Generalized matrix transformation formalism for reflection and transmission of complex optical waves at a plane dielectric interface.

We describe a generalized formalism, addressing the fundamental problem of reflection and transmission of complex optical waves at a plane dielectric interface. Our formalism involves the application of generalized operator matrices to the incident constituent plane-wave fields to obtain the reflected and transmitted fields. This formalism, though physically equivalent to Fresnel formalism, has greater mathematical elegance and computational efficiency as compared to the latter. We utilize exact 3D electric-field expressions, which enable us to seamlessly analyze waves of miscellaneous wavefront shapes and properties using the single formalism, along with appropriately retaining the geometric phase and wavefront curvature information. We demonstrate our formalism by obtaining and analyzing the reflected and transmitted fields in a simulated Gaussian beam model.