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A spectral theory for planar dielectric waveguides

University of British Columbia

The problem of electromagnetic wave propagation across the junction of two similar planar dielectric waveguides is analysed, within the Kirchhoff approximation, by expanding the field into transverse variations of all possible modes. It is proven that the expansion can represent any solution for any planar dielectric waveguide. The spectral function is introduced into the representation, and this helps resolve some of the theoretical problems in passing from the limit of closed waveguides to open waveguides. Using the spectral function and the Gel'fand-Levitan integral equation some new exact solutions to novel dielectric planar waveguides can be found. Examples of waveguiding by total internal reflection or by Bragg reflection (which are physically very different processes) can be generated by changing a single parameter in the formulation. Usually the representation for an open dielectric waveguide requires the matrix spectral function. However the Gel'fand-Levitan reconstruction is defined for scalar spectral functions. A technique for constructing the spectral matrix and the scattering solutions from two spectral functions is demonstrated. This technique uses a variational formulation of a scattering experiment. The connection between a dielectric structure and the characteristics of propagation on it is obscure. However the connection between these characteristics and the spectral function is much clearer. It is sometimes possible to make predictions about the properties of the waveguide by looking at its spectral function only. Since the connection between the spectral function and the dielectric structure is well established by inverse spectral theory, introducing the spectral function has been of help in establishing the desired connection between the dielectric structure and the characteristics of propagation on it. Such considerations suggest one of the above waveguides is sensitive to small perturbations and could be used as an electro-optic modulator. Detailed calculations confirm the hypothesis.

Text

2011-01-19

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http://hdl.handle.net/2429/30690

Electrical and Computer Engineering

University of British Columbia

Graduate