Extended axial irradiances: Barker rings

For extending focal depth we employ a set of transparent concentric rings, which are coded with the Barker sequences of length L. At the neighborhood of the paraxial focal plane, these transparent masks generate an axial uniform distribution, which is modulated with sinusoidal variations. For imaging applications, one can extend focal depth if the Barker length is congruent to unity modulo 4. And, for optical trapping, a bottle neck irradiance distribution is generated if the Barker length is congruent to three modulo 4.