Fourier modal method for relief gratings with oblique boundary conditions

Coordinate transformation method for modeling three-dimensional photonic structures with curved boundaries

The coordinate transformation method (C method) is a powerful tool for modeling photonic structures with curved boundaries of discontinuities. As a modal method upon the Fourier basis, the C method has superior computational efficiency and rich physical intuitiveness compared to other full-wave numerical methods. But presently the C method is limited to two-dimensional (2D) structures if the boundaries between adjacent z-invariant layers are of generally different profiles [with (x,y,z) being the Cartesian coordinate]. Here we report a nontrivial extension of the C method to the general case of three-dimensional (3D) structures with curved boundaries of different profiles between adjacent layers. This extension drastically enlarges the applicability of the C method to the various interesting structures in nanophotonics and plasmonics. The extended 3D-C method adopts a hybrid coordinate transformation which includes not only the z-direction coordinate transformation in the classical C method but also the x- and y-direction matched coordinates adopted in the Fourier modal method (FMM), so as to exactly model the curved boundaries in all the three directions. The method also incorporates the perfectly matched layers (PMLs) for aperiodic structures and the adaptive spatial resolution (ASR) for enhancing the convergence. A modified numerically-stable scattering-matrix algorithm is proposed for solving the equations of boundary condition between adjacent z-invariant layers, which are derived via a transformation of the full 3D covariant field-components between the different curvilinear coordinate systems defined by the different-profile top and bottom boundaries of each layer. The validity of the extended 3D-C method is tested with several numerical examples.

Direct S-matrix calculation for diffractive structures and metasurfaces

The paper presents a derivation of analytical components of S matrices for arbitrary planar diffractive structures and metasurfaces in the Fourier domain. The attained general formulas for S-matrix components can be applied within both formulations in the Cartesian and curvilinear metric. A numerical method based on these results can benefit from all previous improvements of the Fourier domain methods. In addition, we provide expressions for S-matrix calculation in the case of periodically corrugated layers of two-dimensional materials, which are valid for arbitrary corrugation depth-to-period ratios. As an example, the derived equations are used to simulate resonant grating excitation of graphene plasmons and the impact of a silica interlayer on corresponding reflection curves.

High order integral equation method for diffraction gratings

Conventional integral equation methods for diffraction gratings require lattice sum techniques to evaluate quasi-periodic Green's functions. The boundary integral equation Neumann-to-Dirichlet map (BIE-NtD) method in Wu and Lu [J. Opt. Soc. Am. A 26, 2444 (2009)], [J. Opt. Soc. Am. A 28, 1191 (2011)] is a recently developed integral equation method that avoids the quasi-periodic Green's functions and is relatively easy to implement. In this paper, we present a number of improvements for this method, including a revised formulation that is more stable numerically, and more accurate methods for computing tangential derivatives along material interfaces and for matching boundary conditions with the homogeneous top and bottom regions. Numerical examples indicate that the improved BIE-NtD map method achieves a high order of accuracy for in-plane and conical diffractions of dielectric gratings.

Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.

Fourier-matching pseudospectral modal method for diffraction gratings

A Fourier-matching pseudospectral modal method [PSMM(f)] is developed for analyzing lamellar diffraction gratings or grating stacks. A Chebyshev pseudospectral method is first used to accurately calculate the eigenmodes of the grating layers, and then the Fourier coefficients are matched at the interfaces between the layers. Compared with an existing pseudospectral modal method based on point matching, the PSMM(f) is more robust and accurate. The method performs better than the standard Fourier modal method for gratings involving metals.