In general, a spin-polarized light beam cannot couple its spin angular momentum (SAM) with intrinsic orbital angular momentum (IOAM) without spin reversal. Here we find that nonlinear media can give the spin-polarized photon an IOAM, as they travel in the media due to the nonlinear susceptibility along the transmission direction, which does not require spin reversal. To characterize this SAM-to-IOAM conversion process, we establish an evolution ray equation for photons carrying IOAM by reference to the Schrödinger equation. We further reveal the inherent physics of such a phenomenon from a full-wave perspective and find that the vortex generation originates from the nonlinear-induced geometric phase.