| Summary: | The electromagnetic diffraction problem is formulated in terms of either the electric or magnetic Hertz potential. This approach is equivalent to traditional methods based on the vector form of Green's theorem, but it is less widely known. The components of the Hertz potentials are independent, and each satisfies a scalar wave equation. The formal solutions for these components are therefore given by two equations referred to as the Rayleigh formulas, which are familiar from scalar diffraction theory. A physical interpretation of the Rayleigh solution shows that the diffracted wave may be thought of as a superposition of elementary, electromagnetic Huygens wavelets. Depending on the type of Green's function that is chosen, these wavelets have the same form as fields radiated by dipoles of different orientations (D-theory) or by special types of quadrupoles (Q-theory). Using techniques which are well known from scalar theory, it is shown that the diffracted wave can be represented as an angular spectrum of electromagnetic plane waves, and that this description is equivalent to the Q-theory approach. The use of approximate, Kirchhoff-type boundary conditions in the Hertz potential formalism is investigated. When these boundary conditions are used in the D-theory, the diffracted wave is found to be identical with the results of more traditional theories that apply the boundary conditions directly to the fields in the aperture. Using these boundary conditions in the Q-theory yields different results, because they are applied to the Hertz potentials rather than to the fields themselves. The differences between the two approaches are most apparent when the aperture is small in comparison with the wavelength. To determine which theory is more appropriate for Kirchhoff-type boundary conditions, an experiment to measure the diffraction from subwavelength-diameter pinholes is performed. The Q-theory shows better agreement with the results. It is also determined that the best agreement is obtained when the magnetic rather than electric Hertz potential is used.
|
|---|
