Aperiodic Fourier Modal Method Research Articles

Domain Decomposition Spectral Method Applied to Modal Method: Direct and Inverse Spectral Transforms

We introduce a Domain Decomposition Spectral Method (DDSM) as a solution for Maxwell's equations in the frequency domain. It will be illustrated in the framework of the Aperiodic Fourier Modal Method (AFMM). This method may be applied to compute the electromagnetic field diffracted by a large-scale surface under any kind of incident excitation. In the proposed approach, a large-size surface is decomposed into square sub-cells, and a projector, linking the set of eigenvectors of the large-scale problem to those of the small-size sub-cells, is defined. This projector allows one to associate univocally the spectrum of any electromagnetic field of a problem stated on the large-size domain with its footprint on the small-scale problem eigenfunctions. This approach is suitable for parallel computing, since the spectrum of the electromagnetic field is computed on each sub-cell independently from the others. In order to demonstrate the method's ability, to simulate both near and far fields of a full three-dimensional (3D) structure, we apply it to design large area diffractive metalenses with a conventional personal computer.

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.

NUMERICAL STUDY OF A PHOTONIC JET WITH APERIODIC FOURIER MODAL METHOD AND EXPERIMENTAL VALIDATION

This paper proposes to use an Aperiodic Fourier Modal Method (A-FMM) to model an outgoing photonic jet from a dielectric loaded waveguide ended by a tip with a specific shape. The proposed method has several advantages. First of all, the method is fast, which allows to manage optimization investigations. Secondly, the study of excitation (and more particularly the impact of plane wave excitation) can be examined precisely. Using our modelling technique, we show, in comparison with an actual optimized elliptical tip, that an optimized rectangular tip improves energy concentration by 8% and reduces the calculation time by a factor of 10. Furthermore, A-FMM allows to show that plane wave excitation modifies the spatial distribution of the jet, especially in the case of T E polarization. This can explain the differences observed, in previous works, where only fundamental mode excitation was used in the modelling. To validate these general results, prototypes have been realized, and measurements in the microwave regime have been compared favorably with simulation results.

Use of aperiodic Fourier modal method for calculating complex-frequency eigenmodes of long-period photonic crystal slabs.

We present an iterative method for calculating complex-frequency eigenmodes of photonic crystal slabs with 1D periodicity based on the aperiodic Fourier modal method. By comparison with the known methods, we show that the proposed method is efficient for studying resonant properties of long-period photonic crystal slabs and diffraction gratings. We demonstrate that the method can be used to calculate the eigenmodes of the structures with periods up to at least 500λ. We discuss different aspects of the mode calculation, including convergence of the method, mode field and dispersion analysis. Potential applications of the presented method include investigation of periodic structures with defects and of quasiperiodic and random structures within the super-cell approach.

Resonant-state expansion of light propagation in nonuniform waveguides

A new rigorous approach for precise and efficient calculation of light propagation along non-uniform waveguides is presented. Resonant states of a uniform waveguide, which satisfy outgoing-wave boundary conditions, form a natural basis for expansion of the local electromagnetic field. Using such an expansion at fixed frequency, we convert the wave equation for light propagation in a non-uniform waveguide into an ordinary second-order matrix differential equation for the expansion coefficients depending on the coordinate along the waveguide. We illustrate the method on several examples of non-uniform planar waveguides and evaluate its efficiency compared to the aperiodic Fourier modal method and the finite element method, showing improvements of one to four orders of magnitude. A similar improvement can be expected also for applications in other fields of physics showing wave phenomena, such as acoustics and quantum mechanics.

Fully vectorial modeling of cylindrical microresonators with aperiodic Fourier modal method.

An approach for fully vectorial modeling of cylindrical resonators based on the aperiodic Fourier modal method is proposed. The approach allows efficient and accurate calculations of the resonant wavelength, the quality factor, and the field of the whispering-gallery modes of rotationally symmetric resonators. The validity and the efficiency of the method are confirmed by the converged numerical results with microdisk and microring resonators as examples in comparison with other approaches.

Quantitative coupled-mode model for a metal-dielectric-metal waveguide with a side-coupled cavity.

The Fabry-Perot model is proposed to analyze the wavelength-selective transmission behaviors of the metal-dielectric-metal waveguide with a rectangular side-coupled cavity. The guided modes propagating in the waveguide and the cavity are extracted by the aperiodic Fourier modal method (a-FMM). The scattering coefficients that appeared in the model are calculated by the a-FMM and the normal-mode theory. The applications of such structure in the wavelength-selective filter and the refractive index sensor are also discussed. Our model is shown to accurately predict the fully vectorial data and thus can provide reliable and quantitative analysis of this kind of device.

Controlling the surface plasmon excitation efficiency using dielectric magneto-optical cavity

The diffraction of a plane wave by a magneto-optical cavity located on a metal interface is investigated theoretically. We show that the excitation of localized eigenmodes of the cavity allows one to efficiently excite the surface plasmon polariton (SPP). We exploit the polarization and symmetry properties of the cavity modes to propose an efficient approach for controlling the SPP intensity through an external magnetic field. The presented theoretical predictions are in good agreement with the rigorous computations based on the generalized Lorentz reciprocity theorem and aperiodic Fourier modal method. The magnetization-induced relative intensity variation of the excited SPP demonstrated in numerical computations varies from a few percent to 100% depending on the polarization of the incident wave. Such large modulation opens new possibilities for efficient SPP modulation with considered magneto-optical intensity effect.

On the complexity of aperiodic Fourier modal methods for finite periodic structures

The Fourier modal method (FMM) is based on Fourier expansions of the electromagnetic field and is inherently built for infinitely periodic structures. When the infinite periodicity assumption is not realistic, the finiteness of the structure has to be incorporated into the model. In this paper we discuss the recent extensions of the FMM for finite periodic structures and analyze their complexity both with respect to the main discretization parameter Nˆ as well as with respect to the number of periods R. We show that among the three FMM-based approaches able to represent finiteness, the aperiodic Fourier modal method with alternative discretization has the lowest computational cost given by either O(Nˆ3log2R) or O(Nˆ2R) depending on the values of Nˆ and R. This result demonstrates that the method is highly suited for rigorous modeling of scattering from large periodic structures. For instance, for Nˆ=100 and R<1000 the complexity of the aperiodic Fourier modal method with alternative discretization is comparable to the complexity of the standard FMM.

Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing

A cascaded-grating coupling scheme for unidirectionally coupling and launching surface plasmons from free-space light with ultra-broadband and wide-angle efficiency is proposed and theoretically investigated, and the potential applications in sensing are discussed. The cascaded sub-gratings of different periods couple the incident light over a relatively wide wavelength and angular range into surface plasmon polaritons (SPPs). After traveling a short distance, the SPPs interact with the sensed medium, and then are coupled back into freely propagating light through the nanoslit in the metal film. By observing the changes of SPP-mediated light coupled out through the slit, broadband sensing is achieved. A quantitative theoretical model is developed for performance prediction and validated with aperiodic Fourier modal method. Proof-of-principle demonstrations show that the spectral full-width-half-maximum reaches 400 nm and meanwhile the angular width is up to 25°, both of which are almost four times compared to the reference single-periodic gratings. A wavelength sensitivity as high as 852 nm RIU−1 (RIU, refractive index units) and a relative intensity change up to ∼5 × 104% RIU−1 are obtained.

Near- to far-field transformation in the aperiodic Fourier modal method

This paper addresses the task of obtaining the far-field spectrum for a finite structure given the near-field calculated by the aperiodic Fourier modal method in contrast-field formulation (AFMM-CFF). The AFMM-CFF efficiently calculates the solution to Maxwell's equations for a finite structure by truncating the computational domain with perfectly matched layers (PMLs). However, this limits the far-field solution to a narrow strip between the PMLs. The Green's function for layered media is used to extend the solution over the whole super- and substrate. The approach is validated by applying it to the problem of scattering from a cylinder for which the analytical solution is available. Moreover, a numerical study is conducted on the accuracy of the approximate far-field computed with the super-cell Fourier modal method by using the AFMM-CFF with near- to far-field transformation as a reference.

Bidirectional Polarization Splitting Couplers Based on Dielectric-Coated Metallic Gratings

A new bidirectional polarization splitting coupler based on a metal thin film perforated with periodic slits and coated with dielectric is proposed and theoretically investigated. The back-side incident light coupled to the metal-dielectric-air waveguide is split and launched into TM plasmonic and TE photonic modes simultaneously. The operation principle is clarified and theoretically illustrated using the fully vectorial aperiodic Fourier modal method. Polarization extinction ratios better than 15 dB and crosstalk less than 15 dB are achieved for both polarizations over C-band ranging from 1530 to 1570 nm.

Compact Bidirectional Polarization Splitting Antenna

A compact bidirectional polarization splitting antenna (BPSA) composed of a patterned metallic structure coated by a thin dielectric film is proposed and theoretically investigated. Using a polarization-selective array of grooves, the backside illuminated light transmitted through the central nanoslit is split and coupled into TM- and TE-polarized modes supported by the metal-dielectric-air (MDA) configuration. The operation principle of the structure is clarified and theoretically illustrated by utilizing the fully vectorial aperiodic Fourier modal method (a-FMM). Numerical simulations show that insertion losses (ILs) less than 4 dB, polarization extinction ratios (PERs) better than 18 dB, and crosstalk (CR) less than -18 dB are achieved for both polarizations in the wavelength range 1510-1570 nm. The structure will find potential applications in highly integrated polarization diversity systems and photonic integrated circuit.

Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method

The aperiodic Fourier modal method in contrast-field formulation is a numerical discretization and solution technique for solving scattering problems in electromagnetics. Typically, spectral discretization is used in the finite periodic direction and spatial discretization in the orthogonal direction. In the light of the fact that the structures of interest often have a large width-to-height ratio and that the two discretization approaches have different computational complexities, we propose exchanging the directions for spatial and spectral discretization. Moreover, if the scatterer has repeating patterns, swapping the discretization directions facilitates the reuse of previous computations. Therefore, the new method is suited for scattering from objects with a finite number of periods, such as gratings, memory arrays, metamaterials, etc. Numerical experiments demonstrate a considerable reduction of the computational costs in terms of time and memory. For a specific test case considered in this paper, the new method (based on alternative discretization) is 40 times faster and requires 100 times less memory than the method based on classical discretization.

Unidirectionally optical coupling from free space into silicon waveguide with wide flat-top angular efficiency

A grating coupling scheme from free-space light into silicon waveguide with a remarkable property of wide flat-top angular efficiency is proposed and theoretically investigated. The coupling structure is composed of two cascaded gratings with a proper distance between their peak angular efficiencies. A quantitative semi-analytical theory based on coupled-mode models is developed for performance prediction and validated with the fully vectorial aperiodic Fourier modal method (a-FMM). With the theory, wide flat-top angular response is achieved and the conditions are pointed out. Proof-of-principle demonstrations show that the -1 dB angular width, a figure of merit to evaluate the flat-top performance, is broadened to almost 3 to 4 times, and meanwhile the -3 dB angular width, i.e., angular-full-width-half-maximum (AFWHM), is widened to nearly more than twice, compared with the reference gratings composed of the same number of periodic defects. We believe this work will find applications in biological or chemical sensing and novel optical devices.

Application of the three-dimensional aperiodic Fourier modal method using arc elements in curvilinear coordinates

This paper deals with a full vectorial generalization of the aperiodic Fourier modal method (AFMM) in cylindrical coordinates. The goal is to predict some key characteristics such as the bending losses of waveguides having an arbitrary distribution of the transverse refractive index. After a description of the method, we compare the results of the cylindrical coordinates AFMM with simulations by the finite-difference time-domain (FDTD) method performed on an S-bend structure made by a 500 nm × 200 nm silicon core (n=3.48) in silica (n=1.44) at a wavelength λ=1550 nm, the bending radius varying from 0.5 up to 2 μm. The FDTD and AFMM results show differences comparable to the variations obtained by changing the parameters of the FDTD simulations.

Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation

The Fourier modal method (FMM) is a method for efficiently solving Maxwell’s equations with periodic boundary conditions. In order to apply the FMM to nonperiodic structures, perfectly matched layers need to be placed at the periodic boundaries, and the Maxwell equations have to be formulated in terms of a contrast (scattered) field. This reformulation modifies the structure of the resulting linear systems and makes the direct application of available stable recursion algorithms impossible. We adapt the well-known S-matrix algorithm for use with the aperiodic FMM in contrast-field formulation. To this end, stable recursive relations are derived for linear systems with nonhomogeneous structure. The stability of the algorithm is confirmed by numerical results.

Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures

This paper extends the area of application of the Fourier modal method (FMM) from periodic structures to aperiodic ones, in particular for plane-wave illumination at arbitrary angles. This is achieved by placing perfectly matched layers at the lateral sides of the computational domain and reformulating the governing equations in terms of a contrast field that does not contain the incoming field. As a result of the reformulation, the homogeneous system of second-order ordinary differential equations from the original FMM becomes non-homogeneous. Its solution is derived analytically and used in the established FMM framework. The technique is demonstrated on a simple problem of planar scattering of TE-polarized light by a single rectangular line.

Design of a compact static Fourier transform spectrometer in integrated optics based on a leaky loop structure

A compact static Fourier transform spectrometer for integrated optics is proposed. It is based on a plane leaky loop structure combined with a plane waveguide. The interference pattern produced in the loop structure leaks outside of it and is guided in the plane waveguide to the photodetector array. This configuration allows one to control the shape of the field pattern at the end of the plane waveguide. A large fringe pattern with a high interference fringe contrast is obtained. A two-dimensional model based on an aperiodic Fourier modal method is used to modelize the coupling between the bent and the plane waveguides, completed with the Helmholtz-Kirchhoff propagation. This concept gives access to plan and compact spectrometers requiring only a single low-cost realization process step. The simulation has been done to realize a spectrometer in glass integrated optics (Deltalambda=6.1 nm at 1500 nm).

Efficient photonic mirrors for semiconductor nanowires

Using a fully vectorial frequency-domain aperiodic Fourier modal method, we study nanowire metallic mirrors and their photonic performance. We show that the performance of standard quarter-wave Bragg mirrors at subwavelength diameters is surprisingly poor, while engineered metallic mirrors that incorporate a thin dielectric adlayer may offer reflectance larger than 90% even for diameters as small as lambda/5.