Phase-dependence of inversionless gain in a Doppler broadened open four-level system

The effect of the relative phase (?) between the probe and driving fields on the gain without inversion (GWI) in a Doppler broadened open quasi Λ-type four level atomic system with vacuum induced coherence (VIC) for both co- and counter-propagating probe and driving fields cases is studied. It is shown that: (1) GWI and the probe detuning region in which GWI exists are very sensitive to variation of the relative phase; when values of the other parameters keep unvarying, by adjusting value of ?, the largest GWI can be obtained. (2) The Doppler width (D) also has dramatically modulation role on the phase-dependent GWI. When value of D is smaller, the value of ? which corresponds to the largest value of GWI is about π, when value of D is large enough, it is about π/2. (3) GWI varies periodically with ? varying, the period is 2π, but the concrete variation rule is closely related to value of D. (4) In the co-propagating case we can obtain much larger GWI than that in the counter-propagating case.