Propagation properties of radial Lorentz array beam in free space

The analytical expression for the propagation of a radial Lorentz array beam with an initial phase distribution is derived and used to study the propagation properties of the coherently combined beam in free space. For the case of the radial Lorentz array beam with a certain different initial phase distribution, the intensity profile of the resulting beam becomes a dark-hollow one, whereas for the case of the array beam with an equal initial phase distribution, the center of the intensity profile is a bright spot. The beam width and power in the bucket of the resulting beam depend on the waist width of the Lorentz beamlet, and on the initial phase distribution and radius of the radial array. The results are illustrated by numerical examples. When the radius of the radial array is larger an