Abstract

The responses of two-level atoms and of sodium atoms to trains of repetitively pulsed resonance radiation are computed. It is found that the optimum backscatter from mesospheric sodium for a given irradiance is attained when the pulse trains are phase modulated and when the lengths of individual pulses are in the 500–700-ps range, corresponding to a transform-limited bandwidth somewhat smaller than the Doppler-broadened hyperfine-structure pattern of the D2 line.

© 1992 Optical Society of America

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References

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  1. R. A. Humphreys, C. A. Primmerman, L. C. Bradley, and J. Herrmann, “Atmospheric-turbulence measurements using a synthetic beacon in the mesospheric sodium layer,” Opt. Lett. 16, 1367–1369 (1991).
    [CrossRef] [PubMed]
  2. T. H. Jeys, A. A. Brailove, and A. Mooradian, “Sum frequency generation of sodium resonance radiation,” Appl. Opt. 28, 2588–2591 (1989).
    [CrossRef] [PubMed]
  3. T. H. Jeys, “Development of a mesospheric sodium laser beacon for atmospheric adaptive optics,” Lincoln Lab. J. 4, 133 (1991).
  4. Work on FEL lasers for this purpose has been done by R. A. Temkin of the Massachusetts Institute of Technology and L. Thode of the Los Alamos National Laboratory, among others.
  5. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990).
  6. G. F. Thomas, “Pulse train single-photon induced optical Ramsey fringes,” Phys. Rev. A 35, 5060–5063 (1987).
    [CrossRef] [PubMed]
  7. S. Feneuille, M. G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 1976–2009 (1976).
    [CrossRef]
  8. T. W. Hänsen, “Multiple coherent interactions,” in Laser Spectroscopy III, J. L. Hall and J. L. Carlsten, eds. (Springer-Verlag, Berlin, 1977) pp. 149–153.
  9. R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764(1977).
    [CrossRef]
  10. M. M. Salour and C. Cohen-Tannoudji, “Observation of Ramsey’s interference fringes in the profile of Doppler-free two-photon resonance,” Phys. Rev. Lett. 38, 757–760 (1977).
    [CrossRef]
  11. M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
    [CrossRef]
  12. R. R. Parenti, Lincoln Laboratory, Massachusetts Institute of Technology Lexington, Mass. 02173 (personal communication, 1990).
  13. R. J. Temkin, Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, Mass. 02139 (personal communication, 1990).
  14. P. W. Milonni and L. Thode, “Theory of mesospheric sodium fluorescence excited by pulse trains,” Appl. Opt. 31, 785–800 (1992).
    [CrossRef] [PubMed]
  15. The derivation of the optical Bloch equations can be found in many places; see, for example, Ref. 5, Sec. 8.4.
  16. W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–283 (1972).
    [CrossRef]
  17. T. H. Jeys, “Sum frequency mixing of frequency modulated laser radiation,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper THN3.
  18. A. Mooradian, “Use of spatial time-division repetition rate multiplication of mode-locked laser pulses to generate microwave radiation from optoelectronic switches,” Appl. Phys. Lett. 45, 494–496 (1984).
    [CrossRef]
  19. P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).
  20. A. Kastler, “Optical methods of atomic orientation and of magnetic resonance,” J. Opt. Soc. Am. 47, 460–465 (1957).
    [CrossRef]
  21. W B. Hawkins, “Orientation and alignment of sodium atoms by means of polarized resonance radiation,” Phys. Rev. 98, 478–486 (1955).
    [CrossRef]
  22. W. Lange, J. Luther, B. Nottbeck, and H. W. Schröder, “High-resolution fluorescence spectroscopy by use of a cw dye laser,” Opt. Commun. 8, 157–159 (1973).
    [CrossRef]
  23. H. C. Torrey, “Transient nutations in nuclear magnetic resonance,” Phys. Rev. 76, 1059–1068 (1949).
    [CrossRef]
  24. G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
    [CrossRef]
  25. See, for example, G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, New York, 1980), Sec. 5.5.
  26. See, for example, L. Allen and J. H. Eberly, Optical Resonances and Two-Level Atoms (Wiley, New York, 1975), Chap. 3.
  27. A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
    [CrossRef]
  28. L. C. Bradley, “Atomic excitation by a pulse train I: two-level atoms”;“Atomic excitation by a pulse train II: sodium,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 66.

1992 (1)

1991 (2)

1989 (2)

T. H. Jeys, A. A. Brailove, and A. Mooradian, “Sum frequency generation of sodium resonance radiation,” Appl. Opt. 28, 2588–2591 (1989).
[CrossRef] [PubMed]

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

1987 (1)

G. F. Thomas, “Pulse train single-photon induced optical Ramsey fringes,” Phys. Rev. A 35, 5060–5063 (1987).
[CrossRef] [PubMed]

1984 (1)

A. Mooradian, “Use of spatial time-division repetition rate multiplication of mode-locked laser pulses to generate microwave radiation from optoelectronic switches,” Appl. Phys. Lett. 45, 494–496 (1984).
[CrossRef]

1977 (3)

A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764(1977).
[CrossRef]

M. M. Salour and C. Cohen-Tannoudji, “Observation of Ramsey’s interference fringes in the profile of Doppler-free two-photon resonance,” Phys. Rev. Lett. 38, 757–760 (1977).
[CrossRef]

1976 (1)

S. Feneuille, M. G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 1976–2009 (1976).
[CrossRef]

1973 (2)

M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
[CrossRef]

W. Lange, J. Luther, B. Nottbeck, and H. W. Schröder, “High-resolution fluorescence spectroscopy by use of a cw dye laser,” Opt. Commun. 8, 157–159 (1973).
[CrossRef]

1972 (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–283 (1972).
[CrossRef]

1968 (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[CrossRef]

1957 (1)

1955 (1)

W B. Hawkins, “Orientation and alignment of sodium atoms by means of polarized resonance radiation,” Phys. Rev. 98, 478–486 (1955).
[CrossRef]

1949 (1)

H. C. Torrey, “Transient nutations in nuclear magnetic resonance,” Phys. Rev. 76, 1059–1068 (1949).
[CrossRef]

Allen, L.

See, for example, L. Allen and J. H. Eberly, Optical Resonances and Two-Level Atoms (Wiley, New York, 1975), Chap. 3.

Andrä, J.

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

Bradley, L. C.

R. A. Humphreys, C. A. Primmerman, L. C. Bradley, and J. Herrmann, “Atmospheric-turbulence measurements using a synthetic beacon in the mesospheric sodium layer,” Opt. Lett. 16, 1367–1369 (1991).
[CrossRef] [PubMed]

L. C. Bradley, “Atomic excitation by a pulse train I: two-level atoms”;“Atomic excitation by a pulse train II: sodium,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 66.

Brailove, A. A.

Cohen-Tannoudji, C.

M. M. Salour and C. Cohen-Tannoudji, “Observation of Ramsey’s interference fringes in the profile of Doppler-free two-photon resonance,” Phys. Rev. Lett. 38, 757–760 (1977).
[CrossRef]

Ducloy, M.

M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
[CrossRef]

Eberly, J. H.

See, for example, L. Allen and J. H. Eberly, Optical Resonances and Two-Level Atoms (Wiley, New York, 1975), Chap. 3.

Eckstein, J.

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764(1977).
[CrossRef]

Feneuille, S.

S. Feneuille, M. G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 1976–2009 (1976).
[CrossRef]

Hänsch, T. W.

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764(1977).
[CrossRef]

Hänsen, T. W.

T. W. Hänsen, “Multiple coherent interactions,” in Laser Spectroscopy III, J. L. Hall and J. L. Carlsten, eds. (Springer-Verlag, Berlin, 1977) pp. 149–153.

Happer, W.

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–283 (1972).
[CrossRef]

Hawkins, W B.

W B. Hawkins, “Orientation and alignment of sodium atoms by means of polarized resonance radiation,” Phys. Rev. 98, 478–486 (1955).
[CrossRef]

Herrmann, J.

Humphreys, R. A.

Jeys, T. H.

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

T. H. Jeys, A. A. Brailove, and A. Mooradian, “Sum frequency generation of sodium resonance radiation,” Appl. Opt. 28, 2588–2591 (1989).
[CrossRef] [PubMed]

T. H. Jeys, “Sum frequency mixing of frequency modulated laser radiation,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper THN3.

Kastler, A.

Kersebom, T.

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

Krüger, E.

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

Lange, W.

W. Lange, J. Luther, B. Nottbeck, and H. W. Schröder, “High-resolution fluorescence spectroscopy by use of a cw dye laser,” Opt. Commun. 8, 157–159 (1973).
[CrossRef]

Luther, J.

W. Lange, J. Luther, B. Nottbeck, and H. W. Schröder, “High-resolution fluorescence spectroscopy by use of a cw dye laser,” Opt. Commun. 8, 157–159 (1973).
[CrossRef]

Milonni, P. W.

Mooradian, A.

T. H. Jeys, A. A. Brailove, and A. Mooradian, “Sum frequency generation of sodium resonance radiation,” Appl. Opt. 28, 2588–2591 (1989).
[CrossRef] [PubMed]

A. Mooradian, “Use of spatial time-division repetition rate multiplication of mode-locked laser pulses to generate microwave radiation from optoelectronic switches,” Appl. Phys. Lett. 45, 494–496 (1984).
[CrossRef]

Nölle, H.

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

Nottbeck, B.

W. Lange, J. Luther, B. Nottbeck, and H. W. Schröder, “High-resolution fluorescence spectroscopy by use of a cw dye laser,” Opt. Commun. 8, 157–159 (1973).
[CrossRef]

Oliver, G.

S. Feneuille, M. G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 1976–2009 (1976).
[CrossRef]

Omont, A.

A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

Parenti, R. R.

R. R. Parenti, Lincoln Laboratory, Massachusetts Institute of Technology Lexington, Mass. 02173 (personal communication, 1990).

Primmerman, C. A.

Salour, M. M.

M. M. Salour and C. Cohen-Tannoudji, “Observation of Ramsey’s interference fringes in the profile of Doppler-free two-photon resonance,” Phys. Rev. Lett. 38, 757–760 (1977).
[CrossRef]

Schmand, J.

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

Schröder, H. W.

W. Lange, J. Luther, B. Nottbeck, and H. W. Schröder, “High-resolution fluorescence spectroscopy by use of a cw dye laser,” Opt. Commun. 8, 157–159 (1973).
[CrossRef]

Schweighofer, M. G.

S. Feneuille, M. G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 1976–2009 (1976).
[CrossRef]

Shore, B. W.

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990).

Steuter, B.

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

Strang, G.

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[CrossRef]

See, for example, G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, New York, 1980), Sec. 5.5.

Strohmeier, P.

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

Teets, R.

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764(1977).
[CrossRef]

Temkin, R. J.

R. J. Temkin, Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, Mass. 02139 (personal communication, 1990).

Thode, L.

Thomas, G. F.

G. F. Thomas, “Pulse train single-photon induced optical Ramsey fringes,” Phys. Rev. A 35, 5060–5063 (1987).
[CrossRef] [PubMed]

Torrey, H. C.

H. C. Torrey, “Transient nutations in nuclear magnetic resonance,” Phys. Rev. 76, 1059–1068 (1949).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

A. Mooradian, “Use of spatial time-division repetition rate multiplication of mode-locked laser pulses to generate microwave radiation from optoelectronic switches,” Appl. Phys. Lett. 45, 494–496 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. B (1)

S. Feneuille, M. G. Schweighofer, and G. Oliver, “Response of a two-level system to a narrow-band light excitation completely modulated in amplitude,” J. Phys. B 9, 1976–2009 (1976).
[CrossRef]

Lincoln Lab. J. (1)

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

Opt. Commun. (2)

P. Strohmeier, T. Kersebom, E. Krüger, H. Nölle, B. Steuter, J. Schmand, and J. Andrä, “Na-beam deceleration by a mode-locked laser,” Opt. Commun.73, 451–454 (1989).

W. Lange, J. Luther, B. Nottbeck, and H. W. Schröder, “High-resolution fluorescence spectroscopy by use of a cw dye laser,” Opt. Commun. 8, 157–159 (1973).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (2)

H. C. Torrey, “Transient nutations in nuclear magnetic resonance,” Phys. Rev. 76, 1059–1068 (1949).
[CrossRef]

W B. Hawkins, “Orientation and alignment of sodium atoms by means of polarized resonance radiation,” Phys. Rev. 98, 478–486 (1955).
[CrossRef]

Phys. Rev. A (2)

G. F. Thomas, “Pulse train single-photon induced optical Ramsey fringes,” Phys. Rev. A 35, 5060–5063 (1987).
[CrossRef] [PubMed]

M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
[CrossRef]

Phys. Rev. Lett. (2)

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764(1977).
[CrossRef]

M. M. Salour and C. Cohen-Tannoudji, “Observation of Ramsey’s interference fringes in the profile of Doppler-free two-photon resonance,” Phys. Rev. Lett. 38, 757–760 (1977).
[CrossRef]

Prog. Quantum Electron. (1)

A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

Rev. Mod. Phys. (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–283 (1972).
[CrossRef]

SIAM J. Numer. Anal. (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[CrossRef]

Other (10)

See, for example, G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, New York, 1980), Sec. 5.5.

See, for example, L. Allen and J. H. Eberly, Optical Resonances and Two-Level Atoms (Wiley, New York, 1975), Chap. 3.

L. C. Bradley, “Atomic excitation by a pulse train I: two-level atoms”;“Atomic excitation by a pulse train II: sodium,” in Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 66.

T. H. Jeys, “Sum frequency mixing of frequency modulated laser radiation,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper THN3.

The derivation of the optical Bloch equations can be found in many places; see, for example, Ref. 5, Sec. 8.4.

R. R. Parenti, Lincoln Laboratory, Massachusetts Institute of Technology Lexington, Mass. 02173 (personal communication, 1990).

R. J. Temkin, Plasma Fusion Center, Massachusetts Institute of Technology, Cambridge, Mass. 02139 (personal communication, 1990).

T. W. Hänsen, “Multiple coherent interactions,” in Laser Spectroscopy III, J. L. Hall and J. L. Carlsten, eds. (Springer-Verlag, Berlin, 1977) pp. 149–153.

Work on FEL lasers for this purpose has been done by R. A. Temkin of the Massachusetts Institute of Technology and L. Thode of the Los Alamos National Laboratory, among others.

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990).

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Figures (21)

Fig. 1
Fig. 1

Variation of components of w over a single period at steady state for several values of the parameters. Note differences in scale: soliid curves, w1; dashed curves, w 2 / 2; dotted curves, w 3 / 2; ϕ = π/4. (a) PA = 0.3π, (ΩT = 0; (b) PA = 0.3π, (ΩT =π; (c) PA = 2πT = 0; (d) PA = π, ΩT = 0.

Fig. 2
Fig. 2

Mean response of Doppler-broadened ensemble of two-level atoms: PRF, 100 MHz; pulse length, 700 ps. (a) PA = 0.2π, (b) PA = 0.6π, (c) PA = 1.0π, (d) PA = 1.4π, (e) PA = 1.8π.

Fig. 3
Fig. 3

Backscatter as a function of irradiance: pulse length, 700 ps; no phase modulation; PRF, 100 MHz; Doppler-broadened ensemble. The points are for equal increments of 0.48π in pulse area.

Fig. 4
Fig. 4

Mean response of Doppler-broadened ensemble with phase modulation at 30 MHz: amplitude, 0.75π rad; PRF, 100 MHz; pulse length, 700 ps. (a) PA = 0.2π, (b) PA = 0.6π, (c) PA = 1.0π.

Fig. 5
Fig. 5

Backscatter from Maxwellian distribution of two-level atoms as a function of incident irradiance: PRF, 100 MHz; pulse length, 500 ps. The atoms are taken to be fully pumped sodium atoms. The points correspond to equal increments in pulse area, with the last point corresponding to a pulse area of 1.2π. The theoretical limit corresponds to an atom’s spending half of its time in the excited state and radiating in a dipole pattern. Crosses, no phase modulation; circled points, with phase modulation.

Fig. 6
Fig. 6

Example of mean response of a Doppler-broadened ensemble with random phase modulation as described in the text: PRF, 63 MHz; pulse length, 50 ps; PA = 0.4π. Note that the shorter pulse length characteristic of FEL pulses gives greater response in the wings than that in Fig. 4.

Fig. 7
Fig. 7

Effect of light pressure on the two-level atom in the absence of phase modulation. The sharp peaks of the initial excitation cause large oscillations in the radial velocity distribution after 104 pulses, with deep holes at the positions of the original peaks. The small peaks remaining in the excitation are due to inefficient excitation at the peaks of the velocity distribution. Note the overall red shift of the velocity distribution.

Fig. 8
Fig. 8

Effects of light pressure on the two-level atom in the presence of 30-MHz phase modulation. Compared with those in Fig. 7, the oscillations in the initial excitation and in the final velocity distribution are much less pronounced, and the final excitation is quite smooth. The red shift is more pronounced because the overall excitation is more efficient.

Fig. 9
Fig. 9

Time evolution of backscatter as a result of light pressure.

Fig. 10
Fig. 10

Schematic level diagram for the D2 transition in sodium, showing transitions allowed under illumination by right circularly polarized radiation.

Fig. 11
Fig. 11

Doppler-broadened hfs of the sodium D2 transition. The splitting is almost entirely due to the splitting in the 2S1/2 ground state.

Fig. 12
Fig. 12

Populations of four of the upper levels as a function of pulse number: pulse length, 500 ps; PRF, 100 MHz; PA = 0.7π. Single velocity group.

Fig. 13
Fig. 13

Populations of four of the upper levels as a function of time: phase modulation, 30 MHz; pulse length, 500 ps; PRF, 100 MHz; PA = 0.48π. Maxwellian velocity distribution, central frequency at centroid.

Fig. 14
Fig. 14

Populations of four lower levels for the same conditions as in Fig. 13.

Fig. 15
Fig. 15

Populations of four lower levels as a function of time; same conditions as Fig. 14, longer time interval. The 10-pulse oscillations have been averaged out. The central frequency is at centroid.

Fig. 16
Fig. 16

Same as Fig. 15, with the central frequency at the main component.

Fig. 17
Fig. 17

Backscatter as a function of macropulse length for conditions of Figs. 15 and 16, showing greater backscatter for central frequency at the main component.

Fig. 18
Fig. 18

Populations of four lower levels as a function of time: phase modulation, 30 MHz; pulse length, 25 ps; PRF, 100 MHz; PA = 0.47π. Maxwellian velocity distribution; central frequency at centroid.

Fig. 19
Fig. 19

Comparison of backscatter with three pulse lengths: duration of macropulse, 100 μs; PRF, 100 MHz; phase modulation, 30 MHz. Central frequency at main component.

Fig. 20
Fig. 20

Weighted backscatter as a function of peak irradiance; micropulse length 500 ps, macropulse length 100 μs; PRF, 100 MHz; phase modulated at 30 MHz. For the same conditions but with micropulse length 700 ns the curves are almost identical for I0 ≤ 6 W/cm2.

Fig. 21
Fig. 21

Comparison of backscatter as a function of time for illumination with linearly polarized and circularly polarized light; micropulse length, 500 ps; PRF, 100 MHz; phase modulation, 30 MHz; PA = 0.48π; central frequency at centroid.

Tables (3)

Tables Icon

Table 1 Pu (Transpose) × 601/2

Tables Icon

Table 2 Pl (Transpose) × 601/2

Tables Icon

Table 3 G (Branching Ratios × 60)

Equations (39)

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

ρ ˙ = i ( H ρ ρ H ) + decay terms ,
ρ = [ ρ u u ρ u l ρ l u ρ l l ] ,
w ˙ = Aw ,
w = ( ρ u u , 2 1 / 2 ( ρ u l + ρ l u ) , i 2 1 / 2 ( ρ l u ρ u l ) , ρ l l ) T ,
A 1 = γ [ 1 0 0 0 0 ½ 0 0 0 0 ½ 0 1 0 0 0 ] ,
A 2 = Ω [ 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 ] ,
A 3 = 2 1 / 2 U ( t ) [ 0 sin ϕ ( t ) cos ϕ ( t ) 0 sin ϕ ( t ) 0 0 sin ϕ ( t ) cos ϕ ( t ) 0 0 cos ϕ ( t ) 0 sin ϕ ( t ) cos ϕ ( t ) 0 ] .
( w T w ) 1 / 2 = [ i ( w i ) 2 ] 1 / 2 .
ρ = [ ρ u u ρ u l ρ l u ρ l l ] ,
A 1 = γ [ I u 0 0 0 0 ½ I ul 0 0 0 0 ½ I u l 0 G 0 0 0 ] .
A 2 = [ 0 0 0 0 0 0 Ω 0 0 Ω 0 0 0 0 0 0 ] .
A 3 = 2 1 / 2 U ( t ) [ 0 P u sin ϕ ( t ) P u cos ϕ ( t ) 0 P u T sin ϕ ( t ) 0 0 P l T sin ϕ ( t ) P u T cos ϕ ( t ) 0 0 P l T cos ϕ ( t ) 0 P l sin ϕ ( t ) P l cos ϕ ( t ) 0 ] .
w ( t ) = B ( t ) w ( 0 ) .
w ( m T ) = Bw [ ( m 1 ) T ] .
B 1 = exp τ ( A 1 + A 2 ) = exp τ A 1 exp τ A 2 ( since A 1 and A 2 commute ) = [ I u exp ( γ τ ) 0 0 0 0 exp [ ( γ τ ) / 2 ] cos Ω τ exp [ ( γ τ ) / 2 ] sin Ω τ 0 0 exp [ ( γ τ ) / 2 ] sin Ω τ exp [ ( γ τ ) / 2 ] cos Ω τ 0 [ 1 exp ( γ τ ) G 0 0 I 1 ] ,
S ( I 0 ) = 2 π n 0 B [ I ( r ) ] r d r = π n a 2 0 I 0 B ( I ) d I / I photons / s ,
B 2 exp ( t A 1 / 2 ) exp t ( A 2 + A 3 ) exp ( t A 1 / 2 ) .
R 0 = [ 0 1 1 0 ] .
R = [ 0 0 0 0 0 0 0 0 0 0 0 λ 0 0 λ 0 ] ,
exp t R = [ 1 0 0 0 0 1 0 0 0 0 cos λ t sin λ t 0 0 sin λ t cos λ t ] .
t 2 = ( 2 1 / 2 sin θ , cos θ cos ϕ , cos θ , 2 1 / 2 sin ϕ ) T .
( A 2 + A 3 ) [ t 13 t 14 t 23 t 24 t 33 t 34 t 43 t 44 ] = λ [ t 13 t 14 t 23 t 24 t 33 t 34 t 43 t 44 ] R 0 = λ [ t 14 t 13 t 24 t 23 t 34 t 33 t 44 t 43 ] .
( t 3 , t 4 ) = [ 2 1 / 2 cos θ , sin θ cos ϕ , θ sin ϕ , 2 1 / 2 cos θ 0 , sin ϕ , cos ϕ , 0 ] T .
n exp ( t n + 1 / 2 t n + 1 d t A D ) exp t n t n + 1 d t ( A 2 + A H ) ( t n t n + 1 / 2 d t A D ) ,
t n + 1 / 2 t n + 3 / 2 d t A D = 0 ,
T N [ n = 1 N exp ( t R n ) T n 1 T n 1 ] T 0 1
T n 1 T n 1 = [ 1 0 0 0 0 cos ( θ n θ n 1 ) sin ( θ n θ n 1 ) 0 0 sin ( θ n θ n 1 ) cos ( θ n θ n 1 ) 0 0 0 0 1 ] .
1 . ( 1 + 1 + 1 + 1 ) 2 ( M = 3 , M = 2 ) , 2 . ( 2 + 3 + 3 + 2 ) 2 ( M = 2 , M = 1 ) , 3 . ( 3 + 5 + 5 + 2 ) 2 ( M = 1 , M = 0 ) , 4 . ( 4 + 6 + 6 + 2 ) 2 ( M = 0 , M = 1 ) , 5 . ( 3 + 3 + 3 + 1 ) 2 ( M = 1 , M = 2 ) ,
w ˙ = A 2 w ,
w ˙ = A 3 ( t ) w ,
w ( t ) = ( exp t A 2 ) w ( 0 ) ,
w ( t ) = [ exp 0 t d t A 3 ( t ) ] w ( 0 ) .
exp Δ t 2 A 2 { n [ exp n Δ t ( n + 1 ) Δ t d t A 3 ( t ) ] exp Δ t A 2 } exp Δ t 2 A 2 .
( exp Δ t 2 A 2 ) T { Π n [ exp n Δ t ( n + 1 ) Δ t d t R ( t ) ] T 1 ( exp Δ t A 2 ) T } × T 1 ( exp Δ t 2 A 2 ) .
t 1 = ( n u + n 1 ) 1 / 2 ( 1 u T , 0 u l , 0 u l , 1 l T ) T ,
T 2 = [ 0 u l × l I u l cos ϕ I u l sin ϕ 0 l × u l ] ,
( P u T P u + P l T P 1 ) t m = λ 2 t m ,
T i = [ ( 1 / λ i ) P u t m i 0 0 t m i sin ϕ 0 t m i cos ϕ ( 1 / λ i ) P l t m i 0 ] .
[ cos λ i β sin λ i β sin λ i β cos λ i β ]

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