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Science China Mathematics

MOLECULAR 2-PHOTON LINE SHAPES AND ELIMINATION OF THEIR BACKGROUNDS

Ia H

Published Jan 1, 1985• 0 citations
OpenAlex

Abstract

Several paths to populate the final state of the 2-photon transition have been analyzed in this paper. It is pointed out that the contributions by the two packets of molecules with particular velocities to the Lorentzian peak will gradually dominate over the line shape as the offset of the intermediate state reduces. With more intense laser beams, the population of the two packets of molecules in the final state tends to be saturated, which causes the decrease of the peak ratio. By using the equations of motion for the density matrix, it has been proved theoretically that the actual population of the intermediate state results in the deformed line shape. We tried two methods to eliminate the Doppler-broadening backgrounds of the 2-photon transitions and enhanced the spectral resolution. The results discussed in this paper might promise their applications to the study of molecular high-lying states.

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SATURATION ASPECTS OF TWO-PHOTON SPECTROSCOPY OF $Na_{2}$ MOLECULES

J. P. Woerdman, Martin F. H. Schuurmans

We report on the saturation behavior of the $Na_{2}$ two-photon transition at 602.17 $^nm^{1,2}$ This transition has a dominant near-resonant intermediate level; the off-resonance $\\Delta$ is of the order of the Doppler width $\\Delta \\nu_{D}$. The dominant contribution to the Doppler-free two-photon line then cornea from two velocity packets of molecules which are Doppler-shifted into resonance. This results in some novel experimentally confirmed aspect, of the two-photon absorption: (i) the ratio of the amplitudes of the Doppler-free and Doppler-broadened two-photon lines is much larger than in the case $\\Delta \\gg \\Delta \\nu_{D}$, and (ii) the Doppler-free line saturates much stronger than the Doppler-broadened line. The resonant velocity packets of molecules are dominantly saturated on the ground-to-intermediate-level transition. The saturation power density is $\\sim 0.5 W/cm^{2}$; this value is consistent with a recent measurement of the Franck-Condon factors $involved^{3}$

•Jan 1, 1978•0 citations

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Theory of Optical-Microwave Double Resonance I. Fundamentals and Double Resonance of Microwave Detection

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Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state

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The line shape and strength of two-photon absorption in an atomic vapor are studied both theoretically and experimentally for the case of absorption of unequal-frequency photons. Special attention is given to the situation in which the intermediate state is resonant or nearly resonant. It is shown that for the resonant case, maximum absorption rates and Doppler-free spectra are simultaneously obtained. Either counterpropagating or copropagating beams can be used. With counterpropagating beams the two-photon linewidths are quite insensitive to power broadening of the intermediate state. With copropagating beams the two-photon spectra can also reveal the intermediate-state structure with a resolution determined mainly by the intermediate-state lifetime (nearly as well as can be done using any spectroscopic technique). The theoretical analysis provides a closed-form solution for the line shape and strength of the two-photon spectrum. In general the line shape is composed of two features. One is Doppler broadened and predominates when the intermediate state is nonresonant. The second feature is Doppler free, and it predominates with a resonant intermediate state. Detailed comparisons of experimental measurements of the line shape and strength of two-photon absorption in Na vapor are made with the theory and found to be in excellent agreement. Signal strengths for the onresonance case are 6 \ifmmode\times\else\texttimes\fi{} ${10}^{9}$ larger than the case of equal-frequency photons with no loss in resolution.

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Recoil effects on Doppler-free lineshapes in non-linear spectroscopy

J.H. Shirley

Previously derived equations describing recoil effects in saturated absorption by a gas of two-level atoms or molecules are solved numerically for the case of standing plane-wave excitation and large Doppler broadening. The numerical techniques and accuracy estimates are discussed briefly. The lineshapes obtained are well approximated by the superposition of two Lorentzian functions whose width, heights, and separation parametrise the lineshape. The dependence of these parameters on saturation, recoil splitting, and relaxation rates is described and compared with corresponding results in the rate equation approximation and in the weak-probe-beam approximation. Saturation broadening depends strongly on these latter approximations, but very little on recoil splitting. When multiphoton processes are retained in the equations, increasing saturation causes a slight decrease in the fitted separation of the peaks. The ratio of the heights of the two Lorentzians is closely related to the ratio of relaxation rates for the level populations.

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Two-photon spectroscopy. II. Effects of residual Doppler broadening

Rainer Salomaa, Stig Stenholm

For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1976•41 citations
Several paths to populate the final state of the 2-photon transition have been analyzed in this paper. It is pointed out that the contributions by the two packets of molecules with particular velocities to the Lorentzian peak will gradually dominate over the line shape as the offset of the intermediate state reduces. With more intense laser beams, the population of the two packets of molecules in the final state tends to be saturated, which causes the decrease of the peak ratio. By using the equations of motion for the density matrix, it has been proved theoretically that the actual population of the intermediate state results in the deformed line shape. We tried two methods to eliminate the Doppler-broadening backgrounds of the 2-photon transitions and enhanced the spectral resolution. The results discussed in this paper might promise their applications to the study of molecular high-lying states.
SATURATION ASPECTS OF TWO-PHOTON SPECTROSCOPY OF Na2Na_{2}Na2​ MOLECULES
We report on the saturation behavior of the Na2Na_{2}Na2​ two-photon transition at 602.17 nm1,2^nm^{1,2}nm1,2 This transition has a dominant near-resonant intermediate level; the off-resonance Delta\\DeltaDelta is of the order of the Doppler width DeltanuD\\Delta \\nu_{D}DeltanuD​. The dominant contribution to the Doppler-free two-photon line then cornea from two velocity packets of molecules which are Doppler-shifted into resonance. This results in some novel experimentally confirmed aspect, of the two-photon absorption: (i) the ratio of the amplitudes of the Doppler-free and Doppler-broadened two-photon lines is much larger than in the case DeltaggDeltanuD\\Delta \\gg \\Delta \\nu_{D}DeltaggDeltanuD​, and (ii) the Doppler-free line saturates much stronger than the Doppler-broadened line. The resonant velocity packets of molecules are dominantly saturated on the ground-to-intermediate-level transition. The saturation power density is sim0.5W/cm2\\sim 0.5 W/cm^{2}sim0.5W/cm2; this value is consistent with a recent measurement of the Franck-Condon factors involved3involved^{3}involved3
The effects of different laser line shapes on the intensity and spectrum of resonance fluorescence from a two-level atom are investigated. The laser model considered is that of an ideal single-mode laser operating high above threshold, with constant amplitude and undergoing phase-frequency fluctuations analogous to Brownian motion. As the correlation time of the frequency fluctuations increases from zero to infinity, the laser line shape changes from Lorentzian to Gaussian in a continuous way. It is shown that for intermediate and strong fields, the average intensity of fluorescence in the case of a resonant broadband Lorentzian line shape is higher than that in the case of a Gaussian line shape with the same bandwidth and total power. This is in contrast to the weak-field case where the higher peak power of the Gaussian line shape makes it more effective than the Lorentzian line shape. It is also shown that in the case of a nonzero frequency correlation time (non-Lorentzian line shape) the intensity of fluorescence undergoes non-Markovian fluctuations. In relation to the spectrum of resonance fluorescence, it is shown that as the line shape is varied from Lorentzian to Gaussian the following changes take place: In the case of off-resonance excitation, the asymmetry of the spectrum decreases. In the case of resonant excitation, the center-peak to side-peak height ratio for the triplet structure increases. Moreover, the recently predicted center-line dip, which develops in the spectrum in the case of broadband excitation when the Rabi frequency and the bandwidth are nearly equal, becomes increasingly deeper.
The effect of optical pumping on the lineshapes and the peak values of microwave absorption line is calculated by a density matrix method in the Doppler limit approximation. The calculated double resonance signal consists of two terms. One is proportional to the unperturbed population difference of the optical transition and has a Lorentzian lineshape at weak microwave saturation. In this term the two-photon processes cause broadening of the line. The other is proportional to the population difference of the microwave transition and gives the higher-order dynamic Stark effect. At moderate optical saturation the former term is much greater than the latter. At strong optical saturation the latter term becomes noticeable and lineshape anomaly occurs. The peak values of the double resonance signals are also calculated and compared with an experimental result.
The line shape and strength of two-photon absorption in an atomic vapor are studied both theoretically and experimentally for the case of absorption of unequal-frequency photons. Special attention is given to the situation in which the intermediate state is resonant or nearly resonant. It is shown that for the resonant case, maximum absorption rates and Doppler-free spectra are simultaneously obtained. Either counterpropagating or copropagating beams can be used. With counterpropagating beams the two-photon linewidths are quite insensitive to power broadening of the intermediate state. With copropagating beams the two-photon spectra can also reveal the intermediate-state structure with a resolution determined mainly by the intermediate-state lifetime (nearly as well as can be done using any spectroscopic technique). The theoretical analysis provides a closed-form solution for the line shape and strength of the two-photon spectrum. In general the line shape is composed of two features. One is Doppler broadened and predominates when the intermediate state is nonresonant. The second feature is Doppler free, and it predominates with a resonant intermediate state. Detailed comparisons of experimental measurements of the line shape and strength of two-photon absorption in Na vapor are made with the theory and found to be in excellent agreement. Signal strengths for the onresonance case are 6 \ifmmode\times\else\texttimes\fi{} 109{10}^{9}109 larger than the case of equal-frequency photons with no loss in resolution.
Previously derived equations describing recoil effects in saturated absorption by a gas of two-level atoms or molecules are solved numerically for the case of standing plane-wave excitation and large Doppler broadening. The numerical techniques and accuracy estimates are discussed briefly. The lineshapes obtained are well approximated by the superposition of two Lorentzian functions whose width, heights, and separation parametrise the lineshape. The dependence of these parameters on saturation, recoil splitting, and relaxation rates is described and compared with corresponding results in the rate equation approximation and in the weak-probe-beam approximation. Saturation broadening depends strongly on these latter approximations, but very little on recoil splitting. When multiphoton processes are retained in the equations, increasing saturation causes a slight decrease in the fitted separation of the peaks. The ratio of the heights of the two Lorentzians is closely related to the ratio of relaxation rates for the level populations.
For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.