We demonstrate a dispersion-tuned fiber optical parametric oscillator (FOPO)-based swept source with a sweep rate of $40\mathrm{kHz}$ and a wavelength tuning range of $109\mathrm{nm}$ around $1550\mathrm{nm}$ . The cumulative speed exceeds 4,000,000 nm/s. The FOPO is pumped by a sinusoidally modulated pump, which is driven by a clock sweeping linearly from 1 to $1.0006\mathrm{GHz}$ . A spool of dispersion-compensating fiber is added inside the cavity to perform dispersion tuning. The instantaneous linewidth is $0.8\mathrm{nm}$ without the use of any wavelength selective element inside the cavity. $1\mathrm{GHz}$ pulses with pulse width of $150\mathrm{ps}$ are generated.