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Journal of the Optical Society of America B

Phase shifts and intensity dependence in frequency-modulation spectroscopy

H.-R. Xia, J. I. Cirac, Stephen D. Swartz, Bern Kohler, Daniel S. Elliott, J. L. Hall, P. Zoller

Published May 1, 1994• 6 citations
Publisher

Abstract

We discuss experimental observations of an induced phase shift of the rf signal produced in frequency-modulated saturated-absorption measurements under the conditions of large probe-beam intensities. This phase shift can be understood in terms of the phase difference between the phase modulation of the optical driving field and the steady-state macroscopic polarization of the medium. We also present theoretically calculated spectra that are based on the optical Bloch equations and that are in excellent agreement with the experimental results.

DOI
10.1364/josab.11.000721

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We discuss experimental observations of an induced phase shift of the rf signal produced in frequency-modulated saturated-absorption measurements under the conditions of large probe-beam intensities. This phase shift can be understood in terms of the phase difference between the phase modulation of the optical driving field and the steady-state macroscopic polarization of the medium. We also present theoretically calculated spectra that are based on the optical Bloch equations and that are in excellent agreement with the experimental results.
We discuss experimental observations of an induced phase shift of the rf signal produced in FM saturated absorption measurements under the conditions of large probe beam intensities. This phase shift can be understood in terms of the phase difference between the phase modulation of the optical driving field and the steady state macroscopic polarization of the medium. This phase difference is relatively insensitive to detuning of the laser from the atomic resonance, such that essentially the same spectrum can be observed by changing the rf detection phase. We also present theoretically calculated spectra which are based on the optical Bloch equations, and which are in excellent agreement with the experimental results.
We conducted a theoretical and experimental study of lineshape in modulation transfer spectroscopy for ⁸⁷Rb atoms. When a linearly polarized pump beam, modulated at an angular frequency of Ω, overlaps in parallel with an unmodulated linearly polarized probe beam, combined modulated probe beams are generated via nonlinear interaction with atoms. The detected modulation transfer signals are calculated by numerically solving the complete optical Bloch equations for the ⁸⁷Rb atoms without the use of any phenomenological parameters. We find that the calculated results are in good agreement with experimental results.
Optical non-linearities and spectroscopy in fields that are strongly modulated and also intense are examined by obtaining the solution of the optical Bloch equations in modulated fields as a matrix continued fraction. Numerical results for emission and absorption in such fields are presented. The saturation optical susceptibilities and the different types of spectra show resonances at the Rabi frequency if the modulation index is small. With increase in this index, additional resonant features appear at submultiples of the Rabi frequency. These additional features become more and more prominent as the modulation index increases further. For modulated fields on resonance, a dressed-atom formulation is developed to explain the results obtained from the numerical evaluation of the continued fraction. The results are also applicable to non-linear spectroscopic measurements such as Raman gain when the probe field is modulated.
The dynamical behavior of a two-level system in an intense frequency-modulated field is obtained without recourse to the weak-modulation assumption. Solutions to the optical Bloch equations are expressed in terms of continued fractions involving 2\ifmmode\times\else\texttimes\fi{}2 matrices. Such matrix-continued fractions are numerically evaluated to obtain a wide variety of absorption and fluorescence spectra in frequency-modulated fields. Results in various special cases involving combinations of weak fields, weak modulation, strong modulation, and strong fields are given. Analytical results for weak fields but with an arbitrary degree of modulation are also presented. The spectra exhibit additional resonances for strong modulation. The main resonance shifts and broadens considerably due to strong modulation and saturation effects.
A spatial dependence of the demodulated signal over the cross-section of a laser beam used for saturation spectroscopy has been observed. If the detector is not exposed to the full beam this spatial dependence causes a distortion of the line-shape. In extreme cases this distortion can lead to frequency shifts as large as 15% of the full width of the resonance. A model is presented in which the self-focusing effect acts to separate spatially the different frequency components of the modulated laser beam and thus allows for a spatial inhomogeneity of the demodulated signal. Numerical evaluation of the model agrees well with the experimental data provided a strong enhancement of the beam convergence in the absorption medium is applied.