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Journal of Applied Physics

Effects of truncated Gaussian-beam variations in laser saturation spectroscopy

Murray Sargent

Published Jan 1, 1977• 6 citations
Publisher

Abstract

The effects of possibly truncated Gaussian-beam variations in probe and saturator-wave intensities are studied analytically and numerically for saturation spectroscopy of homogeneously and inhomogeneously broadened systems. The measured saturation intensity is shown to be about 2.5 times greater for Gaussian beams than for uniform saturation. Systems characterized by the short dipole lifetime limit (T2≪T1) are treated, revealing sizeable modifications to the depths of coherent pulsation dips and Bennet holes. The equal lifetime case (T2=T1) is studied as well, showing substantial influence on the dynamic Stark effect.

DOI
10.1063/1.323365

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Annual Meeting Optical Society of America•Jan 1, 1989•0 citations
The effects of possibly truncated Gaussian-beam variations in probe and saturator-wave intensities are studied analytically and numerically for saturation spectroscopy of homogeneously and inhomogeneously broadened systems. The measured saturation intensity is shown to be about 2.5 times greater for Gaussian beams than for uniform saturation. Systems characterized by the short dipole lifetime limit (T2≪T1) are treated, revealing sizeable modifications to the depths of coherent pulsation dips and Bennet holes. The equal lifetime case (T2=T1) is studied as well, showing substantial influence on the dynamic Stark effect.
The effects of Gaussian-beam averaging with various apertures are derived for three-wave mixing in nonlinear two-level media. The model studies the phase transition from plane-wave to pure Gaussian saturation. Applications to phase conjugation, beat-frequency spectroscopy, and laser–optical-bistability instability phenomena are considered. The Gaussian beams are seen to reduce or wash out dynamic Stark splittings observed for plane-wave saturation. For strong saturation, the side-mode gain/absorption coefficient versus detuning yields slightly larger gain for the homogeneously broadened laser instability but substantially curtailed gain for the corresponding absorptive optical-bistability instability.
Reference 1 derives the two-photon probe-absorption coefficient for a homogeneously broadened medium subjected to an arbitrarily intense saturator wave. We have generalized this theory to allow for Doppler broadening with a counterrunning probe and saturator waves. We have carried out integrals over a Lorentzian Doppler distribution analytically and have specialized to a number of limits, such as extreme Doppler broadening. In this limit, we find the simple probe absorption coefficient α 1 ≃ 2 α 0 ( γ D 1 I 2 − i ω s T 1 ) , where α0 is the unsaturated two-photon absorption coefficient, I2 is the saturator intensity, γ is the two-photon coherence decay rate constant, ω s is the Stark shift parameter, T1 is the population difference decay time, and D 1 = 1 / [ γ+ i ( ω+ω s I 2 − ν 2 − ν 1 ) ] . For vanishing Stark shifts (ω s = 0), this is the standard two-photon probe absorption coefficient, but for large ω s , the complex Lorentzian D 1 becomes detuned. In the homogeneously broadened limit, the absorption spectrum resembles the one-photon case. In this paper we show the transition between these two limits.
This paper presents a calculation of the depth of the saturation dip observed when laser radiation of a Gaussian TEM00{\mathrm{TEM}}_{00}TEM00​ mode is absorbed in a thermal molecular beam or in a very-low-pressure gas. Transit-time broadening is assumed to be the only broadening mechanism, with neglect of collisions and other relaxations. There is no restriction to small saturation parameters. The absorption experienced by one molecule has been calculated via the optical Bloch equations. Averaging over all relevant parameters of the molecular trajectory and the laser field followed. To get a reference for the saturation, from which to measure the dip depth, simply twice the absorption from a running-wave field has been taken, which therefore also had to be calculated in addition to the calculation of the absorption in a standing wave. Preliminary experiments have been carried out for comparison, and the data agree very well with the calculated shape of the curve and the absolute values of the calculated dip depth. From the laser-power scaling factor, necessary to fit the experimental data to the universal theoretical curve, the dipole moment can be derived without having to know such hard-to-determine quantities as absorber density, partition function, collision relaxation rate, or even the diameter of the laser beam.
Only for optically thin layers and plane waves may the intensity of a light wave be unambiguously expressed in terms of the nonlinear susceptibility of the medium. In aperture limited light beams the saturation parameter and the Stark shifts are local variables, while the output intensity of the pump and probe beams is an integral characteristic of the transmission process. The output spectrum is affected by diffraction, nonlinear refraction, and saturated absorption. The nonlinear refraction is a cause of a spectral line asymmetry which is well- known in saturation spectroscopy. In addition to usual line broadening, the saturated absorption causes specific beam effects, related to the formation of a self-induced transparency channel. The diffraction of the beam due to this self-induced aperture may result in the enhancement of the on-axis intensity, which is not a result of the self-induced lens effects. All these factors are taken into account by the mathematical model suggested in the present work. The propagation of the near-resonance saturating pump and non-saturating probe beams of various initial profiles through a homogeneously broadened two-level gas is simulated numerically. The calculations carried out make it possible to analyze the contribution of the propagation phenomena, mentioned above, to the output spectra of the beams.
The influence of the high-pump-field intensity saturation and dynamic Stark effects on the resonant nearly degenerate four-wave-mixing signal (${\mathrm{\ensuremath{\omega}}}_{3}$=2${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$) is studied in a frequency space defined by the pump- and probe-field frequencies, ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, respectively. Variable inhomogeneous widths have been considered ranging from extremely narrow (homogeneous distribution) to extremely wide. Remarkable differences between nearly homogeneously broadened systems and extreme inhomogeneous ones are exhibited. It is shown, for the situation of homogeneous broadening, that the spectra carried on with fixed probe and fixed pump frequency values have substantial differences in shape. These differences disappear in the situation of extreme inhomogeneous broadening. Fixed-pump-frequency resonant and fixed-probe-frequency nonresonant spectra are proposed as very convenient tools to yield useful evidence about the width of the inhomogeneous molecular distribution.
Polarization and Rayleigh type optical mixing spectra of a homogeneously and inhomogeneously broadened two-level system are calculated. We show that the nonlinear signal has contributions from the population pulsation and saturation effect. In the homogeneous system these effects can induce an asymmetric spectrum when the probe frequency is different from the resonant value. In the inhomogeneous system the spectral response is always symmetric independent of the probe frequency value. Line saturation and dynamic stark splitting are substantially reduced in the inhomogeneous case. The mentioned differences can be used to identify the type of broadening of a given spectral line.