Keller’s theorem revisited

Abstract

Keller's theorem relates the components of the macroscopic dielectric response of a binary twodimensional composite system with those of the reciprocal system obtained by interchanging its components. We present a derivation of the theorem that, unlike previous ones, does not employ the common assumption that the response function relates an irrotational to a solenoidal field and that is valid for dispersive and dissipative anisotropic systems. We show that the usual statement of Keller's theorem in terms of the conductivity is strictly valid only at zero frequency and we obtain a new generalization for finite frequencies. We develop applications of the theorem to the study of the optical properties of systems such as superlattices, 2D isotropic and anisotropic metamaterials and random media, to test the accuracy of theories and computational schemes, and to increase the accuracy of approximate calculations.

Recursive Calculation of the Optical Response of Multicomponent Metamaterials

“…This expression was later proved by Milton . However, as argued in the study by Ortiz and Mochán, we expect that the correct expression for finite frequencies is that written in terms of the dielectric response, i.e.,…”

“…To further test our result (Equation (30)), we will show below that they satisfy a generalization of Keller's theorem [47] for multicomponent metamaterials, which we prove in a simple (limited) form below. Consider a 2D metamaterial with three or more components A, B, C, … , each characterized by a dielectric function ϵ A , ϵ B , ϵ C , … Then, we write its dielectric function as…”

“…To further test our result (Equation ), we will show below that they satisfy a generalization of Keller's theorem for multicomponent metamaterials, which we prove in a simple (limited) form below. Consider a 2D metamaterial with three or more components A , B , C , … , each characterized by a dielectric function , , , … Then, we write its dielectric function as where we introduced characteristic functions , , … , that take the value 1 when r lies within the corresponding region A , B , C … and 0 otherwise.…”

“…This expression was later proved by Milton. [50] However, as argued in the study by Ortiz and Mochán, [47] we expect that the correct expression for finite frequencies is that written in terms of the dielectric response, i.e.,…”

“…This is Keller's theorem for multicomponent metamaterials in 2D. A somewhat more general proof of this theorem may be obtained [47] by realizing that different projections of the macroscopic permittivity may be obtained by either applying a homogenization procedure to 1=ϵ or directly to ϵ. [51,61] We remark that Keller's theorem for two components has been useful in the past to test the convergence of homogenization theories for 2D systems.…”