  Laguerre-Gauss (LG) modes represent an orthonormal basis set of solutions of
the paraxial wave equation. LG are characterized by two integer parameters $n$
and $\ell$ that are related to the radial and azimuthal profile of the beam.
The physical dimension of the mode is instead determined by the beam waist
parameter $w_0$: only LG modes with the same $w_0$ satisfy the orthogonality
relation. Here, we derive the scalar product between two LG modes with
different beam waists and show how this result can be exploited to derive
different expansions of a generic beam in terms of LG modes. In particular, we
apply our results to the recently introduced Circular Beams, by deriving a
previously unknown expansion. We finally show how the waist parameter must be
chosen in order to optimize such expansion.
