Abstract

A radiometric model for the propagation of coherence is formulated, which greatly simplifies the determination of the cross-spectral density and of the spectral degree of coherence of the field at an arbitrary distance from any planar, secondary, quasi-homogeneous source. The radiance function, which plays a central role in this model, satisfies all the postulates of traditional radiometry.

© 1994 Optical Society of America

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References

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  1. L. S. Dolin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 244 (1964).
  2. V. I. Tatarskii, The Effects of the Turbulant Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), Sec. 63.
  3. E. Collett, J. T. Foley, E. Wolf, J. Opt. Soc. Am. 67, 465 (1977).
    [CrossRef]
  4. H. M. Pedersen, J. Opt. Soc. Am. A 8, 176 (1991); H. M. Pedersen, J. Opt. Soc. Am. A 9, 1626 (1992); .
    [CrossRef]
  5. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 60.
  6. E. Wolf, J. Opt. Soc. Am. 72, 343 (1982); J. Opt. Soc. Am. A 3, 76 (1986).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986), Sec. 3.7.
  8. E. W. Marchand, E. Wolf, J. Opt. Soc. Am. 62, 379 (1972).
    [CrossRef]
  9. A. Walther, J. Opt. Soc. Am. 58, 1256 (1968).
    [CrossRef]
  10. Equivalent but formally different generalizations of Walther’s definition, valid at points throughout the half-space z > 0, are considered in Refs. 11 and 12. The function Bν(r, s) defined by Eq. (8) may be regarded as an analog of the Wigner distribution function well known in quantum mechanics.13
  11. M. Bastians, J. Opt. Soc. Am. A 3, 1227 (1986).
    [CrossRef]
  12. A. T. Friberg, Appl. Opt. 25, 4547 (1986).
    [CrossRef] [PubMed]
  13. K. Imre, E. Ozizmer, M. Rosenblum, P. Zweifel, J. Math. Phys. 8, 1097 (1967).
    [CrossRef]
  14. A. T. Friberg, J. Opt. Soc. Am. 69, 192 (1979).
    [CrossRef]
  15. K. Kim, E. Wolf, J. Opt. Soc. Am. A 4, 1233 (1987).
    [CrossRef]
  16. W. H. Carter, E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
    [CrossRef]
  17. E. Wolf, W. H. Carter, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf (Plenum, New York, 1978), p. 415.

1991 (1)

1987 (1)

1986 (2)

1982 (1)

1979 (1)

1977 (2)

1972 (1)

1968 (1)

1967 (1)

K. Imre, E. Ozizmer, M. Rosenblum, P. Zweifel, J. Math. Phys. 8, 1097 (1967).
[CrossRef]

1964 (1)

L. S. Dolin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 244 (1964).

Bastians, M.

Carter, W. H.

W. H. Carter, E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
[CrossRef]

E. Wolf, W. H. Carter, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf (Plenum, New York, 1978), p. 415.

Collett, E.

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 60.

Dolin, L. S.

L. S. Dolin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 244 (1964).

Foley, J. T.

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986), Sec. 3.7.

Imre, K.

K. Imre, E. Ozizmer, M. Rosenblum, P. Zweifel, J. Math. Phys. 8, 1097 (1967).
[CrossRef]

Kim, K.

Marchand, E. W.

Ozizmer, E.

K. Imre, E. Ozizmer, M. Rosenblum, P. Zweifel, J. Math. Phys. 8, 1097 (1967).
[CrossRef]

Pedersen, H. M.

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 60.

Rosenblum, M.

K. Imre, E. Ozizmer, M. Rosenblum, P. Zweifel, J. Math. Phys. 8, 1097 (1967).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulant Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), Sec. 63.

Walther, A.

Wolf, E.

Zweifel, P.

K. Imre, E. Ozizmer, M. Rosenblum, P. Zweifel, J. Math. Phys. 8, 1097 (1967).
[CrossRef]

Appl. Opt. (1)

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

L. S. Dolin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 244 (1964).

J. Math. Phys. (1)

K. Imre, E. Ozizmer, M. Rosenblum, P. Zweifel, J. Math. Phys. 8, 1097 (1967).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (3)

Other (5)

E. Wolf, W. H. Carter, in Coherence and Quantum Optics IV, L. Mandel, E. Wolf (Plenum, New York, 1978), p. 415.

V. I. Tatarskii, The Effects of the Turbulant Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), Sec. 63.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 60.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986), Sec. 3.7.

Equivalent but formally different generalizations of Walther’s definition, valid at points throughout the half-space z > 0, are considered in Refs. 11 and 12. The function Bν(r, s) defined by Eq. (8) may be regarded as an analog of the Wigner distribution function well known in quantum mechanics.13

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Figures (4)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Illustrating notation relating to formula (10).

Fig. 3
Fig. 3

Illustrating the meaning of the symbols appearing in Eqs. (19 and (20).

Fig. 4
Fig. 4

Behavior of the modulus of the spectral degree of coherence at pairs of points P1 and P2 on the normal through the center of a uniform, circular, secondary, planar, quasi-homogeneous, Lambertian source σ, with ka = 1.2 × 105, kz1 = 1.2 × 107.

Equations (19)

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W ( r 1 , r 2 , ν ) = [ U ( r 1 , ν ) ] * U ( r 2 , ν ) ,
U ( r , ν ) = a ( s , ν ) exp ( ik s r ) d 2 s ,
s z = + 1 s 2 when s 2 1 ,
= + s 2 1 when s 2 > 1 ,
W ( r 1 , r 2 , ν ) = A ( s 1 , s 2 , ν ) × exp [ ik ( s 2 r 2 s 1 r 1 ) ] d 2 s 1 d 2 s 2 ,
A ( s 1 , s 2 , ν ) = a * ( s 1 , ν ) a ( s 2 , ν )
s = ( s 1 + s 2 ) / 2 , s = s 2 s 1 .
W ( r 1 , r 2 , ν ) = d 2 s d 2 s A ( s s / 2 , s + s / 2 , ν ) exp { i k [ s ( r 2 r 1 ) + s ( r 2 + r 1 ) / 2 ] } ,
B ν ( r , s ) = s z s 2 4 A ( s s / 2 , s + s / 2 ) × exp ( ik s r ) d 2 s .
W ( r 1 , r 2 , ν ) = ( 2 π ) B ν ( r 1 + r 2 2 , s ) × exp [ iks ( r 2 r 1 ) ] d Ω ,
B ν ( r , s ) = k 2 s z S ( 0 ) ( ρ 0 , ν ) g ˜ ( 0 ) ( k s , ν ) when s Ω P = 0 when s Ω P } ,
g ˜ ( 0 ) ( f , ν ) = 1 ( 2 π ) 2 σ g ( 0 ) ( ρ , ν ) exp ( if ρ ) d 2 ρ
r ¯ = ( r 1 + r 2 ) / 2
W ( r 1 , r 2 , ν ) = k 2 σ S ( 0 ) ( ρ ¯ 0 , ν ) g ˜ ( k s , ν ) ( z ¯ R ¯ ) 2 × exp [ i k s ( r 2 r 1 ) ] d 2 ρ ¯ 0 R ¯ 2 ,
B ν ( r ¯ , s ) = 1 2 π S ( 0 ) ( ν ) when s Ω p ¯ = 0 when s Ω p ¯ } .
W ( r 1 , r 2 , ν ) = 1 2 π S ( 0 ) ( ν ) Ω p ¯ exp [ ik s ( r 2 r 1 ) ] .
s ( r 2 r 1 ) = ( z 2 z 1 ) cos α ,
W ( z 1 z 2 , ν ) = 1 2 π S ( 0 ) ( ν ) 0 2 π d ϕ 0 α ¯ sin α × exp [ ik ( z 2 z 1 ) cos α ] d α ,
μ ( z 1 z 2 , ν ) W ( z 1 , z 2 , ν ) [ W ( z 1 , z 2 , ν ) ] 1 / 2 [ W ( z 1 , z 2 , ν ) ] 1 / 2 = sin 2 ( α ¯ / 2 ) sin 2 ( α ¯ 1 / 2 ) sin 2 ( α ¯ 2 / 2 ) { sin [ k ( z 2 z 1 ) sin 2 ( α ¯ / 2 ) [ k ( z 2 z 1 ) sin 2 ( α ¯ / 2 ) } × exp [ ik ( z 2 z 1 ) cos 2 ( α ¯ / 2 ) ] ,

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