The goal of this research was to develop concepts for affordable, fully functional arrays of superconducting quadrupoles for multi-beam transport and focusing in heavy ion fusion (HIF) accelerators. Previous studies by the Virtual National Laboratory (VNL) collaboration have shown that the multi-beam transport system (consisting of alternating gradient quadrupole magnets, a beam vacuum system, and the beam monitor and control system) will likely be one of the most expensive and critical parts of such an accelerator. This statement is true for near-term fusion research accelerators as well as accelerators for the ultimate goal of power production via inertial fusion. For this reason, research on superconducting quadrupole arrays is both timely and important for the inertial fusion energy (IFE) research program. This research will also benefit near-term heavy ion fusion facilities such as the Integrated Research Experiment (IRE) and/or the Integrated Beam Experiment (IBX).We considered a 2-prong approach that addresses the needs of both the nearer and longer term requirements of the inertial fusion program. First, we studied the flat coil quadrupole design that was developed by LLNL; this magnet is 150 mm long with a 50 mm aperture and thus is suitable for near term experiments that require magnets of a small length to aperture ratio. Secondly, we studied the novel double-helix quadrupole (DHQ) design in a small (3 x 3) array copfiguration; this design can provide an important step to the longer term solution of low-cost, easy to manufacture array constructions. Our Phase I studies were performed using the AMPERES magneto static analysis software. Consideration of these results led to plans for future magnet R&D construction projects.. The first objective of Phase I was to develop the concept of a superconducting focusing array that meets the specific requirements of a heavy ion fusion accelerator. Detailed parameter studies for such quadrupole arrays were performed. Based on these studies, the primary magnet parameters and the general features required for a complete array system (including vacuum and cryostat) were identified. Basic system concepts were formulated to serve as guides for future development work.A related issue was to compare the applicability and benefits of two different magnet technologies for use in such a quadrupole array. Analytical studies were performed for each of the two coil designs, a flat coil based on an HCX quadrupole designed by LLNL and a doublehelix quadrupole designed by AML. These studies have confirmed the feasibility of using either of the two coil designs in a small array.Advanced Magnet Laboratory, June 2003 DOE SBIR Phase I Final Report: DE-FG02-02ER83359