Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings (original) (raw)
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Journal of the Optical Society of America A, 2008
Modal methods often used to model lamellar gratings that include infinitely or highly conducting metallic parts encounter numerical instabilities in some situations. In this paper, the origin of these numerical instabilities is determined, and then a stable algorithm solving this problem is proposed. In order to complete this analysis, the different geometries that can be handled without numerical instabilities are clearly defined. Numerical tests of the exact modal method implemented with the proposed solution are also presented. A test of convergence shows the efficiency of the method while the comparison with the fictitious sources method shows its accuracy.
Local transformation leading to an efficient Fourier modal method for perfectly conducting gratings
Journal of the Optical Society of America A, 2014
We present an efficient Fourier modal method for wave scattering by perfectly conducting gratings (in the two polarizations). The method uses a geometrical transformation, similar to the one used in the C-method, that transforms the grating surface into a flat surface, thus avoiding to question the Rayleigh hypothesis; also, the transformation only affects a bounded inner region that naturally matches the outer region; this allows applying a simple criterion to select the ingoing and outgoing waves. The method is shown to satisfy reciprocity and energy conservation, and it has an exponential rate of convergence for regular groove shapes. Besides, it is shown that the size of the inner region, where the solution is computed, can be reduced to the groove depth, that is, to the minimal computation domain.
Spurious-free analysis of two-dimensional low-loss metallic gratings
Journal of Optics, 2016
Transmission line formulation is used to analyze two-dimensional low-loss metallic gratings at optical frequencies when plasmonic waves propagate in the structure. This method, like the Fourier modal method, suffers from numerical instabilities when applied to such structures. A systematic approach to avoid these instabilities is presented. These numerical artifacts are attributed to the violation of Li's inverse rule and the appearance of higher-order spurious modes. In this paper, a new approach is proposed to identify and to greatly reduce the effect of these spurious modes based on the accuracy by which these modes are satisfying the conservation of momentum. Furthermore, the proposed scheme conserves power and improves the convergence, i.e., reduces the required truncation order.
On error estimation in the Fourier Modal Method for diffractive gratings
The Fourier Modal Method (FMM, also called the Rigorous Coupled Wave Analysis, RCWA) is a numerical discretization method which is often used to calculate a scattered field from a periodic diffraction grating. For 1D periodic gratings in FMM the electromagnetic field is presented by a truncated Fourier series expansion in the direction of the grating periodicity. The grating's material properties are assumed to be piece-wise constant (called slicing), and next per slice the scattered field is approximated by a truncated Fourier series expansion. The truncation representation of the scattered field and the piece-wise constant approximation of the grating's material properties cause the error in FMM. This paper presents an analytical estimate/bound for the FMM error caused by slicing.
Modal method in deep metal-dielectric gratings: the decisive role of hidden modes
Journal of the Optical Society of America A, 2006
The modal method is well adapted for the modeling of deep-groove, high-contrast gratings of short period, possibly involving metal parts. Yet problems remain in the case of the TM polarization in the presence of metal parts in the corrugations: whereas most of the diffraction features are explained by the interplay of an astonishingly small number of true propagating and low-order evanescent modes, the exact solution of the diffraction problem requires the contribution of two types of evanescent modes that are usually overlooked. We investigate the nature and the role of these modes and show that metal gratings can be treated exactly by the modal method.
Antennas and Propagation Society …, 2005
Due to the filtering and diffractive effects of their periodic structures, dielectric gratings have found different applications in controlling the propagation of electromagnetic waves. Different numerical and semi-analytical techniques, such as method of moments, modal analysis and transverse resonance method, were proposed for solving dielectric grating structures [1]-[3]. The main advantage of semi-analytical techniques compared with the numerical techniques is the less computational effort to obtain the characteristics of the grating structure. Modal analysis showed significant advantage over the transverse resonance approach for modeling one-dimensional grating structure where the complexity of the former does not increase with the number of dielectric slabs present in the unit cell as it is usually happens in the later approach [1]. Another important advantage of modal analysis compared with the transverse resonance method is that it can be extended to two-dimensional structure as it is used for solving a waveguide filled with inhomogeneous materials [4]. The present work extends the modal analysis of one-dimensional dielectric grating to the case of two-dimensional dielectric grating.
A modal model for diffraction gratings
Journal of Modern Optics, 2003
A description of an algorithm for a rather general modal grating calculation is presented. Arbitrary profiles, depth, and permittivity are allowed. Gratings built up from subgratings are allowed, as are coatings on the sidewalls of lines, and arbitrary complex structure. Conical angles and good conductors are supported.
Use of grating theories in integrated optics
Journal of the Optical Society of America A, 2001
], two of the present authors proposed extending the domain of applicability of grating theories to aperiodic structures, especially the diffraction structures that are encountered in integrated optics. This extension was achieved by introduction of virtual periodicity and incorporation of artificial absorbers at the boundaries of the elementary cells of periodic structures. Refinements and extensions of that previous research are presented. Included is a thorough discussion of the effect of the absorber quality on the accuracy of the computational results, with highly accurate computational results being achieved with perfectly matched layer absorbers. The extensions are concerned with the diversity of diffraction waveguide problems to which the method is applied. These problems include two-dimensional classical problems such as those involving Bragg mirrors and grating couplers that may be difficult to model because of the length of the components and three-dimensional problems such as those involving integrated diffraction gratings, photonic crystal waveguides, and waveguide airbridge microcavities. Rigorous coupled-wave analysis (also called the Fourier modal method) is used to support the analysis, but we believe that the approach is applicable to other grating theories. The method is tested both against available numerical data obtained with finite-difference techniques and against experimental data. Excellent agreement is obtained. A comparison in terms of convergence speed with the finite-difference modal method that is widely used in waveguide theory confirms the relevancy of the approach. Consequently, a simple, efficient, and stable method that may also be applied to waveguide and grating diffraction problems is proposed.