A&A 510, A20 (2010) Astronomy DOI: 10.1051/0004-6361/200913108 & c ESO 2010 Astrophysics Optimization of cw sodium laser guide star efficiency R. Holzlöhner1, S. M. Rochester2, D. Bonaccini Calia1,D.Budker2,J.M.Higbie3, and W. Hackenberg1 1 Laser Systems Department, European Southern Observatory (ESO), Karl-Schwarzschild-Str. 2, 85748 Garching b. München, Germany, http://www.eso.org/sci/facilities/develop/lgsf/ e-mail: [rholzloe;dbonacci;whackenb]@eso.org 2 University of California Berkeley, Department of Physics, Berkeley, CA 94720-7300, USA e-mail: [simonr;budker]@berkeley.edu 3 Bucknell University, Department of Physics, 701 Moore Avenue, Lewisburg, PA 17837, USA e-mail: [email protected] Received 12 August 2009 / Accepted 1 October 2009 ABSTRACT Context. Sodium laser guide stars (LGS) are about to enter a new range of laser powers. Previous theoretical and numerical methods are inadequate for accurate computations of the return flux, hence for the design of the next-generation LGS systems. Aims. We numerically optimize the cw (continuous wave) laser format, in particular, the light polarization and spectrum. Methods. Using Bloch equations, we simulate the mesospheric sodium atoms, including Doppler broadening, saturation, collisional relaxation, Larmor precession, and recoil, taking all 24 sodium hyperfine states into account and 100–300 velocity groups. Results. LGS return flux is limited by “three evils”: Larmor precession due to the geomagnetic field, atomic recoil due to radiation pressure, and transition saturation. We study their impact and show that the return flux can be boosted by repumping (simultaneous excitation of the sodium D2aandD2b lines with 10−20% of the laser power in the latter). Conclusions. We strongly recommend the use of circularly polarized lasers and repumping. As a rule of thumb, the bandwidth of laser radiation in MHz (at each line) should approximately equal the launched laser power in Watts divided by six, assuming a diffraction-limited spot size. Key words. instrumentation: adaptive optics – methods: numerical – atmospheric effects – telescopes 1. Introduction Careful dimensioning of the required laser power and op- timization of the laser format (spectrum, polarization, spot size) Laser guide stars (LGS) are becoming essential in providing are needed. Numerical simulations are necessary since the exper- artificial beacons for adaptive optics (AO) in large telescopes imental situation has so far been unsatisfactory: the sodium layer (Hubin 2009). The generation of 8–10 m class telescopes, such and atmospheric parameters often fluctuate rapidly (Thomas as the Very Large Telescope (VLT) on Cerro Paranal, Chile, or 2008; Pfrommer 2009), few reliable powerful lasers at 589 nm the Keck telescopes on Mauna Kea, has been retrofitted with have been available up to now, and LGS sky experiments lack LGS and is still operated in many observing programs with natu- commonly agreed on measurement standards. Sodium cells can- + ral guide stars or no AO at all. The upcoming 30 m telescopes, not easily simulate mesospheric conditions. This work provides such as the 42-m European Extremely Large Telescope (E-ELT) optimization rules for the case of continuous wave (cw) lasers or the Thirty Meter Telescope (TMT), by contrast, are being de- over a wide range of laser powers. signed from the start as adaptive telescopes and will require LGS 2 2 in nearly all of their science operations. At the same time, sens- Sodium LGS take advantage of the 3 S 1/2−3 P3/2 dipole − 2 ing subaperture sizes of 20 50 cm and near-kHz AO frame rates, transition, known as the D2 line. The sodium S ground state designed to provide high Strehl ratios in the infrared or even consists of two hyperfine multiplets with 8 magnetic substates in the visible, require unprecedented LGS brightness and hence combined, separated by 1.772 GHz, splitting the D2 line into the laser power. D2aandD2b transition groups, corresponding respectively to the Sodium (Na) LGS at 589 nm are most commonly used be- F = 1andF = 2 Na ground states, where F is the total atomic cause of the large fluorescence cross section-abundance prod- angular momentum quantum number. The four 2P multiplets uct of ca. 10−11 cm2 × 4 × 1013 m−2, a fluorescence wave- (F = 0 ...3) are separated by only 16, 34, and 60 MHz (16 mag- length in the visible, and the high altitude of the sodium layer netic substates, see Ungar 1989). At mesospheric temperatures around 80−100 km compared e.g. to Rayleigh LGS, (Happer near 190 K, the D2aandD2b lines are Doppler broadened to 1994, Table 1), which allows one to sense a large fraction of about 1 GHz each, giving rise to the characteristic double-hump the turbulent atmosphere column above the telescope with a absorption profile (see for instance Bradley 1992, Fig. 11, or small number of guide stars. Unfortunately, powerful diffraction- Fig. 1 of this work). When the D2a transition is excited by cir- limited laser beams at 589 nm are quite expensive to produce cularly polarized light at high irradiance (i.e. optical power den- because of the lack of solid-state materials that amplify 589 nm sity, unit W/m2), a large fraction of atoms is pumped into the or 1178 nm light. Recently, ESO has demonstrated a frequency- (S, F = 2, m = ±2) substate, and the atoms cycle on the transi- doubled narrow-band Raman fiber laser emitting 25 W at 589 nm tion to the (P, F = 3, m = ±3) substate, with m the magnetic (Taylor 2009), and we expect this technology to significantly im- quantum number, so that sodium effectively becomes a two-level prove the experimental situation. system. This situation is twofold desirable because of the large Article published by EDP Sciences Page 1 of 14 A&A 510, A20 (2010) transition cross section and the directional return light peak to- groups can be properly coupled, including their atomic polariza- wards the laser emitter and the telescope. Numerous studies of tion exchange (using Bloch equations, such coupling requires optical pumping have been conducted, e.g. by Happer (1972) the simultaneous solution of sets of 242-dimensional equations and McClelland (1985), and the process is well understood. per velocity group, which we describe in this work for the first The effective sodium return flux depends on the environmen- time in application to sodium LGS, to our knowledge). tal conditions such as collisions with constituent gases, temper- A serious disadvantage of rate equations is that they rely on ature, and the geomagnetic field. At higher irradiance, the atoms steady-state atomic cross sections, limiting the scope to the sim- are driven away from thermal equilibrium, and it is necessary ulation of events slow compared to the sodium transition life- to take effects into account such as saturation, optical pumping, time, (Milonni 1992), and the combination of the excitation- and the recoil caused by radiation pressure. Understanding the emission (Rabi) cycling with Larmor precession is hard to complicated interplay of these effects and obtaining quantitative implement correctly. Furthermore, when compared to atomic values of the fluorescence efficiency requires numerical simula- Bloch equations, rate equations neglect the coherences (off- tions. A commonly used method is the solution of the density diagonal terms of the density matrix that describe transverse matrix evolution (Bloch equation) of a multilevel atom. atomic polarization), which is problematic when modeling Milonni & Thode (1992) simplified the D2 scheme to a Larmor precession or linearly polarized light (the latter even for 2-level Bloch model which they solve in time domain. Bradley B = 0). Finally, Monte Carlo rate-equations need considerable (1992) simulated the full 24-state density matrix, exciting the CPU time in order to converge. sodium by a train of short (nanosecond range) laser pulses like There are also approaches that mix aspects of Bloch and Milonni, using Runge-Kutta integration for one pulse period rate-equation codes, such as BEACON, which has been adapted and exploiting the periodicity. Linear and circular light polar- to model two-step sodium excitation for polychromatic LGS, izations were treated. However, both works neglect the geomag- (Bellanger 2004). BEACON neglects atomic collisions and re- netic field. Morris (1994) studied frequency-modulated pulses coil. Guillet de Chatellus (2008) reports that it requires on the over a wide range of pulse durations and linewidths up to 3 GHz, order of 24 h per run on a 2.6 GHz processor, and that the agree- hence spanning the entire Doppler broadened D2 line, employing ment with a pure rate-equation model can be good in certain time domain integration, for both linearly and circularly polar- cases. ized light. In the end of his paper, he briefly estimates the impact Throughout this paper, we highlight what may be called the of the geomagnetic field, but he does not include Larmor preces- “three evils” of sodium LGS, ordered by decreasing importance sion terms into his Bloch equations. 1. Larmor precession, To our knowledge, Milonni (1998, 1999) has published 2. Recoil (radiation pressure), the most advanced and detailed Bloch-equation simulation of 3. transition saturation (stimulated emission), sodium LGS to date, later generalized by Telle (2006, 2008). In his 1998 article, Milonni treats the cases of laser pulses that and we suggest ways of mitigating them. Larmor precession are short, comparable, and long compared to the 2P lifetime is powerful enough to thwart optical pumping if the angle be- of τ = 16.24 ns, using numerical solution methods similar to tween the beam and the field lines is large (Drummond 2007), Bradley’s. His 1999 publication deals with cw excitation only (Moussaoui 2009). The average 50-kHz redshift that one in- and introduces spin relaxation and Larmor precession into the curs per spontaneous emission due to atomic recoil can lead Bloch equations for the first time.