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Physical Review A

Theory of two-photon Lamb dips

Fujio Shimizu

Published Sep 1, 1974• 26 citations
Publisher

Abstract

A semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R\"omheld, and Oka.

DOI
10.1103/physreva.10.950

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Theory of two-photon Lamb dips

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A semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R\omheld, and Oka.

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The line shape of a pair of Doppler-broadened closely spaced transitions sharing a common lower level and under the effect of a standing-wave monochromatic radiation field is studied. The eight steady-state optical Bloch equations are solved analytically to obtain the polarization by retaining terms up to the third power in the electric-field amplitude. The absorption coefficient derived for the standing-wave case consists of two Lamb dips for the two allowed transitions at ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$ and a crossover resonance dip at the intermediate frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. The Lamp-dip peak frequencies show intensity-dependent shifts. Numerical computations are presented for different values of Rabi frequencies and relaxation linewidth parameters, which are small compared to the frequency difference \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{2}$-${\mathrm{\ensuremath{\omega}}}_{1}$. The computed line shape and frequency shift are compared with the observed results in the Balmer ${\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}$ line of hydrogen and the infrared Zeeman spectra of methane.

Physical Review A•Apr 1, 1992•29 citations
A semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R"omheld, and Oka.
A semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R\omheld, and Oka.
A previously developed diagrammatic method for calculating the nonlinear response of a dilute gas at near resonance with strong classical radiation fields is extended to include velocity-dependent homogeneous and inhomogeneous broadening phenomena. The method is applied to two-level saturated-absorption (Lamb-dip) spectroscopy. Expressions are obtained for the collision-broadened Lamb dip probed by a weak field in the presence of an arbitrarily strong saturating field and are compared to the usual infinite-Doppler-width, weak-saturation expression. We study the effects of velocity and momentum transfer on the line-shape parameters and discuss the effect of phase-changing and strong inelastic collisions on the intensity.
A three-dimensional theory for the resonant interaction of electromagnetic waves with a gas of two-level atoms is formulated in terms of macroscopic variables. The theory is utilized to find the steady-state attenuation of a plane wave in the presence of another plane wave running in the opposite direction with different amplitude. Contributions are included from the reflection of the oppositely running wave by an induced standing-wave inhomogeneity in the population inversion of the medium. The resulting attenuation and reflection coefficients are expressed as velocity integrals of continued fractions. Correspondence is made with existing gas-laser theories, yielding the formulation of a high-intensity ring-laser theory. Analytic approximations for the coefficients are presented for the Doppler-limit cases of both waves weak, one wave weak, and negligible reflection (rate-equation approximation). More-general cases have been calculated numerically. The attenuation coefficients exhibit a Lamb-dip feature. The relative depth of the dip increases rapidly with power at low saturation levels, slowly at high saturation, and is greater in the attenuation of the weaker wave. The width of the dip is nonlinearly power broadened. The shape of the dip is very nearly Lorentzian, except for one special case at high power in which the line splits. The propagation equations for the two waves are integrated over long absorption paths. A large resulting attenuation increases the relative size of the dip while decreasing the power broadening.
The molecular response in laser intracavity experiments where molecules interact simultaneously with a radio frequency field and an infrared one is analyzed theoretically. In the quantized field approach the molecule–radio frequency interaction is treated exactly. For the laser intensity the density matrix equations are solved either in a third-order perturbation treatment or through a matrix continued fraction analysis. The resonance line shapes in different two-level and three-level configurations are analyzed numerically.
The line shape of a pair of Doppler-broadened closely spaced transitions sharing a common lower level and under the effect of a standing-wave monochromatic radiation field is studied. The eight steady-state optical Bloch equations are solved analytically to obtain the polarization by retaining terms up to the third power in the electric-field amplitude. The absorption coefficient derived for the standing-wave case consists of two Lamb dips for the two allowed transitions at ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$ and a crossover resonance dip at the intermediate frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. The Lamp-dip peak frequencies show intensity-dependent shifts. Numerical computations are presented for different values of Rabi frequencies and relaxation linewidth parameters, which are small compared to the frequency difference \ensuremath{\Delta}=${\mathrm{\ensuremath{\omega}}}_{2}$-${\mathrm{\ensuremath{\omega}}}_{1}$. The computed line shape and frequency shift are compared with the observed results in the Balmer ${\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}$ line of hydrogen and the infrared Zeeman spectra of methane.