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Physical Review A

Polarization of photon echoes from S F6 molecules

C. V. Heer, Robert J. Nordstrom

Published Feb 1, 1975• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.11.536

Related papers

Photon echo measurements in S F6 and Si F4

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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.

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Addendum to "Stimulated Electric Polarization and Photon Echoes"

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Physical Review A•Aug 1, 1970•2 citations

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Physical Review•Aug 15, 1948•2 citations

Photon Echoes in Gases

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

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.

Physical Review Letters•May 13, 1968•201 citations
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 \ensuremathβ\ensuremath{\beta}\ensuremathβ between the linearly polarized pulses produced a linearly polarized echo at angle \ensuremathϕ\ensuremath{\phi}\ensuremathϕ with the second pulse, and \ensuremathϕ\ensuremath{\phi}\ensuremathϕ rotated in the same sense as \ensuremathβ\ensuremath{\beta}\ensuremathβ; 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 Ja=Jb\ifmmode±\else\textpm\fi1{J}_{a}={J}_{b}\ifmmode\pm\else\textpm\fi{}1Ja​=Jb​\ifmmode±\else\textpm\fi1 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​, Jb\ifmmode±\else\textpm\fi1{J}_{b}\ifmmode\pm\else\textpm\fi{}1Jb​\ifmmode±\else\textpm\fi1 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.
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 nnn and ${n}^{\ensuremath{'}}$ 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 $\ensuremath{\alpha}$ is over the (2Ja+1)+(2Jb+1)(2{J}_{a}+1)+(2{J}_{b}+1)(2Ja​+1)+(2Jb​+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< 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.
The unitary transformation from linearly to elliptically polarized states of photons is determined in a scheme of the second quantization (\textsection{}1). It is given by ${\mathbf{e}}_{l}={\mathbf{e}}_{1}cos\ensuremath{\phi}+i{\mathbf{e}}_{2}sin\ensuremath{\phi}$, ${\mathbf{e}}_{r}=i{\mathbf{e}}_{1}sin\ensuremath{\phi}+{\mathbf{e}}_{2}cos\ensuremath{\phi}$ where e1{\mathbf{e}}_{1}e1​ and e2{\mathbf{e}}_{2}e2​ are orthogonal unit vectors corresponding to linearly polarized states of photons and el{\mathbf{e}}_{l}el​ and er{\mathbf{e}}_{r}er​ are those corresponding to left and right elliptically polarized states of photons. A general formula for a probability of an emission of light from an atom is deduced on the basis of the Dirac electron theory so as to include up to both a magnetic dipole and an electric quadrupole term (\textsection{}2). Its general form is improved as compared with the hitherto obtained one. The general theory is applied to electric dipole and quadrupole and magnetic dipole spectral lines (\textsection{}3 and \textsection{}4). Polarization states of the Zeeman components corresponding to all combinations of magnetic and orbital magnetic quantum numbers of an atom are completely determined for an arbitrary given direction of an observation on the basis of quantum mechanics.
Photon echoes following two pulses of CO2{\mathrm{O}}_{2}O2​ laser radiation incident on SF6{\mathrm{F}}_{6}F6​ gas have been observed. Homogeneous relaxation times for SF6{\mathrm{F}}_{6}F6​ colliding with SF6{\mathrm{F}}_{6}F6​, He, Ne, and H2{\mathrm{H}}_{2}H2​ are shown to be easily measurable from the decay of photon-echo amplitude.