On the radiation from the boundary of diffracting apertures and obstacles
Neglecting the obliquity factor, which is justified when one is considering only small angle diffraction, it is shown that the surface integral which is usually employed for the determination of the disturbance at any point can be easily converted into a line integral along the boundary of the diffracting screen. The formulæ thus obtained show that with either an aperture or an obstacle the illumination in the region of shadow can be completely represented as the effect of radiations arising from the boundary, while in the region of light the disturbance due to the direct light is superposed on this. The phase of the boundary radiation is determined by the region (of light or shadow) to which the ray towards the point of observation proceeds from the boundary, being opposite to that of the incident light in the former case, and being the same in the latter case. It is however shown that this leads to no discontinuity in the illumination as the point of observation passes from the region of light into the region of shadow. The boundary radiation can again be effectively replaced by the radiations arising from a finite number of point-sources situated on the boundary called ‘poles’, for which the path to the observation pointvia the boundary is a maximum or a minimum. The phase of the resultant disturbance due to regions of the boundary including and lying on either side of a pole is shown to lead over or lag behind that of the wave from the pole by the quantity π/4, according as the pole is one of maximum or minimum path. Applying these ideas to the diffraction pattern of a circular disc, it is shown that the calculated radii of the rings in the region of shadow agree well with those deduced from Lommel’s theory.