By Avijit Lahiri
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Additional info for Basic Optics I-II
After stating the uniqueness theorem in electromagnetic theory in the next section, I will introduce a number of simple solutions to the field equations which turn out to be useful in optics, and in electromagnetic theory in general. 1 The uniqueness theorem Let us consider a region V in space bounded by a closed surface S, within which the Maxwell equations are satisfied. Let the field vectors be given at time t = 0. Further, let the field vectors satisfy the boundary condition that the tangential component of the electric intensity (Et) equals a given vector function (possibly time dependent) on the boundary surface S for all t ≥ 0 (recall that the tangential component is given by × E at points on S, where stands for the unit normal, which is commonly chosen to be the outward drawn one with respect to the interior of V, at any given point of S).
An alternative approach for describing the harmonically varying fields would be to make use of the electromagnetic potentials φ and A. , in the absence of the fictitious magnetic charges and currents. The potentials φ and A, as defined in sec. 3 are, however, not symmetric with respect to the electric and magnetic field vectors, and their definition is, moreover, not consistent with two of the Maxwell equations (the equations for curl E and div B) in the presence of magnetic charge- and current densities.
In the case of a linear anisotropic medium, the susceptibility is in the nature of a tensor, in terms of which the permittivity tensor is defined in an analogous manner. For a nonlinear medium, on the other hand, the polarization P depends on the electric intensity E in a nonlinear manner, giving rise to novel effects in optics. 6b) where χM is the magnetic susceptibility of the medium. 6c) where μr (= 1 + χM) is the relative permeability. In this book we will not have occasion to refer to magnetic anisotropy or magnetic nonlinearity.