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An approximate technique for the integration on the equations of motion in a free-electron laser

Приближенный метод интегрирования уравнений движения в свободном электронном лазере

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Il Nuovo Cimento B (1971-1996)

Summary

We describe an approximation we have used to compute gain, saturation and electron statistics at high power in the Stanford free-electron laser.

Riassunto

Si descrive un'approssimazione usata per calcolare il guadagno, la saturazione e la statistica degli elettroni ad alta potenza nel laser a elettroni liberi di Stanford.

Резюме

Мы описываем приближенный метод, который исполязуется для вычисления усиления, насыщения и статистики электронов при больших мощностях в свободном электронном лазере с Стенфорде.

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Abbreviations

e :

electron charge (4.8·10−10 statcoulomb)

m :

electron mass (9.11·10−28 g)

c :

speed of light (3·1010 cm s−1)

ℏ:

Planck's constant (1.05·10−27 erg s)

K :

Boltzmann's constant (1.38·10−16 erg K−1)

r :

exact electron position

r a :

approximate electron position calculated by using the approximation described in the text for the analysis of the electron motion in the presence of the optical field

δr=r−r a :

position error

\(\delta r_1 = (\delta r \cdot \hat z)\hat z\) :

longitudinal position error

δr ⊥=δr−δr ⊥ :

transverse position error

\(\beta= \dot r/c\) :

exact normalized electron velocity

β0 :

normalized electron velocity when the optical field is absent

\(\hat \beta _0 = \beta _0 /|\beta _0 |\) :

unit vector for normalized velocity when the optical field is absent

βa :

approximate normalized velocity, calculated by using the approximation described in the text to analyze the electron motion in the presence of the optical field

\(\beta _\parallel = (\beta _a \cdot \hat z)\hat z\) :

normalized approximate longitudinal velocity

β⊥=βa−β∥ :

normalized approximate transverse velocity

δβ=β−βa :

normalized velocity error

\(\delta \beta _\parallel = (\delta \beta \cdot \hat z)\hat z\) :

normalized longitudinal-velocity error

δβ⊥=δβ−δβ‖ :

normalized transverse-velocity error

P :

exact electron momentum (g cm s−1)

P 0 :

electron momentum when the optical field is absent

P a :

approximate electron momentum calculated by using the approximation described in the text for the analysis of the electron motion in the presence of the optical field

δp=p−p a :

momentum error

\(\delta p_\parallel = (\delta p \cdot \hat z)\hat z\) :

longitudinal-momentum error

δp ⊥=δp−δp‖:

transverse-momentum error

\(\gamma = \sqrt {p^2 + m^2 c^2 } /mc\) :

exact normalized electron mass-energy

γa :

approximate electron mass-energy, calculated by using the approximation described in the text for the analysis of the electron motion in the presence of the optical field

δγ = γ-γa :

error in normalized energy.

ω:

optical frequency (s−1)

k :

optical wave vector (cm−1)

\(\hat k = k/\left| k \right|\) :

unit wave vector

E :

optical electric field (statvolt cm−1)

B :

optical magnetic field (G)

E 0 :

polarization vector for optical electric field

S lab :

optical power density (erg cm−2 s−1)

λq :

magnet period (cm)

L :

length (cm)

B 0 :

static magnetic field (G)

B 0 :

polarization vector for the static magnetic field

A :

vector potential for the static magnetic field

\(A_ \bot = A - (A \cdot \hat z)\hat z\) :

transverse component of vector potential.

t :

dummy variable for time

T :

total interaction time (s)

K D :

Debye wave number

n 0 :

electron density (cm−3)

τ:

temperature (K).

References

  1. J. M. J. Madey andD. A. G. Deacon: inCo-Operative Effects in Matter and Radiation, edited byC. M. Bowden, D. W. Howgate andH. H. Robl (New York, N. Y., 1977), p. 313.

  2. R. M. Phillips:IRE Trans. Elect. Dev.,1, 231 (1960).

    Article  ADS  Google Scholar 

  3. F. A. Hopf, P. Meystre, M. O. Scully andW. H. Louisell:Opt. Comm.,18, 413 (1976);F. A. Hopf, P. Meystre, M. O. Scully andW. H. Louisell:Phys. Rev. Lett.,37, 1342 (1976);H. Al-Abawi, F. A. Hopf andP. Meystre:Phys. Rev. A,16, 666 (1977).

    Article  ADS  Google Scholar 

  4. W. B. Colson:Phys. Lett.,64 A, 190 (1977).

    Article  ADS  Google Scholar 

  5. A. Bambini andA. Renieri:Nuovo Cimento,21 B, 399 (1978).

    Google Scholar 

  6. V. N. Baier andA. I. Milstein:Phys. Lett.,65 A, 319 (1978).

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Physics Department and High Energy Physics Laboratory, Stanford University, 94305, Stanford, Cal.

    J. M. J. Madey, D. A. G. Deacon, L. R. Elias & T. I. Smith

Authors
  1. J. M. J. Madey
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  2. D. A. G. Deacon
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  3. L. R. Elias
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  4. T. I. Smith
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Additional information

Traduzione a cura della Redazione.

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Madey, J.M.J., Deacon, D.A.G., Elias, L.R. et al. An approximate technique for the integration on the equations of motion in a free-electron laser. Nuov Cim B 51, 53–69 (1979). https://doi.org/10.1007/BF02743696

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  • DOI: https://doi.org/10.1007/BF02743696

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