B-spline Modal Method Research Articles

Modal method for second harmonic generation in nanostructures

A modal method for the study of second harmonic (SH) generation in nanostructures is depicted both for plane waves and focused beam illumination. It is based on the B-spline modal method, and relies on the undepleted pump approximation. The nonlinear polarization induced by an incident plane wave or a focused beam generates a source term at the doubled frequency. The latter is divided into a finite number of sub-sources, and the SH field is subsequently computed by integration of these sub-source contributions.

Electromagnetic modelization of spherical focusing on a one-dimensional grating thanks to a conical B-spline modal method.

Focusing light onto nanostructures thanks to spherical lenses is a first step in enhancing the field and is widely used in applications. Nonetheless, the electromagnetic response of such nanostructures, which have subwavelength patterns, to a focused beam cannot be described by the simple ray tracing formalism. Here, we present a method for computing the response to a focused beam, based on the B-spline modal method adapted to nanostructures in conical mounting. The eigenmodes are computed in each layer for both polarizations and are then combined for the computation of scattering matrices. The simulation of a Gaussian focused beam is obtained thanks to a truncated decomposition into plane waves computed on a single period, which limits the computation burden.

Efficient implementation of B-spline modal method for lamellar gratings

The B-spline modal method (BMM) as applied to lamellar gratings analysis is revisited, and a new implementation is presented. The main difference with our previous work is that we now take into account discontinuities by putting a spline with a degenerate knot on them. Our new approach is compared with other implementations of the BMM and is shown to be superior in terms of numerical convergence.

B-spline modal method: A polynomial approach compared to the Fourier modal method

A detailed analysis of the B-spline Modal Method (BMM) for one- and two-dimensional diffraction gratings and a comparison to the Fourier Modal Method (FMM) is presented. Owing to its intrinsic capability to accurately resolve discontinuities, BMM avoids the notorious problems of FMM that are associated with the Gibbs phenomenon. As a result, BMM facilitates significantly more efficient eigenmode computations. With regard to BMM-based transmission and reflection computations, it is demonstrated that a novel Galerkin approach (in conjunction with a scattering-matrix algorithm) allows for an improved field matching between different layers. This approach is superior relative to the traditional point-wise field matching. Moreover, only this novel Galerkin approach allows for an competitive extension of BMM to the case of two-dimensional diffraction gratings. These improvements will be very useful for high-accuracy grating computations in general and for the analysis of associated electromagnetic field profiles in particular.

Reduced scattering-matrix algorithm for high-density plasmonic structures

We describe a method to compute S-matrix interface terms using a selection of eigenmodes. When solving the modal equation, the computation of left and right eigenvectors leads to rectangular eigenmodes matrices. Expressions of S-matrix interface terms are then expressed so as to allow for a significant reduction of the computation cost. The reduction is even further decreased in the case of the B-spline modal method, which deals with sparse matrices. Its convergence is illustrated on a high-density plasmonic structure and compared to a full modal method.

Fast modal method for subwavelength gratings based on B-spline formulation

We present a B-spline modal method for analyzing a stack of complex structured layers. Thanks to a B-spline approximation of the field, we solve the Maxwell equations. Diffraction calculation is based on the scattering matrices algorithm. We prove a good convergence of this method. Moreover, B-spline approximation results in very sparse matrices, which are used to hasten the computation of eigenmodes. A method for cleaning the inverted sparse matrix is also presented.