  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 constituent plane wave fields. We derive these matrices and
describe the complete formalism by implementing these matrices. This formalism,
though physically equivalent to Fresnel formalism, has greater mathematical
elegance and computational efficiency as compared to the latter. We utilize
exact 3D expressions of the constituent plane wavevectors and electric fields
of the incident, reflected and transmitted waves, which enable us to seamlessly
analyse plane waves, paraxial and non-paraxial beams, highly diverging and
tightly focused beam-fields as well as waves of miscellaneous wavefront-shapes
and properties using the single formalism. The exact electric field expressions
automatically include the geometric phase information; while we retain the
wavefront curvature information by using appropriate multiplicative factors. We
demonstrate our formalism by obtaining the reflected and transmitted fields in
a simulated Gaussian beam model. Finally, we briefly discuss how our
generalized formalism is capable of analysing the reflection-transmission
problem of a very large class of complex optical waves -- by referring to some
novel works from the current literature as exemplary cases.
