Abstract

The polychromatic laser guide star (PLGS) is one of the solutions proposed to extend the sky coverage by large telescopes to 100% by enabling a complete knowledge of all perturbation orders of the wavefront. The knowledge of the tip–tilt is deduced from the monitoring of the chromatic components of the PLGS, from 330nm to the visible or near infrared. Here we study the original scheme to create the PLGS by resonant excitation of the mesospheric sodium by two pulsed lasers (tens of kilohertz repetition rate, tens of watts average power, tens of nanoseconds pulse duration), at 589 and 569nm, respectively. The efficiency of this process is investigated numerically by means of both Bloch equation and rate equation models. The influence of numerous laser parameters is studied. In the best case, having optimized all laser parameters, the return flux at 330nm should not exceed 7×104photonssm2 for 2×18W laser average power at the mesosphere. This maximum is obtained for a modeless laser whose spot diameter corresponds to 4 times the diffraction limit. For a diffraction-limited spot, the return flux falls down to 4×104photonssm2.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  28. M. Schöck, R. Foy, J. P. Pique, P. Chevrou, N. Ageorges, A. Petit, V. Bellanger, H. Fews, F. C. Foy, C. Högemann, M. Laubscher, O. Peillet, P. Segonds, M. Tallon, and J. M. Weulersse, "PASS-2: quantitative photometric measurements of the polychromatic laser guide star," Proc. SPIE 4007, 296-307 (2000).
    [CrossRef]
  29. H. Guillet de Chatellus and J. P. Pique, "Bilan du flux retourné par les étoiles laser polychromatiques," presented at the 10th Conference on Lasers and Quantum Optics (COLOQ 10), Grenoble, France, July 2-5, 2007.
  30. A. E. Siegman, Lasers (University Science Books, 1986), pp. 663-667.
  31. R. Foy, CRAL/Observatoire de Lyon, 9 avenue Charles André, 69561 Saint Genis-Laval, France (personal communication, 2006).
  32. R. Foy, M. Tallon, I. Tallon-Bosc, E. Thiébaut, J. Vaillant, F. C. Foy, D. Robert, H. Friedman, F. Biraben, G. Grynberg, J. P. Gex, A. Mens, A. Migus, J. M. Weulersse, and D. J. Butler, "Photometric observations of a polychromatic laser guide star," J. Opt. Soc. Am. A 17, 2236-2242 (2000).
    [CrossRef]
  33. R. Foy, J. P. Pique, V. Bellanger, P. Chevrou, A. Petit, C. Högemann, L. Noethe, M. Schöck, J. Girard, M. Tallon, E. Thiébaut, J. Vaillant, F. C. Foy, and M. Van Dam, "Feasibility study of the polychromatic laser guide star," Proc. SPIE 4839, 484-491 (2003).
    [CrossRef]

2006 (3)

J. P. Pique, I. Moldovan, and V. Fesquet, "Concept for polychromatic laser guide stars: one-photon excitation of 4P3/2 level of a sodium atom," J. Opt. Soc. Am. A 23, 2817-2827 (2006).
[CrossRef]

P. L. Wizinovitch, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. Van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, "The W. M. Keck observatory laser guide star adaptive optics system: overview," 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]

2004 (3)

L. P. Yatsenko, B. W. Shore, and K. Bergmann, "Theory of a frequency-shifted feedback laser," Opt. Commun. 236, 183-202 (2004).
[CrossRef]

J. Drummond, J. Telle, C. Denman, P. Hillman, and A. Tuffli, "Photometry of a sodium laser guide star at the Starfire Optical Range," Publ. Astron. Soc. Pac. 116, 278-289 (2004).
[CrossRef]

V. Bellanger, A. Courcelle, A. Petit, "A program to compute the two-step excitation of mesospheric sodium atoms for the Polychromatic Laser Guide Star Project," Comput. Phys. Commun. 162, 143-150 (2004).
[CrossRef]

2003 (3)

J. P. Pique and S. Farinotti, "Efficient modeless laser for a mesospheric sodium laser guide star," J. Opt. Soc. Am. B 20, 2093-2102 (2003).
[CrossRef]

J. Biegert and J. C. Diels, "Feasibility study to create a polychromatic guidestar in atomic sodium," Phys. Rev. A 67, 043403 (2003).
[CrossRef]

R. Foy, J. P. Pique, V. Bellanger, P. Chevrou, A. Petit, C. Högemann, L. Noethe, M. Schöck, J. Girard, M. Tallon, E. Thiébaut, J. Vaillant, F. C. Foy, and M. Van Dam, "Feasibility study of the polychromatic laser guide star," Proc. SPIE 4839, 484-491 (2003).
[CrossRef]

2002 (1)

M. Schöck, R. Foy, M. Tallon, L. Noethe, and J.-P. Pique, "Performance analysis of polychromatic laser guide stars used for wavefront tilt sensing," Mon. Not. R. Astron. Soc. 337, 910-920 (2002).
[CrossRef]

2000 (5)

G. Froc, E. Rosencher, B. Attal-Trétout, and V. Michau, "Photon return analysis of a polychromatic laser guide star," Opt. Commun. 178, 405-409 (2000).
[CrossRef]

N. Ageorges and N. Hubin, "Atmospheric sodium monitor for Laser Guide Star Adaptive Optics," Astron. Astrophys. Suppl. Ser. 144, 533-540 (2000).
[CrossRef]

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

M. Schöck, R. Foy, J. P. Pique, P. Chevrou, N. Ageorges, A. Petit, V. Bellanger, H. Fews, F. C. Foy, C. Högemann, M. Laubscher, O. Peillet, P. Segonds, M. Tallon, and J. M. Weulersse, "PASS-2: quantitative photometric measurements of the polychromatic laser guide star," Proc. SPIE 4007, 296-307 (2000).
[CrossRef]

R. Foy, M. Tallon, I. Tallon-Bosc, E. Thiébaut, J. Vaillant, F. C. Foy, D. Robert, H. Friedman, F. Biraben, G. Grynberg, J. P. Gex, A. Mens, A. Migus, J. M. Weulersse, and D. J. Butler, "Photometric observations of a polychromatic laser guide star," J. Opt. Soc. Am. A 17, 2236-2242 (2000).
[CrossRef]

1995 (1)

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, and M. Tallon, "The polychromatic artificial sodium star: a new concept for correcting the atmospheric tilt," Astron. Astrophys. Suppl. Ser. 111, 569-578 (1995).

1994 (1)

1993 (1)

S. Balle, I. C. M. Littler, K. Bergmann, and F. Kowalski, "Frequency shifted feedback dye laser operating at a small shift frequency," Opt. Commun. 102, 166-174 (1993).
[CrossRef]

1992 (2)

I. C. M. Littler, S. Balle, and K. Bergmann, "The cw modeless laser: spectral control, performance data and build-up dynamics," Opt. Commun. 88, 514-522 (1992).
[CrossRef]

F. Rigaut, E. Gendron, "Laser guide star in adaptive optics: the tilt determination problem," Astron. Astrophys. 261, 677 (1992).

1988 (1)

1987 (1)

L. Thompson and C. Gardner, "Experiments on laser guide stars at Mauna Kea Observatory for Adaptive Optics in Astronomy," Nature 328, 229-235 (1987).
[CrossRef]

1985 (2)

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

P. Ewart, "A modeless, variable bandwidth, tunable laser," Opt. Commun. 55, 124-126 (1985).
[CrossRef]

1983 (1)

R. C. Hilborn, "Erratum: Einstein coefficients, cross sections, f values, dipole moments and all that," Am. J. Phys. 51, 471 (1983).
[CrossRef]

1982 (1)

R. C. Hilborn, "Einstein coefficients, cross sections, f values, dipole moments and all that," Am. J. Phys. 50, 982-986 (1982).
[CrossRef]

Am. J. Phys. (2)

R. C. Hilborn, "Einstein coefficients, cross sections, f values, dipole moments and all that," Am. J. Phys. 50, 982-986 (1982).
[CrossRef]

R. C. Hilborn, "Erratum: Einstein coefficients, cross sections, f values, dipole moments and all that," Am. J. Phys. 51, 471 (1983).
[CrossRef]

Astron. Astrophys. (2)

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

F. Rigaut, E. Gendron, "Laser guide star in adaptive optics: the tilt determination problem," Astron. Astrophys. 261, 677 (1992).

Astron. Astrophys. Suppl. Ser. (2)

R. Foy, A. Migus, F. Biraben, G. Grynberg, P. R. McCullough, and M. Tallon, "The polychromatic artificial sodium star: a new concept for correcting the atmospheric tilt," Astron. Astrophys. Suppl. Ser. 111, 569-578 (1995).

N. Ageorges and N. Hubin, "Atmospheric sodium monitor for Laser Guide Star Adaptive Optics," Astron. Astrophys. Suppl. Ser. 144, 533-540 (2000).
[CrossRef]

Comput. Phys. Commun. (1)

V. Bellanger, A. Courcelle, A. Petit, "A program to compute the two-step excitation of mesospheric sodium atoms for the Polychromatic Laser Guide Star Project," Comput. Phys. Commun. 162, 143-150 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Mon. Not. R. Astron. Soc. (2)

M. Schöck, R. Foy, M. Tallon, L. Noethe, and J.-P. Pique, "Performance analysis of polychromatic laser guide stars used for wavefront tilt sensing," Mon. Not. R. Astron. Soc. 337, 910-920 (2002).
[CrossRef]

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Nature (1)

L. Thompson and C. Gardner, "Experiments on laser guide stars at Mauna Kea Observatory for Adaptive Optics in Astronomy," Nature 328, 229-235 (1987).
[CrossRef]

Opt. Commun. (5)

G. Froc, E. Rosencher, B. Attal-Trétout, and V. Michau, "Photon return analysis of a polychromatic laser guide star," Opt. Commun. 178, 405-409 (2000).
[CrossRef]

P. Ewart, "A modeless, variable bandwidth, tunable laser," Opt. Commun. 55, 124-126 (1985).
[CrossRef]

I. C. M. Littler, S. Balle, and K. Bergmann, "The cw modeless laser: spectral control, performance data and build-up dynamics," Opt. Commun. 88, 514-522 (1992).
[CrossRef]

S. Balle, I. C. M. Littler, K. Bergmann, and F. Kowalski, "Frequency shifted feedback dye laser operating at a small shift frequency," Opt. Commun. 102, 166-174 (1993).
[CrossRef]

L. P. Yatsenko, B. W. Shore, and K. Bergmann, "Theory of a frequency-shifted feedback laser," Opt. Commun. 236, 183-202 (2004).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

J. Biegert and J. C. Diels, "Feasibility study to create a polychromatic guidestar in atomic sodium," Phys. Rev. A 67, 043403 (2003).
[CrossRef]

Proc. SPIE (3)

R. Foy, J. P. Pique, V. Bellanger, P. Chevrou, A. Petit, C. Högemann, L. Noethe, M. Schöck, J. Girard, M. Tallon, E. Thiébaut, J. Vaillant, F. C. Foy, and M. Van Dam, "Feasibility study of the polychromatic laser guide star," Proc. SPIE 4839, 484-491 (2003).
[CrossRef]

M. Schöck, R. Foy, J. P. Pique, P. Chevrou, N. Ageorges, A. Petit, V. Bellanger, H. Fews, F. C. Foy, C. Högemann, M. Laubscher, O. Peillet, P. Segonds, M. Tallon, and J. M. Weulersse, "PASS-2: quantitative photometric measurements of the polychromatic laser guide star," Proc. SPIE 4007, 296-307 (2000).
[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]

Publ. Astron. Soc. Pac. (2)

P. L. Wizinovitch, D. Le Mignant, A. H. Bouchez, R. D. Campbell, J. C. Y. Chin, A. R. Contos, M. A. Van Dam, S. K. Hartman, E. M. Johansson, R. E. Lafon, H. Lewis, P. J. Stomski, and D. M. Summers, "The W. M. Keck observatory laser guide star adaptive optics system: overview," Publ. Astron. Soc. Pac. 118, 297-309 (2006).
[CrossRef]

J. Drummond, J. Telle, C. Denman, P. Hillman, and A. Tuffli, "Photometry of a sodium laser guide star at the Starfire Optical Range," Publ. Astron. Soc. Pac. 116, 278-289 (2004).
[CrossRef]

Other (7)

B. Cagnac and J. P. Faroux, Lasers (CNRS Editions, 2002), pp. 495-498.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988), pp. 224-227.

F.Rodier, ed., Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

V. Bellanger, "Etude de l'interaction laser sodium dans le cadre du projet de l'Etoile Laser Polychromatique pour l'Optique Adaptive" Ph.D. thesis (University of Paris VI, 2002).

H. Guillet de Chatellus and J. P. Pique, "Bilan du flux retourné par les étoiles laser polychromatiques," presented at the 10th Conference on Lasers and Quantum Optics (COLOQ 10), Grenoble, France, July 2-5, 2007.

A. E. Siegman, Lasers (University Science Books, 1986), pp. 663-667.

R. Foy, CRAL/Observatoire de Lyon, 9 avenue Charles André, 69561 Saint Genis-Laval, France (personal communication, 2006).

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Figures (18)

Fig. 1
Fig. 1

Overview of the two-photon PLGS. The chromatic components of the PLGS are monitored by the telescope (top). The scheme is based on the two-photon resonant excitation of the mesospheric sodium (bottom). The Doppler-hyperfine widths of the transitions are given in gigahertz.

Fig. 2
Fig. 2

Generic multilevel system. All states can decay through both radiative and nonradiative processes.

Fig. 3
Fig. 3

Plot of the return fluorescence flux given by the BEACON code for both models of the modeless laser: the CCM and the NFSR model (Gaussian time profile, top-hat spatial profile).

Fig. 4
Fig. 4

Energy diagram and relaxation pathways of the two-photon excitation in the REM.

Fig. 5
Fig. 5

Return flux of the PLGS with respect to the average laser power. Scatter, BEACON and REM data; dotted curves, best-case limits; dashed–dotted curves, asymptotes in the linear regime.

Fig. 6
Fig. 6

Return flux of the PLGS with respect to the spot size. Scatter, BEACON and REM data; dotted curves, best-case limits; dashed–dotted curves, asymptotes in the linear regime. The dashed curve represents the figure of merit of the PLGS process in arb. units (see Subsection 2B). The solid vertical line at 0.05 m 2 corresponds to a diffraction-limited spot through a 50 cm projector.

Fig. 7
Fig. 7

Return flux of the PLGS given by BEACON with respect to the spot size for different repetition rates. The laser’s average power is constant (laser power= 2 × 18 W , circular polarization, modeless laser, 80 ns Gaussian time shape, top-hat spatial profile).

Fig. 8
Fig. 8

Return flux of the PLGS given by BEACON with respect to the laser intensity (per laser) for different repetition rates (spot size= 0.5 m 2 , circular polarization, modeless laser, 80 ns Gaussian time shape, top-hat spatial profile).

Fig. 9
Fig. 9

Return flux of the PLGS given by BEACON with respect to the spot size for different laser powers (repetition rate= 30 kHz , circular polarization, modeless laser, 80 ns Gaussian time shape, top-hat spatial profile).

Fig. 10
Fig. 10

Return flux of the PLGS given by BEACON with respect to the laser intensity (per laser) for different laser spot sizes (repetition rate= 30 kHz , circular polarization, modeless laser, 80 ns Gaussian time shape, top-hat spatial profile).

Fig. 11
Fig. 11

Spectra corresponding to different phase modulations with respect to the Doppler-hyperfine lines of the transitions at 589 and 569 nm . The bottom right curve is the spectrum for a laser at 569 nm , modulated at 125 MHz with an intensity equal to 225 MHz . All other graphs concern the double phase modulation of the first laser. All PM frequencies are 180 and 300 MHz . The associated intensities are given in parentheses. Note that all spectra are centered on the F = 2 line of the D 2 line except the bottom left one, centered on the D 2 line.

Fig. 12
Fig. 12

Return flux at 330 nm given by BEACON with respect to the laser spot diameter for different PM functions. Laser parameters: 2 × 15 W , 50 ns , linear polarization, Gaussian time profile, spatial top-hat. All PM frequencies are 180 and 300 MHz . The associated intensities are given in parentheses. All values are given in megahertz. The mention of D 2 or F = 2 indicates the position of the central frequency of the phase modulation.

Fig. 13
Fig. 13

Return fluxes at 589 and 330 nm for the cases treated in [6]. The light gray (red online) scatter data are the numerical values of the flux given in the paper. In this work we took the laser characteristics of ELPOA 1 (circles), ELPOA 2 (squares), and ELPOA 3 (triangles). See text and conclusion for details.

Fig. 14
Fig. 14

Return flux at 330 nm versus spot size for different values of the FSR (REM code).

Fig. 15
Fig. 15

Comparison of the return flux for square and Gaussian time profiles (BEACON and REM codes).

Fig. 16
Fig. 16

Comparison of the return flux for square and Gaussian spatial profiles (BEACON and REM codes). Both time profiles are Gaussian.

Fig. 17
Fig. 17

Evolution of the return flux at 330 nm with respect to the spot diameter. Both modeless and PM lasers are presented with linear and circular polarizations. The parameters of the phase modulation are 180 MHz ( 260 MHz ) and 300 MHz ( 200 MHz ) at 589 nm and 125 MHz ( 225 MHz ) at 569 nm (BEACON code).

Fig. 18
Fig. 18

Return flux at 589 and 330 nm with respect to the spot size. In the first case (diamonds), the modeless laser at 589 nm is centered on the F = 2 lobe of the D 2 line ( FWHM = 1 GHz ) . In the second case (triangles), it covers the whole D 2 line with a 3 GHz FWHM. In both cases the second laser is modeless with a 1 GHz FWHM centered on the transition at 569 nm .

Tables (2)

Tables Icon

Table 1 Numerical Values and Writing Conventions Used throughout the Paper

Tables Icon

Table 2 Comparison of the Return Flux at 330 nm Given in [6] with the BEACON Results a

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

Φ λ = f d N a S D 2 p ,
A ( m b m a ) = k 3 3 π ε 0 j b , m b D q j a , m a 2 .
A b a = m a A ( m b m a ) = k 3 3 π ε 0 m a , q j a 1 m a q j b m b 2 = k 3 3 π ε 0 j b D j a 2 2 j b + 1 .
A b a i = B r b a i 1 τ b = B r b a i Γ b r a d i a t i v e ,
j b D j a i = 3 π λ 3 ε 0 ( 2 j b + 1 ) B r b a i ( 2 π ) 3 τ b .
μ b a i ¯ = j b D j a i 3 ( 2 j a i + 1 ) .
Γ a i b = Γ a i r a d i a t i v e + Γ b r a d i a t i v e + Γ c o l l i s i o n s + ,
σ a i b = 2 π Γ a i b σ 0 .
σ 0 = g b 4 g a i λ 2 A b a i ,
σ a i b = 1 2 π g b g a i λ 2 B r b a i τ a i τ a i + τ b .
E ( t ) = E 0 + e ( t t 0 ) 2 e i Δ 2 τ r ( t n τ s ) 2 e i ω m a x ( t n τ s ) ,
t 0 = 2 π τ r Δ δ υ 2 ln 2 , τ s = 2 π Δ ,
N 1 ( t , r , ν ) d t = N 1 ( t , r , ν ) + σ 1 ( ν ν ) Φ 1 L ( t , r , ν ) d ν ( absorption ) + N 3 ( t , r , ν ) τ 31 + N 2 ( t , r , ν ) τ 21 + N 5 ( t , r , ν ) τ 51 ( spontaneous emission ) + N 3 ( t , r , ν ) g 1 g 3 + σ 1 ( ν ν ) Φ 1 L ( t , r , ν ) d ν , ( stimulated emission )
N 2 ( t , r , ν ) d t = N 4 ( t , r , ν ) τ 42 N 2 ( t , r , ν ) τ 21 ,
N 3 ( t , r , ν ) d t = N 1 ( t , r , ν ) + σ 1 ( ν ν ) Φ 1 L ( t , r , ν ) d ν N 3 ( t , r , ν ) g 1 g 3 + σ 1 ( ν ν ) Φ 1 L ( t , r , ν ) d ν N 3 ( t , r , ν ) + σ 2 ( ν ν ) Φ 2 L ( t , r , ν ) d ν + N 6 ( t , r , ν ) g 3 g 6 + σ 2 ( ν ν ) Φ 2 L ( t , r , ν ) d ν N 3 ( t , r , ν ) τ 31 + N 4 ( t , r , ν ) τ 43 + N 6 ( t , r , ν ) τ 63 ,
N 4 ( t , r , ν ) d t = N 5 ( t , r , ν ) τ 54 N 4 ( t , r , ν ) τ 42 N 4 ( t , r , ν ) τ 43 ,
N 5 ( t , r , ν ) d t = N 6 ( t , r , ν ) τ 65 N 5 ( t , r , ν ) τ 54 N 5 ( t , r , ν ) τ 51 ,
N 6 ( t , r , ν ) d t = N 3 ( t , r , ν ) + σ 2 ( ν ν ) Φ 2 L ( t , r , ν ) d ν N 6 ( t , r , ν ) τ 63 N 6 ( t , r , ν ) τ 65 N 6 ( t , r , ν ) g 3 g 6 + σ 2 ( ν ν ) Φ 2 L ( t , r , ν ) d ν .
i = 1 6 N i ( t , r , ν ) = N D ( ν ) ,
σ i ( ν ) = σ 0 i ( Γ i 2 ) 2 ( ν 0 i ν ) 2 + ( Γ i 2 ) 2 .
Φ L ( t , r , ν ) = φ ( t ) δ ( r ) g ( ν ) .
Φ 589 = ( 1 + τ l a s e r τ b ) S f d N a 4 π D 2 .
Φ 330 = 1 9 S f d N a 4 π D 2 .
φ 589 = I ¯ 1 S Γ 1 δ υ 1 1 ω 1 ,
p = σ 01 φ 589 .
n N a = S d N a δ υ 1 Γ 1 D .
Φ 589 = n N a 4 π D 2 p = σ 01 d N a 4 π D 2 I ¯ 1 ω 1 Γ 1 Γ 1 D .
n 3 P 3 2 = σ 01 I ¯ 1 f Γ 1 Γ 1 D 1 ω 1 d N a .
φ 569 = 1 f I ¯ 2 S Γ 2 Γ 2 D 1 ω 2 .
p = τ 3 P 3 2 τ l a s e r σ 02 φ 569 .
Φ 330 = 1 9 d N a 4 π D 2 τ 3 P 3 2 τ l a s e r 1 f σ 01 σ 02 I ¯ 1 ω 1 I ¯ 2 ω 2 Γ 1 Γ 1 D Γ 2 Γ 2 D 1 S .
Δ X Δ Y Σ Φ 330 .
Φ = S f F ( E p u l s e ) ,
Φ = S f F ( I ¯ S f ) .
Φ ( S , k f , I ¯ ) = S k f F ( I ¯ S k f ) = Φ ( k S , f , I ¯ ) .
Φ ( S , k f , I ¯ ) = k Φ ( S , f , I ¯ k ) ,
Φ ( S , f , k I ¯ ) = S f F ( k I ¯ S f ) = k ( S k ) f F ( I ¯ ( S k ) f ) = k Φ ( S k , f , I ¯ ) .
Φ ( k S , f , I ¯ ) = k S f F ( I ¯ k S f ) = k Φ ( S , f , I ¯ k ) .

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