Filamentation of femtosecond laser pulses at 800 nm provokes many spectacular phenomena like long range propagation, broadband THz generation, pulse self-compression, guided electric discharges and lasing effects. In particular, a coherent forward emission at 391 nm is observed that corresponds to a transition between levels B and X of the singly ionized Nitrogen molecule. Interpretation of this cavity-free air lasing is highly controversial. Several mechanisms like net electronic population inversion, rotational population inversion or Raman-like amplification were proposed but none of them is fully consistent and describes properly the gain properties. We have developed a model of lasing without population inversion in a V scheme arrangement. It is supported by experimental evidence for the long lasting coherent polarizations AX and BX required for such a V scheme. We show that the temporal shape of the lasing emission and its dependence with gas pressure can be well restituted theoretically. In addition, we present new supplementary measurements using consecutive twin femtosecond 800 nm pump pulses. A reduction of the global lasing signal at 391 by a factor ~ 1000 is observed when the gas is pumped with the twin pulses. This spectacular effect is observed over a range of delay between the twin pulses of several ps and can be interpreted in the frame of the V scheme.
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