Physical Review A

Optical properties of photon echoes stimulated by three frequencies

C. V. Heer, Richard L. Sutherland

Published May 1, 1979• 11 citations

Abstract

A gaseous atomic or molecular system with four energy levels ${E}_{a}<{E}_{b}<{E}_{c}<{E}_{d}$ which is stimulated by resonant laser pulses at frequencies ${\ensuremath{\omega}}_{1}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{ba}}$, ${\ensuremath{\omega}}_{2}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{cb}}$, ${\ensuremath{\omega}}_{3}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{dc}}$ is considered. The six photon echoes which are stimulated by first and second pulses at frequencies ${\ensuremath{\omega}}_{1}$, ${\ensuremath{\omega}}_{2}$, ${\ensuremath{\omega}}_{3}$ are discussed. For wave fronts which are described by ${e}^{i(\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t)}$ the echo phase is described by ${\ensuremath{\phi}}_{E}=\ensuremath{\Sigma}(2{\ensuremath{\phi}}_{m}^{\mathrm{II}}\ensuremath{-}{\ensuremath{\phi}}_{m}^{\mathrm{I}})$ at the sum frequency ${\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}+{\ensuremath{\omega}}_{3}$ with $m=1,2,3$, at ${\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}$ with $m=1,2$, at ${\ensuremath{\omega}}_{1}$ with $m=1$, etc. For waves in the $z$ direction where $\ensuremath{\phi}=\ensuremath{\kappa}(x,y,z)+kz$ the gradient of the phase yields the wave-vector direction ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{E}=\ensuremath{\Sigma}(2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{m}^{\mathrm{II}}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{m}^{\mathrm{I}})$ in a physically thin sample. A probe pulse at ${\ensuremath{\omega}}_{2}$ and ${\ensuremath{\omega}}_{3}$ after the second pulse can generate a probe-pulse echo at ${\ensuremath{\omega}}_{1}$ with phase ${\ensuremath{\phi}}_{1}=\ensuremath{\Sigma}(2{\ensuremath{\phi}}_{m}^{\mathrm{II}}\ensuremath{-}{\ensuremath{\phi}}_{m}^{\mathrm{I}})\ensuremath{-}{\ensuremath{\phi}}_{2}^{S}\ensuremath{-}{\ensuremath{\phi}}_{3}^{S}$, etc., for other frequency arrangements. Transverse phases ${\ensuremath{\kappa}}_{m}(\mathrm{xy})$ can be added and subtracted, but no application is suggested for this novel feature. When all ${\ensuremath{\kappa}}_{m}$ except one are zero, phase conjugation or phase-adaptive optics can occur.

DOI: 10.1103/physreva.19.2026

Related papers

DEGENERATE PHOTON ECHOES

Masaharu Mitsunaga, M. K. Kim, Ravinder Kachru

Jul 18, 19880 citations

Photon Echoes in Gases

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

Physical ReviewMar 10, 1969150 citations

Multipulse photon echoes in LaF3 : Pr3+

Lynden E. Erickson

Physical Review ANov 1, 19924 citations

Echoes in gaseous media: A generalized theory of rephasing phenomena

THOMAS W. MOSSBERG, Ravinder Kachru, Sven R. Hartmann +1 more

Physical Review ANov 1, 1979164 citations

Polarization of photon echoes from S F6 molecules

C. V. Heer, Robert J. Nordstrom

Physical Review AFeb 1, 197520 citations

Storage and phase conjugation of light pulses using stimulated photon echoes

N. W. Carlson, William Randall Babbitt, THOMAS W. MOSSBERG

We experimentally demonstrate that three-excitation-pulse (i.e., stimulated) photon echo signals can be backward-propagating, phase-conjugate replicas of the second echo-excitation pulse. Working on the 555.6-nm absorption line of atomic Yb vapor, echo signals, effectively 0.5% as intense as the second excitation pulse, are shown to reproduce image information carried by the second excitation pulse even when wave-front distorters are employed.

Optics LettersDec 1, 198344 citations

Physical Review A

Optical properties of photon echoes stimulated by three frequencies

C. V. Heer, Richard L. Sutherland

Published May 1, 1979• 11 citations

Publisher

Abstract

A gaseous atomic or molecular system with four energy levels

{E}_{a}<{E}_{b}<{E}_{c}<{E}_{d}

which is stimulated by resonant laser pulses at frequencies

{\ensuremath{\omega}}_{1}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{ba}}

{\ensuremath{\omega}}_{2}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{cb}}

{\ensuremath{\omega}}_{3}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{dc}}

is considered. The six photon echoes which are stimulated by first and second pulses at frequencies

{\ensuremath{\omega}}_{1}

{\ensuremath{\omega}}_{2}

{\ensuremath{\omega}}_{3}

are discussed. For wave fronts which are described by

{e}^{i(\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t)}

the echo phase is described by

{\ensuremath{\phi}}_{E}=\ensuremath{\Sigma}(2{\ensuremath{\phi}}_{m}^{\mathrm{II}}\ensuremath{-}{\ensuremath{\phi}}_{m}^{\mathrm{I}})

at the sum frequency

{\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}+{\ensuremath{\omega}}_{3}

with

m=1,2,3

, at

{\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}

with

m=1,2

, at

{\ensuremath{\omega}}_{1}

with

m=1

, etc. For waves in the

z

direction where

\ensuremath{\phi}=\ensuremath{\kappa}(x,y,z)+kz

the gradient of the phase yields the wave-vector direction

{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{E}=\ensuremath{\Sigma}(2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{m}^{\mathrm{II}}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{m}^{\mathrm{I}})

in a physically thin sample. A probe pulse at

{\ensuremath{\omega}}_{2}

and

{\ensuremath{\omega}}_{3}

after the second pulse can generate a probe-pulse echo at

{\ensuremath{\omega}}_{1}

with phase

{\ensuremath{\phi}}_{1}=\ensuremath{\Sigma}(2{\ensuremath{\phi}}_{m}^{\mathrm{II}}\ensuremath{-}{\ensuremath{\phi}}_{m}^{\mathrm{I}})\ensuremath{-}{\ensuremath{\phi}}_{2}^{S}\ensuremath{-}{\ensuremath{\phi}}_{3}^{S}

, etc., for other frequency arrangements. Transverse phases

{\ensuremath{\kappa}}_{m}(\mathrm{xy})

can be added and subtracted, but no application is suggested for this novel feature. When all

{\ensuremath{\kappa}}_{m}

except one are zero, phase conjugation or phase-adaptive optics can occur.

DOI: 10.1103/physreva.19.2026

Related papers

DEGENERATE PHOTON ECHOES

Masaharu Mitsunaga, M. K. Kim, Ravinder Kachru

The three-pulse stimulated photon echo has been considered as one of the most fundamental transient four wave mixing processes1, where each of the three excitation pulses induces a single elementary interaction process between a laser field and an inhomogeneously broadened two-level system., resulting in a burst of the fourth wave, i.e., echo. Now if four or more optical pulses are applied to a medium, a number of photon echoes should be expected, in general. At low excitation intensities where the perturbation theory is applicable, however, the dominant nonlinear process is still the three-pulse stimulated echo and consequently each echo signal is generated only by three of the preceding pulses. It is rather a remarkable result that the echo intensity is unaffected by the presence of all the rest of the pulses other than its three generators. With this in mind, it is theoretically and experimentally shown that by applying a single optical pulse (#1) propagating along k1 followed by N pairs of counter-propagating pulses (#2j (k2j) and #3j− (k3j = −k2j); j=1,…N) in various directions, it is possible to obtain N photon echoes along the direction -k1 opposite to the first pulse #1. We will call this degenerate photon echoes, degenerate in. the sense that the echoes are spatially overlapping and temporally simultaneous. When N=l, this scheme is reduced to the ordinary backward photon echo2,3. The doubly degenerate case (N=2) is illustrated in Fig. 1. The j-th echo will be generated by the set of pulses (#1, #2j, #3j) and appear at t = t3j + t2j − t1 where t1, i2j and t3j are the arrival times of the pulses #1, #2j and #3j, respectively.The three-pulse stimulated photon echo has been considered as one of the most fundamental transient four wave mixing processes1, where each of the three excitation pulses induces a single elementary interaction process between a laser field and an inhomogeneously broadened two-level system., resulting in a burst of the fourth wave, i.e., echo. Now if four or more optical pulses are applied to a medium, a number of photon echoes should be expected, in general. At low excitation intensities where the perturbation theory is applicable, however, the dominant nonlinear process is still the three-pulse stimulated echo and consequently each echo signal is generated only by three of the preceding pulses. It is rather a remarkable result that the echo intensity is unaffected by the presence of all the rest of the pulses other than its three generators. With this in mind, it is theoretically and experimentally shown that by applying a single optical pulse (#1) propagating along k1 followed by N pairs of counter-propagating pulses (#2j (k2j) and #3j− (k3j = −k2j); j=1,…N) in various directions, it is possible to obtain N photon echoes along the direction -k1 opposite to the first pulse #1. We will call this degenerate photon echoes, degenerate in. the sense that the echoes are spatially overlapping and temporally simultaneous. When N=l, this scheme is reduced to the ordinary backward photon echo2,3. The doubly degenerate case (N=2) is illustrated in Fig. 1. The j-th echo will be generated by the set of pulses (#1, #2j, #3j) and appear at t = t3j + t2j − t1 where t1, i2j and t3j are the arrival times of the pulses #1, #2j and #3j, respectively.

Jul 18, 19880 citations

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

Multipulse photon echoes in LaF3 : Pr3+

Lynden E. Erickson

Multipulse photon echoes are observed from ${\mathrm{LaF}}_{3}$ : ${\mathrm{Pr}}^{3+}$ and are shown to be primarily stimulated echoes. The various contributions to the echoes were measured using a three-pulse photon-echo sequence that resolved the image of the echo generated by the first two pulses, and the echo generated by the second and third pulses and that of the first and third pulses from the stimulated echo. By use of a theoretical model of Shoemaker [in Laser and Coherence Spectroscopy, edited by J. I. Steinfeld (Plenum, New York, 1978)] and relaxation theory of Hu and Hartmann [Phys. Rev. B 9, 1 (1974)], the image echo contributions to the third echo is shown to be less than 10% of the total for the pulse widths and Rabi frequencies used in the measurements. As a result, multipulse photon echoes will find limited use in measuring coherence decay until very short high-intensity pulses are used to excite the echoes.Multipulse photon echoes are observed from ${\mathrm{LaF}}_{3}$:${\mathrm{Pr}}^{3+}$ and are shown to be primarily stimulated echoes. The various contributions to the echoes were measured using a three-pulse photon-echo sequence that resolved the image of the echo generated by the first two pulses, and the echo generated by the second and third pulses and that of the first and third pulses from the stimulated echo. By use of a theoretical model of Shoemaker [in Laser and Coherence Spectroscopy, edited by J. I. Steinfeld (Plenum, New York, 1978)] and relaxation theory of Hu and Hartmann [Phys. Rev. B 9, 1 (1974)], the image echo contributions to the third echo is shown to be less than 10% of the total for the pulse widths and Rabi frequencies used in the measurements. As a result, multipulse photon echoes will find limited use in measuring coherence decay until very short high-intensity pulses are used to excite the echoes.

Physical Review ANov 1, 19924 citations

Echoes in gaseous media: A generalized theory of rephasing phenomena

THOMAS W. MOSSBERG, Ravinder Kachru, Sven R. Hartmann +1 more

The primary objective of this paper is to provide a formalism to predict and characterize the echoes which may be produced in a gaseous sample by an arbitrary sequence of one- or two-frequency laser excitation pulses which successively transfer atomic population between two or more atomic energy levels. The resulting echoes can occur at times which depend on the relative frequencies of the excitation pulses as well as on the times at which the pulses were applied. Using an idealized model of the excitation pulses and the atoms, the authors derive a simple transformation equation which provides the density matrix of a single atom after a general excitation sequence. Equations are derived which utilize the single-atom density matrix to predict the properties of the echoes (if any) which result. The means by which the sample of atoms "remembers" the information necessary to produce different types of echoes is discussed, and it is then shown how a study of the echo can provide information concerning the atomic relaxation processes which tend to destroy this information. It is observed that trilevel echoes can be used to determine the relaxation characteristics of superposition states between energy levels coupled by two-photon transitions. In the case of certain other echoes (such as the three-excitation-pulse stimulated echo) the authors discuss the heretofore unappreciated fact that the sample remembers the "echo information" via a nonthermal velocity distribution of the atoms in one specific atomic state. In these cases a study of the echo behavior can provide information pertaining to the relaxation processes affecting only the one atomic state. The formalism developed is used to predict the properties of several echo effects. First, to provide a connection with previous work, the well-known photon echo is briefly discussed. Then important new properties of the three-excitation-pulse-stimulated echo are described. Finally, three different types of trilevel echo are analyzed. Each of the trilevel echoes discussed has widespread applicability in the study of relaxation processes.The primary objective of this paper is to provide a formalism to predict and characterize the echoes which may be produced in a gaseous sample by an arbitrary sequence of one- or two-frequency laser excitation pulses which successively transfer atomic population between two or more atomic energy levels. The resulting echoes can occur at times which depend on the relative frequencies of the excitation pulses as well as on the times at which the pulses were applied. Using an idealized model of the excitation pulses and the atoms, the authors derive a simple transformation equation which provides the density matrix of a single atom after a general excitation sequence. Equations are derived which utilize the single-atom density matrix to predict the properties of the echoes (if any) which result. The means by which the sample of atoms "remembers" the information necessary to produce different types of echoes is discussed, and it is then shown how a study of the echo can provide information concerning the atomic relaxation processes which tend to destroy this information. It is observed that trilevel echoes can be used to determine the relaxation characteristics of superposition states between energy levels coupled by two-photon transitions. In the case of certain other echoes (such as the three-excitation-pulse stimulated echo) the authors discuss the heretofore unappreciated fact that the sample remembers the "echo information" via a nonthermal velocity distribution of the atoms in one specific atomic state. In these cases a study of the echo behavior can provide information pertaining to the relaxation processes affecting only the one atomic state. The formalism developed is used to predict the properties of several echo effects. First, to provide a connection with previous work, the well-known photon echo is briefly discussed. Then important new properties of the three-excitation-pulse-stimulated echo are described. Finally, three different types of trilevel echo are analyzed. Each of the trilevel echoes discussed has widespread applicability in the study of relaxation processes.

Physical Review ANov 1, 1979164 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

Storage and phase conjugation of light pulses using stimulated photon echoes

N. W. Carlson, William Randall Babbitt, THOMAS W. MOSSBERG

Optics LettersDec 1, 198344 citations

A gaseous atomic or molecular system with four energy levels

{E}_{a}<{E}_{b}<{E}_{c}<{E}_{d}

which is stimulated by resonant laser pulses at frequencies

{\ensuremath{\omega}}_{1}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{ba}}

{\ensuremath{\omega}}_{2}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{cb}}

{\ensuremath{\omega}}_{3}\ensuremath{\simeq}{\ensuremath{\omega}}_{\mathrm{dc}}

is considered. The six photon echoes which are stimulated by first and second pulses at frequencies

{\ensuremath{\omega}}_{1}

{\ensuremath{\omega}}_{2}

{\ensuremath{\omega}}_{3}

are discussed. For wave fronts which are described by

{e}^{i(\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t)}

the echo phase is described by

{\ensuremath{\phi}}_{E}=\ensuremath{\Sigma}(2{\ensuremath{\phi}}_{m}^{\mathrm{II}}\ensuremath{-}{\ensuremath{\phi}}_{m}^{\mathrm{I}})

at the sum frequency

{\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}+{\ensuremath{\omega}}_{3}

with

m=1,2,3

, at

{\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}

with

m=1,2

, at

{\ensuremath{\omega}}_{1}

with

m=1

, etc. For waves in the

z

direction where

\ensuremath{\phi}=\ensuremath{\kappa}(x,y,z)+kz

the gradient of the phase yields the wave-vector direction

{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{E}=\ensuremath{\Sigma}(2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{m}^{\mathrm{II}}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{m}^{\mathrm{I}})

in a physically thin sample. A probe pulse at

{\ensuremath{\omega}}_{2}

and

{\ensuremath{\omega}}_{3}

after the second pulse can generate a probe-pulse echo at

{\ensuremath{\omega}}_{1}

with phase

{\ensuremath{\phi}}_{1}=\ensuremath{\Sigma}(2{\ensuremath{\phi}}_{m}^{\mathrm{II}}\ensuremath{-}{\ensuremath{\phi}}_{m}^{\mathrm{I}})\ensuremath{-}{\ensuremath{\phi}}_{2}^{S}\ensuremath{-}{\ensuremath{\phi}}_{3}^{S}

, etc., for other frequency arrangements. Transverse phases

{\ensuremath{\kappa}}_{m}(\mathrm{xy})

can be added and subtracted, but no application is suggested for this novel feature. When all

{\ensuremath{\kappa}}_{m}

except one are zero, phase conjugation or phase-adaptive optics can occur.

{\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.

Multipulse photon echoes are observed from

{\mathrm{LaF}}_{3}

{\mathrm{Pr}}^{3+}

and are shown to be primarily stimulated echoes. The various contributions to the echoes were measured using a three-pulse photon-echo sequence that resolved the image of the echo generated by the first two pulses, and the echo generated by the second and third pulses and that of the first and third pulses from the stimulated echo. By use of a theoretical model of Shoemaker [in Laser and Coherence Spectroscopy, edited by J. I. Steinfeld (Plenum, New York, 1978)] and relaxation theory of Hu and Hartmann [Phys. Rev. B 9, 1 (1974)], the image echo contributions to the third echo is shown to be less than 10% of the total for the pulse widths and Rabi frequencies used in the measurements. As a result, multipulse photon echoes will find limited use in measuring coherence decay until very short high-intensity pulses are used to excite the echoes.

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.