An Integro-Differential Equation Formulation for the Ideal Veselago Lens

The 2-D problem of finding the electric field for a line-source illuminating a lossless Veselago lens (VL) is cast as an integro-differential equation (IDE). The integral corresponds to a surface integral of the total tangential magnetic field on the boundaries of the lens. The IDE is obtained using an incident/scattered-field decomposition such that the VL is considered to be the scatterer. The new formulation is applied in five distinct regions: the illuminating-source region on the left side of the lens, two regions within the lens separated by a power-transfer plane, and two regions to the right of the lens, also separated by a power-transfer plane. The formulation is shown to be consistent with previously derived analytic expressions for the fields. In addition, the new IDE formulation provides a physically intuitive picture of the focusing behavior of the lens, specifically, showing how the induced contrast sources at the boundaries determine the resulting focused field. It is shown how the line integral contributions to the scattered field on the left side of the lens are zero, and how the scattered field produces the expected focus point singularities on the interior and far-side of the lens.