Foldy-Wouthuysen transformation and a quasiparaxial approximation scheme for the scalar wave theory of light beams

The Foldy-Wouthuysen iterative diagonalization technique is applied to the Helmholtz equation to obtain a Hamiltonian description of the propagation of a monochromatic quasiparaxial light beam through a system in which the refractive index $n(x,y,z)$ varies about a background value $n_0$ such that $|n(x,y,z)-n_0| \ll n_0$. This technique is presented as an alternative to the conventional method of series expansion of the radical. Besides reproducing all the traditional quasiparaxial terms, this method leads to additional terms in the optical Hamiltonian.