To explore the new energy frontier, a new generation of particle accelerators is needed. Muon colliders are a promising alternative, if muon cooling can be made to work. Muons are 200 times heavier than electrons, so they produce less synchrotron radiation, and they behave like point particles. However, they have a short lifetime of 2.2 µs and the beam is more difficult to cool than an electron beam. The Muon Accelerator Program (MAP) was created to develop concepts and technologies required by a muon collider. An important effort has been made in the program to design and optimize a muon beam cooling system. The goal is to achieve the small beam emittance required by a muon collider. This work explores a final ionization cooling system using magnetic quadrupole lattices with a low enough β region to cool the beam to the required limit with available low Z absorbers. Dr. Alakabha Datta, and Dr. Hailin Sang, for all of the guidance. Also, many thanks to Dr. Terrence Lee Hart for his help. His discussions were valuable and his guidance in the simulation code was essential for this work. iv