The subject of obtaining the solution of Maxwell's equations in a boundary value problem has kept scientists occupied for more than one hundred years. It is one of the few `classic' problems that continue to attract the attention of researchers. Many contemporary problems of interest involve our knowledge of the exact solution to these equations. However, the complexity of these equations has defied analytic solutions for all but the simplest cases.
To state the problem more correctly, one needs to understand the length-scales involved. In a scattering problem for example there are two lengths involved: R, the physical size of the scattering object and lambda, the wavelength of electromagnetic waves in question. When R is much larger than lambda, the problem is readily solved. The method of solution, called geometrical optics, has been known for at least a few hundred years. In fact we do not even need to invoke Maxwell's equations directly. The other limiting case is when R is much smaller than lambda. This regime is called the quasistatic limit or nano-optics. Although mathematically involved, the solution in the quasistatic limit is obtained by solving Laplace's equation. The problem is far more complicated when R is approximately equal to lambda. It is in this regime that one is required to solve Maxwell's equations exactly.
The problem involves the solution of a set of partial differential equations (The Maxwell Equations) for the coupled vector fields for the electric field, E and the magnetic field, H. Although the general theoretical aspects of these equations are fairly well understood at the moment, there are still some formidable mathematical or computational problems to overcome in the area of solving the time-varying boundary value problem. Whereas the problem of acoustic (scalar fields) diffraction has been solved in a number of cases such as the strip, elliptic cylinder, hyperbolic cylinder, wedge, prolate and oblate spheroids, etc., the corresponding solution for the vector fields are still by and large unconquered. Often the solution is worked out only in the quasistatic approximation by solving Laplace's equation (homogenous case).
This thesis introduces the notion of vector basis functions as a solution to Maxwell's equations and proves that they form a complete set in a given coordinate system. The proof on completeness is followed by a discussion on the relevant boundary conditions in a scattering problem. By proving that an arbitrary plane wave can be expanded in this basis set, it is shown that any incoming light source can be modelled (since any source can be modelled as a superposition of plane waves).
The four movies shown below
highlight the continuity of the fields when modifications are
made to the scattering object. For more information please contact
Dr. Naomi Halas.