Integral equation formulation for reflection by a mirror

When light is incident on a mirror, it induces a current density on its surface. This surface current density emits radiation, which is the observed reflected field. We consider a monochromatic incident field with an arbitrary spatial dependence, and we derive an integral equation for the Fourier-transformed surface current density. This equation contains the incident electric field at the surface as an inhomogeneous term. The incident field, emitted by a source current density in front of the mirror, is then represented by an angular spectrum, and this leads to a solution of the integral equation. From this result we derive a relation between the surface current density and the current density of the source. It is shown with examples that this approach provides a simple method for obtaining the surface current density. It is also shown that with the solution of the integral equation, an image source can be constructed for any current source, and as illustration we construct the images of electric and magnetic dipoles and the mirror image of an electric quadrupole. By applying the general solution for the surface current density, we derive an expression for the reflected field as an integral over the source current distribution, and this may serve as an alternative to the method of images.