Abstract

Relativistic formulation is presented of the second-order classical coherence theory of the electromagnetic field in the presence of random sources. It is shown that the dynamics of the second-order correlation tensors may be reduced to a simpler form in terms of certain tensor potentials. The gauge transformations of these tensor potentials are considered. In particular, two gauges of special interest are found, analogous to the Lorentz gauge and the radiation gauge in Maxwell’s electromagnetic theory. Some interesting relations and conservation laws are presented.

© 1969 Optical Society of America

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Corrections

D. Dialetis, "Errata: Relativistic Correlation Theory of the Electromagnetic Field in the Presence of Random Sources," J. Opt. Soc. Am. 59, 1008-1008 (1969)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-59-8-1008

References

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  1. For an account of classical coherence theory, see for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., Ch. X.
  2. E. Wolf, in Proc. Symposium on Astronomical Optics, Z. Kopal, Ed. (North-Holland Publishing Co., Amsterdam, 1956).
  3. P. Roman and E. Wolf, Nuovo Cimento 17, 462 (1960).
    [Crossref]
  4. P. Roman and E. Wolf, Nuovo Cimento 17, 477 (1960).
    [Crossref]
  5. D. Gabor, J. Inst. Elec. Engrs. 93, 429 (1946).
  6. E. Wolf, Nuovo Cimento 12, 884 (1954).
    [Crossref]
  7. E. Wolf, Proc. Royal Soc. (London) A230, 246 (1955).
    [Crossref]
  8. P. Roman, Nuovo Cimento 20, 759 (1961).
    [Crossref]
  9. P. Roman, Nuovo Cimento 22, 1005 (1961).
    [Crossref]
  10. E. Wolf, in Proceedings of the Symposium in Optical Masers, J. Fox, Ed. (John Wiley & Sons, Inc., New York, 1963), p. 35.
  11. M. Beran and G. Parrent, J. Opt. Soc. Am. 52, 98 (1962).
    [Crossref]
  12. A. Kujawski, Nuovo Cimento 44, 326 (1966).
    [Crossref]
  13. For the definition of the analytic signal see, for example, Refs. 4 and 11.
  14. We assume here that the operations of differentiation and that of taking the ensemble average commute.
  15. See, for example, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw–Hill Book Co., New York, 1965), p. 34, or S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper and Row, New York, 1962), p. 180.
  16. A proof is given in D. Dialetis, “Correlation Theory of the Electromagnetic Field in the Presence of Random Sources” (thesis), University of Rochester, Rochester, N. Y. (1967).
  17. See, for example, A. Messiah, Quantum Mechanics, Vol. II (John Wiley & Sons, Inc., New York, 1962), p. 1020.

1966 (1)

A. Kujawski, Nuovo Cimento 44, 326 (1966).
[Crossref]

1962 (1)

1961 (2)

P. Roman, Nuovo Cimento 20, 759 (1961).
[Crossref]

P. Roman, Nuovo Cimento 22, 1005 (1961).
[Crossref]

1960 (2)

P. Roman and E. Wolf, Nuovo Cimento 17, 462 (1960).
[Crossref]

P. Roman and E. Wolf, Nuovo Cimento 17, 477 (1960).
[Crossref]

1955 (1)

E. Wolf, Proc. Royal Soc. (London) A230, 246 (1955).
[Crossref]

1954 (1)

E. Wolf, Nuovo Cimento 12, 884 (1954).
[Crossref]

1946 (1)

D. Gabor, J. Inst. Elec. Engrs. 93, 429 (1946).

Beran, M.

Bjorken, J. D.

See, for example, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw–Hill Book Co., New York, 1965), p. 34, or S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper and Row, New York, 1962), p. 180.

Born, M.

For an account of classical coherence theory, see for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., Ch. X.

Dialetis, D.

A proof is given in D. Dialetis, “Correlation Theory of the Electromagnetic Field in the Presence of Random Sources” (thesis), University of Rochester, Rochester, N. Y. (1967).

Drell, S. D.

See, for example, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw–Hill Book Co., New York, 1965), p. 34, or S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper and Row, New York, 1962), p. 180.

Gabor, D.

D. Gabor, J. Inst. Elec. Engrs. 93, 429 (1946).

Kujawski, A.

A. Kujawski, Nuovo Cimento 44, 326 (1966).
[Crossref]

Messiah, A.

See, for example, A. Messiah, Quantum Mechanics, Vol. II (John Wiley & Sons, Inc., New York, 1962), p. 1020.

Parrent, G.

Roman, P.

P. Roman, Nuovo Cimento 20, 759 (1961).
[Crossref]

P. Roman, Nuovo Cimento 22, 1005 (1961).
[Crossref]

P. Roman and E. Wolf, Nuovo Cimento 17, 477 (1960).
[Crossref]

P. Roman and E. Wolf, Nuovo Cimento 17, 462 (1960).
[Crossref]

Wolf, E.

P. Roman and E. Wolf, Nuovo Cimento 17, 462 (1960).
[Crossref]

P. Roman and E. Wolf, Nuovo Cimento 17, 477 (1960).
[Crossref]

E. Wolf, Proc. Royal Soc. (London) A230, 246 (1955).
[Crossref]

E. Wolf, Nuovo Cimento 12, 884 (1954).
[Crossref]

For an account of classical coherence theory, see for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., Ch. X.

E. Wolf, in Proc. Symposium on Astronomical Optics, Z. Kopal, Ed. (North-Holland Publishing Co., Amsterdam, 1956).

E. Wolf, in Proceedings of the Symposium in Optical Masers, J. Fox, Ed. (John Wiley & Sons, Inc., New York, 1963), p. 35.

J. Inst. Elec. Engrs. (1)

D. Gabor, J. Inst. Elec. Engrs. 93, 429 (1946).

J. Opt. Soc. Am. (1)

Nuovo Cimento (6)

A. Kujawski, Nuovo Cimento 44, 326 (1966).
[Crossref]

E. Wolf, Nuovo Cimento 12, 884 (1954).
[Crossref]

P. Roman, Nuovo Cimento 20, 759 (1961).
[Crossref]

P. Roman, Nuovo Cimento 22, 1005 (1961).
[Crossref]

P. Roman and E. Wolf, Nuovo Cimento 17, 462 (1960).
[Crossref]

P. Roman and E. Wolf, Nuovo Cimento 17, 477 (1960).
[Crossref]

Proc. Royal Soc. (London) (1)

E. Wolf, Proc. Royal Soc. (London) A230, 246 (1955).
[Crossref]

Other (8)

E. Wolf, in Proceedings of the Symposium in Optical Masers, J. Fox, Ed. (John Wiley & Sons, Inc., New York, 1963), p. 35.

For an account of classical coherence theory, see for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., Ch. X.

E. Wolf, in Proc. Symposium on Astronomical Optics, Z. Kopal, Ed. (North-Holland Publishing Co., Amsterdam, 1956).

For the definition of the analytic signal see, for example, Refs. 4 and 11.

We assume here that the operations of differentiation and that of taking the ensemble average commute.

See, for example, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw–Hill Book Co., New York, 1965), p. 34, or S. S. Schweber, An Introduction to Relativistic Quantum Field Theory (Harper and Row, New York, 1962), p. 180.

A proof is given in D. Dialetis, “Correlation Theory of the Electromagnetic Field in the Presence of Random Sources” (thesis), University of Rochester, Rochester, N. Y. (1967).

See, for example, A. Messiah, Quantum Mechanics, Vol. II (John Wiley & Sons, Inc., New York, 1962), p. 1020.

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Equations (109)

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F 0 l ( x ) = - F 0 l ( x ) = E l ( x ) ,
F l m ( x ) = F l m ( x ) = - l m r H r ( x ) ,
j μ ( x ) = { ρ ( x ) , ( 1 / c ) j ˜ l ( x ) } .
F μ ν α β ( x 1 , x 2 ) = F μ ν ( x 1 ) F α β * ( x 2 ) ,
C μ ν λ ( x 1 , x 2 ) = F μ ν ( x 1 ) j λ * ( x 2 ) ,
D λ μ ν ( x 1 , x 2 ) = j λ ( x 1 ) F μ ν * ( x 2 ) ,
J λ ρ ( x 1 , x 2 ) = j λ ( x 1 ) j ρ * ( x 2 ) ,
F μ ν α β * ( x 1 , x 2 ) = F α β μ ν ( x 2 , x 1 ) ,
C μ ν λ * ( x 1 , x 2 ) = D λ μ ν ( x 2 , x 1 ) ,
D λ μ ν * ( x 1 , x 2 ) = C μ ν λ ( x 2 , x 1 ) ,
J λ ρ * ( x 1 , x 2 ) = J ρ λ ( x 2 , x 1 ) .
F μ ν α β ( x 1 , x 2 ) = - F ν μ α β ( x 1 , x 2 ) = - F μ ν β α ( x 1 , x 2 ) = F ν μ β α ( x 1 , x 2 ) ,
C μ ν λ ( x 1 , x 2 ) = - C ν μ λ ( x 1 , x 2 ) ,
D λ μ ν ( x 1 , x 2 ) = - D λ ν μ ( x 1 , x 2 ) .
μ 1 F μ ν ( x 1 ) = 4 π j ν ( x 1 ) ,
ρ σ μ ν σ 1 F μ ν ( x 1 ) = 0 ,
μ 1 = / x 1 μ ,
μ 1 F μ ν α β ( x 1 , x 2 ) = 4 π D ν α β ( x 1 , x 2 ) ,
ρ σ μ ν a 1 F μ ν α β ( x 1 , x 2 ) = 0 ,
μ 1 C μ ν λ ( x 1 , x 2 ) = 4 π J ν λ ( x 1 , x 2 ) ,
ρ σ μ ν σ 1 C μ ν λ ( x 1 , x 2 ) = 0.
μ 1 F μ ν α β ( x 1 , x 2 ) = 4 π C α β ν * ( x 2 , x 1 ) ,
μ 1 C μ ν λ ( x 1 , x 2 ) = 4 π J ν λ ( x 1 , x 2 ) ,
ρ σ μ ν σ 1 F μ ν α β ( x 1 , x 2 ) = 0 ,
ρ σ μ ν σ 1 C μ ν λ ( x 1 , x 2 ) = 0.
μ 2 F α β μ ν ( x 1 , x 2 ) = 4 π D ν α β * ( x 2 , x 1 ) ,
μ 2 D λ μ ν ( x 1 , x 2 ) = 4 π J λ ν ( x 1 , x 2 ) ,
ρ σ μ ν σ 2 F α β μ ν ( x 1 , x 2 ) = 0 ,
ρ σ μ ν σ 2 D λ μ ν ( x 1 , x 2 ) = 0 ,
μ 2 = / x 2 μ .
λ 1 J λ ρ ( x 1 , x 2 ) = ρ 2 J λ ρ ( x 1 , x 2 ) = 0 ,
λ 2 C μ ν λ ( x 1 , x 2 ) = 0.
J μ ν ( x 1 , x 2 ) = 1 ( 4 π ) 2 λ 1 ρ 2 F μ λ ν ρ ( x 1 , x 2 ) ,
μ 1 F μ ν α β ( x 1 , x 2 ) = 0 ,
ρ σ μ ν σ 1 F μ ν α β ( x 1 , x 2 ) = 0 ,
C α β ν ( x 1 , x 2 ) = D ν α β ( x 1 , x 2 ) = J ν λ ( x 1 , x 2 ) = 0.
Θ ( k 0 ) = 1 2 [ 1 + ( k 0 ) ] ,
( k 0 ) = 1 , if k 0 > 0 - 1 , if k 0 < 0.
f μ ν α β ( k 1 , k 2 ) Θ ( k 1 0 ) Θ ( k 2 0 ) = F μ ν α β ( x 1 , x 2 ) exp [ i ( k 1 x 1 - k 2 x 2 ) ] d 4 x 1 d 4 x 2 ,
F μ ν α β ( x 1 , x 2 ) = μ 1 α 2 a ν β ( x 1 , x 2 ) - μ 1 β 2 a ν α ( x 1 , x 2 ) - ν 1 α 2 a μ β ( x 1 , x 2 ) + ν 1 β 2 a μ α ( x 1 , x 2 ) ,
C μ ν λ ( x 1 , x 2 ) = μ 1 ν λ ( x 1 , x 2 ) - ν 1 μ λ ( x 1 , x 2 ) ,
a μ ν * ( x 1 , x 2 ) = a ν μ ( x 2 , x 1 ) .
D λ μ ν ( x 1 , x 2 ) = C μ ν λ * ( x 2 , x 1 ) = μ 2 ν λ * ( x 2 , x 1 ) - ν 2 μ λ * ( x 2 , x 1 ) ,
a μ ν ( x 1 , x 2 ) a μ ν ( x 1 , x 2 ) = a μ ν ( x 1 , x 2 ) + μ 1 χ ν ( x 1 , x 2 ) + ν 2 ψ μ ( x 1 , x 2 ) ,
μ ν ( x 1 , x 2 ) μ ν ( x 1 , x 2 ) = μ ν ( x 1 , x 2 ) + μ 1 ϕ ν ( x 1 , x 2 ) ,
ρ μ λ ν μ 1 λ 1 Ω ( x 1 , x 2 ) = 0 ,
1 ρ ρ 1 a μ ν ( x 1 , x 2 ) = 4 π ν μ * ( x 2 , x 1 ) ,
ρ 1 a ρ ν ( x 1 , x 2 ) = 0 ,
δ 1 ρ ρ 1 μ ν ( x 1 , x 2 ) = 4 π J μ ν ( x 1 , x 2 ) ,
ρ 1 ρ ν ( x 1 , x 2 ) = 0 ,
ρ 2 μ ρ ( x 1 , x 2 ) = 0 ,
a μ ν ( R ) ( x 1 , x 2 ) = a μ ν ( x 1 , x 2 ) + μ 1 χ ν ( x 1 , x 2 ) + ν 2 χ μ * ( x 2 , x 1 ) ,
μ ν ( R ) ( x 1 , x 2 ) = μ ν ( x 1 , x 2 ) + μ 1 ϕ ν ( x 1 , x 2 ) ,
χ ν ( x 1 , x 2 ) = 1 4 π 0 1 a 0 ν ( x , x 1 0 ; x 2 ) x 1 - x d 3 x + 1 2 1 ( 4 π ) 2 ν 2 0 1 0 2 × a 00 ( x , x 1 0 ; x , x 2 0 ) x 1 - x x 2 - x d 3 x d 3 x ,
ϕ ν ( x 1 , x 2 ) = 1 4 π 0 1 0 ν ( x , x 1 0 ; x 2 ) x 1 - x d 3 x .
a μ ν ( R ) * ( x 1 , x ˙ 2 ) = a ν μ ( R ) ( x 2 , x 1 ) ,
a 0 ν ( R ) ( x 1 , x 2 ) = ν 0 ( R ) * ( x 2 ; x , x 1 0 ) x 1 - x d 3 x ,
a ν 0 ( R ) ( x 1 , x 2 ) = a 0 ν ( R ) * ( x 2 , x 1 ) = ν 0 ( R ) ( x 1 ; x , x 2 0 ) x 2 - x d 3 x ,
0 ν ( R ) ( x 1 , x 2 ) = J 0 ν ( x , x 1 0 ; x 2 ) x 1 - x d 3 x .
1 ρ ρ 1 a m l ( R ) ( x 1 , x 2 ) = m r s r 1 s p q p l q ( R ) * ( x 2 ; x , x 1 0 ) x 1 - x d 3 x ,
m 1 a m l ( R ) ( x 1 , x 2 ) = 0 ,
1 ρ ρ 1 m ν ( R ) ( x 1 , x 2 ) = m r s r 1 s p q p J q ν ( x , x 1 0 ; x 2 ) x 1 - x d 3 x ,
m 1 m ν ( R ) ( x 1 , x 2 ) = 0 ,
2 ν m ν ( R ) ( x 1 , x 2 ) = 0.
A 0 ( x ) = ρ ( x , x 0 ) x - x d 3 x ,
l A l ( x ) = 0 ,
ρ ρ A l ( x ) = 1 c l r s r s p q p j ˜ q ( x , x 0 ) x - x d 3 x .
a μ ν ( x 1 , x 2 ) = A μ ( x 1 ) A ν * ( x 2 ) ,
μ ν ( x 1 , x 2 ) = A μ ( x 1 ) j ν * ( x 2 ) ,
1 ρ ρ 1 a μ ν ( x 1 , x 2 ) = 0 ,
ρ 1 a ρ ν ( x 1 , x 2 ) = 0 ,
a 0 ν ( R ) ( x 1 , x 2 ) = a ν 0 ( R ) ( x 1 , x 2 ) = 0 ,
1 ρ ρ 1 a m l ( R ) ( x 1 , x 2 ) = 0 ,
m 1 a m l ( R ) ( x 1 , x 2 ) = 0.
E i j ( x 1 , x 2 ) = E i ( x 1 ) E j * ( x 2 ) ,
H i j ( x 1 , x 2 ) = H i ( x 1 ) H j * ( x 2 ) ,
M i j ( x 1 , x 2 ) = E i ( x 1 ) H j * ( x 2 ) ,
N i j ( x 1 , x 2 ) = H i ( x 1 ) E j * ( x 2 ) ,
U i ( x 1 , x 2 ) = E i ( x 1 ) ρ * ( x 2 ) ,
V i ( x 1 , x 2 ) = H i ( x 1 ) ρ * ( x 2 ) ,
X i j ( x 1 , x 2 ) = c - 1 E i ( x 1 ) j ˜ j * ( x 2 ) ,
Y i j ( x 1 , x 2 ) = c - 1 H i ( x 1 ) j ˜ j * ( x 2 ) .
E i j ( x 1 , x 2 ) = 0 1 0 2 a i j ( x 1 , x 2 ) - 0 1 j 2 a i 0 ( x 1 , x 2 ) - i 1 0 2 a 0 j ( x 1 , x 2 ) + i 1 j 2 a 00 ( x 1 , x 2 ) ,
H i j ( x 1 , x 2 ) = δ i j [ m 1 m 2 a l l ( x 1 , x 2 ) - m 1 l 2 a l m ( x 1 , x 2 ) ] - j 1 i 2 a l l ( x 1 , x 2 ) + j 1 l 2 a l i ( x 1 , x 2 ) + l 1 i 2 a j l ( x 1 , x 2 ) - m 1 m 2 a j i ( x 1 , x 2 ) ,
M i j ( x 1 , x 2 ) = j l m l 2 [ i 1 a 0 m ( x 1 , x 2 ) - 0 1 a i m ( x 1 , x 2 ) ] ,
N i j ( x 1 , x 2 ) = i l m l 1 [ j 2 a m 0 ( x 1 , x 2 ) - 0 2 a m j ( x 1 , x 2 ) ] ,
U i ( x 1 , x 2 ) = 0 1 i 0 ( x 1 , x 2 ) - i 1 00 ( x 1 , x 2 ) ,
V i ( x 1 , x 2 ) = - i l m l 1 m 0 ( x 1 , x 2 ) ,
X i j ( x 1 , x 2 ) = - 0 1 i j ( x 1 , x 2 ) + i 1 0 j ( x 1 , x 2 ) ,
Y i j ( x 1 , x 2 ) = i l m l 1 m j ( x 1 , x 2 ) .
U i ( x 1 , x 2 ) = ( 4 π ) - 1 l 2 E i l ( x 1 , x 2 ) ,
V i ( x 1 , x 2 ) = ( 4 π ) - 1 l 2 N i l ( x 1 , x 2 ) ,
X i j ( x 1 , x 2 ) = ( 4 π ) - 1 [ - 0 2 E i j ( x 1 , x 2 ) + i l q l 2 M i q ( x 1 , x 2 ) ] ,
Y i j ( x 1 , x 2 ) = ( 4 π ) - 1 [ - 0 2 N i j ( x 1 , x 2 ) + j l q l 2 H i q ( x 1 , x 2 ) ] .
T μ α ( x 1 , x 2 ) = - 1 4 π [ g ν β F μ ν α β ( x 1 , x 2 ) + 1 4 g μ ν g α β ν ρ σ τ g ρ λ β λ γ δ F σ τ γ δ ( x 1 , x 2 ) ] ,
U μ α ( x 1 , x 2 ) = ( 8 π ) - 1 [ g α τ τ ν ρ σ F μ ν ρ σ ( x 1 , x 2 ) - g μ τ τ ν ρ σ F ρ σ α ν ( x 1 , x 2 ) ] .
ρ 1 T ρ α ( x 1 , x 2 ) = - C α β β * ( x 2 , x 1 ) ,
ρ 2 T α ρ ( x 1 , x 2 ) = - C α β β ( x 1 , x 2 ) ,
ρ 1 U ρ α ( x 1 , x 2 ) = 1 2 α ν ρ σ C ρ σ ν * ( x 2 , x 1 ) ,
ρ 2 U α ρ ( x 1 , x 2 ) = - 1 2 α ν ρ σ C ρ σ ν ( x 1 , x 2 ) .
ρ T ρ α ( x , x ) = - [ C α β β ( x , x ) + C α β β * ( x , x ) ] ,
ρ U ρ α ( x , x ) = - 1 2 g α τ τ ν ρ σ [ C ρ σ ν ( x , x ) - C ρ σ ν * ( x , x ) ] ,
ρ Ω ( x , x ) = ρ 1 Ω ( x 1 , x 2 ) x 1 = x 2 = x + ρ 2 Ω ( x 1 , x 2 ) x 1 = x 2 = x ,
W ( x ) = 1 2 T 00 ( x , x ) = ( 8 π ) - 1 [ E i i ( x , x ) + H i i ( x , x ) ] ,
l ( x ) = - ( 1 / 2 c ) T 0 l ( x , x ) = ( 8 π c ) - 1 l i j [ M i j ( x , x ) - N i j ( x , x ) ] .
U μ α ( x , x ) 0 ,
ρ T ρ α ( x , x ) = - C α β β ( x , x ) ,
C 0 β β ( x , x ) = X i i ( x , x ) = ( 1 / c ) E i ( x ) j ˜ i ( x ) ,
C l β β ( x , x ) = U l ( x , x ) + l m n Y n m ( x , x ) = ρ ( x ) E l ( x ) + ( 1 / c ) l m n J ˜ m ( x ) H n ( x ) .