A multimode optical fiber includes: (i) a graded index glass core having a radius R1 in the range of 20 microns to 50 microns, a maximum relative refractive index Δ1MAX in the range between 0.5% and 3%; a graded index having a profile with (a) by an alpha (α) parameter wherein 1.9≦α≦2.2, and (b) a deviation from the alpha profile in at least one region of the core, such that the difference in the refractive index delta of the core from that determined by the core alpha value, at the radius R1 is less than 0.001, and (ii) a cladding surrounding and in contact with the core, wherein the fiber has an bandwidth greater than 5000 MHz-km at a wavelength λ where λ≧800 nm.