Non-Quasi-Static Eigenstates of Maxwell’s Equations in a Two-Constituent Composite Medium and their Application to a Calculation of the Local Eletric Field of a Time Dependent Point Charge

In conventional optics the image is formed only by the propagating waves and the information encoded in the evanescent waves is lost. This limits the resolution which is inversely proportional to the wavelength of the light. A flat slab with a negative refractive index metamaterial can focus at a point the radiation from a point source [1]. Such a slab can also amplify evanescent waves and thus enable the generation of an image by both propagating and non propagating waves, theoretically leading to unlimited resolution [2]. Experiments motivated by this theory have achieved an enhanced resolution image [3].

The imaging of an electric point charge in a composite structure composed of a ε1 slab surrounded by a ε₂ medium, was recently analyzed by expanding the local electric potential in a series of the quasi-static eigenfunctions of the composite structure. This analysis yielded exact one dimensional integral expressions for the quasi-static electric potential of a point charge in that system [4]. Numerical evaluations of those integrals, using realistic values for physical parameters like the electric permittivities and the slab thickness, revealed some surprising effects among which is that the maximum concentration of the electric field occurs not at the geometric optics foci but at the interfaces between the negative permittivity slab and the surrounding medium [5].

Here we describe an exact calculation of the local electric field E(r) for the full Maxwell`s equations where μ=1 everywhere in the system. For this purpose we first calculate all the eigenstates of Maxwell`s equations for the composite structure. The eigenvalues appear as special, non-physical values of ε₁ when ε₂ is given. These eigenstates are then used to develop an exact expansion for the physical values of E(r) in the system for the case of a time dependent point charge q·e^-iωt in the ε₂ medium.

References

[1] V. Veselago, Usp. Fiz. Nauk 92, 517 (1967) [V. Veselago, Phys. Usp. 10, 509– 514 (1968)].

[2] J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).

[3] N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).

[4] D. J. Bergman, Phys. Rev. A 89, 015801 (2014).

[5] A. Farhi and D. J. Bergman, Phys. Rev. A 90, 013806 (2014).

asaffarhi@post.tau.ac.il