Comparison of exact and asymptotic results for the focusing of electromagnetic waves through a plane interface

Recently, exact Kirchhoff solutions and the corresponding asymptotic solutions for the focusing of electromagnetic waves through a plane interface between two different dielectrics were reported. But the computation of exact results takes a long time because it requires the quadruple integration of a rapidly oscillating integrand. By using asymptotic techniques to perform two of the integrations, one can reduce the computing time dramatically. Therefore it is important to establish the accuracy and the range of validity of the asymptotic technique. To that end, we compare the exact and the asymptotic results for high-aperture, near-field focusing systems with a total distance from the aperture to the focal point of a few wavelengths and with a distance from the aperture to the interface as small as a fraction of a wavelength. The systems examined have f-numbers in the range from 0.6 to 0.9 and Fresnel numbers in the range from 0.4 to 3.5. Our results show that the accuracy of the asymptotic method increases with the aperture-interface distance when the aperture-focus distance is kept fixed and that it increases with the aperture-focus distance when the aperture-interface distance is kept fixed. To an accuracy of 7.8%, the asymptotic techniques are valid for aperture-interface distances as small as 0.5lambda as long as the total distance from the aperture to the focal point exceeds 8lambda. It is also shown that an accuracy of better than 1% can be obtained for the same aperture-interface distance of 0.5lambda and for interface-observation-point distances as small as 0.1lambda as long as the total distance from the aperture to the focal point exceeds 12lambda. By use of the asymptotic technique the computing time is reduced by a factor of 10(3).