Someone argues: "The Helmholtz-Kirchhoff integral theorem is exact; Kirchhoff's…
PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Interference and diffraction · Interference and diffraction
What an Instrument Can See, Ch Scalar diffraction theory, derived
Problem
Someone argues: "The Helmholtz-Kirchhoff integral theorem is exact; Kirchhoff's boundary conditions merely supply reasonable estimates of U and ∂U/∂n over the closed surface. So wherever those estimates happen to be good, for instance well away from the aperture edges, the resulting Fresnel-Kirchhoff diffraction formula must there give the exact diffracted field." Assess this claim. Specifically: state what would follow, by the general behaviour of solutions of the Helmholtz equation, if both U=0 and ∂U/∂n=0 held simultaneously over any finite open patch of a surface bounding a source-free region, and use this to explain why the Fresnel-Kirchhoff formula can never give the exact field of the screen problem, however good the individual estimates of U and ∂U/∂n might separately be. Finish by stating in one sentence how the Rayleigh-Sommerfeld construction avoids the difficulty.