^{13}$${\mathrm{H}}_{3}$F relaxes primarily by rotational energy transfer.

Physical Review LettersSep 6, 1971338 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

Photon echoes following two pulses of C

{\mathrm{O}}_{2}

laser radiation incident on S

{\mathrm{F}}_{6}

gas have been observed. Homogeneous relaxation times for S

{\mathrm{F}}_{6}

colliding with S

{\mathrm{F}}_{6}

, He, Ne, and

{\mathrm{H}}_{2}

are shown to be easily measurable from the decay of photon-echo amplitude.

{\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.

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.

Coherent interactions between short C

{\mathrm{O}}_{2}

laser pulses and a resonant absorbing S

{\mathrm{F}}_{6}

gas medium have been examined carefully, using a laser cavity-dumped rectangular pulse with a width of 20 nsec and input intensities from 10 to

{10}^{4}

{\mathrm{cm}}^{2}

. Parameters studied included the width, shape, and delay time of the transmitted pulse after propagation through several absorption lengths of S

{\mathrm{F}}_{6}

cell. In addition to these measurements, we have observed two new phenomena, namely, optical free induction decay and edge echo, which are induced by this short rectangular C

{\mathrm{O}}_{2}

laser input pulse. Results indicate that although level degeneracy in S

{\mathrm{F}}_{6}

may be involved, the simple two-level model can account qualitatively for most observed phenomena, with the exception that the observed shape of a $2\ensuremath{\pi}$ pulse deviates significantly from the symmetric hyperbolic-secant function predicted by the simple two-level theory. In the $\ensuremath{\pi}$ pulse region, we have demonstrated that the free-induction-decay tail can be valuable for studies of atomic and molecular collision processes.

^{13}\mathrm{C}

{\mathrm{H}}_{3}

F 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

^{13}\mathrm{C}

{\mathrm{H}}_{3}

F-

^{13}\mathrm{C}

{\mathrm{H}}_{3}

F collision, i.e., $\ensuremath{\Delta}u=85$ cm/sec, thereby justifying the weak-collision model. The total elastic collision cross section is 430

{\mathrm{\AA{}}}^{2}

. It follows that elastic collisions lead to velocity thermalization in a time of \ensuremath{\sim}5 sec when the

^{13}\mathrm{C}

{\mathrm{H}}_{3}

F pressure is 1 mTorr. A comparison is also made of the

^{13}\mathrm{C}

{\mathrm{H}}_{3}

F dephasing time ${\ensuremath{\tau}}_{2}$ in a coherent Raman beat decay, which is independent of velocity-changing collisions, with the longitudinal decay time

{T}_{1}

. Here,

{T}_{1}

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

{T}_{1}

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

^{13}\mathrm{C}

{\mathrm{H}}_{3}

F transverse dephasing time

{T}_{2}

without the effect of elastic collisions while being sensitive to "phase interrupting collisions." Again, we find that

{T}_{2}={T}_{1}

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.

Photon echo and optical nutation have been easily observed in

{\mathrm{C}}^{13}

{\mathrm{H}}_{3}

F and N

{\mathrm{H}}_{2}

D by applying Stark pulses which shift the molecular levels into resonance with cw laser radiation. Numerical computations of the nutation effect agree with observation, and echo characteristics closely follow predictions of existing theories. The $T_{2}^{}{}_{}{}^{\ensuremath{'}}$ pressure dependence, from infrared-echo measurements, indicates that

{\mathrm{C}}^{13}

{\mathrm{H}}_{3}

F relaxes primarily by rotational energy transfer.

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.