Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Applied Optics

Theory of mesospheric sodium fluorescence excited by pulse trains

Peter W. Milonni, L.E. Thode

Published Feb 20, 1992• 34 citations
Publisher

Abstract

We describe a two-level model for the mesospheric sodium resonance fluorescence excited by a train of pulses. For pulse durations that are short compared with the 16-ns natural lifetime, population-rate equations are inadequate and must be replaced by density-matrix (Bloch) equations. We briefly contrast these two approaches and discuss several issues associated with pulse-train excitation. Analytical approximations to averages over atomic velocities and the transverse spatial profile of the laser are described. Estimates of return photon numbers for trains of pulses, ranging in duration from tens of picoseconds to half a nanosecond, are made and these estimates are compared with those of more conventional long-pulse lidar schemes. Roughly similar photon numbers are predicted for both long and short pulses whenever their average intensities are comparable. For average intensities, less than ~ 10 W/cm(2), trains of asymptotically equal to 0.5-ns pulses yield greater photon returns than trains of asymptotically equal to 30-ps pulses. For larger intensities the reverse can be true.

DOI
10.1364/ao.31.000785

Related papers

Modeling the response of mesospheric sodium to pulsed-laser excitation

Joschua Hellemeier, Paul Hickson, Lucas Labadie

line by nanosecond time scale pulsed lasers is described. By numerically integrating transition rates in the sodium hyperfine structure, the return flux per sodium atom is predicted as a function of laser power. The simulation should be useful for studies of mesospheric sodium and adaptive optics. Applications include the estimation of sodium column density from lidar return flux, and of laser guide star brightness for different pulsed laser formats. The simulation assumes that the pulse repetition frequency is sufficiently low (smaller than a few kilohertz) that atomic collisions restore local thermodynamic equilibrium between pulses. It is also assumed that the pulse length is short compared to the Larmor precession time scale. The numerical results are well-approximated by a simple analytic model for a three-level atom. The number of emitted photons is found to be primarily dependent on the product of the length of the laser pulse and the energy density.

Journal of the Optical Society of America A•Jul 19, 2017•2 citations

Efficient excitation of a mesospheric sodium laser guide star by intermediate-duration pulses

J. R. Morris

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.

Journal of the Optical Society of America A•Feb 1, 1994•53 citations

Pulse-train excitation of sodium for use as a synthetic beacon

Lee C. Bradley

The responses of two-level atoms and of sodium atoms to trains of repetitively pulsed resonance radiation are computed. It is found that the optimum backscatter from mesospheric sodium for a given irradiance is attained when the pulse trains are phase modulated and when the lengths of individual pulses are in the 500–700-ps range, corresponding to a transform-limited bandwidth somewhat smaller than the Doppler-broadened hyperfine-structure pattern of the D2 line.

Journal of the Optical Society of America B•Oct 1, 1992•47 citations

Excitation of an atom by a train of short pulses

Richard J. Temkin

A general, analytic theory is presented of the excitation of a two-level atom by a series of short optical laser pulses. The atom is assumed to be isolated (collisionless) with radiative decay from upper to lower state at rate γ. The laser pulses are assumed to have a duration tp such that γtp ≪ 1 (the transient regime), and the spacing between pulses tr is such that tr ≫ tp. In this regime the atomic density matrix is calculated as a function of time during each optical pulse. An analytic expression for the atomic state (Bloch vector) after n optical pulses is also derived. For large n the atomic state converges to a repeating (steady state) cycle of pump excitation followed by radiative decay. The excited-state population in the steady state can be large but only if the spacing between pulses tr is small so that γtr ≲ 0.5. These results indicate that lasers that produce trains of pulses, such as mode-locked conventional lasers and free-electron lasers, can be effective in optical pumping of atoms.

Journal of the Optical Society of America B•May 1, 1993•33 citations

Response of the atmospheric sodium layer to short optical pulse excitation

Bruce J. Herman, Marek J. Konopnicki

Some of the adaptive optics systems contemplated to ameliorate the effects of the atmospheric turbulence utilize guide stars. Guide stars are artificial beacons formed in the atmosphere by the interaction of a laser with matter in the atmosphere. Sodium guide stars are formed as a result of laser interaction with the sodium layer in the upper atmosphere. A small rf FEL is one of the candidates considered as a potential source capable of creating a sodium beacon. The pulse format of such a laser consists of trains of short optical pulses. The duration of an individual pulse is typically shorter than that of relevant relaxation times in the medium. The separation of the pulses may be longer or of the order of the relevant atomic memory time scale. To properly describe interactions of the atmospheric sodium layer with a laser characterized by such a pulse format one must utilize the optical Bloch equations (OBE). The OBE have been solved numerically using a simplified two-level model, and the resulting sodium layer response has been characterized. Additionally, an analytic solution has been obtained for the case in which both the population and coherence relaxation times are equal. The model is capable of providing useful insights.

Annual Meeting Optical Society of America•Jan 1, 1991•0 citations

Population-inversion map of the mesospheric sodium ladder: continuous-wave and pulsed pumping schemes for directed emission

Yucheng Yang, Chunyang Lei, Kai Guo +2 more

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.

arXiv:2608.09037•Aug 10, 2026•0 citations
We describe a two-level model for the mesospheric sodium resonance fluorescence excited by a train of pulses. For pulse durations that are short compared with the 16-ns natural lifetime, population-rate equations are inadequate and must be replaced by density-matrix (Bloch) equations. We briefly contrast these two approaches and discuss several issues associated with pulse-train excitation. Analytical approximations to averages over atomic velocities and the transverse spatial profile of the laser are described. Estimates of return photon numbers for trains of pulses, ranging in duration from tens of picoseconds to half a nanosecond, are made and these estimates are compared with those of more conventional long-pulse lidar schemes. Roughly similar photon numbers are predicted for both long and short pulses whenever their average intensities are comparable. For average intensities, less than ~ 10 W/cm(2), trains of asymptotically equal to 0.5-ns pulses yield greater photon returns than trains of asymptotically equal to 30-ps pulses. For larger intensities the reverse can be true.
line by nanosecond time scale pulsed lasers is described. By numerically integrating transition rates in the sodium hyperfine structure, the return flux per sodium atom is predicted as a function of laser power. The simulation should be useful for studies of mesospheric sodium and adaptive optics. Applications include the estimation of sodium column density from lidar return flux, and of laser guide star brightness for different pulsed laser formats. The simulation assumes that the pulse repetition frequency is sufficiently low (smaller than a few kilohertz) that atomic collisions restore local thermodynamic equilibrium between pulses. It is also assumed that the pulse length is short compared to the Larmor precession time scale. The numerical results are well-approximated by a simple analytic model for a three-level atom. The number of emitted photons is found to be primarily dependent on the product of the length of the laser pulse and the energy density.
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.
A general, analytic theory is presented of the excitation of a two-level atom by a series of short optical laser pulses. The atom is assumed to be isolated (collisionless) with radiative decay from upper to lower state at rate γ. The laser pulses are assumed to have a duration tp such that γtp ≪ 1 (the transient regime), and the spacing between pulses tr is such that tr ≫ tp. In this regime the atomic density matrix is calculated as a function of time during each optical pulse. An analytic expression for the atomic state (Bloch vector) after n optical pulses is also derived. For large n the atomic state converges to a repeating (steady state) cycle of pump excitation followed by radiative decay. The excited-state population in the steady state can be large but only if the spacing between pulses tr is small so that γtr ≲ 0.5. These results indicate that lasers that produce trains of pulses, such as mode-locked conventional lasers and free-electron lasers, can be effective in optical pumping of atoms.
Some of the adaptive optics systems contemplated to ameliorate the effects of the atmospheric turbulence utilize guide stars. Guide stars are artificial beacons formed in the atmosphere by the interaction of a laser with matter in the atmosphere. Sodium guide stars are formed as a result of laser interaction with the sodium layer in the upper atmosphere. A small rf FEL is one of the candidates considered as a potential source capable of creating a sodium beacon. The pulse format of such a laser consists of trains of short optical pulses. The duration of an individual pulse is typically shorter than that of relevant relaxation times in the medium. The separation of the pulses may be longer or of the order of the relevant atomic memory time scale. To properly describe interactions of the atmospheric sodium layer with a laser characterized by such a pulse format one must utilize the optical Bloch equations (OBE). The OBE have been solved numerically using a simplified two-level model, and the resulting sodium layer response has been characterized. Additionally, an analytic solution has been obtained for the case in which both the population and coherence relaxation times are equal. The model is capable of providing useful insights.
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.