繞射之線積分理論

Abstract

本篇論文探討以沿著孔洞邊緣的線積分來表示一個純量波經過孔洞之後的繞射行為，並探討不同線積分的形式所能給出的不同物理詮釋。

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 optics wave 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. Though providing us with an alternative perspective on the diffraction phenomena, this decomposition theory 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 to be more rigorous this decomposition formulation must be slightly modified by taking into account the correct boundary conditions. In this thesis, we derive a slightly different decomposition formulation that avoids the discontinuity, and also we deform the functional form of the line integral into another one that mimics the ordinary reflection behavior of waves, and finally, all these works are done based on the mathematically consistent boundary conditions. In the appendix, we digress a little to see how to solve diffraction problems subject to 'physical' boundary conditions, which best describe the diffraction phenomena in the real world.