Lower NA in large-mode-area fibers enables better single-mode operation and larger core diameters. Fiber NA has traditionally been limited to 0.06, mostly due to the control tolerance in the fabrication process. It has been recognized recently that transverse mode instability is a major limit to average power scaling in fiber lasers. One effective method to mitigate this limit is to operate nearer to the single-mode regime. Lower fiber NA is critical in this since it allows relatively larger core diameters which is the key to mitigate the limits imposed by nonlinear effects. We have developed a fabrication process of ytterbium-doped silica glass which is capable of highly accurate refractive index control and sufficient uniformity for LMA fibers. This process is also capable of large-volume production. It is based on a significant amount of post-processing once the fiber preforms are made. We have demonstrated 30/400 and 40/400 LMA fibers with a NA of ~0.028 operating very close to the single-mode regime. The second-order mode cuts off at ~1.2μm and ~1.55µm respectively. We have also studied issues related to bend losses due to the low NA and further optimization of LMA fibers.