Analytical Expressions for the Mutual Coupling of Loop Antennas Valid From the RF to Optical Regimes

Arrays of circular loop antennas are commonly employed at radio frequencies for communications and geophysical sensing, while also holding enormous potential in the optical regime for applications such as solar energy harvesting. Exact analytical expressions exist for predicting the mutual coupling between a variety of antennas, including dipoles and slots. However, due to the complexity of the integrals involved, analytical expressions for evaluating the coupling between loop antennas have not been previously available. This paper presents straightforward analytical expressions for efficient calculation of the coupling between two circular loops at arbitrary locations. The theory is extended to the optical regime by taking into account the dispersion and loss of the material comprising the loop antenna. These analytical expressions provide insight into the physics underlying the mutual coupling phenomenon. Along with the approximate analytical expressions, a useful pseudo-analytical representation is developed which is more exact, especially in the near-field regime, and can be easily and efficiently evaluated in MATLAB via numerical integration. It is shown that full-wave simulations for a two-element array of nanoloops can take up to six hours, while the corresponding analytical and pseudo-analytical implementations derived here take less than a minute.