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Materials with negative permittivity and permeability can overcome the diffraction limit, thereby making the sub-wavelength imaging possible. In this study, we analyze the effects of gradient index on a half-infinite perfect lens. We assume that the sharp interface between the vacuum and the negative-index material is replaced by a smooth transition profile such that the index gradually changing from positive to negative. Interestingly, we find that if the graded index profile is modeled by a tanh function, we can have closed-form analytical solutions for this problem, which is a distinct advantage as numerical solutions are not accurate for evanescent waves with large transverse wave vectors. By analyzing the analytical formulas we confirm that a nonzero total absorption can occur even for a near-zero absorption coefficient in the steady-state limit and the image plane contains multiple sub-wavelength images of an object.

Optical waves in a gradient negative-index lens of a half-infinite length