An early result of optical focusing theory is the Lommel field, resulting from a uniformly illuminated lens; the dark rings in the focal plane, the Airy rings, have been recognized as phase singularities. On the other hand, it is well known that Gaussian illumination leads to a Gaussian beam in the focal region without phase singularities. We report a theoretical and experimental study of the transition between the two cases. Theoretically, we studied this transition both within and outside the paraxial limit by means of diffraction theory. We show that in the gradual transition from uniform toward Gaussian illumination, the Airy rings reorganize themselves by means of a creation/annihilation process of the singularities. The most pronounced effect is the occurrence of extra dark rings (phase singularities) in front of and behind the focal plane. We demonstrate theoretically that one can bring these rings arbitrarily close together, thus leading to structures on a scale arbitrarily smaller than 1 wavelength, although at low intensities. Experimentally, we have studied the consequences of the reorganization process in the paraxial limit at optical wavelengths. To this end, we developed a technique to measure the three-dimensional intensity (3D) distribution of a focal field. We applied this technique in the study of truncated Gaussian beams; the experimentally obtained 3D intensity distributions confirm the existence and the reorganization of extra dark rings outside the focal plane.
© 1998 Optical Society of America
G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, "Airy pattern reorganization and subwavelength structure in a focus," J. Opt. Soc. Am. A 15, 884-899 (1998)