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Physical Review A

Stimulated Electric Polarization and Photon Echoes

C. V. Heer, Ronald H. Kohl

Published Mar 1, 1970• 11 citations
Publisher

Abstract

The change in the wave function of an atom or molecule which interacts with saturating incident radiation is shown to have a time evolution operator $U(t, 0)=\ensuremath{\Sigma}|n)({n}^{\ensuremath{'}}|B(n\ensuremath{\alpha}){B}^{*}({n}^{\ensuremath{'}}\ensuremath{\alpha})\ifmmode\times\else\texttimes\fi{}{e}^{\ensuremath{-}i[\ensuremath{\xi}(n\ensuremath{\alpha})+\frac{E(n)}{\ensuremath{\hbar}}]t}$, where the sum over $n$ and ${n}^{\ensuremath{'}}$ is over both the ($2{J}_{a}+1$) values of ${m}_{a}$ and ($2{J}_{b}+1$) values of ${m}_{b}$, and the sum over $\ensuremath{\alpha}$ is over the $(2{J}_{a}+1)+(2{J}_{b}+1)$ modes indicated by the index $\ensuremath{\alpha}$. The eigenvalues $\ensuremath{\xi}(n\ensuremath{\alpha})$ and their eigenvectors $B(n\ensuremath{\alpha})$ depend on the intensity and polarization of the incident radiation, produce a modulation term in the electric polarization $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}$ of the molecule, and give rise to anomalous polarization in the stimulated radiation. This unitary operator is used to discuss the radiation stimulated by two pulses or photon echoes. Echoes from elliptic pulses are discussed for $J<~2$, and the linear-linear sequence is compared with the theory of Gordon, Wang, Patel, Slusher, and Tomlinson. Echoes from linear-circular and circular-linear pulse sequences are discussed in detail.

DOI
10.1103/physreva.1.693

Related papers

Polarization of photon echoes from S F6 molecules

C. V. Heer, Robert J. Nordstrom

We measured the polarization of the photon echo which was stimulated in a gas of S${\mathrm{F}}_{6}$ molecules by either linear-linear, circular-linear, or linear-circular pulse sequences from the $P(14)$, $P(16)$, $P(18)$, $P(20)$, $P(22)$, and $P(24)$ C${\mathrm{O}}_{2}$ laser lines near 10.4 \ensuremath{\mu}m. For $P(16)$ a linear-linear pulse sequence with angle $\ensuremath{\beta}$ between the linearly polarized pulses produced a linearly polarized echo at angle $\ensuremath{\phi}$ with the second pulse, and $\ensuremath{\phi}$ rotated in the same sense as $\ensuremath{\beta}$; a left-circular-linear sequence produced an echo with right elliptic polarization, and a linear-left-circular sequence produced an echo with left-circular polarization. For all other lines a linear-linear sequence produced an echo polarized along the second pulse, and results were similar to the measurement of Gordon et al. on $P(20)$. For $P(14)$, $P(18)$, and $P(20)$ a circular-linear sequence produced an echo linearly polarized along the second pulse. For $P(14)$ through $P(20)$ a linear-left-circular sequence produced a left-circular echo. A theory is developed for these pulse sequences for arbitrary values of angular momentum ${J}_{a}$, and a simplification found for large ${J}_{a}$. For laser radiation in direction $Y$ and the linear polarization of the second pulse along $Z$, the polarization vector for a linear-linear sequence is approximately $\ensuremath{\mp}(\frac{1}{2})\stackrel{^}{X}sin\ensuremath{\beta}+\stackrel{^}{Z}cos\ensuremath{\beta}$, and for a left-circular-linear sequence $\ifmmode\pm\else\textpm\fi{}(\frac{i}{2})\stackrel{^}{X}+\stackrel{^}{Z}$, where the upper sign refers to ${J}_{a}={J}_{b}\ifmmode\pm\else\textpm\fi{}1$ or the $P$ and $R$ branches and the lower sign to ${J}_{a}={J}_{b}$ or $Q$ branch. For a linear-left-circular sequence the echo is left circular for ${J}_{a}={J}_{b}$, ${J}_{b}\ifmmode\pm\else\textpm\fi{}1$ or the $P$, $Q$, and $R$ branches. The general features of the linear-linear sequence are similar to those suggested by Gordon et al. when a sign ambiguity is considered. Only $P(16)$ was in good agreement with theory. The $\ensuremath{-}(\frac{1}{2})\stackrel{^}{X}sin\ensuremath{\beta}$ or the $+(\frac{i}{2})\stackrel{^}{X}$ terms were not observed for the other lines.

Physical Review A•Feb 1, 1975•20 citations

Addendum to "Stimulated Electric Polarization and Photon Echoes"

Ronald H. Kohl, C. V. Heer

The theory presented earlier demonstrates in a very simple fashion that regardless of the degree of saturation, the degree of degeneracy, or the displacement from line center, two circularly polarized pulses of opposite polarization will not produce a photon echo; and circular and linear polarization form very special canonical states of polarization.

Physical Review A•Aug 1, 1970•2 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.

Physical Review A•Aug 1, 1977•13 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.

Physical Review•Mar 10, 1969•150 citations

Photon echo generated in ytterbium vapor by linear and circular polarizations of resonant radiation pulses

Хворостов Евгений Борисович, Гольдорт Вениамин Гершевич, В. Н. Ищенко +4 more

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Laser Physics Letters•Jan 31, 2011•7 citations

Phase of radiation from stimulated emission

Masahiko HIROOKA

With the help of the quantum theory of radiation, it is shown that the stimulated radiation from an atom has the same direction and phase dependence as the incident stimulating radiation.

International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz)•Dec 7, 2017•0 citations
The change in the wave function of an atom or molecule which interacts with saturating incident radiation is shown to have a time evolution operator U(t,0)=\ensuremathΣ∣n)(n\ensuremath′∣B(n\ensuremathα)B∗(n\ensuremath′\ensuremathα)\ifmmode×\else\texttimes\fie\ensuremath−i[\ensuremathξ(n\ensuremathα)+E(n)\ensuremathℏ]tU(t, 0)=\ensuremath{\Sigma}|n)({n}^{\ensuremath{'}}|B(n\ensuremath{\alpha}){B}^{*}({n}^{\ensuremath{'}}\ensuremath{\alpha})\ifmmode\times\else\texttimes\fi{}{e}^{\ensuremath{-}i[\ensuremath{\xi}(n\ensuremath{\alpha})+\frac{E(n)}{\ensuremath{\hbar}}]t}U(t,0)=\ensuremathΣ∣n)(n\ensuremath′∣B(n\ensuremathα)B∗(n\ensuremath′\ensuremathα)\ifmmode×\else\texttimes\fie\ensuremath−i[\ensuremathξ(n\ensuremathα)+\ensuremathℏE(n)​]t, where the sum over nnn and n\ensuremath′{n}^{\ensuremath{'}}n\ensuremath′ is over both the (2Ja+12{J}_{a}+12Ja​+1) values of ma{m}_{a}ma​ and (2Jb+12{J}_{b}+12Jb​+1) values of mb{m}_{b}mb​, and the sum over \ensuremathα\ensuremath{\alpha}\ensuremathα is over the (2Ja+1)+(2Jb+1)(2{J}_{a}+1)+(2{J}_{b}+1)(2Ja​+1)+(2Jb​+1) modes indicated by the index \ensuremathα\ensuremath{\alpha}\ensuremathα. The eigenvalues \ensuremathξ(n\ensuremathα)\ensuremath{\xi}(n\ensuremath{\alpha})\ensuremathξ(n\ensuremathα) and their eigenvectors B(n\ensuremathα)B(n\ensuremath{\alpha})B(n\ensuremathα) depend on the intensity and polarization of the incident radiation, produce a modulation term in the electric polarization P\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}P\ensuremath→​ of the molecule, and give rise to anomalous polarization in the stimulated radiation. This unitary operator is used to discuss the radiation stimulated by two pulses or photon echoes. Echoes from elliptic pulses are discussed for J< 2J<~2J< 2, and the linear-linear sequence is compared with the theory of Gordon, Wang, Patel, Slusher, and Tomlinson. Echoes from linear-circular and circular-linear pulse sequences are discussed in detail.
We measured the polarization of the photon echo which was stimulated in a gas of SF6{\mathrm{F}}_{6}F6​ molecules by either linear-linear, circular-linear, or linear-circular pulse sequences from the P(14)P(14)P(14), P(16)P(16)P(16), P(18)P(18)P(18), P(20)P(20)P(20), P(22)P(22)P(22), and P(24)P(24)P(24) CO2{\mathrm{O}}_{2}O2​ laser lines near 10.4 \ensuremath{\mu}m. For P(16)P(16)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)P(20)P(20). For P(14)P(14)P(14), P(18)P(18)P(18), and P(20)P(20)P(20) a circular-linear sequence produced an echo linearly polarized along the second pulse. For P(14)P(14)P(14) through P(20)P(20)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 Ja{J}_{a}Ja​, and a simplification found for large Ja{J}_{a}Ja​. For laser radiation in direction YYY and the linear polarization of the second pulse along ZZZ, 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 PPP and RRR branches and the lower sign to Ja=Jb{J}_{a}={J}_{b}Ja​=Jb​ or QQQ branch. For a linear-left-circular sequence the echo is left circular for Ja=Jb{J}_{a}={J}_{b}Ja​=Jb​, ${J}_{b}\ifmmode\pm\else\textpm\fi{}1$ or the PPP, QQQ, and RRR 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)P(16)P(16) was in good agreement with theory. The $\ensuremath{-}(\frac{1}{2})\stackrel{^}{X}sin\ensuremath{\beta}$ or the $+(\frac{i}{2})\stackrel{^}{X}$ terms were not observed for the other lines.
The polarization of photon echoes stimulated by linear-linear, circular-linear, and linear-circular excitation sequences has been studied in SF6{\mathrm{F}}_{6}F6​ with the P(12)P(12)P(12) through P(26)P(26)P(26) laser lines of CO2{\mathrm{O}}_{2}O2​ near 10.6 \ensuremath{\mu}m and in SiF4{\mathrm{F}}_{4}F4​ with the P(28)P(28)P(28) through P(34)P(34)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)P(16)P(16) in SF6{\mathrm{F}}_{6}F6​ and in the opposite sense as $\ensuremath{\beta}$ with the other lines in SF6{\mathrm{F}}_{6}F6​, and with the lines in SiF4{\mathrm{F}}_{4}F4​. A left-circular-linear sequence produces a right-elliptic echo with P(16)P(16)P(16) in SF6{\mathrm{F}}_{6}F6​, 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-JJJ 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 PPP- or RRR-branch transition, and the plus sign to a QQQ-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 PPP- or RRR-branch transition, and the negative sign to a QQQ-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 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 SF6{\mathrm{F}}_{6}F6​ excited by pulses of CO2{\mathrm{O}}_{2}O2​ laser radiation near 10.6 \ensuremath{\mu}. The analysis demonstrates that for the $|{\ensuremath{\Delta}}_{j}|=1$ transitions (PPP and RRR 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 (QQQ 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.