Abstract

A radiometric framework is described for modeling the propagation of nonparaxial scalar fields of any degree of coherence past planar boundaries (or composite interfaces) between homogeneous, isotropic nonabsorptive media in three dimensions. The transformation is shown to be, to lowest order, that predicted by classical radiometry but potentially including a Goos–Hänchen shift. Higher-order corrections take the form of coefficients multiplied by derivatives of the basic estimate. The accuracy of the radiometric term, along with second-order derivative corrections, are examined for Gaussian Schell-model fields of varying width and states of coherence. This technique is found to work well for most such fields but to fail in reflection for fields with significant total-internally-reflected components.

© 2009 Optical Society of America

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References

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  1. L. S. Dolin, “Beam description of weakly inhomogeneous wave fields,” Izv. VUZ Radiofiz. 7, 559-562 (1964).
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [CrossRef]
  3. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  4. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710-1716 (1979).
    [CrossRef]
  5. M. A. Alonso, “Radiometry and wide-angle wave fields III: partial coherence,” J. Opt. Soc. Am. A 18, 2502-2511 (2001).
    [CrossRef]
  6. 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]
  7. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), pp. 193-199.
  8. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 13-25.
  9. L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects (Gordon & Breach, 1996), pp. 11-19.
  10. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192-198 (1979).
    [CrossRef]
  11. A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386-1393 (1992).
    [CrossRef]
  12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), 170-176.
  13. 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]
  14. S. Cho, Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA, J. C. Petruccelli, and M. A. Alonso are preparing a manuscript to be called “Wigner functions for paraxial and nonparaxial fields.”
  15. J. C. Petruccelli and M. A. Alonso, “Ray-based coherence propagation,” J. Opt. Soc. Am. A 25, 1395-1405 (2008).
    [CrossRef]
  16. 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]
  17. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 75-77.
  18. Note that in , the prefactor |n¯u¯z/nuz(u¯z)|1/2 was mistakenly not included.
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 276-287.

2008 (1)

2007 (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 (2)

1992 (1)

1979 (2)

1968 (1)

1964 (1)

L. S. Dolin, “Beam 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]

Agarwal, G. S.

Alonso, M. A.

Apresyan, L. A.

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

Bastiaans, M. J.

Born, M.

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

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 13-25.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 75-77.

Cho, S.

S. Cho, Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA, J. C. Petruccelli, and M. A. Alonso are preparing a manuscript to be called “Wigner functions for paraxial and nonparaxial fields.”

Dolin, L. S.

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

Foley, J. T.

Friberg, A. T.

Kravtsov, Yu. A.

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

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 276-287.

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

Petruccelli, J. C.

J. C. Petruccelli and M. A. Alonso, “Ray-based coherence propagation,” J. Opt. Soc. Am. A 25, 1395-1405 (2008).
[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]

S. Cho, Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA, J. C. Petruccelli, and M. A. Alonso are preparing a manuscript to be called “Wigner functions for paraxial and nonparaxial fields.”

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.

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386-1393 (1992).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 276-287.

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

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

Izv. VUZ Radiofiz. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

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]

Phys. Rev. (1)

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

Other (8)

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

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 13-25.

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

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

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983), pp. 75-77.

Note that in , the prefactor |n¯u¯z/nuz(u¯z)|1/2 was mistakenly not included.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), pp. 276-287.

S. Cho, Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA, J. C. Petruccelli, and M. A. Alonso are preparing a manuscript to be called “Wigner functions for paraxial and nonparaxial fields.”

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Figures (7)

Fig. 1
Fig. 1

Change of variables from u 1 , 2 to u and w ( u , ξ ) indicated in Eq. (6).

Fig. 2
Fig. 2

Change of the ray direction u upon striking an interface at z = 0 . Also included are the unit vectors e ̂ θ and e ̂ ϕ that, along with the ray direction, define an orthormal basis.

Fig. 3
Fig. 3

Error measures R Rad ( θ ¯ ) (solid curves) and R 2 ( θ ¯ ) (dashed curves) in transmission for n 1 = 1 , n 2 = 1.4 .

Fig. 4
Fig. 4

Error measures R Rad ( θ ¯ ) (solid curves) and R 2 ( θ ¯ ) (dashed curves) in reflection for n 1 = 1 , n 2 = 1.4 .

Fig. 5
Fig. 5

Error measures R Rad ( θ ¯ ) (solid curves) and R 2 ( θ ¯ ) (dashed curves) in transmission for n 1 = 1.4 , n 2 = 1 .

Fig. 6
Fig. 6

Error measures R Rad ( θ ¯ ) (solid curves) and R 2 ( θ ¯ ) (dashed curves) in reflection for n 1 = 1.4 , n 2 = 1 .

Fig. 7
Fig. 7

Spectral density calculated using the radiometric estimate and second-order corrections to the radiance for n 1 = 1 to n 2 = 1.4 of a GSM field with σ = 0.5 and ε = σ 4 . The spectral density contributions from S 2 S Rad may be negative, and therefore the zero contour is highlighted in black. All figures are normalized to S Rad ( 0 , 0 , 0 ) = 1 in transmission.

Equations (66)

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u B ( r , u ) = 0 .
B ( r , u ) = M ( P u r , u ) ,
I ( r ) = 4 π B ( r , u ) d Ω u = 4 π M ( P u r , u ) d Ω u ,
U ( r ) = k n 2 π 4 π A ( u ) exp ( i k n u r ) d Ω u ,
W ( r 1 , r 2 ) = U * ( r 1 ) U ( r 2 ) = ( k n 2 π ) 2 4 π A ( u 1 , u 2 ) exp [ i k n ( u 2 r 2 u 1 r 1 ) ] d Ω u 1 d Ω u 2 ,
u 1 , 2 = u cos α 2 w ( u , ξ ) sin α 2 ,
S ( r ) = 1 n 4 π M ( 0 ) ( P u r , u ) d Ω u ,
M ( j ) ( l , u ) = n 3 ( k 2 π ) 2 2 π 0 π A [ u cos α 2 w ( u , ξ ) sin α 2 , u cos α 2 + w ( u , ξ ) sin α 2 ] × exp [ 2 i k n l w ( u , ξ ) sin α 2 ] sin α cos j α 2 d α d ξ ,
n = n 1 , n ¯ = { n 2 , in transmission n 1 , in reflection } ,
u = u i , u ¯ = { u t , in transmission u r , in reflection } .
u ( u ¯ ) = μ 1 ( u ¯ + τ z ̂ u ¯ z 2 + μ 2 1 ) ,
B ¯ ( r , u ¯ ) = ρ ¯ 2 ( u ¯ ) n ¯ 2 n 2 B [ r , u ( u ¯ ) ] ,
M ¯ ( l ¯ , u ¯ ) = ρ ¯ 2 ( u ¯ ) n ¯ 2 n 2 M [ L GO ( l ¯ , u ¯ ) , u ( u ¯ ) ] ,
e ̂ θ ( u ) = P u z ̂ | P u z ̂ | = u z u z ̂ 1 u z 2 , e ̂ ϕ ( u ) = z ̂ × u 1 u z 2 .
L GO = u z u ¯ z l ¯ θ ¯ e ̂ θ ( u ) + l ¯ ϕ ¯ e ̂ ϕ ( u ) ,
v w ( u , ξ ) sin α 2 ,
M ( l , u ) = n 3 ( k π ) 2 D A ( u 1 v 2 v , u 1 v 2 + v ) exp ( 2 i k n l v ) d 2 v ,
A ¯ ( u ¯ ) = T ¯ 1 2 ( u ¯ z ) exp [ i ψ ¯ ( u ¯ z ) ] A [ u ( u ¯ ) ] ,
T ¯ 1 2 ( u ¯ z ) = | n ¯ u ¯ z n u z ( u ¯ z ) | 1 2 ρ ¯ ( u ¯ z )
M ¯ ( l ¯ , u ¯ ) = n ¯ 3 ( k π ) 2 D ¯ exp { i [ 2 k n ¯ l ¯ v ¯ + ψ ¯ ( u ¯ z 1 v ¯ 2 + v ¯ z ) ψ ¯ ( u ¯ z 1 v ¯ 2 v ¯ z ) ] } × A [ u ( u ¯ 1 v ¯ 2 v ¯ ) , u ( u ¯ 1 v ¯ 2 + v ¯ ) ] × T ¯ 1 2 ( u ¯ z 1 v ¯ 2 v ¯ z ) T ¯ 1 2 ( u ¯ z 1 v ¯ 2 + v ¯ z ) d 2 v ¯ .
M ¯ Rad [ l ¯ , u ¯ ( θ ¯ , ϕ ¯ ) ] = ρ ¯ 2 n ¯ 2 n 2 M { L [ l ¯ , u ¯ ( θ ¯ , ϕ ¯ ) ] , u [ u ¯ ( θ ¯ , ϕ ¯ ) ] } ,
L ( l ¯ , u ¯ ) L GO ( l ¯ , u ¯ ) 2 s ¯ c ψ c ¯ e ̂ θ ( u ) .
M ¯ ( l ¯ , u ¯ ) = [ 1 + i = 2 j = 0 i l = 0 j 2 m = 0 l C j , i j , m , l m ( u ¯ z ) ( i + l ) l ¯ θ ¯ j l ¯ ϕ ¯ i j θ ¯ m ϕ ¯ l m ] M ¯ Rad ( l ¯ , u ¯ ) ,
C 2 , 0 , 0 , 0 = 2 c 2 c ¯ [ T ¯ 1 2 T ¯ 1 2 ( c ¯ 4 + μ 2 1 ) + μ 2 c 2 c ¯ s ¯ 2 ( T ¯ 1 2 2 T ¯ 1 2 T ¯ 1 2 ) ] s 2 ( μ 2 1 ) 2 T ¯ 1 2 2 8 k 2 n 2 c 4 c ¯ 2 T ¯ 1 2 2 ,
C 0 , 2 , 0 , 0 = 2 μ 2 c 4 c ¯ T ¯ 1 2 + s 2 ( μ 2 1 ) 2 T ¯ 1 2 8 k 2 n 2 c 2 c ¯ 2 T ¯ 1 2 ,
C 2 , 0 , 1 , 0 = ( μ 2 1 ) s ¯ 8 k 2 n 2 c 2 c ¯ ,
C 0 , 2 , 1 , 0 = ( μ 2 1 ) s ¯ 8 k 2 n 2 c ¯ ,
W ( r 1 , , r 2 , ) = I r exp [ a ( | r 1 , | 2 + | r 2 , | 2 ) + 2 b r 1 , r 2 , ] ,
A ( u 1 , u 2 ) = ( k n 2 π ) 2 u 1 , z u 2 , z W ( r 1 , , r 2 , ) × exp [ i k n ( r 1 , u 1 , r 2 , u 2 , ) ] d 2 r 1 , d 2 r 2 , ,
A ( u 1 , u 2 ) = n 2 k 2 I u u 1 , z u 2 , z × exp [ n 2 ( | u 1 , | 2 + | u 2 , | 2 2 σ 2 + | u 2 , u 1 , | 2 2 ε 2 ) ] ,
R Rad ( θ ¯ ) = Log 10 { Max l ¯ [ | Δ M ¯ Rad ( l ¯ , θ ¯ ) | ] Max [ | M ¯ ( l ¯ , θ ¯ ) | ] } ,
R 2 ( θ ¯ ) = Log 10 { Max l ¯ [ | | M ¯ ( l ¯ , θ ¯ ) M ¯ 2 ( l ¯ , θ ¯ ) | | ] Max [ | M ¯ ( l , θ ¯ ) | ] } ,
( u 1 v 2 ± v ) z ̂ 0 .
v θ 2 u z 2 + v ϕ 2 = 1 ,
( u ¯ 1 v ¯ 2 ± v ¯ ) z ̂ 1 μ 2 ,
v ¯ ϕ ¯ 2 μ 2 + [ v ¯ θ ¯ ± ( 1 u ¯ z 2 ) ( 1 μ 2 ) ] 2 μ 2 u ¯ z 2 = 1 .
( u 1 v 2 ± v ) z ̂ 1 μ 2 .
μ 2 u z 2 [ v θ ± ( 1 u z 2 ) ( 1 μ 2 ) ] 2 + μ 2 v ϕ 2 = 1 .
( u ¯ 1 v ¯ 2 ± v ¯ ) z ̂ 0 ,
v ¯ θ ¯ 2 u ¯ z 2 + v ¯ ϕ ¯ 2 = 1 .
v ̃ ϕ 2 + u z 2 u ¯ z 4 [ v ̃ θ ± u ¯ z ( 1 μ 2 ) ( 1 u ¯ z 2 ) μ u z ] 2 = 1 .
μ 2 v ̃ ϕ 2 + μ 2 u z 2 u ¯ z 4 v ̃ θ 2 = 1 .
u ( u ¯ 1 v ¯ 2 ± v ¯ ) = μ 1 ( u ¯ 1 v ¯ 2 ± v ¯ ) + z ̂ [ u z ± v ¯ z u ¯ z μ 2 u z + v ¯ z 2 v ¯ 2 u ¯ z 2 2 μ 2 u z v ¯ z 2 u ¯ z 2 2 μ 4 u z 3 ] + O ( v ¯ 3 ) .
v ̃ ( v ¯ ) = μ 1 [ v ¯ + v ¯ z u ¯ z μ u z z ̂ ]
u ( u ¯ 1 v ¯ 2 ± v ¯ ) = u 1 v ̃ 2 ± v ̃ + a + O ( v ̃ 3 ) ,
v ¯ z = μ 2 u z u ¯ z v ̃ z , v ¯ = μ v ̃ + μ 2 u z μ u ¯ z u ¯ z v ̃ z z ̂ ,
v ¯ 2 = μ 2 ( v ̃ 2 + μ 2 1 u ¯ z 2 v ̃ z 2 ) ,
d 2 v ¯ = μ 2 u z u ¯ z d 2 v ̃ .
a = u 1 μ 2 2 [ v ̃ ϕ 2 + 1 + ( μ 2 1 ) c ¯ 2 μ 2 c ¯ 2 v ̃ θ 2 ] + z ̂ μ 2 1 2 c ( v ̃ ϕ 2 + 1 c ¯ 2 v ̃ θ 2 ) ,
A [ u ( u ¯ 1 v ¯ 2 v ¯ ) , u ( u ¯ 1 v ¯ 2 + v ¯ ) ] = ( 1 + a u ¯ ) A ( u 1 v ̃ 2 v ̃ , u 1 v ̃ 2 + v ̃ ) + O ( v ̃ 3 ) ,
a u ¯ = ( 1 μ 2 ) s ¯ 2 c ¯ ( v ̃ ϕ 2 + 1 c ¯ 2 v ̃ θ 2 ) θ ¯ .
T ¯ 1 2 ( u ¯ z 1 v ¯ 2 v ¯ z ) T ¯ 1 2 ( u ¯ z 1 v ¯ 2 + v ¯ z ) = T ¯ 1 2 2 μ 2 T ¯ 1 2 T ¯ 1 2 c ¯ v ̃ ϕ 2 + c 2 μ 2 c ¯ 2 [ s ¯ 2 ( T ¯ 1 2 T ¯ 1 2 T ¯ 1 2 2 ) c ¯ T ¯ 1 2 T ¯ 1 2 ] v ̃ θ 2 + O ( v ̃ 3 ) ,
exp { i [ ψ ¯ ( u ¯ z 1 v ¯ 2 + v ¯ z ) ψ ¯ ( u ¯ z 1 v ¯ 2 v ¯ z ) ] } = exp ( 2 i μ c c ¯ s ¯ ψ ¯ v ̃ θ ) + O ( v ̃ 3 ) .
L ( l ¯ , u ¯ ) = [ c c ¯ l ¯ θ ¯ e ̂ θ ( u ) + l ¯ ϕ ¯ e ̂ ϕ ( u ) ] 2 s ¯ c ψ ¯ c ¯ e ̂ θ ( u ) .
[ T ¯ 1 2 2 μ 2 T ¯ 1 2 T ¯ 1 2 c ¯ v ̃ ϕ 2 + μ 2 c 2 [ s ¯ 2 ( T ¯ 1 2 T ¯ 1 2 T ¯ 1 2 2 ) c ¯ T ¯ 1 2 T ¯ 1 2 ] c ¯ 2 v ̃ θ 2 + T ¯ 1 2 2 a u ¯ ] × A ( u 1 v ̃ 2 v ̃ , u 1 v ̃ 2 + v ̃ ) exp ( 2 i k n L v ̃ ) + O ( v ̃ 3 ) .
v ̃ θ 2 exp ( 2 i k n L v ̃ ) = c ¯ 2 4 k 2 n 2 c 2 2 l ¯ θ ¯ 2 exp ( 2 i k n L v ̃ ) ,
v ̃ ϕ 2 exp ( 2 i k n L v ̃ ) = 1 4 k 2 n 2 2 l ¯ ϕ ¯ 2 exp ( 2 i k n L v ̃ ) ,
M ¯ ( l ¯ , u ¯ ) = n ¯ 2 ρ ¯ 2 n 2 { 1 + μ 2 T ¯ 1 2 c ¯ 4 k 2 n 2 T ¯ 1 2 2 l ¯ ϕ ¯ 2 + μ 2 ( s ¯ 2 T ¯ 1 2 2 + c ¯ T ¯ 1 2 T ¯ 1 2 s ¯ 2 T ¯ 1 2 T ¯ 1 2 ) 4 k 2 n 2 T ¯ 1 2 2 2 l ¯ θ ¯ 2 + ( μ 2 1 ) s ¯ 8 k 2 n 2 c ¯ 3 l ¯ ϕ ¯ 2 θ ¯ + ( μ 2 1 ) s ¯ 8 k 2 n 2 c 2 c ¯ 3 l ¯ θ ¯ 2 θ ¯ + } × n 3 ( k π ) 2 D ¯ A ( u 1 v ̃ 2 v ̃ , u 1 v ̃ 2 + v ̃ ) exp ( 2 i k n L v ̃ ) d 2 v ̃ ,
ρ ¯ 2 θ ¯ = θ ¯ ρ ¯ 2 + s ¯ [ ( 1 μ 2 ) T ¯ 1 2 + 2 μ 2 c 2 c ¯ T ¯ 1 2 ] μ 2 c 2 c ¯ T ¯ 1 2 ρ ¯ 2 ,
M ¯ ( l ¯ , u ¯ ) { 1 + C 2 , 0 , 0 , 0 ( u ¯ ) 2 l ¯ θ ¯ 2 + C 0 , 2 , 0 , 0 ( u ¯ ) 2 l ¯ ϕ ¯ 2 + C 2 , 0 , 1 , 0 ( u ¯ ) 3 l ¯ θ ¯ 2 θ ¯ + C 0 , 2 , 1 , 0 ( u ¯ ) 3 l ¯ ϕ ¯ 2 θ ¯ + } M ¯ Rad ( l ¯ , u ¯ ) ,
u ( u ¯ 1 v ¯ 2 v ¯ ) = u 1 , 2 + a + O ( v ̃ 3 ) ,
A [ u ( u ¯ 1 v ¯ 2 v ¯ ) ] = A ( u 1 , 2 ) + a u 1 , 2 A ( u 1 , 2 ) + O ( v ̃ 3 ) ,
u i = e ̂ θ ( u i ) θ i + e ̂ ϕ ( u i ) sin θ i ϕ i .
a u 1 , 2 A ( u 1 , 2 ) = a e ̂ θ ( u ) θ 1 , 2 A ( u 1 , 2 ) + O ( v ̃ 3 ) .
θ 1 , 2 = μ 2 u z sin θ i u ¯ z sin θ ¯ 1 v ̃ 2 θ ¯ = μ c c ¯ θ ¯ + O ( v ̃ ) .
a u 1 , 2 A ( u 1 , 2 ) = ( 1 μ 2 ) s ¯ 2 c ¯ ( v ̃ ϕ 2 + 1 c ¯ 2 v ̃ θ 2 ) × θ ¯ A ( u 1 v ̃ 2 v ̃ ) + O ( v ̃ 3 ) .

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