Success Stories

The Challenge

Rigorous Coupled Wave Analysis (RCWA), also known as the Fourier Modal Method (FMM), is a frequency domain, eigenmode expansion, spectral method for rigorously solving Maxwell’s equations of electromagnetism in periodic and artifcially aperiodic, wavelength and subwavelength scale geometries. This fully vectorial computational technique is widely used for modeling the steady-state properties of grating and photonic crystal based wave guiding and diffractive integrated optical devices in both two and three spatial dimensions.

Eigenmode expansion/Scattering Matrix (S-Matrix) techniques are some of the most efficient means of rigorously solving the field and power distributions within certain classes of primarily periodic wavelength scale integrated optical devices, especially when compared to other widely used methods (FDTD, FEM, BPM), but as the geometric complexity and size of a single device increases or if performing geometric optimizations or tolerance studies are of interest, the memory requirements and computation time needed to perform a device study can quickly become impractical for the resources of a single desktop.

In a recent application of this method, researchers at the University of Central Florida ’s College of Optics and Photonics explored using an RCWA/S-Matrix technique with Perfectly Matched Layer (PML) boundary conditions to calculate the power flow throughout a high refractive index contrast photonic-wire S-bend waveguide. The S-bend geometry is approximated with 101 slices, and each layer uses 251 spatial harmonic basis functions to calculate the guided modes and radiation modes of each slice. Consequently, 103 separate layer, complex-valued, asymmetric eigenmode problems of size 251 by 251 are solved and 103 separate individual layer complex-valued scattering matrices of size 251 by 251 are created.

Opportunity for Task-Parallel Computation
This scenario of propagating energy through an asymmetric, non-periodic structure could make use of task parallelism in order to calculate the individual layer eigenproblems and scattering matrices, as well as using a domain decomposition/binary-tree approach to separate the device into separate sections and then processing each section in a relatively independent fashion. In such a setting, the necessary scattering matrix product operations could be performed in parallel for each section.

Furthermore, for performing geometric or spectral parameter scans or optimizations, the task-parallel computations can be integrated to efficiently study the nature of narrow spectral resonances over broadband regions or to perform multi-parameter optimization / evolutionary algorithm studies.

Opportunity for Data-Parallel Computations
Data parallel operations may also become necessary if the number of longitudinal slices in the system is increased, thereby increasing the number of unique eigenmode problems and scattering matrices to be stored. Furthermore, as the transverse size of the computational window is increased, the number of system basis functions also needs to be increased to maintain convergence in the layer eigenmode problems, thereby increasing the storage requirements for all matrices in the system.

Additionally, if the 2D problem is transformed into a 3D problem, by the introduction of a second transverse dimension, the Fourier harmonic grid utilized in each eigenmode problem then becomes a 2D lattice. Depending on the window sizes in each transverse spatial dimension, the total number of basis functions needed to maintain eigenmode convergence may jump from the hundreds into the thousands. Furthermore, the eigenmode problem that is to be solved in the case with two transverse dimensions is a 2Nx2N matrix problem, doubling or quadrupling the storage requirements of every matrix in the system.

In three-dimensional device studies, when hundreds or thousands of basis functions may be needed to obtain convergence in a single layer’s eigenmodes, or for 2D/3D continuously varying longitudinal geometries that may require hundreds of unique individual layer eigenmode problems, individual layer scattering matrices, and intermediary scattering matrices, a distributed memory system must be used to store data that could not otherwise fit within a single desktop’s memory.

Star-P Solution

Together, the data and task parallel capabilities of Star-P can dramatically improve the efficiency and capabilities of an RCWA/S-Matrix computation. In a recent application of Star-P to the RCWA/S-Matrix algorithm, the properties of a novel high refractive index dual grating output coupler were studied extensively. The task parallel properties of Star-P were integrated with an RCWA/S-Matrix code that had previously been written purely in MATLAB®®, reducing run times from 4-6 hours on the desktop, to less than 30 minute runs on a 16-processor server.

Very fine scale geometric parameter scans were performed over large scale parameter ranges that simultaneously show both narrowband and broadband features of the system response. By choosing optimal geometric properties and avoiding potential geometric problems that were discovered, an efficient, all-dielectric, unidirectional, dual grating output coupler was designed. Results from this study can be viewed in the following paper: http://www.opticsexpress.org/abstract.cfm?id=125451

Summary & Metrics

• Straightforward parallelization of data- and task-parallel computation
• 11X speed-up on 16-processor server
• Ability to explore the parameter space and fine-tune the design

A similar set of task parallel codes were used to explore the parameter space in the design of multiple wavelength resonant grating filters having broadened angular acceptance at a single angle of incidence. Results from this study can be viewed at the following location:

http://www.opticsexpress.org/abstract.cfm?id=138977

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