Split-field Finite-difference Time-domain Research Articles

The Unconditionally Stable Associated Hermite FDTD Method for 3-D Periodic Structure Analysis

In this letter, the unconditionally stable associated hermite (AH) finite-difference time-domain (FDTD) method is extended to analyze three-dimensional (3-D) periodic structure. By combining the 3-D AH FDTD method and split-field technique, a 13-points coefficient matrix equation to eliminate phase shift is obtained with the corrected nonzero elements. To deal with the overflow at the periodic boundaries, the coefficient matrix is updated by the Floquet theory as long as the treatment of the angular points at the periodic boundary. The updating equation of the source term is derived by the total-field/scattered-field formula. The proposed method is verified by calculating the reflection coefficients of the infinite dielectric slab and the Jerusalem Cross FSS structure, respectively. The former shows a higher accuracy compared with the analytical solution with the average relative difference below <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$-$</tex-math></inline-formula> 60 dB, and reflects less sensitive to the angle of incidence. And the later represents 7 times efficiency than that of the split-field FDTD method while keeping a sound accuracy.

Accurate, Efficient and Rigorous Numerical Analysis of 3D H-PDLC Gratings.

This work presents recent results derived from the rigorous modelling of holographic polymer-dispersed liquid crystal (H-PDLC) gratings. More precisely, the diffractive properties of transmission gratings are the focus of this research. This work extends previous analysis performed by the authors but includes new features and approaches. More precisely, full 3D numerical modelling was carried out in all analyses. Each H-PDLC sample was generated randomly by a set of ellipsoid geometry-based LC droplets. The liquid crystal (LC) director inside each droplet was computed by the minimisation of the Frank elastic free energy as a function of the applied electric field. The analysis carried out considered the effects of Frank elastic constants , and ; the anchoring strength ; and even the saddle-splay constant . The external electric field induced an orientation of the LC director, modifying the optical anisotropy of the optical media. This effect was analysed using the 3D split-field finite-difference time-domain (SF-FDTD) method. In order to reduce the computational costs due to a full 3D tensorial analysis, a highly optimised method for high-performance computing solutions (HPC) was developed. The influences of the anchoring and voltage on the diffraction efficiencies were investigated, showing the potential of this approach.

Simplified physical modeling of parallel-aligned liquid crystal devices at highly non-linear tilt angle profiles

In recent works, we demonstrated the accuracy and physical relevance of a highly simplified reverse-engineering analytical model for a parallel-aligned liquid crystal on silicon devices (PA-LCoS). Both experimental measurements and computational simulations applying the rigorous split-field finite difference time domain (SF-FDTD) technique led to this conclusion in the low applied voltages range. In this paper, we develop a complete rigorous validation covering the full range of possible applied voltages, including highly non-linear liquid crystal (LC) tilt angle profiles. We demonstrate the applicability of the model for spectral and angular retardation calculations, of interest in spatial light modulation applications. We also show that our analytical model enables the calculation of the retardance for novel PA-LC devices as a function of the LC compound and cell gap, becoming an appealing alternative to the usual numerical approaches for PA-LC devices design.

Numerical Analysis of H-PDLC Using the Split-Field Finite-Difference Time-Domain Method.

In this work, an accurate numerical modeling of the diffraction properties of transmission holographic polymer dispersed liquid crystal (H-PDLC) gratings is presented. The method considers ellipsoid geometry-based liquid crystal (LC) droplets with random properties regarding size and location across the H-PLDC layer and also the non-homogeneous orientation of the LC director within the droplet. The direction of the LC director inside the droplets can be varied to reproduce the effects of the external voltage applied in H-PDLC-based gratings. From the LC director distribution in the droplet, the permittivity tensor is defined, which establishes the optical anisotropy of the media, and it is used for numerically solving the light propagation through the system. In this work, the split-field finite-difference time-domain method (SF-FDTD) is applied. This method is suited for accurately analyzing periodic media, and it considers spatial and time discretisation of Maxwell’s equations. The scheme proposed here is used to investigate the influence on the diffraction properties of H-PDLC as a function of the droplets size and the bulk fraction of LC dispersed material.

Thin-slot formalism for the split-field FDTD analysis of periodic structure

An expression of the thin-slot formalism is presented to alleviate the gridding of the split-field finite-difference time-domain (FDTD) solution for periodic structure. The varying auxiliary-field ( , ) and split-field ( , ) distributions near the slots are analytically derived from the varying field ( , ). The update equations for the split-field FDTD are obtained by incorporating those varying field distributions into the split-field equations in integral form. A frequency selective surface (FSS) structure is applied to verify the proposed method. The results indicate that the computational efficiency is improved.

Split-field finite-difference time-domain method for second-harmonic generation in two-dimensionally periodic structures

The split-field finite-difference time-domain method is extended to second-harmonic generation in two-dimensionally periodic structures. Making use of the full coefficient-tensor formalism, a coupled nonlinear system of equations, which must be solved at each update of the electromagnetic field, is developed. The accuracy of the method is verified by comparing the results to well-known one-dimensional problems. The results for L-shaped arrays are compared with results obtained with the Fourier modal method.

Study of Periodic Structures at Oblique Incidence by Radial Point Interpolation Meshless Method

This paper presents a novel split-field radial point interpolation meshless (RPIM) scheme for the study of periodic structures under oblique incidence condition. The time delay in the transverse plane is eliminated with the introduction of a set of auxiliary variables. With the application of split-field technique to the standard RPIM method, a leap-frog formulation is derived. Numerical experiment is implemented to demonstrate the effectiveness of the code and the obtained result is compared with result of split-field finite-difference time domain method.

A 2D FDTD Algorithm for Whole-Hemisphere Incidence on Periodic Media

We present a modified version of the 2D split-field finite difference time domain (FDTD) method which enables efficient simulation of periodic structures. Our algorithm allows for broadband, whole-hemisphere oblique incidence sources with structures that are inhomogeneous in permittivity, conductivity, and permeability. The structures considered are of finite extent in one dimension, periodic in a second orthogonal dimension, and uniform (or homogeneous) in a third dimension. With prior FDTD methods, this required a full 3D simulation space. In this work, we reduce the modeling space from a 3D grid to a 2D grid, while still allowing incident waves to be oblique with respect to that dimension. We derive this new algorithm beginning with a complete source definition that allows for arbitrary polarization and incidence direction. The key update equations are found, and we also give a method for finding the full vectorial far-field orders from the simulation output. We validate the method by simulating an etalon, a Bragg grating, and a photonic band gap structure.

Acceleration of split-field finite difference time-domain method for anisotropic media by means of graphics processing unit computing

The implementation of split-field finite difference time domain (SF-FDTD) applied to light-wave propagation through periodic media with arbitrary anisotropy method in graphics processing units (GPUs) is described. The SF-FDTD technique and the periodic boundary condition allow the consideration of a single period of the structure reducing the simulation grid. Nevertheless, the analysis of the anisotropic media implies considering all the electromagnetic field components and the use of complex notation. These aspects reduce the computational efficiency of the numerical method compared with the isotropic and nonperiodic implementation. Specifically, the implementation of the SF-FDTD in the Kepler family of GPUs of NVIDIA is presented. An analysis of the performance of this implementation is done, and several applications have been considered in order to estimate the possibilities provided by both the formalism and the implementation into GPU: binary phase gratings and twisted-nematic liquid crystal cells. Regarding the analysis of binary phase gratings, the validity of the scalar diffraction theory is evaluated by the comparison of the diffraction efficiencies predicted by SF-FDTD. The analysis for the second order of diffraction is extended, which is considered as a reference for the transmittance obtained by the SF-FDTD scheme for periodic media.

Tensorial split-field finite-difference time-domain approach for second- and third-order nonlinear materials

The split-field finite-difference time-domain (SF-FDTD) method for one-dimensionally periodic structures is extended to include the coefficient-tensor description of second- and third-order nonlinear-optical media. A set of nonlinear equations related to the split-field values of the electric field is established. An iterative fixed-point approach for solving the coupled nonlinear system of equations needed to update the electric field components in the SF-FDTD is then developed. The third-order nonlinear susceptibility dispersion is also considered by means of the Raman effect and its implementation in the SF-FDTD scheme. Different scenarios are considered in order to verify the reliability of the method for simulating second- and third-order nonlinear-optical media. First, second-harmonic generation and its efficiency are investigated in a homogeneous layer with and without the quasi-phase-matching technique. Second, the nonlinear dispersion is analyzed by means of the generation of solitons in Kerr media due to the Raman effect. Last, a set of binary phase gratings with nonlinear pillars is considered under oblique incidence. Here the nonlinear refractive index is generated by different physical mechanisms modeled with the nonscalar third-order susceptibility.

Enhanced Optical Transmission Through Metallic Holes Array: Role of TE Polarization in SPP Excitation

Optical transmission through double-layer metallic subwavelength holes array is studied under oblique incidence by split-field finite-difference time-domain method. Both TM and TE polarizations are investigated. It is proved that the transmission peaks can also be observed for TE polarization due to the excitation of surface plasmon polaritons (SPP) through diffraction orders. By changing the incident angle, these transmission peaks follow the SPP wavelength shift. The field profiles, even for the field components not present in the incident field, clearly show the SPP excitation. The mechanism of enhanced transmission will be fully discussed.

Study of periodic dispersive structures using splitfield FDTD method

The dispersive periodic structures are simulated by the split-field finite difference time domain (FDTD) method. According to the Floquet theorem, a set of auxiliary elements are introduced into the FDTD iteration to deal with electromagnetic simulation of oblique incidence on periodic structures, by combining the periodic and absorption boundary condition. We here extend the split-field method to the study of periodic dispersive structures by combining the Z transformation method. The iterative equations of the Drude dispersive model are also provided. By Comparing analytical and other numerical results, the efficiency and wide applicability of our method are demonstrated.

Comprehensive three-dimensional split-field finite-difference time-domain method for analysis of periodic plasmonic nanostructures: near- and far-field formulation

The three-dimensional split-field finite-difference time-domain (SF-FDTD) method is combined with the total-field–scattered-field method for injecting a plane wave. A formulation is derived for calculating the incidence transformed fields of SF-FDTD on a one-dimensional auxiliary grid. The resulting fields obtained in the scattered zone are used to calculate the far fields, based on a proposed fully time-domain near-to-far-field transformation. The far-field information is used to calculate the extinction cross section of the periodic structure under oblique incidence. To analyze metallic periodic structures, a formulation with a reduced number of variables is proposed based on the auxiliary differential equation method for dispersive media.

The WLP-FDTD Method for Periodic Structures With Oblique Incident Wave

A new unconditionally stable finite-difference time-domain (FDTD) method for periodic structures is presented, which is based on the field transformation method and the weighted Laguerre polynomials FDTD (WLP-FDTD). The proposed method uses a field transformation to remove the time gradient across the grid, and then uses the concept of the WLP-FDTD to get the implicit relationship between the transformed field variables. It holds the advantages of the WLP-FDTD, can eliminate the restriction of the Courant-Friedrich-Levy (CFL) stability condition. Compared with other field transform methods, the new method needn't to do special treatment for the additional terms. It appears to be much more efficient than the other field transformation FDTD method for solving periodic structures with fine structures and large incident angle. To verify the accuracy and the efficiency of the proposed method, we compare the results of the Split-Field FDTD method with the proposed method.

Split-field FDTD method for oblique incidence study of periodic dispersive metallic structures

The study of periodic structures illuminated by a normally incident plane wave is a simple task that can be numerically simulated by the finite-difference time-domain (FDTD) method. On the contrary, for off-normal incidence, a widely modified algorithm must be developed in order to bypass the frequency dependence appearing in the periodic boundary conditions. After recently implementing this FDTD algorithm for pure dielectric materials, we here extend it to the study of metallic structures where dispersion can be described by analytical models. The accuracy of our code is demonstrated through comparisons with already-published results in the case of 1D and 3D structures.