Optical gain in rotationally excited nitrogen molecular ions

We pump low-pressure nitrogen gas with ionizing femtosecond laser pulses at 1.5 $\mu \mathrm{m}$ wavelength. The resulting rotationally excited ${\mathrm{N}}_2^{+}$ molecular ions generate directional, forward-propagating stimulated and isotropic spontaneous emissions at 428 nm wavelength. Through high-resolution spectroscopy of these emissions, we quantify rotational population distributions in the upper and lower emission levels. We show that these distributions are shifted with respect to each other, which has a strong influence on the transient optical gain in this system. Although we find that electronic population inversion exists in our particular experiment, we show that sufficient dissimilarity of rotational distributions in the upper and lower emission levels could, in principle, lead to gain without net electronic population inversion.

Figure 1

Left: A diagram of the $B^2{\mathrm{\Sigma }}_u^{+}$ and $X^2{\mathrm{\Sigma }}_g^{+}$ electronic energy manifolds of the ${\mathrm{N}}_2^{+}$ molecular ion. Integers ${\nu }^{\prime }$ and ${\nu }^{″}$ index different vibrational levels within the upper and lower electronic manifolds, respectively. Right: An expanded diagram of vibrational energy manifolds $v^{\prime }=0$ and $v^{″}=1$ within the upper and lower electronic energy manifolds. $N^{\prime }$ and $N^{″}$ are the the nuclear-angular-momentum quantum numbers in the upper and lower emission states, respectively. In this Rapid Communication, different spectral lines within P and R emission branches are numbered by the quantum number $N^{″}$ of the lower rotational state involved in the particular transition.

Figure 2

Experimental data and a fit for spontaneous emission detected from the side, in the case of pumping 4 torr of pure nitrogen with 2 mJ, 60 fs laser pulses at 1500 nm wavelength. The tabulated values of the wave number for different rotational transitions [14] are marked with vertical dashed lines. The corresponding values of the rotational quantum number $N^{″}$ in the lower emission state are shown next to the individual lines. Spontaneous emission peaks at $N^{″}=6$.

Figure 3

Same as in Fig. 2, but for the stimulated emission that is self-seeded by the spectrally broadened third harmonic of the pump and measured in the forward direction. The R-branch emission, which comprises the transitions with the change of the nuclear-angular-momentum quantum number $N$ of $-1$, is shown in the main figure. The inset shows the full emission spectrum including both P- and R-branch emissions. Stimulated emission peaks at the $N^{″}$ value of 12, which is larger than the peak value for spontaneous emission. Based on the ratio of the stimulated-emission signals on the adjacent spectral lines in the R branch, we estimate the peak value of the total gain in the 1-mm-long interaction zone as 64, corresponding to the lower-bound estimate for the peak exponential gain per unit length on the strongest R-branch line of about 40 ${\mathrm{cm}}^{-1}$.

Figure 4

(a) Rotational population distributions in the upper and lower emission levels of the ${\mathrm{N}}_2^{+}$ molecular ion, inferred from fitting of the experimental data. Allowed transitions that produce the strongest spontaneous and stimulated emissions are marked with dashed and solid arrows, respectively. (b) Rotational distributions calculated according to Eqs. (1) and (2) with $N_B=5$ and $N_X=0$ and negative net electronic population inversion [the ratio of total populations in the $B^2{\mathrm{\Sigma }}_u^{+}$(0) and $X^2{\mathrm{\Sigma }}_g^{+}$(1) states equals 0.75]. Population inversion and gain still exist on a subset of rotational transitions with large values of the rotational quantum number $N$.