Sergio Bleda

Educational Software for Interference and Optical Diffraction Analysis in Fresnel and Fraunhofer Regions Based on MATLAB GUIs and the FDTD Method

Interference and diffraction of light are elementary topics in optics. The aim of the work presented here is to develop an accurate and cheap optical-system simulation software that provides a virtual laboratory for studying the effects of propagation in both time and space for the near- and far-field regions. In laboratory sessions, this software can let optical engineering undergraduates simulate many optical systems based on thin slits. The numerical method used is the finite-difference time-domain (FDTD) method that has been successfully applied in many engineering fields. Using this numerical method, the irradiance distribution can be successfully evaluated in different planes far from the simulation grid without degrading performance. In addition, an easy-to-use MATLAB GUI handles all the parameters of the FDTD simulation and computes theoretical values of irradiance for both the Fresnel and Fraunhofer regions. Therefore, by using this software, the student is able to analyze the behavior of the Fresnel and Fraunhofer expressions as a function of the distance. This distance is defined as the space between the slits plane and the plane that contains the virtual screen on which the irradiance pattern is represented.

Performance analysis of SSE and AVX instructions in multi-core CPUs and GPU computing on FDTD scheme for solid and fluid vibration problems

In this work a unified treatment of solid and fluid vibration problems is developed by means of the Finite-Difference Time-Domain (FDTD). The scheme here proposed takes advantage from a scaling factor in the velocity fields that improves the performance of the method and the vibration analysis in heterogenous media. Moreover, the scheme has been extended in order to simulate both the propagation in porous media and the lossy solid materials. In order to accurately reproduce the interaction of fluids and solids in FDTD both time and spatial resolutions must be reduced compared with the set up used in acoustic FDTD problems. This aspect implies the use of bigger grids and hence more time and memory resources. For reducing the time simulation costs, FDTD code has been adapted in order to exploit the resources available in modern parallel architectures. For CPUs the implicit usage of the advanced vectorial extensions (AVX) in multi-core CPUs has been considered. In addition, the computation has been distributed along the different cores available by means of OpenMP directives. Graphic Processing Units have been also considered and the degree of improvement achieved by means of this parallel architecture has been compared with the highly-tuned CPU scheme by means of the relative speed up. The speed up obtained by the parallel versions implemented were up to 3 (AVX and OpenMP) and 40 (CUDA) times faster than the best sequential version for CPU that also uses OpenMP with auto-vectorization techniques, but non includes implicitely vectorial instructions. Results obtained with both parallel approaches demonstrate that massive parallel programming techniques are mandatory in solid-vibration problems with FDTD.

Comparison of simplified theories in the analysis of the diffraction efficiency in surface-relief gratings

In this work a set of simplified theories for predicting diffraction efficiencies of diffraction phase and triangular gratings are considered. The simplified theories applied are the scalar diffraction and the effective medium theories. These theories are used in a wide range of the value Λ/λ and for different angles of incidence. However, when 1 ≤ Λ/λ ≤ 10, the behaviour of the diffraction light is difficult to understand intuitively and the simplified theories are not accurate. The accuracy of these formalisms is compared with both rigorous coupled wave theory and the finite-difference time domain method. Regarding the RCWT, the influence of the number of harmonics considered in the Fourier basis in the accuracy of the model is analyzed for different surface-relief gratings. In all cases the FDTD method is used for validating the results of the rest of theories. The FDTD method permits to visualize the interaction between the electromagnetic fields within the whole structure providing reliable information in real time. The drawbacks related with the spatial and time resolution of the finite-difference methods has been avoided by means of massive parallel implementation based on graphics processing units. Furthermore, analysis of the performance of the parallel method is shown obtaining a severe improvement respect to the classical version of the FDTD method.

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

Abstract. 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.

Performance analysis of the FDTD method applied to holographic volume gratings: Multi-core CPU versus GPU computing

The finite-difference time-domain method (FDTD) allows electromagnetic field distribution analysis as a function of time and space. The method is applied to analyze holographic volume gratings (HVGs) for the near-field distribution at optical wavelengths. Usually, this application requires the simulation of wide areas, which implies more memory and time processing. In this work, we propose a specific implementation of the FDTD method including several add-ons for a precise simulation of optical diffractive elements. Values in the near-field region are computed considering the illumination of the grating by means of a plane wave for different angles of incidence and including absorbing boundaries as well. We compare the results obtained by FDTD with those obtained using a matrix method (MM) applied to diffraction gratings. In addition, we have developed two optimized versions of the algorithm, for both CPU and GPU, in order to analyze the improvement of using the new NVIDIA Fermi GPU architecture versus highly tuned multi-core CPU as a function of the size simulation. In particular, the optimized CPU implementation takes advantage of the arithmetic and data transfer streaming SIMD (single instruction multiple data) extensions (SSE) included explicitly in the code and also of multi-threading by means of OpenMP directives. A good agreement between the results obtained using both FDTD and MM methods is obtained, thus validating our methodology. Moreover, the performance of the GPU is compared to the SSE+OpenMP CPU implementation, and it is quantitatively determined that a highly optimized CPU program can be competitive for a wider range of simulation sizes, whereas GPU computing becomes more powerful for large-scale simulations.

Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions

Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.

Development of a unified FDTD-FEM library for electromagnetic analysis with CPU and GPU computing

The present paper describes an optimized C++ library for the study of electromagnetics. The implementation is based on the Finite-Difference Time-Domain method for transient analysis, and the Finite Element Method for electrostatics. Both methods share the same core and are optimized for CPU and GPU computing. To illustrate its running, FEM method is applied for solving Laplace’s equation analyzing the relation between surface curvature and electrostatic potential of a long cylindrical conductor, whereas FDTD is applied for analyzing Thin Film Filters at optical wavelengths. Furthermore, a comparison of the performance of both CPU and GPU versions is analyzed as a function of the grid size simulation. This approach allows the study of a wide range of electromagnetic problems taking advantage of the benefits of each numerical method and the computing power of the modern CPUs and GPUs.

Analysis of periodic anisotropic media by means of split-field FDTD method and GPU computing

The implementation of the Split-Field Finite Difference Time-Domain (SP-FDTD) method in Graphics Pro- cessing Units is described in this work. This formalism is applied to light wave propagation through periodic media with arbitrary anisotropy. The anisotropic media is modeled by means of a permittivity tensor with non-diagonal elements and absorbing boundary conditions are also considered. The split-field technique and the periodic boundary condition allow to consider a single period of the structure reducing the simulation grid. Nevertheless, the analysis of 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 to the isotropic and non-periodic implementation. With the upcoming of the new generation of General-Purpose Computing on Graphics Units many scientific applications have been accelerated and others are being developed into this new parallel digital computing architecture. Specifically, the implementation of the SP-FDTD in the Fermi 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. The formalism has been used for analyzing different structures and phenomena: binary phase gratings and twisted-nematic liquid crystal cells. The numerical predictions obtained by means of the FDTD method here implemented are compared with theoretical curves achieving good results, thus validating the accuracy and the potential of the implementation.

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.

Multi-GPU and multi-CPU accelerated FDTD scheme for vibroacoustic applications

Abstract The Finite-Difference Time-Domain (FDTD) method is applied to the analysis of vibroacoustic problems and to study the propagation of longitudinal and transversal waves in a stratified media. The potential of the scheme and the relevance of each acceleration strategy for massively computations in FDTD are demonstrated in this work. In this paper, we propose two new specific implementations of the bi-dimensional scheme of the FDTD method using multi-CPU and multi-GPU, respectively. In the first implementation, an open source message passing interface (OMPI) has been included in order to massively exploit the resources of a biprocessor station with two Intel Xeon processors. Moreover, regarding CPU code version, the streaming SIMD extensions (SSE) and also the advanced vectorial extensions (AVX) have been included with shared memory approaches that take advantage of the multi-core platforms. On the other hand, the second implementation called the multi-GPU code version is based on Peer-to-Peer communications available in CUDA on two GPUs (NVIDIA GTX 670). Subsequently, this paper presents an accurate analysis of the influence of the different code versions including shared memory approaches, vector instructions and multi-processors (both CPU and GPU) and compares them in order to delimit the degree of improvement of using distributed solutions based on multi-CPU and multi-GPU. The performance of both approaches was analysed and it has been demonstrated that the addition of shared memory schemes to CPU computing improves substantially the performance of vector instructions enlarging the simulation sizes that use efficiently the cache memory of CPUs. In this case GPU computing is slightly twice times faster than the fine tuned CPU version in both cases one and two nodes. However, for massively computations explicit vector instructions do not worth it since the memory bandwidth is the limiting factor and the performance tends to be the same than the sequential version with auto-vectorisation and also shared memory approach. In this scenario GPU computing is the best option since it provides a homogeneous behaviour. More specifically, the speedup of GPU computing achieves an upper limit of 12 for both one and two GPUs, whereas the performance reaches peak values of 80 GFlops and 146 GFlops for the performance for one GPU and two GPUs respectively. Finally, the method is applied to an earth crust profile in order to demonstrate the potential of our approach and the necessity of applying acceleration strategies in these type of applications.