I present an analysis of a cascade laser system consisting of an ensemble of incoherently pumped multilevel atoms confined in a doubly resonant cavity. When the cavity is tuned onto resonances with both of the one-photon transitions, steady-state simultaneous lasing on two cascade transitions may be established from the initial population inversion created by the incoherent pumping. When the cascade lasing is initialized, a two-photon coherence is induced and is built up with the increasing lasing intensities. The generated coherence is reacted back to the system and results in light amplification without inversion on one transition. But on the other transition, the two-photon coherence induces light attenuation. Depending on the ratio of the cavity decay rates for the two cascade lasing fields, the system in the steady state may exhibit simultaneously both lasing without population inversion and lasing with population inversion. That is, one laser field is generated from light amplification by coherence, while the other is generated by stimulated emission from population inversion. This is the first model system in which both an inversionless laser and an inversion laser may be realized simultaneously. When two cavity modes are tuned off the corresponding one-photon resonances and the frequency summation of the two modes are kept at the two-photon resonance, the model evolves into a nondegenerate two-photon laser system. Because the doubly resonant nature of the cavity, the two-photon detuning from intermediate states can be made very small, so that the nondegenerate two-photon gain is substantially enhanced with respect to the degenerate two-photon gain. Thus the system is made more favorable for achieving the nondegenerative two-photon lasing. Several interesting aspects of this cascade system are discussed. Close correlation between the two lasing fields is expected because the photon emission sequences in the cascade laser. The degree of correlation depends on the ratio of the photon loss rates of the two cavity modes. The inversionless and inversion lasers simultaneously generated from the cascade transitions may exhibit interesting statistical properties. It has been shown that the optical coherence associated with the light amplification may lead to novel statistical properties, such as reduced laser line-widths and amplitude squeezing, in inversionless lasers. It is expected that the two cascade lasing modes may have complementary photon statistics. Several real atomic systems that may be used to study the cascade one- and two-photon emission are presented, and experimental progress is described.
