Comparison between effects of Doppler broadening on pure and non-pure inversionless gains with frequency up-conversion

A study is made of the effects of Doppler broadening onpure gain without inversion, which means that neither one-photonnor two-photon inversions are allowed, and non-pure gain withoutinversion, which means that one-photon inversion does not occurbut two-photon inversion is present, in a closed Λ-typethree-level system with incoherent pumping. It is shown that whenthe driving field is resonant but the probe field is not, in acertain range of Doppler width, for the case of the lower degree offrequency up-conversion, generally, pure gain without inversionincreases monotonically and non-pure gain without inversion doesnot monotonically increase or decrease with increasing Doppler width;for the case of the higher degree of frequencyup-conversion, pure gain without inversion decreasesmonotonically but non-pure gain without inversion cannot beproduced. In the case of two-photon resonance, in some range ofDoppler width, pure gain without inversion does notmonotonically increase or decrease while non-pure gain withoutinversion decreases monotonically with Doppler width increasing.Finally, an experimental scheme for examining our theoretical resultis given.