Asymptotic field of evanescent waves of a Hertzian dipole

The asymptotic expansion of the electromagnetic field associated with plane waves of the evanescent type in the transverse magnetic (TM) field of an electric point-source dipole has been obtained for large axial and off-axial distances of a half-space. (1) The point-source-dipole field bandlimited to the exclusion of all evanescent waves does not satisfy strong radiation conditions of order r−2 at almost all field points in every direction as the radial distance . (2) The point-source-dipole field that contains the uniform-plane-wave components plus a bandlimited low-frequency part of the evanescent-wave spectrum satisfies approximately strong radiation conditions of order r−2 as for fixed (x2 + y2)1/2. (3) The point-source-dipole field has to be unbandlimited so as to include all evanescent-wave components in order to satisfy strong radiation conditions of order r−2 as for any fixed value of z.