New exact solutions to the paraxial wave equation are obtained in the form of a product of Laguerre polynomials, Bessel functions, and Gaussian functions. In the limit of large Laguerre-Gaussian beam size, the Bessel factor dominates and the solution sets reduce to the modes of closed resonators, hollow metal waveguides, and dielectric waveguides. In the opposite limit the solutions reduce to Laguerre-Gaussian modes of open resonators and graded-index waveguides. While these solutions are valid for electromagnetic waves traveling through free space, they are also valid for propagation through circularly symmetric optical systems representable by ABCD matrices as well.

An interesting feature of the new solution set is the existence of three mode indices, where only two are required for an orthogonal expansion. As an example, Laguerre-Gaussian beam propagation through an optical system that contains a Bessel-like amplitude filter is discussed..