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With quantum Bloch equations and sodium cell based experiment, we systematically resolved optical repumping mechanism of sodium atoms, a path to brighter sodium laser guide star (SLGS) that would be accepted by worldwide observatories. Besides the former studies, we detailed the population distribution of sodium atoms with and without repumping, which makes the repumping mechanism easy to understand. Experimental results based on a buffer gas free sodium cell and a single frequency laser implies that the optimum repumping frequency offset is 1712 MHz, and the repumping power fraction should be 10–18%. The light intensity depended character of repumping is also carried out. We learned that future SLGS pumped by higher effective light intensity would benefit more from repumping. To the best of our knowledge, this is the first systematic investigation of repumping mechanism using a single mode sum-frequency laser, which gives thorough support for previous numerical work and shows that the lab bench experiment could be used as an intermediate link between theoretical modeling and on-sky test.
© 2016 Optical Society of America
1. Introduction
Sodium laser guide stars (SLGS) based Adaptive Optics (AO) has become the essential tool for large ground based telescopes to compensate atmospheric turbulence induced wave-front distortion [1,2]. And the next generation giant telescopes such as the upcoming European Extremely Large Telescope (E-ELT) [3], Giant Magellan Telescope (GMT) [4] and the Thirty Meter Telescope (TMT) [5] are all initially designed as SLGS AO assisted telescopes. However, it must be noted that the performance of AO systems are still limited by insufficient SLGS brightness [2]. Better understanding of laser interaction with sodium atom assembly and the related laser parameters optimization are critical to SLGS efficiency improvement. This kind of research has been evolved since SLGS born but not ended until now [6–9]. Heretofore one of the achievements of SLGS mechanism study is optical repumping, that, compared with only exciting the D2a line, simultaneous exciting D2a and D2b lines with 10-20% of laser power in the latter would significantly enhance SLGS brightness [10, 11], especially under high intensity pumping [12].
Although many approaches have been proposed to enhance SLGS brightness and exciting efficiency theoretically [12–15], experimental data and systematically parametric study are rare due to the high cost of on-sky test, and besides the cost, complexity induced by uncontrollable factors such laser beam dithering and atmospheric turbulence further adds uncertainty to the field experiments. So, between fully theoretical modeling and the last on-sky experimental test, lab based experimental study would bridge this large gap and hence provide a valuable tool for SLGS optimization. H. G. de Chatellus et al. [16] and O. Lardiere et al. [17] give good examples that lab bench experiments could bring valuable information before the on-sky test.
In this paper, we built a lab bench to study optical repumping mechanism of sodium atoms based on a 589 nm sum-frequency single mode laser which is the heart of the lab bench. Accompanied by a tunable electro-optic phase modulator (EOM), the laser can pump the D2a and D2b lines simultaneously. Although the parametric condition of the sodium cell used here is not the same as the atmospheric sodium layer’s, the results are also meaningful due to the similar features of repumping. Theoretically, to make the brightness enhancing mechanism of repumping easy to understand, the population distribution is numerically calculated by solving steady state Bloch equations in section 2. Experimental setup and results with varying offset frequency, repumping power fraction and light intensity are presented in section 3. The difference between sodium cell and atmosphere sodium layer is discussed in section 4.
2. Theoretical analysis
2.1 Sodium D2 line data
The sodium D2 line data is well summarized by D. A. Steck in [18] and only several key features are described here. As illustrated in Fig. 1, sodium D2 line involves 24 magnetic sub-states. The two ground hyperfine states are separated by 1771.6 MHz while the four upper hyperfine states are separated by 15.8, 34.3, and 58.3 MHz.
2.2 Bloch equations
We begin with density matrix equation, which is equal to the Schrodinger equation [19],
where H = H0 + HE + HB is the total Hamiltonian with H0 the unperturbed Hamiltonian, HE the electric-dipole interaction and HB the magnetic-field Hamiltonian. The relaxation matrix is a diagonal matrix conclude decay rate of each state. For ground level, each sub state undergoes relaxation due to collision, as well as atoms leaving the light beam (transit relaxation). While for upper level, the relaxation includes spontaneous decay in addition. For a buffer gas free sodium cell, collision between sodium atoms is rare, so only transit relaxation is considered here and its rate is noted as γ. Corresponding to relaxation process, Λ represents repopulation process to make sure the number of atoms in the system conserved. Here Λ is ρ independent because “new atoms” that enter the system have random polarization state. This situation is valid in a buffer gas free sodium cell because the state of sodium atoms is essentially completely randomized when the atoms strike the glass wall of the cell and then return to the system. At the same time, the spontaneous decay process also repopulates the ground sub states.All elements on the right side of Eq. (1) are time independent except the electric-dipole interaction. But the optical frequencies can also be removed entirely under rotating wave approximation as described in [12] although there are two light frequencies when both D2a and D2b lines are excited by two laser fields. This makes it easy to find steady state density matrix by setting the left part of Eq. (1) to be zero and solving algebraic equations. In this paper we use a well established robust Mathematica package, named Atomic Density Matrix [20], to generate the Hamiltonian and the system of Liouville equations.
2.3 Steady-state population distribution
To illustrate the repumping effect, we gave an example of steady-state population distributions of sodium atoms. Originally, the atoms are supposed to be in thermal equilibrium at temperature of 70 °C (343 K), and their velocities distribute according to Maxwell-Boltzmann rule. And the average velocity is 562 m/s. The light field is supposed to be single frequency with σ+ circularly polarization and the light intensity is 500 W/m2. For a beam with 1 mm2 of spot area, its diameter is 1.13 mm. So the average time that atoms transit through the beam is 2 µs and the transit relaxation rate is γ = 1/(2 µs). The magnetic field is 0.42 G with 40.5°zenith angle between the light propagating direction. Considering the natural width of sodium atoms is 10 MHz and the Doppler width is 1.4 GHz, we solved 3000 frequency velocity groups with 1 MHz each step to get fine resolution of Doppler shifts.
Figure 2(a) shows the occupation density for different states without repumping. The upper most black curve is for (3S1/2, F = 1) state and the red curve is for (3S1/2, F = 2) state. It is obvious that a large proportion of atoms are trapped in (3S1/2, F = 1) state especially when atoms are resonant with the light field oscillating between (3S1/2, F = 2) and (3P3/2, F’ = 2 or F’ = 1) states. Optical pumping can relief the situation when the atoms are resonant with the light field oscillating between (3S1/2, F = 2) and (3P3/2, F = 3′), because some atoms have been pumped to (3S1/2, F = 2, m = 2) state (blue curve) before falling to (3S1/2, F = 1) state and can only transit between (3S1/2, F = 2, m = 2) and (3P3/2, F’ = 3, m’ = 3) (magenta curve).
Fig. 2 Occupation density of different states in sodium atoms pumped by 500 W/m2 light intensity. (a) Without repumping. (b) “(R)” represents 14% of light power repumping and dash lines illustrate states without repumping.
The comparison of repumping with the situation discussed above is showed in Fig. 2(b). The solid (repumping) and dash (no repumping) lines present the population distributions of sodium atoms pumped by the same light intensity. Here 14% of light power with 1.713 GHz shifted frequency is used to pump the D2b line. It is noticeable that although only a small proportion of light power is applied to repump the atoms, the number of atoms at (3S1/2, F = 2) state increases significantly when the atoms resonant with the light, while the D2a line transition keeps its high efficiency because most laser power is still exciting it. As a result, the upper state occupation density at the frequency with 0 GHz detuning is increased by a factor of 1.2 without changing the total light intensity.
3. Experiment
3.1 Experimental setup
Figure 3 illustrates the experimental setup. The 589 nm light source is a 1064 + 1319 nm sum-frequency laser with 10 mW output power (fluctuating less than ± 0.3% in half an hour), single frequency and near diffraction limited beam quality. The wavelength is tuned to 589.15905 nm (fluctuating less than 20 MHz in 5 minutes) and monitored by the wave meter. A KTP crystal based electro-optic phase modulator (EOM), whose resonant frequency is tunable between 1.6 and 1.8 GHz, is employed to produce 1.7 GHz sidebands. And the power fractions of sidebands are measured by a Fabry-Perot interferometer (F-P) as showed in Fig. 4. Then a polarizing beam splitter (PBS) is employed to purify the polarization of the light beam before the electro-optic amplitude modulator (AM). As the polarization of the light changes with the voltage applied to the AM, the power of the light reflected by the latter PBS becomes controllable. After that, about 0.5% of the light power is split out to a Si detector to monitor the power of the light in the sodium cell. Before the cell, the polarization of the light is converted from linear to circular by a λ/4 wave plate. At the place where fluorescence is measured, the effect area of the light beam is 1 mm2 (86.5% of barrel power in Gaussian beam measured by a beam profiler) and the magnetic field is 0.42 G with 40.5°zenith angle between the light propagating direction, which is measured by a vector magnetometer. The sodium cell is buffer gas free with 75 cm of length and 25 cm of diameter. And the cell is heated by an electric heating oven to 70 ± 0.1 °C. Though higher temperature means higher atom density and brighter fluorescence, the atom is abundant enough at the temperature of 70 °C for the measurement of the relative brightness of the resonant fluorescence in this experiment because signals corresponding to fluorescence are far larger (>100 times) than the dark noise of the amplified photomultiplier. About 1 cm away from the incident window, we open a 0.8 × 2.4 cm2 side slit to observe the fluorescence from the cell. After all, fluorescence from the slit is collected by a lens with 2.5 cm diameter and detected by an amplified photomultiplier with 7.1 × 106 max gain. Signals from the F-P, Si detector and Photomultiplier are collected by a 4 channels digital oscilloscope (not showed in the picture).
Fig. 3 Experimental setup. (M: mirror; EOM: electro-optic phase modulator; BS: beam splitter; F-P: Fabry-Perot interferometer; PBS: polarizing beam splitter; AM: amplitude modulator; λ/4: λ/4wave plate.)
Figure 4 shows the repumping sidebands measured by the F-P. The free spectral range of the F-P is 1.5 GHz with 7.5 MHz of resolution. As 1.7 GHz splitting overtakes the free spectral range, so sidebands appear at the points 0.2 GHz away from the central lines. Though phase modulator makes two first order sidebands and even second order sidebands, only one of the first order sidebands couples to the D2b line. So the power of other sidebands is subtracted in latter data processing. For the example in Fig. 4, the peak of the zero band, first and second order sidebands are 2.06, 0.468 and 0.023 respectively. First of all, all bands added together are 3.042. And then, the zero band and one of the first order sidebands added together are 2.528. Thus the effective light power fraction is 2.528/3.042 = 83.1%. At last, the repumping power fraction is 0.468/2.528 = 18.5%.
To monitor the light intensity in the sodium cell in real time, we calibrated Si detector’s response to monitor the laser power in front of the cell beforehand. As the size of the beam spot is unchanged, it is convenient to monitor the light intensity indirectly through the Si detector.
From Eq. (1) in [18], the vapor pressure Pv is 7.09 × 10−9 torr at temperature of 70 °C which leads to a vapor density with ρv of 2.00 × 10−8 atoms/cm3 according to ρv = Pv/(kT). As the temperature fluctuates in ± 0.1 °C, the vapor density as well as the fluorescence fluctuates about 1%. This is the main uncertainty in the experiment.
It must be pointed out that, instead of the absolute value, we only measured the variation of the fluorescence when the repumping offset frequency, repumping power fraction or light intensity changed. So signals can only be compared with each other in the same picture. Although direct comparison between our results and the return flux or return photons in [10, 12] cannot be made because it is out of the scope of this paper, the optimum repumping offset frequency or power fraction and the enhancing times of fluorescence brightness brought by repumping in different situations can be compared because they can be learned from the variation of the fluorescence.
3.2 Repumping frequency offset
Keeping the light intensity at 500 W/m2 and the repumping power fraction to be 12%, tuning the offset frequency from 1695 to 1730 MHz with 2.5 MHz each step, the resonant scattering fluorescence signals with different offset frequencies are measured and illustrated in Fig. 5.
Fig. 5 Fluorescence signals with different repumping offset frequencies. The light intensity is 500 W/m2, and the repumping power fraction is 12%.
The signal peaks at about 1712 MHz and drops no more than 5% at both 1705 and 1720 MHz. Considering the initial light frequency is resonant with the transition between (3S1/2, F = 2) and (3P3/2, F’ = 3), the light with 1712 MHz frequency shifted is very close to the transition between (3S1/2, F = 1) and (3P3/2, F’ = 2). The optimum repumping offset frequency we got here agrees with the on-sky test result in [10], and we got more accurate value due to more simple conditions. It has been pointed out in [12] that the most efficient repumping offset frequency should lie between the (3S1/2, F = 1) to (3P3/2, F’ = 1,2) transition frequencies minus the (3S1/2, F = 2) to (3P3/2, F’ = 3) transition frequency. Our result agrees with this assertion although the optimum repumping offset frequency is 1717 MHz in [12]. It should be pointed out that when the light intensity is changed between 100 and 500 W/m2 we got the same optimum repumping offset frequency in this experiment. So we only give the example with 500 W/m2 here, a light intensity can easily be reached by the long pulsed SLGS [11].
3.3 Repumping power fraction
The signals with different repumping power fractions are showed in Fig. 6. Where the repumping frequency offset is 1713 MHz.
Fig. 6 Fluorescence signals with different repumping power fractions. The repumping frequency offset is 1713 MHz and the light intensity is (a) 200 W/m2 and (b) 500 W/m2 respectively.
At first, both signals with 200 and 500 W/m2 of light intensity increase with the repumping power fraction, and peak at about 14% of repumping power fraction. But the latter signal with higher light intensity declines slower than the former one when the repumping power fraction increases further. Maybe because at higher light intensity, saturation becomes severe while there are still many atoms trapped in (3S1/2, F = 1) state with 14% of laser power repumping, as showed in Fig. 2(b), then higher repumping power fraction will relieve the saturation and release more atoms from (3S1/2, F = 1) state. It should be pointed out that, compared with the peak point, signals with 10% and 18% repumping power fraction reduce less than 4% in both situations. Besides, the fluctuation characteristics of these two lines agree well with the theoretical work in [12] though many parameters are different.
3.4. Light intensity
To observe the light intensity dependence feature, the light power as well as the intensity feeding into the cell are changed slowly from 50 to 550 W/m2 in 10 seconds by the triangle wave Driven AM, the fluorescence signals with different light intensities are measured and presented in Fig. 7. The upper line illustrates the result with 14% of light power repumping while the lower line displays the result without repumping.
Fig. 7 Fluorescence signals with 14% of light power repumping (the upper line) and without repumping (the lower line).
Repumping does not show any benefit when the light intensity is below about 100 W/m2. But as the light intensity increases further, signals with repumping are stronger than those without repumping. At 50 W/m2 light intensity, signals with and without repumping are similar. As the light intensity increases to 500 W/m2, signal without repumping only climbs to 0.485 while signal with repumping rises to 0.603, which is enhanced by a factor of 1.24 due to repumping. The enhancing times is better than our numerical result in Fig. 2, maybe because the transit relaxation rate we estimated is faster than the actual one. In section 2, the transit time is calculated with effective area, but in fact, some light still exists outside this area and also contributes to down-pumping.
4. Discussion
In our experiment, repumping does not show as effective as simulation predictions [12], nor the on sky test [10]. The main reason lies in the different spin damping mechanism of sodium atoms between sodium cell and atmospheric sodium layer. In the sodium cell, the polarization of the sodium atoms in light field damps mainly due to transit relaxation, while in atmosphere sodium layer the main channel is spin-exchange collisions. In our experiment, the transit relaxation rate is about 1/(2 µs). So atoms might be replaced by “new atoms” before gathered in (3S1/2, F = 1) state if the light intensity is too low. But the spin damping in atmospheric sodium layer is far slower. The spin-exchange rate is estimated to be about 1/(500 µs) [12]. So repumping would perform better in the real SLGS.
Nevertheless, the brightness enhancing mechanism of repumping is similar in both situations. When atoms in the system are trapped in (3S1/2, F = 1) state due to down pumping, and transit relaxation and spin-exchange collisions cannot release these atoms as fast as it trapped, repumping provides an additional high efficient channel to bring these trapped atoms back to (3S1/2, F = 2) state. So the higher light intensity and the slower spin damping, then the more atoms would be trapped and the better repumping would perform.
5. Conclusion
The steady-state population distribution of sodium atoms is calculated through solving Bloch equations. The numerical results show that the upper state occupation density of sodium atoms can be boosted by repumping without changing the total light intensity because simultaneous exciting the D2a and D2b lines can make full use of the large transition cross section of D2a line and release atoms trapped in (3S1/2, F = 1) state at the same time.
Experimental data indicates that the optimum repumping offset frequency and repumping power fraction are 1712 MHz and 14% respectively. It means that the (3S1/2, F = 1) to (3P3/2, F’ = 2) transition can provide a high efficient channel to repump atoms back to (3S1/2, F = 2) state. And the repumping effect with different light intensity implies that future SLGS pumped by higher light intensity might benefit more from repumping.
Although differences exist between sodium cell and atmospheric sodium layer, the similar repumping characteristics with the on-sky test and theoretical modeling show that the lab bench experiment could possibly bridge the gap between theoretical modeling and on-sky test. Other brightness enhancing mechanism is also under testing on this sodium cell based lab bench platform.
Acknowledgments
We are grateful to Prof. Yong Bo and his colleagues (TTIPC, CAS) for assistance in building the 589 nm laser.
References and links
1. P. Wizinowich, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Chin, A. R. Contos, M. A. van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, D. M. Summers, C. G. Brown, P. M. Danforth, C. E. Max, and D. M. Pennington, “The WM Keck Observatory laser guide star adaptive optics system: overview,” Astron. Soc. Pac. 118(840), 297–309 (2006). [CrossRef]
2. P. Wizinowich, “Progress in laser guide star adaptive optics and lessons learned,” Proc. SPIE 8447, 84470D (2012). [CrossRef]
3. T. Fusco, S. Meimon, Y. Clenet, M. Cohen, H. Schnetler, J. Paufique, V. Michau, J. P. Amans, D. Gratadour, C. Petit, C. Robert, P. Jagourel, E. Gendron, G. Rousset, J.-M. Conan, and N. Hubin, “ATLAS: the E-ELT laser tomographic adaptive optics system,” Proc. SPIE 7736, 77360D (2010). [CrossRef]
4. P. M. Hinz, G. Brusa, V. Vaitheeswaran, T. McMahon, T. Connors, R. Knox, A. Bouchez, and M. Montoya, “Design and predicted performance of the GMT ground-layer adaptive optics mode,” Proc. SPIE 8447, 84473R (2012). [CrossRef]
5. S. M. Rochester, A. Otarola, C. Boyer, D. Budker, B. Ellerbroek, R. Holzloehner, and L. Wang, “Modeling of pulsed-laser guide stars for the Thirty Meter Telescope project,” J. Opt. Soc. Am. B 29(8), 2176–2188 (2012). [CrossRef]
6. W. Happer, G. MacDonald, C. Max, and F. Dyson, “Atmospheric-turbulence compensation by resonant optical backscattering from the sodium layer in the upper atmosphere,” J. Opt. Soc. Am. A 11(1), 263–276 (1994). [CrossRef]
7. P. W. Milonni, H. Fearn, J. M. Tell, and R. Q. Fugate, “Theory of continuous-wave excitation of the sodium beacon,” J. Opt. Soc. Am. A 16, 11 (1999).
8. E. Kibblewhite, “The physics of the sodium laser guide star: Predicting and enhancing the photon returns,” in Advanced Maui Optical and Space Surveilance Technologies Conference (2009).
9. R. Rampy, D. Gavel, S. M. Rochester, and R. Holzlöhner, “Toward optimization of pulsed sodium laser guide stars,” J. Opt. Soc. Am. B 32(12), 2425–2434 (2015). [CrossRef]
10. C. Denman, G. Moore, J. Drummond, M. Eickhoff, P. Hillman, J. Telle, S. Novotny, and R. Fugate, “Two-frequency sodium guide star excitation at the Starfire Optical Range,” in The Central for Adaptive Optics (CFAO) Workshop (2006).
11. K. Jin, K. Wei, L. Feng, Y. Bo, J. Zuo, M. Li, H. Fu, X. Dai, Q. Bian, J. Yao, C. Xu, Z. Wang, Q. Peng, X. Xue, X. Cheng, C. Rao, Z. Xu, and Y. Zhang, “Photon return on-sky test of pulsed sodium laser guide star with D2b repumping,” Astron. Soc. Pac. 127(954), 749–756 (2015). [CrossRef]
12. R. Holzloehner, S. M. Rochester, D. B. Calia, D. Budker, J. M. Higbie, and W. Hackenberg, “Optimization of cw sodium laser guide star efficiency,” Astron. Astrophys. 510, A20 (2010).
13. J. R. Morris, “Efficient excitation of a mesospheric sodium laser guide star by intermediate-duration pulses,” J. Opt. Soc. Am. A 11(2), 832–845 (1994). [CrossRef]
14. J.-P. Pique and S. Farinotti, “Efficient modeless laser for a mesospheric sodium laser guide star,” J. Opt. Soc. Am. B 20(10), 2093–2101 (2003). [CrossRef]
15. P. D. H. Thomas, J. Kane, and C. A. Denman, “Pulsed laser architecture for enhancing backscatter from sodium,” Proc. SPIE 2048, 20483G (2014).
16. H. Guillet de Chatellus, I. Moldovan, V. Fesquet, and J.-P. Pique, “Suppression of Rayleigh scattering noise in sodium laser guide stars by hyperfine depolarization of fluorescence,” Opt. Express 14(24), 11494–11505 (2006). [CrossRef] [PubMed]
17. O. Lardiere, R. Conan, C. Bradley, K. Jackson, and G. Herriot, “A laser guide star wavefront sensor bench demonstrator for TMT,” Opt. Express 16(8), 5527–5543 (2008). [CrossRef] [PubMed]
18. D. A. Steck, “Sodium D Line Data,” http://steck.us/alkalidata.
19. M. Auzinsh, D. Budker, and S. M. Rochester, Optically Polarized Atoms Understanding Light-Atom Interactions (Oxford University, 2010).
20. S. M. Rochester, “Atomic Density Matrix package for Mathematica, version 12.06.21,” http://budker.berkeley.edu/ADM/.
