Reformulation of the Optical Equivalence Theorem in Terms of the LAGUERRE Polynomials

The “diagonal” weighting function Φ(α) of the density matrix expressed in the coherent states is examined being decomposed in the base of the Laguerre polynomials. A criterion of the convergence in L 2 ‐space is obtained. This decomposition is also studied as a generalized function. The equivalence of the FOCK and SUDARSHAN representations of the density matrix is proved using present formulae. Some theorems from the moment theory are applied and general assumptions are given under which the Φ‐representation is an ordinary non‐negative function or a generalized function whose support is composed of a finite number of points. The results are demonstrated on the examples of the thermal and coherent fields, their superposition and on the Fock state of the field. The decomposition of the Φ‐function in terms of the LAGUERRE polynomials is useful for approximative determination of the Φ‐function from a finite number of moments.