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Applied Optics

Linewidth in saturated absorption spectroscopy for two-level atoms: an empirical formula

Geol Moon, Heung‐Ryoul Noh

Published May 7, 2018• 1 citation
Publisher

Abstract

We present an empirical formula for linewidth in saturated absorption spectroscopy for two-level atoms with cycling transition lines, taking the coherence term in the Doppler-broadened limit into account. The full width at half-maximum is obtained as (1+(1+as0)b)γt, where s0 is the on-resonance saturation parameter, γt is the transverse decay rate, and a and b are the parameters that depend on γt. We find that as γt increases, a and b approach 1 and 1/2, respectively. These are the typical values in the case without the coherence term.

DOI
10.1364/ao.57.003881

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We present an empirical formula for linewidth in saturated absorption spectroscopy for two-level atoms with cycling transition lines, taking the coherence term in the Doppler-broadened limit into account. The full width at half-maximum is obtained as (1+(1+as0)b)γt, where s0 is the on-resonance saturation parameter, γt is the transverse decay rate, and a and b are the parameters that depend on γt. We find that as γt increases, a and b approach 1 and 1/2, respectively. These are the typical values in the case without the coherence term.
We present the analytic solutions of the saturated absorption spectrum for an atomic medium composed of two-level atoms and describe the effects of saturation and optical pumping on the spectrum. The background-subtracted Doppler-free spectrum is Lorentzian-shaped with an amplitude and width given by 1 − {1 + s 0 [1 + (γ 2 t /2)]} −1/2 and γ t [1 + {1 + s 0 [1 + (γ 2 t /2)]} 1/2 ], respectively, where γ t is the transverse decay rate, γ 2 is the decay rate to energy levels other than the ground state, s 0 is the saturation parameter and t is the interaction time of the atoms with laser beams. Analytic results are compared with numerical results and show good agreement with one another. This paper is intended for graduate students who have started to study basic atomic physics.
The line shapes of saturation spectroscopy in optically thick two-level media are calculated for atomic and molecular transitions. Line narrowing and distortion effects due to propagation are presented. For Doppler-broadened media it is shown that, due to multiphoton processes, the natural linewidth limits the ultimate narrowing that can be reached. Thus, up to ${\ensuremath{\alpha}}_{0}L=20,$ only when the rate-equation approximation is valid subhomogeneous linewidths are expected.
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.
We present an analytical study of susceptibility in pump–probe laser spectroscopy for an atomic medium composed of two-level atoms in the Doppler-broadened limit. We derive an accurate analytical formula for susceptibility up to the first (arbitrary) order at the Rabi frequency of the probe (pump) beam. This is expressed in a succinct single term rather than as a complicated summation, as presented in previous papers. We also derive analytical solutions for the linewidth and peak value of the absorption spectrum of the probe beam.
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.
We consider in this paper the Doppler-broadened absorption by two-level molecules of a gas from a weak traveling electromagnetic plane wave in the presence of an oppositely traveling strong plane wave at the same frequency. Here the absorption spectrum exhibits a dip that broadens and deepens as the gas is increasingly saturated by the strong wave. This saturation effect has been employed in the recent experiments on the fine structure of atomic hydrogen performed by H\"ansch, Shahin, and Schawlow (HSS). The form, linewidth, and strength of saturation dips in the fine-structure lines are studied for experimental parameters appropriate to the hydrogen experiment. In addition, saturation line-shape curves are calculated for experimental situations in which the strengths of the stronger field give rise to stronger or weaker saturation of the gas medium and varying degrees of Doppler broadening exist. The calculation of shape of the saturation dip and absorption curves for a Doppler-broadened gas of two-level atoms is made exactly; in effect, the strong field is treated as arbitrarily strong by avoiding a perturbation-theory treatment in favor of the exact solutions for a driven two-level system. The general behavior found is in accord with intuition and experimental observations by HSS. There is an optimum saturation power which gives a prominent dip and beyond which the dip suffers "power broadening."