Within the framework of a unified approach, we derive several identical representations of the realistic (and ideal) pattern functions
fk,l necessary to reconstruct the density-matrix element ${\ensuremath{\varrho}}_{\mathrm{kl}}$ in the Fock basis from realistic quadrature distributions measured with a nonideal balanced homodyne detector of overall efficiency $\ensuremath{\eta}.$ To this end we first establish a biorthogonality relation for the pattern functions from which the form of the Fourier-transformed pattern functions can be easily read off. Then a different handling of the inverse Fourier transform yields several identical representations for the desired pattern functions provided that $\ensuremath{\eta}>1/2.$ In particular, we derive a representation of the realistic pattern functions in terms of a finite weighted sum over scaled ideal ones. Compared to a previously given form, this expression has a nice analytical structure, reveals some general features of the realistic pattern functions, and is particularly well suited for their numerical computation. From this representation sum relations for the pattern functions follow. With their help we show that both a smoothed Wigner function and a generalized moment generating function can be directly sampled from realistic quadrature distributions. The corresponding sampling functions are basically shifted and/or scaled versions of the ideal pattern functions
f0,0 and
f0,k, respectively. As a by-product we derive the sampling functions needed to reconstruct the density matrix from the Fourier-transformed realistic quadrature distribution, i.e., from the s-ordered characteristic function with $s=\ensuremath{-}(1\ensuremath{-}\ensuremath{\eta})/\ensuremath{\eta}.$
This paper describes the model of a compound Poisson distribution for photon statistics with regard to their bunching in Fock states, thermal states, and others. The method of generating functions is used to calculate the probability distributions, moments, and correlation functions. The parameters of conditional states arising from the subtraction of photons by splitting the beam are determined. The problem of state reconstruction with regard to quadrature quantum measurements is considered. The study is aimed at developing high-precision methods for generating and controlling optical quantum states.