Fast integral methods for integrated optical systems simulations: a review

Abstract

Boundary integral equation methods (BIM) or simply integral methods (IM) in the context of optical design and simulation are rigorous electromagnetic methods solving Helmholtz or Maxwell equations on the boundary (surface or interface of the structures between two materials) for scattering or/and diffraction purposes. This work is mainly restricted to integral methods for diffracting structures such as gratings, kinoforms, diffractive optical elements (DOEs), micro Fresnel lenses, computer generated holograms (CGHs), holographic or digital phase holograms, periodic lithographic structures, and the like. In most cases all of the mentioned structures have dimensions of thousands of wavelengths in diameter. Therefore, the basic methods necessary for the numerical treatment are locally applied electromagnetic grating diffraction algorithms. Interestingly, integral methods belong to the first electromagnetic methods investigated for grating diffraction. The development started in the mid 1960ies for gratings with infinite conductivity and it was mainly due to the good convergence of the integral methods especially for TM polarization. The first integral equation methods (IEM) for finite conductivity were the methods by D. Maystre at Fresnel Institute in Marseille: in 1972/74 for dielectric, and metallic gratings, and later for multiprofile, and other types of gratings and for photonic crystals. Other methods such as differential and modal methods suffered from unstable behaviour and slow convergence compared to BIMs for metallic gratings in TM polarization from the beginning to the mid 1990ies. The first BIM for gratings using a parametrization of the profile was developed at Karl-Weierstrass Institute in Berlin under a contract with Carl Zeiss Jena works in 1984–1986 by A. Pomp, J. Creutziger, and the author. Due to the parametrization, this method was able to deal with any kind of surface grating from the beginning: whether profiles with edges, overhanging non-functional profiles, very deep ones, very large ones compared to wavelength, or simple smooth profiles. This integral method with either trigonometric or spline collocation, iterative solver with <i>O(N²) </i>complexity, named IESMP, was significantly improved by an efficient mesh refinement, matrix preconditioning, Ewald summation method, and an exponentially convergent quadrature in 2006 by G. Schmidt and A. Rathsfeld from Weierstrass-Institute (WIAS) Berlin. The so-called modified integral method (MIM) is a modification of the IEM of D. Maystre and has been introduced by L. Goray in 1995. It has been improved for weak convergence problems in 2001 and it was the only commercial available integral method for a long time, known as PCGRATE. All referenced integral methods so far are for in-plane diffraction only, no conical diffraction was possible. The first integral method for gratings in conical mounting was developed and proven under very weak conditions by G. Schmidt (WIAS) in 2010. It works for separated interfaces and for inclusions as well as for interpenetrating interfaces and for a large number of thin and thick layers in the same stable way. This very fast method has then been implemented for parallel processing under Unix and Windows operating systems. This work gives an overview over the most important BIMs for grating diffraction. It starts by presenting the historical evolution of the methods, highlights their advantages and differences, and gives insight into new approaches and their achievements. It addresses future open challenges at the end.