Study of singular radius and surface boundary constraints in refractive   beam shaper design
  This paper presents analysis of important issues associated with the design of refractive laser beam shaping systems. The concept of singular radius is introduced along with solutions to minimize its adverse effect on shaper performance. In addition, the surface boundary constraint is discussed in detail. This study provides useful guidelines to circumvent possible design errors that would degrade the shaper quality or add undesired complication to the system. 
C. Liu, S. Zhang