Fourier Modal Method Research Articles

Improved Fourier Modal Method Analyzing 2-D Ultrathin Periodic Structures Without Solving Eigenvalues

Intensive computation for solving eigenvalues of large eigenmatrix restricted the development of Fourier modal method (FMM) in analyzing periodic structures, especially 2-D ultrathin periodic structures. To alleviate the problem of intensive computation in the traditional FMM algorithm when analyzing 2-D ultrathin periodic structures, an efficient FMM algorithm without solving eigenvalues is proposed in this letter. Based on the first-order Taylor Expansion, the exponential term matrix in the traditional algorithm can be replaced by two simple diagonal matrices, which avoids the large dimensional matrix inversion process, and the newly formed execution equation is only related to the eigenmatrix. Two numerical examples show that the results of the traditional FMM algorithm, improved FMM algorithm and CST simulation, are highly consistent. Compared with the traditional algorithm, the improved algorithm can greatly alleviate the computing-intensive problems, and the improved algorithm can significantly reduce the CPU time by more than 90% and reduce computation memory by about 70%.

Resonant mode approximation of the scattering matrix of photonic crystal slabs near several Wood-Rayleigh anomalies

The resonant mode approximation of the scattering matrix is considered for calculating the optical properties of multilayered periodic structures within the formalism of the Fourier-modal method for two diffraction thresholds in close proximity of the spectral-angular range of interest. The developed approximation opens up possibilities for the fast calculation of the scattering matrix of these structures when describing the integral characteristics of spectra and dispersion curves containing high-Q resonances, such as bound states in the continuum.

Dynamical Control of Multilayer Spacetime Structures Using Extended Fourier Modal Method

We introduce two-dimensional space plus time (2D+1) structure and numerically investigate it using a developed multilayer simulation framework, for the first time. The new structure is consisting of crossed grating with time-varying permittivity which is inspired from 1D+1. In this regard, we extend Fourier Modal Method (FMM) in a general approach for spacetime multilayer states. Our proposed framework is fast, robust, and powerful compared to various finite difference methods. We use the scattering matrix technique to develop the proposed spacetime simulation method for multilayer structures using a non-uniform stack of layers. The method is perfectly suitable to investigate the spatiotemporal effects of surfaces/metasurfaces which covers both the transverse electric and magnetic (TE & TM) polarizations. The results show more freedom to control the optical outcomes of the multilayer considering two spatial periodicities in addition to the modulation frequency to reach nonreciprocity as one of the main consequences of the proposed structure. Moreover, we investigate the condition and limitation of breaking the Lorentz rule for spacetime structures. 2D+1 structure is more controllable than the 1D+1 due to its greater ability to adjust spatial manipulation in addition to temporal variations to reach nonreciprocity applications, digital coding, beam steering, etc.

Polarization modulation by surface plasmons in Young's double-slit setup

A simple phenomenological model, justified by rigorous electromagnetic analysis, is used to examine far-field polarization modulation introduced by a metallic screen with two subwavelength slits in uniform, partially polarized illumination. The accuracy of this intuitive model is assessed numerically, by employing the Fourier modal method, as regards the wavelength, the screen metal, and the slit depth, width, and separation. The Stokes parameters and the degree of polarization are evaluated paraxially in the far zone of the slits. We demonstrate that a uniform, partially linearly polarized incident beam can be rendered unpolarized by the nanoslits. Our work provides insight into the pivotal role that surface plasmons have in polarization modulation in photonics.

Towards a metasurface adapted to hyperspectral imaging applications: from subwavelength design to definition of optical properties.

We numerically demonstrate the capability of a single metasurface to simultaneously separate and focus spectral features in accordance with the specifications of a pushbroom hyperspectral imager. This is achieved through the dispersion engineering of a library of two-level TiO2 nano-elements. Sommerfeld integrals are used to confirm our numerical simulations provided by our solver based on Fourier modal method. As a proof of concept, a metasurface with a 175 µm diameter is designed to be compatible with hyperspectral imaging over a spectral range of ± 50 nm around 650 nm with a spectral resolution of 8.5 nm and a field of view of 8° around the normal incidence (angular resolution of 0.2°).

Fourier modal method for moiré lattices

In recent years twisted bi-layers of 2D materials became very popular in the field due to the possibility to totally change their electronic properties by simple rotation. At the same time, in the wide field of photonic crystals, this idea still remains almost untouched, and only some particular problems were considered. One of the reasons is the computational difficulty of the accurate consideration of Moir\'e superlattices that appear due to the superimposition of misaligned lattices. Indeed, the unit cell of the complex lattice is typically much larger than the original crystals and requires much more computational resources for the computations. Here, we propose a careful adaptation of the Fourier modal method in the form of the scattering matrices for the description of twisted 1D gratings' stacks. Our approach allows us to consider sublattices in close vicinity to each other and account for their interaction via the near-field. In the developed numerical scheme, we utilize the fact that each sublattice is only 1D-periodic and therefore simpler than the resulting 2D superlattice, as well as the fact that even a small gap between the lattices filters out high Fourier harmonics due to their evanescent origin. This accelerates the computations from 1 up to 3 and more orders of magnitude for typical structures depending on the number of harmonics. This paves the way for rigorous study of almost any photonic crystals of the proposed geometry and demonstration of specific Moir\'e-associated effects.

Photonic Bound States in the Continuum in Si Structures with the Self‐Assembled Ge Nanoislands

AbstractGermanium self‐assembled nanoislands and quantum dots are very prospective for CMOS‐compatible optoelectronic integrated circuits but their photoluminescence (PL) intensity is still insufficient for many practical applications. Here, it is demonstrated experimentally that the PL of Ge nanoislands in silicon photonic crystal slabs (PCS) with hexagonal lattice can be dramatically enhanced due to the involvement in the emission process of the bounds states in the continuum. These high‐Q photonic resonances allow to achieve PL resonant peaks with the quality factor as high as 2200 and with the peak PL enhancement factor of more than two orders of magnitude. The corresponding integrated PL enhancement is demonstrated to be more than one order of magnitude. This effect is studied theoretically by the Fourier modal method in the scattering matrix form. The symmetry of the quasi‐normal guided modes in the PCS is described in terms of group theory. This work paves the way toward a new class of optoelectronic components compatible with silicon technology.

Semi-Analytical Computation of a Quasi-Static Field Induced by a 3-D Eddy Current Probe in Anisotropic Material With Rough Interfaces

This article presents a numerical model dedicated to the computation of a quasi-static field induced by a 3-D eddy current probe in unidirectional composite material presenting delamination. The specimen is homogeneously anisotropic and the flaw is represented by a local deformation in the planar geometry. The curvilinear coordinate method (CCM) is used to solve Maxwell's equations in covariant form. Based on the Fourier modal method, the semi-analytical approach consists of presenting the electromagnetic field components as eigenmode expansion in a new non-orthogonal coordinate system fitting the geometrical perturbation. Then, interface conditions can be analytically included in order to evaluate the unknown amplitude of eigenmodes. To take into account the contribution of lower interfaces, a recursive and stable scattering matrix algorithm is implemented. The numerical validation shows a good agreement between simulation results and finite element (FE) data.

Simple reformulation of the coordinate transformation method for gratings with a vertical facet or overhanging profile.

We reformulate the coordinate transformation method (C method) for gratings with a vertical facet or overhanging profile (overhanging gratings), in which no tensor concept is involved, only the knowledge of elementary mathematics and Maxwell's equations in the rectangular coordinate system is used, and we provide a detailed recipe for programming. This formulation is easy to understand and implement. It adopts the strategy of a rotating coordinate system from Plumey et al. [J. Opt. Soc. Am. A14, 610 (1997)JOAOD60740-323210.1364/JOSAA.14.000610] and expresses it with the method of changing variables from Li et al. [Appl. Opt.38, 304 (1999)APOPAI0003-693510.1364/AO.38.000304]. We investigate several typical overhanging gratings by the reformulated C method, and we validate and compare the results with the Fourier modal method, which shows that it is superior, especially for metal deep smooth gratings. This reformulation can facilitate the research in light couplers for optical engineers.

Matched coordinates for the analysis of 1D gratings.

The Fourier modal method (FMM) is certainly one of the most popular and general methods for the modeling of diffraction gratings. However, for non-lamellar gratings it is associated with a staircase approximation of the profile, leading to poor convergence rate for metallic gratings in TM polarization. One way to overcome this weakness of the FMM is the use of the fast Fourier factorization (FFF) first derived for the differential method. That approach relies on the definition of normal and tangential vectors to the profile. Instead, we introduce a coordinate system that matches laterally the profile and solve the covariant Maxwell's equations in the new coordinate system, hence the name matched coordinate method (MCM). Comparison of efficiencies computed with MCM with other data from the literature validates the method.

Analysis of the influence of antireflective coatings on the diffraction efficiency of diffractive optical elements.

Diffractive optical elements (DOEs) may be used as a single optical component with dedicated functionality (e.g. focusing), or as one of the elements in complex hybrid optical systems to reduce the number of the elements and improve imaging quality. It is desirable to cover DOE's surface with antireflective coating. In this paper, we show the results of simulations using the Fourier modal method and measure the efficiency of a diffractive lens with antireflective (AR) coating. Results were compared with a previously proposed model of AR covered DOE by Mao et al. [Opt. Express25, 11673 (2017)10.1364/OE.25.011673]. Contrary to the previously published model, we have not observed a shift of the efficiency curve. Our findings prove that AR coatings improve the efficiency of DOEs and it is not necessary to take them into account when calculating optimal profile height.

A new modal method for the grating diffraction

A new method whose basis function expressed by that of a interface is proposed to analysis the grating diffraction, which can solve the physical defects of the Rayleigh method and avoid the coordinate transformation of the Coordinate transformation method. This method is named as the Huygens modal method. At the same time, the Huygens modal method can simplify the eigen matrix equation and the traditional Fourier modal method can be summarized as a special case of the Huygens modal method. In the end, a general form of the H-method of a grating diffraction is derived.

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.

Highly Angular Tolerant Transmission Filters for Narrow-Band Image Sensors

A dielectric transmittance filter composed of subwavelength grating sandwiched between two few-layers distributed Bragg reflectors (DBRs) is proposed with the aim of being compatible with CMOS technology and to be tunable by lithographic means of the grating pattern without the need of thickness changes, in the broad spirit of metamaterials. The DBR mirrors form a Fabry-Perot (FP) cavity whose resonant frequency can be tuned by changing the effective refractive index of the cavity, here, by tailoring the in-plane filling factor of the grating. The structure has been studied and designed by performing numerical simulations using Fourier Modal Method (FMM). This filter proves to have high broad angular tolerance up to ±30˚. This feature is crucial for evaluating the spectral performance of narrow-band filters especially the so-called Ambient light sensors (ALS). By analyzing the transmittance spectral distributions in the band diagram, it is found that the angular tolerance is due to coupling between the FP and the guided mode inside the cavity in analogy to resonances occurring within multimode periodic waveguides in a different context.

Comparison of absorption simulation in semiconductor nanowire and nanocone arrays with the Fourier modal method, the finite element method, and the finite-difference time-domain method

For the design of nanostructured semiconductor solar cells and photodetectors, optics modelling can be a useful tool that reduces the need of time-consuming and costly prototyping. We compare the performance of three of the most popular numerical simulation methods for nanostructure arrays: the Fourier modal method (FMM), the finite element method (FEM) and the finite-difference time-domain (FDTD) method. The difference between the methods in computational time can be three orders of magnitude or more for a given system. The preferential method depends on the geometry of the nanostructures, the accuracy needed from the simulations, whether we are interested in the total, volume-integrated absorption or spatially resolved absorption, and whether we are interested in broadband or narrowband response. Based on our benchmarking results, we provide guidance on how to choose the method.

Normal vector approach to Fourier modal scattering from planar periodic photonic structures

Fourier modal expansion techniques are very well suited for electromagnetic (EM) scattering in planar periodic stratified media. These techniques expand the transverse EM fields in Bloch modes in each strata unit-cell, but do not rigorously enforce EM boundary conditions at interior material interfaces which gives rise to slow convergence of the polarized S-matrix values with increasing Fourier series truncation order. We present a normal vector (NV) Fourier modal scattering approach that uses the gradient of the real permittivity in the unit-cell to generate a local NV resulting in anisotropic eigenmodes of Maxwell's equations that satisfy interior boundary conditions. The convergence of this expansion method is examine numerically for our analytic NV derived from planar primitive geometry elements for various example structures. The Fourier modal NV method is then derived for the planar cavity boundary value problem and the cavity eigenmodes and density of states is determined from the analytic continuation of the secular determinant. A complete scattering method for highly dispersive materials and sub-wavelength structures has been presented that is adaptable to planar EM cavity problems using fluctuation electrodynamics formalism.

Nanoparticle lattices with bases: Fourier modal method and dipole approximation

The utilization of periodic structures such as photonic crystals and metasurfaces is common for light manipulation at nanoscales. One of the most widely used computational approaches to consider them and design effective optical devices is the Fourier modal method (FMM) based on Fourier decomposition of electromagnetic fields. Nevertheless, calculating of periodic structures with small inclusions is often a difficult task, since they induce lots of high-$k_\parallel$ harmonics that should be taken into account. In this paper, we consider small particle lattices with bases and construct their scattering matrices via discrete dipole approximation (DDA). Afterwards, these matrices are implemented in FMM for consideration of complicated layered structures. We show the performance of the proposed hybrid approach by its application to a lattice, which routes left and right circularly polarized incident light to guided modes propagating in opposite directions. We also demonstrate its precision by spectra comparison with finite element method (FEM) calculations. The high speed and precision of this approach enable the calculation of angle-dependent spectra with very high resolution in a reasonable time, which allows resolving narrow lines unobservable by other methods.

Highly improved convergence approach incorporating edge conditions for scattering analysis of graphene gratings

This research developed an effective and efficient approach for improving the slow convergence in the scattering analysis of a one-dimensional graphene grating, made of a periodic array of parallel graphene strips, illuminated by a TM-polarized plane wave. Specifically, the electric fields over the graphene strips and slit regions in a unit cell are individually expressed as an expansion of local basis functions inherently satisfying edge conditions. Interestingly, convergence rate is highly improved compared to the customary and modified Fourier modal method. Additionally, with the aid of local basis functions, the Gibbs phenomenon occurring at both edges of graphene strip can be removed.

Aperiodic differential method associated with FFF: an efficient electromagnetic computational tool for integrated optical waveguides modelization.

A reformulation of the differential theory associated with fast Fourier factorization used for periodic diffractive structures is presented. The incorporation of a complex coordinate transformation in the propagation equations allows the modeling of semi-infinite open problems through an artificially periodized space. Hence, the outgoing wave conditions of an open structure must be satisfied. On the other hand, the excitation technique must be adjusted to adapt with guided structures. These modifications turn the differential theory into an aperiodic tool used with guided optical structure. Our method is verified through numerical results and comparisons with the aperiodic Fourier modal method showing enhanced convergence and accuracy, especially when complex-shaped photonic guided devices are considered.

Motion-free TSOM using a deformable mirror.

Through-focus scanning optical microscopy (TSOM) is a model-based optical metrology method that involves the scanning of a target through the focus of an optical microscope. Unlike a conventional optical microscope that directly extracts the diffraction-limited optical information from a single in-focus image, the TSOM method extracts nanometer scale sensitive information by matching the target TSOM data/image to reference TSOM data/images that are either experimentally or computationally collected. Therefore, the sensitivity and accuracy of the TSOM method strongly depends on the similarities between the conditions in which the target and reference TSOM images are taken or simulated, especially the lateral instability during through-focus scanning. As a remedy to the lateral instability, we proposed the application of adaptive optics to the through-focus scanning operation and initially developed a closed-loop system with a tip/tilt mirror and a Shack-Hartmann sensor, with which we were able to keep the plane position within peak-to-valley (PV) 33 nm. We then further developed a motion-free TSOM tool reducing the instability down to practically zero by the replacement of the tip/tilt mirror with a deformable mirror that performs through-focus scanning by deforming its mirror surface. The motion-free TSOM tool with a × 50 (NA 0.55) objective lens could provide a scanning range of up to ± 25 µm with a minimum step of 25 nm at a maximum update rate of 4 kHz. The tool was demonstrated to have a recognition accuracy of < 4 nm for critical dimension (CD) values in the range of 60 ∼ 120 nm with a reference TSOM image library generated by a Fourier modal method matching various observations conditions.