The RELEVANCE OF HANLE EFFECT ON NA AND FE LIDARS

A laser resonant scattering process involves two steps, excitation and emission. That emission occurs spontaneously is well accepted. That the atoms involved in the emission are excited coherently by a laser beam leading to a nonisotropic angular distribution of emission (an antenna pattern) is not well known. The difference between coherent and incoherent excitation leads to the Hanle effect. In this paper, I discuss the physics of Hanle effect, and its influences on the backward scattering intensity of Na, K, and Fe atomic transitions and the associated Na and Fe resonant fluorescence lidar systems.


INTRODUCTION
It is well known that the dipole moment responsible for non-resonant Rayleigh scattering is coherently excited by a laser beam. In a resonant scattering process, the scattered photons are spontaneous emissions from atoms excited during the process. That such atoms, quite different from those involved in a fluorescence light, are also coherently excited is not as well known. Unlike incoherent excitation in the case of a fluorescence light, which leads to isotropic emission, the coherent excitation by a laser beam leads to angular distribution of emitted (or scattered) light, an antenna pattern if you wish. The difference between the two leads to the Hanle effect. We discuss the physics and formulation of the Hanle effect, and apply it to the backscattering of Na, K, and Fe atoms, and in turn treat the associated consequences on the associated resonance fluorescence lidar signals.

METHODOLOGY
A semiclassical/quantum mechanical treatment aimed at lidar applications was recently presented [1]. The resonant scattering cross-section from initial state | f ! via excited state | F ! to the final state | f ' ! is given in (16) of the paper. By respectively, transition wavelength, excited state degeneracy, ground state degeneracy, Einstein A coefficient from excited to initial state, fractional transition rate, and absorption line-shape. The backscattering strength factor C fFf ' (ê;ê ') is responsible for the angular distribution in question and it depends on the incident and scattering polarizations, ê andê ' .
Referring to the basic scattering coordinate and geometry shown in Fig. 3 of She et al. [1], there are 4 cases denoting as: C e e C e e C e e { as given in (B8a) and (B8c) of [1]. To allow for the possibility of more than one emission final state, we define the differential cross section, summing all allowed emission we have the desired differential cross-section as Eq. (2): We denote the quantity in the square bracket as the distribution function of the fluorescence intensity, To make a complex discussion manageable, we simplify our discussion to a judiciously selected case with incident polarization as , as in the textbook discussion of classical scattering, giving the fluorescence intensity as Eq. (3): Here the coefficient is given in terms of the 6-j coefficient in (16) of [1].
With the above simplification, we now consider two cases for pathedagogy reason and lidar application.

Physics demonstrated in a simple system
We now apply these formulae to the simplest quantum structure without spin-orbit or hyperfine interaction, and substitute f f ' 0 and F 1 into Eq. (3) the 6-j coefficients for (2) ' fFf B . As a result, we obtain 1 3 g , D 10 1and (2) 101 5 / 9 B , leading to the fluorescence intensity functions, , an angular distribution agreeing with the classical results in textbook [2]. This clearly demonstrated the difference from the spontaneous emission distribution that the Hanle effect is not of quantum origin. Rather, it is because of the excitation of the atomic system by a laser-like source, which coherently prepares the mixed excited sub-states before fluorescence emission. Indeed, the Hanle effect has a classical explanation and is most prominent when the excited sub-states are degenerate, i.e., when there is no Zeeman splitting at B = 0; thereby, it is an effective tool for measuring the lifetime of atomic excited states [3].

Hanle effect of Na, K and Fe transitions
In order to apply the formulation to atoms of practical interest, we need to evaluate this factor for the transitions of Na, K, and Fe atoms. Though tedious, this can be down in a straightforward manner. The results for Na and K are given in Table 2

Radiation (antenna) patterns of laser induced fluorescence from Na and Fe atoms
Using the recipe presented for the simple case of zpolarized incident beam, we compute and plot the radiation patterns for Na atom in Fig. 3.1.a and that for Fe atom in Fig. 3.1.b below.
The backward values, l l

Relevance to resonance fluorescence lidars
We consider the relevance of Hanle effect and apply the results outlined in Section 3.1 to Na and Fe fluorescence lidars. We do not discuss K lidar in part because its similarity with Na lidar and smaller atmospheric abundance. The simplest lidar (often broadband) for measuring metal density uses a transmitting laser beam. In this case, the Hanle effect could amount to up top 12 % density error. For atmospheric temperature and wind measurements, typically, intensity ratios are invoked. If the two or three frequencies in the ratio(s) are from the same atomic transition, as in the 3-frequency Na lidar [5], then the correction of the Hanle effect does not affect the temperature or wind retrieval. However, the accompanying Na density determination will still be affected. A different situation exists in the temperature determination with iron Boltzmann lidar, in which, the intensity ratio of 374 nm and 372 nm transitions is used. In this case, unless the multiplicative factors on the intensity of the two lines due to the Hanle effect are the same, they should be included. Fortunately, in this case, the multiplicative factors are nearly the same, respectively 1.077 and 1.087, ignoring the Hanle effect is justifiable.
We had only discussed the Hanle effect at B = 0. This is also justifiable, not only because the earth magnetic field is small compare to the internal magnetic field in the atom [5], also the Hanle effect is largest at B=0.
Note added in proof: A more extensive version of this paper has since been published [7].