Abstract

We describe the quantum initiation for free-electron lasers and derive the collective instability threshold starting from the electron position and the momentum quantum fluctuations. We obtain the Glauber distribution P(α) for the field, that is, a displaced Gaussian, which represents the superposition of a coherent field (stimulated emission) and an incoherent one (spontaneous emission). We define and discuss the concept of superradiance for free-electron lasers, both in a classical framework and at a quantum statistical level, stating the conditions under which the effect should be observed.

© 1985 Optical Society of America

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  1. S. F. Jacobs, G. T. Moore, H. S. Pilloff, M. Sargent, M. O. Scully, R. Spitzer, eds., Free-Electron Generators of Coherent Radiation (Addison-Wesley, Reading, Mass., 1982);S. Martellucci, N. A. Chester, eds., Free Electron Lasers (Plenum, New York, 1983).
  2. L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
    [CrossRef]
  3. J. R. Pierce, Traveling Wave Tubes (Van Nostrand, New York, 1950).
  4. N. M. Kroll, W. A. McMullin, “Stimulated emission from relativistic electrons passing through a spatially periodic transverse magnetic field,” Phys. Rev. A 17, 300–308 (1978);I. B. Bernstein, J. L. Hirschfield, “Amplification on a relativistic electron beam in a spatially periodic transverse magnetic field,” Phys. Rev. A 20, 1661–1670 (1979);P. Sprangle, C. M. Tang, W. H. Manheimer, “Nonlinear theory of free-electron lasers and efficiency enhancement,” Phys. Rev. A 21, 302–308 (1980);A. Gover, P. Sprangle, “A unified theory of magnetic bremmstrahlung, electrostatic bremsstrahlung, Compton–Raman scattering and Cherenkov–Smith–Purcell free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1196–1215 (1981);G. Dattoli, A. Marino, A. Renieri, F. Romanelli, “Progress in the Hamiltonian picture of the free-electron laser,” IEEE J. Quantum Electron. QE-17, 1371–1386 (1981);C. C. Shih, A. Yariv, “Inclusion of space-charge effects with Maxwell’s equations in the single-particle analysis of free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1387–1394 (1981).Instabilities of a different kind with respect to the collective instability considered here have been discussed, e.g., in W. B. Colson, R. A. Friedman, “Synchrotron instability for long pulses in free electron laser oscillators,” Opt. Commun. 46, 37–42 (1983);V. A. Buts, V. V. Ognivenko, “Stochastic instability of the motion of particles in free-electron lasers,” JETP Lett. 38, 525–528 (1983).
    [CrossRef]
  5. V. L. Bratman, N. S. Ginzburg, M. I. Petelin, “Nonlinear theory of stimulated wave scattering by relativistic electron beams,” Sov. Phys. JETP 49, 469–475 (1979);L. A. Vainshtein, “Type-O relativistic electron devices,” Sov. Phys. Tech. Phys. 24, 625–633 (1979);Y. L. Bolomolov, V. L. Bratman, N. S. Ginzburg, M. I. Petelin, A. D. Yanakovsky, “Nonstationary generation in free-electron lasers,” Opt. Commun. 36, 209–212 (1981).
    [CrossRef]
  6. R. Bonifacio, C. Pellegrini, L. M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun. 50, 373 (1984).
    [CrossRef]
  7. J. B. Murphy, C. Pellegrini, R. Bonifacio, “Collective instability of a free-electron laser including space charge and harmonics,” Brookhaven National Laboratory Rep. BNL-34156 (1984).
  8. J M. J. Madey, “Simulated emission of bremsstrahlung in a periodic magnetic field,” J. Appl. Phys. 42, 1906–1913 (1971).
    [CrossRef]
  9. F. Hopf, P. Meystre, G. T. Moore, M. O. Scully, “Nonlinear theory of free-electron devices,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 41–114;N. M. Kroll, “The free-electron laser as a traveling-wave amplifier,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 115–156;W. B. Colson, “One-body analysis of free-electron lasers,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 157–196.
  10. G. Dattoli, A. Renieri, “FEL quantum aspects,” in J. Phys. (Paris) 44, 126–136 (1983);W. Becker, J. K. McIver, “Quantum description of free-electron lasers,” J. Phys. (Paris) 44, 289–311 (1983).
    [CrossRef]
  11. R. Bonifacio, F. Casagrande, “Instabilities and quantum initiation in the free-electron laser,” Opt. Commun. 50, 251–255 (1984).
    [CrossRef]
  12. A classical analysis of the FEL oscillator startup has been carried out in Ref. 13 that introduces an incoherent contribution to the driving current in the Maxwell equations.
  13. P. Sprangle, C. M. Tang, I. B. Bernstein, “Initiation of a pulsed-beam free-electron-laser oscillator,” Phys. Rev. Lett. 50, 1775–1778 (1983);“Evolution of spontaneous and coherent radiation in the free-electron laser oscillator,” Phys. Rev. A 28, 2300–2309 (1983).
    [CrossRef]
  14. J. M. J. Madey, D. A. G. Deacon, “Free electron lasers,” in Cooperative Effects in Matter and Radiation, C. M. Bowden, D. W. Howgate, H. R. Robl, eds. (Plenum, New York, 1977), pp. 313–334.
    [CrossRef]
  15. R. Bonifacio, F. Casagrande, “A model for superradiance and superfluorescence in free-electron lasers,” Lett. Nuovo Cimento 37, 39–47 (1983).
  16. C. Pellegrini, “Physics of the free-electron laser,” in Free-Electron Lasers, S. Martellucci, N. A. Chester, eds. (Plenum, New York, 1983), pp. 91–119;W. B. Colson, “Free-electron wave and particle dynamics,” Free-Electron Lasers, S. Martellucci, N. A. Chester, eds. (Plenum, New York, 1983), pp. 189–209.
  17. R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic transition of a free-electron laser Hamiltonian model,” Opt. Commun. 40, 219–223 (1982).
    [CrossRef]
  18. R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic effects in a Hamiltonian model of the free-electron laser,” in Evolution of Order and Chaos, H. Haken, ed. (Springer-VerlagBerlin, 1982).
    [CrossRef]
  19. R. Bonifacio, F. Casagrande, G. Casati, S. Celi, “Chaotic and cooperative effects in a free-electron laser Hamiltonian model,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 801–810.
  20. A. Bambini, A. Renieri, “The free electron laser: a single particle classical model,” Lett. Nuovo Cimento 21, 399–404 (1978).
    [CrossRef]
  21. W. Becker, J. K. McIver, “Fully quantized many-particle theory of a free-electron laser,” Phys. Rev. A 27, 1030–1043 (1983).
    [CrossRef]
  22. R. J. Glauber, in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp 1–43.

1984 (2)

R. Bonifacio, C. Pellegrini, L. M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun. 50, 373 (1984).
[CrossRef]

R. Bonifacio, F. Casagrande, “Instabilities and quantum initiation in the free-electron laser,” Opt. Commun. 50, 251–255 (1984).
[CrossRef]

1983 (4)

P. Sprangle, C. M. Tang, I. B. Bernstein, “Initiation of a pulsed-beam free-electron-laser oscillator,” Phys. Rev. Lett. 50, 1775–1778 (1983);“Evolution of spontaneous and coherent radiation in the free-electron laser oscillator,” Phys. Rev. A 28, 2300–2309 (1983).
[CrossRef]

R. Bonifacio, F. Casagrande, “A model for superradiance and superfluorescence in free-electron lasers,” Lett. Nuovo Cimento 37, 39–47 (1983).

G. Dattoli, A. Renieri, “FEL quantum aspects,” in J. Phys. (Paris) 44, 126–136 (1983);W. Becker, J. K. McIver, “Quantum description of free-electron lasers,” J. Phys. (Paris) 44, 289–311 (1983).
[CrossRef]

W. Becker, J. K. McIver, “Fully quantized many-particle theory of a free-electron laser,” Phys. Rev. A 27, 1030–1043 (1983).
[CrossRef]

1982 (1)

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic transition of a free-electron laser Hamiltonian model,” Opt. Commun. 40, 219–223 (1982).
[CrossRef]

1979 (1)

V. L. Bratman, N. S. Ginzburg, M. I. Petelin, “Nonlinear theory of stimulated wave scattering by relativistic electron beams,” Sov. Phys. JETP 49, 469–475 (1979);L. A. Vainshtein, “Type-O relativistic electron devices,” Sov. Phys. Tech. Phys. 24, 625–633 (1979);Y. L. Bolomolov, V. L. Bratman, N. S. Ginzburg, M. I. Petelin, A. D. Yanakovsky, “Nonstationary generation in free-electron lasers,” Opt. Commun. 36, 209–212 (1981).
[CrossRef]

1978 (2)

A. Bambini, A. Renieri, “The free electron laser: a single particle classical model,” Lett. Nuovo Cimento 21, 399–404 (1978).
[CrossRef]

N. M. Kroll, W. A. McMullin, “Stimulated emission from relativistic electrons passing through a spatially periodic transverse magnetic field,” Phys. Rev. A 17, 300–308 (1978);I. B. Bernstein, J. L. Hirschfield, “Amplification on a relativistic electron beam in a spatially periodic transverse magnetic field,” Phys. Rev. A 20, 1661–1670 (1979);P. Sprangle, C. M. Tang, W. H. Manheimer, “Nonlinear theory of free-electron lasers and efficiency enhancement,” Phys. Rev. A 21, 302–308 (1980);A. Gover, P. Sprangle, “A unified theory of magnetic bremmstrahlung, electrostatic bremsstrahlung, Compton–Raman scattering and Cherenkov–Smith–Purcell free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1196–1215 (1981);G. Dattoli, A. Marino, A. Renieri, F. Romanelli, “Progress in the Hamiltonian picture of the free-electron laser,” IEEE J. Quantum Electron. QE-17, 1371–1386 (1981);C. C. Shih, A. Yariv, “Inclusion of space-charge effects with Maxwell’s equations in the single-particle analysis of free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1387–1394 (1981).Instabilities of a different kind with respect to the collective instability considered here have been discussed, e.g., in W. B. Colson, R. A. Friedman, “Synchrotron instability for long pulses in free electron laser oscillators,” Opt. Commun. 46, 37–42 (1983);V. A. Buts, V. V. Ognivenko, “Stochastic instability of the motion of particles in free-electron lasers,” JETP Lett. 38, 525–528 (1983).
[CrossRef]

1976 (1)

L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

1971 (1)

J M. J. Madey, “Simulated emission of bremsstrahlung in a periodic magnetic field,” J. Appl. Phys. 42, 1906–1913 (1971).
[CrossRef]

Bambini, A.

A. Bambini, A. Renieri, “The free electron laser: a single particle classical model,” Lett. Nuovo Cimento 21, 399–404 (1978).
[CrossRef]

Becker, W.

W. Becker, J. K. McIver, “Fully quantized many-particle theory of a free-electron laser,” Phys. Rev. A 27, 1030–1043 (1983).
[CrossRef]

Bernstein, I. B.

P. Sprangle, C. M. Tang, I. B. Bernstein, “Initiation of a pulsed-beam free-electron-laser oscillator,” Phys. Rev. Lett. 50, 1775–1778 (1983);“Evolution of spontaneous and coherent radiation in the free-electron laser oscillator,” Phys. Rev. A 28, 2300–2309 (1983).
[CrossRef]

Bonifacio, R.

R. Bonifacio, C. Pellegrini, L. M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun. 50, 373 (1984).
[CrossRef]

R. Bonifacio, F. Casagrande, “Instabilities and quantum initiation in the free-electron laser,” Opt. Commun. 50, 251–255 (1984).
[CrossRef]

R. Bonifacio, F. Casagrande, “A model for superradiance and superfluorescence in free-electron lasers,” Lett. Nuovo Cimento 37, 39–47 (1983).

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic transition of a free-electron laser Hamiltonian model,” Opt. Commun. 40, 219–223 (1982).
[CrossRef]

R. Bonifacio, F. Casagrande, G. Casati, S. Celi, “Chaotic and cooperative effects in a free-electron laser Hamiltonian model,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 801–810.

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic effects in a Hamiltonian model of the free-electron laser,” in Evolution of Order and Chaos, H. Haken, ed. (Springer-VerlagBerlin, 1982).
[CrossRef]

J. B. Murphy, C. Pellegrini, R. Bonifacio, “Collective instability of a free-electron laser including space charge and harmonics,” Brookhaven National Laboratory Rep. BNL-34156 (1984).

Bratman, V. L.

V. L. Bratman, N. S. Ginzburg, M. I. Petelin, “Nonlinear theory of stimulated wave scattering by relativistic electron beams,” Sov. Phys. JETP 49, 469–475 (1979);L. A. Vainshtein, “Type-O relativistic electron devices,” Sov. Phys. Tech. Phys. 24, 625–633 (1979);Y. L. Bolomolov, V. L. Bratman, N. S. Ginzburg, M. I. Petelin, A. D. Yanakovsky, “Nonstationary generation in free-electron lasers,” Opt. Commun. 36, 209–212 (1981).
[CrossRef]

Casagrande, F.

R. Bonifacio, F. Casagrande, “Instabilities and quantum initiation in the free-electron laser,” Opt. Commun. 50, 251–255 (1984).
[CrossRef]

R. Bonifacio, F. Casagrande, “A model for superradiance and superfluorescence in free-electron lasers,” Lett. Nuovo Cimento 37, 39–47 (1983).

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic transition of a free-electron laser Hamiltonian model,” Opt. Commun. 40, 219–223 (1982).
[CrossRef]

R. Bonifacio, F. Casagrande, G. Casati, S. Celi, “Chaotic and cooperative effects in a free-electron laser Hamiltonian model,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 801–810.

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic effects in a Hamiltonian model of the free-electron laser,” in Evolution of Order and Chaos, H. Haken, ed. (Springer-VerlagBerlin, 1982).
[CrossRef]

Casati, G.

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic transition of a free-electron laser Hamiltonian model,” Opt. Commun. 40, 219–223 (1982).
[CrossRef]

R. Bonifacio, F. Casagrande, G. Casati, S. Celi, “Chaotic and cooperative effects in a free-electron laser Hamiltonian model,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 801–810.

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic effects in a Hamiltonian model of the free-electron laser,” in Evolution of Order and Chaos, H. Haken, ed. (Springer-VerlagBerlin, 1982).
[CrossRef]

Celi, S.

R. Bonifacio, F. Casagrande, G. Casati, S. Celi, “Chaotic and cooperative effects in a free-electron laser Hamiltonian model,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 801–810.

Dattoli, G.

G. Dattoli, A. Renieri, “FEL quantum aspects,” in J. Phys. (Paris) 44, 126–136 (1983);W. Becker, J. K. McIver, “Quantum description of free-electron lasers,” J. Phys. (Paris) 44, 289–311 (1983).
[CrossRef]

Deacon, D. A. G.

J. M. J. Madey, D. A. G. Deacon, “Free electron lasers,” in Cooperative Effects in Matter and Radiation, C. M. Bowden, D. W. Howgate, H. R. Robl, eds. (Plenum, New York, 1977), pp. 313–334.
[CrossRef]

Elias, L. R.

L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Fairbank, W. M.

L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Ginzburg, N. S.

V. L. Bratman, N. S. Ginzburg, M. I. Petelin, “Nonlinear theory of stimulated wave scattering by relativistic electron beams,” Sov. Phys. JETP 49, 469–475 (1979);L. A. Vainshtein, “Type-O relativistic electron devices,” Sov. Phys. Tech. Phys. 24, 625–633 (1979);Y. L. Bolomolov, V. L. Bratman, N. S. Ginzburg, M. I. Petelin, A. D. Yanakovsky, “Nonstationary generation in free-electron lasers,” Opt. Commun. 36, 209–212 (1981).
[CrossRef]

Glauber, R. J.

R. J. Glauber, in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp 1–43.

Hopf, F.

F. Hopf, P. Meystre, G. T. Moore, M. O. Scully, “Nonlinear theory of free-electron devices,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 41–114;N. M. Kroll, “The free-electron laser as a traveling-wave amplifier,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 115–156;W. B. Colson, “One-body analysis of free-electron lasers,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 157–196.

Kroll, N. M.

N. M. Kroll, W. A. McMullin, “Stimulated emission from relativistic electrons passing through a spatially periodic transverse magnetic field,” Phys. Rev. A 17, 300–308 (1978);I. B. Bernstein, J. L. Hirschfield, “Amplification on a relativistic electron beam in a spatially periodic transverse magnetic field,” Phys. Rev. A 20, 1661–1670 (1979);P. Sprangle, C. M. Tang, W. H. Manheimer, “Nonlinear theory of free-electron lasers and efficiency enhancement,” Phys. Rev. A 21, 302–308 (1980);A. Gover, P. Sprangle, “A unified theory of magnetic bremmstrahlung, electrostatic bremsstrahlung, Compton–Raman scattering and Cherenkov–Smith–Purcell free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1196–1215 (1981);G. Dattoli, A. Marino, A. Renieri, F. Romanelli, “Progress in the Hamiltonian picture of the free-electron laser,” IEEE J. Quantum Electron. QE-17, 1371–1386 (1981);C. C. Shih, A. Yariv, “Inclusion of space-charge effects with Maxwell’s equations in the single-particle analysis of free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1387–1394 (1981).Instabilities of a different kind with respect to the collective instability considered here have been discussed, e.g., in W. B. Colson, R. A. Friedman, “Synchrotron instability for long pulses in free electron laser oscillators,” Opt. Commun. 46, 37–42 (1983);V. A. Buts, V. V. Ognivenko, “Stochastic instability of the motion of particles in free-electron lasers,” JETP Lett. 38, 525–528 (1983).
[CrossRef]

Madey, J M. J.

J M. J. Madey, “Simulated emission of bremsstrahlung in a periodic magnetic field,” J. Appl. Phys. 42, 1906–1913 (1971).
[CrossRef]

Madey, J. M. J.

L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

J. M. J. Madey, D. A. G. Deacon, “Free electron lasers,” in Cooperative Effects in Matter and Radiation, C. M. Bowden, D. W. Howgate, H. R. Robl, eds. (Plenum, New York, 1977), pp. 313–334.
[CrossRef]

McIver, J. K.

W. Becker, J. K. McIver, “Fully quantized many-particle theory of a free-electron laser,” Phys. Rev. A 27, 1030–1043 (1983).
[CrossRef]

McMullin, W. A.

N. M. Kroll, W. A. McMullin, “Stimulated emission from relativistic electrons passing through a spatially periodic transverse magnetic field,” Phys. Rev. A 17, 300–308 (1978);I. B. Bernstein, J. L. Hirschfield, “Amplification on a relativistic electron beam in a spatially periodic transverse magnetic field,” Phys. Rev. A 20, 1661–1670 (1979);P. Sprangle, C. M. Tang, W. H. Manheimer, “Nonlinear theory of free-electron lasers and efficiency enhancement,” Phys. Rev. A 21, 302–308 (1980);A. Gover, P. Sprangle, “A unified theory of magnetic bremmstrahlung, electrostatic bremsstrahlung, Compton–Raman scattering and Cherenkov–Smith–Purcell free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1196–1215 (1981);G. Dattoli, A. Marino, A. Renieri, F. Romanelli, “Progress in the Hamiltonian picture of the free-electron laser,” IEEE J. Quantum Electron. QE-17, 1371–1386 (1981);C. C. Shih, A. Yariv, “Inclusion of space-charge effects with Maxwell’s equations in the single-particle analysis of free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1387–1394 (1981).Instabilities of a different kind with respect to the collective instability considered here have been discussed, e.g., in W. B. Colson, R. A. Friedman, “Synchrotron instability for long pulses in free electron laser oscillators,” Opt. Commun. 46, 37–42 (1983);V. A. Buts, V. V. Ognivenko, “Stochastic instability of the motion of particles in free-electron lasers,” JETP Lett. 38, 525–528 (1983).
[CrossRef]

Meystre, P.

F. Hopf, P. Meystre, G. T. Moore, M. O. Scully, “Nonlinear theory of free-electron devices,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 41–114;N. M. Kroll, “The free-electron laser as a traveling-wave amplifier,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 115–156;W. B. Colson, “One-body analysis of free-electron lasers,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 157–196.

Moore, G. T.

F. Hopf, P. Meystre, G. T. Moore, M. O. Scully, “Nonlinear theory of free-electron devices,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 41–114;N. M. Kroll, “The free-electron laser as a traveling-wave amplifier,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 115–156;W. B. Colson, “One-body analysis of free-electron lasers,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 157–196.

Murphy, J. B.

J. B. Murphy, C. Pellegrini, R. Bonifacio, “Collective instability of a free-electron laser including space charge and harmonics,” Brookhaven National Laboratory Rep. BNL-34156 (1984).

Narducci, L. M.

R. Bonifacio, C. Pellegrini, L. M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun. 50, 373 (1984).
[CrossRef]

Pellegrini, C.

R. Bonifacio, C. Pellegrini, L. M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun. 50, 373 (1984).
[CrossRef]

J. B. Murphy, C. Pellegrini, R. Bonifacio, “Collective instability of a free-electron laser including space charge and harmonics,” Brookhaven National Laboratory Rep. BNL-34156 (1984).

C. Pellegrini, “Physics of the free-electron laser,” in Free-Electron Lasers, S. Martellucci, N. A. Chester, eds. (Plenum, New York, 1983), pp. 91–119;W. B. Colson, “Free-electron wave and particle dynamics,” Free-Electron Lasers, S. Martellucci, N. A. Chester, eds. (Plenum, New York, 1983), pp. 189–209.

Petelin, M. I.

V. L. Bratman, N. S. Ginzburg, M. I. Petelin, “Nonlinear theory of stimulated wave scattering by relativistic electron beams,” Sov. Phys. JETP 49, 469–475 (1979);L. A. Vainshtein, “Type-O relativistic electron devices,” Sov. Phys. Tech. Phys. 24, 625–633 (1979);Y. L. Bolomolov, V. L. Bratman, N. S. Ginzburg, M. I. Petelin, A. D. Yanakovsky, “Nonstationary generation in free-electron lasers,” Opt. Commun. 36, 209–212 (1981).
[CrossRef]

Pierce, J. R.

J. R. Pierce, Traveling Wave Tubes (Van Nostrand, New York, 1950).

Renieri, A.

G. Dattoli, A. Renieri, “FEL quantum aspects,” in J. Phys. (Paris) 44, 126–136 (1983);W. Becker, J. K. McIver, “Quantum description of free-electron lasers,” J. Phys. (Paris) 44, 289–311 (1983).
[CrossRef]

A. Bambini, A. Renieri, “The free electron laser: a single particle classical model,” Lett. Nuovo Cimento 21, 399–404 (1978).
[CrossRef]

Schwettman, H. A.

L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Scully, M. O.

F. Hopf, P. Meystre, G. T. Moore, M. O. Scully, “Nonlinear theory of free-electron devices,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 41–114;N. M. Kroll, “The free-electron laser as a traveling-wave amplifier,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 115–156;W. B. Colson, “One-body analysis of free-electron lasers,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 157–196.

Smith, T. I.

L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Sprangle, P.

P. Sprangle, C. M. Tang, I. B. Bernstein, “Initiation of a pulsed-beam free-electron-laser oscillator,” Phys. Rev. Lett. 50, 1775–1778 (1983);“Evolution of spontaneous and coherent radiation in the free-electron laser oscillator,” Phys. Rev. A 28, 2300–2309 (1983).
[CrossRef]

Tang, C. M.

P. Sprangle, C. M. Tang, I. B. Bernstein, “Initiation of a pulsed-beam free-electron-laser oscillator,” Phys. Rev. Lett. 50, 1775–1778 (1983);“Evolution of spontaneous and coherent radiation in the free-electron laser oscillator,” Phys. Rev. A 28, 2300–2309 (1983).
[CrossRef]

J. Appl. Phys. (1)

J M. J. Madey, “Simulated emission of bremsstrahlung in a periodic magnetic field,” J. Appl. Phys. 42, 1906–1913 (1971).
[CrossRef]

J. Phys. (Paris) (1)

G. Dattoli, A. Renieri, “FEL quantum aspects,” in J. Phys. (Paris) 44, 126–136 (1983);W. Becker, J. K. McIver, “Quantum description of free-electron lasers,” J. Phys. (Paris) 44, 289–311 (1983).
[CrossRef]

Lett. Nuovo Cimento (2)

R. Bonifacio, F. Casagrande, “A model for superradiance and superfluorescence in free-electron lasers,” Lett. Nuovo Cimento 37, 39–47 (1983).

A. Bambini, A. Renieri, “The free electron laser: a single particle classical model,” Lett. Nuovo Cimento 21, 399–404 (1978).
[CrossRef]

Opt. Commun. (3)

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic transition of a free-electron laser Hamiltonian model,” Opt. Commun. 40, 219–223 (1982).
[CrossRef]

R. Bonifacio, F. Casagrande, “Instabilities and quantum initiation in the free-electron laser,” Opt. Commun. 50, 251–255 (1984).
[CrossRef]

R. Bonifacio, C. Pellegrini, L. M. Narducci, “Collective instabilities and high-gain regime in a free-electron laser,” Opt. Commun. 50, 373 (1984).
[CrossRef]

Phys. Rev. A (2)

N. M. Kroll, W. A. McMullin, “Stimulated emission from relativistic electrons passing through a spatially periodic transverse magnetic field,” Phys. Rev. A 17, 300–308 (1978);I. B. Bernstein, J. L. Hirschfield, “Amplification on a relativistic electron beam in a spatially periodic transverse magnetic field,” Phys. Rev. A 20, 1661–1670 (1979);P. Sprangle, C. M. Tang, W. H. Manheimer, “Nonlinear theory of free-electron lasers and efficiency enhancement,” Phys. Rev. A 21, 302–308 (1980);A. Gover, P. Sprangle, “A unified theory of magnetic bremmstrahlung, electrostatic bremsstrahlung, Compton–Raman scattering and Cherenkov–Smith–Purcell free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1196–1215 (1981);G. Dattoli, A. Marino, A. Renieri, F. Romanelli, “Progress in the Hamiltonian picture of the free-electron laser,” IEEE J. Quantum Electron. QE-17, 1371–1386 (1981);C. C. Shih, A. Yariv, “Inclusion of space-charge effects with Maxwell’s equations in the single-particle analysis of free-electron lasers,” IEEE J. Quantum Electron. QE-17, 1387–1394 (1981).Instabilities of a different kind with respect to the collective instability considered here have been discussed, e.g., in W. B. Colson, R. A. Friedman, “Synchrotron instability for long pulses in free electron laser oscillators,” Opt. Commun. 46, 37–42 (1983);V. A. Buts, V. V. Ognivenko, “Stochastic instability of the motion of particles in free-electron lasers,” JETP Lett. 38, 525–528 (1983).
[CrossRef]

W. Becker, J. K. McIver, “Fully quantized many-particle theory of a free-electron laser,” Phys. Rev. A 27, 1030–1043 (1983).
[CrossRef]

Phys. Rev. Lett. (2)

L. R. Elias, W. M. Fairbank, J. M. J. Madey, H. A. Schwettman, T. I. Smith, “Observation of stimulated emission of radiation by relativistic electrons in a spatially periodic transverse magnetic field,” Phys. Rev. Lett. 36, 717–720 (1976);D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

P. Sprangle, C. M. Tang, I. B. Bernstein, “Initiation of a pulsed-beam free-electron-laser oscillator,” Phys. Rev. Lett. 50, 1775–1778 (1983);“Evolution of spontaneous and coherent radiation in the free-electron laser oscillator,” Phys. Rev. A 28, 2300–2309 (1983).
[CrossRef]

Sov. Phys. JETP (1)

V. L. Bratman, N. S. Ginzburg, M. I. Petelin, “Nonlinear theory of stimulated wave scattering by relativistic electron beams,” Sov. Phys. JETP 49, 469–475 (1979);L. A. Vainshtein, “Type-O relativistic electron devices,” Sov. Phys. Tech. Phys. 24, 625–633 (1979);Y. L. Bolomolov, V. L. Bratman, N. S. Ginzburg, M. I. Petelin, A. D. Yanakovsky, “Nonstationary generation in free-electron lasers,” Opt. Commun. 36, 209–212 (1981).
[CrossRef]

Other (10)

J. R. Pierce, Traveling Wave Tubes (Van Nostrand, New York, 1950).

J. B. Murphy, C. Pellegrini, R. Bonifacio, “Collective instability of a free-electron laser including space charge and harmonics,” Brookhaven National Laboratory Rep. BNL-34156 (1984).

F. Hopf, P. Meystre, G. T. Moore, M. O. Scully, “Nonlinear theory of free-electron devices,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 41–114;N. M. Kroll, “The free-electron laser as a traveling-wave amplifier,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 115–156;W. B. Colson, “One-body analysis of free-electron lasers,” in Novel Sources of Coherent Radiation, S. F. Jacobs, M. Sargent, M. O. Scully, eds. (Addison-Wesley, Reading, Mass., 1978), pp. 157–196.

J. M. J. Madey, D. A. G. Deacon, “Free electron lasers,” in Cooperative Effects in Matter and Radiation, C. M. Bowden, D. W. Howgate, H. R. Robl, eds. (Plenum, New York, 1977), pp. 313–334.
[CrossRef]

A classical analysis of the FEL oscillator startup has been carried out in Ref. 13 that introduces an incoherent contribution to the driving current in the Maxwell equations.

R. Bonifacio, F. Casagrande, G. Casati, “Cooperative and chaotic effects in a Hamiltonian model of the free-electron laser,” in Evolution of Order and Chaos, H. Haken, ed. (Springer-VerlagBerlin, 1982).
[CrossRef]

R. Bonifacio, F. Casagrande, G. Casati, S. Celi, “Chaotic and cooperative effects in a free-electron laser Hamiltonian model,” in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), pp. 801–810.

C. Pellegrini, “Physics of the free-electron laser,” in Free-Electron Lasers, S. Martellucci, N. A. Chester, eds. (Plenum, New York, 1983), pp. 91–119;W. B. Colson, “Free-electron wave and particle dynamics,” Free-Electron Lasers, S. Martellucci, N. A. Chester, eds. (Plenum, New York, 1983), pp. 189–209.

R. J. Glauber, in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), pp 1–43.

S. F. Jacobs, G. T. Moore, H. S. Pilloff, M. Sargent, M. O. Scully, R. Spitzer, eds., Free-Electron Generators of Coherent Radiation (Addison-Wesley, Reading, Mass., 1982);S. Martellucci, N. A. Chester, eds., Free Electron Lasers (Plenum, New York, 1983).

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Figures (1)

Fig. 1
Fig. 1

FEL collective instability: normalized intensity |a|2 versus normalized time τ (δ < δτ, N = 8).

Equations (91)

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B w = B w [ sin ( k w z ) x ̂ + cos ( k w z ) ŷ ] ,
E = E 0 [ sin ( k L z ω L t + ϕ 0 ) x ̂ + cos ( k L z ω L t + ϕ 0 ) ŷ ] ,
V R = c k L / ( k L + k w ) ,
ω L = 2 ω w γ R 2 / ( 1 + κ 2 ) 2 ω w ( γ ) R 2 ,
κ = e B w / m 0 c ω w , γ = ( 1 β 2 ) 1 / 2 , β = υ / c ,
θ i = ( k L + k w ) z i ω L t , α = i E 0 exp ( i ϕ 0 ) / ( 4 π n γ 0 m 0 c 2 ) 1 / 2 , n = N / V ,
d θ i / d t = ω w ( 1 γ i 2 ) ( i = 1 , , N ) ,
d γ i / d t = 2 ω w ρ 3 / 2 [ exp ( i θ i ) / γ i + c.c . ] ( i = 1 , , N ) ,
d / d t = 2 ω w ρ 3 / 2 N 1 i = 1 N exp ( i θ i ) / γ i + i Δ 0 ,
θ i = θ i Δ 0 t , Δ 0 = ω w ( γ 0 2 γ R 2 ) / γ 0 2 , γ i = γ i / γ 0 , = α exp ( i Δ 0 t ) , ω w = ( γ R 2 / γ 0 2 ) ω w , ρ = [ ( κ / 4 ) ( Ω p / ω w ) ] 2 / 3 , Ω p = ω p / γ 0 3 / 2 , ω p = ( 4 π e 2 n / m 0 ) 1 / 2 .
m 0 c 2 i = 1 N γ i + ( | E 0 | 2 / 4 π ) V = const .
d / d t = 2 ω w ρ 3 / 2 N 1 i = 1 N exp ( i θ i ) / γ i + i Δ 0 k ,
k = c ( 1 ( β ) 0 ) / L e c / 2 ( γ ) 0 2 L e ,
Δ t = L e / c ( 1 ( β ) 0 ) 2 ( γ ) 0 2 L e / c .
L e ( 1 ( β ) 0 ) L w N w λ L .
k 1 L w / c or L e ( 1 ( β ) 0 ) L w N w λ L
( γ i γ 0 ) / γ 0 1 ,
η i = γ i 1 a = N τ = 2 ω w t , w = ρ 3 / 2 / N δ 0 = Δ 0 / 2 ω w ( γ 0 γ R ) / γ R , k ¯ = k / 2 ω w ,
d θ i / d τ = η i ,
d η i / d τ = w [ a exp ( i θ i ) + c.c . ] ,
d a / d τ = w J = 1 N exp ( i θ J ) + ( i δ 0 k ¯ ) a .
a ( 0 ) = 0 , η i ( 0 ) = 0 , J = 1 N exp ( i θ J 0 ) = 0 .
θ i ( t ) = θ i ( t ) θ i 0 ,
d Θ / d τ = P , d P / d τ = w ¯ a , d a / d τ = i w ¯ Θ + i δ 0 a ,
Θ = N 1 / 2 i = 1 N exp ( i θ i 0 ) θ ¯ i , P = N 1 / 2 i = 1 N exp ( i θ i 0 ) η i ,
w ¯ = w N = ρ 3 / 2 ,
i = 1 N exp ( i m θ i 0 ) = 0 ( m = 1 , 2 , ) .
Θ ( τ ) = exp ( i λ τ ) Θ 0 ,
δ > δ T 3 / 2 2 / 3 , δ δ 0 / ρ ,
η N 1 J = 1 N exp ( i θ J ) , 0 | η | 1 ,
d θ i / d τ = η i ,
d η i / d τ = w [ a exp ( i θ i ) + c.c . ] ,
d a / d τ = w J = 1 N exp ( i θ J ) k ¯ a .
a = ( w / k ¯ ) J = 1 N exp ( i θ J ) .
d 2 θ i / d τ 2 = d η i / d τ = 2 ( w 2 / k ¯ ) J = 1 N cos ( θ i θ J ) .
p ( t ) m 0 c 2 ( d / d t ) i = 1 N ( γ i γ 0 ) = ( B w 2 / 4 π ) c [ ( 2 λ w ) 2 / S e ] ( r e N | η ( t ) | ) 2 ,
d 2 Θ / d τ 2 = d P / d τ = i ( 2 ω w / k ) ω ¯ 2 Θ .
Θ ( t ) exp ( λ s t ) , λ s = 2 w ¯ ω w ( ω w / k ) 1 / 2 .
k 1 λ s 1 L w / c .
L e L c ( λ L N w L e ) 2 / 3 ,
L c = 2 2 / 3 ( λ L / 2 π ρ ) .
A = A L + A w , A L = A L ê exp [ i ( k L z ω L t ) ] + c.c . , A w = i A w [ ê exp ( i k w z ) c.c . ] , A L = ( E 0 / 2 k L ) exp ( i ϕ 0 ) , A w = B w / 2 k w , ê = ( x ̂ + i ŷ ) / 2 .
m = m 0 1 + κ 2 ,
H = i = 1 N p 2 / 2 + i w [ + i = 1 N exp ( i θ i ) + i = 1 N exp ( i θ i ) ] δ 0 a + ,
θ i = k z i δ 0 τ p i = p i / k σ 0 , [ θ i , p ¯ J ] = i δ i j , = a exp ( i δ 0 τ ) [ , + ] = 1 , δ 0 = m ( υ 0 υ R ) / k ω = k 2 / 2 m , τ = 2 ω t w = g / 2 ω .
d θ i / d τ = p i ,
d p i / d τ = w [ + exp ( i θ i ) + exp ( i θ i ) ] ,
d / d τ = w J = 1 N exp ( i θ J ) + i δ 0 .
i = 1 N p i + k + a = const .
δ θ i = θ i θ i 0 ( i = 1 , , N )
Θ ̂ = N 1 / 2 J = 1 N exp ( i θ J 0 ) δ θ J , P ̂ = N 1 / 2 J = 1 N exp ( i θ J 0 ) p J
J = 1 N exp ( i m θ J 0 ) = 0 ( m = 1 , 2 , ) .
d Θ / d τ = P ̂ d P ̂ / d τ = w ¯ , d / d τ = i w ¯ Θ ̂ + i δ 0 ,
w ¯ = w N .
[ Θ ̂ , P ̂ + ] = i [ Θ ̂ , Θ ̂ + ] = [ Θ ̂ , P ̂ ] = [ P ̂ , P ̂ + ] = 0 .
Ĥ = P ̂ + P ̂ + w ¯ ( + Θ ̂ + Θ ̂ + ) δ 0 + a .
Θ ̂ + Θ ̂ 0 = N 1 J = 1 N ( δ θ J ) 2 0 = σ θ 2 ( 0 ) ,
Θ ̂ + Θ ̂ 0 = σ θ 2 ( 0 ) N | η 0 | 2 .
λ 3 δ λ 2 + 1 = 0 ,
λ = λ / ρ , δ = δ 0 / ρ , ρ = ( w ¯ ) 2 / 3 .
a + a ( τ ¯ ) = ρ | f 1 ( τ ¯ ) | 2 Θ ̂ + Θ ̂ 0 + [ | f 2 ( τ ¯ ) | 2 / ρ ] P + P 0 + i [ f 1 ( τ ¯ ) f 2 * ( τ ¯ ) Θ ̂ + P 0 hc ] + | f 3 ( τ ¯ ) | 2 a + a 0 ,
τ ¯ = ρ τ , f 1 = i = 1 3 f 1 i exp ( i λ i τ ¯ ) , f 2 = i = 1 3 ( f 1 i / λ i 2 ) exp ( i λ i τ ¯ ) , f 3 = i = 1 3 λ i f 1 i exp ( i λ i τ ¯ ) , f 1 i = λ i [ ( λ J λ i ) ( λ i λ k ) ] 1 ( i j k = 1 , 2 , 3 ) .
a + a ( τ ¯ ) = | g ( τ ¯ ) | 2 + | f 3 ( τ ¯ ) | 2 a + a 0 ,
g = ρ σ θ f 1 ( σ p / ρ ) f 2 .
n = n s p + n s t ,
n s p = | g | 2 = σ θ 2 ρ | f 1 | 2 + ( σ p 2 / ρ ) | f 2 | 2 ( 1 / 2 ) ( f 1 f 2 * + f 1 * f 2 ) ,
n s t = | f 3 | 2 n 0 .
χ N ( ζ ) = exp [ | ζ | 2 n 0 i ( ζ * f 3 * δ α 0 * + ζ f 3 δ α 0 ) ] ,
P ( α , τ ¯ ) = ( π n s p ) 1 exp ( | α α ( τ ¯ ) | 2 / n s p ) ,
α ( τ ¯ ) = f 3 ( τ ¯ ) δ α 0 .
σ 2 ( n ) = n s p ( n s p + 1 ) + n s t + 2 n s t n s p .
n ( τ ¯ ) ( 1 / 9 ) ( σ θ 2 ρ + σ P 2 / ρ 1 / 2 + n 0 ) exp ( 3 τ ¯ ) .
Θ ̂ + Θ ̂ ( τ ¯ ) ( ρ ) 1 [ n ( 0 ) + 1 / 9 ] exp ( 3 τ ¯ ) ,
τ ¯ D ( 1 / 3 ) ln { N ρ / [ n ( 0 ) + 1 / 9 ] } ( 1 / 3 ) ln N c ,
( τ ¯ D ) max = ( 1 / 3 ) ln ( 6 N ρ ) .
( 1 / 3 ) ln ( 6 ρ N ) 2 ω ρ L w / c .
( 1 / 3 ) ln ( 6 ρ N γ 0 m c 2 / ω L ) 4 π ρ ( γ R 2 / γ 0 2 ) N w .
d / d τ = w J = 1 N exp ( i θ J ) + ( i δ 0 k ¯ ) ,
k ¯ = ( 2 ω ) 1 ( c / L e ) .
d Θ ̂ / d τ = P ̂ ,
d P ̂ / d τ = w ¯ ,
d / d τ = i w ¯ Θ ̂ k ̂ .
d 2 Θ ̂ / d τ 2 = d P ̂ / d τ = i ( w ¯ 2 / k ¯ ) Θ ̂ ,
d W e / d τ = i [ P ̂ + P ̂ , W e ] + ( w ¯ 2 / k ¯ ) ( [ Θ ̂ , W e Θ ̂ + ] + [ Θ ̂ W e , Θ ̂ + ] ) .
d Θ ̂ / d τ = P ̂ , d P ̂ / d τ = i ( w ¯ 2 / k ¯ ) Θ ̂ .
Θ ̂ + Θ ̂ ( t ) = ( 1 / 4 ) [ σ θ 2 + 2 ( ω / λ s ) 2 σ p 2 + ( ω / λ s ) ] × exp ( 2 λ s t ) ,
λ s = 2 w ¯ ω ( ω L e / c ) 1 / 2 .
t D ( λ s ) 1 ln [ N / Θ ̂ + Θ ̂ ( 0 ) ] 1 / 2 .
( t D ) max = ( λ s ) 1 ln { [ 4 / ( 1 + 2 ) ] N ( λ s / ω ) } 1 / 2 .
( λ s ) 1 ln { [ 4 / ( 1 + 2 ) ] N ( λ s / ω s ) } 1 / 2 L w / c .
ln { [ 8 / ( 1 + 2 ) ] ρ 3 / 2 ( k L L e ) 1 / 2 ( N γ 0 m c 2 / ω L ) } 4 π ρ 3 / 2 ( k L L e ) 1 / 2 N w .

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