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Journal of the Optical Society of America B

Recursion-relation analysis for optical pumping to polarize nuclei by a sequence of short laser pulses

Takashi Nakajima

Published Aug 5, 2011• 1 citation
Publisher

Abstract

We carry out the recursion-relation analysis for the optical pumping scheme to polarize nuclei by a sequence of short laser pulses and show that such analysis agrees very well with the numerical solution of density matrix equations. A small difference between the results obtained by the numerical solution of density matrix equations and recursion-relation analysis arises from the contribution of the very small populations remaining in excited states when the next pulse arrives, and the difference becomes smaller if the pulse interval is chosen to be much longer than the spontaneous lifetime of the excited states.

DOI
10.1364/josab.28.002123

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We carry out the recursion-relation analysis for the optical pumping scheme to polarize nuclei by a sequence of short laser pulses and show that such analysis agrees very well with the numerical solution of density matrix equations. A small difference between the results obtained by the numerical solution of density matrix equations and recursion-relation analysis arises from the contribution of the very small populations remaining in excited states when the next pulse arrives, and the difference becomes smaller if the pulse interval is chosen to be much longer than the spontaneous lifetime of the excited states.
The physics of short-pulse polarization spectroscopy (PS) and the diagnostic potential for quantitative measurements of species concentration are investigated by direct numerical integration (DNI) of the time-dependent density matrix equations for a multistate system. The effects of laser power, collision rates, and Doppler broadening on the short-pulse PS signal generation process are investigated by systematically varying these parameters in the numerical calculations. It is found that the use of a short-pulse laser (laser pulse width τL<characteristic collision time τC) significantly decreases the collision-rate dependence of the PS signal compared to the long-pulse laser case (τL>τC), even for a nonsaturating pump beam. For a saturating pump beam, the short-pulse PS signal is found to be nearly independent of collision rate. Increasing the collision rate by a factor of 100 (from 108 to 1010 s−1), the calculated PS signal strength decreases by only a factor of 2 for a 100-ps pump laser at high intensity. This insensitivity of the PS signal to the collision rate in the medium enhances greatly the potential for quantitative application of the technique for concentration measurements in reacting flows. The underlying physics of the short-pulse PS is explored by studying the effects of collision rate, Doppler broadening, and the pump laser intensity on the temporal profile of the Zeeman state populations and the coherences between the Zeeman states.
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A general, analytic theory is presented of the excitation of a two-level atom by a series of short optical laser pulses. The atom is assumed to be isolated (collisionless) with radiative decay from upper to lower state at rate γ. The laser pulses are assumed to have a duration tp such that γtp ≪ 1 (the transient regime), and the spacing between pulses tr is such that tr ≫ tp. In this regime the atomic density matrix is calculated as a function of time during each optical pulse. An analytic expression for the atomic state (Bloch vector) after n optical pulses is also derived. For large n the atomic state converges to a repeating (steady state) cycle of pump excitation followed by radiative decay. The excited-state population in the steady state can be large but only if the spacing between pulses tr is small so that γtr ≲ 0.5. These results indicate that lasers that produce trains of pulses, such as mode-locked conventional lasers and free-electron lasers, can be effective in optical pumping of atoms.
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The effects of multifrequency-mode laser radiation on polarization-spectroscopy signal generation are investigated by direct numerical integration of the time-dependent density-matrix equations. The numerical solution of the density-matrix equations allows us to incorporate a physically reasonable model for pulsed dye-laser radiation in our analysis of the laser–resonance interaction. The laser radiation is modeled as the sum of electric fields from a finite number of modes that are assumed to have random pulse-to-pulse phases and exponentially distributed amplitudes. Calculations are performed for a homogeneously broadened resonance (only collisional broadening) and for a resonance that is both collision and Doppler broadened. The effect of the multimode laser radiation on polarization-spectroscopy line shapes and saturation curves is investigated for different values of the laser bandwidth and mode spacing and resonance collision and Doppler widths. The saturation parameter for the resonance is strongly dependent on the ratio of the laser bandwidth to the resonance collision width when the laser bandwidth is greater than the collision width. The pulse-to-pulse fluctuations in polarization-spectroscopy signal levels are found to decrease substantially for saturating pump intensities. The inclusion of the multimode laser structure into our density-matrix equations represents a significant advance in modeling the nonlinear interaction of laser radiation with atomic or molecular resonances.
Excitation dynamics for pulsed optical excitation are described with the density-matrix equations and the rate equations for a two-level system. A critical comparison of the two descriptions is made with complete and consistent formalisms that are amenable to the modeling of applied laser-diagnostic techniques. General solutions, resulting from numerical integration of the differential equations describing the excitation process, are compared for collisional conditions that range from the completely coherent limit to the steady-state limit, for which the two formalisms are identical. This analysis demonstrates the failure of the rate equations to correctly describe the transient details of the excitation process outside the steady-state limit. However, reasonable estimates of the resultant population are obtained for nonsaturating (linear) excitation. This comparison provides the laser diagnostician with the means to evaluate the appropriate model for excitation through a simple picture of the breakdown of the rate-equation validity.