Laser phase noise remains a limiting factor in many experimental settings, including metrology, time-keeping, as well as quantum optics. Hitherto this issue was addressed at low frequencies ranging from well below 1 Hz to maximally 100 kHz. However, a wide range of experiments, such as, e.g., those involving nanomechanical membrane resonators, are highly sensitive to noise at higher frequencies in the range of 100 kHz to 10 MHz, such as nanomechanical membrane resonators. Here we employ a fiber-loop delay line interferometer optimized to cancel laser phase noise at frequencies around 1.5 MHz. We achieve noise reduction in 300 kHz-wide bands with a peak reduction of more than 10 d13 at desired frequencies, reaching phase noise of less than -160 dB(rad(2)/Hz) with a Ti:Al2O3 laser. These results provide a convenient noise reduction technique to achieve deep ground-state cooling of mechanical motion. (C) 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement