  We analyze the main physical processes in quantum cascade lasers with spatial
separation between the region of photon radiation and LO phonon emission
providing depopulation of the lower level of the optical transition. Our
purpose is to find reasons of reduction of the population inversion at low
photon energy and search for ways of its increase. The expression for the
population inversion is obtained from equation for simplified density matrix.
This allows us, on one hand, to take into account coherence of tunneling
between different levels and, on the other hand, to understand its role in
transition probabilities in a simple way. We come to the conclusion that
population inversion in THz lasers can be significantly increased by
optimization of tunneling matrix element between the two regions and LO phonon
emission time. The optimal value of the matrix element is smaller than its
maximal possible value. As well, the optimal LO phonon emission time is larger
than the time reached at LO phonon resonant emission.
