Functional Analytic Formulation of Fresnel Diffraction

The theory of Fresnel diffraction is developed strictly by means of functional analytic method. Fresnel transforms and inverse Fresnel transforms, which give a basis for Fresnel diffraction, are formulated systematically in terms of Fresnel diffraction operator T( z ). Then, it is shown that the set { T( z )} is a one-parameter group of unitary and factor-type operators from the algebraic and topological properties of T( z ). Furthermore, from the representation with respect to the infinitesimal operator of T( z ), the differential formulation of Fresnel diffraction is obtained. This infinitesimal operator, which is -i k times the paraxial approximate expression of the quantized Hamiltonian operator of the variational problem which is given by Fermat's principle in optics, is closely related to the light ray slope.