Up to now, beam shape coefficients, g(n) or g(n)(m), encoding an illuminating beam in generalized Lorenz-Mie theory have been derived from a priori theoretical electromagnetic descriptions. It is shown that, from intensity measurements in the laboratory, one can measure so-called density matrices associated with the beam shape coefficients. In the case of axisymmetric beams, when the beam is encoded by a set of special beam shape coefficients, g(n), one has to consider one matrix, I-nm. In the general case, i.e. when the beam is encoded by a double set of coefficients, g(n,TM)(m), g(n,TE)(n), one can measure three 4D matrices, M(np)(mq), E(np)(mq), C-np(mq). Measuring such matrices from an actual beam in a laboratory and using them in the density matrix approach to the generalized Lorenz-Mie theory would allow a better characterization of the scattering phenomena occurring when a scatter center is illuminated by an arbitrary-shaped beam, therefore opening up new opportunities for refined particle characterization.