Exercise:
Suppose that you have a beam of light that propagates along the z-axis of a coordinate system along the +z-direction. Ase that at the plane z d the ensity outside a circle of radius R centered at x y is immeasurably small. Also ase that at the plane z we can place an aperture with an arbitrary shape but that blocks all light greater than a distance a from the z-axis. Let Ix y d be the ensity pattern measured at z d with the aperture in place. Suppose now that we replace the aperture with its complement: If we let txy be the transmission function of the original aperture we now exchange this aperture for a new one with a new aperture with transmission function tcx y -tx y. In other words all places where the original aperture blocked light will now allow light to pass and vice versa. We now measure a new ensity pattern I_cx y d at the detector plane. Show using the principle of superposition that Ix y d I_cx y d for x_ + y_ R_.

Solution:
In physics Babinet’s Principle states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam ensity. It was formulated in the s by French physicist Jacques Babinet. Ase B is the original diffracting body and B' is its complement i.e. a body that is transparent where B is opaque and opaque where B is transparent. We can think that the superposition of an object and its complement is equivalent to having empty space thus the of the radiation patterns caused by B and B' must be the same as the radiation pattern of the unobstructed beam. In this case we define Uxyd as the field from the original aperture and U_cxyd as the field from the complement. Outside the circular of radius R we know that according to the principle of superposition U + U_c . Therefore U_c -U i.e. U_c and U have opposite phases. Moreover we can write I_cxyd |U_c|^ |U|^ Ixyd.