Exercise:
Consider a po source of monochromatic light at a distance s from a po detector. In between the source and detector is an opaque thin circular disk with radius R with R gg lambda positioned such that its center is on the midpo of the line connecting the source and detector. The disk is oriented such that the surface is normal to this line. Show using Huygens’ Principle that a non-zero ensity of light reaches the detector and that the part of the phase difference that deps on the positions of the source detector and disk is exactly the phase acquired by glqq least pathgrqq light paths as would be predicted by Fermat’s principle. H: To analytically solve the problem the following approximation beyond Huygens’ Principle is required. For large values of a we can write: _a^inftyfrace^ixxddxapprox fraciae^ia

Solution:
According to Huygen's principle every po on the perimeter of the disc and beyond radiates a spherical wave. In this specific case let rsqrtx^+y^ be the distance from the center of the opaque disk s the distance between the source and the disk equal to the distance from disk to detector and theta the angle from the vertical direction. We can thus write in cylindrical coordinates: UP &propto A__R^inftyrddr_^piddthetafrace^iksqrts^+r^sqrts^+r^frace^iksqrts^+r^sqrts^+r^ pi A_ _R^inftyfracre^iksqrts^+r^s^+r^ddr Now using the substitution uksqrts^+r^ and thus ddufrackrsqrts^+r^ddr we obtain UP &propto pi A_ _ksqrts^+R^^inftyfracue^iuddu For ksqrts^+R^ gg true in view of the requirement Rgg lambda the amplitude UP simplifies to UPpropto frac-ipi A_ksqrts^+R^e^iksqrts^+R^neq Therefore a finite Amplitude UP can be measured/imaged in the center of the shadow area the Arago Spot. Following Fermat's principle the minimum path is sqrts^+R^. The light travelling this optical path accumulates a phase difference of phi e^isqrts^+R^fracpilambda e^iksqrts^+R^ quod erat demonstrandum.