Considerations for the efficient spectral evaluation of reaction integrals associated with separated domains

The authors address some aspects that have to be considered when using a recently proposed contour-deformation technique for the efficient numerical evaluation of reaction integrals arising in the spectral-domain method of moments for testing and expansion functions that exist on separated domains. The method is particularly efficient in the context of microstrip patch antenna elements and arrays where the testing and expansion functions can either be entire-domain functions on different antenna elements or where they are widely separated subdomain functions on the same or different antenna elements. However, the method becomes less efficient for testing and expansion functions that are not so widely separated. It is shown how the separation distance between the testing and expansion functions, in addition to substrate thickness, affects the behaviour of the reaction integrand and how the choice of a more suitable integration contour can simplify the numerical evaluation of the associated integral, especially for closely spaced testing and expansion functions on thick substrates, where the original method becomes less efficient. The minimum separation distance between the domains of the testing and expansion functions, that can be accommodated with this method, is also better defined