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Quantum Fluctuations and Transverse Effects in Superfluorescence

  • F. P. Mattar
  • E. A. Watson
  • H. M. Gibbs
  • H. M. Cormier
  • Y. Claude
  • S. L. Mccall
  • M. S. Feld
Conference paper

Abstract

Superfluorescence[1] (SF) emission profiles are computed using one-way coupled Maxwell-Bloch equations. Transverse effects[2] are included in the full three-spatial-dimension case as well as in the cylindrical-symmetry case. Initiating quantum fluctuations[3–5] are approximated by a random polarization source with a completely random phase and root-mean-square tipping angle of 2/ N, where N is the number of atoms in each volume element. These fluctuations reduce the tail of the output obtained with transverse effects alone[6]. In fact, the fluctuations in output pulse shapes encompass the Cs data of Gibbs, Vrehen and Hikspoors[7]; see Fig. 1. The standard deviation for the delay time is found to be (12.5±4)% for Fresnel number of 0.8 compared with the value (10±2)% recently measured by Vrehen and der Weduwe[8], also for Cs. The Fresnel-number dependence of the standard deviation is shown in Fig. 2; Drummond and Eberly[9] have more extensive results for Fresnel numbers from 1 to 16. Excluded from Fig. 2 were phase-wave[10] fluctuations in which the second peak exceeded the first. Figure 3 shows that the transverse profiles retain little of the on-axis ringing found[2] without fluctuations. Ringing can also be removed by inhomogeneous broadening, but the 32-ns T2* of the Cs experiment has little effect; T2* < 1 ns is needed to eliminate plane-wave ringing[6].

Fig. 1

Intensity integrated over the transverse cylindri?cal coordinate as a function of time for single trajector?ies. (a) Cs data for n 0 0 ≅ 7.6×10 cm−3. (b) Simulation with transverse effects, but no fluctuations: n 0 0 = 1.82 × 1010 cm-3, λ0 = 1.37×10−4 rad, and Fresnel number F=1. (c)−(f) Simulations with transverse effects and fluctuations fol. n 0 0 = 18.2×1010 cm−3, <λ 0 2 > −1/2 = 1.37×10−4 rad, and F=1

Fig. 2

Fresnel-number dependence of the uncertainty in delay time normalized to the average delay. Points are as follows: seven trajectories with n 0 0 = 9.5×100 cm−3 and <λ 0 2 >1/2= 1.89×10−4 rad;o ,n 0 0 = 18×1010 cm−3 and <λ 0 2 >1/2= 1.37 × 10−4 rad, for 13 trajectories for F=1 and for 16 trajectories for F =ύ-1; ∆, experimental value for 468 trajectories. A peak close to F=1 can be argued as follows: for small F, strong diffractive coup?ling reduces fluctuations in the overall output. For large F, so many transverse modes compete that a good average is obtained on every shot. For F≅1, competition of a few modes is maximal, resulting in large fluctuations

Fig. 3

Transverse energy current JT and intensity are plotted iso?metrically for four shots in a statistical ensemble. In some of the shots the phase curvature is such that the associated energy flux flows inwardly, i.e., the transverse energy current is negative, which could lead to self-focusing. Inward energy flow never occurred for simulations using a homogeneous initial tipping angle (without quantum initiation) for any value of the Fresnel number. Here n 0 0 = 9.5×1010 cm−3, F = 1.49, and 104 0 2 >1/2= 2.15, 1.63, 1.79 and 1.16 rad, respectively, from top to bottom.

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References

  1. 1.
    Reviews of SF: M.S. Feld and J.C. MacGillivray, in Coherent Nonlinear Optics, ed. by M. S. Feld and V. S. Letokhov (Springer, Berlin, 1980); M. F. H. Schuurmans, Q. H. F. Vrehen, D. Polder, and H. M. Gibbs, in Advances in Atomic and Molecular Physics, ed. by D. R. Bates and B. Bederson ( Academic, New York, 1981 ) p. 168.Google Scholar
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    For more details see present authors, Phys. Rev. A, to be published (March 1983); F. P. Mattar, ARO Workshop on Coupled Nonlinear Oscillators, Los Alamos National Laboratory, Los Alamos, New Mexico and Los Alamos Nonlinear Optics Conference Proc., ( SPIE, April 1981 ).Google Scholar
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    H. M. Gibbs, Q. H. F. Vrehen and H. M. J. Hikspoors, Phys. Rev. Lett. 39, 547 (1977).ADSCrossRefGoogle Scholar
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Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • F. P. Mattar
    • 1
  • E. A. Watson
    • 2
  • H. M. Gibbs
    • 2
  • H. M. Cormier
    • 3
  • Y. Claude
    • 3
  • S. L. Mccall
    • 4
  • M. S. Feld
    • 5
  1. 1.Polytechnic Institute of New YorkBrooklyn, N.Y. and M.I.T.CambridgeUSA
  2. 2.University of ArizonaTucsonUSA
  3. 3.Université de MontréalMontréalCanada
  4. 4.Bell LaboratoriesMurray HillUSA
  5. 5.M.I.T.CambridgeUSA

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