We present a simplification of the well-known Fourier Modal Method (FMM) equipped with Adaptive Spatial Resolution (ASR) concept for gratings with subwavelength heights. We show that in this case it is possible to compute the scattering matrix of the structure by solving only one eigenvalue problem (instead of three) which reduces the most expensive computational part of the FMMASR algorithm. This approach is very efficient and thus suitable for periodic metasurfaces.The Fourier Modal Method1 is one of the most versatile and efficient methods for modeling diffraction from gratings. This is why it is suitable for modeling metasurfaces based on binary gratings. The problem with this approach is that for metallic structures, it is necessary to truncate the Fourier series representing the fields to a high order implying the use of large matrices (especially in the case of crossed gratings). This situation can be improved by using the ASR concept2 to accelerate the convergence rate and thus reduce the size of the matrices in play. Still, this remains demanding from the computational point of view, for certain applications, because the heart of the FMM is based on the solution of an eigenvalue problem. This later has a cost scaling with the third power of the dimension of the matrices involved. This can be a severe limitation and hinder the use of the FMM.On the other hand, an interesting class of metasurfaces is made of gratings with subwavelength thicknesses. In this special case, it is possible (via a Taylor series expansion of the phase propagation matrix) to simplify the FMMASR is such a way that it is no longer needed to solve any eigenvalue problem3. This proves to be very efficient in reducing the computational cost.After recalling the principle of the FMMASR and the details of the corresponding algorithm, I will show how the assumption of subwavelength height leads to a simplified scattering matrix where it is sufficient to solve one eigenvalue problem. Numerical examples will be presented to support this new algorithm.References1. L. Li, “New formulation of the Fourier Modal Method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758-2767, 1997.2. B. Guizal, et al., “Reformulation of the eigenvalue problem in the Fourier modal method with spatial adaptive resolution,” Opt. Lett. 34, 2790-2792, 2009.3. J. Li, et al., “Efficient Rigorous Coupled-Wave Analysis Without Solving Eigenvalues for Analyzing One-Dimensional Ultrathin Periodic Structures,” IEEE Access, Vol. 8, 198131-198138, 2020.

We present a dramatic simplification of the, well-known, Fourier Modal Method1 (FMM) equipped with the Adaptive Spatial Resolution2 (ASR) concept for gratings with subwavelength heights. We show that, in this case, it is possible to compute the scattering matrix of the grating by solving only one eigenvalue problem (instead of three or even more) which reduces the most expensive computational part of the FMM-ASR algorithm. This approach is very efficient and thus suitable for periodic metasurfaces.References1. G. Granet and B. Guizal, Efficient implementation of the coupled wave method for metallic lamellar gratings in TM polarization, J. Opt. Soc. Am. A, Vol 13, No 5, pp 1019-1023, May 1996.2. B. Guizal, et al., “Reformulation of the eigenvalue problem in the Fourier modal method with spatial adaptive resolution,” Opt. Lett. 34, N° 18, 2790-2792, 2009.

In the context of fluctuational electrodynamics, an approach for the computation of the Casimir-Lifshitz force and the near field radiative heat transfer has been established under a unified formalism involving, as a main ingredient, the S matrices of the interacting objects [1]. Thus an important part of the computational effort relies on the efficient evaluation of these matrices. To show how crucial this step can be and how match it deserves attention, we will consider the simple example of graphene strips gratings. The S matrix of such a structure can be obtained through different approaches. The simplest one is the classical Fourier Modal Method (FMM) [2] where the graphene is taken into account through its surface conductivity. This method is very simple but suffers from a serious drawback for it doesn’t handle properly the singularity of the electric field at the level of the strips edges [2]. Another, commonly used, approach is the well-known Rigorous Coupled Wave Analysis (RCWA) in which the graphene strips are given finite thickness and their 2D conductivity is converted into a dielectric permittivity for the resulting rods. This method is more efficient but at a highest computational cost compared to the classical FMM. For this type of structures, we will show that the best approach is the FMM augmented by a set of Local Basis Functions [3] devised to account for the field singularities at the surface of the grating.Bibliography[1] R. Messina and M. Antezza, Scattering-matrix approach to Casimir-Lifshitz force and heat transfer out of thermal equilibrium between arbitrary bodies, Phys. Rev. A 84, 042102 (2011).[2] A. Khavasi, Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons, Opt. Lett. 38, 3009(2013).[3] Y. Jeyar, M. Antezza and B. Guizal, Electromagnetic scattering by a partially graphene-coated dielectric cylinder: Efficient computation and multiple plasmonic resonances, Phys. Rev. E 107, 025306 (2023).

Abstract Open many-body quantum systems can exhibit intriguing nonequilibrium phases of matter, such as time crystals. In these phases, the state of the system spontaneously breaks the time-translation symmetry of the dynamical generator, which typically manifests through persistent oscillations of an order parameter. A paradigmatic model displaying such a symmetry breaking is the boundary time crystal (BTC), which has been extensively analyzed experimentally and theoretically. Despite the broad interest in these nonequilibrium phases, their thermodynamics and their fluctuating behavior remain largely unexplored, in particular for the case of coupled time crystals. In this work, we consider two interacting BTCs and derive a consistent interpretation of their thermodynamic behavior. We fully characterize their average dynamics and the behavior of their quantum fluctuations, which allows us to demonstrate the presence of quantum and classical correlations in both the stationary and the time-crystal phases displayed by the system. We furthermore exploit our theoretical derivation to explore possible applications of time crystals as quantum batteries, demonstrating their ability to efficiently store energy.

Discrete time crystals (DTCs) are a nonequilibrium phase of matter characterized by the breaking of time-translation symmetry in periodically driven quantum systems. In this work, we present a detailed thermodynamic analysis of a DTC in a one-dimensional spin-1/2 chain coupled to a thermal bath. We derive a master equation from the microscopic model, and we explore key thermodynamic quantities, such as work, heat, and entropy production. Our results reveal that the DTC signature inevitably decays in the presence of environmental noise, but we show that a periodic measurement scheme can mitigate the effects of decoherence, stabilizing the subharmonic oscillations of the DTC for extended periods. These findings provide insights into the robustness of time-crystalline phases and potential strategies for protecting them in experimental settings.

We present a simplification of the Fourier Modal Method equipped with Adaptive Spatial Resolution for shallow gratings. We show that it is possible to compute the S-matrix by solving only one eigenvalue problem, which reduces the most expensive computational part of the algorithm.