Physical Review A

Addendum to "Stimulated Electric Polarization and Photon Echoes"

Ronald H. Kohl, C. V. Heer

Published Aug 1, 1970• 2 citations

Abstract

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.

DOI: 10.1103/physreva.2.549

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Photon echo measurements in S F6 and Si F4

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Physical Review A

Addendum to "Stimulated Electric Polarization and Photon Echoes"

Ronald H. Kohl, C. V. Heer

Published Aug 1, 1970• 2 citations

Publisher

Abstract

DOI: 10.1103/physreva.2.549

Related papers

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.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 AAug 1, 197713 citations

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Alexander V. Evseev, I. V. Evseev, Решетов Владимир Александрович

Optics and SpectroscopyMar 1, 19850 citations

Stimulated Electric Polarization and Photon Echoes

C. V. Heer, Ronald H. Kohl

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.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.

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Physical Review AOct 1, 19702 citations

Photon echo polarization taking into account the reaction of the medium on the exciting light pulses

Igor V. Yevseyev

Physics Letters AJun 1, 19731 citation

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

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.

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.

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.