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

A study is made of the effects of Doppler broadening on pure gain without inversion,which means that neither one-photon nor two-photon inversions are allowed,and non-pure gain without inversion,which means that one-photon inversion does not occur but two-photon inversion is present,in a closed Λ-type three-level system with incoherent pumping.It is shown that when the driving field is resonant but the probe field is not,in a certain range of Doppler width,for the case of the lower degree of frequency up-conversion,generally,pure gain without inversion increases monotonically and non-pure gain without inversion does not monotonically increase or decrease with increasing Doppler width;for the case of the higher degree of frequency up-conversion,pure gain without inversion decreases monotonically but non-pure gain without inversion cannot be produced.In the case of two-photon resonance,in some range of Doppler width,pure gain without inversion does not monotonically increase or decrease while non-pure gain without inversion decreases monotonically with Doppler width increasing.Finally,an experimental scheme for examining our theoretical result is given.