Analytic evaluation of Doppler-averaged ladder-type double-resonance lineshape beyond the weak-pump–weak-probe limit

Derive a closed-form analytic expression for the velocity-averaged upper-state population in ladder-type optical–optical double resonance under the infinite-Doppler-width approximation, namely I(Δω23) = ∫_{−∞}^{∞} ρ33(Δω12 + k_a v_z, Δω23 ± k_b v_z) dv_z, where ρ33 is the steady-state population of level 3 for a three-level ladder system with equal relaxation rate γ. Determine this integral for co- and counter-propagating geometries using the full steady-state solution ρ33 (arbitrary pump and probe strengths) and, at least, using the weak-probe approximation ρ33^{wp}, extending the existing weak-pump–weak-probe result ρ33^{wpp}.

Background

The paper analyzes three-level ladder-type optical–optical double resonance (DR) with equal relaxation rates and considers Doppler broadening by integrating the homogeneous response over the thermal velocity distribution. In the infinite-Doppler-width approximation, the Doppler factor separates, leaving an integral over the steady-state solution ρ33 that governs the probe absorption lineshape.

While the authors obtain analytic Lorentzians in the weakest-field limit (both pump and probe weak, ρ33^{wpp}), they resort to numerical integration for stronger fields. They explicitly state that they could not carry out the analytic evaluation using the full solution ρ33 or even the weak-probe approximation ρ33^{wp}, which would yield more general closed-form lineshapes under strong pump or finite probe conditions.