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.