Theoretical Study on the Rank of Integral Operators for Broadband Electromagnetic Modeling From Static to Electrodynamic Frequencies

To facilitate the broadband modeling of integrated electronic and photonic systems from static to electrodynamic frequencies, we propose an analytical approach to study the rank of the integral operator for electromagnetic analysis, which is valid for an arbitrarily shaped object with an arbitrary electric size. With this analytical approach, we theoretically prove that for a prescribed error bound, the minimal rank of the interaction between two separated geometry blocks in an integral operator, asymptotically, is a constant for 1-D distributions of source and observation points, grows very slowly with electric size as square root of the logarithm for 2-D distributions, and scales linearly with the electric size of the block diameter for 3-D distributions. We thus prove the existence of an error-bounded low-rank representation of both surface- and volume-based integral operators for electromagnetic analysis, irrespective of electric size and object shape. Numerical experiments validated the proposed analytical approach and the resultant findings on the rank of integral operators. This paper provides a theoretical basis for employing and further developing low-rank matrix algebra for accelerating the integral-equation-based electromagnetic analysis from static to electrodynamic frequencies.