An exact treatment of beam optics, starting ab initio from the Maxwell's equations is presented. The starting point of this approach is a matrix representation of the Maxwell's equation in a medium with varying permittivity and permeability. Formal expressions are obtained for the paraxial and leading order aberrating Hamiltonians, without making any assumptions on the form of the varying refractive index. We derive the wavelength-dependent contributions at each order, starting with the lowest-order paraxial Hamiltonian. To illustrate the general theory, we consider the computations of the transfer maps for an axially symmetric graded-index medium. For this system, in the traditional approaches, one gets only six aberrations. In our formalism, we get all the nine aberrations permitted by the axial symmetry. The six aberrations coefficients of the traditional approaches get modified by the wavelength-dependent …