Abstract

An exact radiometric model is proposed for the nonparaxial propagation of fully or partially coherent electromagnetic radiation within anisotropic and/or chiral media in terms of rectilinear ray propagation. The definition of the generalized radiance is motivated by a geometric interpretation, based on the plane-wave decomposition of the field in the medium. The new representations are illustrated through two examples, corresponding to the propagation of two-dimensional fields within birefringent and chiral media.

© 2011 Optical Society of America

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References

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Radiofizika 7, 559–562 (1964).
  3. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  4. A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28 (1981).
    [CrossRef]
  5. S. Abe and J. T. Sheridan, “Wigner optics in the metaxial regime,” Optik 114, 139–141 (2003).
    [CrossRef]
  6. G. Ovchinnikov and V. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiofizika 15, 1419–1421 (1972).
  7. H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991).
    [CrossRef]
  8. R. G. Littlejohn and R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  9. M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).
    [CrossRef]
  10. M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids,” J. Math. Phys. 43, 5857–5871 (2002).
    [CrossRef]
  11. M. Alonso, “Exact description of free electromagnetic wave fields in terms of rays,” Opt. Express 11, 3128–3135 (2003).
    [CrossRef] [PubMed]
  12. M. A. Alonso, “Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space,” J. Opt. Soc. Am. A 21, 2233–2243 (2004).
    [CrossRef]
  13. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487(1999).
    [CrossRef]
  14. C. J. R. Sheppard and K. G. Larkin, “Wigner function for nonparaxial wave fields,” J. Opt. Soc. Am. A 18, 2486–2490 (2001).
    [CrossRef]
  15. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  16. J. C. Petruccelli and M. A. Alonso, “Propagation of partially coherent fields through planar dielectric boundaries using angle-impact Wigner functions I. Two dimensions,” J. Opt. Soc. Am. A 24, 2590–2603 (2007).
    [CrossRef]
  17. J. C. Petruccelli and M. A. Alonso, “Propagation of nonparaxial partially coherent fields across interfaces using generalized radiometry,” J. Opt. Soc. Am. A 26, 2012–2022 (2009).
    [CrossRef]
  18. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
    [CrossRef]
  19. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
    [CrossRef]
  20. 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]

2009 (1)

2007 (1)

2004 (1)

2003 (2)

2002 (2)

M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids,” J. Math. Phys. 43, 5857–5871 (2002).
[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]

2001 (2)

1999 (1)

1993 (3)

1991 (1)

1981 (1)

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28 (1981).
[CrossRef]

1972 (1)

G. Ovchinnikov and V. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiofizika 15, 1419–1421 (1972).

1968 (1)

1964 (1)

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Radiofizika 7, 559–562 (1964).

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

1932 (1)

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

Abe, S.

S. Abe and J. T. Sheridan, “Wigner optics in the metaxial regime,” Optik 114, 139–141 (2003).
[CrossRef]

Alonso, M.

Alonso, M. A.

Chipman, R. A.

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Dolin, L.

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Radiofizika 7, 559–562 (1964).

Forbes, G. W.

Friberg, A. T.

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28 (1981).
[CrossRef]

Hillman, L. W.

Larkin, K. G.

Littlejohn, R. G.

McClain, S. C.

Ovchinnikov, G.

G. Ovchinnikov and V. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiofizika 15, 1419–1421 (1972).

Pedersen, H. M.

Petruccelli, J. C.

Pogosyan, G. S.

M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids,” J. Math. Phys. 43, 5857–5871 (2002).
[CrossRef]

Sheppard, C. J. R.

Sheridan, J. T.

S. Abe and J. T. Sheridan, “Wigner optics in the metaxial regime,” Optik 114, 139–141 (2003).
[CrossRef]

Tatarskii, V.

G. Ovchinnikov and V. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiofizika 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]

Winston, R.

Wolf, K. B.

M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids,” J. Math. Phys. 43, 5857–5871 (2002).
[CrossRef]

K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487(1999).
[CrossRef]

J. Math. Phys. (1)

M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids,” J. Math. Phys. 43, 5857–5871 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991).
[CrossRef]

R. G. Littlejohn and R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).
[CrossRef]

M. A. Alonso, “Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space,” J. Opt. Soc. Am. A 21, 2233–2243 (2004).
[CrossRef]

K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487(1999).
[CrossRef]

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

J. C. Petruccelli and M. A. Alonso, “Propagation of partially coherent fields through planar dielectric boundaries using angle-impact Wigner functions I. Two dimensions,” J. Opt. Soc. Am. A 24, 2590–2603 (2007).
[CrossRef]

J. C. Petruccelli and M. A. Alonso, “Propagation of nonparaxial partially coherent fields across interfaces using generalized radiometry,” J. Opt. Soc. Am. A 26, 2012–2022 (2009).
[CrossRef]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
[CrossRef]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
[CrossRef]

Opt. Acta (1)

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28 (1981).
[CrossRef]

Opt. Commun. (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]

Opt. Express (1)

Optik (1)

S. Abe and J. T. Sheridan, “Wigner optics in the metaxial regime,” Optik 114, 139–141 (2003).
[CrossRef]

Phys. Rev. (1)

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

Radiofizika (2)

L. Dolin, “Beam description of weakly inhomogeneous wave fields,” Radiofizika 7, 559–562 (1964).

G. Ovchinnikov and V. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiofizika 15, 1419–1421 (1972).

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Illustration of the change of variables from the unit vectors u 1 and u 2 to a unit vector u and two angles, α and ϕ. (a) α is the angle between u 1 and u 2 , and u bisects these vectors. Note that u 1 and u 2 then correspond to the two intersections of the following three surfaces: the unit sphere, a plane perpendicular to u, whose distance from the origin is controlled by α, and a plane containing u, whose orientation is controlled by the angle ϕ. (b) The separation between u 1 and u 2 then has length equal to 2 sin α / 2 , and its direction can be parametrized in terms of a unit vector w ( u , ϕ ) , perpendicular to u and at an angle ϕ with respect to an arbitrary reference.

Fig. 2
Fig. 2

Illustration of the change of variables for a birefringent anisotropic medium, where n a ( u ) u parametrizes an ellipsoid and n b ( u ) u parametrizes a sphere, while vectors K p , p ( i ) ( u , α , ϕ ) w are constrained to a plane oriented perpendicular to u. The notation a i = n a ( u i ) u i , b i = n b ( u i ) u i has been adopted in this figure for simplicity, and these vectors are represented by arrows that are i) coplanar with u and ii) point from the origin to points lying in the intersection of the plane with the ellipse and sphere. The vector w ( u , ϕ ) [ K p , p ( 1 ) ( u , α , ϕ ) + K p , p ( 2 ) ( u , α , ϕ ) ] for a single choice of ϕ is denoted in each figure by the dashed arrow. This change of variables is shown graphically here for (a)  p = p = a , (b)  p = p = b , and (c)  p = a , p = b . Note that, for the cases in (a) and (c), the ray direction vector u may point in a direction that does not lie between u 1 and u 2 .

Fig. 3
Fig. 3

Illustration of the change of variables for two-dimensional extraordinary plane-wave propagation in an electrically birefringent, magnetically isotropic medium. The ellipse is described by n ( θ ) u ( θ ) for all θ. The generalized radiance for a ray direction u ( θ ) is given by an integral over all pairs of plane-wave directions corresponding to intersections of the ellipse with line segments (dashed lines) oriented perpendicular to u and restricted to the “forward” (unshaded) half-ellipse.

Fig. 4
Fig. 4

Generalized radiances for the energy density and Poynting vector (magnitude) are shown in the top two figures of each column for a nonparaxial generalization of a Gaussian Schell-model field propagating in two dimensions in (a) free space and (b) a birefringent medium whose optical axis is oriented at an angle of 45 ° with respect to the field’s central angle. Note that the radiances only depend on r through the quantity r = r · w 0 ( θ ) . The bottom figures illustrate the electric energy density profile and the Poynting vector over a region of space. The magnitude of w e over this region is given by the shading of the plot, while the magnitude and direction of the Poynting vector are illustrated by the arrows.

Fig. 5
Fig. 5

Illustration of the change of variables for a chiral medium in two dimensions. The circles represent n p u , with right-hand circular polarization represented by the outer circle and left-hand circular polarization by the inner circle. For p = p , shown in (a) and (b), the dashed lines join two points on the same circle, and, therefore, the geometry is identical to the isotropic case. For p p , shown in (c), the dashed lines join points in different circles. Note that, for each line, there are four such combinations of points. The shorthand notation N p , p = K p , p ( 1 ) + K p , p ( 2 ) is adopted in this figure for simplicity.

Fig. 6
Fig. 6

Components of the energy density (bottom row) and associated generalized radiances (top row) given by Eqs. (38) due to (a) only left-hand circular polarization, (b) only right-hand circular polarization, and (c) the sum of all polarization components for the field described by Eq. (39).

Fig. 7
Fig. 7

Components of the Poynting vector (bottom row) as well as the u (top row) and y ^ (middle row) components of B P due to (a) only left-hand circular polarization, (b) only right-hand circular polarization, and (c) the sum of all polarization components for the field described by Eq. (39). The radiances only depend on r through the quantity r = r · w 0 ( θ ) . In the bottom row, the magnitude and direction of the red arrows illustrate the component of the Poynting vector lying in the x z plane, while the component along the y axis is illustrated by circles, whose the radii are proportional to the magnitude, and whose shading illustrates a component pointing into the page (black) or out of the page (unshaded).

Equations (60)

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u · B ( r , u ) = 0 .
I ( r ) = 4 π B ( r , u ) d Ω u ,
2 U ( r ) + k 2 U ( r ) = 0 ,
U ( r ) = k 2 π 4 π A ( u ) exp ( i k u · r ) d Ω u ,
S ( r ) = U * ( r ) U ( r ) = ( k 2 π ) 2 A * ( u 1 ) A ( u 2 ) exp [ i k r · ( u 2 u 1 ) ] d Ω u 1 d Ω u 2 ,
u 1 , 2 = u cos α 2 w ( u , ϕ ) sin α 2 ,
S ( r ) = B ( r , u ) d Ω u ,
B ( r , u ) = ( k 2 π ) 2 0 2 π 0 π A * [ u cos α 2 w ( u , ϕ ) sin α 2 ] A [ u cos α 2 + w ( u , ϕ ) sin α 2 ] × exp [ 2 i k w ( u , ϕ ) · r sin α 2 ] sin α d α d ϕ .
U i ( r ) = k 2 π A i ( u ) exp ( i k r · u ) d Ω u ,
· U D ( r ) = 0 ,
· U B ( r ) = 0 ,
× U E ( r ) = i k U B ( r ) ,
× U H ( r ) = i k U D ( r ) .
U D ( r ) = k 2 π p = a , b 4 π A p ( u ) e ^ p ( u ) exp [ i k n p ( u ) u · r ] d Ω u ,
U i ( r ) = k 2 π p = a , b 4 π A p ( u ) F i ( u ) · e ^ p ( u ) exp [ i k n p ( u ) u · r ] d Ω u ,
u · F B ( u ) · e ^ p ( u ) = 0 ,
n p ( u ) u × F E ( u ) · e ^ p ( u ) = F B ( u ) · e ^ p ( u ) ,
n p ( u ) u × F H ( u ) · e ^ p ( u ) = e ^ p ( u ) .
S i , i ( r ) = ( k 2 π ) 2 p = a , b p = a , b 4 π A p * ( u 1 ) A p ( u 2 ) [ F i ( u 1 ) · e ^ p ( u 1 ) ] * [ F i ( u 2 ) · e ^ p ( u 2 ) ] × exp { i k [ n p ( u 2 ) u 2 n p ( u 1 ) u 1 ] · r } d Ω u 1 d Ω u 2 ,
w e ( r ) = 1 8 π Re { Tr [ S D , E ( r ) ] } ,
w m ( r ) = 1 8 π Re { Tr [ S B , H ( r ) ] } ,
{ S ( r ) } j = c 4 π Re [ { S E , H ( r ) } l , m ε j , l , m ] ,
S i , i ( r ) = 4 π B i , i ( r , u ) d Ω u ,
( u · ) B i , i ( r , u ) = O ,
S i , i ( r ) = 4 π ( k 2 π ) 2 p = a , b p = a , b V i , i ( u , α , ϕ ; p , p ) A p * ( u 1 ) A p ( u 2 ) × exp { i k [ n p ( u 2 ) u 2 n p ( u 1 ) u 1 ] · r } d α d ϕ d Ω u ,
V i , i ( u , α , ϕ ; p , p ) = | ( u 1 , u 2 ) ( u , α , ϕ ) | [ F i ( u 1 ) · e ^ p ( u 1 ) ] * [ F i ( u 2 ) · e ^ p ( u 2 ) ] ,
n p ( u 1 ) u 1 = u C p , p ( u , α , ϕ ) w ( u , ϕ ) K p , p ( 1 ) ( u , α , ϕ ) ,
n p ( u 2 ) u 2 = u C p , p ( u , α , ϕ ) + w ( u , ϕ ) K p , p ( 2 ) ( u , α , ϕ ) ,
C p , p ( u , α , ϕ ) = C p , p ( u , α , ϕ + π ) 0 ,
K p , p ( 1 ) ( u , α , ϕ ) = K p , p ( 2 ) ( u , α , ϕ + π ) ,
[ C p , p ( u , α , ϕ ) ] 2 + [ K p , p ( 1 ) ( u , α , ϕ ) ] 2 = n p 2 ( u 1 ) ,
[ C p , p ( u , α , ϕ ) ] 2 + [ K p , p ( 2 ) ( u , α , ϕ ) ] 2 = n p 2 ( u 2 ) .
B i , i ( r , u ) = ( k 2 π ) 2 p = a , b p = a , b 2 π 0 π V i , i ( u , α , ϕ ; p , p ) × A p * [ u 1 ( u , α , ϕ ; p , p ) ] A p [ u 2 ( u , α , ϕ ; p , p ) ] × exp { i k [ K p , p ( 1 ) ( u , α , ϕ ) + K p , p ( 2 ) ( u , α , ϕ ) ] w ( u , ϕ ) · r } d α d ϕ .
B e ( r , u ) = 1 8 π Re { Tr [ B D , E ( r , u ) ] } ,
B m ( r , u ) = 1 8 π Re { Tr [ B B , H ( r , u ) ] } ,
{ B P ( r , u ) } j = c 4 π Re [ { B E , H ( r , u ) } l , m ε j , l , m ] .
U E ( r ) = ϵ 1 · U D ( r ) ,
U B ( r ) = μ U H ( r ) ,
ϵ = ( ϵ o 0 0 0 ϵ o 0 0 0 ϵ e ) .
F E ( u ) = ϵ 1 ,
F B ( u ) = μ F H ( u ) = μ [ u × e ^ a ( u ) ] e ^ a * ( u ) n a ( u ) + μ [ u × e ^ b ( u ) ] e ^ b * ( u ) n b ( u ) .
n a ( θ ) = μ w 0 ( θ ) · ϵ 1 · w 0 ( θ ) = n o 2 n e 2 n o 2 sin 2 θ + n e 2 cos 2 θ , n b ( θ ) = n o ,
θ 1 , 2 = arctan 2 { n e sin [ arctan 2 ( n e sin θ , n o cos θ ) α 2 ] , n o cos [ arctan 2 ( n e sin θ , n o cos θ ) α 2 ] } ,
| ( θ 1 , θ 2 ) ( θ , α ) | = 4 n o 3 n e 3 g ( θ , α ) n o 2 cos 2 θ + n e 2 sin 2 θ ,
g ( θ , α ) = 1 n o 4 [ Δ ( θ ) + cos α ] 2 + 2 n o 2 n e 2 [ 1 Δ 2 ( θ ) + sin 2 α ] + n e 4 [ Δ ( θ ) cos α ] 2 ,
Δ ( θ ) = n o 2 cos 2 θ n e 2 sin 2 θ n o 2 cos 2 θ + n e 2 sin 2 θ .
B e [ r , u ( θ ) ] = μ k n o 3 n e 3 4 π 3 ( n o 2 cos 2 θ + n e 2 sin 2 θ ) π π A * { u [ θ 1 ( θ , α ) ] } A { u [ θ 2 ( θ , α ) ] } × g 3 / 2 ( θ , α ) cos α exp [ i k n o n e n o 2 cos 2 θ + n e 2 sin 2 θ 2 sin α 2 w 0 ( θ ) · r ] d α .
A * [ u ( θ 1 ) ] A [ u ( θ 2 ) ] = I 0 exp { q [ u ( θ 2 ) + u ( θ 1 ) ] · u ( θ 0 ) } exp [ Q | u ( θ 2 ) u ( θ 1 ) | 2 ] ,
D ( r ) = ϵ E ( r ) + i G H ( r ) ,
B ( r ) = μ H ( r ) i G E ( r ) ,
F E ( u ) = p = l , r n 0 ϵ n p e ^ p ( u ) e ^ p * ( u ) ,
F B ( u ) = p = l , r i μ σ ¯ p n 0 e ^ p ( u ) e ^ p * ( u ) ,
F H ( u ) = p = l , r i σ ¯ p n p e ^ p ( u ) e ^ p * ( u ) .
e ^ l , r ( θ ) = y ^ ± i w 0 ( θ ) 2 .
θ 1 ( θ , α , σ 1 , p , p ) = θ σ 1 arccos ( μ p , p cos α 2 ) ,
θ 2 ( θ , α , σ 2 , p , p ) = θ + σ 2 arccos ( μ p , p cos α 2 ) ,
| ( θ 1 , θ 2 ) ( θ , α ) | = 1 2 ( 1 + σ 1 σ 2 sin α 2 1 η p , p 2 cos 2 α 2 ) ,
B e ( r , u ) = p = l , r p = l , r B e , p p ( r , u ) ,
B e , p p ( r , u ) = k 2 n 0 256 π 2 ϵ σ 1 = ± 1 σ 2 = ± 1 ( 1 n p + 1 n p ) 0 π ( 1 + σ 1 σ 2 sin α 2 1 η p , p 2 cos 2 α 2 ) × ( 1 + σ ¯ p σ ¯ p η p , p cos 2 α 2 σ 1 σ 2 σ ¯ p σ ¯ p sin α 2 1 η p , p 2 cos 2 α 2 ) × A p * { u [ θ 1 ( θ , α , σ 1 , p , p ) ] } A p { u [ θ 2 ( θ , α , σ 2 , p , p ) ] } × exp [ i k ( σ 1 1 μ p , p 2 cos 2 α 2 + σ 2 1 μ p , p 2 cos 2 α 2 ) w 0 ( θ ) · r ] d α ,
A p * [ u ( θ 1 ) ] A p [ u ( θ 2 ) ] = I 0 exp [ q [ u 2 ( θ 2 ) + u 1 ( θ 1 ) ] · z ^ ] × exp [ Q | u ( θ 2 ) u ( θ 1 ) | 2 ] ,

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