Exact Calculation of the Field due to a Single Fresnel Zone

The consistency of the Fresnel zone construction is customarily checked by verifying that it correctly describes wave propagation from a point source in free space. The calculation, based on an approximate evaluation of the contribution of individual Fresnel zones to the total field, leads to the Fresnel theorem: the total field is just one-half that due to the first zone alone.We show here that if the Fresnel zones are defined on a plane passing through the midpoint of the line joining the source point to the field point, the field due to each Fresnel zone may be calculated exactly. When the Fresnel zones are defined on a plane perpendicular to the line between the source and field points, the contribution of the first zone is equal to the total field multiplied by the factor 1+ (1+π/kR′)−2. Here, k is the wavenumber and R′ is the distance separating source and field points. Thus, for this geometry, the Fresnel theorem holds only in the limit kR′≫1; for arbitrary kR′, the factor quoted must be used.A formula, valid for any Fresnel zone, and for arbitrary orientation of the plane on which the Fresnel zones are defined, is given in the text.