We propose what we believe is a novel method for enabling the complete suppression of noise due to Rayleigh scattering in sodium laser guide star systems by means of selective discrimination between Rayleigh and fluorescence signals based on polarization properties. We show that, contrary to the nearly 100% polarized Rayleigh scattering, fluorescence from the D2 sodium line is strongly depolarized under excitation by a modeless laser. This offers the possibility of completely cancelling the effects of the Rayleigh scattering background while preserving the fluorescence signal to about 40% of its maximal value, leading to an improvement of the signal-to-noise ratio by several orders of magnitude. Both theoretical and experimental data confirm this new proposal.
©2006 Optical Society of America
1.1 Overview of laser guide star developments
Large telescopes have the potential of obtaining high-intensity diffraction-limited images of astronomical objects. However, atmospheric turbulences distort the wavefront, and without compensation, large ground-based telescopes have the same resolution as telescopes that are a few decimeters in diameter. By correcting the aberrations induced by the atmosphere, adaptive optics (AO) systems enable us in principle to reobtain diffraction-limited imaging, provided that a guide light-source in the isoplanatic patch of the telescope is used as a reference for the wavefront correction. Unfortunately, at visible and near-infrared wavelengths, bright stars are required. It turns out that sky coverage is disastrously low, particularly in the visible wavelength range. To overcome this strong limitation, artificial laser guide stars (LGSs) produced by laser-induced fluorescence (LIF) of mesospheric sodium at 589 nm between the levels 3S1/2 and 3P3/2 (D2 line) have been proposed to provide a positionable light source above the turbulent layers of low atmosphere [1, 2]. Real-time monitoring of the wavefront distortions enables us to pilot the adaptive optics loop. However, the apparent direction of the LGS is independent of the so-called “tip-tilt.” This technique cannot give any information on the first order of distortion, and the LGS has to be used in conjunction with a less-intense natural guide star (TTNGS for “tip-tilt natural guide star”), leading to better sky coverage . In the last 10 years, large ground-based telescopes have implemented laser guide stars (Calar Alto, Lick, Keck, Gemini North, ESO/VLT) using various laser solutions and have demonstrated their capabilities. LGS-based AO systems are now used routinely at Lick and have come to be largely used at Keck II. But still some limitations remain.
As mentioned above, the sky coverage is far from being complete, especially in the visible bands. The problem of the LGS-based correction of the tip-tilt can be solved by exciting a polychromatic LGS (PLGS) . This program is currently being implemented in the French Etoile Laser Polychromatique pour l’Optique Adaptative (ELP-OA) program . Another possibility for increasing the field-of-view of the telescope is given by the multi-conjugate adaptive optics (MCAO) program, where several LGSs are created in the vicinity of the object . MCAO also enables us to solve the problem of the cone effect: namely, the guide source being located at a finite range where the return beam does not probe exactly the same volume as the beam coming from an astronomical object at infinity. This is a major limitation for very large (8–10 m diameter) telescopes (VLT) or extremely large (30–42 m) telescopes (ELT) that MCAO should be able to overcome.
1.2 The problem posed by Rayleigh scattering
A major problem posed by LGSs is Rayleigh scattering of the laser beam, primarily due to the nitrogen molecules in the first 35 km, that gives unwanted return photons and parasite noise on the wavefront analyzer. Rayleigh scattering has proved to be particularly problematic when a laser is launched off the axis of the telescope, leading to a broad residual noise spot on the wavefront sensor . The situation where a laser is launched behind the secondary mirror is a little more favorable because of the occultation of the Rayleigh scattering by the secondary mirror . However, MCAO techniques require sending off-axis laser beams to place several LGSs in the field of view of the telescope, leading to critical signal-to-noise ratio issues in VLTs or ELTs. On the side of the ELP-OA program, because of its λ-4 dependence, Rayleigh scattering is at first sight a major barrier for the 330 nm single-laser excitation of the sodium cascade that was recently proposed [9,10]. Therefore, eliminating the Rayleigh-scattering-induced noise is becoming a critical issue for the present development of LGS-based AO systems.
To overcome Rayleigh scattering, one can use the fact that the sampling time of wavefront analysis is required to be about 1 ms (since the coherence time of the atmosphere is close to 10 ms) and is larger than the round trip time of the laser pulse to the sodium layer (600 µs). However, this would impose relatively low-repetition-rate lasers with the constraint of providing enough fluorescence flux without saturating the transition. Such laser sources (1 kHz, 1–3 µs, and 10–20 W average power) have not been built yet. Moreover, most of the LGSs implemented so far are based on continuous wave (CW), quasi-CW, or high-repetition-rate laser excitation, thereby limiting time-gating implementations.
Here we propose another straightforward way to eliminate Rayleigh scattered photons by placing a polarizer on the return path of the light at an angle of 90° from the direction of the polarization of the laser. (We restrict this discussion to the case of linear polarization; we do not consider here the case of optical pumping of the mesospheric sodium by a circularly polarized laser .) Rayleigh scattering presents two components: a frequency unshifted contribution, and two Raman rotational wings due to the rotation of the molecules . We define the polarization rate as the ratio ρ=(I //-I⊥)/(I //+I ⊥), where I // (resp. I ⊥) denotes the intensity of the scattering having a polarization parallel (resp. perpendicular) to the polarization of the laser. Both contributions to Rayleigh scattering give a depolarization rate (i.e., 1-ρ) close to 1.5% in air . About 98.5% of Rayleigh scattering is suppressed by the analyzer as shown in Fig. 1. Moreover, polarization filtering also enables us to eliminate most of the Mie scattering since the polarization of the light scattered by spherical droplets or aerosols is unchanged. A weak depolarization in Mie scattering can arise from the scattering of nonspherical particles.
Polarization filtering can also suppress a significant proportion of the fluorescence flux, depending on the depolarization of the fluorescence light due to intrinsic atomic and laser properties. For instance, for a polarization rate of fluorescence equal to 0 (resp. 100%), the flux collected after the analyzer—the “effective” fluorescence signal defined by the ratio (1-ρ)/2—is equal to 50% (resp. 0) of the total produced fluorescence. Increasing the depolarization is essential in order to keep the effective fluorescence signal as high as possible. On the other hand, it has been shown that increasing the saturation parameter leads to an increase of the polarization rate . Thus the Rayleigh scattering cancellation is closely linked to the saturation parameter of the laser source.
We show that the modeless laser described previously  and currently developed at the Laboratoire de Spectrométrie Physique (LSP) in the frame of the ELP-OA program presents ideal characteristics in terms of polarization rate and Rayleigh cancellation. A rate-equation model has been developed to describe the excitation by a modeless laser of the hyperfine/Doppler structure of the D2 line, and this enables us to account for the large depolarization of the fluorescence signal in the CW and pulsed regimes. Experimental data were performed first in a sodium vapor cell in the CW regime. In the perspective of a pulsed laser-based program such as ELP-OA and the ELT, we also performed experimental measurements at high peak power by running the modeless laser in pulsed mode. In both cases the polarization rate remains below 25%. This result can be compared to the case of a single-mode laser where both previous theoretical density matrix models and experimental data show an increase of the polarization rate up to 60% for high saturation parameters . Therefore, modeless lasers exhibit a significant advantage over conventional single-mode lasers for Rayleigh cancellation. In the conclusion we discuss possible applications of Rayleigh scattering suppression using a modeless laser in both PLGS and fluorescence LIDAR techniques.
2. Polarization rate of the D2 line
2.1 Sodium D2 line
Atomic resonance experiments in sodium have been performed for more than a century . Polarization measurements have played a major role in the validation of early quantum mechanics by evidencing the nuclear spin and validating fine and hyperfine structure descriptions . In particular, the polarization rate of the fluorescence of sodium atoms shined by a broadband source (sodium vapor lamp) should be 60% if only fine structure is taken into account . Hyperfine structure leads to quite different results, giving a polarization rate of about 16% . The introduction of bright single-mode laser sources lead to the observation of power-broadening of the fluorescence and selective wavelength measurements varying the position of the laser line within the D2 line. The polarization rate is found to increase from less than 20% for a low laser power to 60% for highly saturating laser intensity . The polarization is also found to vary with the position of the laser mode within the D2 line [19, 20]. The polarization rate drops from 60% outside the line, to 12% for the Fg=1, and to 23% for the Fg=2. These results are supported by a rate-equation description in the low power limit and by density matrix simulations on the whole laser-intensity range. To our knowledge, no model has been developed yet for a broadband excitation of the hyperfine-Doppler broadened D2 line. In the present section we define the theoretical model that we developed to characterize the polarization rate.
2.2 D2 line polarization: rate equations model
For simplicity, we chose to ignore coherence effects between the Zeeman sublevels and to implement a rate-equation description of the sodium fluorescence. Collisional effects are also supposed to be negligible according to the low mesospheric pressure. Actually, at the laser powers used in LGS techniques, rate-equation models are sufficient for giving accurate results on the evolution of the populations. The D2 line hyperfine structure is recalled in Fig. 2. Figure 3 represents the shape of the D2 line, including hyperfine structure and Doppler broadening.
where , are the populations of the ground and excited sublevels of the D2 line; q=0, +1, -1 for π, and σ+ and σ- polarization of the laser light, respectively. The coefficients W are the induced transition rates and are given by
The transfer coefficients are given in the Wigner 3-j and 6-j notations by
λ is the laser wavelength, h is the Planck constant, c is the speed of light, IL is the laser intensity (expressed in W/m2), and g(ν) and ρ(ν) are the atomic transition and laser light spectral profiles, respectively. Recall that the laser spectral width is taken equal to 3 GHz. g(ν) is the Doppler-broadened hyperfine structure of the D2 line plotted on Fig. 3. τ is the transition lifetime of the D2 transition. Note that we chose to ignore any depolarizing effect such as collisions. This is justified by the short lifetime of the excited level compared to typical collision times at the mesosphere. Initially the population of the ground state is equally distributed among the hyperfine levels Fg. Then the population of the sublevels mFg of Fg=1 and Fg=2 are 1/6 and 1/10, respectively, at t=0. By means of a 4th-order Runge-Kutta algorithm, we calculate the evolution of the populations during the irradiation and we integrate both the π- and the σ-polarized fluorescence (Iπ and Iσ, respectively).
In the pulsed regime, the integration time is given by the duration of the laser pulse, whereas in the CW regime it is given by the average travel time θ of the sodium atom in the cylindric volume shined by the laser. A simple integration leads to , where w is the laser beamwaist (FWHM), m is the mass of the sodium atom, kB is the Boltzmann constant, and T is the temperature. Since the quantization axis is chosen along the polarization of the laser (y), the π photons have their polarization parallel to the incident laser polarization. Their intensity is maximum in the plane Oxz. On the contrary, the σ photons emitted in the plane Oxz are polarized perpendicularly to the incident polarization. Their flux in this plane is equal to half the maximal value Iσ. The polarization rate can be deduced from the relation . Figure 4 presents numerical results obtained by our model in the case of a 3 GHz laser spectrum centered on the Doppler-broadened D2 line at 130°C.
3. Experimental results and discussion
3.1 The ELP-OA modeless laser
The original purpose for developing a modeless laser was to avoid saturation of the sodium atom and to dramatically decrease the return fluorescence yield . The saturation intensity of the D2 sodium line is relatively high (95.4 W/m2), and saturation is not a problem for the CW LGSs developed up to now. However, CW laser solutions are limited by the technology of the pump lasers and do not seem to show further enhancements for the future in terms of power and compactness. On the other hand, pulsed-laser technology is more mature and promising but leads to saturation problems. A reduction in the saturation parameter can be obtained by increasing the laser spectral width. In this case, several velocity classes of the sodium atom are excited, and the saturation parameter per velocity class is decreased. The solution developed by the Keck Observatory combines a single-mode laser with two extracavity acousto-optic modulators (AOMs) to give side modes in the Doppler-hyperfine width of the D2 transition . In the ELP-OA program, a modeless dye laser with an intracavity frequency shifter has been implemented, providing a continuous spectrum over 3 GHz of the D2 line width . The saturation parameter per velocity class is strongly reduced. It has been experimentally shown that the fluorescence flux is increased by a factor of 5 over a conventional single-mode laser and by a factor of 3 over a five-mode modulated laser .
3.2 Experimental setup
The modeless laser developed at the LSP is based on an L-shaped cavity, wherein a Rhodamine 6G jet is pumped by a 3 W CW intracavity frequency-doubled Nd-YVO4 laser (Verdi, Coherent Inc.). An AOM tuned at 40 MHz is placed in the cavity to produce the frequency shift. The cavity is closed on the first order of the AOM diffraction. A Fabry-Perot étalon is used for fine-tuning the laser frequency. The width of the laser spectrum is adjusted to 3 GHz in order to optimize the overlap with the hyperfine-Doppler width of the sodium D2 line. In the CW regime the laser power is close to 130 mW with a 3W pump. In order to reach high peak power and for future implementation of the ELP-OA program, an extra dye cell (Pyrrhomethene) is inserted into the cavity. This cell can be pumped by a 3 W average-power nanosecond frequency-doubled Nd-YAG laser tuned to a 17 kHz repetition rate (Spectra-Physics, Navigator II YHP70). The modeless laser can be used in the CW or in the pulsed regime. The horizontally polarized 589 nm beam (i.e., along y) is slightly focused in a lowpressure sodium cell by means of a 400 mm focal-length lens. The spot size is about 100 µm (FWHM) and the Rayleigh range is longer than the useful length inside the cell. The temperature of the cell is tunable up to 130°C. In a vertical direction perpendicular to the laser beam (z), two pinholes spatially filter the fluorescence photons to a photomultiplier tube (See Fig. 5). The lastter is kept in the regime where the output voltage is proportional to the light flux. A polarizing beam splitter is placed between the two pinholes and can be rotated to select the polarization of the fluorescence photons and to determine the value of the polarization rate. It is important to note that owing to the choice of the quantization axis (y), the fluorescence flux measured in the z-direction is the same as that in the x-direction. Thus the polarization rate measured in our experimental configuration is the same as in the LGS projects where the fluorescence is detected in the direction of the laser beam.
3.3 Polarization rate vs. laser detuning
To check the principle of the experiment, we tune the modeless laser to the D1 sodium line (3S1/2-3P1/2 at 589.593 nm) and we scan the laser across the line by adjusting the intracavity Fabry-Perot étalon. At each position the fluorescence is measured with the analyzer set parallel and perpendicular to the laser polarization. The polarization rate is deduced from the measurements and plotted on Fig. 6 (top). As expected, the D1 line shows a polarization rate equal to zero owing to summation rules on the hyperfine structure of the 3S1/2-3P1/2 transition.
Then the laser line is tuned to the D2 wavelength (588.997 nm), and a similar scanning is performed. Results are presented in Fig. 6 (bottom). The polarization rate remains below 25% and increases when the laser lines exceed the resonance. Note that the polarization rate is slightly higher for a negative detuning (i.e., close to the Fg=2 to Fe=1, 2, 3 transition) than for a positive one (Fg=1 to Fe=0, 1, 2 transition). This trend is to be related to experimental results obtained with a single mode laser far below saturation .
3.4 Polarization rate vs. laser intensity
Then we characterize the evolution of the polarization rate of fluorescence of the D2 line with the laser intensity. The laser line is tuned to the maximum of the fluorescence signal (i.e., the frequency shift is set to zero). The results are presented in Fig. 7. Experimental data have been obtained both in the CW and also in the pulsed regimes. We also plot the curve given by the previously described rate equation model and the results extracted from Ref. . In the latter case, a density matrix model was developed to account for polarization rate measurements on the sodium D2 excited by a CW monochromatic laser centered on the Fg=2-Fe=3 transition. In this case, the polarization rate exceeds 50% for a Rabi frequency equal to 0.5 GHz (2×106 W/m2) and then extends to the limit at 60%. Both pulsed and CW cases show good agreement with the theoretical model in the limit of validity of the rate-equation model. They also prove that the polarization rate remains below 25% over eight decades of the laser power, contrary to a CW single-mode laser excitation.
In the conditions of the ELP-OA program, the fluorescence polarization rate is expected to be about 18%, leading to an effective fluorescence equal to 41% of the total fluorescence. This is close to the optimal value of 50% where the fluorescence is completely depolarized. In terms of Rayleigh cancellation, a modeless laser exhibits almost ideal properties. It is important to note that far below the saturation point a CW single-mode laser also exhibits a weak fluorescence polarization rate. Scanning a single-mode laser across the D2 line, authors of Ref.  obtained a polarization rate varying from 12% for the Fg=1 transitions to 23% for the Fg=2 ones, giving effective fluorescence fluxes equal to 44% and 33% of the total produced fluorescence, respectively. Then the suppression by polarization filtering of the Rayleigh scattering noise in LGSs excited by a CW laser shows good efficiency, although slightly lower than the modeless laser. For efficiency reasons, CW single-mode excitation is usually tuned to the Fg=2 transitions, where the effective fluorescence ratio is 33%.
Finally, one could think of using the D1 line of the sodium atom to produce the LGS, since the transition is completely depolarized and the effective fluorescence ratio is 50%; however, the excitation of the D1 line is half as efficient as the D2, since the oscillator strength of the D1 is half the value of the D2.
In conclusion, a modeless laser covering the whole D2 line appears to be an ideal solution, both in terms of efficiency of excitation and of Rayleigh cancellation.
The possibility of cancelling Rayleigh scattering by means of polarization filtering can be extended to other sodium atomic transitions or other atoms. Again, the only requirement is that the fluorescence is depolarized enough to keep a reasonable amount of return flux. Among the possible depolarizing mechanisms are hyperfine structures and collisions. As an example, we discuss below the possibility of cancelling Rayleigh scattering in the direct excitation of the mesospheric sodium at 330 nm, as proposed recently. Then we extend this possibility to Laser Induced Fluorescence (LIF) for LIDAR applications.
4.1 Polychromatic LGS (ELPOA program)
To create the polychromatic cascade needed to correct the tip-tilt of the atmosphere, we studied the possibility of direct excitation of the 3S1/2-4P3/2 transition using a modeless laser centered at 330 nm (spectral bandwidth: 3.5 GHz) [9, 10]. This solution has a significant advantage over the previously chosen two-photon solution (589+569 nm) . That is, 1 W (in the mesosphere) at 330 nm produces the same return fluorescence flux at 330 nm as 2 * 15 W at 589 and 569 nm, with greater flexibility in terms of laser solutions. For a given return flux at 330 nm, the ratio of the intensity of Rayleigh scattering between the two solutions is . Rayleigh scattering is about 3 times weaker in direct excitation at 330 nm than in the original scheme (589+569 nm). But still, polarization properties also enable us to suppress the Rayleigh-induced noise completely in the direct 330 nm excitation scheme. For the moment, no experiment has been done to measure the polarization rate at 330 nm by a modeless laser. However, early experiments performed using a broadband source report a polarization rate for the 3S1/2-4P3/2 transition equal to 19% . This value should be close to what would be measured using a modeless laser even for a high laser peak power. To support this view, it is worth noting that the 3S1/2-4P3/2 transition has the same hyperfine structure as the 3S1/2-3P3/2 one. Since the laser pulse duration of the ELP-OA program (50 ns) is much smaller than the lifetime of the level 4P3/2 (320 ns), there is no time for significant population redistribution of the 3S1/2 level via the cascade 4P3/2-4S1/2-3P3(1)/2-3S1/2. Moreover, the saturation parameter per velocity class is kept close to 1, excluding significant optical pumping on the 3S1/2-4P3/2 transition. Therefore it can be considered that the polarization rate of the 330 nm fluorescence is close to the case of the D2 line; that is to say, below 20–25%. Then the problem posed by Rayleigh scattering in the direct 330 nm excitation scheme is also completely solvable by polarization filtering, again at the expense of doubling the laser flux.
4.2 Fluorescence LIDAR
Light scattering is a problem in fluorescence LIDAR techniques where resonant fluorescence of low-concentrated species can be hidden by nonresonant scattering. Contrary to the excitation of mesospheric species where the collisions are not significant during the lifetime of the excited level, atmospheric LIDAR measurements at low altitude are dominated by collisional processes. It is well known that collisions lead to the depolarization of the fluorescence of atoms or molecules induced by a polarized laser . In that case, a fluorescence signal can be extracted from the Rayleigh scattering and from most of the Mie scattering using the same polarization filter described above. The signal-to-noise ratio can then be increased dramatically.
In conclusion, we propose the possibility of completely cancelling the effect of Rayleigh scattering produced in LGSs-based AO systems by placing an analyzer on the return path of the fluorescence signal. A dramatic increase of the signal-to-noise ratio is to be expected from this possibility at the expense of an important loss on the exploitable fluorescence flux. This loss decreases with the polarization rate of the fluorescence. In addition to the hyperfine structure of the 3S1/2-3P3/2 that depolarizes the fluorescence, we show that the saturation parameter of the laser has to be kept as weak as possible to avoid optical pumping and polarization of the fluorescence. We demonstrate that the modeless laser, previously developed in the frame of the ELP-OA program to overcome the problem of saturation and to optimize the return flux, offers the minimum fluorescence polarization rate and is the ideal solution to suppress the Rayleigh scattering-induced noise at high peak intensities. We extend the possibility of cancelling the Rayleigh effects by selective filtering based on polarization to the direct excitation of the PLGS at 330 nm, and to LIF and LIDAR techniques as well.
This work was supported by the CNRS/SPM and CNRS/MRCT. We thank Thibaut Vacelet for helpful technical support.
References and links
1. R. Foy and A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).
2. L. Thompson and C. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for Adaptive Optics in Astronomy,” Nature 328, 229–235 (1987). [CrossRef]
3. F. Rigaut and E. Gendron, “Laser guide star in adaptive optics: the tilt determination problem,” Astron. Astrophys. 261, 677–682 (1992).
4. R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, and M. Tallon, “The polychromatic artificial sodium star: a new concept for correcting the atmospheric tilt,” Astron. Astrophys. Suppl. Ser. 111, 569–578 (1995).
5. M. Schöck, J.-P. Pique, A. Petit, P. Chevrou, V. Michau, G. Grynberg, A. Migus, N. Ageorges, V. Bellanger, F. Biraben, R. Deron, H. Fews, F. Foy, C. Högemann, M. Laubscher, D. Muller, C. d’Orgeville, O. Peillet, M. Redfern, R. Foy, P. Segonds, R. Soden, M. Tallon, E. Thiébaut, A. Tokovinin, J. Vaillant, and J.-M. Weulersse, “ELP-OA: measuring the wavefront tilt without a natural guide star,” Proc. SPIE 4125, 41–53 (2000). [CrossRef]
6. R. H. Dicke, “Phase-contrast detection of telescope seeing and their correction,” Astron. J. 198, 605 (1975).
7. P. L. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory laser guide star adaptive optics system: overview,” Pub. Astron. Soc. Pac 118, 297–307 (2006). [CrossRef]
8. D. Bonaccini, ESO, Karl-Schwarzschild-Str. 2, D-85748 Garching bei München (personal communication, 2006).
9. J. P. Pique, I. Moldovan, and V. Fesquet, “Concept of the polychromatic laser guide star: direct one photon excitation of 4P3/2 level of the sodium atom,” J. Opt. Soc. Am A 23 (2006), to appear. [CrossRef]
10. J. P. Pique, I. Moldovan, and V. Fesquet, “New method for atmospheric tip-tilt correction: low power UV modeless laser excitation of mesospheric sodium layer,” Proc. CLEO Europe (2005).
11. J. R. Morris, “Efficient excitation of a mesospheric sodium laser guide star by intermediate-duration pulses,” J. Opt. Soc. Am. A 11, 832–845 (1994). [CrossRef]
12. A. T. Young, “Rayleigh scattering,” Phys. Today, 42–48 (1982). [CrossRef]
14. J. Pique and S. Farinotti, “Efficient modeless laser for a mesospheric sodium laser guide star,” J. Opt. Soc. Am. B 20, 2093–2102 (2003). [CrossRef]
15. R. W. Wood and F. L. Mohler, “Resonance radiation of sodium vapor excited by one of the D lines,” Phys. Rev. 11, 70–81 (1918). [CrossRef]
16. N. P. Heydenburg, L. Larrick, and A. Ellett, “Polarization of sodium resonance radiation and nuclear moment of the sodium atom,” Phys. Rev. 40, 1041–1044 (1932).
17. P. Feofilov, The Physical Basis of Polarized Emission (Consultants Bureau, New York, 1961).
18. L. Larrick, “Nuclear magnetic moments from the polarization of resonance radiation sodium, 32S1/2-32P3/2, 1/2,” Phys. Rev. 46, 581–586 (1934). [CrossRef]
19. H. Hamaguchi, A. D. Buckingham, and M. Kakimoto, “Depolarization of the near-resonant light scattered by atomic sodium,” Opt. Lett. 5, 124–126 (1980). [CrossRef]
20. R. Walkup, A. L. Migdall, and D. E. Pritchard, “Frequency-dependent polarization of light scattered near the Na D2 resonance line,” Phys. Rev. A 25, 3114–3120 (1982). [CrossRef]
21. Y. Nafcha, M. Rosenbluh, P. Tremblay, and C. Jacques, “Coherence-induced population redistribution in optical pumping,” Phys. Rev. A 52, 3116–3124 (1995). [CrossRef]
22. G. Alzetta, S. Gozzini, A. Lucchesini, S. Cartaleva, T. Karaulanov, C. Marinelli, and L. Moi, “Complete electromagnetically induced transparency in sodium atoms excited by a multimode dye laser,” Phys. Rev. A 69, 1–9 (2004). [CrossRef]
23. A. R. Contos, P. L. Wizinovitch, S. K. Hartman, D. Le Mignant, C. R. Neyman, P. J. Stomski, and D. Summers, “Laser guide star adaptive optics at the Keck Observatory,” in Adaptive Optics System Technologies II, P. L. Wizinovitch and D. Bonaccini, eds, Proc. SPIE4839, 370 (2002). [CrossRef]
24. N. P. Heydenburg and A. Ellett, “Nuclear moment of Na, and polarization of resonance radiation,” Bull. Am. Phys. Soc., June 2 (1933).