Modeling of pulsed-laser guide stars for the Thirty Meter ... · using pulsed lasers based on...

Modeling of pulsed-laser guide stars for the Thirty MeterTelescope project

Simon M. Rochester,1,* Angel Otarola,2 Corinne Boyer,2 Dmitry Budker,3,1 Brent Ellerbroek,2

Ronald Holzlöhner,4 and Lianqi Wang2

1Rochester Scientific, LLC, 2041 Tapscott Ave., El Cerrito, California 94530, USA2TMT Observatory Corporation, 1200 East California Boulevard, Mail Code 102-8, Pasadena, California 91125, USA

3Department of Physics, University of California, Berkeley, 366 LeConte Hall MC 7300 Berkeley,California 94720-7300, USA

4European Southern Observatory (ESO), Garching bei München, D-85748, Germany*Corresponding author: [email protected]

Received April 10, 2012; accepted June 4, 2012;posted June 20, 2012 (Doc. ID 166420); published August 1, 2012

The Thirty Meter Telescope (TMT) has been designed to include an adaptive optics system and associated laserguide star (LGS) facility to correct for the image distortion due to Earth’s atmospheric turbulence and achievediffraction-limited imaging. We have calculated the response of mesospheric sodium atoms to a pulsed laser thathas been proposed for use in the LGS facility, including modeling of the atomic physics, the light–atom interac-tions, and the effect of the geomagnetic field and atomic collisions. This particular pulsed-laser format is shown toprovide comparable photon return to a continuous-wave (cw) laser of the same average power; both the cw andpulsed lasers have the potential to satisfy the TMT design requirements for photon return flux. © 2012 OpticalSociety of America

OCIS codes: 350.1260, 300.2530, 010.1080, 010.1310, 010.1350.

1. INTRODUCTION AND MOTIVATIONThe Giant Segmented Mirror Telescope (GSMT) initiative hasbeen ranked among the highest priorities for U.S. astronomyin the last two decadal surveys of astronomy and astrophysics[1,2]. This ranking is motivated by the potential scientificimpact of the sensitivity and diffraction-limited resolution thatit is possible to achieve with a large-aperture telescope. TheThirty Meter Telescope (TMT) project has designed and isplanning the construction of a diffraction-limited, large-aper-ture segmented telescope on the northern slope of the MaunaKea volcano in Hawaii (characteristics of the site are summar-ized in [3]). For such a telescope, adaptive optics (AO) andan associated laser guide star (LGS) system are needed toachieve diffraction-limited capacity.

AO systems are of increasing importance for current [4–8]and future [9–11] ground-based astronomical telescopes. Largeground-based telescopes require AO to correct for the imagedistortion induced by atmospheric turbulence. The use ofAO requires a reference light source, such as a bright naturalstar, in order to measure the effect of the atmosphere on thelight wavefront. Often there is no bright natural star availablewithin the isoplanatic angle from the astronomical target, i.e.,the angle out to which phase variations in the wavefront areless than 1 rad. In this case, a laser can be used to generate anartificial guide star. This can be done by exploiting Rayleigh-backscattered light from a pulsed laser [12], or in the case con-sidered here, resonance fluorescence from sodium atoms inthe mesosphere [13,14] at 80 to 125 km altitude, excited byeither continuous-wave (cw) or pulsed lasers resonant withthe sodium D2 transition at 589 nm. Such techniques were firstproposed for astronomy in the 1980s [15,16]; almost 30 yearslater, LGS systems are available in all 6–10 m-class telescopes.

Many of the key scientific questions to be addressed withTMT [17]—the nature and composition of the universe, theformation of first stars and proto-galaxies, the evolution ofgalaxies and the interplanetary medium, the relation betweenblack holes and their host galaxies, the formation of stars andplanets, the nature of extrasolar planets, and the detection ofextrasolar planets within the habitable zone of their hoststars—require diffraction-limited imaging and thus an LGS-AO system. An example of the need for diffraction-limited ima-ging is observations of the crowded star field close to ourgalactic center. Such observations are technically challenging,and astronomers have devised observation strategies to re-solve the stellar dynamics and accurately locate and weighthe supermassive black hole in the center of our galaxy. Accu-rate determinations of the stellar dynamics have evolved sig-nificantly since the first survey of this star field by means ofspeckle imaging in the late 1990s [18], followed by NaturalGuideStar-AO in the early 2000s [19]; the best results havebeenachieved with the use of LGS-AO in the mid- to late 2000s [20].Yet, there remains the need to improve these observations toresolve stars at angular distances closer to the galactic center.This will allow testing the predictions of general relativity un-der the gravitational field of the supermassive black holehosted in the center of our galaxy. This will be made possibleby a large-aperture, LGS-AO-assisted telescope, such as TMT.

The TMT LGS facility (LGSF) will initially include six lasersto form an asterism consisting of one LGS on-axis and fiveLGS equally spaced on a circle of 35 arcsec angular diameter,with the laser-launch telescope (LLT) located behind the sec-ondary mirror [21]. Using the multiconjugate adaptive-optics(MCAO) approach, the LGSF will be able to provide diffrac-tion-limited imaging in a field-of-view from tens of arcseconds

2176 J. Opt. Soc. Am. B / Vol. 29, No. 8 / August 2012 Rochester et al.

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to about 1–2 arcmin [22]. The LGSF design supports expan-sion to up to nine lasers to form asterisms for other AO modesand radii from 5 to 510 arcsec [23].

The original design for the TMT LGSF called for cw lasers(either solid-state Nd:YAG sum-frequency lasers or Raman fi-ber lasers). Subsequently, TMT has also included the option ofusing pulsed lasers based on diode-pumped solid-state sum-frequency Nd:YAG technology. Such lasers have been devel-oped by scientists at the Technical Institute of Physics andChemistry of the Chinese Academy of Sciences (TIPC). In thisarticle, we investigate the effectiveness of pulsed lasers forLGS, focusing on the TIPC laser format. We also comparethe photon return from the pulsed laser to that of a cw laserof the same average power.

The effectiveness of an LGS system in correcting the turbu-lence-induced aberrations in the image wavefront depends onhaving sufficient photon return from the laser beacons. Themain requirements set for the first-light TMT AO system arethe following: (1) a root-mean-squared (rms) wavefront errorat zenith smaller than 187 nm on-axis and 191 nm over a 17 arc-sec field-of-view, (2) 50% sky coverage at the galactic pole,(3) operation up to a zenith angle of 65°, and (4) operationunder atmospheric turbulence conditions characterized byan atmospheric length scale (Fried length r0 [24]) as low as0.1 m in the direction of observation. The TMT wavefront re-construction algorithm [25] has been tested under various sce-narios and is expected to meet the maximum wavefront errorspecifications (see, for instance, [21]), with a significant mar-gin for error, when using a LGS that provides 900 photo-detected electrons (PDE) per wavefront sensor subaperturein an integration time of 1.25 ms. The specification of 900PDE applies when the laser is pointed in the zenith direction,and includes a rough factor of 2 safety margin to account forvarious effects that have the potential to increase the rmswavefront error, including the effect of directing the laserat larger zenith angles. The photon return scales due to purelygeometrical considerations as the cosine of the zenith angle(i.e., the inverse of the airmass); this scaling is used as a re-ference for comparison with the actual calculated directionaldependence of photon return reported here. Some additionalfactors that affect the actual PDE count are the laser format(power, polarization, spectral characteristics, etc.), various ef-ficiencies in the propagation of the laser light to the launchtelescope, atmospheric absorption, the actual column densityof sodium atoms in the mesosphere, and the relative directionof the local geomagnetic field with respect to the laser beam.Also important is the efficiency of the LGS detection system(i.e., the wavefront sensor).

Optimizing a laser system to maximize the guide starphoton return has been an area of ongoing research for manyyears. Reference [26] investigated the physical aspects affect-ing the performance (i.e., photon flux return) of cw sodiumlasers. The calculation included the important physical effectsof Larmor precession of the sodium atoms in the geomagneticfield, radiation pressure (recoil), and saturation of the opticaltransitions, as well as velocity-changing and spin-randomizingcollisions. This was achieved using a density-matrix calcula-tion with coupled velocity groups. A discussion was given ofvarious mechanisms that are detrimental to photon return;i.e., Larmor precession reduces the efficiency of optical pump-ing, recoil depopulates the velocity classes that interact with

the laser light, and optical saturation leads to stimulated emis-sion, causing photons to be emitted along the propagation ofthe laser rather than back to the telescope.

Pulsed-laser LGS systems, in the general case in which thepulses are not very short or very long, are more challenging tomodel, due to the dynamical processes occurring on differenttime scales. Experiments and modeling efforts have been re-ported in [27–29], among others. However, none of the theo-retical studies to date have included all of the effectsdiscussed above for the cw case, which are all relevant topulse lengths on a 100 μs time scale. In addition, a full char-acterization of a particular pulsed laser depends on its pulselength and pulse repetition rate and how these parameterscompare to collision and relaxation rates and evolution timescales in the system, as well as the multimode and coherencecharacteristics of the laser.

In this paper, we report the results of physical modeling ofLGS with pulsed lasers, including all of the above-mentionedeffects. We focus on the TIPC pulsed laser and compute theexpected photon return at the TMT site as a function of pulselength, pulse repetition rate, angular distance between the la-ser propagation, and the direction of the geomagnetic field. Asthe first field test of the pulsed laser is planned for mid-2012at the location of the Large Zenith Telescope (LZT) operatedby the University of British Columbia in Canada, we also per-form simulations under the conditions at the LZT site. In ad-dition, the photon return of the pulsed laser is compared tothat obtained with a cw laser of similar transmitted power.Briefly, we find that both the cw and pulsed formats appearto be able to meet the design requirements for photon return.The cw and pulsed formats provide comparable photon re-turn, with the differences at the level of the uncertainty inthe model.

Results obtained from the numerical model also allow us toinvestigate various physical mechanisms occurring in the sys-tem that affect the photon return, often by aiding or inhibitingthe generation of ground-state atomic polarization. These me-chanisms can lead to interaction between seemingly unrelatedparameters of the system; for example, we find that the opti-mal laser line width depends on the angle between the lightpropagation direction and the geomagnetic field direction, ap-parently due to the effect that each of these parameters has onthe creation of atomic polarization.

The paper is organized as follows. In Section 2 the systembeing modeled and various physical mechanisms that can in-fluence the photon return are discussed. In Section 3 the de-tails of the Bloch-equation model and the numerical solverare given. Section 4 presents the environmental data andthe geometrical optics results that are used as inputs to themodel. In Section 5 the method of integration over the spatialextent of the beam is described. Section 6 gives the results,and conclusions are given in Section 7.

2. DESCRIPTION OF THE SYSTEMWhen sodium atoms are subject to a strong circularly polar-ized light field resonant with the D2 transition, a number ofphysical processes occur that affect the efficiency at whichlight is spontaneously re-emitted back toward the light source.

Left-circularly polarized light resonant with the D2a F �2 → F 0 transition group induces optical pumping that tendsto transfer the atoms to the jF � 2;m � 2i ground state

Rochester et al. Vol. 29, No. 8 / August 2012 / J. Opt. Soc. Am. B 2177

(Fig. 1). This atomic ground-state polarization is beneficial forphoton return in three ways:

1. The cycling jF � 2;m � 2i → jF 0 � 3;m � 3i transi-tion is the strongest transition in the D2 transition group, in-creasing the effective absorption cross section.

2. Atoms excited on the cycling transition cannot spon-taneously decay to the F � 1 ground state, where they wouldno longer interact with the light.

3. Fluorescence from jF 0 � 3;m � 3i is preferentially di-rected along the light beam, leading to an enhancement ofphoton flux observed at the location of the light source.

Several mechanisms work to counteract the production ofatomic polarization—the relative importance of these is re-lated to the characteristic time scale of each.

The geomagnetic field induces Larmor precession, whichcan act to wash out the atomic polarization. Because circu-larly polarized light produces atomic polarization parallel tothe light propagation direction, precession will be inducedif the magnetic field has a component transverse to the wavevector k (Fig. 2). A magnetic field of 0.33 G, present at theMauna Kea site, corresponds to a Larmor precession fre-quency of ∼220 kHz or a precession time of 4.3 μs, meaningthat this effect is important during the 120 μs pulse producedby the TIPC laser. The detrimental effect of the magnetic fieldis strongly dependent on the direction of light propagationrelative to the magnetic-field direction.

The atomic ground-state polarization is also relaxed byspin-randomization collisions, predominantly with O2 mole-cules [30]. The time scale for these collisions varies withthe O2 density, but can be estimated to be in the 200–5000 μs range [26]. This mechanism can thus weakly affectthe dynamics during a 120 μs pulse. Finally, transit of theNa atoms through the light beam (due to the motion of theatoms or the motion of the beam itself) can also be consideredto be a relaxation mechanism. The net effective atomic velo-city can be estimated to be less than 370 m ∕ s [26], which, for atypical beam size of 20 cm, corresponds to a millisecondtime scale for transit relaxation. This effect can thus generallybe neglected for the pulse lengths in question, although weinclude it in the numerical model for completeness.

The effective loss of atoms by spontaneous decay to theF � 1 ground state can be directly combatted by applying “re-pump” light resonant with the D2b F � 1 → F 0 transitiongroup to transfer the atoms back to the F � 2 ground state[13,26,30]. As the simulation results show, this technique isvital to obtain sufficient photon return fluxes. Alleviation ofthe loss of atoms to the F � 1 state also has the indirect effectof reducing the effect of the magnetic field on the returnflux, as this is one of the worst effects resulting from the de-struction of atomic polarization caused by Larmor precession.

Another process that tends to reduce the photon return fluxfor strong light fields is recoil of the atoms induced by inter-action with the light. Each absorption/spontaneous emissioncycle gives the atom a small momentum kick in the laser beampropagation direction; the cumulative effect of these kicks isto create holes in the Doppler distribution, depleting the atom-ic-velocity groups that are the most strongly resonant with thelight (this is illustrated in Subsection 6.C). Velocity-changingcollisions of the Na atoms with N2 and O2 molecules act toreequilibrate the atomic-velocity distribution, filling in theholes created by recoil. These collisions occur on a time scaleof 15–300 μs [26], depending on gas density, and so can actduring a single laser pulse.

3. DESCRIPTION OF THE MODELA. The Bloch EquationsIn order to calculate the observed fluorescence, the evolutionof the atoms is modeled using the optical Bloch equations forthe atomic density matrix. The density matrix describes themagnitudes of and correlations between basis wave func-tions in the ensemble of Na atoms, in particular, the spin wave

Fig. 1. (Color online) Level diagram of the Na D2 line, showing transitions induced by circularly polarized light resonant with the D2a transition.The widths of the arrows indicate the relative transition strengths.

Fig. 2. (Color online) Precession of the atomic polarization P, origin-ally produced along the light wave vector k, about the magnetic-fieldvector B. If the polarization is continuously generated and has alifetime longer than the Larmor precession time, it will tend to be aver-aged about the magnetic-field direction.


functions of the states involved in the Na D2 transition. In or-der to account for atoms with different Doppler shifts, thedensity matrix is considered to be a function of atomic velo-city along the laser beam propagation direction. (An addi-tional degree of freedom may be included to account forthe multimode structure of the laser line as discussed below.)The calculation is semiclassical in the sense that while theatoms are treated quantum mechanically, the light fieldsare treated classically; thus, the effect of spontaneous decaymust be included phenomenologically. Because the densitymatrix describes all populations of, and coherences between,the 24 Zeeman sublevels making up the ground and excitedstates, the calculation describes, in principle, all saturationand mixing effects for essentially arbitrarily large opticaland magnetic fields. (In practice, certain coherences in thesystem are negligible under normal experimental conditionsand can be neglected in order to increase the computationalefficiency.)

In order to perform numerical calculations, the velocity de-pendence of the density matrix is discretized to describe anappropriate number nv:g: of velocity groups, each with a fixedlongitudinal velocity. Because coherences between atomswith different velocities can be neglected, the complete den-sity matrix ρ can be thought of as a collection of nv:g: separatebut coupled density matrices, each of dimension 24 × 24.

The evolution of the density matrix is given by a general-ization of the Schrödinger equation:

ddt

ρ � 1iℏ

�H; ρ� �Λ�ρ� � β: (1)

Here the atomic level structure and the interaction withexternal fields are described by the total HamiltonianH � H0 �HE �HB, with H0 the Hamiltonian for the unper-turbed energy structure of the atom, HE � −d · E the Hamil-tonian for the interaction of the electric dipole d of the atomwith the electric field E of the light, HB � −μ · B the Hamilto-nian for the interaction of the magnetic moment μ of the atomwith the local magnetic field B, and the square brackets denot-ing the commutator. The termΛ in Eq. (1) represents phenom-enological terms added to account for relaxation processesnot described by the Hamiltonian. In our case these relaxationprocesses include spontaneous decay (omitted from the Ha-miltonian due to the semiclassical approximation), collisionalspin relaxation (“S-damping”) proportional to S2ρ − S · �ρS�[31], where S is the electronic spin-angular-momentum opera-tor, and the exit of atoms from the light beam due to motion ofthe atoms and the beam (transit relaxation). In addition, thereare terms included inΛ to describe changes in atomic velocitydue to collisions and light-induced recoil, as well as an effec-tive relaxation rate for optical coherences that simulates a la-ser spectrum with nonnegligible bandwidth. These terms aredescribed in more detail below. Each relaxation processdescribed by Λ includes a corresponding “repopulation” pro-cess, so that the trace over the density matrix for all velocitygroups is conserved, corresponding to conservation of the to-tal number of atoms. The repopulation process describing theentrance of atoms into the beam is independent of ρ and so iswritten as a separate term β.

Velocity-changing collisions are treated as hard collisions inwhich the velocity of the colliding atom is rethermalized in a

Maxwellian distribution (no speed memory). The internalstate of the atom is assumed to be unchanged.

Light-induced recoil is described phenomenologically bycausing a fraction vr ∕Δvv:g: of the excited-state atoms in eachvelocity group to be transferred upon decay into the next high-er velocity group. Here vr is the recoil velocity andΔvv:g: is thewidth of the particular velocity group. This model relies on thefact that vr � 2.9461 cm ∕ s (equivalent to a Doppler shift of50 kHz) is much smaller than the typical value of Δvv:g:.

Equation (1) supplies a linear system of differential equa-tions for the density-matrix elements, known as the opticalBloch equations. Thinking of ρ as a column vector of nv:g: ×242 density-matrix elements, the Bloch equations can be writ-ten as _ρ � Aρ� b, where A and b are a matrix and a vector,respectively, that are independent of ρ. The vector b corre-sponds to β and A to the rest of the right-hand side of Eq. (1).

The laser light field has a frequency component tuned nearthe D2a transition group, and may have an additional “re-pump” component tuned near the D2b transition group. Thusthe matrix A has components that oscillate at each of thesefrequencies. Under the rotating-wave approximation, theoverall optical frequency is removed from A. However, thebeat frequency between the two light-field components re-mains. This beat frequency can also be removed from theBloch equations in our case: each frequency component inter-acts strongly with one transition group and very weakly withthe other, so the weak coupling can be neglected for eachtransition. If, in addition, the small magnetic-field-inducedmixing between the two hyperfine ground states is neglected,the beat frequency can be entirely removed from the evolutionequations.

The fluorescent photon flux per solid angle emitted in agiven direction can be found from the solution for ρ as theexpectation value of a fluorescence operator [32].

B. Methods for Simulation of Multiple Laser ModesThe spectrum of the TIPC laser is composed of three equallyspaced modes. The proper method for including the effect ofthe modes in the model depends on the details of the time de-pendence of the spectrum, including the coherence timebetween the modes and the pulse-to-pulse fluctuations ofthe mode amplitudes. Because these details were not knownwith certainty, three different methods were compared inthe study.

In the “coherent-modes” method, the three modes are as-sumed to be perfectly coherent and nonfluctuating. The atomsare subject to a field with components oscillating at each ofthe three mode frequencies. This is equivalent to a single field,with a frequency equal to that of the central mode, amplitudemodulated with a modulation frequency equal to the mode se-paration. Using this model, the resulting return-flux signal ex-hibits beats at the modulation frequency. The width of eachlaser mode can be accounted for by including additional re-laxation terms for the atomic optical coherences. We note thateven when the effective line widths are broadened so that theyoverlap, the observed beats remain, because the modes arestill perfectly coherent. Direct integration of the evolutionequations in this situation is computationally slow, becauseof the rapid oscillations in the solution. Therefore, we haveimplemented a Floquet technique in which the explicit appear-ance of the rapid oscillation is removed from the evolution


equations through expansion of the density matrix in a Fourierseries. This results in a coupled system of equations for therelatively slowly evolving Fourier coefficients. These equa-tions can be efficiently integrated to find the time dependenceof the lowest few Fourier harmonics.

In the “mode-hopping” method, the laser frequency is as-sumed to fluctuate between the three mode frequencies ona time scale much shorter than the pulse length. This is accom-plished by writing density matrices describing “regions,” eachsubject to one of the laser modes, and allowing atomic transitbetween the regions. The transit rate corresponds to the widthof each laser mode. In this model, the optical coherences areassumed to be destroyed upon transit between regions, mean-ing that the laser modes can be considered to be effectivelyincoherent. Thus no beats are observed in the return flux.

In the “pulse-averaging” method, large pulse-to-pulse fluc-tuations in the relative mode amplitudes are assumed, so thatduring each pulse the laser power is present on only one of thethree modes, with equal probability. This is accomplished byrunning the simulation three times, once with the light fre-quency equal to each mode frequency, and averaging the re-sults. The modes are effectively incoherent with this method.

C. The Numerical SolverThe density-matrix evolution equations are generated usingthe LGSBloch package for Mathematica, which is basedon the Atomic Density-Matrix package written by Rochester[33]. The system of ordinary differential equations is solvedusing a code written in C and based on the open-sourceODE solver CVODE, from the SUNDIALS package [34]. Thissolver addresses the stiffness of the system using the implicitbackward differentiation formulas (BDFs) [35]. These meth-ods require the solution of a linear system of equations at eachstep of the solver; these solutions are accomplished with apreconditioned Krylov method. (In such a method an initialguess is improved by minimizing the residual over a subspacewith dimension much smaller than that of the full system [35].)The rate of convergence of the method is increased by premul-tiplication with a block-diagonal preconditioner (approximateinverse of A), obtained by setting all terms that connect den-sity-matrix elements from different velocity groups to zero,and then inverting the block for each velocity group. Forthe mode-hopping multimode method, the transit terms thatcouple the density matrices for the three modes are also in-cluded in the preconditioner, using an operator splitting tech-nique that inverts these terms separately. For the coherent-modes Floquet technique, the terms for all of the includedcoupled Fourier coefficients are included in the block for eachvelocity group.

4. INPUT DATAA. Standard Parameters and Environmental ConditionsThe standard parameters used in the model for the TIPC laserare given in Table 1. Environmental parameters and para-meters specific to the Mauna Kea and LZT sites are listedin Table 2. The sodium density as a function of altitude is as-sumed to be described by a Gaussian function centered at93 km with a full width at half maximum (FWHM) of 8 km.This is a rough estimate, as the measured density profile atany given time may deviate significantly from a Gaussian.However, our modeling indicates that the photon return

has only a weak dependence on the width of the density pro-file: increasing the assumed width to 11 km leads to a reduc-tion in photon return of about 1%. For each site, thetemperature and gas densities as a function of altitude inthe sodium layer were obtained from the MSIS-E-90 Atmo-sphere Model [36]; for the Mauna Kea site, these values, alongwith the assumed Na density profile, are plotted in Fig. 3. Col-lision rates and the Doppler width as a function of altitudeare derived from these data and the assumed collision crosssections are given in Table 3.

B. Laser Spot Image ModelTo find the expected laser intensity profile, a geometrical op-tics simulation was performed. The LLT has a diameter of0.4 m and is nominally focused (absent diffraction and turbu-lence effects) at a range of 125 km. The Gaussian laser beamhas a waist diameter (e−2 of the peak intensity) of 0.24 m at theLLT pupil.

We computed the expected laser intensity profile at rangesof 85, 95, 105, 115, and 125 km. This was accomplished bymodeling the propagation of the 589 nm laser light fromthe launch telescope for several instantiations of a singleturbulence screen assumed to be located close to the LLT.The turbulence screen wasmodeled with a von Karman powerspectral density, an outer scale of turbulence [37] set to 30 m,and for various turbulence strengths characterized by theFried lengths r0 � 0.10, 0.15, 0.20, 0.25, and 0.30 m. The TIPCgeometrical beam quality is characterized by a broadeningfactor of M2 � 1.2. This M2 factor is achieved in the lightpropagation model by introducing a slight coma aberration

Table 1. Standard Parameters for the TIPCPulsed Laser

Average laser power 20 WAverage projected laser power

Mauna KeaLZT

12 W18.6 W

Polarization circularNumber of modes 3Mode spacing 150 MHzPulse length 120 μsRepetition rate 800 Hz

Table 2. Standard Parameters for Mauna Kea andLZT sitesa

Mauna Kea LZT

Latitude 19.8° 49.3°Longitude 155.5° 122.6°Observatory altitude 4050 m 395 mB-field 0.334 G 0.525 GNa Larmor frequency 234 kHz 367 kHzB-field zenith angle 126°18’ 160°30’B-field azimuthal angle 9°45’ 16°57’Atmospheric transmission (zenith) 0.84 0.8Fried length (zenith) 21 cm 5 cmNa column density 4. × 1013 m−2

Na centroid altitude 93 kmNa layer thickness 8 kmaThe Fried length is the size of the circular aperture within which the

variance of the wavefront phase aberrations is less than 1 radian [24].


in the LLT mirror and a total wavefront error of 76.5 nm rms.In addition, because the LLT mirror has a fixed focal distanceof f L � 125 km, the laser intensity beam profiles at altitudeslower than 125 km also include the effect of a focus error mod-eled as r2Δh ∕ �2f 2L�, where r is the laser beam radius andΔh isthe distance from the LGS spot to the 125 km focal distance.

The beam profiles generated through the process abovewere interpolated as needed for intermediate values of r0used for simulations at the Mauna Kea site. At the LZT site,the atmospheric turbulence is stronger, with the median tur-bulence conditions better characterized by an r0 of 0.05 m. Inthis case, the laser beam profile was extrapolated linearlydown to r0 � 0.05 m from the profiles available for largervalues of r0.

The LGS spot images are sampled at 0.121” and are 64 pix-els across. An example of the beam profile is given in Fig. 4(a).Fitting the narrower dimension of the beam profiles with aGaussian function yields angular sizes ranging from 0.68” to0.37” FWHM for r0 � 0.05 to 0.3 m.

5. SPATIAL INTEGRATIONIntegration of the observed return flux over the spatial extent(profile and path length) of the laser beam is accomplished infour steps, using a method similar to that of [38]:

1. Construct table of return flux versus intensity and al-titude.

2. For each pixel in the modeled beam profile, propagatelight through sodium layer using the table, keeping track ofabsorption.

3. Integrate return flux over path length.4. Sum over pixels.

We first generate a table of return fluxes as a function oflight intensity and altitude. Parameters that change with alti-tude, plotted in Fig. 3, are N2 and O2 density (affects collisionrates), Na density (multiplicative factor), and temperature(weakly affects collision rates and Doppler width). Sodiumdensity is modeled by a Gaussian with integrated columndensity 4 × 1013 m−2. N2 and O2 density and temperatureare obtained from the MSIS-E-90 Atmosphere Model [36].

Both the return flux Φ�I; z� and total fluorescence into 4π,Φtot�I; z�, are obtained from the model. Return values fromthe model are multiplied by the Na density and area of a0.12” spot (the size of a pixel in the beam profile) at a givenaltitude. This results in values of fluxes per unit path length dℓ.

Light power in the uplink beam is absorbed as it travelsthrough the sodium layer, and reradiated as fluorescence inall directions. For a sub-beam with angular size equal toone pixel in the beam profile, the power absorbed perunit path length dℓ is ℏωdΦtot�I; z� ∕ dℓ, with I � P�ℓ� ∕ �θpℓ�2and z � ℓ ∕X , where P�ℓ� is the power at distance ℓ, θp is

80 90 100 110180

190

200

210

0

2000

4000

1012

1013

1014

N2

O2

Altitude km

Tem

pera

ture

°KN

ade

nsity

cm3

Gas

dens

itycm

3

Fig. 3. (Color online) Temperature and O2 and N2 gas densities as afunction of altitude obtained from the MSIS-E-90 Atmosphere Modelfor the Mauna Kea site, along with the assumed Na density profile.(Note that deviations of the actual Na density profile from a Gaussianare often quite large.)

Table 3. Estimated Values for CrossSectionsa

σNa–N27.15 × 10−15 cm2

σNa–O27 × 10−15 cm2

σSNa–O22.5 × 10−15 cm2

aSee [26] and references therein; note that the factor of1 ∕ 2 described in [26] is incorporated into σSNa–O2

.

Angular separation arcsec

Ang

ular

sepa

ratio

nar

csec

a

b

1

0.5

0

0.5

1

0

0.4

1

0.5

0

0.5

1

0

60.

Fig. 4. (Color online) (a) Interpolated beam profile from geometricaloptics model, given in terms of the fraction of power in each pixel. Thepeak value is the Strehl ratio. (b) Return flux from each pixel in103 photons ∕ s ∕m2.


the angular size of one pixel, X � sec ζ is the airmass (ζ is thezenith angle) [39], and ℏω is the energy of one photon. Thusthe power along the optical path is given by the propagationequation

dP�ℓ� ∕ dℓ � −ℏωdΦtot�P�ℓ� ∕ �θpℓ�2; ℓ ∕X� ∕ dℓ: (2)

We integrate this equation for a range of initial light powers,chosen to cover the range obtained using the given beamprofile and the average laser power.

For the transmission of the return flux through the sodiumlayer, we assume linear absorption and so use the curve ob-tained from Eq. (2) in the low power limit, denoted by T lin�ℓ�.The observed return flux into a detector of unit area per unitpath length emitted from a pixel-sized sub-beam at a distance ℓis then given by

dΦobs�P0; ℓ� ∕ dℓ �TXa

ℓ2T lin�ℓ�dΦ�P�ℓ� ∕ �θpℓ�2; ℓ ∕X� ∕ dℓ; (3)

where P�ℓ� depends on the initial power in the sub-beam, P0,and Ta is the atmospheric transmission at zenith.

Integrating dΦobs�P0; ℓ� ∕ dℓ over the path length, we find theobserved flux from each sub-beam as a function of the initialpower, Φobs�P0�. We then sum the return flux from all of thepixels in the beam profile. An example of the return flux pro-file is shown in Fig. 4(b), the intermediate steps in the integra-tion procedure are illustrated in Subsection 6.D.

6. RESULTSIn Subsections 6.A–6.C, we illustrate various aspects of thesignal using the return flux calculated from a particular loca-tion in the sodium layer, i.e., not integrated over the spatialextent of the beam. Subsection 6.D illustrates the steps takenin the spatial integration procedure, Subsection 6.E presentsthe integrated return-flux results, and Subsection 6.F presentsplots that relate to optimization of the pulse length, repumppower fraction, and laser line width.

A. Time DependenceExamples of the time dependence of photon flux return dur-ing a 120 μs square pulse using the three different methods fortreating the multimode laser spectrum (coherent modes,mode-hopping, and pulse averaging) are shown in Fig. 5. InFig. 5(a), no repump light is applied. Following the initial rapidrise in flux at the beginning of the pulse, the flux begins to falloff. This is primarily due to pumping to the F � 1 groundstate, as can be seen by diverting some of the laser power intothe repump light beam [Fig. 5(b)], which counteracts most ofthe falloff. As throughout, unless otherwise noted, we take10% of the total laser power to be repump light. We also as-sume that the repump light is generated by amplitude modula-tion of the D2 laser field, meaning that an additional 10% of thelaser power is lost into the additional low-frequency sidebandthat is produced. Unless otherwise noted, the repump lightis taken to be of the same polarization as the main light field.The effect of the repump light is examined in more detail inSubsection 6.C.

The rapid beats present when using the coherent-modetreatment can be seen more clearly in Fig. 6, which showsthe initial part of the pulse.

B. Pulse ShapeThe results of Fig. 5 are calculated assuming a perfectlysquare pulse shape. A measurement of the actual pulse shapeproduced by the TIPC laser is shown in Fig. 7. This shape dif-fers from a square pulse in that there are finite rise and falltimes, and also short spikes in the initial part of the pulse.To investigate the effect of these differences, we defined anapproximate analytical model for the measured pulse shape.The model represents the initial spikes as rapid exponentiallydecaying oscillations. While not a completely faithful repro-duction of the measured pulse shape, the accuracy of this re-presentation should be enough to indicate whether the initialspikes have the potential to cause a significant deviation fromthe results obtained using an ideal square pulse.

To model the measured pulse shape, we used a functionalform with the pulse rise given by

0 50 1000

25

50

75

100

w repump

0

25

50

75

100

w o repump

Coherent modesMode hoppingPulse averaging

Flux

103

phot

ons

ssr

atom

Time s

b

a

Fig. 5. (Color online) Instantaneous return flux during a singlepulse as a function of elapsed time, using three different multimodetreatments, both with no repump light and with 10% repump. Ex-ample plots are for Mauna Kea, zenith angle zenith angle � 30°,azimuth angle � 190°, Iavg � 27.5 W ∕m2. The oscillations present inthe coherent-mode treatment are too rapid to be distinguished on thistime scale, and so appear as solid bands.

0 0.1 0.2 0.3 0.40

20

40

60

Time s

Flux

103

phot

ons

ssr

atom

Fig. 6. (Color online) An expanded view of the initial part of the co-herent-modes plot from Fig. 5(b). The oscillations are due to interfer-ence between the laser modes.


1 − e−�t−t0� ∕ τr � aosce−�t−t0� ∕ τosc sin2�ωt ∕ 2�; (4)

where t0 is the pulse start time, τr is the rise time, aosc is theamplitude of the initial oscillations, τosc is the decay time ofthe oscillations, and ω is the oscillation frequency. The pulsefall is somewhat faster than exponential, and so is modeled by

exp�−��t − t1� ∕ τf �p�; (5)

where t1 is the time at the beginning of the fall, τf is the nom-inal fall time, and p ≈ 1.5.

Appropriate parameters were chosen by eye to match thesupplied measured plot, as shown in Fig. 7. Also shown in thefigure is a pulse shape obtained by averaging out the initialoscillations of the formula (4).

The calculated responses to a square pulse, a pulse withinitial oscillations, and a pulse with the initial oscillationsaveraged out are shown in Figs. 8(a), 8(b), and 8(c), respec-tively. The pulses are each 120 μs from the beginning of thepulse to the start of the fall, with the other pulse-shape para-meters chosen to match those given in Fig. 7. The three typesof pulses are each normalized to have the same total pulseenergy. A single-mode laser is assumed for these plots.

The time-averaged flux from the more realistic pulse ishigher than that from the square pulse, apparently becausethe pulse energy is spread out over a longer effective timedue to the pulse fall time. Averaging out the initial oscillationshas almost no effect on the time-averaged flux.

These results indicate that the deviations of the measuredpulse shape from a square pulse have an insignificant effecton the photon return, given the current uncertainty in the

environmental and laser parameters. We therefore will usean ideal square pulse in the following simulations.

C. Atomic-Velocity DependenceThe return flux as a function of the atomic-velocity group nearthe beginning of the laser pulse is shown in Fig. 9. The responseto the three laser modes is seen, with the shape of each peakdetermined by the upper-state hyperfine structure of the D2 line.

0 100 2000

0.25

0.5

0.75

10

0.25

0.5

0.75

1

Time s

Puls

ehe

ight

arb.

units

Fig. 7. (Color online) Laser pulse shapes: (top) the suppliedmeasured pulse shape; (bottom) the pulse-shape model describedin the text (solid line) and the same model with the initial oscillationsaveraged out (dashed line). The parameters used here are t0 � 22 μs,τr � 19 μs, aosc � 1.9, τosc � 12 μs, ω � 2π × 0.6 MHz, t1 � 202 μs,τf � 9.6 μs, and p � 1.5. The overall amplitude is scaled to matchthe measurement. The nominal pulse length is t1 − t0 � 180 μs, longerthan the standard pulse length of 120 μs used in the simulations.

0 50 100 1500

50

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150

200c

0

50

100

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200b

0

50

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200a

Return fluxPulse shape

Flux

106

phot

ons

sm

2

Time s

Avg. flux 6.94 106 ph. s m2



Fig. 8. (Color online) Return flux in response to three different pulseshapes: (a) ideal square pulse, (b) pulse with initial oscillations andfinite rise and fall times, and (c) pulse as in (b) but with initial oscilla-tions averaged out. A single-mode laser is assumed. The nominal pulselengths are 120 μs; the other pulse-shape parameters are as given inthe caption to Fig. 7.

500 0 5000

100

200

300

Doppler shift MHz

Flux

phot

ons

MH

zs

srat

om t 0.23 s

Fig. 9. (Color online) An example of return flux as a function of thelongitudinal atomic velocity (given in terms of the corresponding Dop-pler shift), at a time near the beginning of a laser pulse.


To study the effect of various mechanisms on the dynamicsof the atomic system, it is helpful to plot atomic populations asa function of velocity group at various times during the pulse.Figure 10 shows populations of the F � 1 ground state(bottom row), F � 2 ground state (middle row), and total

ground-state and upper-state populations (top row), as a func-tion of longitudinal atomic velocity using the mode-hoppingtreatment of the laser modes. The left-hand column showsthe populations near the beginning of the pulse. A smallamount of the total population has been transferred fromthe ground state to the upper state, and some atoms have beentransferred from the F � 2 ground state to the F � 1 groundstate by optical pumping. The middle column shows thepopulations at the end of the pulse if no repump light is used.In this case, the majority of the F � 2 atoms have beenpumped into the F � 1 ground state, where they do not inter-act with the laser light. As a result, the number of atoms in theexcited state (and hence the photon return flux) is greatlyreduced. The hole-burning effect of atomic recoil can alsobe seen in the total ground-state population. The right-handcolumn shows what happens if repump light is used. In thiscase, the population of the F � 2 state is largely preserved,as atoms pumped to the F � 1 state are repumped to the F �2 state. When repump light is applied, the effect of recoilbecomes much more dramatic, as each atom is able to interactwith the light many more times during the pulse.

500 0 5000

1

2

500 0 500 500 0 500

0

1

2

0

1

2Total ground state pop.

Total excited state pop.

Doppler shift MHz

Popu

latio

nar

b.un

its

t 0.3 s t 120.0 s w o repump t 120.0 s w repump

F 1

F 2

Fig. 10. (Color online) Populations of the F � 1 ground state (bottom row), F � 2 ground state (middle row), and total ground-state and upper-state populations (top row), as a function of longitudinal atomic velocity for the mode-hopping method. Populations are shown for times near thebeginning of the pulse (left column), at the end of a pulse without repump light (middle column), and at the end of a pulse with repump light (rightcolumn), superimposed over the unperturbed equilibrium populations (dashed lines).

5

5

10

20

30

40

45

50

0.01 1 10085

90

95

100

0.5

0.5

1

23

4

4.5 5

85

90

95

100

Alti

tude

km

Light intensity W m2

b

a

Fig. 11. (Color online) (a) Return flux dΦ�I; z� ∕ dℓ ∕ I in units of109 photons ∕ s ∕ sr ∕m ∕ �W ∕m2� as a function of light intensity and altitude.(b) Total flux dΦtot�I; z� ∕ dℓ ∕ I in units of 109 photons ∕ s ∕m ∕ �W ∕m2�,used to calculate optical absorption in the sodium layer.

Fig. 12. (Color online) Transmission through the sodium layer as afunction of path length and initial light power in a pixel-sized beam.


D. Spatial IntegrationHere we illustrate the steps taken to perform the integration ofthe photon return over the spatial extent of the beam. First,we run the model a number of times assuming a 0.12” pixel-sized spot to construct a table of the return flux dΦ�I; z� ∕ dℓ asa function of altitude and intensity [Fig. 11(a)]. At the sametime, we also construct a table of the total fluorescence[Fig. 11(b)], which is used to calculate the resonant absorptionof light in the sodium layer (Fig. 12).

Accounting for resonant absorption on the uplink anddownlink, we can then construct a table of observed fluores-cence at the detector per unit path length in the sodium layer,dΦobs�P0; ℓ� ∕ dℓ, as a function of distance and initial power ineach pixel (Fig. 13). Integrating over distance, we find the to-tal return per pixel as a function of initial power, Φobs�P0�(Fig. 14). As seen in Fig. 14, the efficiency of return is constantfor low power (the linear regime), begins to rise as power isincreased—due to the beneficial effects of optical pumping—and then falls off due to saturation effects, pumping to theF � 1 state, and recoil.

We then take the beam profile obtained from the geometri-cal optics simulation with the appropriate adjusted Fried para-meter [Fig. 4(a)], and apply Φobs�P0� to each pixel to obtainthe return-flux spot profile [Fig. 4(b)]. Because of the satura-tion effect, the central peak of the return-flux profile is com-pressed relative to that of the beam profile, increasingthe relative prominence of the dimmer features due to the as-tigmatism seen at the top of the profile.

Summing over the pixels of the photon return-flux profilegives the final calculated value for the total return flux.

E. Beam-Integrated Return Flux1. Return Flux as a Function of Zenith AngleThe total return flux integrated over the spatial extent of thebeam is shown in Fig. 15 as a function of zenith angle forpulsed light, using each of the three multimode methods,and also for narrow-band cw light. Positive zenith angles

10

10

20

30

40

50

60

70

80

10 4 0.01 1

100

105

110

115

Initial power in pixel W

Dis

tanc

efr

omde

tect

orkm

Fig. 13. (Color online) Normalized observed return flux per dℓfrom a pixel-sized sub-beam, dΦobs�P0; ℓ� ∕ dℓ ∕P0, in units ofphotons ∕ s ∕m2 ∕m ∕W.

10 5 10 4 10 3 0.01 0.1

200

400

600

800

Inital power W

Nor

mal

ized

flux

103

ph.

sm

2W

Fig. 14. (Color online) Total observed normalized return fluxΦobs�P0� ∕P0 from a pixel-sized beam as a function of initial launchedlight power in that pixel.

60 30 0 30 600

2.5

5

7.5

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1500

Zenith angle ° , azimuth 190°

Ret

urn

flux

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phot

ons

sm

2

Mauna Kea

PDE

1.25

ms,

0.52

m2,0

.48

DE

cwCoherent modesMode hoppingPulse averaging

no repump

10repump

00

5

10

15

Zenith angle °

Ret

urn

flux

106

phot

ons

sm

2

LZT

nore

pum

p10

repu

mp

Fig. 15. (Color online) Beam-integrated return flux for Mauna Kea (as a function of zenith angle) and LZT (at zenith), for pulsed light using each ofthe three multimode methods and for narrow-band cw light. Results are given for both no repump light and for 10% of the light power in the repumpbeam. The vertical axis on the left side is in units of 106 photons ∕ s ∕m2, while the axis on the right side of the plot for Mauna Kea shows photo-detected electrons (PDE) assuming a collection time of 1.25 ms, collection area of 0.5 m2, and 48% detection efficiency. The dashed gray line showsthe design requirement for 900 PDEs at zenith, combined with the nominal dependence on the inverse of the airmass X expected from purelygeometrical considerations. (The optical path length in the sodium layer increases with X , while the detector collection solid angle falls as X−2.) Thehigher return at the LZT site is primarily due to the higher throughput of the laser-launch system and the different direction of the local geomagneticfield compared to that at the Mauna Kea site.


correspond to 190° azimuth, while negative zenith angles cor-respond to 10° azimuth. Results are shown both for no repumplight and for the standard repump fraction (i.e., 10% of the lightpower in the repump beam, 80% of the laser power in the D2alight, and 10% lost).

The three methods for modeling the laser modes give re-sults that vary by ∼15%; this may be taken as an indicatorof the level of uncertainty in the model due to incompleteknowledge of the characteristics of the laser. (By running cal-culations with finer grid spacings and stricter tolerances, wehave found that uncertainties due to numerical approxima-tions in the model are below 5%.) At this level of uncertainty,the narrow-band cw light provides approximately the samephoton return as the pulsed format. These results indicate thatboth pulsed and cw light have the potential to satisfy the TMTdesign requirements as long as repump light is used, and willlikely not satisfy the requirements if repump is not used.

2. Reversed-Polarization RepumpFor technical reasons, it may be easier to produce repumplight with opposite polarization to that of the main D2a light.Figure 16 shows the effect of reversing the polarization of therepump light. Results are shown for a pulsed laser with norepump, with 10% repump, and with 10% repump with oppo-site circular polarization to the D2a light, using the pulse-averaging multimode treatment. Using opposite circularpolarization for repumping incurs a 5–10% penalty, dependingon zenith angle, relative to repumping with the same polariza-tion. Because this is a comparison of the same laser formatand multimode model with itself, an observed difference atthis level may be numerically significant.

3. Dependence on Spot Size for LZT siteTo find the dependence of return flux on spot size for the LZTsite, the beam profile was scaled proportionally from the nom-inal size 0.68” FWHM obtained from the geometrical opticsmodel using the standard Fried parameter r0 � 0.05 m. Whenvarying the beam size from 1 ∕ 2 to three times the standardbeam size, the return flux varies from about 25% below to25% above the value obtained with the standard size (Fig. 17).

F. OptimizationIn this subsection, we plot the integrated return flux for theMauna Kea site as a function of zenith angle and various otherparameters for use in optimization. Results for pulsed and

cw lasers are given; for the pulsed format, we use the pulse-averaging multimode treatment.

Figure 18 shows a contour plot of return flux in units of106 photon ∕ s ∕m2 as a function of zenith angle and pulselength, holding the repetition rate and the average laser powerconstant. The standard amount of repump light (of the samepolarization as the D2a light) is used. The plot runs over therange of pulse lengths that may be accessible to the TIPC la-ser. In this range, the plot shows that longer pulse lengths aregenerally better.

Figure 19 shows a contour plot of return flux as a functionof zenith angle and repump power percentage for the pulsedand cw cases. For Mauna Kea, it appears that 10% repump isoptimal for both pulsed and cw light.

Figure 20 shows a contour plot of return flux as a functionof zenith angle and the width of each laser mode for the pulsedand cw cases. The standard amount of repump light is used.The optimal width depends on the zenith angle, or more pre-cisely, the angle between the light propagation direction andthe magnetic field. When this angle is small, a broader linewidth is advantageous, apparently because the reduced effectof Larmor precession means that less light intensity per velo-city group is required to produce beneficial atomic polariza-tion. When the angle is larger, narrower laser lines areoptimal. For large angles, the optimum line width is reason-ably consistent with the value of ∼20 MHz obtained fromthe formula given in [26] for the cw case. From these results,

60 30 0 30 600

2.5

5

7.5

10

0

500

1000

1500


Ret

urn

flux

106

phot

ons

sm

2

P DE

1.25

ms,

0.52

m2,0

.48

DE

10 repump10 rev. repumpNo repump

Fig. 16. (Color online) Beam-averaged return flux and PDE for apulsed laser without repump, with 10% repump, and with 10% repumpwith opposite circular polarization as the D2a light.

0.5 1 1.5 20

5

10

15

Spot size arcsec

Ret

urn

flux

106

phot

ons

sm

2

Fig. 17. (Color online) Return flux as a function of spot size (FWHM)at the LZT site, using the pulse-averaging method. The dashed grayline shows the spot size obtained from the geometrical optics modelusing the standard Fried parameter r0 � 0.05 m. The beam profilewas directly scaled proportionally smaller and larger to obtain theresults for this plot.

34

55 66

7

60 30 0 30 6050

100

150

200


Puls

ele

ngth

s

Fig. 18. (Color online) Return flux in units of 106 photon ∕ s ∕m2 as afunction of zenith angle and pulse length for the Mauna Kea site. Therepetition rate and average laser power are held constant as the pulselength is varied.


we see that increasing the line width of the modes of thepulsed laser beyond the current assumed value of 15 MHzwould result in reduced flux return for many observationdirections.

7. CONCLUSIONSNumerical simulations of photon return using both a pulsedlaser with specifications provided by TIPC and a narrow-bandcw laser have been performed for the Mauna Kea and LZTsites. The results indicate that, if repump light is used, boththe cw and pulsed formats may be able to meet the designrequirements for photon return. Within the level of uncer-tainty in the model, the cw laser format provides comparablereturn to the pulsed format.

We have also investigated the dependence of the return fluxon parameters such as the pulse length, the fraction of laserpower dedicated to repumping, and the laser line width. Forthe case of the laser line width, we found that the optimal va-lue has a strong dependence on the angle between the lightpropagation direction and the magnetic-field direction.

Future modeling work could use time-resolved measure-ments of the mode structure of the TIPC laser to obtain moreaccurate results. Another avenue of research could be toquantify the impact of changing environmental conditionson the photon return, for example, the sodium-layer densityprofile, optical seeing conditions, and uncertainties in colli-sion rates.

Finally, experimental validation of the model will greatlyaid future development. Published reports of comparison be-tween theoretical models and measurements for Na LGS sys-tems have been limited; there are some results for a cw LGS atthe Starfire Optical Range [40] and for a transportable cw LGSdeveloped by the European Southern Observatory (ESO) [41].Significant information can be obtained even in the absence ofabsolute sodium density measurements: by measuring thedependence of the photon flux on experimentally variableparameters, such as zenith angle, light power, light polariza-tion, and repump power fraction, the accuracy of various as-pects of the model may be verified, providing informationabout the accuracy of the absolute photon return valuesobtained from the simulation.

ACKNOWLEDGMENTSThe TMT Project gratefully acknowledges the support of theTMT collaborating institutions. They are the Association ofCanadian Universities for Research in Astronomy (ACURA),the California Institute of Technology, the University of Cali-fornia, the National Astronomical Observatory of Japan, theNational Astronomical Observatories of China and their con-sortium partners, and the Department of Science and Technol-ogy of India and their supported institutes. This work wassupported as well by the Gordon and Betty Moore Foundation,the Canada Foundation for Innovation, the Ontario Ministry ofResearch and Innovation, the National Research Council ofCanada, the Natural Sciences and Engineering ResearchCouncil of Canada, the British Columbia Knowledge Develop-ment Fund, the Association of Universities for Research in As-tronomy (AURA) and the U.S. National Science Foundation.The authors also acknowledge the invaluable assistance of theTechnical Institute of Physics and Chemistry of the ChineseAcademy of Sciences.

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2

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2

3 4

4

4

56

7 8

0.1

1

10

100

103

Lin

ew

idth

MH

z


cw

Pulsed

Fig. 20. (Color online) Return flux in units of 106 photon ∕ s ∕m2 as afunction of zenith angle and line width of each mode for the MaunaKea site. Upper plot is for pulsed light, and lower plot is for cw. Thenatural line width of the Na D2 transition is ∼10 MHz.

3 4 5

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6

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10

Rep

ump

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cw

Pulsed

Fig. 19. (Color online) Return flux in units of 106 photon ∕ s ∕m2 as afunction of zenith angle and repump power fraction for the Mauna Keasite. Upper plot is for pulsed light, and the lower plot is for cw.


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http://omniweb.gsfc.nasa.gov/vitmo/msis_vitmo.html





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