In the following pages we take a fresh look at the ancient Indian Chakravāla or Cyclic algorithm for solving the Brahmagupta-Fermat-Pell quadratic Diophantine equation in integers taking account of recent developments. This is the oldest general algorithm (1150 CE) for solving this equation. The algorithm can be proved directly in its own terms, following a recent work by A. Bauval, to always lead to a periodic solution in a finite number of steps. We review a slightly modified version of this work. It forms the basis of a reinterpretation of this algorithm in the framework of a reduction theory of binary indefinite integer valued quadratic forms on which the modular group SL(2, Z) acts. The reduction condition of this algorithm are restated in terms of the roots of the quadratic form. The SL(2, Z) action on (reduced) roots of this form furnish semi-regular continued fractions which are periodic and which furnish SL(2, Z) automorphisms of the quadratic form. Very much as in the classical theory of Gauss, this gives solutions of the Brahmagupta-Fermat-Pell equation. We give a proof that the solution at the end of the first cycle is fundamental (positive and least). We also give the conversion to regular continued fractions which involve larger periods. A number of worked out examples are given to illustrate the main points and this for the benefit of the uninitiated reader.

We present an accurate and simple semi analytical model for studying the magneto- plasmonic response of a 1D subwavelength graphene strip grating under an external static mag- netic field. In this model, the graphene sheet is considered as an anisotropic layer with atomic thickness. Under these conditions and in contrast to the previous works, an effective medium approach (EMA) is applied to the graphene permittivity tensor and a rigorous phase correction is required to take into account the periodicity effect. The resonance phenomena occurring into the structure are taken into account in the model using the scattering matrix approach. The re- flection and transmission spectra of the structure are then given as the sum of two contributions, the scattering contribution and the single strip contribution. In order to numerically validate and evaluate the proposed model, the results have been compared with those obtained from the PMM [1, 2] method and from methods published in the literature [3, 4]. This simple approach is useful to better understand graphene surface magnetoplasmons GSMPs and may facilitate the design of various tunable devices based on graphene magnetoplasmonics.

In this work, we propose a novel hybrid magneto plasmonic structure based on graphene and doped InSbto enhance the magneto optical Kerr effect at terahertz frequencies.The structure is composed of a 1D periodic dopedInSb inlayed between a metallic backgate and a dielectric/doped graphene sheet .By computing the optical responseof this structure, w e show an enhanced and large Kerr rotation in a wide range of THz frequencies compared to thatinduced by a single graphene sheet and/or a doped magnetized InSb

Design of the metallic contact grid at the front side of thermophotovoltaic cells is critical. Our study, based on a rigorous approach, investigates the real influence of the front metal contact grid. By modelling this grid by a metallic grating, we show that it can significantly affect the electrical power generated by the cell. Quantitative and qualitative analyses indicate behaviors which are quite different from those predicted by previous simplistic approaches.

We present a numerical approach for the solution of EM scattering from a dielectric cylinder partially covered with graphene. It is based on a classical Fourier-Bessel expansion of the fields inside and outside the cylinder to which we apply the ad-hoc boundary conditions in the presence of graphene. Due to the singular nature of the electric field at the ends of the graphene sheet, we introduce auxiliary boundary conditions to better take this reality into account.

We present a simplification of the Fourier Modal Method (FMM) for crossed gratings with subwavelength heights. We show that in this case it is possible to compute the scattering matrix of the structure without solving the eigenvalue problem which is the most expensive computational part of the FMM algorithm. This approach is very efficient and thus suitable for periodic metasurfaces.

The Fourier Modal Method [1] (FMM) is very popular and efficient approach for modelling diffraction from gratings. It can be applied to graphene gratings either by using the Zero Thickness Model (ZTM) i.e. using directly the optical conductivity of graphene in the boundary conditions, or by using the Finite Thickness Model (FTM) where graphene is seen as a slab of atomic thickness and a relative dielectric permittivity deduced from its optical conduc- tivity. For 1D gratings, the FMM based on the ZTM proves to be very efficient for the transverse electric polarization case (electric filed parallel to the direction of invariance of the strips) but suffers from low convergence in the transverse magnetic polarization case (magnetic filed parallel to the direction of invariance of the strips). This is due to a an inappropriate use of the Fourier factorization rules [1]. The FMM based on the FTM, on the other hand, doesn’t experience such a limitation but at the expense of solving an eigenvalue problem inside the grating which has been given a finite thickness. This is very demanding from the computational point of view because solving an eigenvalue problem has a cost scaling with the third power of the dimension of the matrices in play. This increases the computational cost of the approach especially for crossed gratings. Furthermore, a in a recent work [2], the authors have shown the it is possible to avoid solving this eigenvalue problem if the grating has a deep subwavelength thickness. This condition is exactly fulfilled by graphene under the FTM where it is assumed to have an atomic thickness. I will show that using such a simplification lowers the computational cost of the FMM-FTM while giving reliable and accurate results.