Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions.

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example in Born and Wolf’s treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of Un and Vn type Lommel functions. An alternative solution expresses the distribution in terms of an infinite summation of spherical Bessel functions, and was presented by Nijboer in 1947. In practical calculations however only a finite number of terms from these infinite expansions is actually used to calculate the focal distribution. In this manuscript we compare and contrast these different solutions, paying particular attention to how quickly each solution converges for different a range of spatial locations behind the focusing lens.