Abstract

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

© 2012 Optical Society of America

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2010 (5)

M. Hart, N. M. Milton, C. Baranec, K. Powell, T. Stalcup, D. McCarthy, C. Kulesa, and E. Bendek, “A ground-layer adaptive optics system with multiple laser guide stars,” Nature 466, 727–729 (2010).
[CrossRef]

L. Gilles, L. Wang, and B. L. Ellerbroek, “Impact of laser launch location on the performance of laser tomography and multiconjugate adaptive optics for extremely large telescopes,” Appl. Opt. 49, G114–G119 (2010).
[CrossRef]

L. Wang, A. Otarola, and B. Ellerbroek, “Impact of sodium laser guide star fratricide on multi-conjugate adaptive optics systems,” J. Opt. Soc. Am. A 27, A19–A28 (2010).
[CrossRef]

R. Holzlöhner, S. M. Rochester, D. Bonaccini Calia, D. Budker, J. M. Higbie, and W. Hackenberg, “Optimization of cw sodium laser guide star efficiency,” Astron. Astrophys. 510, A20 (2010).
[CrossRef]

R. Holzlöhner, S. M. Rochester, T. Pfrommer, D. B. Calia, D. Budker, J. M. Higbie, and W. Hackenberg, “Laser guide star return flux simulations based on observed sodium density profiles,” Proc. SPIE 7736, 77360V (2010).
[CrossRef]

2009 (1)

M. Schöck, S. Els, R. Riddle, W. Skidmore, T. Travouillon, R. Blum, E. Bustos, G. Chanan, S. G. Djorgovski, P. Gillett, B. Gregory, J. Nelson, A. Otárola, J. Seguel, J. Vasquez, A. Walker, D. Walker, and L. Wang, “Thirty meter telescope site testing I: overview,” Publ. Astron. Soc. Pac. 121, 384–395 (2009).
[CrossRef]

2008 (4)

B. Ellerbroek, S. Adkins, D. Andersen, J. Atwood, C. Boyer, P. Byrnes, R. Conan, L. Gilles, G. Herriot, P. Hickson, E. Hileman, D. Joyce, B. Leckie, M. Liang, T. Pfrommer, J.-C. Sinquin, J.-P. Veran, L. Wang, and P. Welle, “Progress toward developing the TMT adaptive optical systems and their components,” Proc. SPIE 7015, 70150R (2008).
[CrossRef]

E. Diolaiti, J.-M. Conan, I. Foppiani, M. Lombini, C. Petit, C. Robert, L. Schreiber, P. Ciliegi, E. Marchetti, M. Bellazzini, L. Busoni, S. Esposito, T. Fusco, N. Hubin, F. Quiros-Pacheco, A. Baruffolo, S. D’Odorico, J. Farinato, B. Neichel, R. Ragazzoni, C. Arcidiacono, V. Biliotti, G. Bregoli, G. Cosentino, and G. Innocenti, “A preliminary overview of the multiconjugate adaptive optics module for the E-ELT,” Proc. SPIE 7015, 70150U (2008).
[CrossRef]

P. D. Hillman, J. D. Drummond, C. A. Denman, and R. Q. Fugate, “Simple model, including recoil, for the brightness of sodium guide stars created from cw single frequency fasors and comparison to measurements,” Proc. SPIE 7015, 70150L (2008).
[CrossRef]

A. M. Ghez, S. Salim, N. N. Weinberg, J. R. Lu, T. Do, J. K. Dunn, K. Matthews, M. R. Morris, S. Yelda, E. E. Becklin, T. Kremenek, M. Milosavljevic, and J. Naiman, “Measuring distance and properties of the Milky Way’s central supermassive black hole with stellar orbits,” Astrophys. J. 689, 1044–1062 (2008).
[CrossRef]

2006 (4)

L. Gilles and B. Ellerbroek, “Shack-Hartmann wavefront sensing with elongated sodium laser beacons: centroiding versus matched filtering,” Appl. Opt. 45, 6568–6576 (2006).
[CrossRef]

M. Lloyd-Hart, R. Angel, N. M. Milton, M. Rademacher, and J. Codona, “Design of the adaptive optics systems for GMT,” Proc. SPIE 6272, 62720E (2006).
[CrossRef]

P. L. Wizinowich, D. L. Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. v. 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 W. M. Keck Observatory laser guide star adaptive optics system: oerview,” Publ. Astron. Soc. Pac. 118, 297–309 (2006).
[CrossRef]

M. Boccas, F. Rigaut, M. Bec, B. Irarrazaval, E. James, A. Ebbers, C. d’Orgeville, K. Grace, G. Arriagada, S. Karewicz, M. Sheehan, J. White, and S. Chan, “Laser guide star upgrade of Altair at Gemini North,” Proc. SPIE 6272, 62723L (2006).
[CrossRef]

2005 (1)

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward, “SUNDIALS: suite of nonlinear and differential/algebraic equation solvers,” ACM Trans. Math. Softw. 31, 363–396 (2005).
[CrossRef]

2004 (1)

M. Le Louarn, N. N. Hubin, and R. Arsenault, “Adaptive optics for second-generation VLT instruments,” Proc. SPIE 5490248–259 (2004).
[CrossRef]

2003 (2)

D. T. Gavel, E. L. Gates, C. E. Max, S. S. Olivier, B. J. Bauman, D. M. Pennington, B. A. Macintosh, J. Patience, C. G. Brown, P. M. Danforth, R. L. Hurd, S. A. Severson, and J. P. Lloyd, “Recent science and engineering results with the laser guidestar adaptive optic system at Lick Observatory,” Proc. SPIE 4839, 354–359 (2003).
[CrossRef]

J.-P. Pique and S. Farinotti, “Efficient modeless laser for a mesospheric sodium laser guide star,” J. Opt. Soc. Am. B 20, 2093–2101 (2003).
[CrossRef]

2002 (1)

R. Schödel, T. Ott, R. Genzel, R. Hofmann, M. Lehnert, A. Eckart, N. Mouawad, T. Alexander, M. J. Reid, R. Lenzen, M. Hartung, F. Lacombe, D. Rouan, E. Gendron, G. Rousset, A.-M. Lagrange, W. Brandner, N. Ageorges, C. Lidman, A. F. M. Moorwood, J. Spyromilio, N. Hubin, and K. M. Menten, “A star in a 15.2-year orbit around the supermassive black hole at the centre of the Milky Way,” Nature 419, 694–696 (2002).
[CrossRef]

1998 (1)

1996 (1)

K. Matthews, A. M. Ghez, A. J. Weinberger, and G. Neugebauer, “The first diffraction-limited images from the W. M. Keck Telescope,” Publ. Astron. Soc. Pac. 108, 615–619 (1996).
[CrossRef]

1994 (2)

1991 (2)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144–146 (1991).
[CrossRef]

T. H. Jeys, “Development of a mesospheric sodium laser beacon for atmospheric adaptive optics,” Lincoln Lab. J. 4, 133–150 (1991).

1985 (1)

R. Foy and A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

1966 (1)

Adkins, S.

B. Ellerbroek, S. Adkins, D. Andersen, J. Atwood, C. Boyer, P. Byrnes, R. Conan, L. Gilles, G. Herriot, P. Hickson, E. Hileman, D. Joyce, B. Leckie, M. Liang, T. Pfrommer, J.-C. Sinquin, J.-P. Veran, L. Wang, and P. Welle, “Progress toward developing the TMT adaptive optical systems and their components,” Proc. SPIE 7015, 70150R (2008).
[CrossRef]

Ageorges, N.

R. Schödel, T. Ott, R. Genzel, R. Hofmann, M. Lehnert, A. Eckart, N. Mouawad, T. Alexander, M. J. Reid, R. Lenzen, M. Hartung, F. Lacombe, D. Rouan, E. Gendron, G. Rousset, A.-M. Lagrange, W. Brandner, N. Ageorges, C. Lidman, A. F. M. Moorwood, J. Spyromilio, N. Hubin, and K. M. Menten, “A star in a 15.2-year orbit around the supermassive black hole at the centre of the Milky Way,” Nature 419, 694–696 (2002).
[CrossRef]

Alexander, T.

R. Schödel, T. Ott, R. Genzel, R. Hofmann, M. Lehnert, A. Eckart, N. Mouawad, T. Alexander, M. J. Reid, R. Lenzen, M. Hartung, F. Lacombe, D. Rouan, E. Gendron, G. Rousset, A.-M. Lagrange, W. Brandner, N. Ageorges, C. Lidman, A. F. M. Moorwood, J. Spyromilio, N. Hubin, and K. M. Menten, “A star in a 15.2-year orbit around the supermassive black hole at the centre of the Milky Way,” Nature 419, 694–696 (2002).
[CrossRef]

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, and L. M. Wopat, “Measurement of atmospheric wavefront distortion using scattered light from a laser guide-star,” Nature 353, 144–146 (1991).
[CrossRef]

Andersen, D.

B. Ellerbroek, S. Adkins, D. Andersen, J. Atwood, C. Boyer, P. Byrnes, R. Conan, L. Gilles, G. Herriot, P. Hickson, E. Hileman, D. Joyce, B. Leckie, M. Liang, T. Pfrommer, J.-C. Sinquin, J.-P. Veran, L. Wang, and P. Welle, “Progress toward developing the TMT adaptive optical systems and their components,” Proc. SPIE 7015, 70150R (2008).
[CrossRef]

Angel, R.

M. Lloyd-Hart, R. Angel, N. M. Milton, M. Rademacher, and J. Codona, “Design of the adaptive optics systems for GMT,” Proc. SPIE 6272, 62720E (2006).
[CrossRef]

Arcidiacono, C.

E. Diolaiti, J.-M. Conan, I. Foppiani, M. Lombini, C. Petit, C. Robert, L. Schreiber, P. Ciliegi, E. Marchetti, M. Bellazzini, L. Busoni, S. Esposito, T. Fusco, N. Hubin, F. Quiros-Pacheco, A. Baruffolo, S. D’Odorico, J. Farinato, B. Neichel, R. Ragazzoni, C. Arcidiacono, V. Biliotti, G. Bregoli, G. Cosentino, and G. Innocenti, “A preliminary overview of the multiconjugate adaptive optics module for the E-ELT,” Proc. SPIE 7015, 70150U (2008).
[CrossRef]

Arriagada, G.

M. Boccas, F. Rigaut, M. Bec, B. Irarrazaval, E. James, A. Ebbers, C. d’Orgeville, K. Grace, G. Arriagada, S. Karewicz, M. Sheehan, J. White, and S. Chan, “Laser guide star upgrade of Altair at Gemini North,” Proc. SPIE 6272, 62723L (2006).
[CrossRef]

Arsenault, R.

M. Le Louarn, N. N. Hubin, and R. Arsenault, “Adaptive optics for second-generation VLT instruments,” Proc. SPIE 5490248–259 (2004).
[CrossRef]

Atwood, J.

B. Ellerbroek, S. Adkins, D. Andersen, J. Atwood, C. Boyer, P. Byrnes, R. Conan, L. Gilles, G. Herriot, P. Hickson, E. Hileman, D. Joyce, B. Leckie, M. Liang, T. Pfrommer, J.-C. Sinquin, J.-P. Veran, L. Wang, and P. Welle, “Progress toward developing the TMT adaptive optical systems and their components,” Proc. SPIE 7015, 70150R (2008).
[CrossRef]

Avicola, K.

Baranec, C.

M. Hart, N. M. Milton, C. Baranec, K. Powell, T. Stalcup, D. McCarthy, C. Kulesa, and E. Bendek, “A ground-layer adaptive optics system with multiple laser guide stars,” Nature 466, 727–729 (2010).
[CrossRef]

Baruffolo, A.

E. Diolaiti, J.-M. Conan, I. Foppiani, M. Lombini, C. Petit, C. Robert, L. Schreiber, P. Ciliegi, E. Marchetti, M. Bellazzini, L. Busoni, S. Esposito, T. Fusco, N. Hubin, F. Quiros-Pacheco, A. Baruffolo, S. D’Odorico, J. Farinato, B. Neichel, R. Ragazzoni, C. Arcidiacono, V. Biliotti, G. Bregoli, G. Cosentino, and G. Innocenti, “A preliminary overview of the multiconjugate adaptive optics module for the E-ELT,” Proc. SPIE 7015, 70150U (2008).
[CrossRef]

Bauman, B. J.

D. T. Gavel, E. L. Gates, C. E. Max, S. S. Olivier, B. J. Bauman, D. M. Pennington, B. A. Macintosh, J. Patience, C. G. Brown, P. M. Danforth, R. L. Hurd, S. A. Severson, and J. P. Lloyd, “Recent science and engineering results with the laser guidestar adaptive optic system at Lick Observatory,” Proc. SPIE 4839, 354–359 (2003).
[CrossRef]

Bec, M.

M. Boccas, F. Rigaut, M. Bec, B. Irarrazaval, E. James, A. Ebbers, C. d’Orgeville, K. Grace, G. Arriagada, S. Karewicz, M. Sheehan, J. White, and S. Chan, “Laser guide star upgrade of Altair at Gemini North,” Proc. SPIE 6272, 62723L (2006).
[CrossRef]

Becklin, E. E.

A. M. Ghez, S. Salim, N. N. Weinberg, J. R. Lu, T. Do, J. K. Dunn, K. Matthews, M. R. Morris, S. Yelda, E. E. Becklin, T. Kremenek, M. Milosavljevic, and J. Naiman, “Measuring distance and properties of the Milky Way’s central supermassive black hole with stellar orbits,” Astrophys. J. 689, 1044–1062 (2008).
[CrossRef]

Bellazzini, M.

E. Diolaiti, J.-M. Conan, I. Foppiani, M. Lombini, C. Petit, C. Robert, L. Schreiber, P. Ciliegi, E. Marchetti, M. Bellazzini, L. Busoni, S. Esposito, T. Fusco, N. Hubin, F. Quiros-Pacheco, A. Baruffolo, S. D’Odorico, J. Farinato, B. Neichel, R. Ragazzoni, C. Arcidiacono, V. Biliotti, G. Bregoli, G. Cosentino, and G. Innocenti, “A preliminary overview of the multiconjugate adaptive optics module for the E-ELT,” Proc. SPIE 7015, 70150U (2008).
[CrossRef]

Bendek, E.

M. Hart, N. M. Milton, C. Baranec, K. Powell, T. Stalcup, D. McCarthy, C. Kulesa, and E. Bendek, “A ground-layer adaptive optics system with multiple laser guide stars,” Nature 466, 727–729 (2010).
[CrossRef]

Biliotti, V.

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M. Hart, N. M. Milton, C. Baranec, K. Powell, T. Stalcup, D. McCarthy, C. Kulesa, and E. Bendek, “A ground-layer adaptive optics system with multiple laser guide stars,” Nature 466, 727–729 (2010).
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M. Hart, N. M. Milton, C. Baranec, K. Powell, T. Stalcup, D. McCarthy, C. Kulesa, and E. Bendek, “A ground-layer adaptive optics system with multiple laser guide stars,” Nature 466, 727–729 (2010).
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R. Holzlöhner, S. M. Rochester, T. Pfrommer, D. B. Calia, D. Budker, J. M. Higbie, and W. Hackenberg, “Laser guide star return flux simulations based on observed sodium density profiles,” Proc. SPIE 7736, 77360V (2010).
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R. Schödel, T. Ott, R. Genzel, R. Hofmann, M. Lehnert, A. Eckart, N. Mouawad, T. Alexander, M. J. Reid, R. Lenzen, M. Hartung, F. Lacombe, D. Rouan, E. Gendron, G. Rousset, A.-M. Lagrange, W. Brandner, N. Ageorges, C. Lidman, A. F. M. Moorwood, J. Spyromilio, N. Hubin, and K. M. Menten, “A star in a 15.2-year orbit around the supermassive black hole at the centre of the Milky Way,” Nature 419, 694–696 (2002).
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B. Ellerbroek, S. Adkins, D. Andersen, J. Atwood, C. Boyer, P. Byrnes, R. Conan, L. Gilles, G. Herriot, P. Hickson, E. Hileman, D. Joyce, B. Leckie, M. Liang, T. Pfrommer, J.-C. Sinquin, J.-P. Veran, L. Wang, and P. Welle, “Progress toward developing the TMT adaptive optical systems and their components,” Proc. SPIE 7015, 70150R (2008).
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E. Diolaiti, J.-M. Conan, I. Foppiani, M. Lombini, C. Petit, C. Robert, L. Schreiber, P. Ciliegi, E. Marchetti, M. Bellazzini, L. Busoni, S. Esposito, T. Fusco, N. Hubin, F. Quiros-Pacheco, A. Baruffolo, S. D’Odorico, J. Farinato, B. Neichel, R. Ragazzoni, C. Arcidiacono, V. Biliotti, G. Bregoli, G. Cosentino, and G. Innocenti, “A preliminary overview of the multiconjugate adaptive optics module for the E-ELT,” Proc. SPIE 7015, 70150U (2008).
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This formula for airmass dependence neglects the curvature of the Earth, which is a reasonable assumption for the range of zenith angles (≤60°) considered in this study.

Available at http://omniweb.gsfc.nasa.gov/vitmo/msis\_vitmo.html .

V. I. Tatarski, Wave Propagation in a Turbulent Medium Translated from Russian by R. A. Silverman (McGraw-Hill, 1961).

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Figures (20)

Fig. 1.
Fig. 1.

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.
Fig. 2.

Precession of the atomic polarization P, originally produced along the light wave vector k, about the magnetic-field vector B. If the polarization is continuously generated and has a lifetime longer than the Larmor precession time, it will tend to be averaged about the magnetic-field direction.

Fig. 3.
Fig. 3.

Temperature and O2 and N2 gas densities as a function of altitude obtained from the MSIS-E-90 Atmosphere Model for the Mauna Kea site, along with the assumed Na density profile. (Note that deviations of the actual Na density profile from a Gaussian are often quite large.)

Fig. 4.
Fig. 4.

(a) Interpolated beam profile from geometrical optics model, given in terms of the fraction of power in each pixel. The peak value is the Strehl ratio. (b) Return flux from each pixel in 103photons/s/m2.

Fig. 5.
Fig. 5.

Instantaneous return flux during a single pulse as a function of elapsed time, using three different multimode treatments, both with no repump light and with 10% repump. Example plots are for Mauna Kea, zenith angle zenith angle=30°, azimuth angle=190°, Iavg=27.5W/m2. The oscillations present in the coherent-mode treatment are too rapid to be distinguished on this time scale, and so appear as solid bands.

Fig. 6.
Fig. 6.

An expanded view of the initial part of the coherent-modes plot from Fig. 5(b). The oscillations are due to interference between the laser modes.

Fig. 7.
Fig. 7.

Laser pulse shapes: (top) the supplied measured pulse shape; (bottom) the pulse-shape model described in the text (solid line) and the same model with the initial oscillations averaged out (dashed line). The parameters used here are t0=22μs, τr=19μs, aosc=1.9, τosc=12μs, ω=2π×0.6MHz, t1=202μs, τf=9.6μs, and p=1.5. The overall amplitude is scaled to match the measurement. The nominal pulse length is t1t0=180μs, longer than the standard pulse length of 120 μs used in the simulations.

Fig. 8.
Fig. 8.

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

Fig. 9.
Fig. 9.

An example of return flux as a function of the longitudinal atomic velocity (given in terms of the corresponding Doppler shift), at a time near the beginning of a laser pulse.

Fig. 10.
Fig. 10.

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 the beginning 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 (right column), superimposed over the unperturbed equilibrium populations (dashed lines).

Fig. 11.
Fig. 11.

(a) Return flux dΦ(I,z)/d/I in units of 109photons/s/sr/m/(W/m2) as a function of light intensity and altitude. (b) Total flux dΦtot(I,z)/d/I in units of 109photons/s/m/(W/m2), used to calculate optical absorption in the sodium layer.

Fig. 12.
Fig. 12.

Transmission through the sodium layer as a function of path length and initial light power in a pixel-sized beam.

Fig. 13.
Fig. 13.

Normalized observed return flux per d from a pixel-sized sub-beam, dΦobs(P0,)/d/P0, in units of photons/s/m2/m/W.

Fig. 14.
Fig. 14.

Total observed normalized return flux Φobs(P0)/P0 from a pixel-sized beam as a function of initial launched light power in that pixel.

Fig. 15.
Fig. 15.

Beam-integrated return flux for Mauna Kea (as a function of zenith angle) and LZT (at zenith), for pulsed light using each of the 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 repump beam. The vertical axis on the left side is in units of 106photons/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 0.5m2, and 48% detection efficiency. The dashed gray line shows the design requirement for 900 PDEs at zenith, combined with the nominal dependence on the inverse of the airmass X expected from purely geometrical considerations. (The optical path length in the sodium layer increases with X, while the detector collection solid angle falls as X2.) The higher return at the LZT site is primarily due to the higher throughput of the laser-launch system and the different direction of the local geomagnetic field compared to that at the Mauna Kea site.

Fig. 16.
Fig. 16.

Beam-averaged return flux and PDE for a pulsed laser without repump, with 10% repump, and with 10% repump with opposite circular polarization as the D2a light.

Fig. 17.
Fig. 17.

Return flux as a function of spot size (FWHM) at the LZT site, using the pulse-averaging method. The dashed gray line shows the spot size obtained from the geometrical optics model using the standard Fried parameter r0=0.05m. The beam profile was directly scaled proportionally smaller and larger to obtain the results for this plot.

Fig. 18.
Fig. 18.

Return flux in units of 106photon/s/m2 as a function of zenith angle and pulse length for the Mauna Kea site. The repetition rate and average laser power are held constant as the pulse length is varied.

Fig. 19.
Fig. 19.

Return flux in units of 106photon/s/m2 as a function of zenith angle and repump power fraction for the Mauna Kea site. Upper plot is for pulsed light, and the lower plot is for cw.

Fig. 20.
Fig. 20.

Return flux in units of 106photon/s/m2 as a function of zenith angle and line width of each mode for the Mauna Kea site. Upper plot is for pulsed light, and lower plot is for cw. The natural line width of the Na D2 transition is 10MHz.

Tables (3)

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Table 1. Standard Parameters for the TIPC Pulsed Laser

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Table 2. Standard Parameters for Mauna Kea and LZT sitesa

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Table 3. Estimated Values for Cross Sectionsa

Equations (5)

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ddtρ=1i[H,ρ]+Λ(ρ)+β.
dP()/d=ωdΦtot(P()/(θp)2,/X)/d.
dΦobs(P0,)/d=TaX2Tlin()dΦ(P()/(θp)2,/X)/d,
1e(tt0)/τr+aosce(tt0)/τoscsin2(ωt/2),
exp([(tt1)/τf]p),

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