Systems of cascaded circular apertures may be built to focus electromagnetic waves if the difference (d) between the Fresnel numbers of any two apertures pertaining to two points (A and B) is an even number. With a point source at A, the spherical incident wave will be focused to B. In general, the irradiance at B (principal focus) is roughly proportional to (n±1)2 where n is the number of apertures; the plus sign is used when the Fresnel numbers are odd integers, and minus when even. A system of a few apertures with d = 0 can focus waves over a wide range of frequencies, though the actual irradiance at the principal focus is a function of frequency. A condition to maximize the irradiance at the principal focus has been found. Theoretically such systems have been analyzed by the boundary-diffraction-wave theory generalized by Miyamoto and Wolf. Analytical expressions for the diffraction wave amplitude have been obtained for axial and off-axial points, when the iteration method of Fox and Li is also applicable, the two theoretical results agree very well with each other. Experimental data obtained agree well with the theoretical results.
JOHN W. Y. LIT and RÉAL TREMBLAY, "Boundary-Diffraction-Wave Theory of Cascaded-Apertures Diffraction," J. Opt. Soc. Am. 59, 559-564 (1969)