An Integral Equation Approach to the Theory of Optical Resonance
In a bounded material medium, embedded in vacuuo, solutions to Maxwell’s equations can satisfy the outgoing boundary conditions at infinity and the continuity conditions at the boundary even without the presence of an incident electromagnetic field. Such solutions are broadly classified as natural oscillations which can further be divided into radiative modes and nonradiative modes depending on whether the frequencies are complex or real. Surface waves have complex wave vectors but real frequencies and hence belong to the class of non-radiative natural modes, On the other hand virtual, or resonance, modes have complex frequencies (and hence, in general, have complex wave vectors) and belong to the class of radiative natural modes. Clearly the frequencies of the natural oscillations depend upon the geometry of the material medium as much as upon the different physical parameters of the medium. Natural oscillations satisfy, in addition to the normal dispersion relations (the constraint due to the Maxwell wave equation), another dispersion relation, i.e. the natural mode dispersion relation (also known as the surface wave dispersion relation), a constraint necessary in order that the natural oscillations satisfy the Maxwell continuity conditions at the boundary.
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