Using the Hertz vectors, it is possible to reduce the Maxwell field equations inside index- guided optical devices to two couple scalar partial differential equations. With the proper selection of gauge functions, the mixed partial derivatives with respect to the propagation direction and either transverse direction can be eliminated. This results in a form similar to the equations used for the scalar beam propagation method. However, space-varying functions multiplied by partial derivative operators rule out convenient use of the same Fourier transform methods commonly used for the scalar case. A numerical method for solving time independent propagation of coherent light through guided nonlinear media including vector coupling using these equations is presented.
Carl S. Milsted,
Cyrus D. Cantrell,
"Vector beam propagation using Hertz vectors", Proc. SPIE 1497, Nonlinear Optics and Materials, (1 September 1991); doi: 10.1117/12.46776; https://doi.org/10.1117/12.46776