The Modeling of Time-Structured Multiturn Injection into Fermilab Main   Injector (Microbunch Injection with Parasitic Longitudinal Painting)
  This paper presents the modeling of time-structured multiturn injection for an upgraded Main Injector with the 8-GeV Superconducting RF proton driver, or an ILC-style linac, or a Project-X linac. The Radio-Frequency mismatch between a linac and the upgraded Main Injector will induce parasitic longitudinal painting in RF-phase direction. Several different scenarios with a choice of different RF parameters for single RF system and double RF system in the presence of longitudinal space charge have been investigated. From the studies of microbunch injection with the aid of ESME (2003) numerical simulations, it is found that the dual RF system with a choice of appropriate RF parameters allows us to overcome the space-charge limitation set by beam intensity during the multiturn-injection process. A double RF system with a harmonic ratio (R_H = H_2/H_1) of 2.0 and a voltage ratio (R_V = V_2/V_1) of 0.5 are most favored to reduce both longitudinal and transverse effects of space charge in the Main Injector. 
Phil S. Yoon, David E. Johnson, and Weiren Chou