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Physical Review A

Diagrammatic representation of photon echoes and other laser-induced ordering processes in gases

THOMAS W. MOSSBERG, Sven R. Hartmann

Published Mar 1, 1981• 42 citations
Publisher

Abstract

A simple diagrammatic approach to the description of laser-induced macroscopic ordering processes in gases is developed. We derive a set of simple rules which can be used to construct diagrams---each of which represents the macroscopic properties of a gaseous sample after it has been excited by a particular sequence of short, resonant, laser-excitation pulses. The features of the diagrams which signify various forms of laser-induced macroscopic ordering, e.g., photon echoes, grating echoes, stimulated echoes, standing-wave echoes, trilevel echoes, nonthermal velocity distributions, etc., are discussed. Important differences between the actions of traveling-wave and standing-wave excitation pulses are pointed out. The diagrammatic technique presented here for gases can easily be generalized to describe laser-induced macroscopic ordering in solids as well.

DOI
10.1103/physreva.23.1271

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Echoes in gaseous media: A generalized theory of rephasing phenomena

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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.

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Optical properties of photon echoes stimulated by three frequencies

C. V. Heer, Richard L. Sutherland

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.

Physical Review A•May 1, 1979•11 citations

Standing-Wave—Induced Backward Photon Echoes in Gases

N. W. Carlson, Arjun G. Yodh, THOMAS W. MOSSBERG

It has been found that a standing-wave pulse may be employed to produce an easily detectable, backward-propagating, phase-conjugate "replica" of any traditional gas-phase photon-echo signal. Standing-wave---induced backward two-pulse photon echoes have been observed in lithium vapor while only apertures were used to block laser background. An interesting dependence of backward echo intensity on standing-wave orientation is observed and discussed.

Physical Review Letters•Jul 4, 1983•6 citations
A simple diagrammatic approach to the description of laser-induced macroscopic ordering processes in gases is developed. We derive a set of simple rules which can be used to construct diagrams---each of which represents the macroscopic properties of a gaseous sample after it has been excited by a particular sequence of short, resonant, laser-excitation pulses. The features of the diagrams which signify various forms of laser-induced macroscopic ordering, e.g., photon echoes, grating echoes, stimulated echoes, standing-wave echoes, trilevel echoes, nonthermal velocity distributions, etc., are discussed. Important differences between the actions of traveling-wave and standing-wave excitation pulses are pointed out. The diagrammatic technique presented here for gases can easily be generalized to describe laser-induced macroscopic ordering in solids as well.
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.
A theoretical study is made of the photon echo which appears in a rarefied gaseous medium as a result of successive action of traveling and standing wave pulses, polarized linearly at an angle ψ. This photon echo represents a superposition of two opposite traveling waves, one of which has the polarization of a standing wave pulse, the other has a linear polarization which depends on ψ and on the parameters of a resonant atomic transition. Similar polarization features are exhibited also by coherent radiation from a resonant medium subjected to steady-state fields of the traveling and standing waves separated by some distance in space. The possibility of obtaining a characteristic narrow resonance is discussed.
Photon echoes in gaseous media are explained with the use of a simple heuristic model in which the atoms behave like composite billiard balls. The laser providing the excitation pulses becomes an atom smasher which "splits" the atoms and then by judicious programming puts them back together again. The most general photon-echo reordering process is explained without recourse to formal analysis so that an intuitive feeling is obtained which works equally well for the ordinary two-pulse photon echo, the Raman echo, the trilevel echo, the grating echo, etc. A formal analysis of the billiard-ball echo model is presented in its support.
We predict and observe the periodic reformation of spatial gratings in Na vapor commencing $\frac{\ensuremath{\tau}}{2}$ after the application of the second of two resonant ($\ensuremath{\lambda}=5896$ \AA{}) standing-wave excitation pulses which are themselves separated by $\ensuremath{\tau}$. These gratings have spatial periodicity $\frac{\ensuremath{\lambda}}{2}$, and backscatter our laser. One set of alternate echoes relaxes because of velocity-changing collisions; phase-changing collisions play no role. The other set is amplitude modulated with laser detuning $\ensuremath{\Delta}$ as $cos\ensuremath{\Delta}\ensuremath{\tau}$.
A gaseous atomic or molecular system with four energy levels Ea<Eb<Ec<Ed{E}_{a}<{E}_{b}<{E}_{c}<{E}_{d}Ea​<Eb​<Ec​<Ed​ 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=1,2,3m=1,2,3m=1,2,3, at ${\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}$ with m=1,2m=1,2m=1,2, at ${\ensuremath{\omega}}_{1}$ with m=1m=1m=1, etc. For waves in the zzz 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.