Neutron production via D(d, n)³He nuclear reaction during the interaction of two counter-propagating circularly polarized laser pulses with ultra-thin deuterium target is investigated by particle-in-cell simulation and Monte Carlo method. It is found that the rotation direction and initial relative phase difference of laser electric field vector have important effects on deuterium foil compression and neutron characteristics. The reason is attributed to the net light pressure and the difference in transverse instability development. The highest neutron yield can be obtained by choosing two laser pulses with a relative phase difference of 0 and the same rotation direction of the electric field vector. When the relative phase difference is 0.5π or 1.5π and the rotation direction of electric field vector is different, the neutrons have a directional spatial distribution and the neutron yield only slightly decreases. For left-handed circularly polarized laser pulse and right-handed circularly polarized laser pulse, each with an intensity of 1.23 × 10²¹ W/cm², a pulse width of 33 fs and a relative phase difference of 0.5π, it is possible to produce a pulsed neutron source with a yield of 8.5 × 10⁴ n, production rate of 1.2 × 10¹⁹ n/s, pulse width of 23 fs and good forward direction as well as tunable spatial distribution. Comparing with photonuclear neutron source and beam target neutron source driven by ultraintense laser pulses, the duration of neutron source in our scheme decreases significantly, thereby possessing many potential applications such as neutron nuclear data measurement. Our scheme offers a possible method to obtain a compact neutron source with short pulse width, high production rate and good forward direction.