Linearization of wave equations

We explore the ways to linearize the wave equations. Special emphasis is paid to the Klein-Gordon equation for a spin-0 relativistic particle and the Helmholtz equation governing scalar optics. Owing to the mathematical similarity, both of these equations are linearized using the Feshbach-Villars procedure. Maxwell's equations are linear but coupled and constrained. So, a matrix representation is presented. New formalisms of beam optics are presented using the linearized form of the Helmholtz equations and the Dirac-like matrix form of Maxwell's equations respectively. It is shown that the matrix formulation of the Maxwell optics naturally leads to a unified treatment of beam optics (including aberrations to all orders) and light polarization, from a single parent Hamiltonian. The non-traditional treatments using quantum methodologies lead to wavelength-dependent modifications of the traditional prescriptions. In the limit of low wavelength, the non-traditional prescriptions of both Helmholtz optics and Maxwell optics presented here reproduce the 'Lie algebraic formalism of light beam optics'. The accompanying machinery of the Foldy-Wouthuysen transformation technique is described. From the new prescriptions of light beam optics, it is seen that the Hamilton's optical-mechanical analogy persists in the wavelength-dependent regime.