Consequences of the Angular Spectrum Decomposition of a Focused Beam, including Slower than C Beam Propagation

When dealing with light scattering and propagation of an electromagnetic beam, there are essentially two kinds of expansions which have been used to describe the incident beam (i) a discrete expansion involving beam shape coefficients and (ii) a continuous expansion in terms of an angular spectrum of plane waves. In this paper, we demonstrate that the angular spectrum decomposition readily leads to two important consequences, (i) laser light beams travel in free space with an effective velocity that is smaller than the speed of light c , and (ii) the optical theorem does not hold for arbitrary shaped beams, both in the case of electromagnetic waves and scalar waves, e.g. quantum and acoustical waves.