Stable GSTC Formulation for Maxwell’s Equations

Abstract

We revisit the classical zero-thickness Generalized Sheet Transition Conditions (GSTCs) which are a key tool for efficiently designing metafilms able to control the flow of light in a desired way. It is shown that it is more convenient to use an enlarged formulation of the GSTC in which the original metafilm is replaced by GSTCs that exclude the layer from the physical or computational domain. These new "layer" transition conditions have the same form as their "sheet" analogues hence they do not necessitate additional complications in their use; their advantage is that they provide a well-posed problem hence guaranty the stability of numerical schemes in the timedomain. These assessments are demonstrated for an all-dielectric structure; the effective susceptibility tensors are derived thanks to asymptotic analysis combined with homogenization technique and bounds for the susceptibilities entering the balance of energy are provided. While negative constant susceptibilities appear in the classical zero-thickness GSTCs, their values in the enlarged formulation are always positive which ensure the stability of the effective problem. Validation of the effective model is provided by means of comparison with direct numerics in two and three dimensions.

Susceptibility synthesis of arbitrary shaped metasurfaces

“…However, it is important to keep in mind that even if the presented numerical results may seem promising for future applications of conformal metasurfaces, the susceptibilities obtained through the inversion procedure are not directly linked to physical materials or structures. To alleviate this issue one could for instance consider using susceptibilities coming from the homogenization theory such as in [30,31]. Other synthesis method could also be considered (and adapted to conformal geometries) such as the one producing physically relevant real-valued susceptibilities in [32].…”

“…(C14) These constitutive relations may also be seen as D 0 = ε 0 E 0 + P 0 and B 0 = µ 0 H 0 + M 0 where the polarization P 0 and magnetization M 0 are given by P 0 = ε 0 χ ee,0 {E} and M 0 = µ 0 χ mm,0 {H} (note that in [35,Section 4.3.1] the author directly assume that the polarization and magnetization are given by such relations while we tried here to justify such expressions through equations (C11)-(C12), although a proper justification of these relations is only found through the homogenization theory as it was done in [30,31]). Now, by taking the scalar and cross product of the constitutive relations (C14) with the normal vector n and using eq.…”

Homogenized transition conditions for plasmonic metasurfaces

“…where d is the array spacing and ϕ = S/d 2 is the particle cross section S normalized with d 2 (for several particles in the unit cell, S is the sum of their sections). The effective surface susceptibilities ( χ x , χ y ) are the diagonal terms of the electric susceptibility tensor entering the GSTCs recently derived for Maxwell's equations [24]. In (3), the susceptibilities are defined by introducing a discrete set of real, negative eigenvalues ε n of the periodic PEP and the weights ( χ x n , χ y n ) that are specified in Sec.…”

“…Starting with the pioneering work of Idemen [18], these generalized sheet transition conditions (GSTCs) have been popularized in their most general form. We mention the few works in which the homogenization process leading to GSTCs is explicit [19][20][21][22][23][24].…”

“…They are found as static solutions of source-free electrostatic equations for given real, negative permittivity values [1]. From a homogenization perspective, this means that they will be captured within the framework used for nonresonant particles, as recently proposed for the full Maxwell's equations [24]. This could be sufficient; however, an annoying drawback of the resulting formulation is that the effective susceptibilities depend on the permittivity of the metal.…”