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Y.L. Le Coz (Task Supervisor, lecozy@rpi.edu), R.B. Iverson (Research Associate),
J. Kalyanasundharam (Student, PhD '05)
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute
Troy, NY 121803590
ABSTRACT
We are attempting to create a new, stochastic (MonteCarlo/randomwalk) algorithm that efficiently characterizes 3D electromagnetic wave propagation within materially heterogeneous dielectric domains. This task represents a first critical step towards practical ICCAD algorithms for 3D electromagnetic propagation within massively coupled onchip optical interconnects. Stochastic algorithms generally possess a number of advantages compared with conventional deterministic numerical fieldsolution methods (FEM, BI, spectral). Stochastic algorithms require no numerical mesh; they are computationally efficient in massively coupled, highdimensionality problems; they allow “dialin” solution accuracy; and they usually maintain full parallelism. Our theoretical approach begins with the time Fourier transform of the 3D Maxwell equations, in 4vector potential representation, subject to an initial condition of an injected impulsive “wave packet”. We then, by means of perturbation theory, derive a series of coupled Poisson and heat equations for 4potential temporal impulseresponse (IR) moments. Employing either finite or infinitedomain Green’s function integral solution techniques, we, when necessary, solve the coupled moment equations stochastically. We have discovered, importantly, the existence of, and the value of, nonvanishing “surfaceresidual” terms in our Green’s function second IRmoment solution. The 4potential temporal moments concisely portray the propagative behavior (delay, cross talk, attenuation, dispersion) exhibited by Maxwell’s equations. In addition, we have preliminarily validated our proposed theoretical approach numerically—in the limiting case of a homogeneous dielectric domain. Using a stochastic Monte Carlo integration scheme for a spherically symmetric benchmark problem, we have numerically evaluated the second IR moment of a 4potential component. Benchmark geometry consisted of a spatial and temporal impulsive source located at the origin of a dielectric sphere of unity dimensionless radius; embedded in quasiinfinite cube domain of, possibly, different dielectric value. Magnetic permeability was conveniently normalized to unity. At 10 9 Monte Carlo samples and 2.5 (dimensionless) radius, we obtained an order 1% statistical standard error when reverting to a unity spheretodomain dielectric ratio (the homogeneous case). As the dielectric ratio was decreased below unity, systematic, expected, qualitative variation in numerical solution was additionally observed. Lastly, we, as well, have obtained an associated spherically symmetric benchmark solution for first IR moment and its spatial gradient. Here, dielectric sphere radius was 1.5. The product of magnetic permeability, sphere radius, and spheretodomain dielectricconstant difference was conveniently normalized to the product of 4 p and the domain dielectric constant. The commercial randomwalk solver we used agreed excellently with the known analytical result over a 1.5 to 5.0 radius.
This work has been sponsored in part by the Defense Advanced Research Projects Agency (DARPA); the New York State Office of Science, Technology, and Academic Research (NYSTAR); and the Semiconductor Research Corporation (SRC) Microelectronics Advanced Research Corporation (MARCO).
