This paper describes an idealized magnetic transition that results in optimum performance for perpendicular magnetic recording systems. This transition has a step at its center, and its magnetic moment is the maximum allowed when the perpendicular demagnetizing field at the coating surface is limited to Hc. System performance is modelled by the timing margin for a 10-10error rate at the output of a idealized differentiated read channel  in which white gaussian noise has been inserted at the input. This model closely approximates experimental performance measurements. For ring head read, the performance of the optimum and dual arctangent  perpendicular transitions is compared with longitudinal recording on thin film media. Since the optimum perpendicular model performs only slightly better than longitudinal recording, and is likely to be superior to real transitions written with a ring head, it would appear that perpendicular recording with ring head read offers little advantage. The performance model has been used to evaluate a recently described perpendicular recording system with single pole read/write on two layer media [3, 4], which shows great promise.