A scheme for a quasi-monoenergetic high-flux neutron source with femtosecond duration and highly anisotropic angular distribution is proposed. This scheme is based on bulk acceleration of deuteron ions in an optical trap or density grating formed by two counter-propagating laser pulses at an intensity of $\sim 10^{16}\mathrm{W}/\mathrm{c}{\mathrm{m}}^{\mathrm{2}}$ in a near-critical-density plasma. The deuterons are first pre-accelerated to an energy of tens of keV in the ambipolar fields formed in the optical trap. Their energy is boosted to the MeV level by another one or two laser pulses at an intensity of $\sim 10^{20}\mathrm{W}/\mathrm{c}{\mathrm{m}}^{\mathrm{2}}$, enabling fusion reactions to be triggered with high efficiency. In contrast to previously proposed pitcher–catcher configurations, our scheme can provide spatially periodic acceleration structures and effective collisions between deuterons inside the whole target volume. Subsequently, neutrons are generated directly inside the optical trap. Our simulations show that neutron pulses with energy 2–8 MeV, yield 10¹⁸–10¹⁹n/s, and total number 10⁶–10⁷ in a duration $\sim 400$ fs can be obtained with a 25 μm target. Moreover, the neutron pulses exhibit unique angularly dependent energy spectra and flux distributions, predominantly along the axis of the energy-boosting lasers. Such microsize femtosecond neutron pulses may find many applications, such as high-resolution fast neutron imaging and nuclear physics research.