Abstract

We explore the propagation of the cross-spectral density for scalar and electromagnetic fields based on generalized radiances that are exactly conserved along rays. Two formulas are derived: The first uses all rays to calculate the cross-spectral density exactly, while the second uses only the subset of those rays that pass through a single spatial point to construct an infinite series expression for the cross-spectral density. The evaluation of the truncated series is examined numerically for a variety of fields of varying angular width and coherence and is found to exhibit better convergence to the cross-spectral density when the rays through the centroid between the two observation points are used, when the fields are less coherent, and when the fields are more paraxial. In generalizing the series formula, two new cross-spectral correlations associated with the flux and energy density are examined.

© 2008 Optical Society of America

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    [CrossRef]
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2007 (1)

2005 (1)

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

2004 (1)

2002 (1)

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
[CrossRef]

2001 (4)

1999 (1)

1998 (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

1994 (1)

1991 (2)

1982 (1)

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927-936 (1982).

1981 (2)

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15-31 (1981).
[CrossRef]

J. J. Stamnes, “Focusing of a perfect wave and the airy pattern formula,” Opt. Commun. 37, 311-314 (1981).
[CrossRef]

1975 (1)

1972 (1)

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. VUZ Radiofiz. 15, 1419-1421 (1972).

1968 (1)

1967 (1)

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328-1334 (1967).
[CrossRef]

1964 (1)

L. Dolin, “Ray description of weakly inhomogeneous wave fields,” Izv. VUZ Radiofiz. 7, 559-562 (1964).

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Alonso, M. A.

Apresyan, L. A.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996), pp. 11-19.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996), pp. 122-125.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 193-199.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 918-920.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 572-577.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 13-27.

Carney, P. S.

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Carter, W.

Dolin, L.

L. Dolin, “Ray description of weakly inhomogeneous wave fields,” Izv. VUZ Radiofiz. 7, 559-562 (1964).

Forbes, G. W.

Friberg, A. T.

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927-936 (1982).

Kravtsov, Yu. A.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996), pp. 122-125.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996), pp. 11-19.

Larkin, K. G.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 170-176.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 181-183.

Mehta, C. L.

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328-1334 (1967).
[CrossRef]

Ovchinnikov, G. I.

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. VUZ Radiofiz. 15, 1419-1421 (1972).

Pedersen, H. M.

Petruccelli, J. C.

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Schotland, J. C.

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard and K. G. Larkin, “Wigner function for nonparaxial wave fields,” J. Opt. Soc. Am. A 18, 2486-2490 (2001).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, “Focusing of a perfect wave and the airy pattern formula,” Opt. Commun. 37, 311-314 (1981).
[CrossRef]

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15-31 (1981).
[CrossRef]

Tatarskii, V. I.

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. VUZ Radiofiz. 15, 1419-1421 (1972).

Vicent, L. E.

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
[CrossRef]

Walther, A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Wolf, E.

E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024-2026 (1994).
[CrossRef] [PubMed]

W. Carter and E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067-1071 (1975).
[CrossRef]

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328-1334 (1967).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 170-176.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 193-199.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 181-183.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 918-920.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 572-577.

Wolf, K. B.

Zysk, A. M.

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Izv. VUZ Radiofiz. (2)

L. Dolin, “Ray description of weakly inhomogeneous wave fields,” Izv. VUZ Radiofiz. 7, 559-562 (1964).

G. I. Ovchinnikov and V. I. Tatarskii, “On the relation between coherence theory and the radiative transfer equation,” Izv. VUZ Radiofiz. 15, 1419-1421 (1972).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (9)

Opt. Commun. (2)

J. J. Stamnes, “Focusing of a perfect wave and the airy pattern formula,” Opt. Commun. 37, 311-314 (1981).
[CrossRef]

L. E. Vicent and M. A. Alonso, “Generalized radiometry as a tool for the propagation of partially coherent fields,” Opt. Commun. 207, 101-112 (2002).
[CrossRef]

Opt. Eng. (1)

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927-936 (1982).

Opt. Lett. (1)

Phys. Rev. (2)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. III. Cross-spectral tensors,” Phys. Rev. 161, 1328-1334 (1967).
[CrossRef]

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: A scalar treatment,” Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Other (8)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 181-183.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 193-199.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 13-27.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996), pp. 11-19.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 918-920.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 572-577.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 170-176.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon and Breach, 1996), pp. 122-125.

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Figures (6)

Fig. 1
Fig. 1

The measurement point r b associated with observation points r 1 and r 2 and ray ( l , u ) . The field plotted in (b) corresponds to the “perfect wave” ( j = 0 ) case.

Fig. 2
Fig. 2

Spectral density S ( r ) in two dimensions of beams given by Eq. (26) for (a) a Gaussian-like nearly paraxial beam ( σ = 0.1 ) , (b) a nonparaxial beam ( σ = 0.5 ) , and (c) a nearly circular wave ( σ = 1.0 ) . (Notice the different scale in this case.) The coherence of each beam varies as indicated by the parameter ϵ, from highly incoherent on the left to highly coherent on the right.

Fig. 3
Fig. 3

Comparisons of the cross-spectral density calculations using Eq. (22) for a partially coherent ( ϵ = 2 2.5 ) nonparaxial ( σ = 0.5 ) beam. The exact complex degree of coherence, μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) S ( r 1 ) S ( r 2 ) , is plotted in (a) as a function of r 2 for k r 1 = ( 50 , 50 ) . The grayscale indicates the magnitude, and the black lines indicate contours of 0 phase. The relative error of Eq. (22) up to fourth order in derivatives for calculating the cross-spectral density is plotted in (b) by summing over all rays through the indicated points.

Fig. 4
Fig. 4

Complex degree of coherence μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) S ( r 1 ) S ( r 2 ) , with magnitude represented by the grayscale and contours of 0 phase indicated by the black lines for the three narrow Gaussian-like beams indicated in Fig. 2a. Equation (23) is used to calculate the cross-spectral density, and the relative error of this formula is plotted up to fourth order.

Fig. 5
Fig. 5

Complex degree of coherence μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) S ( r 1 ) S ( r 2 ) , with magnitude represented by the grayscale and contours of 0 phase indicated by the black lines for the three nonparaxial beams indicated in Fig. 2b. Equation (23) is used to calculate the cross-spectral density, and the relative error of this formula is plotted up to fourth order.

Fig. 6
Fig. 6

Complex degree of coherence μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) S ( r 1 ) S ( r 2 ) , with magnitude represented by the grayscale and contours of 0 phase indicated by the black lines for the three nearly circular waves indicated in Fig. 2c. Equation (23) is used to calculate the cross-spectral density, and the relative error of this formula is plotted up to fourth order.

Equations (38)

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u B ( r , u ) = 0
S ( r ) = 4 π B ( r , u ) d Ω u ,
W ( r 1 , r 2 ; ω ) = U * ( r 1 ; ω ) U ( r 2 ; ω ) ,
W ( r 1 , r 2 ) = 4 π B ( r ¯ , u ) exp ( i k u δ r ) d Ω u ,
U ( r ) = k 2 π 4 π A ( u ) exp ( i k r u ) d Ω u ,
A ( u 1 , u 2 ) = A * ( u 1 ) A ( u 2 ) .
W ( r 1 , r 2 ) = ( k 2 π ) 2 4 π 4 π A ( u 1 , u 2 ) exp [ i k ( r 2 u 2 r 1 u 1 ) ] d Ω u 1 d Ω u 2 .
u 1 , 2 = u cos α 2 w ( u , ϕ ) sin α 2 ,
W ( r 1 , r 2 ) = ( k 2 π ) 2 4 π 0 π 0 2 π A [ u cos α 2 w ( u , ϕ ) sin α 2 , u cos α 2 + w ( u , ϕ ) sin α 2 ] × exp { i k [ δ r u cos α 2 + 2 r ¯ w ( u , ϕ ) sin α 2 ] } sin α d ϕ d α d Ω u .
M ( j ) ( l , u ) = ( k 2 π ) 2 0 2 π 0 π A [ u cos α 2 w ( u , ϕ ) sin α 2 , u cos α 2 + w ( u , ϕ ) sin α 2 ] × exp [ 2 i k l w ( u , ϕ ) sin α 2 ] sin α cos j α 2 d α d ϕ ,
l = P u r = r u ( u r ) ,
B ( j ) ( r , u ) = M ( j ) ( P u r , u ) = ( k 2 π ) 2 0 2 π 0 π A [ u cos α 2 w ( u , ϕ ) sin α 2 , u cos α 2 + w ( u , ϕ ) sin α 2 ] × exp [ 2 i k r w ( u , ϕ ) sin α 2 ] sin α cos j α 2 d α d ϕ ,
F ( r ) : = 1 2 i k [ U * ( r ) U ( r ) U ( r ) U * ( r ) ] = 4 π u B ( 1 ) ( r , u ) d Ω u ,
H ( r ) : = 1 2 c [ U * ( r ) U ( r ) + k 2 U * ( r ) U ( r ) ] = 1 c 4 π B ( 2 ) ( r , u ) d Ω u .
A [ u cos α 2 w ( u , ϕ ) sin α 2 , u cos α 2 + w ( u , ϕ ) sin α 2 ] cos j α 2 = R 2 M ( j ) ( l , u ) exp [ 2 i k l w ( u , ϕ ) sin α 2 ] d 2 l = R 2 B ( j ) ( l , u ) exp [ 2 i k l w ( u , ϕ ) sin α 2 ] d 2 l ,
W ( r 1 , r 2 ) = 4 π R 2 B ( j ) ( l , u ) U p ( j ) [ P u ( r 1 + r 2 ) 2 l , u ( r 2 r 1 ) ] d 2 l d Ω u ,
U p ( j ) ( ρ , z ) = k 2 π 0 π 2 cos ( 1 j ) θ exp ( i k z cos θ ) × J 0 ( k ρ sin θ ) sin θ d θ .
W ( r 1 , r 2 ) = ( k 2 π ) 2 4 π 0 2 π 0 π A [ u cos α 2 w ( u , ϕ ) sin α 2 , u cos α 2 + w ( u , ϕ ) sin α 2 ] × exp [ 2 i k ( r s ) w ( u , ϕ ) sin α 2 ] cos j α 2 × [ exp ( i k u δ r cos α 2 ) cos j α 2 ] sin α d α d ϕ d Ω u .
exp [ 2 i k ( r s ) w ( u , ϕ ) sin α 2 ] = exp ( s ) exp [ 2 i k r w ( u , ϕ ) sin α 2 ] ,
exp ( s ) = n = 0 ( s ) n n !
exp ( i k u δ r cos α 2 ) cos j α 2 = exp ( i k u δ r 1 sin 2 α 2 ) ( 1 sin 2 α 2 ) j 2 = exp ( i k u δ r ) [ 1 + j i k u δ r 2 sin 2 α 2 ] + [ j ( j + 2 ) i ( 1 + 2 j ) k u δ r k 2 ( u δ r ) 2 8 sin 4 α 2 + ] .
sin 2 n α 2 exp [ 2 i k r w ( u , ϕ ) sin α 2 ] = ( 1 ) n ( 2 k ) 2 n 2 n exp [ 2 i k r w ( u , ϕ ) sin α 2 ] .
W ( r 1 , r 2 ) = exp ( s ) 4 π exp ( i k u δ r ) [ 1 + i k u δ r j 8 k 2 2 + j ( j + 2 ) i ( 1 + 2 j ) k u δ r k 2 ( u δ r ) 2 128 k 4 4 + ] B ( j ) ( r , u ) d Ω u .
W ( r 1 , r 2 ) = 4 π exp ( i k u δ r ) [ 1 + i k u δ r j 8 k 2 ¯ 2 ] [ + j ( j + 2 ) i ( 1 + 2 j ) k u δ r k 2 ( u δ r ) 2 128 k 4 ¯ 4 + ] × B ( j ) ( r ¯ , u ) d Ω u ,
W ( r 1 , r 2 ) = 4 π B 0 sin ( k δ r ) k δ r ,
W ( r 1 , r 2 ) = 4 π B 0 { [ j 0 ( k δ r ) j 1 ( k δ r ) k δ r ] I + j 2 ( k δ r ) δ r 2 δ r δ r } ,
A ( u 1 , u 2 ) = 1 2 π I 0 ( σ 2 ) exp [ ( u 2 u 1 ) z ̂ 2 σ 2 ] exp ( u 1 u 2 1 2 ϵ 2 ) ,
R n ( r 1 , r 2 ) = W ( r 1 , r 2 ) W n ( r 1 , r 2 ) W ( r 1 , r 2 ) .
F c ( r 1 , r 2 ) = 1 2 i k U * ( r 1 ) 2 U ( r 2 ) 1 U * ( r 1 ) U ( r 2 ) = 1 2 i k ( 2 1 ) W ( r 1 , r 2 ) ,
H c ( r 1 , r 2 ) = 1 2 c U * ( r 1 ) U ( r 2 ) + k 2 1 U * ( r 1 ) 2 U ( r 2 ) = 1 2 c ( 1 + k 2 1 2 ) W ( r 1 , r 2 ) .
( i 2 + k 2 ) F c ( r 1 , r 2 ) = 0 ,
( i 2 + k 2 ) H c ( r 1 , r 2 ) = 0 .
1 F c ( r 1 , r 2 ) = 2 F c ( r 1 , r 2 ) = i k c H c ( r 1 , r 2 ) .
δ W ( r 1 , r 2 ) = i k F c ( r 1 , r 2 ) ,
δ F c ( r 1 , r 2 ) = i k δ 2 W ( r 1 , r 2 ) = i k c H c ( r 1 , r 2 ) ,
¯ F c = i k ¯ δ W ( r 1 , r 2 ) = 0 .
Θ ( j ) ( r 1 , r 2 ) = 4 π μ ( j ) exp ( i k u δ r ) [ 1 + i k u δ r + n j 8 k 2 ¯ 2 ] + [ ( j n ) ( j n + 2 ) i ( 1 + 2 j 2 n ) k u δ r k 2 ( u δ r ) 2 128 k 4 ¯ 4 + ] B ( j ) ( r ¯ , u ) d Ω u ,
Θ ( n ) ( r 1 , r 2 ) = 4 π R 2 μ ( j ) B ( j ) ( l , u ) U p ( j n ) [ P u ( r 1 + r 2 ) 2 l , u ( r 2 r 1 ) ] d 2 l d Ω u .

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