Finite-Difference Time-Domain Method

What is FDTD? 

The Finite-Difference Time-Domain (FDTD) method is a rigorous and powerful tool for modeling nano-scale optical devices. FDTD solves Maxwell’s equations directly without any physical approximation, and the maximum problem size is limited only by the extent of the computing power available.

How does FDTD work and what problem does it solve?

The FDTD method solves Maxwell’s equations on a mesh and computes E and H at grid points spaced Δx, Δy, and Δz apart, with E and H interlaced in all three spatial dimensions. FDTD includes the effects of scattering, transmission, reflection, absorption, etc. FDTD is a time-domain solution, but frequency analysis is also possible through the use of the Fast Fourier Transform (FFT) and the Discrete Fourier Transform (DFT).

FDTD Yee Cell of dimension Ax, Ay, Az. | Synopsys

FDTD Yee Cell of dimension Ax, Ay, Az. [3]

When would you use FDTD compared to other techniques?

FDTD can simulate any structure where Maxwell’s equations describe the necessary physics. Typical applications for this method include: LEDs, solar cells, filters, optical switches, semiconductor-based photonic devices, sensors, nano- and micro-lithography, nonlinear devices, and meta-materials (negative index of refraction). Read about FullWAVE FDTD for details about additional applications.

FDTD Simulation of Y-branch PBG splitter | Synopsys

FDTD Simulation of Y-branch PBG splitter

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What software is used to model FDTD?

Synopsys offers several photonic solutions tools that employ the FDTD method.

Synopsys' FullWAVE simulation software, part of RSoft Photonic Device Tools, employs FDTD to perform a full-vector simulation of photonic structures. Its award-winning, innovative design and feature set has made FullWAVE the market leader among optical device simulation tools, with a cutting-edge implementation of a mature FDTD algorithm that allows for a wide range of simulation and analysis capabilities. For a wide range of integrated and nano-optic devices, FullWAVE has applications such as LED extraction analysis, diffractive optical element (DOE) design, PIC/Custom PDK element design, nanophotonics, and meta-materials design.

OptoDesigner, part of the Synopsys PIC Design Suite, has photonic-aware physical layout capabilities to synthesize photonic integrated circuit (PIC) layouts for fabrication. OptoDesigner propagation simulation modules combine powerful simulations with next-generation design capabilities. Designers can choose the simulation module that best suits their component design. The 2D FDTD Simulations Module simulates fully time-dependent propagation in smaller structures. In this module, 2D FDTD solves the full set of time-dependent Maxwell’s equations by discretizing time using Courant–Friedrichs–Lewy condition to prevent divergence in the solution. The algorithm provides a robust solution for all-axis field propagation analysis.

 

FullWAVE FDTD Example: Modeling a Surface-Plasmon-Based Spatial Multiplexer

The speed of intra-chip and inter-chip connection is one of the main bottlenecks to achieving faster computer chip performance. Routing the signals through surface-plasmon-based waveguides provides one possible way to achieve faster optical connection speeds; these waveguides are compact, not bound by the diffraction limit, and can be easily integrated with both optical and electronic technologies. 

One basic challenge facing the adoption of plasmon guides within electrical chips is the excitation of the plasmons from external sources. Simulating this effect requires a rigorous full-vector modeling environment that provides accurate solutions for arbitrary device geometries containing both metallic and nonmetallic components. FullWAVE FDTD is the ideal tool to meet this need. FullWAVE provides a full-vector solution to Maxwell’s equations and allows engineers to use complex material definitions, arbitrary device geometries, non-uniform grids, and sophisticated measurement techniques to create new plasmonic devices and fine-tune existing designs for specific applications. The design parameters of the structure can also be perturbed in FullWAVE FDTD to allow the study of manufacturing tolerances on device performance.

The surface-plasmon-based spatial multiplexer studied in Figure 1 consists of a multiplexing switch that steers light toward one of several subwavelength metal-strip waveguides. Several 3D FullWAVE FDTD simulations were performed at a fixed wavelength, at various incident angles of the illumination to determine the optimal angles at which light is coupled into each of the three metal-strip waveguides.

Figure 1: Schematic of the surface plasmon spatial multiplexer | Synopsys

Figure 1: Schematic of the surface-plasmon spatial multiplexer

Figure 2: Simulation results showing the amplitude of the Ey field on the surface of the metal film:   a) Normal incidence light (shown above) is coupled into the central metalstrip waveguide  b) Angled incident light is coupled into one of the side metal strip waveguides | Synopsys

Figure 2: Simulation results showing the amplitude of the Ey field on the surface of the metal film:  
a) Normal incidence light (shown above) is coupled into the central metal-strip waveguide 
b) Angled incident light is coupled into one of the side metal-strip waveguides

The use of surface plasmon resonances in chip interconnects can allow for much faster chip performance. As shown in this example, rigorous simulation software like FullWAVE FDTD provides the necessary tools to study all factors that contribute to the design of a surface plasmon device.

For more examples, see these application notes:

References:

  1. J.P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic waves” J. Comput. Phys., 114, 185 (1994) 
  2. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Norwood, MA, 1995) 
  3. K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media” IEEE Trans. Antennas Propagat., AP-14, 302 (1966)
  4. A. Imre et al. "Multiplexing surface plasmon polaritons on nanowires" Applied Physics Letters 91 083115 (2007).