Precise-Integration Time-Domain Formulation for Optical Periodic Media

A numerical formulation based on the precise-integration time-domain (PITD) method for simulating periodic media is extended for overcoming the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the periodic boundary conditions are implemented, permitting the simulation of a wide range of periodic optical media, i.e., gratings, or thin-film filters. Furthermore, the complete tensorial derivation for the permittivity also allows simulating anisotropic periodic media. Numerical results demonstrate that PITD is reliable and even considering anisotropic media can be competitive compared to traditional FDTD solutions. Furthermore, the maximum allowable time-step size has been demonstrated to be much larger than that of the CFL limit of the FDTD method, being a valuable tool in cases in which the steady-state requires a large number of time-steps.

Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches

We describe the emergence of topological singularities in periodic media within the Ginzburg–Landau model and the core-radius approach. The energy functionals of both models are denoted by $$E_{\varepsilon ,\delta }$$ E ε , δ , where $$\varepsilon $$ ε represent the coherence length (in the Ginzburg–Landau model) or the core-radius size (in the core-radius approach) and $$\delta $$ δ denotes the periodicity scale. We carry out the $$\Gamma $$ Γ -convergence analysis of $$E_{\varepsilon ,\delta }$$ E ε , δ as $$\varepsilon \rightarrow 0$$ ε → 0 and $$\delta =\delta _\varepsilon \rightarrow 0$$ δ = δ ε → 0 in the $$|\log \varepsilon |$$ | log ε | scaling regime, showing that the $$\Gamma $$ Γ -limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter $$\begin{aligned} \lambda =\min \Bigl \{1,\lim _{\varepsilon \rightarrow 0} {|\log \delta _\varepsilon |\over |\log \varepsilon |}\Bigr \} \end{aligned}$$ λ = min { 1 , lim ε → 0 | log δ ε | | log ε | } (upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than $$\varepsilon ^\lambda $$ ε λ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than $$\varepsilon ^\lambda $$ ε λ the concentration process takes place “after” homogenization.

The geometric average of curl-free fields in periodic geometries

In periodic homogenization problems, one considers a sequence ( u η ) η {(u^{\eta})_{\eta}} of solutions to periodic problems and derives a homogenized equation for an effective quantity u ^ {\hat{u}} . In many applications, u ^ {\hat{u}} is the weak limit of ( u η ) η {(u^{\eta})_{\eta}} , but in some applications u ^ {\hat{u}} must be defined differently. In the homogenization of Maxwell’s equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced in [G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris 347 2009, 9–10, 571–576]; it associates to a curl-free field Y ∖ Σ ¯ → ℝ 3 {Y\setminus\overline{\Sigma}\to\mathbb{R}^{3}} , where Y is the periodicity cell and Σ an inclusion, a vector in ℝ 3 {\mathbb{R}^{3}} . In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces.