Abstract

The transverse cooling of a beam of sodium atoms in an axisymmetric light field formed by a reflecting axicon is studied. It is shown that transverse cooling leads to a decrease in angular divergence (collimation) of the atomic beam. The transverse velocities of the beam are reduced from 5.5 × 102 to 1.6 × 102 cm/sec, which corresponds to the decrease in effective transverse temperature of the beam from T = 42 to T = 3.3 mK. The spatial and velocity distributions of the atomic beam are calculated numerically. It is found that theory and experiment are in good agreement.

© 1985 Optical Society of America

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References

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  1. W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Nat. Bur. Stand. U.S. Spec. Publ. 653(1983); “Laser-cooled and trapped atoms” Prog. Quantum Electron. 8, 314 (1984).
  2. V. G. Minogin, V. S. Letokhov, Laser Light Pressure on Atoms (Nauka, Moscow, to be published).
  3. V. I. Balykin, V. S. Letokhov, V. I. Mishin, “Observation of cooling of free sodium atoms in resonant laser field with scanning frequency,” Pis’ma Zh. Eksp. Teor. Fiz. 29, 614 (1979); “Cooling of Na atoms by resonant laser radiation,” Zh. Eksp. Teor. Fiz. 78, 1376 (1980).
  4. S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, “Radiative slowing down and monochromatization of a sodium atoms beam in counter-moving laser beam,” Zh. Eksp. Teor. Fiz. 82, 1429 (1982).
  5. W. D. Phillips, H. Metcalf, “Laser deceleration of atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
    [CrossRef]
  6. V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Intense stationary flow of cold atoms formed by laser deceleration of atomic beam,” Opt. Commun. 49, 248 (1984); “Formation of an intense stationary beam of cold atoms by laser deceleration of an atomic beam,” Zh. Eksp. Teor. Fiz. 86, 2019 (1984).
    [CrossRef]
  7. R. Blatt, W. Ertmer, J. L. Hall, “Cooling of an atomic beam with frequency-sweep techniques,” in Laser-Cooled and Trapped Atoms, W. D. Phillips, ed., Nat. Bureau Stand. U.S. Spec. Publ.653(1983), pp. 142–153.
  8. G. I. Budker, A. N. Skrinsky, “The electron cooling and new possibilities in elementary particle physics,” Usp. Fiz. Nauk. 124, 561, (1978); A. Septier, Focusing of Charged Particles (Academic, New York, 1967), Vols. I and II.
  9. V. I. Balykin, V. S. Letokhov, V. G. Minogin, T. V. Zyeva, “Collimation of atomic beams by resonant laser radiation pressure,” Appl. Phys. B 35, 149 (1984).
    [CrossRef]
  10. V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Radiative collimation of atomic beam by laser-induced two-dimensional cooling,” Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984).
  11. T. W. Hänsch, A. W. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68 (1975).
    [CrossRef]
  12. V. S. Letokhov, V. G. Minogin, B. D. Pavlik, “Cooling and trapping of atoms and molecules by resonant light field,” Zh. Eksp. Teor. Fiz. 72, 1328 (1977).
  13. V. G. Minogin, “Kinetic theory of atomic scattering by a resonant standing light wave,” Zh. Eksp. Teor. Fiz. 80, 2231 (1981).
  14. H. R. Gray, R. M. Whitley, C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978).
    [CrossRef] [PubMed]
  15. M. L. Citron, H. R. Cray, C. W. Cabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. 16, 1507 (1977).
    [CrossRef]
  16. V. I. Balykin, A. I. Sidorov, “A high-power two-frequency cw dye laser,” Kvantovaya Elektron. 11, 2001 (1984).
  17. I. E. Björkholm, R. R. Freeman, A. Ashkin, O. B. Pearson, “Experimental observation of the influence of the quantum fluctuations of resonance-radiation pressure,” Opt. Lett. 5, 111 (1980).
    [CrossRef] [PubMed]

1984 (4)

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Intense stationary flow of cold atoms formed by laser deceleration of atomic beam,” Opt. Commun. 49, 248 (1984); “Formation of an intense stationary beam of cold atoms by laser deceleration of an atomic beam,” Zh. Eksp. Teor. Fiz. 86, 2019 (1984).
[CrossRef]

V. I. Balykin, V. S. Letokhov, V. G. Minogin, T. V. Zyeva, “Collimation of atomic beams by resonant laser radiation pressure,” Appl. Phys. B 35, 149 (1984).
[CrossRef]

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Radiative collimation of atomic beam by laser-induced two-dimensional cooling,” Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984).

V. I. Balykin, A. I. Sidorov, “A high-power two-frequency cw dye laser,” Kvantovaya Elektron. 11, 2001 (1984).

1983 (1)

W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Nat. Bur. Stand. U.S. Spec. Publ. 653(1983); “Laser-cooled and trapped atoms” Prog. Quantum Electron. 8, 314 (1984).

1982 (2)

S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, “Radiative slowing down and monochromatization of a sodium atoms beam in counter-moving laser beam,” Zh. Eksp. Teor. Fiz. 82, 1429 (1982).

W. D. Phillips, H. Metcalf, “Laser deceleration of atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

1981 (1)

V. G. Minogin, “Kinetic theory of atomic scattering by a resonant standing light wave,” Zh. Eksp. Teor. Fiz. 80, 2231 (1981).

1980 (1)

1979 (1)

V. I. Balykin, V. S. Letokhov, V. I. Mishin, “Observation of cooling of free sodium atoms in resonant laser field with scanning frequency,” Pis’ma Zh. Eksp. Teor. Fiz. 29, 614 (1979); “Cooling of Na atoms by resonant laser radiation,” Zh. Eksp. Teor. Fiz. 78, 1376 (1980).

1978 (2)

G. I. Budker, A. N. Skrinsky, “The electron cooling and new possibilities in elementary particle physics,” Usp. Fiz. Nauk. 124, 561, (1978); A. Septier, Focusing of Charged Particles (Academic, New York, 1967), Vols. I and II.

H. R. Gray, R. M. Whitley, C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978).
[CrossRef] [PubMed]

1977 (2)

M. L. Citron, H. R. Cray, C. W. Cabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. 16, 1507 (1977).
[CrossRef]

V. S. Letokhov, V. G. Minogin, B. D. Pavlik, “Cooling and trapping of atoms and molecules by resonant light field,” Zh. Eksp. Teor. Fiz. 72, 1328 (1977).

1975 (1)

T. W. Hänsch, A. W. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68 (1975).
[CrossRef]

Andreev, S. V.

S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, “Radiative slowing down and monochromatization of a sodium atoms beam in counter-moving laser beam,” Zh. Eksp. Teor. Fiz. 82, 1429 (1982).

Ashkin, A.

Balykin, V. I.

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Radiative collimation of atomic beam by laser-induced two-dimensional cooling,” Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984).

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Intense stationary flow of cold atoms formed by laser deceleration of atomic beam,” Opt. Commun. 49, 248 (1984); “Formation of an intense stationary beam of cold atoms by laser deceleration of an atomic beam,” Zh. Eksp. Teor. Fiz. 86, 2019 (1984).
[CrossRef]

V. I. Balykin, V. S. Letokhov, V. G. Minogin, T. V. Zyeva, “Collimation of atomic beams by resonant laser radiation pressure,” Appl. Phys. B 35, 149 (1984).
[CrossRef]

V. I. Balykin, A. I. Sidorov, “A high-power two-frequency cw dye laser,” Kvantovaya Elektron. 11, 2001 (1984).

S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, “Radiative slowing down and monochromatization of a sodium atoms beam in counter-moving laser beam,” Zh. Eksp. Teor. Fiz. 82, 1429 (1982).

V. I. Balykin, V. S. Letokhov, V. I. Mishin, “Observation of cooling of free sodium atoms in resonant laser field with scanning frequency,” Pis’ma Zh. Eksp. Teor. Fiz. 29, 614 (1979); “Cooling of Na atoms by resonant laser radiation,” Zh. Eksp. Teor. Fiz. 78, 1376 (1980).

Björkholm, I. E.

Blatt, R.

R. Blatt, W. Ertmer, J. L. Hall, “Cooling of an atomic beam with frequency-sweep techniques,” in Laser-Cooled and Trapped Atoms, W. D. Phillips, ed., Nat. Bureau Stand. U.S. Spec. Publ.653(1983), pp. 142–153.

Budker, G. I.

G. I. Budker, A. N. Skrinsky, “The electron cooling and new possibilities in elementary particle physics,” Usp. Fiz. Nauk. 124, 561, (1978); A. Septier, Focusing of Charged Particles (Academic, New York, 1967), Vols. I and II.

Cabel, C. W.

M. L. Citron, H. R. Cray, C. W. Cabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. 16, 1507 (1977).
[CrossRef]

Citron, M. L.

M. L. Citron, H. R. Cray, C. W. Cabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. 16, 1507 (1977).
[CrossRef]

Cray, H. R.

M. L. Citron, H. R. Cray, C. W. Cabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. 16, 1507 (1977).
[CrossRef]

Ertmer, W.

R. Blatt, W. Ertmer, J. L. Hall, “Cooling of an atomic beam with frequency-sweep techniques,” in Laser-Cooled and Trapped Atoms, W. D. Phillips, ed., Nat. Bureau Stand. U.S. Spec. Publ.653(1983), pp. 142–153.

Freeman, R. R.

Gray, H. R.

Hall, J. L.

R. Blatt, W. Ertmer, J. L. Hall, “Cooling of an atomic beam with frequency-sweep techniques,” in Laser-Cooled and Trapped Atoms, W. D. Phillips, ed., Nat. Bureau Stand. U.S. Spec. Publ.653(1983), pp. 142–153.

Hänsch, T. W.

T. W. Hänsch, A. W. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68 (1975).
[CrossRef]

Letokhov, V. S.

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Radiative collimation of atomic beam by laser-induced two-dimensional cooling,” Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984).

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Intense stationary flow of cold atoms formed by laser deceleration of atomic beam,” Opt. Commun. 49, 248 (1984); “Formation of an intense stationary beam of cold atoms by laser deceleration of an atomic beam,” Zh. Eksp. Teor. Fiz. 86, 2019 (1984).
[CrossRef]

V. I. Balykin, V. S. Letokhov, V. G. Minogin, T. V. Zyeva, “Collimation of atomic beams by resonant laser radiation pressure,” Appl. Phys. B 35, 149 (1984).
[CrossRef]

S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, “Radiative slowing down and monochromatization of a sodium atoms beam in counter-moving laser beam,” Zh. Eksp. Teor. Fiz. 82, 1429 (1982).

V. I. Balykin, V. S. Letokhov, V. I. Mishin, “Observation of cooling of free sodium atoms in resonant laser field with scanning frequency,” Pis’ma Zh. Eksp. Teor. Fiz. 29, 614 (1979); “Cooling of Na atoms by resonant laser radiation,” Zh. Eksp. Teor. Fiz. 78, 1376 (1980).

V. S. Letokhov, V. G. Minogin, B. D. Pavlik, “Cooling and trapping of atoms and molecules by resonant light field,” Zh. Eksp. Teor. Fiz. 72, 1328 (1977).

V. G. Minogin, V. S. Letokhov, Laser Light Pressure on Atoms (Nauka, Moscow, to be published).

Metcalf, H.

W. D. Phillips, H. Metcalf, “Laser deceleration of atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

Minogin, V. G.

V. I. Balykin, V. S. Letokhov, V. G. Minogin, T. V. Zyeva, “Collimation of atomic beams by resonant laser radiation pressure,” Appl. Phys. B 35, 149 (1984).
[CrossRef]

S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, “Radiative slowing down and monochromatization of a sodium atoms beam in counter-moving laser beam,” Zh. Eksp. Teor. Fiz. 82, 1429 (1982).

V. G. Minogin, “Kinetic theory of atomic scattering by a resonant standing light wave,” Zh. Eksp. Teor. Fiz. 80, 2231 (1981).

V. S. Letokhov, V. G. Minogin, B. D. Pavlik, “Cooling and trapping of atoms and molecules by resonant light field,” Zh. Eksp. Teor. Fiz. 72, 1328 (1977).

V. G. Minogin, V. S. Letokhov, Laser Light Pressure on Atoms (Nauka, Moscow, to be published).

Mishin, V. I.

V. I. Balykin, V. S. Letokhov, V. I. Mishin, “Observation of cooling of free sodium atoms in resonant laser field with scanning frequency,” Pis’ma Zh. Eksp. Teor. Fiz. 29, 614 (1979); “Cooling of Na atoms by resonant laser radiation,” Zh. Eksp. Teor. Fiz. 78, 1376 (1980).

Pavlik, B. D.

V. S. Letokhov, V. G. Minogin, B. D. Pavlik, “Cooling and trapping of atoms and molecules by resonant light field,” Zh. Eksp. Teor. Fiz. 72, 1328 (1977).

Pearson, O. B.

Phillips, W. D.

W. D. Phillips, H. Metcalf, “Laser deceleration of atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

Schawlow, A. W.

T. W. Hänsch, A. W. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68 (1975).
[CrossRef]

Sidorov, A. I.

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Radiative collimation of atomic beam by laser-induced two-dimensional cooling,” Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984).

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Intense stationary flow of cold atoms formed by laser deceleration of atomic beam,” Opt. Commun. 49, 248 (1984); “Formation of an intense stationary beam of cold atoms by laser deceleration of an atomic beam,” Zh. Eksp. Teor. Fiz. 86, 2019 (1984).
[CrossRef]

V. I. Balykin, A. I. Sidorov, “A high-power two-frequency cw dye laser,” Kvantovaya Elektron. 11, 2001 (1984).

Skrinsky, A. N.

G. I. Budker, A. N. Skrinsky, “The electron cooling and new possibilities in elementary particle physics,” Usp. Fiz. Nauk. 124, 561, (1978); A. Septier, Focusing of Charged Particles (Academic, New York, 1967), Vols. I and II.

Stroud, C. R.

H. R. Gray, R. M. Whitley, C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218 (1978).
[CrossRef] [PubMed]

M. L. Citron, H. R. Cray, C. W. Cabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. 16, 1507 (1977).
[CrossRef]

Whitley, R. M.

Zyeva, T. V.

V. I. Balykin, V. S. Letokhov, V. G. Minogin, T. V. Zyeva, “Collimation of atomic beams by resonant laser radiation pressure,” Appl. Phys. B 35, 149 (1984).
[CrossRef]

Appl. Phys. B (1)

V. I. Balykin, V. S. Letokhov, V. G. Minogin, T. V. Zyeva, “Collimation of atomic beams by resonant laser radiation pressure,” Appl. Phys. B 35, 149 (1984).
[CrossRef]

Kvantovaya Elektron. (1)

V. I. Balykin, A. I. Sidorov, “A high-power two-frequency cw dye laser,” Kvantovaya Elektron. 11, 2001 (1984).

Laser-Cooled and Trapped Atoms (1)

W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Nat. Bur. Stand. U.S. Spec. Publ. 653(1983); “Laser-cooled and trapped atoms” Prog. Quantum Electron. 8, 314 (1984).

Opt. Commun. (2)

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Intense stationary flow of cold atoms formed by laser deceleration of atomic beam,” Opt. Commun. 49, 248 (1984); “Formation of an intense stationary beam of cold atoms by laser deceleration of an atomic beam,” Zh. Eksp. Teor. Fiz. 86, 2019 (1984).
[CrossRef]

T. W. Hänsch, A. W. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68 (1975).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

M. L. Citron, H. R. Cray, C. W. Cabel, C. R. Stroud, “Experimental study of power broadening in a two-level atom,” Phys. Rev. 16, 1507 (1977).
[CrossRef]

Phys. Rev. Lett. (1)

W. D. Phillips, H. Metcalf, “Laser deceleration of atomic beam,” Phys. Rev. Lett. 48, 596 (1982).
[CrossRef]

Pis’ma Zh. Eksp. Teor. Fiz. (2)

V. I. Balykin, V. S. Letokhov, V. I. Mishin, “Observation of cooling of free sodium atoms in resonant laser field with scanning frequency,” Pis’ma Zh. Eksp. Teor. Fiz. 29, 614 (1979); “Cooling of Na atoms by resonant laser radiation,” Zh. Eksp. Teor. Fiz. 78, 1376 (1980).

V. I. Balykin, V. S. Letokhov, A. I. Sidorov, “Radiative collimation of atomic beam by laser-induced two-dimensional cooling,” Pis’ma Zh. Eksp. Teor. Fiz. 40, 251 (1984).

Usp. Fiz. Nauk. (1)

G. I. Budker, A. N. Skrinsky, “The electron cooling and new possibilities in elementary particle physics,” Usp. Fiz. Nauk. 124, 561, (1978); A. Septier, Focusing of Charged Particles (Academic, New York, 1967), Vols. I and II.

Zh. Eksp. Teor. Fiz. (3)

S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, “Radiative slowing down and monochromatization of a sodium atoms beam in counter-moving laser beam,” Zh. Eksp. Teor. Fiz. 82, 1429 (1982).

V. S. Letokhov, V. G. Minogin, B. D. Pavlik, “Cooling and trapping of atoms and molecules by resonant light field,” Zh. Eksp. Teor. Fiz. 72, 1328 (1977).

V. G. Minogin, “Kinetic theory of atomic scattering by a resonant standing light wave,” Zh. Eksp. Teor. Fiz. 80, 2231 (1981).

Other (2)

V. G. Minogin, V. S. Letokhov, Laser Light Pressure on Atoms (Nauka, Moscow, to be published).

R. Blatt, W. Ertmer, J. L. Hall, “Cooling of an atomic beam with frequency-sweep techniques,” in Laser-Cooled and Trapped Atoms, W. D. Phillips, ed., Nat. Bureau Stand. U.S. Spec. Publ.653(1983), pp. 142–153.

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Figures (13)

Fig. 1
Fig. 1

The scheme of radiative collimation of an atomic beam. a: 1, Source of atoms; 2, atomic beam; 3, collimating radiation; 4, exicon. b: Velocity-distribution narrowing by collimation.

Fig. 2
Fig. 2

Structure of the D1 line of a Na atom. a, The mode frequencies red shifted about the transition frequencies. b, The transitions between the magnetic sublevels of the ground and excited states for the case of linearly polarized laser radiation. c, The case of circularly polarized radiation.

Fig. 3
Fig. 3

The structure of the D2 line of a Na atom. a, The mode frequencies red shifted about the transition frequencies. b, The transitions between the magnetic sublevels in the case of linearly polarized laser radiation.

Fig. 4
Fig. 4

The dependence of the fluorescence signal of Na atoms excited by the schemes in Fig. 2 (curve 1) and Fig. 3 (curve 2) on laser-radiation intensity. G = I/Isat. Isat is the atomic-transition saturation intensity.

Fig. 5
Fig. 5

Scheme of an experimental setup for radiative collimation of atomic beam: 1, two-mode dye laser; 2, probe single-mode laser; PM’s, photomultipliers; P, prism; M, mechanical modulator.

Fig. 6
Fig. 6

Detection of the spatial distribution of an atomic beam exposed to single-mode radiation in an axicon. The beam is not narrowed (no collimation). An increase in signal is due to optical pumping to the sublevel F = 2.

Fig. 7
Fig. 7

Atomic-beam collimation by laser radiation. Curves 1 and 2 denote the spatial distribution of atoms at the sublevel F = 2 and at the both sublevels, respectively; curve 3 is the distribution of atoms in a beam after it is irradiated by laser field.

Fig. 8
Fig. 8

The dependence of the atomic intensity in the center of an atomic beam on laser-frequency detuning at different values of the collimating laser power.

Fig. 9
Fig. 9

The dependence of the optimal detuning of collimating laser frequency on laser intensity.

Fig. 10
Fig. 10

Atomic-beam decollimation.

Fig. 11
Fig. 11

The dependence of the atomic intensity in the center of a beam on the longitudinal velocity of the atoms being collimated.

Fig. 12
Fig. 12

The relative position of 1, the source of the atoms; 2, the diaphragm; and 3, the axicon. 0102 is the axis of symmetry. 0′ is the point source shifted off the axis 0102.

Fig. 13
Fig. 13

The spatial and velocity distributions of atoms in a beam for P = 45 mW and Δν = −3.8γ. 1, The theoretical spatial and velocity distributions of atoms in the detection region in the absence of collimating field. 2, The theoretical spatial distribution of atoms in the detection region after the beam has interacted with the collimating field. 3, The experimental spatial distribution of atoms in the detection region after the beam has interacted with the collimating field. 4, The theoretical velocity distribution of atoms after the collimating field is switched on.

Equations (9)

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

F = e ^ ρ k γ G ( L + - L - ) / [ 1 + G ( L + + L - ) ] .
L ± = γ 2 / [ ( Ω ± k v ρ ) 2 + γ 2 ] ,
F = - β m v ρ ,
β = 4 k 2 m | Ω γ | G ( 1 + Ω 2 / γ 2 ) - 1 ( 1 + Ω 2 / Ω 2 + 2 G ) - 1
T min = γ ( γ / Ω + Ω / γ ) / 2 k B ,
Δ φ min = ( 2 k B T min ) 1 / 2 / v ¯ z m 1 / 2 = ( γ / m ) 1 / 2 / v ¯ z .
Δ φ min = λ ( γ / e v ¯ z ) 1 / 2 .
t w + v r w = - p ( F w ) ,
v ˙ ρ = F ( ρ , v ρ ) / m + v t 2 / ρ ,             v ˙ t = - v ρ v t / ρ , v ˙ z = 0 ,             v ρ = ρ ˙ ,             v t = φ ˙ ρ ,             v z = z ˙ .

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