Abstract

Synchrotron radiation has been used as a source for spectroscopic experiments in many countries. In this paper, the statistical properties of a light pulse emitted by an electron pulse in a storage ring or a synchrotron are studied in a classical formalism. The field is shown to be gaussian for every wavelength. The coherence time of the filtered synchrotron radiation is calculated.

© 1974 Optical Society of America

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References

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  1. R. P. Godwin, Springer Tracts Mod. Phys. 51, 1 (1969); K. Codling, Rep. Prog. Phys. 36, 541 (1973).
    [Crossref]
  2. Proceedings of the International Symposium on Synchrotron Radiation Uses, edited by I. Munro and G. Marr (Publ. Daresbury Nuclear Phys. Lab., Daresbury, 1973), Rapport 26.
  3. R. J. Glauber, in Quantum Optics and Electronics edited by C. de Witt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965); L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Ch. X. See also the first of Ref. 3, p. 69.
  5. For simplification, this function is assumed to be scalar and not vectorial. This can be understood by supposing that one polarization only is considered.
  6. By chaotic field, we mean a stationary gaussian field. See, for instance, B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).
    [Crossref]
  7. A. A. Sokolov and I. M. Ternov, Synchrotron Radiation (Pergamon, New York, 1968), and references therein.
  8. J. Schwinger, Phys. Rev. 75, 1912 (1949); Phys. Rev. D 7, 1696 (1973).
    [Crossref]
  9. R. J. Glauber, Phys. Rev. 84, 395 (1951).
    [Crossref]
  10. M. Sands, Phys. Rev. 97, 470 (1956).
    [Crossref]
  11. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  12. H. Bruck, Accélérateurs circulaires de particules (Presses Universitaires de France, Paris, 1966).
  13. A. A. Kolomenskii and A. N. Lebedev, Zh. Tekh, Fiz. 32, 1237 (1962) [Sov. Phys.-Tech. Phys. 7, 913 (1963)].
  14. L. V. Iogansen and M. S. Rabinovitch, Zh. Eksp. Teor. Fiz. 35, 1013 (1958); Zh. Eksp. Teor. Fiz. 37, 118 (1959) [Sov. Phys.–JETP 35, 708 (1958); 37, 83 (1960)]. I. L. Zel’Manov, A. S. Kompaneets, and Yu S. Sayasov, Dokl. Akad. Nauk. SSSR 143, 72 (1962) [Sov. Phys.-Doklady 7, 201 (1962)].
  15. P. Goldreich and D. A. Keeley, Astrophys. J. 170, 463 (1971).
    [Crossref]
  16. M. S. Livingstone and J. P. Blewett, Particle Accelerators (McGraw–Hill, New York, 1962).
  17. See Eqs. (14–62) and (14–68) of Ref. 13, Sec. 14.5,A (ω)=(e28π2c)12∑i∫-∞+∞exp{iω(t+Ri(t)c)}×(ni×(ni-βi)×β˙i)K-2 dt=∑i∫-∞+∞fi(t) exp{iω(t+Ri(t)c)} dt.The complex amplitude of the field for the given frequency ω can be written as a time Fourier transform of a functionA (t)=∑ifi(t) exp{iωRi(t)c},where Ri(t) is the distance from 0 of the i th electron at the same time t, for every i.
  18. Synchronous electron located in the middle of the bunch.
  19. The authors are grateful to Dr. H. Zyngier for having pointed this out.

1971 (1)

P. Goldreich and D. A. Keeley, Astrophys. J. 170, 463 (1971).
[Crossref]

1970 (1)

By chaotic field, we mean a stationary gaussian field. See, for instance, B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).
[Crossref]

1969 (1)

R. P. Godwin, Springer Tracts Mod. Phys. 51, 1 (1969); K. Codling, Rep. Prog. Phys. 36, 541 (1973).
[Crossref]

1962 (1)

A. A. Kolomenskii and A. N. Lebedev, Zh. Tekh, Fiz. 32, 1237 (1962) [Sov. Phys.-Tech. Phys. 7, 913 (1963)].

1958 (1)

L. V. Iogansen and M. S. Rabinovitch, Zh. Eksp. Teor. Fiz. 35, 1013 (1958); Zh. Eksp. Teor. Fiz. 37, 118 (1959) [Sov. Phys.–JETP 35, 708 (1958); 37, 83 (1960)]. I. L. Zel’Manov, A. S. Kompaneets, and Yu S. Sayasov, Dokl. Akad. Nauk. SSSR 143, 72 (1962) [Sov. Phys.-Doklady 7, 201 (1962)].

1956 (1)

M. Sands, Phys. Rev. 97, 470 (1956).
[Crossref]

1951 (1)

R. J. Glauber, Phys. Rev. 84, 395 (1951).
[Crossref]

1949 (1)

J. Schwinger, Phys. Rev. 75, 1912 (1949); Phys. Rev. D 7, 1696 (1973).
[Crossref]

Blewett, J. P.

M. S. Livingstone and J. P. Blewett, Particle Accelerators (McGraw–Hill, New York, 1962).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Ch. X. See also the first of Ref. 3, p. 69.

Bruck, H.

H. Bruck, Accélérateurs circulaires de particules (Presses Universitaires de France, Paris, 1966).

Glauber, R. J.

R. J. Glauber, Phys. Rev. 84, 395 (1951).
[Crossref]

R. J. Glauber, in Quantum Optics and Electronics edited by C. de Witt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965); L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

Godwin, R. P.

R. P. Godwin, Springer Tracts Mod. Phys. 51, 1 (1969); K. Codling, Rep. Prog. Phys. 36, 541 (1973).
[Crossref]

Goldreich, P.

P. Goldreich and D. A. Keeley, Astrophys. J. 170, 463 (1971).
[Crossref]

Iogansen, L. V.

L. V. Iogansen and M. S. Rabinovitch, Zh. Eksp. Teor. Fiz. 35, 1013 (1958); Zh. Eksp. Teor. Fiz. 37, 118 (1959) [Sov. Phys.–JETP 35, 708 (1958); 37, 83 (1960)]. I. L. Zel’Manov, A. S. Kompaneets, and Yu S. Sayasov, Dokl. Akad. Nauk. SSSR 143, 72 (1962) [Sov. Phys.-Doklady 7, 201 (1962)].

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Keeley, D. A.

P. Goldreich and D. A. Keeley, Astrophys. J. 170, 463 (1971).
[Crossref]

Kolomenskii, A. A.

A. A. Kolomenskii and A. N. Lebedev, Zh. Tekh, Fiz. 32, 1237 (1962) [Sov. Phys.-Tech. Phys. 7, 913 (1963)].

Lebedev, A. N.

A. A. Kolomenskii and A. N. Lebedev, Zh. Tekh, Fiz. 32, 1237 (1962) [Sov. Phys.-Tech. Phys. 7, 913 (1963)].

Livingstone, M. S.

M. S. Livingstone and J. P. Blewett, Particle Accelerators (McGraw–Hill, New York, 1962).

Picinbono, B.

By chaotic field, we mean a stationary gaussian field. See, for instance, B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).
[Crossref]

Rabinovitch, M. S.

L. V. Iogansen and M. S. Rabinovitch, Zh. Eksp. Teor. Fiz. 35, 1013 (1958); Zh. Eksp. Teor. Fiz. 37, 118 (1959) [Sov. Phys.–JETP 35, 708 (1958); 37, 83 (1960)]. I. L. Zel’Manov, A. S. Kompaneets, and Yu S. Sayasov, Dokl. Akad. Nauk. SSSR 143, 72 (1962) [Sov. Phys.-Doklady 7, 201 (1962)].

Rousseau, M.

By chaotic field, we mean a stationary gaussian field. See, for instance, B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).
[Crossref]

Sands, M.

M. Sands, Phys. Rev. 97, 470 (1956).
[Crossref]

Schwinger, J.

J. Schwinger, Phys. Rev. 75, 1912 (1949); Phys. Rev. D 7, 1696 (1973).
[Crossref]

Sokolov, A. A.

A. A. Sokolov and I. M. Ternov, Synchrotron Radiation (Pergamon, New York, 1968), and references therein.

Ternov, I. M.

A. A. Sokolov and I. M. Ternov, Synchrotron Radiation (Pergamon, New York, 1968), and references therein.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Ch. X. See also the first of Ref. 3, p. 69.

Astrophys. J. (1)

P. Goldreich and D. A. Keeley, Astrophys. J. 170, 463 (1971).
[Crossref]

Phys. Rev. (3)

J. Schwinger, Phys. Rev. 75, 1912 (1949); Phys. Rev. D 7, 1696 (1973).
[Crossref]

R. J. Glauber, Phys. Rev. 84, 395 (1951).
[Crossref]

M. Sands, Phys. Rev. 97, 470 (1956).
[Crossref]

Phys. Rev. A (1)

By chaotic field, we mean a stationary gaussian field. See, for instance, B. Picinbono and M. Rousseau, Phys. Rev. A 1, 635 (1970).
[Crossref]

Springer Tracts Mod. Phys. (1)

R. P. Godwin, Springer Tracts Mod. Phys. 51, 1 (1969); K. Codling, Rep. Prog. Phys. 36, 541 (1973).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

L. V. Iogansen and M. S. Rabinovitch, Zh. Eksp. Teor. Fiz. 35, 1013 (1958); Zh. Eksp. Teor. Fiz. 37, 118 (1959) [Sov. Phys.–JETP 35, 708 (1958); 37, 83 (1960)]. I. L. Zel’Manov, A. S. Kompaneets, and Yu S. Sayasov, Dokl. Akad. Nauk. SSSR 143, 72 (1962) [Sov. Phys.-Doklady 7, 201 (1962)].

Zh. Tekh, Fiz. (1)

A. A. Kolomenskii and A. N. Lebedev, Zh. Tekh, Fiz. 32, 1237 (1962) [Sov. Phys.-Tech. Phys. 7, 913 (1963)].

Other (11)

A. A. Sokolov and I. M. Ternov, Synchrotron Radiation (Pergamon, New York, 1968), and references therein.

M. S. Livingstone and J. P. Blewett, Particle Accelerators (McGraw–Hill, New York, 1962).

See Eqs. (14–62) and (14–68) of Ref. 13, Sec. 14.5,A (ω)=(e28π2c)12∑i∫-∞+∞exp{iω(t+Ri(t)c)}×(ni×(ni-βi)×β˙i)K-2 dt=∑i∫-∞+∞fi(t) exp{iω(t+Ri(t)c)} dt.The complex amplitude of the field for the given frequency ω can be written as a time Fourier transform of a functionA (t)=∑ifi(t) exp{iωRi(t)c},where Ri(t) is the distance from 0 of the i th electron at the same time t, for every i.

Synchronous electron located in the middle of the bunch.

The authors are grateful to Dr. H. Zyngier for having pointed this out.

Proceedings of the International Symposium on Synchrotron Radiation Uses, edited by I. Munro and G. Marr (Publ. Daresbury Nuclear Phys. Lab., Daresbury, 1973), Rapport 26.

R. J. Glauber, in Quantum Optics and Electronics edited by C. de Witt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965); L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Ch. X. See also the first of Ref. 3, p. 69.

For simplification, this function is assumed to be scalar and not vectorial. This can be understood by supposing that one polarization only is considered.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

H. Bruck, Accélérateurs circulaires de particules (Presses Universitaires de France, Paris, 1966).

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Figures (4)

Fig. 1
Fig. 1

O is the observation point; A is the emission point of the electron whose trajectory is the full curve.

Fig. 2
Fig. 2

The full curve is the trajectory of the synchronous electron. The dashed curve is the trajectory of the electron that is subject to betatron oscillations.

Fig. 3
Fig. 3

x and x′ are two observation points; A and A′ are the corresponding emission points for the synchronous electron. AA′ = l.

Fig. 4
Fig. 4

Time correlation function for the monochromatic filtered field of frequency ω. (a) for ωωL; (b) for ω < ωL. In both cases, point 0 corresponds to zero for the y axis.

Equations (89)

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G 2 ( r 1 , r 2 , t 1 , t 2 ) = E ( r 1 , t 1 ) E * ( r 2 , t 2 ) ,
G n ( r 1 , , r n ; t 1 , , t n ) = E ( * ) ( r 1 , t 1 ) E ( * ) ( r n , t n ) ,
G n A ( r 1 r n ; t 1 t n ) = E ( * ) ( r 1 , t 1 ) · · E ( * ) ( r n , t n ) .
Z 1 Z 2 n = α β Z α 1 Z β 1 Z α n Z β n ,
E A * ( r 1 , t 1 ) E A ( r 2 , t 2 ) = E B * ( r 1 , t 1 ) E B ( r 2 , t 2 ) ,
G 4 ( r 1 r 1 r 2 r 2 ; t 1 t 1 t 2 t 2 ) = I ( r 1 , t 1 ) I ( r 2 , t 2 ) = E ( r 1 , t 1 ) E * ( r 1 , t 1 ) E ( r 2 , t 2 ) E * ( r 2 , t 2 ) ,
j ( x , t ) = e v ( t ) δ ( x - r ( t ) ) ,
a ( x , t ) = 1 c j ( x , t ) R ( t ) δ ( t + R ( t ) c - t ) d x d t ,
R ( t ) - R ( t 0 ) v ( t - t 0 ) .
θ c ( ω ) = 1 γ ( ω c ω ) 1 3 ,
A B = ρ θ c
Δ θ v ~ 10 - 4 rad .
A B = ρ [ θ c ( λ ) + Δ θ ] .
δ t = A B ( λ ) c γ 2 = ρ c γ 2 { θ c ( λ ) + Δ θ } .
N r = 5 2 3 × γ 137 .
N r = 11.2             photons / rad .
Δ τ = ρ c N r , Δ τ ~ 0.33 × 10 - 9 s .
N = 1 Δ τ ~ 3 × 10 9 photons / s .
t A - t B = - 1 c [ R A ( t A ) - R B ( t B ) ] - x ρ c · O A ,
δ = R A ( t A ) - R B ( t A ) = x ρ O A ( 1 - v c ) ,
1 - v / c = 1 / 2 γ 2 .
δ x ρ / ( 0 A × 2 γ 2 ) .
[ 1 T 0 t - T 0 t T ( θ ) d θ if T 0 is the electron period ] ,
d 2 T ¯ d t 2 + Γ d T ¯ d t + Ω 2 T ¯ = g ( t ) ,
Ω 2 = c 2 ρ 2 × V 0 cos φ 0 2 π E 0 ( 1 - n ) , Γ = 3 - 4 n 1 - n × W 0 E 0 , W 0 = 2 3 c e 0 2 ρ 2 ( E 0 m 0 c 2 ) 4 ,
g ( t ) = a i δ ( t - t i ) - g ¯ .
T ¯ ( t ) = a Ω i i exp { - Γ ( t - t i ) } sin Ω ( t - t i ) - K ,
K = a d Ω 0 exp { - Γ θ 2 } sin Ω θ d θ .
T ¯ ( t ) = R ( t - θ ) ( θ ) d N ( θ ) - K , R ( t ) = a Ω exp { - Γ t 2 } sin Ω t .
τ R = 2 π Ω ,
d τ R = N τ R ~ 3 × 10 3 .
σ T = L / c 10 - 9 s .
Δ T ( τ ) = T ( t + τ ) - T ( t ) .
δ l / l = δ E / E 0 ( 1 - n ) .
Δ T ( τ ) / τ = δ E / E 0 ( 1 - n ) .
σ τ = τ σ E / E 0 ( 1 - n ) ,
σ E 2.07 × 10 5 eV ,             E 0 = 0.54 × 10 9 MeV ,             n = 0.5 ,
σ τ = 0.79 × 10 - 3 τ .
Δ T ( Δ t ) = 2 × 10 - 14 s ,
A ( x , t ) = ( c 4 π ) 1 2 R E ( x , t ) ,
A ( x , ω ) = 1 2 π - + A ( x , t ) e i ω t d t .
d I ( x , ω ) d Ω = A ( x , ω ) 2 + A ( x , - ω ) 2 = 2 A ( x , ω ) 2 .
F ( x , t ) = e i ω t A ( x , ω ) g ( ω ) d ω .
F ˜ ( Δ , t ) = Δ d x d ω e i ω t A ( x , ω ) g ( ω ) .
F 0 ( x , t ) = e i ω t A ( x , ω ) ,
A ( x , ω ) = i = 1 N A i ( x , ω ) .
A i ( x , t ) = e ( 4 π c ) 1 2 ( n i × [ ( n i - β i ) × β ˙ i ] K i 3 ) ret .
K i ( t ) = 1 - n i ( t ) · β i ( t ) , β i ( t ) = v i ( t ) c ,
A i ( x , ω ) = ( e 2 8 π 2 c ) 1 2 - + exp { i ω ( t + R i ( t ) c ) } × n i × [ ( n i - β i ) × β ˙ i ] K i 2 d t .
f ( R i ( t ) ) = n i × [ ( n i - β i ) × β ˙ i ] K i 2 , k = ( e 2 8 π 2 c ) 1 2 .
A i ( x , ω ) = exp { i ω ( T i - T 1 ) } A 1 ( x , ω ) ,
A ( x , ω ) = A 1 ( x , ω ) i = 1 N exp { i ω ( T i - T 1 ) } .
A ( x , ω ) = A 1 ( x , ω ) i = 1 N exp { i ω T i } .
A ( x , ω ) = A 1 ( x , ω ) N exp { - σ T 2 2 ω 2 } .
A ( x , ω ) = A 1 ( x , ω ) i = 1 N exp { i ω T i } ,
A ( x , ω ) = A 1 ( x , ω ) i = 1 N exp { i ω T i } ,
σ τ = a τ .
A ( x , ω ) A * ( x , ω ) = C ω ( τ ) = i = 1 N exp ( i ω T i ) j = 1 N exp ( - i ω T j ) A 1 ( x , ω ) A 1 * ( x , ω ) + i = 1 N exp { i ω ( T i - T i ) } A 1 ( x , ω ) A 1 * ( x , ω ) ,
C ω ( τ ) = ( N 2 exp ( - σ T 2 ω 2 ) + N exp { - a 2 τ 2 ω 2 2 } ) × A 1 ( x , ω ) A 1 * ( x , ω ) .
σ T 2 ω 2 = ln N ,
ω 4 × 10 9 Hz = ω L .
t c = ( a ω ) - 1 = C 2 π a ,
t c 202 C .
t c 3.34 × 10 - 13 s .
l c = c t c 0.1 mm .
C ω ( Δ ω , τ ) = A ( x , ω ) A * ( x , ω ) = i = 1 N e i ω T i j = 1 N e - i ω T i A 1 ( x , ω ) A 1 * ( x , ω ) + i = 1 N e i ω T i - i ω T i A 1 ( x , ω ) A 1 * ( x , ω ) = A ( x , ω ) A * ( x , ω ) + i = 1 N e - i Δ ω T i + i ω ( T i - T i ) A 1 ( x , ω ) A 1 * ( x , ω ) ,
Δ ω = ω - ω .
i = 1 N exp { - i Δ ω T i + i ω ( T i - T i ) } = N ( 2 π σ T σ τ ) - 1 exp { - T i 2 2 σ T 2 - ( T i - T i ) 2 2 σ τ 2 - i Δ ω T i + i ω ( T i - T i ) } d T i d ( T i - T i ) = N exp { - σ T 2 2 ( ω - ω ) 2 - σ τ 2 ω 2 2 } ,
C ω ( Δ ω , τ ) = A 1 ( x , ω ) A 1 * ( x , ω + Δ ω ) × [ N 2 exp { - σ T 2 ω 2 2 - σ T 2 ( ω + Δ ω ) 2 2 } + N exp { - σ T 2 Δ ω 2 2 - σ τ 2 ( ω + Δ ω ) 2 2 } ] .
σ T 2 ω ( ω + Δ ω ) ln N .
C ( t , τ ) = A ( x , t ) A * ( x , t + τ ) = exp { i ( ω - ω ) t + i ω τ } [ A ( x , ω ) A * ( x , ω ) + N exp { - σ T 2 2 ( ω - ω ) 2 - σ τ 2 ω 2 2 } × A 1 ( x , ω ) A 1 * ( x , ω ) d ω d ω ] .
C ( t , τ ) N 2 2 π σ T 2 ( δ t A ¯ 1 ( x ) ) 2 exp { - t 2 2 σ T 2 } + N ( δ t A ¯ 1 ( x ) ) 2 2 π σ T ( δ t 2 + σ τ 2 ) 1 2 exp ( - t 2 2 σ T 2 ) .
C ˜ ω ( τ ) = F ( x , t ) F * ( x , t + τ ) = e i ( ω - ω ) t - i ω τ g ( ω ) g * ( ω ) A ( x , ω ) A * ( x , ω ) d ω d ω = e i ( ω - ω ) t - i ω τ g ( ω ) g * ( ω ) C ω ( Δ ω , τ ) d ω d ω = F ( x , t ) F * ( x , t + τ ) + N A 1 ( x , ω ) × A 1 * ( x , ω + Δ ω ) exp { - σ T 2 Δ ω 2 2 - σ τ 2 ( ω + Δ ω ) 2 2 } × g ( ω ) g * ( ω ) exp { i ( ω - ω ) t - i ω τ } d ω d ω .
N g ( ω 0 ) 2 e - i ω 0 τ A 1 ( x , ω 0 ) A 1 * ( x , ω 0 ) exp { - σ τ 2 ω 0 2 2 } ,
F ( x , t ) F * ( x , t + τ ) = e i ω t - i ω ( t + τ ) g ( ω ) g * ( ω ) A ( x , ω ) A * ( x , ω ) d ω d ω = F ( x , t ) F * ( x , t + τ ) + N A 1 * ( x , ω ) A 1 ( x , ω + Δ ω ) × e i Δ ω t - i ω τ exp { - σ T 2 Δ ω 2 2 } g ( ω + Δ ω ) g * ( ω ) d ω d ( Δ ω ) = N 2 A 1 ( x , ω 0 ) 2 t G 2 σ T 2 + t G 2 × exp { - t 2 σ T 2 + t G 2 - ω 0 2 t G 2 σ T 2 σ T 2 + t G 2 } + N A 1 ( x , ω 0 ) 2 t G ( 2 σ T 2 + t G 2 ) 1 2 × exp { - 1 2 2 t 2 + 2 t τ + τ 2 t G - 2 ( t G 2 + σ T 2 ) 2 σ T 2 + t G 2 } .
F ( x , t ) F * ( x , t + τ ) N 2 A 1 ( x , ω 0 ) 2 t G 2 σ T 2 exp { - ω 0 2 t G 2 - t 2 σ T 2 } + N A 1 ( x , ω 0 ) 2 t G σ T 2 e - i ω 0 τ exp { - 1 2 t 2 σ T 2 - t τ 2 σ T 2 - τ 2 4 t G 2 } .
A i ( x , ω ) = k e i ω T i - + e i ω ( t + r i ( t ) / c ) f ( r i ( t ) ) d t .
r i ( 0 ) = r j ( 0 ) .
δ l i ( t ) = v t δ E i E 0 ( 1 - n ) = Δ v i t ,
r i ( t ) - r 1 ( t ) = Δ v i t ,
A i ( x , ω ) = k e i ω T i - + exp { i ω ( t + r 1 ( t ) c + Δ v i c t ) } × f ( r 1 ( t ) + Δ v i ( t ) ) d t .
f ( r 1 ( t ) + Δ v i t ) f ( r 1 ( t ) ) ,
A i ( x , ω ) = k e i ω T i - + exp { i ω ( t + r 1 ( t ) c + Δ v i c t ) } f ( r 1 ( t ) ) d t = e i ω T i - + e i ω t k exp { i ω r 1 ( t ) c } × f ( r 1 ( t ) ) exp { + i ω c Δ v i t } d t .
A i ( x , ω ) = e i ω ( T i - T 1 ) A 1 ( x , ω - ω ) δ ( ω - ω c Δ v i ) d ω ,
A 1 ( x , ω ) = e i ω T 1 e i ω t k exp { i ω r 1 ( t ) c } f ( r 1 ( t ) ) d t .
A i ( x , ω ) = e i ω ( T i - T 1 ) A 1 ( x , ω - ω Δ v i c ) .
v σ E [ E 0 ( 1 - n ) ] - 1             ( 2 × 10 5 ms in the case of ACO ) .
A(ω)=(e28π2c)12i-+exp{iω(t+Ri(t)c)}×(ni×(ni-βi)×β˙i)K-2dt=i-+fi(t)exp{iω(t+Ri(t)c)}dt.
A(t)=ifi(t)exp{iωRi(t)c},