Revisiting self-seeding mechanism by generating vector ultraviolet N$_2^{+}$ lasing
An intense femtosecond laser pulse can generate ultraviolet air lasing, offering a promising remote light source. A long-standing hypothesis is whether it is seeded by a self-generated spectral component, such as the second harmonic that is inevitably produced by the plasma gradient. Here, we report the generation of both radially and azimuthally polarized N$_2^+$ lasing driven by a single 800-nm cylindrical vector beam. Meanwhile, the same vector pump was applied to drive the generation of vector second harmonics in plasma. The radially polarized pump produces radially polarized second harmonics while the azimuthally polarized pump yields no second harmonic generation owing to the radial direction of plasma gradient. The absence of the azimuthally polarized second harmonic rules out the hypothesis of self-seeding by second harmonics, as both radially and azimuthally polarized N$_2^+$ lasing are observed with comparable intensities. By characterizing the spatial phase distribution of vector 391-nm lasing, we concluded that its phase is synchronized with the pump. These results suggest that amplified spontaneous emissions are the origin of N$_2^+$ lasing under the most common condition of low gas pressure, which was effectively demonstrated by theoretical simulations. Our work provides a promising method for remotely generating vector ultraviolet light sources.