Abstract

A relationship is established between two correlation coefficients (complex degrees of coherence) frequently used in the theory of partially coherent optical fields. One of them characterizes correlation in the space – time domain, the other in the space – frequency domain. Some relations obtained previously by ad hoc methods are shown to readily follow from our general formula.

© 1995 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford1980).
  2. E. Wolf, in International Trends in Optics, J. W. Goodman, eds. (Academic, San Diego, Calif., 1991), p. 221.
  3. E. Wolf, in Proceedings of Symposium on Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, eds. (North-Holland, Amsterdam, 1992), p. 113.
  4. E. Wolf, in Recent Developments in Quantum Optics, R. Ingua, ed. (Plenum, New York, 1993), p. 369.
    [CrossRef]
  5. L. Mandel, E. Wolf, J. Opt. Soc. Am. 66, 529 (1976).
    [CrossRef]
  6. E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
    [CrossRef]
  7. As shown in Ref. 8, the spectral degree of coherence μ12 does not change on filtering and hence we do not need to distinguish between μ12(+)(ω) and μ12(ω).
  8. E. Wolf, Opt. Lett. 8, 250 (1983).
    [CrossRef] [PubMed]
  9. L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961).
    [CrossRef]

1983 (1)

1982 (1)

1976 (1)

1961 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford1980).

Mandel, L.

Wolf, E.

E. Wolf, Opt. Lett. 8, 250 (1983).
[CrossRef] [PubMed]

E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
[CrossRef]

L. Mandel, E. Wolf, J. Opt. Soc. Am. 66, 529 (1976).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford1980).

E. Wolf, in International Trends in Optics, J. W. Goodman, eds. (Academic, San Diego, Calif., 1991), p. 221.

E. Wolf, in Proceedings of Symposium on Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, eds. (North-Holland, Amsterdam, 1992), p. 113.

E. Wolf, in Recent Developments in Quantum Optics, R. Ingua, ed. (Plenum, New York, 1993), p. 369.
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Lett. (1)

Other (5)

As shown in Ref. 8, the spectral degree of coherence μ12 does not change on filtering and hence we do not need to distinguish between μ12(+)(ω) and μ12(ω).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford1980).

E. Wolf, in International Trends in Optics, J. W. Goodman, eds. (Academic, San Diego, Calif., 1991), p. 221.

E. Wolf, in Proceedings of Symposium on Huygens’ Principle 1690–1990: Theory and Applications, H. Blok, H. A. Ferwerda, H. K. Kuiken, eds. (North-Holland, Amsterdam, 1992), p. 113.

E. Wolf, in Recent Developments in Quantum Optics, R. Ingua, ed. (Plenum, New York, 1993), p. 369.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Behavior of the absolute value of the equal-time degree of coherence γ12(0) ≡ γ(r2r1, 0), with the choice Γ/ω0 = 10−4. The scale factor λ0 on the horizontal axis is the wavelength associated with the frequency ω0 (λ0 = 2πc/ω0).

Equations (33)

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γ 12 ( τ ) = Γ 12 ( τ ) Γ 11 ( 0 ) Γ 22 ( 0 ) .
Γ 12 ( τ ) = V * ( r 1 , t ) V ( r 2 , t + τ )
μ 12 ( ω ) = W 12 ( ω ) W 11 ( ω ) W 22 ( ω ) .
W 12 ( ω ) 1 2 π Γ 12 ( τ ) exp ( i ω τ ) d τ
W 11 ( ω ) = S 1 ( ω ) = 1 2 π Γ 11 ( τ ) exp ( i ω τ ) d τ
W 22 ( ω ) = S 2 ( ω ) = 1 2 π Γ 22 ( τ ) exp ( i ω τ ) d τ
γ 12 ( τ ) μ 12 ( ω ) = Γ 12 ( τ ) W 12 ( ω ) S 1 ( ω ) Γ 11 ( 0 ) S 2 ( ω ) Γ 22 ( 0 ) .
Γ 11 ( 0 ) = 0 S 1 ( ω ) d ω .
S 1 ( ω ) Γ 11 ( 0 ) S 2 ( ω ) Γ 22 ( 0 ) = s 1 ( ω ) s 2 ( ω ) ,
s j ( ω ) = S j ( ω ) S j ( ω ) d ω ( j = 1 , 2 )
γ 12 ( τ ) W 12 ( ω ) Γ 12 ( τ ) = s 1 ( ω ) s 2 ( ω ) μ 12 ( ω ) .
γ 12 ( τ ) = s 1 ( ω ) s 2 ( ω ) μ 12 ( ω ) exp ( i ω τ ) d ω .
μ 12 ( ω ) = 1 2 π 1 s 1 ( ω ) s 2 ( ω ) γ 12 ( τ ) exp ( i ω τ ) d τ .
s 2 ( ω ) = s 1 ( ω ) = s ( ω ) .
γ 12 ( τ ) = μ ˜ 12 ~ 0 s ( ω ) exp ( i ω τ ) d ω .
μ ˜ 12 ~ = γ 12 ( 0 ) .
γ 12 ( τ ) = γ 12 ( 0 ) exp ( i ω 0 τ ) , | τ | 1 / Δ ω ,
γ 12 ( τ ) = μ ˜ 12 ~ exp ( i ω 0 τ ) , | τ | 1 / Δ ω .
μ ˜ 12 ~ = γ 12 ( + ) ( 0 ) .
μ 12 ( ω ) = η 12 exp ( i ω τ 12 ) ,
γ 12 ( τ ) = η 12 0 s ( ω ) exp [ i ω ( τ τ 12 ) ] d ω
γ 12 ( τ ) = η 12 Γ 11 ( 0 ) 0 S ( ω ) exp [ i ω ( τ τ 12 ) ] d ω = η 12 Γ 11 ( τ τ 12 ) Γ 11 ( 0 ) .
γ 12 ( τ ) = η 12 γ 11 ( τ τ 12 ) .
γ 12 ( τ 12 ) = η 12 .
γ 12 ( τ ) = γ 12 ( τ 12 ) γ 11 ( τ τ 12 ) .
μ 12 ( ω ) = γ 12 ( τ 12 ) exp ( i ω τ 12 ) ,
s 2 ( ω ) = s 1 ( ω ) = 1 Γ 2 π exp [ ( ω ω 0 ) 2 / 2 Γ 2 ] , ( Γ / ω 0 1 ) ,
μ 12 ( ω ) = exp [ i k ( r 2 r 1 ) ] ,
γ 12 ( τ ) = exp [ 1 2 Γ 2 ( τ r / c ) 2 ] × exp [ i ω 0 ( τ r / c ) ] ,
r = r 2 r 1 .
γ 12 ( 0 ) = exp ( 1 2 Γ 2 t 2 ) exp ( i ω 0 t ) ,
t = ( r 2 r 1 ) / c .
γ 12 ( t ) = 1.

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