Graphical solution of recursion relations in the lossless micromaser context: I. Diagonal formulas and the photon distribution

The theory of the lossless micromaser can be developed using a non-steady-state approach which also takes account of the Stark-shift of the driving transition caused by virtual transitions to other atomic levels. The nature of the recursive pumping of the cavity field by repeated injection of pump atoms can be manifested by a general recursion relationship for the photon distribution. This recursion formula simplifies enormously for cases in which the atoms are injected in a purely excited (or ground) state as well as for incoherent excited pump atoms. For such cases an elegant analytical solution by a graphical method is introduced in the paper.