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arXiv:2608.23272

Which transition can be used in sodium mirrorless lasing for mesospheric magnetometry?

Yucheng Yang, Chunyang Lei, Kai Guo, Chi Peng

Published Aug 24, 2026• 0 citations
arXivHTMLPDF

Abstract

Directed mirrorless lasing from the mesospheric sodium layer has been proposed as a way to overcome the isotropy of laser-guide-star fluorescence, with demonstrated cell-scale analogues and a demonstrated stand-off magnetometry application. Several transition paths on the Na ladder compete for the same pump photons, and which of them can sustain a population inversion---and under what pumping format---is usually settled by numerical scans of a multi-level rate model. Here the same question is answered in closed form, and three design-relevant results follow from the atomic data alone. (1) A cascade inversion on $u\to l$ requires $τ_u/τ_l>(g_u/g_l)b_{ul}$. The inequality reproduces the full continuous-wave classification---admitting $4P_{3/2}\to4S_{1/2}$ (2.21~\textmu m), $4P_{3/2}\to3D_{5/2}$ (9.1~\textmu m) and the fine-structure companion $4S_{1/2}\to3P_{1/2}$ (1138~nm), excluding $4D_{5/2}\to4P_{3/2}$ (2.34~\textmu m) and $3D_{5/2}\to3P_{3/2}$ (819~nm)---and shows the 2.34~\textmu m line to miss by only 9\%, which is why it is transiently accessible. (2) Velocity selectivity reverses the naive scheme ranking because a Doppler-averaged treatment, exact for one-step pumping, underestimates two-step pumping by a participation factor of order the Doppler-to-natural width ratio, $\sim$$10^{2}$; equal division of power between two pump beams is exactly optimal. (3) The transient window on 2.34~\textmu m closes after $\sim$2$τ(4P)\approx210$~ns, set by the reservoir lifetime rather than by the pump. Each conclusion is traceable to a lifetime, a branching ratio or a linewidth, and transfers to another species without rerunning a model.

DOI
10.48550/arxiv.2608.23272

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Directed mirrorless lasing from the mesospheric sodium layer has been proposed as a way to overcome the isotropy of laser-guide-star fluorescence, with demonstrated cell-scale analogues and a demonstrated stand-off magnetometry application. Several transition paths on the Na ladder compete for the same pump photons, and which of them can sustain a population inversion---and under what pumping format---is usually settled by numerical scans of a multi-level rate model. Here the same question is answered in closed form, and three design-relevant results follow from the atomic data alone. (1) A cascade inversion on u→lu\to lu→l requires τu/τl>(gu/gl)bulτ_u/τ_l>(g_u/g_l)b_{ul}τu​/τl​>(gu​/gl​)bul​. The inequality reproduces the full continuous-wave classification---admitting 4P3/2→4S1/24P_{3/2}\to4S_{1/2}4P3/2​→4S1/2​ (2.21~\textmu m), 4P3/2→3D5/24P_{3/2}\to3D_{5/2}4P3/2​→3D5/2​ (9.1~\textmu m) and the fine-structure companion 4S1/2→3P1/24S_{1/2}\to3P_{1/2}4S1/2​→3P1/2​ (1138nm), excluding 4D5/2→4P3/24D_{5/2}\to4P_{3/2}4D5/2​→4P3/2​ (2.34\textmu m) and 3D5/2→3P3/23D_{5/2}\to3P_{3/2}3D5/2​→3P3/2​ (819nm)---and shows the 2.34\textmu m line to miss by only 9%, which is why it is transiently accessible. (2) Velocity selectivity reverses the naive scheme ranking because a Doppler-averaged treatment, exact for one-step pumping, underestimates two-step pumping by a participation factor of order the Doppler-to-natural width ratio, \sim$$10^{2}; equal division of power between two pump beams is exactly optimal. (3) The transient window on 2.34~\textmu m closes after ∼\sim∼2τ(4P)≈210τ(4P)\approx210τ(4P)≈210~ns, set by the reservoir lifetime rather than by the pump. Each conclusion is traceable to a lifetime, a branching ratio or a linewidth, and transfers to another species without rerunning a model.
Directed mirrorless lasing from the mesospheric sodium layer has been proposed as a way to overcome the isotropy of laser guide star fluorescence, with demonstrated cell-scale analogues and a demonstrated stand-off magnetometry application. Several transition paths on the Na ladder compete for the same pump photons. We build a ten-level rate-equation model of the ladder from NIST transition probabilities and evaluate every electric-dipole line under four continuous-wave pumping schemes, both in a Doppler-averaged treatment and in a velocity-selective treatment appropriate for the collision-poor mesosphere. Three design-relevant results emerge. At practically accessible continuous-wave irradiances the column-gain exponents remain far below unity, consistent with published feasibility estimates; the value of the classification is to identify which lines, schemes, and pulse formats merit further study.
Mirrorless lasing from the mesospheric sodium layer could transform laser-guide-star emission into a directional, gain-driven source. Here we convert this possibility into quantitative requirements, using a ten-level velocity-resolved rate-equation model and an exact compression--irradiance equivalence. Continuous-wave operation remains below threshold across the accessible parameter space. The transient 4D5/2-4P3/2 channel at 2339 nm reaches threshold at pulsed irradiances of 104-105 W/m-2, given six- to ten-fold velocity compression. The obstacle is therefore not laser power but momentum transfer: spontaneous scattering delivers at most one recoil per event at a bounded rate, leaving chirp-driven compression an order of magnitude short. This gap defines the technological challenge for mesospheric mirrorless lasing.
Mirrorless lasing from the mesospheric sodium layer could transform laser-guide-star emission into a directional, gain-driven source. Here we convert this possibility into quantitative requirements, using a ten-level velocity-resolved rate-equation model and an exact compression--irradiance equivalence. Continuous-wave operation remains below threshold across the accessible parameter space. The transient 4D5/2-4P3/2 channel at 2339 nm reaches threshold at pulsed irradiances of 104-105 Wm-2, given six- to ten-fold velocity compression. The obstacle is therefore not laser power but momentum transfer: spontaneous scattering delivers at most one recoil per event at a bounded rate, leaving chirp-driven compression an order of magnitude short. This gap defines the technological challenge for mesospheric mirrorless lasing.
population-inverted transition in Na vapors that are two-photon excited with resonant laser light at 589 and 569 nm. Our study of power and atom-number-density threshold characteristics shows that lasing occurs at sub-10 mW total power of the applied laser light. The observed 6 mrad divergence is defined mainly by the aspect ratio of the gain region. We find that mirrorless lasing at 2.21 μm is magnetic field and polarization dependent and this may be useful for remote magnetometry. The presented results could help determine the requirements for obtaining directional return from sodium atoms in the mesosphere.
We demonstrate backward-directed continuous-wave (cw) emission at 2.21 {\mu}m generated on the 4P3/2-4S1/2 population-inverted transition in Na vapors two-photon excited with resonant laser light at 589 and 569 nm. Our study of power and atom-number-density threshold characteristics shows that lasing occurs at sub-10 mW total power of the applied laser light. The observed 6 mrad divergence is defined mainly by the aspect ratio of the gain region. We find that mirrorless lasing at 2.21 {\mu}m is magnetic field and polarization dependent that may be useful for remote magnetometry. The presented results could help determine the requirements for obtaining directional return from sodium atoms in the mesosphere
Calculations of backscatter emission of mesospheric sodium atoms in a laser guide star that is excited by pulses ranging from 30-ns to 0.9-μs duration are described. The efficient use of such pulses at saturating irradiance values is shown to require ~3 GHz of spectral broadening to provide access to the full absorption spectrum of the D2 line. The broadening is provided by frequency modulation. A set of density matrices was used to account for all 24 hyperfine states and inhomogeneous Doppler broadening. At the broadband (3-GHz) saturation irradiance of 4 W/cm2, both linearly and circularly polarized laser beams are shown to produce emission rates exceeding 60% of the maximum possible rate-equation rate for the 0.9-μs pulses. As expected, circular polarization produced more backscatter than did linear polarization, but the enhancement never exceeded 1/3 in the calculations that are reported. A brief estimate of state precession in the Earth’s magnetic field suggests that achieving even this enhancement will require that the time scale for optical pumping be held to less than 1 μs, which will require the use of irradiances greater than 0.7 W/cm2 and spectral coverage of the full 3-GHz hyperfine-plus-Doppler absorption profile, at least until most of the population is pumped out of the F = 1 ground states.