A perfectly conducting flat screen occupies half of the x-y plane (i.e., x < 0).=”” a=”” plane=”” wave=”” of=”” intensity=”” i0=”” and=”” wave=”” number=”” ?o=”” is=”” incident=”” along=”” the=”” z=”” axis=”” from=”” the=”” region=”” z=””>< 0.=”” discuss=”” the=”” values=”” of=”” the=”” diffracted=”” fields=”” in=”” the=”” plane=”” parallel=”” to=”” the=”” x-y=”” plane=”” defined=”” by=”” z=”Z”> 0. Let the coordinates of the observation point be (X, 0, Z).
(a) Show that, for the usual scalar Kirchhoff approximation and in the limit Z >> X and ?^?kZ >> 1, the diffracted field is

Where ( = (k/2Z)1/2X.
(b) Show that the intensity can be written
I = |I^|2 = I0/2[(C (() + ?1)2 + (S (() + ?1)2]
Where C(() and S(() are the so-called Fresnel integrals. Determine the asymptotic behavior of I for ( large and positive (illuminated region) and ( large and negative (shadow region). What is the value of I at X = 0? Make a sketch of I as a function of X for fixed Z.