  This paper discusses the use of the Riemann-Silberstein vector to solve the
source-free Maxwell's equations and obtains novel analytical solutions. The
solving process naturally leads to the spinor form of the source-free Maxwell's
equations. Several powerful theorems are established to solve this spinor form
equation. The Waveguide Solution Theorem provides an elegant way to solve
waveguide problems, while The Schrodinger Solution Theorem connects the
Maxwell's equations with the two-dimensional Schrodinger equation. By utilizing
The Schrodinger Solution Theorem, a precise formula for spatiotemporal
diffraction of the Maxwell's equations is derived, which allows for the
reconstruction of electromagnetic waves throughout space and time based on the
field distribution on the diffraction screen. And finally, by studying some
relevant contour integrals, two additional simple but beautiful mathematical
theorems that are necessarily satisfied by electromagnetic field in any
source-free region are derived.
