Eigenstates of Maxwell’s equations in the quasistatic regime were used recently to calculate the response of a Veselago Lens1 to the field produced by a time dependent point electric charge.2, 3 More recently, this approach was extended to calculate the non-quasistatic response of such a lens. This necessitated a calculation of the eigenstates of the full Maxwell equations in a flat slab structure where the electric permittivity ϵ1 of the slab differs from the electric permittivity ϵ2 of its surroundings while the magnetic permeability is equal to 1 everywhere.4 These eigenstates were used to calculate the response of a Veselago Lens to an oscillating point electric dipole source of electromagnetic (EM) waves. A result of these calculations was that, although images with subwavelength resolution are achievable, as first predicted by John Pendry,5 those images appear not at the points predicted by geometric optics. They appear, instead, at points which lie upon the slab surfaces. This is strongly connected to the fact that when ϵ1/ϵ2 = −1 a strong singularity occurs in Maxwell’s equations: This value of ϵ1/ϵ2 is a mathemetical accumulation point for the EM eigenvalues.6 Unfortunately, many physicists are unaware of this crucial mathematical property of Maxwell’s equations. In this article we describe how the non-quasistatic eigenstates of Maxwell’s equations in a composite microstructure can be calculated for general two-constituent microstructures, where both ϵ and μ have different values in the two constituents.
An exact calculation of the local electric field E(r) is described for the case of an external current or plane wave source in a setup of an E1, μ1 slab in an E2, μ2 medium. For this purpose we first calculate all the general eigenstates of the full Maxwell equations. These eigenstates are then used to develop an exact expansion for the physical values of E(r) in the system characterized by physical values of E1, E2, μ1, and μ2. Results are compared with those of a previous calculation of the local field where μ = 1 everywhere. Numerical results are shown for the eigenvalues in practically important configurations where attaining an optical image with sub-wavelength resolution has practical significance. We show that the k ≫ k2 components are enhanced for the TM field when E1/E2 = −1 and for the TE field when μ1/μ2 = −1 where the enhancement of the evanescent waves starts from lower k values as we approach a setup with both E1/E2 = −1 and μ1/μ2 = −1. We also show that the eigenfunctions for the setup where μ = 1 everywhere correspond to configurations of 3D phased arrays.
An exact calculation of the local electric field E(r) is described for the case of a monochromatic (∼ e−iωt) source or incident field in an ∈1, ∈2 composite structure. For this purpose we expand the local electric field E(r) in a complete set of eigenstates of the full Maxwell equations. The eigenvalues appear as special, non-physical values of ∈1 when ∈2 is given. These eigenstates are then used to write an exact expansion for the physical values of E(r) in the system characterized by physical values of ∈1(ω) and ∈2(ω). The application of this approach to the analysis of a Veselago Lens is discussed. In that case the ∈1 constituent has the shape of a flat slab in the otherwise uniform ∈2 constituent. The eigenstates of the full Maxwell equations for this structure are easy to find. This will allow an in depth analysis of the Veselago Lens to be developed, including the possibility of attaining an optical image with sub-wavelength resolution
In conventional optics the image is formed only by the propagating waves and the information encoded in the evanescent waves is lost, leading to limited resolution. A negative refractive index slab can amplify evanescent waves and enable the generation of an image by both propagating and non propagating waves, theoretically leading to unlimited resolution. Here we analyze the imaging of an oscillating dipole in a composite structure composed of an (epsilon)1 slab surrounded by an (epsilon)2 medium, where μ = 1 everywhere. For this purpose we calculate all the eigenstates and eigenvalues of the full Maxwell equations for the composite structure and develop an exact expansion for the local electric field E(r) in the system. Then we calculate the intensity and resolution for various permittivity values. We show that only the low order modes contribute to the expansion of the electric field which enables an efficient calculation of the physical quantities.
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