An optical fiber with low attenuation and methods of making same are disclosed. The optical fiber has a core, an inner cladding surround the core, and an outer cladding surrounding the inner cladding.The outer cladding is chlorine-doped such that the relative refractive index varies as a function of radius. The radially varying relative refractive index profile of the outer cladding reduces excess stress in the core and inner cladding, which helps lower fiber attenuation while also reducing macrobend and microbend loss. A process of fabricating the optical fiber includes doping an overclad soot layer of a soot preform with chlorine and then removing a portion of the chlorine dopant from an outermost region of the overclad soot layer. The soot preform with the modified chlorine dopant profile is then sintered to form a glass preform, which can then be used for drawing the optical fiber.