arXiv subjects

Ehsan Faghihifar

Publications and source records attributed to Ehsan Faghihifar.

4 recordsLinked to original sources

arXiv2026

Bounds on Propagation Constants in Dielectric Gratings

We establish analytical bounds on the normalized longitudinal propagation constants $β=k_z/k_0$ of modes in lossless dielectric diffraction gratings. If $ε_{\max}$ denotes the maximum relative permittivity, we show that regular modes ($β^2\in\mathbb{R}$) satisfy $β^2\leε_{\max}$ despite the lack of a scalar dispersion relation. We further show that the so-called ghost modes ($β^2\notin\mathbb{R}$), admitted by the non-Hermitian nature of the associated eigenproblem, satisfy $|\mathrm{Re}(β)|\le\sqrt{ε_{\max}}/2$. Both bounds are obtained directly from the Fourier-series formulation of the fully coupled vectorial eigenproblem.

physics.optics↗

arXiv2026

Boundedness of Left Half-Plane Eigenvalues for Coefficient-Coupled Sturm--Liouville Problems with Application to Fourier Modal Methods

We study a class of Sturm--Liouville problems of the form \[ -(p\,y')' + q\,y = λ\, w\, y, \] on a finite interval with complex-valued coefficients, where $p$ and $w$ are piecewise smooth, the ratio $w/p$ is real and positive, and $q$ is bounded. We prove that all eigenvalues in the open left half-plane are contained in a bounded set, which implies that only finitely many eigenvalues lie in this region. This stands in contrast to the known unboundedness, for real-valued coefficients, when $p$ or $w$ changes sign independently. A canonical instance of this class, with $w=p$, arises in transverse-magnetic (TM) diffraction by metallic lamellar gratings, a benchmark problem in computational photonics, central to the development of modal methods. In Fourier modal methods, in particular, the emergence of spurious modes with unbounded propagation constants (eigenvalues), rooted in discretization of sign-changing coefficients, leads to notorious convergence difficulties. These modes cannot be excluded \textit{a priori}, since genuine eigenvalues are not constrained by conventional bounds in this regime. Nevertheless, our result shows that the physical eigenvalues remain bounded, providing a rigorous criterion for identifying spurious modes in low-loss metallic gratings.

math.NA↗

arXiv2021

Exclusive robustness of Gegenbauer method to truncated convolution errors

Spectral reconstructions provide rigorous means to remove the Gibbs phenomenon and accelerate the convergence of spectral solutions in non-smooth differential equations. In this paper, we show the concurrent emergence of truncated convolution errors could entirely disrupt the performance of most reconstruction techniques in the vicinity of discontinuities. They arise when the Fourier coefficients of the product of two discontinuous functions, namely $f=gh$, are approximated via truncated convolution of the corresponding Fourier series, i.e. $\hat{f}_k\approx \sum_{|\ell|\leqslant N}{\hat{g}_\ell\hat{h}_{k-\ell}}$. Nonetheless, we numerically illustrate and rigorously prove that the classical Gegenbauer method remains exceptionally robust against this phenomenon, with the reconstruction error still diminishing proportional to $\mathcal{O}(N^{-1})$ for the Fourier order $N$, and exponentially fast regardless of a constant. Finally, as a case study and a problem of interest in grating analysis whence the phenomenon initially was noticed, we demonstrate the emergence and practical resolution of truncated convolution errors in grating modes, which constitute the basis of Fourier modal methods.

physics.comp-ph↗

arXiv2021

Fast estimation of propagation constants in crossed gratings

Fourier-based modal methods are among the most effective numerical tools for the accurate analysis of crossed gratings. However, leading to computationally expensive eigenvalue equations significantly restricts their applicability, particularly when large truncation orders are required. The resultant eigenvalues are the longitudinal propagation constants of the grating and play a key role in applying the boundary conditions, as well as in the convergence and stability analyses. This paper aims to propose simple techniques for the fast estimation of propagation constants in crossed gratings, predominantly with no need to solve an eigenvalue equation. In particular, we show that for regular optical gratings comprised of lossless dielectrics, nearly every propagation constant appears on the main diagonal of the modal matrix.

physics.optics↗