Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Focusing of Photon Echoes

C. V. Heer

Published May 1, 1973• 12 citations
Publisher

Abstract

It is shown that the radius of curvature ${R}^{\ensuremath{'}\ensuremath{'}}$ of the photon echo is related to the radius of curvature $R$ of the first pulse and ${R}^{\ensuremath{'}}$ of the second pulse by $\frac{1}{{R}^{\ensuremath{'}\ensuremath{'}}}=\frac{2}{{R}^{\ensuremath{'}}}\ensuremath{-}\frac{1}{R}$ and yields a focusing effect for the photon echo.

DOI
10.1103/physreva.7.1635

Related papers

Focusing and phase conjugation of photon echoes in Na vapor

N. C. Griffen, C. V. Heer

The focusing and phase conjugation of photon echoes was observed in Na vapor. If the radius of curvature of the first pulse was R1 and the second pulse R2, the radius of curvature of the echo was given by 1/RE=2/R2−1/R1 and the echo center of curvature or focus followed from the phase φE(r) =2φ2(r)−φ1(r) in the Na cell.

Applied Physics Letters•Nov 15, 1978•23 citations

Focusing of Carr-Purcell photon echoes, and collisional effects

C. V. Heer

Since for echo formation the off-diagonal density matrix element ${\ensuremath{\sigma}}_{\mathrm{ab}}$ evolves from ${\ensuremath{\sigma}}_{\mathrm{ba}}={\ensuremath{\sigma}}_{\mathrm{ab}}^{*}$ of the previous pulse, the eikonal $\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t$ reverses sign with each pulse. If the radius of curvature of the first $\frac{\ensuremath{\pi}}{2}$ pulse is ${R}_{1}$, the radius of curvature of the subsequent $\ensuremath{\pi}$ pulses ${R}_{2}$, ${R}_{3}$, etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{{R}_{E1}}=\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{{R}_{E2}}=\frac{2}{{R}_{3}}\ensuremath{-}\frac{2}{{R}_{2}}+\frac{1}{{R}_{1}}$, and for subsequent Carr-Purcell echoes by $\frac{1}{{R}_{E3}}=\frac{2}{{R}_{4}}\ensuremath{-}\frac{2}{{R}_{3}}+\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, etc. The influence of motion, adiabatic collisions, and velocity-changing collisions on the decay of echo amplitude is considered for curved wave fronts. Since the change in position from the constant-velocity position is important in the decay, a statistical theory for small displacement and its relationship to the forward scattering cross section is given. These results with ${r}^{\ensuremath{-}3}$ and ${r}^{\ensuremath{-}6}$ interactions between colliding molecules are compared with the experimental data for ${\mathrm{CH}}_{3}$F and with an ${r}^{\ensuremath{-}6}$ potential with the data for ${\mathrm{SF}}_{6}$ and ${\mathrm{SiF}}_{4}$.

Physical Review A•May 1, 1976•14 citations

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

The Long-Lived Photon Echo Response Locking Effect in the Presence of External Non-Resonant Laser Pulses with a Different Spatial Orientation

L. A. Nefedev, É. I. Khakimzyanova, Guzel I. Garnaeva

The correlation of the inhomogeneous broadening of the resonance transition at different time intervals and the efficiency of long-lived photon echo response locking by the action of standing waves of non-resonant laser pulses are considered.It is shown that the long-lived photon echo response locking effect may be observed even at the angles of the relative orientation of the non-resonant standing wave laser pulses of less than one degree, due to the change in the correlation coefficient of inhomogeneous broadening at time intervals between the first and the second and after the third resonant laser pulses.

Optics and Photonics Journal•Jan 1, 2013•4 citations

Photon Echoes and Degeneration of the Resonant Energy Levels

Leonid A. Nefediev, В. В. Самарцев, A. I. Sirasiev

The influence of the degeneration of the resonant energy levels on the photon echo intensity is investigated.

Spectroscopy Letters•Jan 1, 1974•2 citations

Photon echo in superradiant media

Ashat M. Basharov, N. V. Znamenskiı̆

It was shown that if an additional emission of a superradiance pulse by a medium occurs after an initial exciting pulse, two-pulse photon echoes are formed at a different frequencies and at a new time as compared to an ordinary medium. Conditions for the spatial synchronism of these photon echo were found.

Bulletin of the Russian Academy of Sciences Physics•Jul 1, 2010•0 citations
It is shown that the radius of curvature R\ensuremath′\ensuremath′{R}^{\ensuremath{'}\ensuremath{'}}R\ensuremath′\ensuremath′ of the photon echo is related to the radius of curvature RRR of the first pulse and R\ensuremath′{R}^{\ensuremath{'}}R\ensuremath′ of the second pulse by 1R\ensuremath′\ensuremath′=2R\ensuremath′\ensuremath−1R\frac{1}{{R}^{\ensuremath{'}\ensuremath{'}}}=\frac{2}{{R}^{\ensuremath{'}}}\ensuremath{-}\frac{1}{R}R\ensuremath′\ensuremath′1​=R\ensuremath′2​\ensuremath−R1​ and yields a focusing effect for the photon echo.
Since for echo formation the off-diagonal density matrix element ${\ensuremath{\sigma}}_{\mathrm{ab}}$ evolves from ${\ensuremath{\sigma}}_{\mathrm{ba}}={\ensuremath{\sigma}}_{\mathrm{ab}}^{*}$ of the previous pulse, the eikonal $\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t$ reverses sign with each pulse. If the radius of curvature of the first $\frac{\ensuremath{\pi}}{2}$ pulse is R1{R}_{1}R1​, the radius of curvature of the subsequent $\ensuremath{\pi}$ pulses R2{R}_{2}R2​, R3{R}_{3}R3​, etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{{R}_{E1}}=\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{{R}_{E2}}=\frac{2}{{R}_{3}}\ensuremath{-}\frac{2}{{R}_{2}}+\frac{1}{{R}_{1}}$, and for subsequent Carr-Purcell echoes by $\frac{1}{{R}_{E3}}=\frac{2}{{R}_{4}}\ensuremath{-}\frac{2}{{R}_{3}}+\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, etc. The influence of motion, adiabatic collisions, and velocity-changing collisions on the decay of echo amplitude is considered for curved wave fronts. Since the change in position from the constant-velocity position is important in the decay, a statistical theory for small displacement and its relationship to the forward scattering cross section is given. These results with ${r}^{\ensuremath{-}3}$ and ${r}^{\ensuremath{-}6}$ interactions between colliding molecules are compared with the experimental data for CH3{\mathrm{CH}}_{3}CH3​F and with an ${r}^{\ensuremath{-}6}$ potential with the data for SF6{\mathrm{SF}}_{6}SF6​ and SiF4{\mathrm{SiF}}_{4}SiF4​.
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.
The correlation of the inhomogeneous broadening of the resonance transition at different time intervals and the efficiency of long-lived photon echo response locking by the action of standing waves of non-resonant laser pulses are considered.It is shown that the long-lived photon echo response locking effect may be observed even at the angles of the relative orientation of the non-resonant standing wave laser pulses of less than one degree, due to the change in the correlation coefficient of inhomogeneous broadening at time intervals between the first and the second and after the third resonant laser pulses.