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.     

Differential theory of gratings made of anisotropic materials

Arbitrary profiled gratings made with anisotropic materials are discussed; the anisotropic character concerns electric and/or magnetic properties. Our aim is to avoid the use of the staircase approximation of the profile, whose convergence is questionable. A coupled first-order differential-equation set is derived by taking into account Li's remarks about Fourier factorization [J. Opt. Soc. Am. A 13, 1870 (1996)], but the present formulation shows that, in return for a convenient form of the differential system, it is possible to use only the intuitive Laurent rule. Our method, when applied to the simpler case of isotropic gratings, is shown to be consistent with that of previous studies. Moreover, from the numerical point of view, the convergence of our formulation for an anisotropic grating is faster than that of the conventional differential method.     

Improved method for computing of light-matter interaction in multilayer corrugated structures: comment

The method recently proposed by Korovin [J. Opt. Soc. Am. A25, 394 (2008)] for modeling multilayer diffraction gratings is in fact the well-known Rayleigh-Fourier method. Many remarks in the above reference in comparing the proposed method and the C method are biased and inaccurate.     

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.     

Constrained least squares Fourier modal method for computing scattering from metallic binary gratings

In a recent article [J. Opt. Soc. Am. A27, 1694 (2010)], we proposed a rectangular truncation method to mitigate the convergence problems arising from the boundary matching conditions of a binary metallic grating. The proposed method may underestimate the total power in the scattered field for certain grating parameters. In this article, we extend this method to preserve the total power by introducing appropriate constraints and solving the resulting problem as a constrained least squares minimization problem. We provide examples to show that the new method provides a convergent solution for both lossy and lossless binary metallic gratings while preserving the total power.     

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation

A numerical implementation and generalized conservation properties of a formulation for calculating wave propagation through stacked gratings comprising metallic and dielectric cylinders are presented. The basic formulation of the method was given in a companion paper [J. Opt. Soc. Am. A. 17, 2165 (2000)]. Here, details of the numerical implementation of the method are discussed and are illustrated for the ensemble average of a strongly scattering structure with refractive index and radius disorder. Also presented are a comprehensive treatment of energy conservation and generalized phase relations, as well as reciprocity.     

Study of the differential theory of lamellar gratings made of highly conducting materials

Differential theory is said to be difficult to apply to surface-relief gratings made of metals with very high conductivity even though the formulation follows Li's Fourier factorization rules. Recently, Popov et al. [J. Opt. Soc. Am. 21, 199 (2004)] pointed out this difficulty and explained that its origin is related to the inversion of Toeplitz matrices constructed by the permittivity distribution inside the groove region. The current paper provides information about the differential theory for highly conducting gratings and considers the numerical instability problems. A stable calculation for lossless gratings is described, based on the extrapolation technique with the assumption of small losses.     

Grating theory: new equations in Fourier space leading to fast converging results for TM polarization

Using theorems of Fourier factorization, a recent paper [J. Opt. Soc. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are not converging everywhere in TM polarization. They turn out to be converging everywhere only at the limit of infinitely low modulated gratings. We derive new truncated equations and implement them numerically. The computed efficiencies turn out to converge about as fast as in the TE-polarization case with respect to the number of Fourier harmonics used to represent the field. The fast convergence is observed on both metallic and dielectric gratings with sinusoidal, triangular, and lamellar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a new wide range of applications of the method, concerning not only gratings used in TM polarization but also conical diffraction, crossed gratings, three-dimensional problems, nonperiodic objects, rough surfaces, photonic band gaps, nonlinear optics, etc. The formulation also concerns the TE polarization case for a grating ruled on a magnetic material as well as gratings ruled on anisotropic materials. The method developed is applicable to any theory that requires the Fourier analysis of continuous products of discontinuous periodic functions; we propose to call it the fast Fourier factorization method.     

Numerical study on the spectroscopic ellipsometry of lamellar gratings made of lossless dielectric materials

The spectroscopic ellipsometry of lamellar gratings made of lossless dielectric materials is studied numerically by using the rigorous coupled-wave method with the use of Li's Fourier factorization rules [J. Opt. Soc. Am. A 13, 1870 (1996)], which are known to improve the convergence on the analyses of metallic gratings. Numerical results show that the calculation method also provides fast convergence on lossless gratings, and accurate values of the ellipsometric angles are obtained in very short computation times. Moreover, estimation of grating parameters is investigated by using a cost function defined by the average distance on the Poincaré sphere, and it is shown that the computation required for accurate estimation is possible in reasonable computation time.     

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method

We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green's function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.     

Multiscale stochastic approach for phase screens synthesis

Simulating the turbulence effect on ground telescope observations is of fundamental importance for the design and test of suitable control algorithms for adaptive optics systems. In this paper we propose a multiscale approach for efficiently synthesizing turbulent phases at very high resolution. First, the turbulence is simulated at low resolution, taking advantage of a previously developed method for generating phase screens [J. Opt. Soc. Am. A 25, 515 (2008)]. Then, high-resolution phase screens are obtained as the output of a multiscale linear stochastic system. The multiscale approach significantly improves the computational efficiency of turbulence simulation with respect to recently developed methods [Opt. Express 14, 988 (2006)] [J. Opt. Soc. Am. A 25, 515 (2008)] [J. Opt. Soc. Am. A 25, 463 (2008)]. Furthermore, the proposed procedure ensures good accuracy in reproducing the statistical characteristics of the turbulent phase.     

Rigorous and efficient grating-analysis method made easy for optical engineers

The coordinate-transformation-based differential method of Chandezon et al. [J. Opt. (Paris) 11, 235 (1980); J. Opt. Soc. Am. 72, 839 (1982)] (the C method) is one of the simplest and most versatile methods for modeling surface-relief gratings. However, to date it has been used by only a small number of people, probably because, traditionally, elementary tensor theory is used to formulate the method. We reformulate the C method without using any knowledge of tensor, thus, we hope, making the C method more accessible to optical engineers.     

Field singularities at lossless metal-dielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings

I extend a previous work [J. Opt. Soc. Am. A, 738 (2011)] on field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings to the case of arbitrary metallic wedge angles. Simple criteria are given that allow one knowing the lossless permittivities and the arbitrary wedge angles to determine if the electric field at the edges is nonsingular, can be regularly singular, or can be irregularly singular without calculating the singularity exponent. Furthermore, the knowledge of the singularity type enables one to predict immediately if a numerical method that uses Fourier expansions of the transverse electric field components at the edges will converge or not without making any numerical tests. All conclusions of the previous work about the general relationships between field singularities, Fourier representation of singular fields, and convergence of numerical methods for modeling lossless metal-dielectric gratings have been reconfirmed.     

Fast and accurate modeling of waveguide grating couplers

A boundary variation method for the analysis of both infinite periodic and finite aperiodic waveguide grating couplers in two dimensions is introduced. Based on a previously introduced boundary variation method for the analysis of metallic and transmission gratings [J. Opt. Soc. Am. A 10, 2307, 2551 (1993)], a numerical algorithm suitable for waveguide grating couplers is derived. Examples of the analysis of purely periodic grating couplers are given that illustrate the convergence of the scheme. An analysis of the use of the proposed method for focusing waveguide grating couplers is given, and a comparison with a highly accurate spectral collocation method yields excellent agreement and illustrates the attractiveness of the proposed boundary variation method in terms of speed and achievable accuracy.     

Symmetries of cross-polarization diffraction coefficients of gratings

In a recent paper [J. Opt. Soc. Am. A 16, 1108 (1999)] Logofatu et al. demonstrated by experimental and numerical evidence that the 0th-order cross-polarization (s to p and p to s) reflection coefficients of isotropic, symmetrical, surface-relief gratings in conical mount are identical. Here an analytical proof is given to show that the observed identity is merely a manifestation of the electromagnetic reciprocity theorem for the 0th-order diffraction of symmetrical gratings. The above result is further generalized to bianisotropic gratings, to the 0th-order cross-polarization transmission coefficients, and to the mth-order reflection and transmission coefficients when the wave vector of the incident plane wave and the negative of the wave vector of the mth reflected order are symmetrical with respect to the plane perpendicular to the grating grooves.     

Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings

We mathematically prove and numerically demonstrate that the source of the convergence problem of the analytical modal method and the Fourier modal method for modeling some lossless metal-dielectric lamellar gratings in TM polarization recently reported by Gundu and Mafi [J. Opt. Soc. Am. A 27, 1694 (2010)] is the existence of irregular field singularities at the edges of the grating grooves. We show that Fourier series are incapable of representing the transverse electric field components in the vicinity of an edge of irregular field singularity; therefore, any method, not necessarily of modal type, using Fourier series in this way is doomed to fail. A set of precise and simple criteria is given with which, given a lamellar grating, one can predict whether the conventional implementation of a modal method of any kind will converge without running a convergence test.     

Efficiency of the human observer for detecting a Gaussian signal at a known location in non-Gaussian distributed lumpy backgrounds

A previous study [J. Opt. Soc. Am. A22, 3 (2005)] has shown that human efficiency for detecting a Gaussian signal at a known location in non-Gaussian distributed lumpy backgrounds is approximately 4%. This human efficiency is much less than the reported 40% efficiency that has been documented for Gaussian-distributed lumpy backgrounds [J. Opt. Soc. Am. A16, 694 (1999) and J. Opt. Soc. Am. A18, 473 (2001)]. We conducted a psychophysical study with a number of changes, specifically in display-device calibration and data scaling, from the design of the aforementioned study. Human efficiency relative to the ideal observer was found again to be approximately 5%. Our variance analysis indicates that neither scaling nor display made a statistically significant difference in human performance for the task. We conclude that the non-Gaussian distributed lumpy background is a major factor in our low human-efficiency results.     

Differential theory: application to highly conducting gratings

The recently developed fast Fourier factorization method, which has greatly improved the application range of the differential theory of gratings, suffers from numerical instability when applied to metallic gratings with very low losses. This occurs when the real part of the refractive index is small, in particular, smaller than 0.1-0.2, for example, when silver and gold gratings are analyzed in the infrared region. This failure can be attributed to Li's "inverse rule" [L. Li, J. Opt. Soc. Am. A 13, 1870 (1996)] as shown by studying the condition number of matrices that have to be inverted. Two ways of overcoming the difficulty are explored: first, an additional truncation of the matrices containing the coefficients of the differential system, which reduces the numerical problems in some cases, and second, an introduction of lossier material inside the bulk, thus leaving only a thin layer of the highly conducting metal. If the layer is sufficiently thick, this does not change the optical properties of the system but significantly improves the convergence of the differential theory, including the rigorous coupled-wave method, for various types of grating profiles.     

Morphology of the nonspherically decaying radiation generated by a rotating superluminal source: reply to comment

The fact that the formula used by Hannay in the preceding Comment [J. Opt. Soc. Am. A25, 2165 (2008)] is "from a standard text on electrodynamics" neither warrants that it is universally applicable nor that it is unequivocally correct. We have explicitly shown [J. Opt. Soc. Am. A25, 543 (2008)] that, since it does not include the boundary contribution toward the value of the field, the formula in question is not applicable when the source is extended and has a distribution pattern that rotates faster than light in vacuo. The neglected boundary term in the retarded solution to the wave equation governing the electromagnetic field forms the basis of diffraction theory. If this term were identically zero, for the reasons given by Hannay, the diffraction of electromagnetic waves through apertures on a surface enclosing a source would have been impossible.