Line-Integral Representations of the Diffraction of Scalar Fields
  Traditionally, the diffraction of a scalar wave satisfying Helmholtz equation through an aperture on an otherwise black screen can be solved approximately by Kirchhoff's integral over the aperture. Rubinowicz, on the other hand, was able to split the solution into two parts: one is the geometrical part that appears only in the geometrical illuminated region, and the other representing the reflected wave is a line-integral along the edge of the aperture. However, this decomposition is not entirely satisfactory in the sense that the two separated fields are discontinuous at the boundary of the illuminated region. Also, the functional form of the line-integral is not what one would expect an ordinary reflection wave should be due to some confusing factors in the integrand. Finally, the boundary conditions on the screen imposed by Kirchhoff's approximation are mathematically inconsistent, and therefore, rigorously, this decomposition formulation must be slightly modified by taking into account the correct B.C.s. In this thesis, we use the consistent boundary conditions to derive a slightly different decomposition formula which shows that the behavior of the diffracted wave at the edge is exactly just like an ordinary reflection-realizing the conjecture of Thomas Young in the 18th century. We also derived another decomposition formula which avoids mathematical discontinuity encountered by Rubinowicz. In the last section we demonstrate that our solution is consistent with that obtained by Sommerfeld in the rigorous 2-D plane-wave diffraction problem, so our formulation in this sense may describe more accurately the behavior of diffracted wave near the edge of the aperture than Kirchhoff's formula. 
Yi-Chuan Lu
physics.class-ph