Scalar diffraction theory

Scalar diffraction theory approximates light by one complex field satisfying a scalar wave equation. Aperture propagation is represented by an integral of secondary contributions, with vector polarization of waves and detailed electromagnetic edge conditions suppressed. The approximation is useful for many apertures large compared with wavelength, but is not an exact substitute for Maxwell equations at arbitrary apertures.

In linear scalar diffraction theory, complementary aperture transmissions $t$ and $1-t$ give complex fields $U_t+U_{1-t}=U_0$, where $U_0$ is the unobstructed field under identical illumination. Away from the direct beam in Fraunhofer diffraction, their diffracted fields differ by a sign and have equal intensity. Intensities do not generally add: in particular, complementary on-axis Fresnel diffraction patterns need not have equal intensity. The ideal scalar statement does not incorporate vector boundary details at a real conducting edge.