Metamaterials with *extreme parameters* (e.g., very small or very large components of the constitutive
tensors) are of interest in many important applications.
For instance, metallo-dielectric multilayers characterized by extreme anisotropy
convert evanescent spectral components with large transverse wavenumbers into propagating waves, thereby
exhibiting intriguing subwavelength focusing effects that are of great interest in imaging applications.
More recently, the use of obliquely layered structures has been proposed in order to achieve simple
image manipulation (lateral displacement) with subwavelength resolution.

With the exception of few cases for which analytical approximations of the Green’s function can be worked out, the imaging properties (e.g., resolution) of the above configurations need to be assessed numerically.

In a study in collaboration with Nader Engheta (University of Pennsylvania) and
Andrea Alù (University of Texas at Austin)
[1], we studied in detail a slab of uniaxial
*epsilon-near-zero* metamaterial with tilted optical axis, as schematized in the figure top panel
(also shown are the typical elliptic and hyperbolic equifrequency contours in the global
and rotated spectral reference systems).

In particular, we derived a closed-form analytical solution for the Green’s function in terms of special cylindrical functions. These functions can be efficiently computed via well-established numerical schemes, yielding computational savings up to nearly three orders of magnitudes by comparison with brute-force numerical quadrature of the corresponding spectral integrals. Moreover, in the near-field parameter range of interest, they can be conveniently approximated in terms of simple analytical functions. Validation and calibration of our results against a numerical-integration-based reference solution confirmed the applicability of our solution to subwavelength imaging scenarios with low-loss constitutive parameters that are within reach of current (e.g., gain-assisted) technologies. We have employed this solution to analyze the imaging properties of anisotropic epsilon-near-zero metamaterial slabs varying the design parameters and tilt angle. The figure bottom panel illustrates some representative results, in terms of intensity maps (within and beyond the image plane), for the optical axis angle varying from 0 to 75° with step of 15°.

Our results shed new light on the subwavelength imaging properties of epsilon-near-zero metamaterial slabs and allow tailoring their properties without the need of extensive numerical simulations.

## Relevant papers

- Castaldi, G., Savoia, S., Galdi, V., Alù, A., & Engheta, N. (2012). Analytical study of subwavelength imaging by uniaxial epsilon-near-zero metamaterial slabs.
*Physical Review B**86*(11), 115123.We discuss the imaging properties of uniaxial epsilon-near-zero metamaterial slabs with possibly tilted optical axis, analyzing their subwavelength focusing properties as a function of the design parameters. We derive in closed analytical form the associated two-dimensional Green’s function in terms of special cylindrical functions. For the near-field parameter ranges of interest, we are also able to derive a small-argument approximation in terms of simpler analytical functions. Our results, validated and calibrated against a full-wave reference solution, expand the analytical tools available for computationally efficient and physically incisive modeling and design of metamaterial-based subwavelength imaging systems.

@article{IJ101_PRB_86_115123_2012, title = {Analytical study of subwavelength imaging by uniaxial epsilon-near-zero metamaterial slabs}, author = {Castaldi, Giuseppe and Savoia, Silvio and Galdi, Vincenzo and Al\`u, Andrea and Engheta, Nader}, journal = {Physical Review B}, volume = {86}, issue = {11}, pages = {115123}, numpages = {10}, year = {2012}, month = sep, doi = {10.1103/PhysRevB.86.115123} }