Generalized parabolic nondiffracting beams of two orders

In this paper we consider a generalization of standard nondiffracting parabolic beams. The proposed generalized beams have two orders: a continuous parameter a, as in standard beams, and the new parameter is an integer index m. Physically, the last parameter is equal to the number of rotated repetitions of the structure of the original angular spectrum on the total circle in the frequency space. Theoretical investigation shows that for a=0 the beams are real functions and have a symmetry of order 2m. If a≠0 the beams will be real functions only for odd values of m. Moreover, in this case the beams have a symmetry of order m, while for even values of m the order of symmetry is 2m. The results of numerical simulation confirm these conclusions. Examples of generalized traveling parabolic waves, which are formed on the basis of generalized static parabolic beams, are also given.