# Atomic transitions for adaptive optics

Rui Yang  School of Physics and Astronomy, Yunnan University, South Section,
East Outer Ring Road, Chenggong District, Kunming, 650500, China  Department
of Physics and Astronomy, University of British Columbia, 6224 Agriculture
Road, Vancouver BC, V6T 1Z1,Canada  Joschua Hellemeier  Department of Physics
and Astronomy, University of British Columbia, 6224 Agriculture Road,
Vancouver BC, V6T 1Z1,Canada  Paul Hickson  Department of Physics and
Astronomy, University of British Columbia, 6224 Agriculture Road, Vancouver
BC, V6T 1Z1,Canada  National Astronomical Observatories, Chinese Academy of
Sciences, A20 Datun Road, Chaoyang District, Beijing, China  Corresponding
author<EMAIL_ADDRESS>
###### Abstract

Adaptive optics systems using sodium laser guide stars are widely employed at
major astronomical observatories. It is natural to ask whether other atomic
species might offer advantages. In this paper we review all abundant atoms and
ions in the upper atmosphere, including Na, Fe, Mg+, Si+, Ca+, K and also the
non-metallic species N, N+, O, H, considering their potential for adaptive
optics. Return fluxes for all transitions that can be excited using either one
or two wavelengths were computed. We also considered multi-wavelength
emission, comparing the performance of different transitions for polychromatic
laser guide star (PLGS) adaptive optics. We find that of all the mesospheric
metals, Na is the most suitable for both monochromatic and polychromatic laser
guide stars, providing about six times more return flux than the best
transitions in Fe. For high-altitude observatories, excitation at 330 nm in Na
should give the highest PLGS performance. Atomic O, N and N+ have strong
transitions and very high abundances in the mesosphere. This makes them
potential candidates for the generation of intense laser guide stars by
amplified spontaneous emission, if a suitable excitation process can be
demonstrated. Direct excitation by CW lasers is impractical as all transitions
from the ground state are beyond the atmospheric cutoff. Nevertheless, it may
be possible using high-power pulsed lasers.

††journal: josab

##  1 Introduction

Adaptive optics (AO) systems are employed by many ground-based telescopes to
compensate atmospheric turbulence, thereby greatly improving the resolution of
astronomical images [1]. Many of these systems employ laser guide star (LGS)
beacons, which allow sensing of atmospheric turbulence in any desired
direction, increasing sky coverage [2]. Large telescopes typically employ
resonant LGS systems in which transitions in mesospheric atom are excited,
typically the $D_{2}$ transition of neutral atomic sodium. Although other
chemical species have been considered [3, 4], sodium is preferred due to its
relatively-high abundance and strong resonance lines.

Despite the impressive success of AO systems employing sodium LGS, there are
several reasons to reconsider other atoms and ions. The performance of such AO
systems is limited by the return flux of the LGS, due to photon shot noise.
Advances have been made both in increasing laser power, and improving the
efficiency of sodium LGS [5, 6, 7, 8, 9]. For example, simulations indicate
that repumping atoms from the $D_{2b}$ ground state, in combination with
circular polarization, can increase the return flux by a factor of $3-4$ [6].
Nevertheless, any technique that might further increase the return flux would
be valuable.

Several new techniques that might potentially achieve high-intensity LGS are
discussed in [10] . One possible approach is amplified spontaneous emission
(ASE) [11]. In this idea, laser pumping of an abundant atmospheric atomic or
molecular species could produce a population inversion that would amplify
radiation generated by spontaneous emission. A critical requirement for ASE is
a suitable transition for which the optical depth $\tau$, defined as the
integral of the absorption coefficient along the propagation path, is negative
and has an absolute value that exceeds unity. This is a consequence of the
equation of radiative transfer, from which it follows that the gain in
intensity is proportional to $\exp(-\tau)$.

Another motivation for considering alternatives to sodium is the problem of
atmospheric tilt indeterminacy [12]. Backscattered photons follow the same
path as the transmitted beam, so the apparent position of the LGS is
independent of wavefront tilt induced by the atmosphere. A number of
techniques have been proposed to break this degeneracy (see, for example, [13,
14] and references therein). A promising approach is that of polychromatic
laser guide stars (PLGS) [15, 16]. In this technique, a laser is used to
generate emission at two or more widely-separated wavelengths. Because of
differential refraction, the backscattered light propagates along different
paths through the atmosphere. Thus one need not rely on natural guide stars to
sense the atmospheric tilt. The differential positions of the PLGS at
different wavelengths is small and must be measured with high precision.
Creating sufficiently-intense polychromatic emission is a challenge.

There is also a related problem of focus indeterminacy, as the sodium centroid
altitude varies on short timescales [17]. This makes it difficult to separate
atmospheric focus variations from other focus errors. This problem primarily
affects very-large telescopes as the wavefront error due to focus variations
increases in proportion to the square of the telescope aperture diameter [18].

The present paper examines atoms and ions in the upper atmosphere that might
be useful for AO. We consider the abundance and distribution of all relevant
species and compute the relative return flux for transitions that can be
excited by ground-based lasers. Our aim is to identify potential transitions
for monochromatic and PLGS, and to assess the feasibility of ASE.

##  2 Mesospheric atoms and parameters of interest

Meteoric ablation is the primary source of metallic species in the mesosphere
and lower thermosphere (MLT) region of the atmosphere, at altitudes between 80
and 120 km. In this altitude range, the atmospheric density decreases from
$10^{-5}$ to $10^{-8}$ kg m-3 and collision rates are in the range
$10^{3}-10^{5}$ s-1. This is also the coldest region of the atmosphere. At
mid-latitude sites, the temperature at the mesopause ranges from $178-192$ K,
at an altitude of $85-102$ km, depending on the season [19].

Typical values of the column density $N$, altitude of the centroid of the
density distribution, and the RMS width (vertical extent) for each species are
listed in Table 1. For sodium, we use recent data by Gardner [20] to update
earlier values listed by Happer [3]. The centroid altitude and RMS width for
non-metallic species were computed from the MSIS-90 atmospheric model [21].
These were evaluated at midnight, January 1 for latitude $+30^{\circ}$ and
longitude $0^{\circ}$.

The major meteoric species are Na, Fe, Mg and Si. There are also two less-
abundant species, Ca and K. These metals exist as layers of atoms between
about 80 and 105 km and occur primarily as atomic ions at higher altitudes.
Below 85 km they form compounds: oxides, hydroxides, and carbonates, which
polymerize into nanometer-sized meteoric smoke particles. Above 85 km,
photolysis of N2, O2 and H2O by extreme ultraviolet (EUV) radiation leads to
high concentrations of nitrogen, oxygen and hydrogen atoms. These atoms attack
metallic compounds such as hydroxides and oxides, reducing them and thereby
returning the metals to their atomic phase.

Table 1: Parameters of mesospheric atoms and ions Species |  $N$ | Altitude | RMS width | Sources
---|---|---|---|---
|  m-2 | km | km |
Na |  $4.0$$\times 10^{13}$ | $91.5$ | $4.6$ | [20]
Fe |  $10.2$$\times 10^{13}$ | $88.3$ | $4.5$ | [20]
Mg |  $1$$\times 10^{13}$ | $89.4$ | $4.2$ | [23, 22]
Mg+ |  $8$$\times 10^{13}$ | $94.6$ | $7.0$ | [23, 22]
Si+ |  $4$$\times 10^{13}$ | $111.0$ | $10.1$ | [24]
Ca |  $3.4$$\times 10^{11}$ | $90.5$ | $3.5$ | [20]
Ca+ |  $7.2$$\times 10^{11}$ | $95.0$ | $3.6$ | [20]
K |  $4.5$$\times 10^{11}$ | $91.0$ | $4.7$ | [20]
N |  $1$$\times 10^{18}$ | $223.0$ | $60.2$ | [25, 21]
N+ |  $1$$\times 10^{18}$ | - | - | [25]
O |  $6.5$$\times 10^{21}$ | $110.3$ | $28.7$ | [26, 21]
H |  $2.3$$\times 10^{18}$ | $95.6$ | $38.9$ | [27, 21]

In a typical AO application, a laser is used to excite atoms from the ground
state to an excited state. The atom then returns to the ground state via
spontaneous and stimulated emission. Only spontaneous emission contributes to
backscattered radiation since stimulated emission propagates in the same
direction as the laser radiation. The LGS return flux can be computed from the
equation of radiative transfer. For an optically-thin medium (optical depth
$\tau\ll 1$) the specific intensity $I_{\nu}$ (W m-2 Hz-1 sr-1) of the
backscattered radiation is given by [10]

$I_{\nu}(\nu)=\frac{h\nu}{4\pi}NA_{21}x\varphi(\nu)=\frac{2h\nu}{\lambda^{2}}N\sigma_{21}(\nu)x.$
(1)

Here $\lambda$ is the wavelength, $\nu=c/\lambda$ is the frequency, $c$ is the
speed of light in vacuum and $N$ is the column density of atoms of this
species. $A_{21}$ is the Einstein A coefficient for the transition and the
function $\varphi(\nu)$ is the mean line profile, normalized so that its
integral over frequency $\nu$ is unity. The dimensionless parameter $x$ is the
occupation fraction of the upper state, i.e. the equilibrium fraction of atoms
found in this state under continuous excitation by a laser having specific
energy density $\rho_{\nu}$, and $\sigma_{21}(\nu)$ is the cross section for
stimulated emission. The second equality follows from the relation between
stimulated and spontaneous emission, $B_{21}$ = $\lambda^{3}A_{21}/8\pi h$,
and the definition

$\sigma_{21}(\nu)\equiv\frac{h\nu}{c}B_{21}\varphi_{\nu}(\nu)=\frac{\lambda^{2}}{8\pi}A_{21}\varphi(\nu).$
(2)

Here the subscript 1 denotes the lower (ground) state, 2 denotes the upper
state, and $B_{21}$ is the Einstein B coefficient for stimulated emission.

For a CW laser having beam area $\mathcal{A}$ in the MLT, the return flux
$\Phi$, in units of photons s-1 m-2 is related to the intensity by

$\Phi=\frac{\mathcal{A}}{z^{2}}\int_{0}^{\infty}\frac{I_{\nu}(\nu)}{h\nu}d\nu,$
(3)

where $z$ is the line-of-sight distance to the LGS.

In the mesosphere, Doppler broadening is the dominant line-broadening process.
As the temperature varies within the MLT, the line profile is more precisely
written as

$\varphi(\nu)=\frac{1}{N}\int_{0}^{\infty}n(z)\varphi(\nu,z)dz,$ (4)

where $n(z)$ is the number density of the atomic species under consideration
at distance $z$ along the line of sight. In practice, the function
$\varphi(\nu)$ is well-represented by a Gaussian

$\varphi(\nu)=\frac{2}{w}\sqrt{\frac{\ln 2}{\pi}}e^{-4\ln
2(\nu-\nu_{0})^{2}/w^{2}},$ (5)

where $w$ is the line width at half maximum intensity. It depends on the
characteristic temperature $T$ in the MLT and the mass $m$ of the atomic
species,

$w=\frac{1}{\lambda_{0}}\left(\frac{8k_{B}T\ln 2}{m}\right)^{1/2}$ (6)

where $k_{B}$ is Boltzmann’s constant. Adopting $T=180$ K results in a width
for the sodium D line of 1.02 GHz.

The absorption cross section $\sigma_{12}(\nu)$ is given by

$\sigma_{12}(\nu)=\frac{g_{2}}{g_{1}}\sigma_{21}(\nu),$ (7)

where $g_{1}$ and $g_{2}$ are the statistical weights of the lower and upper
state, respectively. Tabulated values of cross sections
$\sigma_{21}\equiv\sigma_{21}(\nu_{0})$ are computed at the line center
$\nu_{0}$.

Substituting Eqs. (1) and (2) into Eq. (3), we obtain

$\displaystyle\Phi$ $\displaystyle=\frac{\mathcal{A}}{4\pi
z^{2}}NA_{21}\int_{0}^{\infty}x(\nu)\varphi(\nu)d\nu$
$\displaystyle\simeq\frac{\mathcal{A}}{4\pi
z^{2}}NA_{21}x(\nu_{0})\varphi(\nu_{0})\Delta\nu,$ (8)

where $\Delta\nu$ is the laser line width. The second equality follows because
the line width for a CW laser, and therefore the frequency range over which
$x$ is nonzero, is typically much smaller than the Doppler width. From this we
see that the atomic quantity that best characterizes the LGS return flux, is

$\varepsilon=NA_{21}x(\nu_{0}).$ (9)

We shall call $\varepsilon$ the _return flux coefficient_ for the transition.
It has units of photons s-1 m-2.

Also of interest is the dimensionless product $N\sigma_{21}$, which determines
the maximum possible optical depth. Specifically, $\tau$ is bounded by
$-N\sigma_{21}\leq\tau\leq N\sigma_{12}$ [10]. Therefore, ASE is not possible
if $N\sigma_{21}<1$.

Several other physical processes also play a role in the response of atoms to
laser excitation. Atomic collisions effectively reset atoms that have been
excited, either returning them to the ground state, or causing a transition to
a different excited state. The latter process makes the atom unavailable for
excitation by the laser. At the same time, collisions impart a velocity to the
atom, generally pushing it out of the velocity interval that can be excited by
the laser (the “velocity class”). In equilibrium this is balanced by
collisions that move atoms into the velocity class. Those atoms are in the
ground state as the collision energies, typically $\sim 0.02$ eV, are not
sufficient to excite the atom from the ground state. Therefore any individual
atom is effectively “reset” to the ground state on a time scale equal to the
mean time between collisions.

Collision rates vary by species and altitude. Rates for the species of
interest are calculated in Appendix A, for a range of altitudes.

In order to be useful for AO, a transition should have a high spontaneous
transition rate to the lower state (Einstein $A$ coefficient). Also, as most
atoms will be in the ground state, there should be a path to the upper state
from the ground state, with a high transition probability (Einstein $B$
coefficient). As the Einstein coefficients are related, it is sufficient to
consider transitions that have a large $A$ coefficient.

A second consideration is that the upper state should not have many
transitions to metastable states, which have low $A$ coefficients. In such a
situation, atoms can become trapped in those states, reducing the number of
atoms available for the desired transition. Ultimately, atoms trapped in
metastable states will be reset at the collision rate.

The performance of PLGS will critically depend on the wavelengths and return
flux of the available transitions. For amplitude $\theta$ of a Zernike mode
(except for piston), the ratio of the difference in amplitude $\Delta\theta$
at different wavelengths, is related to the ratio of refracting indexes by
[15]

$p\equiv\frac{\theta}{\Delta\theta}=\frac{n_{\lambda_{\text{obs}}}-1}{n_{\lambda_{1}}-n_{\lambda_{2}}},$
(10)

where $n_{\lambda_{1}}$ and $n_{\lambda_{2}}$ are the refractive indexes at
wavelengths of the measurement and $n_{\lambda_{obs}}$ is the refractive index
of the wavelength at which the amplitude of the Zernike mode is determined.
The dimensionless ratio $p$ is called the _penalty factor_. Smaller values of
$p$ correspond to greater PLGS performance. We take $\lambda_{\text{obs}}=500$
nm as the reference wavelength.

The impact of return flux and penalty factor can be captured in a single merit
function. The signal-to-noise ratio for the measurement of the atmospheric
wavefront tilt is proportional to the square root of the return flux and
inversely proportional to the penalty factor [15]. Therefore,
$\varepsilon^{1/2}/p$ is a measure of relative PLGS performance. To make this
dimensionless, we divide by the same quantity evaluated for a reference
system. We thus define the PLGS merit function

$q\equiv\frac{p_{0}}{p}\left(\frac{\varepsilon}{\varepsilon_{0}}\right)^{1/2},$
(11)

where $\varepsilon_{0}$ and $p_{0}$ are reference values. For these we use the
330.298/2208.370 nm transitions in Na I, which is the highest-performance
system employing single-photon excitation. In all cases, the return flux
coefficient used is that of the weaker line.

When discussing atomic transitions we use spectroscopic notation and refer to
Fe as Fe I, Fe+ as Fe II, etc. We refer to transitions having wavelengths
greater than 300 nm as “visible” transitions. Shorter-wavelengths are greatly
attenuated by the atmosphere so we refer to those as “vacuum-ultraviolet”
transitions. We refer to transitions having an Einstein $A$ coefficient
greater than $10^{6}$ s-1 as “strong transitions”, and “weak transitions”
otherwise. This distinction is somewhat arbitrary but serves to generally
separate permitted lines from forbidden lines. An excitation scheme that uses
two different transitions to reach an excited state will be called a “two-
step” process. One that uses two photons to excite a single transition, using
a virtual intermediate state, will be called a “two-photon” process.

Transition wavelengths described in the text and listed in the tables in the
following sections are in air for wavelengths greater than 185 nm and in
vacuum otherwise.

##  3 Analysis and Results

The data set employed for this analysis is the atomic database of the National
Institute of Standards and Technology (NIST) [28]. Complete data for all
transitions were downloaded and a computer program was written to apply
various criteria to identify transitions of interest for AO. We examined all
excited states that could be reached from the ground state by at most two
successive excitations using wavelengths greater than 300 nm. For species in
which no such transitions are possible, two-photon excitation was considered.
For every such state, all possible downward transitions were examined, and the
excitation fraction $x$ of the excited state was computed according to the
methods presented in Appendix B. A specific energy density of
$\rho_{\nu}=10^{-12}$ J m-1 Hz-1 was assumed for the laser excitation. This is
at the high end of range of energy densities that can be achieved with current
AO lasers. A complete listing of all such transitions that have $A\geq 10^{7}$
s-1 is available from the authors upon request. From this and the data of
Table 1, the return flux coefficient was computed. For PLGS, the penalty
factor $p$ and merit function $q$ were computed for all downward transitions.

Our results are shown in Tables 2 – 9. Table 2 lists all transitions, for all
species, that have a return flux coefficient $\varepsilon>10^{18}$ photons s-1
m-2. In this, and subsequent tables, $\lambda_{\text{ex}}$ refers to the
excitation wavelength(s) employed to reach the excited state, and $\lambda$ is
the wavelength of the downward transition. Unless otherwise specified, values
of $p$ are computed with respect to the smallest wavelength.

We now discuss individual atomic species in more detail.

###  3.1 Sodium

Sodium is most-often employed for the generation of LGS. Neutral sodium is
used as it is more abundant than ionized forms. Averaged over all seasons and
latitudes, the observed ratio of Na+ to Na is between 0.2 and 0.25 [29]. An
energy level diagram of neutral sodium, indicating the lifetimes and
wavelengths for selected transitions is shown in Fig. 1. For simplicity, fine-
structure splitting of the levels is not shown.

At visible wavelengths, Na I has four strong transitions from the ground
state, at 330.237, 330.298, 588.995 and 589.592 nm. The latter two form the
sodium D doublet. A fifth transition, at 388.390 nm, has a very-small rate
($A=6.95\times 10^{-4}$ s-1).

The transition between the 3S1/2 ground state and the 3P3/2 excited state, the
D2 line, has been of interest for adaptive optics since 1985 [2]. Laser-
induced excitation followed by spontaneous emission produces the backscattered
photons of the LGS. The LGS return flux is limited by the low abundance and
small absorption cross section, $N\sigma_{12}\simeq 3$%. In practice, the D2b
line, which involves only transitions between the $F=2$ lower and $F=1,2$ and
3 upper sub-levels, is used for AO. A CW laser can excite 3/4 of the available
atoms to the upper state, corresponding to the ratio $15/20$ of the number of
magnetic substates in the upper levels to the total. Current AO lasers are
sufficiently powerful that the excitation fraction closely approaches this
limit. However, only a fraction of the total number of sodium atoms are
available for excitation because of limited spectral overlap as the laser
linewidth is typically much smaller than the sodium linewidth. This limitation
might be overcome with further development of efficient broad-band modeless
pulsed lasers [30].

Figure 1: Low-lying energy levels of the sodium atom and transitions of
interest.  Table 2. Transitions for a monochromatic LGS having $\lambda\geq
300$ nm, $\lambda_{\text{ex}}\geq 150$ nm and $\log(\varepsilon)>18.3$.
---
Species | $\lambda_{\text{ex}}$ | $\lambda$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
| (nm) | (nm) |  |
Na I | $588.995$ | $588.995$ | $21.20$ | $-1.50$
| $589.592$ | $589.592$ | $21.07$ | $-1.50$
| $330.237$ | $2205.647$ | $20.00$ | $-0.75$
| $330.237$ | $588.995$ | $19.83$ | $-1.50$
| $330.298$ | $2208.370$ | $19.82$ | $-0.75$
| $330.237$ | $1140.377$ | $19.82$ | $-1.19$
| $330.298$ | $588.995$ | $19.65$ | $-1.50$
| $330.298$ | $1140.377$ | $19.65$ | $-1.19$
| $330.237$ | $330.237$ | $19.61$ | $-3.61$
| $330.237$ | $589.592$ | $19.52$ | $-1.50$
| $330.237$ | $1138.144$ | $19.52$ | $-1.49$
| $330.298$ | $330.298$ | $19.44$ | $-3.61$
| $330.298$ | $589.592$ | $19.37$ | $-1.50$
| $330.298$ | $1138.144$ | $19.35$ | $-1.49$
| $285.281$ | $5426.880$ | $19.20$ | $-0.25$
| $285.281$ | $588.995$ | $19.18$ | $-1.50$
| $285.301$ | $5434.170$ | $18.98$ | $-0.25$
| $285.301$ | $588.995$ | $18.95$ | $-1.50$
| $285.281$ | $1140.377$ | $18.93$ | $-1.19$
| $285.281$ | $1074.645$ | $18.91$ | $-2.65$
| $285.281$ | $589.592$ | $18.87$ | $-1.50$
| $285.281$ | $616.075$ | $18.78$ | $-2.54$
| $268.034$ | $588.995$ | $18.71$ | $-1.50$
| $285.301$ | $1140.377$ | $18.71$ | $-1.19$
| $285.301$ | $1074.930$ | $18.68$ | $-2.65$
| $285.301$ | $589.592$ | $18.67$ | $-1.50$
| $285.281$ | $3414.261$ | $18.64$ | $-0.45$
| $285.281$ | $1138.144$ | $18.63$ | $-1.49$
| $268.034$ | $10\,807.960$ | $18.61$ | $0.14$
| $285.301$ | $616.075$ | $18.56$ | $-2.54$
| $285.281$ | $2205.647$ | $18.51$ | $-0.75$
| $285.281$ | $615.423$ | $18.49$ | $-2.84$
| $268.043$ | $588.995$ | $18.45$ | $-1.50$
| $285.301$ | $3414.261$ | $18.42$ | $-0.45$
| $268.034$ | $1140.377$ | $18.41$ | $-1.19$
| $285.301$ | $1138.144$ | $18.41$ | $-1.49$
| $268.034$ | $589.592$ | $18.40$ | $-1.50$
| $268.043$ | $10\,823.050$ | $18.36$ | $0.14$
| $259.387$ | $588.995$ | $18.34$ | $-1.50$
| $285.281$ | $3407.755$ | $18.34$ | $-0.75$
| $330.237$ | $9090.560$ | $18.33$ | $-0.57$
| $330.237$ | $819.482$ | $18.33$ | $-1.15$
| $268.034$ | $2440.182$ | $18.32$ | $-2.08$
| $268.034$ | $864.993$ | $18.31$ | $-3.44$
Fe I | $371.993$ | $371.993$ | $20.40$ | $-2.08$
| $385.991$ | $385.991$ | $18.72$ | $-2.26$
| $344.061$ | $344.061$ | $18.49$ | $-2.16$
| $367.991$ | $373.713$ | $18.18$ | $-2.14$
| $296.690$ | $373.486$ | $18.17$ | $-1.33$
| $216.677$ | $349.710$ | $18.13$ | $-2.42$
| $302.064$ | $302.064$ | $18.13$ | $-1.68$
| $261.871$ | $381.584$ | $18.08$ | $-1.21$
| $302.064$ | $382.042$ | $18.07$ | $-1.43$
| $344.061$ | $349.057$ | $18.05$ | $-2.59$
| $213.202$ | $340.746$ | $18.04$ | $-1.62$
| $437.593$ | $646.271$ | $18.04$ | $-3.59$
| $319.166$ | $516.749$ | $18.03$ | $-2.43$
Ca I | $422.673$ | $422.673$ | $19.72$ | $-3.34$
K I | $766.490$ | $766.490$ | $19.05$ | $-3.21$
| $769.896$ | $769.896$ | $18.92$ | $-3.21$
Mg I | $202.582$ | $1710.866$ | $18.15$ | $-1.55$
| $202.582$ | $1182.819$ | $18.15$ | $-1.63$

Quantitative results for single-photon excitation in Na I are shown in Table
2. In Table 3 we list all transitions in Na I that are available with two-step
excitation and have a return flux coefficient $\varepsilon>10^{20}$ photons
s-1 m-2. There are many, but even the strongest is two times weaker that the
sodium D2 line.

Of particular interest for PLGS are transitions from the 4P3/2 state [16].
This state may be populated directly from the ground state by absorption of a
330 nm photon. De-excitation occurs through two pathways. Approximately 29%
decay directly to the ground state, emitting a 330 nm photon and 70% decay to
the 4S1/2 state, emitting a 2206 nm photon. That state can decay to the ground
state via two pathways. About 67% decay via the 3P3/2 state, emitting 1140 and
589 nm photons. The remaining decay via the 3P1/2 state emitting 1138 and 590
nm photons. About 1.5% of the atoms in the 4P3/2 state return to the ground
state via the 3D5/2 state, emitting 9093, 819 and 589 nm photons.

Table 3. Na I transitions for a monochromatic LGS employing two-step
excitation having $\lambda\geq 300$ nm and $\log(\varepsilon)>20$.
---
$\lambda$ | $\lambda_{\text{ex}}$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
(nm) | (nm) |  |
$819.482$ | 588.995+819.482 | $21.00$ | $-1.15$
$818.326$ | 589.592+818.326 | $20.89$ | $-1.23$
$818.326$ | 588.995+819.479 | $20.68$ | $-1.23$
$589.592$ | 588.995+819.479 | $20.68$ | $-1.50$
$1140.377$ | 589.592+1138.144 | $20.31$ | $-1.19$
$588.995$ | 589.592+1138.144 | $20.31$ | $-1.50$
$1140.377$ | 588.995+1140.377 | $20.22$ | $-1.19$
$568.820$ | 588.995+568.820 | $20.19$ | $-2.26$
$819.479$ | 589.592+818.326 | $20.19$ | $-1.93$
$588.995$ | 589.592+818.326 | $20.19$ | $-1.50$
$588.995$ | 330.237+2337.915 | $20.18$ | $-1.50$
$568.263$ | 589.592+568.263 | $20.10$ | $-2.34$
$568.820$ | 330.237+2337.915 | $20.08$ | $-2.26$
$1138.144$ | 589.592+1138.144 | $20.01$ | $-1.49$

It is also possible to use a 330 nm photon to excite directly to the 4P1/2
state, rather than the 4P3/2 state, with a comparable transition rate. That
state decays directly to the ground state with a 29% probability, to the 4S1/2
state with a 70% probability, and to the 3D3/2 state with a 1.6% probability,
producing a 9140 nm photon. The 3D3/2 state decays to ground state via the
3P1/2 state, producing 818 and 590 nm photons, or via the 3P3/2 state,
producing 819 and 589 nm photons.

We searched for all possible PLGS line combinations employing single-photon
excitation in Na I, with the condition that they have a return flux greater
than $10^{17}$ photons s-1 m-2 (four orders of magnitude weaker than the Na D2
lines) and a penalty factor less than 50. The results are shown in Table 4.
The only possibilities are the two already mentioned.

Table 4. Na I transitions for a PLGS. All transitions having $q>0.4$ and
$\lambda_{\text{ex}}\geq 300$ nm are listed.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) |  |  | (nm) |  | (nm) |
$330.237$ | $1.00$ | $18.59$ | $330.237$ | $19.61$ | $2205.647$ | $20.00$
| $0.94$ | $19.74$ | $330.237$ | $19.61$ | $1140.377$ | $19.82$
| $0.84$ | $19.74$ | $330.237$ | $19.61$ | $1138.144$ | $19.52$
| $0.72$ | $25.86$ | $330.237$ | $19.61$ | $588.995$ | $19.83$
| $0.65$ | $25.84$ | $330.237$ | $19.61$ | $589.592$ | $19.52$
$330.298$ | $0.82$ | $18.60$ | $330.298$ | $19.44$ | $2208.370$ | $19.82$
| $0.77$ | $19.75$ | $330.298$ | $19.44$ | $1140.377$ | $19.65$
| $0.69$ | $19.75$ | $330.298$ | $19.44$ | $1138.144$ | $19.35$
| $0.59$ | $25.87$ | $330.298$ | $19.44$ | $588.995$ | $19.65$
| $0.54$ | $25.85$ | $330.298$ | $19.44$ | $589.592$ | $19.37$

Several transitions involving two-step excitation have been discussed in the
literature. The processes of 3S${}_{1/2}\to$ 3P${}_{3/2}\to$ 4D5/2 ($589+569$
nm) [15], 3S${}_{1/2}\to$ 3P${}_{3/2}\to$ 4S1/2 ($589+1140$ nm) [31] and
3S${}_{1/2}\to$ 3P${}_{3/2}\to$ 3D5/2 ($589+819.7$ nm) [32]. Excitation via
the $3P_{1/2}$ state is also possible but with somewhat lower transition
rates.

Atoms excited to the 3P levels can be further excited to the 4D, 4S or 3D
levels by absorption of visible or infrared light, provided that this is done
within a few ns. Sodium can also be excited from the ground state to the 4D5/2
level by absorption of two 578 nm photons. This is a non-linear process
involving a virtual intermediate state, requiring picosecond pulses and a
laser irradiance on the order of $10^{10}$ W m-2 [31].

All possible combinations employing two-step excitation in Na I are listed in
Table 5. The selection criteria is $q>0.4$. The strongest of these is
$589+569$ nm excitation producing lines at 569 and 2334 nm. These transitions
would provide a SNR that is 0.79 times the single-photon excitation
transitions shown in Table 4. That is roughly equivalent to a decrease to 65%
in LGS return flux.

Table 5. Na I transitions for a PLGS employing two-step excitation. All
transitions that have $q>0.4$ and $\lambda_{\text{ex}}\geq 300$ nm are listed.
The last line for each excitation scheme shows the return flux coefficient for
the first excitation wavelength.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) |  |  | (nm) |  | (nm) |
588.995 | $0.79$ | $18.55$ | $330.237$ | $19.41$ | $2337.915$ | $19.95$
\+ 568.820 | $0.79$ | $18.59$ | $330.237$ | $19.41$ | $2205.647$ | $19.79$
| $0.75$ | $19.74$ | $330.237$ | $19.41$ | $1140.377$ | $19.62$
| $0.67$ | $19.74$ | $330.237$ | $19.41$ | $1138.144$ | $19.32$
| $0.55$ | $26.69$ | $330.237$ | $19.41$ | $568.820$ | $20.19$
| $0.51$ | $25.84$ | $330.237$ | $19.41$ | $589.592$ | $19.32$
| $0.45$ | $60.81$ | $568.820$ | $20.19$ | $2337.915$ | $19.95$
|  |  | $588.995$ | $20.78$ |  |
589.592 | $0.71$ | $18.56$ | $330.298$ | $19.32$ | $2334.843$ | $19.86$
\+ 568.263 | $0.71$ | $18.60$ | $330.298$ | $19.32$ | $2208.370$ | $19.70$
| $0.67$ | $19.75$ | $330.298$ | $19.32$ | $1140.377$ | $19.60$
| $0.66$ | $19.75$ | $330.298$ | $19.32$ | $1138.144$ | $19.30$
| $0.59$ | $18.56$ | $330.298$ | $19.32$ | $2337.896$ | $19.16$
| $0.51$ | $25.87$ | $330.298$ | $19.32$ | $588.995$ | $19.82$
| $0.49$ | $18.60$ | $330.298$ | $19.32$ | $2205.647$ | $19.00$
| $0.49$ | $26.71$ | $330.298$ | $19.32$ | $568.819$ | $19.40$
| $0.49$ | $26.73$ | $330.298$ | $19.32$ | $568.263$ | $20.10$
| $0.41$ | $60.69$ | $568.263$ | $20.10$ | $2334.843$ | $19.86$
|  |  | $589.592$ | $20.65$ |  |
588.995 | $0.57$ | $18.56$ | $330.298$ | $19.13$ | $2334.843$ | $19.67$
\+ 568.819 | $0.57$ | $18.60$ | $330.298$ | $19.13$ | $2208.370$ | $19.51$
| $0.54$ | $19.75$ | $330.298$ | $19.13$ | $1140.377$ | $19.41$
| $0.53$ | $19.75$ | $330.298$ | $19.13$ | $1138.144$ | $19.11$
| $0.48$ | $18.56$ | $330.298$ | $19.13$ | $2337.896$ | $18.97$
| $0.41$ | $25.85$ | $330.298$ | $19.13$ | $589.592$ | $19.98$
| $0.40$ | $18.60$ | $330.298$ | $19.13$ | $2205.647$ | $18.81$
| $0.40$ | $26.71$ | $330.298$ | $19.13$ | $568.819$ | $19.21$
| $0.40$ | $26.73$ | $330.298$ | $19.13$ | $568.263$ | $19.91$
|  |  | $588.995$ | $20.96$ |  |
330.237 | $0.51$ | $18.56$ | $330.298$ | $19.03$ | $2334.843$ | $19.57$
\+ 2337.896 | $0.51$ | $18.60$ | $330.298$ | $19.03$ | $2208.370$ | $19.41$
| $0.51$ | $18.60$ | $330.298$ | $19.03$ | $2205.647$ | $19.64$
| $0.48$ | $19.75$ | $330.298$ | $19.03$ | $1140.377$ | $19.66$
| $0.48$ | $19.75$ | $330.298$ | $19.03$ | $1138.144$ | $19.36$
| $0.43$ | $18.56$ | $330.298$ | $19.03$ | $2337.896$ | $18.87$
|  |  | $330.237$ | $19.25$ |  |
588.995 | $0.50$ | $126.58$ | $589.592$ | $20.68$ | $818.326$ | $20.68$
\+ 819.479 | $0.23$ | $126.21$ | $589.592$ | $20.68$ | $819.479$ | $19.99$
|  |  | $588.995$ | $20.91$ |  |
589.592 | $0.50$ | $83.39$ | $588.995$ | $20.31$ | $1140.377$ | $20.31$
\+ 1138.144 | $0.35$ | $83.50$ | $588.995$ | $20.31$ | $1138.144$ | $20.01$
|  |  | $589.592$ | $20.87$ |  |
588.995 | $0.49$ | $18.27$ | $330.237$ | $18.99$ | $5426.880$ | $19.01$
\+ 498.281 | $0.49$ | $18.28$ | $330.237$ | $18.99$ | $5013.920$ | $19.30$
| $0.49$ | $18.59$ | $330.237$ | $18.99$ | $2205.647$ | $19.37$
| $0.47$ | $19.09$ | $330.237$ | $18.99$ | $1477.974$ | $19.49$
| $0.46$ | $19.74$ | $330.237$ | $18.99$ | $1140.377$ | $19.30$
| $0.46$ | $19.74$ | $330.237$ | $18.99$ | $1138.144$ | $18.99$
|  |  | $588.995$ | $20.78$ |  |
589.592 | $0.47$ | $18.37$ | $330.237$ | $18.94$ | $3414.261$ | $19.49$
\+ 615.423 | $0.47$ | $18.37$ | $330.237$ | $18.94$ | $3407.755$ | $19.19$
| $0.46$ | $18.59$ | $330.237$ | $18.94$ | $2208.370$ | $19.03$
| $0.46$ | $18.59$ | $330.237$ | $18.94$ | $2205.647$ | $19.33$
| $0.44$ | $19.74$ | $330.237$ | $18.94$ | $1140.377$ | $19.33$
| $0.44$ | $19.74$ | $330.237$ | $18.94$ | $1138.144$ | $19.03$
|  |  | $589.592$ | $20.86$ |  |
588.995 | $0.44$ | $18.37$ | $330.237$ | $18.88$ | $3414.261$ | $19.42$
\+ 616.075 | $0.44$ | $18.37$ | $330.237$ | $18.88$ | $3407.755$ | $19.12$
| $0.43$ | $18.59$ | $330.237$ | $18.88$ | $2208.370$ | $18.96$
| $0.43$ | $18.59$ | $330.237$ | $18.88$ | $2205.647$ | $19.26$
| $0.41$ | $19.74$ | $330.237$ | $18.88$ | $1140.377$ | $19.26$
| $0.40$ | $19.74$ | $330.237$ | $18.88$ | $1138.144$ | $18.96$
|  |  | $588.995$ | $21.03$ |  |
589.592 | $0.44$ | $18.28$ | $330.298$ | $18.88$ | $5434.170$ | $18.92$
\+ 497.854 | $0.44$ | $18.29$ | $330.298$ | $18.88$ | $5007.670$ | $19.21$
| $0.43$ | $18.60$ | $330.298$ | $18.88$ | $2208.370$ | $19.26$
| $0.42$ | $19.10$ | $330.298$ | $18.88$ | $1476.749$ | $19.40$
| $0.40$ | $19.75$ | $330.298$ | $18.88$ | $1140.377$ | $19.28$
| $0.40$ | $19.75$ | $330.298$ | $18.88$ | $1138.144$ | $18.98$
|  |  | $589.592$ | $20.66$ |  |
330.237 | $0.40$ | $60.81$ | $568.820$ | $20.08$ | $2337.915$ | $19.85$
\+ 2337.915 |  |  | $330.237$ | $19.26$ |  |

###  3.2 Iron

The average iron abundances near the density peak at 90 km is two orders of
magnitude higher than that of most other mesospheric metals, and twice that of
sodium. Although its cross section is small, the key product $N\sigma_{21}$ is
still larger than that of all other metallic elements except sodium. The
abundance ratio Fe${}^{+}/$Fe is approximately 0.2 above 90 km [33].

Atomic iron has the largest number of spectral lines of all mesospheric
species. The iron-group elements have complex transitions with thousands of
lines from the vacuum ultraviolet to the infrared. An energy-level diagram for
Fe I is shown in Fig. 2. The ground state is split into three fine-structure
levels. Only the lowest-energy a5D4 state is populated in thermal equilibrium
in the mesosphere. There are five strong visible-wavelength transitions from
this ground state, at 344.061, 367.991, 371.993, 382.444 and 385.991 nm. Most
of these can decay to the other two ground states. The z5D3 state, reached by
the 372 nm transition from the ground state, has a moderately-strong
($A=1.15\times 10^{6}$ s-1) downward transition to the metastable a5F4 state.

Figure 2: Selected states and transitions of Fe I.

The strongest transition in Fe I using single-photon excitation has a return
flux coefficient of $2.51\times 10^{20}$ photons s-1 m-2 (Table 2). This is
six times weaker than the sodium D2 line. Two-step excitation (Table 6)
provides more possibilities, but these are all weaker by at least an order of
magnitude.

Fe I has combinations of transitions which have promising wavelengths and
penalty factors for PLGS. However, for one-step and two-step excitation, $q$
does not exceed $q>0.18$ for any combination of wavelengths (Tables 7 and 8).

Fe II has no transitions from the ground state for wavelengths greater than
260 nm. This rules out excitation by ground-based lasers unless two-photon
excitation is used. That would require pulsed lasers with very high
irradiance.

Table 6. Fe I transitions for a monochromatic LGS employing two-step
excitation. All transitions that have $\log(\varepsilon)>19$ and
$\lambda_{\text{ex}}\geq 300$ nm are listed.
---
$\lambda$ | $\lambda_{\text{ex}}$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
(nm) | (nm) |  |
355.492 | 437.593+355.492 | $19.32$ | $-1.21$
314.399 | 385.991+314.399 | $19.31$ | $-1.73$
381.764 | 371.993+381.764 | $19.29$ | $-2.37$
380.198 | 371.993+380.198 | $19.08$ | $-2.70$
Table 7. Fe I transitions for a PLGS employing one-step excitation. All
transitions that have $q>0.1$ and $\lambda_{\text{ex}}\geq 300$ nm are listed.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) |  |  | (nm) |  | (nm) |
319.323 | 0.10 | 26.39 | 319.323 | $17.90$ | 517.160 | $17.90$

###  3.3 Magnesium

Magnesium is one of the more abundant meteoritic constituents, 9.6% by mass,
so that meteoric ablation should inject large quantities of this metal into
the MLT region [22]. Compared to other meteoric metals, magnesium has the
largest ionization fraction, with a ratio Mg+/Mg in the range 4 to 12 [22].

Mg I has six strong transitions from the ground state, all in the vacuum
ultraviolet. The strongest of these goes to the 31P1 state, with a wavelength
of 285.213 nm and a rate ($A_{21}=4.91\times 10^{8}$ s-1) that is an order of
magnitude greater than that of the sodium D lines. The only other downward
transitions from this upper state are forbidden lines that have negligible
rates. In vacuum, this would be a strong candidate for LGS, however, the
wavelength is beyond the atmospheric cutoff.

Table 8. Fe I transitions for a PLGS employing two-step excitation. All
transitions that have a de-excitation $q>0.1$ and $\lambda_{\text{ex}}\geq
300$ nm are listed. In brackets in the last line for each excitation scheme
the return flux coefficient for the first excitation wavelength is shown.
---
$\lambda_{\text{ex}}$ | $q$ | $p$ | $\lambda_{1}$ | $\log(\varepsilon_{1})$ | $\lambda_{2}$ | $\log(\varepsilon_{2})$
(nm) |  |  | (nm) |  | (nm) |
437.593 | $0.12$ | $25.41$ | $361.207$ | $18.17$ | $973.857$ | $18.01$
\+ 361.207 | $0.11$ | $27.33$ | $371.993$ | $18.85$ | $973.857$ | $18.01$
|  |  | $437.593$ | $16.06$ |  |
385.991 | $0.12$ | $40.78$ | $371.993$ | $18.49$ | $561.564$ | $18.49$
\+ 532.418 | $0.10$ | $44.57$ | $371.993$ | $18.49$ | $532.418$ | $18.38$
|  |  | $385.991$ | $18.18$ |  |
Figure 3: Selected states and transitions of Mg II.

Mg II has two transitions from the ground state that have rates greater than
$10^{8}$ s-1. These go to the 32P1/2 and 32P3/2 states, with wavelengths of
280.271 and 279.553 nm respectively. Both of these upper states return
directly to the ground state, with no other significant transitions. Given the
high abundance of Mg+ (twice that of Na), these would be very good candidates
for LGS, but they are beyond the atmospheric cutoff.

Mg II does have strong emission lines in the visible region [34] (Fig.3).
However, populating these levels from the ground state requires photons having
wavelengths of 124 nm or less. This probably rules out Mg+ for ground-based
adaptive optics.

###  3.4 Silicon

Silicon is one of the most abundant elements in cosmic dust, comprising around
11% by mass, and meteoric ablation injects a significant amount of Si into the
atmosphere [24]. However, neutral Si is oxidized to SiO very rapidly by
reaction with O2 at the altitude of maximum ablation, around 90 km. Si+, SiO,
and Si(OH)4 are predicted to be the major silicon species above 80 km. Below
97 km the dominant sink is Si(OH)4. The column density of Si+ varies
seasonally by an order of magnitude and peaks at $4.0\times 10^{9}$ cm-2
during the summer [24].

Figure 4: Selected states and transitions of Si II.

Singly-ionized Silicon has several strong visible-light transitions, as shown
in Fig. 4 [35]. However, excitation from the ground state requires UV photons.
The ground state is a doublet (32P1/2, 32P3/2), separated by 0.0356 eV. This
is comparable to the mean particle energy of $3kT/2\simeq 0.023$ eV, so both
states will be populated. Transitions can occur from the lower ground state to
the 4P1/2 state, followed by a 637 nm visible-light transition, but this
requires a 123-nm photon. The lowest-energy transitions from either ground
state to the 4P levels requires a 123-nm photon. There is a transition from
the ground state to the 32P3/2 state, but this is quite weak
($A_{21}=2.54\times 10^{6}$ s-1). This likely rules out SI II as a viable
candidate for AO, and indeed no transitions met our thresholds for inclusion.

###  3.5 Calcium

Lidar measurements of calcium showed a peak density of $2\times 10^{7}$ atoms
m-3, about 200 times lower than the typical sodium atom density and 400 times
lower than that of iron [36]. The annual average Ca+/Ca column density ratio
is 2.4, the second largest ratio after magnesium. In comparison, the ratios of
Na+/Na and Fe+/Fe are only about 0.2 [37]. Ca+ is the dominant form above 90
km altitude, peaking near 105 km.

Ca I has a strong transition from the ground state to the 31P1 state with a
rate of $2.18\times 10^{8}$ s-1 and a wavelength of 422.673 nm. The upper
state returns directly to the ground state, with no other transitions. The
only problem here is the low abundance. As can be seen in Table 2, the return
flux coefficient is about 30 times smaller than that of the sodium D2 line.

Table 9. Ca I transitions for a monochromatic LGS employing two-step
excitation having wavelength $\lambda\geq 300$ nm and return-flux coefficient
$\log(\varepsilon)>19$.
---
$\lambda$ | $\lambda_{\text{ex}}$ | $\log(\varepsilon)$ | $\log(N\sigma_{21})$
(nm) | (nm) |  |
$585.745$ | 422.673+585.745 | 19.07 | -3.43
$430.774$ | 657.278+430.774 | 19.01 | -3.35

Fig. 5 shows the relevant energy levels and transitions of Ca+ [38].
Excitation from the ground state to the 4P1/2 and 4P3/2 levels is possible
using 397 nm and 393 nm photons, respectively. These states decay quickly to
the ground state and also, with 7% probability, to the 3D3/2 and 3D5/2 states,
emitting 866 nm, 850 nm and 854 nm photons. These states are metastable, so
most of the atoms excited from the ground state using a UV laser will become
trapped in these states. No lines of Ca II met the criteria for inclusion in
Table 2.

Figure 5: Selected states and transitions of Ca II.

###  3.6 Potassium

Potassium has relatively low abundance in the upper atmosphere, having a
column density that is about two orders of magnitude less than that of Na. For
neutral potassium atoms, many transitions covering the visible light range
have high transition probabilities. The relevant energy levels and transitions
are shown as Fig. 6. Its persistent lines are 769.896, 766.490, 404.414 and
404.721 nm. Other strong lines having slightly smaller transition
probabilities occur at higher energy levels. Direct transitions to and from
the ground state are possible at 766.490 and 769.896 nm. These have cross
sections that are comparable to that of the sodium D2 line. However, the
relatively low column density of potassium results in return flux coefficients
that are two orders of magnitude smaller.

Figure 6: Selected states and transitions of K I. Figure 7: Energy level
diagram of N I (left) and N II (right).

Potassium can also be excited directly from the ground state to the 5P1/2 and
5P3/2 levels. These have a path back to the ground state via the 3D states and
4P states, producing strong lines at 3139.265, 3160.163, 1169.024 and 1176.967
nm, in addition to the 766.490 and 769.896 nm doublet. This would be an
attractive system for PLGS if it was not for the relatively low abundance.

###  3.7 Nitrogen

In the mesosphere, nitrogen molecules are ionized and dissociated by energetic
electrons and protons, producing N${}_{2}^{+}$ by direct ionization and N+ by
dissociative ionization. Secondary electrons associated with ionization
produce N in both the ground and 2D states. At 80 km atomic nitrogen has a
peak density of $2\times 10^{8}$ cm-3, falling to $10^{6}$ cm-3 at 70 km [25].
The estimated column density of atomic nitrogen is $\sim 1\times 10^{14}$
cm-2.

The most relevant energy levels and transitions of atomic NI and singly-
ionized NII are shown in Fig.7. It can be seen that NII has many more strong
emission lines in the visible range than does NI. Furthermore, nitrogen-
discharge experiments [39] show that the optical emission spectrum in the
visible and UV regions, is quite complex with multiple-peaks and many blended
lines.

Direct transitions from the ground state require vacuum-ultraviolet photons,
even using two-photon excitation. This likely rules out nitrogen for ground-
based AO systems.

###  3.8 Oxygen

Atomic oxygen (O) is a fundamental component in chemical aeronomy of Earth’s
MLT region extending from approximately 50 km to over 100 km altitude.
Primarily, O is generated through photolysis of molecular oxygen by UV
radiation. The peak atomic oxygen concentrations, $\sim 6.0-6.5\times 10^{17}$
m-3, are found at an altitude of approximately 95 km [26]. Assuming an atomic
oxygen column width of 10 km in the mesosphere, the estimated column density
of atomic oxygen is $\sim 6.0-6.5\times 10^{21}$m-2.

Energy level diagrams for O and O+ are shown in Fig. 8. The triplet at $\sim
130$ nm has been observed in the cometary spectra from above the Earth’s
atmosphere [40]. These lines arise when O atoms in the ground state are
excited to the 5S1 state by solar photons. In the visible spectrum region,
several astrophysically-important forbidden transitions exist. For O, these
are the red doublet at 630.030 nm and 636.377 nm, arising from transitions
between the 2D2 and the 2P2 and the 2P1 levels respectively, and the green
line at 557.734 nm from the 2D2 and 2S0 levels. These are magnetic dipole and
electric quadrupole transitions respectively. These forbidden oxygen emission
lines likely arise from atoms produced directly in the excited 2S0 or 2D2
states by photo-dissociation of parent molecules such as H2O, CO and CO2. In
O+, the doublet at 372.602 and 372.882 nm arises from transitions from the
2D3/2 and 2D5/2 levels to the 2S3/2 ground state. As with nitrogen,
$N\sigma_{21}$ is high for permitted transitions in oxygen, but low for the
forbidden transitions.

In O, strong visible lines arise from transitions between the 3P and 3S
levels, and also the 5S1 level. Visible lines can also occur from states above
the 5S1 energy level. Each line is a doublet or triplet due to hyperfine
structure. However, all these lines require vacuum-ultraviolet photons for
excitation from the ground state.

In O+, strong visible lines result from transitions from the 3P and 3D levels
to the 3P levels, and also the 3S1/2 and 3P3/2 levels. In this case there are
no direct transitions from the ground state to the upper states involved, and
transitions to the lower states require photons of wavelength 54 nm or less.

Figure 8: Strong resonance transitions in O I (left) and O II (right).

###  3.9 Hydrogen

Atomic hydrogen (H) is also a fundamental component in the photochemistry and
energy balance of the mesopause region between approximately 80 and 100 km
altitude. H is generated primarily by photolysis of water vapor and
participates in a highly-exothermic reaction with ozone. During the day,
atomic hydrogen concentrations peak in excess of $2.25\times 10^{14}$ m-3 at
higher latitudes near 85 km [27]. The estimated column density of atomic
hydrogen is $\sim 2\times 10^{18}$ m-2.

Hydrogen and hydrogen-like ions have familiar Lyman, Balmer, and Paschen
spectral series. Hydrogen emission spectral lines in the visible range include
Balmer lines at 410.173, 434.047, 486.134, and 656.283 nm wavelengths,
corresponding to transitions to the $n=2$ level. All energy levels for a given
principal quantum number $n$ are degenerate. Transitions from the ground state
require vacuum-ultraviolet photons having wavelengths of 122 nm or less.

###  3.10 Carbon

Meteoric ablation injects a significant amount of carbon into the mesosphere.
However, atomic carbon is a very short-lived species. It is stabilized in
various multi-atomic structures having different molecular configurations. The
carbon monoxide (CO) mixing ratio increases steeply with altitude in the
mesosphere and in the lower part of the thermosphere is a result of the
photolysis of carbon dioxide (CO2) [41]. Additionally, as the chemical
lifetime of CO is much longer than the characteristic dynamical times, this
gas is an excellent tracer of dynamics due to its strong 4.7 $\mu$m
transition.

The most common oxidation state of carbon in inorganic compounds is $+4$,
while $+2$ is found in carbon monoxide. The four outer electrons of carbon are
valence electrons, so its ionization energies are much higher than those of
most other elements. Short wavelength UV photons are needed to excite the
atoms from the ground state, making excitation using a ground-based laser
unfeasible. Also, there is little information about the detailed distribution
and abundance of carbon and their ions in mesosphere, so carbon and related
species are not considered further here.

##  4 Discussion

It can be seen from Table 2, that the sodium $D_{2}$ line provides the highest
return flux of all possible transitions, including two-step excitation
schemes. This is due to its relatively-large natural abundance and large cross
section compared to other metal atoms. Mg+ and Si+ have stronger lines, but
lack an effective scheme for excitation from the ground state. Although the
column density of Fe is twice that of sodium, its lower cross-section results
in a return flux that is one to two orders of magnitude smaller.

In contrast, the Mg+ lines at 280.271 and 279.553 nm have transition rates
that are an order of magnitude greater than that of the sodium D lines, and
the column density of Mg+ is twice that of sodium. These would be very strong
lines if they were not beyond the atmospheric cutoff. Similarly, Si+ has
strong transitions, and a column density comparable to that of Na, but vacuum-
ultraviolet excitation is required.

Potassium, like Ca+, has a column density that is smaller than that of sodium
by two orders of magnitude, so its return flux is small.

As with monochromatic LGS, Na yields the highest performance for PLGS from
both one-step and two-step excitation. According to our results, the next-
ranked element, Fe, provides at most 10% of the SNR of the best-suited
transitions in Na. If atmospheric absorption can be compensated by
sufficiently-high laser power the one-step excitation at 330 nm yields a
higher SNR than the two-step excitation at $569+589$ nm. At an altitude of
4200 m, the atmospheric transmission at 330 nm is about 60% of the
transmission at 600 nm. The most-promising combinations of transitions for
PLGS include de-excitations at 330 nm, however atmospheric absorption will
decrease the return flux at 330 nm. For two-step excitation, an alternative
approach could be to combine the return flux from the D1 and D2 line at 589
nm. For an excitation at $589+569$ nm, the photons at 589 nm can only be used
if it was ensured that the photons were emitted from fully-coincident laser
spots in the mesosphere. Even though the two-step excitation adds wavelengths
at about 3410 and 2338 nm, the PLGS performance does not exceed that of the
one-step excitation, since the penalty factor does not decrease significantly
for infrared wavelengths.

Our analysis used statistical weights provided by NIST, which give the number
of hyperfine structure states. Magnetic sub-states arising from Zeeman
splitting in an external magnetic field are not considered. This is a
limitation to our approach. Excitation with optimized laser polarization,
which can provide a greater return flux, can be assessed by employing a more-
detailed treatment of single transitions using Bloch-equations. Nevertheless,
we believe that the simple approach that we employ is useful for a first
assessment of potential transitions.

For a species in which essentially all atoms are in the ground state, the
optical depth at the line center is equal to the product of column density,
$\tau_{0}=N\sigma_{12}(\nu_{0})=g_{2}N\sigma_{21}(\nu_{0})/g_{1}$. From the
values listed in Table C1, we see that all transitions of mesospheric metals
have optical depth less than 0.2, so the atmosphere is optically thin
($\tau<1$). In contrast, the abundant species of N I and O I have $\tau_{0}\gg
1$ for most permitted transitions, which means that the atmosphere is opaque
at those wavelengths.

The optical depth for stimulated emission is closely related to the optical
depth for absorption. In a fully-excited medium, $\tau\sim-\sigma_{21}N$.
Significant amplification by stimulated emission can only occur in a medium
for which $-\tau\gg 1$. To achieve an optical depth of order unity,
$N_{2}A_{21}\geq 10^{23}$ m-2s-1 is required. For a typical transition rate,
$A_{21}\sim 10^{7}$ s-1, the column density of the medium must therefore
exceed $10^{16}$ m-2 to achieve ASE. This would appear to rule out all
metallic species in the mesosphere.

An interesting new development is the use of high-powered lasers to create a
plasma in the atmosphere. Rairoux et al. [42] have demonstrated ASE in
backscattered fluorescence of nitrogen from filaments generated by intense
ultra-fast Ti:sapphire laser pulses propagating over a distance of up to
several km. As it propagates along the filament, the backscattered radiation
is amplified by a population inversion resulting from direct ionization. The
laser excitation leads to plasma formation and the consequence is a white-
light laser pulse. Backscattered fluorescence from N2 molecules and ions shows
the exponential increase with increasing filament length expected for ASE. It
has the potential to generate very bright LGS, but a terawatt femtosecond
Ti:sapphire laser system is required to produce the $10^{17}$ W/m2 intensity
that is necessary to generate non-linear effects and plasma. Nevertheless, the
potential of a white-light N2 ASE for adaptive optics might well be realized
as laser technology continues to improve.

Other factors, in addition to LGS return flux, also affect AO performance.
With large-aperture telescopes, LGS appear elongated for subapertures of the
pupil that are offset from the axis of the laser launch telescope. This
increases noise and reduces the wavefront-sensing accuracy. In this regard, an
atomic species that has a high centroid altitude and small vertical extent
would be favored. In practice, the altitude is limited by the rapidly
decreasing atmospheric density and there is not a large difference between the
atomic species listed in Table 1. If an LGS can be created by a high-power
pulsed laser, an important consideration will be to limit the extent of the
emitting region along the line of sight.

##  5 Summary and conclusions

We have reviewed transitions of atoms and ions in the upper atmosphere of
interest for AO. Besides Na, which is used extensively by current systems, Fe,
Mg, Si, Ca, K all have potential uses. Iron has the highest abundance and the
largest number of transitions. However, its return flux is less than that of
sodium due to lower cross sections.

Our results confirm the general presumption of the LGS community that Fe is
not as well-suited as Na for PLGS. However, we find that one-step excitation
at 330 nm may be more suitable than two-step excitation at $569+589$ nm.
Detailed simulations, using Bloch equations and including the effects of
Zeeman splitting, would be a useful next step to evaluate the potential of
these excitation schemes.

Mg+ and Si+ have high abundance and several very-strong visible-light
transitions. However they require vacuum ultraviolet photons for excitation
that appear to be implausible even with two-photon excitation.

Amplified spontaneous emission using metallic species appears to be unfeasible
because of the small optical depths. A high abundance is required for ASE, so
nitrogen and oxygen are likely the only atoms that could be used.

##  Appendix A Atomic collision rates in the mesosphere and lower thermosphere

We wish to estimate the collision rates for various metallic atoms and ions in
the mesosphere and lower thermosphere (MLT) region. The atmospheric
composition in the MLT is essentially the same as in the troposphere,
consisting almost entirely of three species, N2, O2 and Ar. Their fractions,
by volume, are approximately 78%, 21% and 1% respectively. The species of
interest to us are Na, Fe, Ca, Si, Mg, K, N, O, H and their ions.

The collision rate $R$, for a given atom, is given in terms of the collision
cross-section Q and the number densities $n$ for the dominant atomic and
molecular species. For atom $i$, interacting with field atoms or molecules
$j$,

$R_{i}=\sum_{j}n_{j}Q_{ij}v_{ij},$ (A1)

where $v_{ij}$ is the relative velocity. For a Maxwell-Boltzmann distribution
at temperature $T$, the RMS relative velocity of an atom or molecule of mass
$m_{i}$ and one of mass $m_{j}$ is,

$v_{ij}=\sqrt{v_{i}^{2}+v_{j}^{2}}=\sqrt{\frac{8kT}{\pi\mu_{ij}}},$ (A2)

where $\mu_{ij}=m_{i}m_{j}/(m_{i}+m_{j})$ is the reduced mass. The low-energy
collision cross section between two atoms is well-approximated by the hard-
sphere model,

$Q_{ij}=\pi(r_{i}+r_{j})^{2},$ (A3)

where $r_{i}$ and $r_{j}$ are the atomic radii. For a diatomic molecule $j$,
the situation is more complicated. Modelling the molecule by two spheres of
radius $r_{j}$ in contact, the appropriate geometrical cross section is the
projected area $A$ of two spheres of radius $r=r_{i}+r_{j}$, having centres
separated by a distance $d$. The separation depends on the angle $\theta$
between the axis of the molecule and the velocity vector,
$d=2r_{j}\sin\theta$. To find the mean cross-section we average over this
angle, assuming an isotropic velocity distribution. From plane geometry,

$A=2\pi
r^{2}-2r^{2}\arccos\left(\frac{d}{2r}\right)+\frac{d}{2}\sqrt{4r^{2}-d^{2}}.$
(A4)

The frequency with which a particular angle $\theta$ appears is proportional
to $\sin\theta$, so

$\displaystyle Q_{ij}$
$\displaystyle=\frac{4}{\pi}\int_{0}^{\pi/2}A\sin^{2}\theta d\theta,$
$\displaystyle=2\pi r^{2}+\frac{r^{4}+2r^{2}r_{j}^{2}-3r_{j}^{4}}{\pi
r_{j}^{2}}\text{arcsinh}\left(\frac{r_{j}}{\sqrt{r^{2}-r_{j}^{2}}}\right)$
$\displaystyle\quad-\frac{r^{3}-3rr_{j}^{2}}{\pi
r_{j}}+\frac{32r_{j}r}{3\pi\sqrt{2\pi}}{{}_{3}F_{2}}\left[2,\frac{1}{2},\frac{1}{2};\frac{5}{2},\frac{3}{2};\left(\frac{r_{j}}{r}\right)^{2}\right],$
(A5)

where ${}_{3}F_{2}$ is a generalized hypergeometric function.

Atomic radii are taken from [43]. For the atmospheric density and temperature
the MSIS-E-90 atmospheric model was used [21]. These were evaluated, for a
latitude of $30^{\circ}$ ond longitude of $0^{\circ}$, at midnight on the
first day of each month, and the average over a full year was computed. The
resulting mean cross sections for the atoms and ions of interest, colliding
with N2, O2 and Ar, are listed in Table A1. The computed collision rates are
listed in Tables A2 and A3. For comparison, Holzlöhner et al. [6] estimated a
Na – O2 collision rate of 1/(35 $\mu$s) at an altitude of 92 km, which is
within a factor of two of our estimate.

Table A1. Mean collision cross sections.
---
Atom | mass | radius | cross section
| (amu) | (nm) | (nm2)
|  |  | N2 | O2 | Ar
Na | 22.990 | 0.191 | 0.565 | 0.550 | 0.277
Fe | 55.845 | 0.127 | 0.315 | 0.303 | 0.171
Mg | 24.305 | 0.162 | 0.443 | 0.429 | 0.226
Si | 28.085 | 0.109 | 0.257 | 0.247 | 0.145
Ca | 40.078 | 0.197 | 0.593 | 0.577 | 0.288
K | 39.098 | 0.237 | 0.790 | 0.771 | 0.370
N | 14.007 | 0.070 | 0.152 | 0.144 | 0.097
O | 15.999 | 0.066 | 0.143 | 0.135 | 0.093
Ar | 39.948 | 0.106 | 0.248 | 0.238 | 0.141
H | 1.008 | 0.070 | 0.152 | 0.144 | 0.097
Table A2. Collision rates, in units of $10^{2}$ s-1, for metallic species.
---
altitude | Ca | Fe | K | Mg | Na | Si
(km) |  |  |  |  |  |
$75$ | 2449 | 1219 | 3283 | 2062 | 2675 | 1150
$80$ | 1108 | 551.7 | 1486 | 933.3 | 1211 | 520.7
$85$ | 484.1 | 241.0 | 649.1 | 407.7 | 528.8 | 227.5
$90$ | 203.7 | 101.4 | 273.1 | 171.5 | 222.5 | 95.70
$95$ | 82.66 | 41.15 | 110.8 | 69.60 | 90.28 | 38.84
$100$ | 32.65 | 16.26 | 43.77 | 27.49 | 35.65 | 15.34
$105$ | 13.08 | 6.515 | 17.54 | 11.01 | 14.28 | 6.147
$110$ | 5.662 | 2.821 | 7.590 | 4.766 | 6.181 | 2.661
$115$ | 2.740 | 1.365 | 3.673 | 2.306 | 2.991 | 1.288
$120$ | 1.489 | 0.742 | 1.996 | 1.253 | 1.625 | 0.700
Table A3. Collision rates, in units of $10^{2}$ s-1, for non-metallic species.
---
altitude | Ar | H | N | O
(km) |  |  |  |
$75$ | 1023 | 2592 | 833.3 | 748.7
$80$ | 463.0 | 1173 | 377.1 | 338.8
$85$ | 202.3 | 512.3 | 164.7 | 148.0
$90$ | 85.10 | 215.5 | 69.30 | 62.27
$95$ | 34.54 | 87.46 | 28.13 | 25.27
$100$ | 13.64 | 34.54 | 11.11 | 9.982
$105$ | 5.468 | 13.83 | 4.451 | 4.000
$110$ | 2.367 | 5.984 | 1.926 | 1.731
$115$ | 1.146 | 2.895 | 0.932 | 0.838
$120$ | 0.623 | 1.573 | 0.507 | 0.455

##  Appendix B Transition rates in a multi-level atom

In a multi-level atom, with states $j$, $j=0,1,\cdots,n-1$, the fraction of
atoms $x_{j}$ for which the electron is in state $j$ is determined by $n$ rate
equations

$\dot{x}_{j}=\Gamma_{jk}x_{k}.$ (B1)

Here $\Gamma_{jk}$ specifies the net transition rate between states $j$ and
$k$. These equations are not linearly independent because the occupations
fractions are related by the constraint

$\sum x_{k}=1.$ (B2)

This condition can be used to eliminate one of the variables, $x_{0}$ say,
which results in a set of $n-1$ linearly-independent inhomogeneous equations

$\dot{x}_{j}=M_{jk}x_{k}+b_{j},\quad j=1,2,\cdots,n-1.$ (B3)

The quantities $M_{jk}$ and $b_{j}$ depend on transition rates and the
radiation energy densities at the transition wavelengths. If the radiation
energy densities can be regarded as independent of time, these quantities are
constant and the equations are readily solved using standard techniques. The
general solution is

$\boldsymbol{x}=\sum
c_{k}\boldsymbol{a}_{k}e^{\lambda_{k}t}-M^{-1}\boldsymbol{b},$ (B4)

where $\lambda_{k}$ and $c_{k}$ are the eigenvalues and eigenvectors of $M$.

The steady-state solution is found by equating the left hand side of Eqn (B3)
to zero,

$\boldsymbol{x}=-M^{-1}\boldsymbol{b},$ (B5)

Funding. Yunnan Provincial Department of Education; Natural Sciences and
Engineering Research Council of Canada (RGPIN-2019-04369); Chinese Academy of
Sciences, CAS President’s International Fellowship Initiative (2017VMA0013).

Acknowledgement. We thank Profs. D. Budker and R. Holzlöhner for comments on
an earlier version of the manuscript. PH thanks the National Astronomical
Observatories, Chinese Academy of Sciences for hospitality during a sabbatical
visit.

Disclosures. The authors declare no conflict of interests.

Data availability. Data underlying the results presented in this paper are
available in Refs. [21] and [28]. Selected data for strong transitions in the
atomic species discussed here are available from the authors upon request.

## References

  * [1] J. M. Beckers, Ann. Rev. Astron. Astrophys., 31, 13 (1993).
  * [2] R. Foy and A. Labeyrie, Astron. Astrophys. 152, L29 (1985).
  * [3] W. Happer, G. J. MacDonald, C. E. Max and F. J. Dyson, J. Opt. Soc. Am. A, 11, 263 (1994).
  * [4] J. M. Telle, Opt. Soc. Am. Tech. Dig. Ser. 13, 100 (1996).
  * [5] T. W. Fan, T. H. Zhou, and Y. Feng, Sci. Rep. 6,19859 (2016).
  * [6] a R. Holzlöhner, S. M. Rochester, D. B. Calia, D. Budker, J. M. Higbie, and W. Hackenberg, Astron. Astrophys. 510, A20 (2010).
  * [7] P. W. Milonni, H. Fearn, J. M. Tell and R. Q. Fugate, J. Opt. Soc. Am. A 16, 11 (1999).
  * [8] R. Rampy, D. Gavel, S. M. Rochester and R. Holzlöhner, J. Opt. Soc. Am. B 32, 2425 (2015).
  * [9] F. P. Bustos, R. Holzlöhner, S. Rochester, D. B. Calia, J. Hellemeier and D. Budker, J. Opt. Soc. Am. B 37, 1208 (2020).
  * [10] P. Hickson, J. Hellemeier and R. Yang, Opt. Lett., 46, 1792 (2021).
  * [11] F. P. Bustos, A. Akulshin, R. Holzlöhner, et al., Proc. SPIE 10703, 107030R-1 (2018).
  * [12] F. Rigaut and E. Gendron, Astron. Astrophys. 261, 677 (1992).
  * [13] S. Esposito, Proc. SPIE. 3353, 468 (1998).
  * [14] R. Ragazzoni, “Laser Guide Star Adaptive Optics for Astronomy,” Ed. N. Ageorges and C. Dainty, NATO ASI Series C, 551, 125 (2000).
  * [15] R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, and M. Tallon, Astron. Astrophys. Suppl. 111, 569 (1995).
  * [16] J. P. Pique, I. C. Moldovan, and V. Fesquet, J. Opt. Soc. Am. A 23, 2817 (2006).
  * [17] Pfrommer, T. and Hickson, P., JOSA A, 27, A97 (2010)
  * [18] D. S. Davis, P. Hickson, G. Herriot, and C-Y She, Optics Let., 31, 3369 (2006).
  * [19] C. Y. She, S. Chen, Z. Hu, J. Sherman, J. D. Vance, V. Vasoli, M. A. White, J. Yu, and D. A. Krueger, Geophys. Res. Let. 27, 3289 (2000).
  * [20] C. S. Gardner, App. Opt. 43, 4941 (2004).
  * [21] A. E. Hedin, J. Geophys. Res. 96, 1159 (1991), available online at https://ccmc.gsfc.nasa.gov/modelweb
  * [22] C. L. Whalley, J. C. Gomez-Martin, T. G. Wright, et al., Phys. Chem. Chem. Phys., 13, 6352 (2011).
  * [23] J. M. C. Plane, Air Force Research Laboratory Report AFRL-AFOSR-UK-TR-2010-0005, 22 pp, (2010).
  * [24] J. M. C. Plane, J. C. Gomez-Martin, W. Feng, and D. Janches, J. Geophys. Res. Atmos. 121, 3718 (2016).
  * [25] A. C. Aikin and H. J. P. Smith, Phys. Chem. Earth (C) 25, 203 (2000).
  * [26] M. G. Mlynczak, L. A. Hunt, J. C. Mast, et al., J. Geophys. Res. Atmos. 118, 5724 (2013).
  * [27] M. G. Mlynczak, L. A. Hunt, B. T. Marshall, et al., J. Geophys. Res. Atmos. 119, 3516 (2014).
  * [28] A. Kramida, Y. Ralchenko, J. Reader, J. and NIST ASD Team (2020). NIST Atomic Spectra Database (ver. 5.8), [Online]. Available: https://physics.nist.gov/asd [2020, November 17]. National Institute of Standards and Technology, Gaithersburg, MD. DOI: https://doi.org/10.18434/T4W30F
  * [29] D. R. Marsh, D. Janches, W. H Feng, and J. M. C. Plane, J. Geophys. Res. Atmos. 118,11442 (2013).
  * [30] J-P Pique and S. Farinotti, JOSA B 20, 2093 (2003).
  * [31] J. Biegert and J. C. Diels, Phys. Rev. A 67, 043403 (2003).
  * [32] C. Xu, C. Guo, H-B. Yu, et al., Appl. Phys. B 123, 94 (2017).
  * [33] W. H. Feng, D. R. Marsh, M. P. Chipperfield, et al., J. Geophys. Res. Atmos. 118, 9456 (2013).
  * [34] D. E. Kelleher and L. I. Podobedova J. Phys. Chem. Ref. Data 37, 267 (2008).
  * [35] D. E. Kelleher and L. I. Podobedova, J. Phys. Chem. Ref. Data 37, 1285 (2008).
  * [36] J. M. C. Plane, Nature Chem. 3, 900 (2011).
  * [37] E. Kopp, J. Geophys. Res. 102, 9667 (1997).
  * [38] C. Maurer, C. Becher, C. Russo, J. Eshner, and R. Blatt, New J. Phys. 6, 94 (2004).
  * [39] S. Conti, P. I. Porshnev, A. Fridman, et al., Exp. Therm. Fluid Sci. 14, 79 (2001).
  * [40] A. L. Cochran and W. D. Cochran, Icarus 154, 381 (2001).
  * [41] K. U. Grossmann, O. Gusev, and P. Knieling, J. Atmos. Solar-Terr. Phys. 68, 1764 (2006).
  * [42] P. Rairoux, H. Schillinger, S. Niedermeier, et al., Appl. Phys. B 71, 573 (2000).
  * [43] A. N. Cox, “Allen’s Astrophysical Quantities,” Springer, NY (1999).