Efficient homogenization procedure for the calculation of optical properties of 3D nanostructured composites

Abstract

We present a very efficient recursive method to calculate the effective optical response of metamaterials made up of arbitrarily shaped inclusions arranged in periodic 3D arrays. We apply it to dielectric particles embedded in a metal matrix with a lattice constant much smaller than the wavelength of the incident field, so that we may neglect retardation and factor the geometrical properties from the properties of the materials. If the conducting phase is continuous the low frequency behavior is metallic, and if the conducting paths are thin, the high frequency behavior is dielectric. Thus, extraordinary-transparency bands may develop at intermediate frequencies, whose properties may be tuned by geometrical manipulation.

Magnetic Response of Metamaterials

“…The analogy of the macroscopic wave operator with the Green's function allow us to apply the well‐known Haydock method to recursively generate an orthonormal basis in which our operator can be expressed as a tridiagonal matrix . Choosing an initial state |0⟩ normalized under the metric and corresponding to a plane wave with wavevector k and polarization e , one can generate new states by recursively acting with our “Hamiltonian.” Given the states |0⟩, |1⟩, |2⟩…| n ⟩, we generate the state | n + 1⟩ from where the Haydock coefficients a n and b n are chosen to satisfy …”

“…Bb g can be expressed as a tridiagonal matrix. [30,34,[38][39][40] Choosing an initial state |0i normalized under the metric b g and corresponding to a plane wave with wavevector k and polarization e, one can generate new states by recursively acting with our "Hamiltonian." Given the states |0i, |1i, |2i.…”

Keller’s theorem revisited

“…where we identified the longitudinal projector  L from equation (20) and the macroscopic dielectric function from equation (18). We invert both sides to obtain…”

“…with a moderate damping characterized by the mean collision frequency γ=0.01ω p . We calculate ò M and  M for these systems employing an efficient procedure [19,20,[25][26][27]] based on HRM [28] and implemented in the Photonic computational package [29].…”

“…In section 3 we develop some applications of the theory. Namely, we show that the normal and parallel response functions of a superlattice are determined one from the other; we test the compliance of effective medium theories to Keller's condition; we test the accuracy of an efficient computational scheme based on Haydock's recursive method (HRM) calculation [19][20][21] for the calculation of the macroscopic response of periodic systems; we discuss the relation among the dielectric resonances of a system and that of its reciprocal system and we explore the corresponding microscopic fields [22]; we test the accuracy of numerical computations for ensemble members of disordered systems; and we illustrate how Keller's theorem may be used to increase the accuracy of rough approximate theories. Finally, section 4 is devoted to conclusions.…”