Continuous-Time Algorithms for Solving Maxwell's Equations using Analog Circuits

In this paper, we propose solutions to Maxwell's equations that can be computed using analog computers. Spatially-discrete time-continuous (SDTC) algorithms running on analog computers can be potentially faster and more energy-efficient than fully-discrete numerical solvers. The implementations of fully-discrete partial differential equation (PDE) solvers on high speed digital processors, such as graphics processing units (GPUs), take many clock cycles to compute a single temporal frame of the update equation and thus have relatively low equivalent bandwidths. Our approach is to directly implement temporal recursions in continuous-time by using analog circuits. Such circuits can have bandwidths that greatly exceed the equivalent bandwidths of GPUs. In particular, we propose two analog computing methods that compute the SDTC solutions to Maxwell's equations. In addition to Maxwell's equations, such platforms can be used to accelerate other hard computational problems that involve PDEs derived from continuous-time systems. In continuous-time in Laplace domain (CTLD) method (first approach), the spatial domain partial derivatives in the governing PDE are approximated using discrete finite differences, while applying the Laplace transformation along the time dimension. The resulting spatially-discrete time-continuous update equation is utilized to design an analog circuit that can compute the continuous-time solution. The all-pass delay approximate (APDA) method (second approach) replaces the discrete-time difference operators in the standard finite difference time domain (FDTD) cell (Yee cell) using continuous-time delay operators, which can be realized using analog all-pass filters. Both methods have been simulated using ideal analog circuits in Cadence Spectre for the Dirichlet, Neumann, and radiation boundary conditions. The performance of the proposed methods have been quantified using i) mean squared differences between the results and fully-discrete FDTD simulations, and ii) the noise to signal energy ratio. The CTLD and APDA methods are able to compute the solutions to Maxwell's equations with a noise energy to signal energy ratio γ better than -26 dB and -19 dB, respectively. Both methods have been extended to design analog circuits that compute the continuous-time solution of the 1-D and 2-D wave equations. The CTLD-based 1-D and 2-D analog wave equation solvers are able to compute the solutions with γ better than -72 dB and -60 dB, respectively. The APDA-based 1-D wave equation solver is simulated with a dominant-pole model (which better approximates the non-ideal circuit behavior) along with a propagation delay compensation technique. The non-ideal analog models compute the solution with a difference smaller than -13 dB (in terms of γ). Experimental results from a simplified board-level low-frequency implementation are also presented. The key challenges toward CMOS implementations of the proposed solvers are identified and briefly discussed with possible solutions.