Abstract

Propagation theories of partially coherent electromagnetic fields based on coherent mode decomposition or separated-coordinate mode decomposition are proposed. With the proposed propagation theories, various powerful theories for the propagation of fully coherent electromagnetic fields can be used for the propagation of partially coherent electromagnetic fields. The proposed theories are applicable to any propagation problem of partially coherent electromagnetic fields governed by linear Maxwell equations. Some examples are provided to illustrate the validity of the proposed theories.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
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    [CrossRef]
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    [CrossRef]
  11. J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
    [CrossRef]
  12. R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1964), pp. 319-320.
  13. O. Korotkova and E. Wolf, "Spectral degree of coherence of a random three-dimensional electromagnetic field," J. Opt. Soc. Am. A 21, 2382-2385 (2004).
    [CrossRef]

2006

M. A. Alonso, O. Korotkova, and E. Wolf, "Propagation of the electric correlation matrix and the van Cittert-Zernike theorem for random electromagnetic fields," J. Mod. Opt. 57, 969-978 (2006).
[CrossRef]

2005

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

T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, "Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume," Phys. Rev. E 71, 036618 (2005).
[CrossRef]

2004

2003

2002

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

1996

1995

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

1985

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.

1975

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

1964

R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1964), pp. 319-320.

Alonso, M. A.

M. A. Alonso, O. Korotkova, and E. Wolf, "Propagation of the electric correlation matrix and the van Cittert-Zernike theorem for random electromagnetic fields," J. Mod. Opt. 57, 969-978 (2006).
[CrossRef]

Blomstedt, K.

T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, "Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume," Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Carney, P. S.

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

Duan, K.

Friberg, A. T.

T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, "Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume," Phys. Rev. E 71, 036618 (2005).
[CrossRef]

J. Tervo, T. Setala, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.

Gori, F.

Guattari, G.

Korotkova, O.

M. A. Alonso, O. Korotkova, and E. Wolf, "Propagation of the electric correlation matrix and the van Cittert-Zernike theorem for random electromagnetic fields," J. Mod. Opt. 57, 969-978 (2006).
[CrossRef]

O. Korotkova and E. Wolf, "Spectral degree of coherence of a random three-dimensional electromagnetic field," J. Opt. Soc. Am. A 21, 2382-2385 (2004).
[CrossRef]

Li, L.

Lindberg, J.

T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, "Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume," Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Lü, B.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1964), pp. 319-320.

Mandel, L.

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

Piquero, G.

Santarsiero, M.

Schotland, J. C.

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

Setala, T.

T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, "Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume," Phys. Rev. E 71, 036618 (2005).
[CrossRef]

J. Tervo, T. Setala, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
[CrossRef]

Simon, R.

Tervo, J.

T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, "Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume," Phys. Rev. E 71, 036618 (2005).
[CrossRef]

J. Tervo, T. Setala, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
[CrossRef]

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

Turunen, J.

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

Wolf, E.

M. A. Alonso, O. Korotkova, and E. Wolf, "Propagation of the electric correlation matrix and the van Cittert-Zernike theorem for random electromagnetic fields," J. Mod. Opt. 57, 969-978 (2006).
[CrossRef]

O. Korotkova and E. Wolf, "Spectral degree of coherence of a random three-dimensional electromagnetic field," J. Opt. Soc. Am. A 21, 2382-2385 (2004).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Zysk, A. M.

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

J. Mod. Opt.

M. A. Alonso, O. Korotkova, and E. Wolf, "Propagation of the electric correlation matrix and the van Cittert-Zernike theorem for random electromagnetic fields," J. Mod. Opt. 57, 969-978 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

Phys. Rev. E

T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, "Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume," Phys. Rev. E 71, 036618 (2005).
[CrossRef]

Phys. Rev. Lett.

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

Other

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

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

J. W. Goodman, Statistical Optics (Wiley, 1985), Chap. 5.

R. K. Luneberg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1964), pp. 319-320.

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Figures (2)

Fig. 1
Fig. 1

Scheme of the example in Subsection 3A.

Fig. 2
Fig. 2

Reflectance R of the numerical example in Subsection 3A.

Equations (104)

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W ( e ) ( r 1 , r 2 , ν ) = E ( r 1 , ν ) E + ( r 2 , ν ) = [ E p ( r 1 , ν ) E q * ( r 2 , ν ) ] ,
W ( h ) ( r 1 , r 2 , ν ) = H ( r 1 , ν ) H + ( r 2 , ν ) = [ H p ( r 1 , ν ) H q * ( r 2 , ν ) ] ,
W ( m ) ( r 1 , r 2 , ν ) = E ( r 1 , ν ) H + ( r 2 , ν ) = [ E p ( r 1 , ν ) H q * ( r 2 , ν ) ] , p , q { x , y , z } ,
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = E ( i ) ( ρ 1 , ν ) ( E ( i ) ( ρ 2 , ν ) ) + = [ E p ( i ) ( ρ 1 , ν ) ( E q ( i ) ( ρ 2 , ν ) ) * ] , p , q { x , y } ,
× E ( r , ν ) = i 2 π ν μ H ( r , ν ) ,
× H ( r , ν ) = i 2 π ν ϵ E ( r , ν ) , b . c . E ( i ) ( ρ , ν ) ,
E ( r , ν ) = P ( e ) ( r , ρ , ν ) E ( i ) ( ρ , ν ) ,
H ( r , ν ) = P ( h ) ( r , ρ , ν ) E ( i ) ( ρ , ν ) .
E x ( r , ν ) = 1 2 π G ( r , ρ , ν ) z E x ( i ) ( ρ , ν ) d 2 ρ ,
E y ( r , ν ) = 1 2 π G ( r , ρ , ν ) z E y ( i ) ( ρ , ν ) d 2 ρ ,
E z ( r , ν ) = 1 2 π [ G ( r , ρ , ν ) x E x ( i ) ( ρ , ν ) + G ( r , ρ , ν ) y E y ( i ) ( ρ , ν ) ] d 2 ρ ,
H ( r , ν ) = ( i 2 π ν μ ) 1 × E ( r , ν ) ,
P ( e ) ( r , ρ , ν ) = [ 1 2 π d 2 ρ G ( r , ρ , ν ) z 0 0 1 2 π d 2 ρ G ( r , ρ , ν ) z 1 2 π d 2 ρ G ( r , ρ , ν ) x 1 2 π d 2 ρ G ( R , ρ , ν ) y ] ,
P ( h ) ( r , ρ , ν ) = 1 i 2 π ν μ [ 1 2 π d 2 ρ 2 G ( r , ρ , ν ) x y 1 2 π d 2 ρ [ 2 G ( r , ρ , ν ) y 2 + 2 G ( r , ρ , ν ) z 2 ] 1 2 π d 2 ρ [ 2 G ( r , ρ , ν ) x 2 + 2 G ( r , ρ , ν ) z 2 ] 1 2 π d 2 ρ 2 G ( r , ρ , ν ) x y 1 2 π d 2 ρ 2 G ( r , ρ , ν ) y z 1 2 π d 2 ρ 2 G ( r , ρ , ν ) x z ] .
( W ( e ) ( i ) ( ρ 2 , ρ 1 , ν ) ) + = W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) ,
f + ( ρ 1 ) W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) f ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 = f + ( ρ 1 ) E ( i ) ( ρ 1 , ν ) ( E ( i ) ( ρ 2 , ν ) ) + f ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 = f + ( ρ ) E ( i ) ( ρ , ν ) d 2 ρ 2 0 f ( ρ ) = ( f x ( ρ ) , f y ( ρ ) ) ,
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = n λ n ( ν ) E n ( i ) ( ρ 1 , ν ) ( E n ( i ) ( ρ 2 , ν ) ) + ,
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) E n ( i ) ( ρ 2 , ν ) d 2 ρ 2 = λ n ( ν ) E n ( i ) ( ρ 1 , ν ) ,
( E m ( i ) ( ρ , ν ) ) + E n ( i ) ( ρ , ν ) d 2 ρ = δ m n ,
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = n λ ̃ n ( ν ) E ̃ n ( i ) ( ρ 1 , ν ) ( E ̃ n ( i ) ( ρ 2 , ν ) ) + ,
W p q ( e ) ( r 1 , r 2 , ν ) = 1 E p ( r 1 , ν ) E q * ( r 2 , ν ) = 2 [ k P p k ( e ) ( r 1 , ρ 1 , ν ) E k ( i ) ( ρ 1 , ν ) ] [ i P q l ( e ) ( r 2 , ρ 2 , ν ) E l ( ( i ) ( ρ 2 , ν ) ] * = 3 [ k P p k ( e ) ( r 1 , ρ 1 , ν ) E k ( i ) ( ρ 1 , ν ) ] [ l ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * ( E l ( i ) ( ρ 2 , ν ) ) * ] = 4 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * E k ( i ) ( ρ 1 , ν ) ( E l ( i ) ( ρ 2 , ν ) ) * = 5 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * E k ( i ) ( ρ 1 , ν ) ( E l ( i ) ( ρ 2 , ν ) ) * = 6 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * W k l ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = 7 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * [ n λ n ( ν ) E n k ( i ) ( ρ 1 , ν ) ( E n l ( i ) ( ρ 2 , ν ) ) * ] = 8 n λ n ( ν ) [ k P p k ( e ) ( r 1 , ρ 1 , ν ) E n k ( i ) ( ρ 1 , ν ) ] [ l ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * ( E n l ( i ) ( ρ 2 , ν ) ) * ] = 9 n λ n ( ν ) [ k P p k ( e ) ( r 1 , ρ 1 , ν ) E n k ( i ) ( ρ 1 , ν ) ] [ l P q l ( e ) ( r 2 , ρ 2 , ν ) E n l ( i ) ( ρ 2 , ν ) ] * = 10 n λ n ( ν ) E n p ( r 1 , ν ) E n q * ( r 2 , ν ) ,
W ( e ) ( r 1 , r 2 , ν ) = 1 E ( r 1 , ν ) ( E ( r 2 , ν ) ) + = 2 [ P ( e ) ( r 1 , ρ 1 , ν ) E ( i ) ( ρ 1 , ν ) ] [ P ( e ) ( r 2 , ρ 2 , ν ) E ( i ) ( ρ 2 , ν ) ] + = 3 , 4 P ( e ) ( r 1 , ρ 1 , ν ) [ E ( i ) ( ρ 1 , ν ) ( E ( i ) ( ρ 2 , ν ) ) + ] ( P ( e ) ( r 2 , ρ 2 , ν ) ) + = 5 P ( e ) ( r 1 , ρ 1 , ν ) E ( i ) ( ρ 1 , ν ) ( E ( i ) ( ρ 2 , ν ) ) + ( P ( e ) ( r 2 , ρ 2 , ν ) ) + = 6 P ( e ) ( r 1 , ρ 1 , ν ) W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) ( P ( e ) ( r 2 , ρ 2 , ν ) ) + = 7 P ( e ) ( r 1 , ρ 1 , ν ) [ n λ n ( ν ) E n ( i ) ( ρ 1 , ν ) ( E n ( i ) ( ρ 2 , ν ) ) + ] ( P ( e ) ( r 2 , ρ 2 , ν ) ) + = 8 , 9 n λ n ( ν ) [ P ( e ) ( r 1 , ρ 1 , ν ) E n ( i ) ( ρ 1 , ν ) ] [ P ( e ) ( r 2 , ρ 2 , ν ) E n ( i ) ( ρ 2 , ν ) ] + = 10 n λ n ( ν ) E n ( r 1 , ν ) E n + ( r 2 , ν ) ,
E n ( r , ν ) = P ( e ) ( r , ρ , ν ) E n ( i ) ( ρ , ν ) .
W ( h ) ( r 1 , r 2 , ν ) = n λ n ( ν ) H n ( r 1 , ν ) H n + ( r 2 , ν ) ,
W ( m ) ( r 1 , r 2 , ν ) = n λ n ( ν ) E n ( r 1 , ν ) H n + ( r 2 , ν ) ,
H n ( r , ν ) = P ( h ) ( r , ρ , ν ) E n ( i ) ( ρ , ν ) .
× E n ( r , ν ) = i 2 π ν μ H n ( r , ν ) ,
× H n ( r , ν ) = i 2 π ν ϵ E n ( r , ν ) , b.c. E n ( i ) ( ρ , ν ) .
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = n c n ( ν ) E n ( > ) ( i ) ( ρ 1 , ν ) ( E n ( < ) ( i ) ( ρ 2 , ν ) ) + ,
W ( e ) ( r 1 , r 2 , ν ) = P ( e ) ( r 1 , ρ 1 , ν ) W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) ( P ( e ) ( r 2 , ρ 2 , ν ) ) + = P ( e ) ( r 1 , ρ 1 , ν ) [ n c n ( ν ) E n ( > ) ( i ) ( ρ 1 , ν ) ( E n ( < ) ( i ) ( ρ 2 , ν ) ) + ] ( P ( e ) ( r 2 , ρ 2 , ν ) ) + = n c n ( ν ) [ P ( e ) ( r 1 , ρ 1 , ν ) E n ( > ) ( i ) ( ρ 1 , ν ) ] [ P ( e ) ( r 2 , ρ 2 , ν ) E n ( < ) ( i ) ( ρ 2 , ν ) ] + = n c n ( ν ) E n ( > ) ( r 1 , ν ) ( E n ( < ) ( r 2 , ν ) ) + ,
E n ( > ) ( r , ν ) = P ( e ) ( r , ρ , ν ) E n ( > ) ( i ) ( ρ , ν ) ,
E n ( < ) ( r , ν ) = P ( e ) ( r , ρ , ν ) E n ( < ) ( i ) ( ρ , ν ) .
W ( h ) ( r 1 , r 2 , ν ) = n c n ( ν ) H n ( > ) ( r 1 , ν ) ( H n ( < ) ( r 2 , ν ) ) + ,
W ( m ) ( r 1 , r 2 , ν ) = n c n ( ν ) E n ( > ) ( r 1 , ν ) ( H n ( < ) ( r 2 , ν ) ) + ,
H n ( > ) ( r , ν ) = P ( h ) ( r , ρ , ν ) E n ( > ) ( i ) ( ρ , ν ) ,
H n ( < ) ( r , ν ) = P ( h ) ( r , ρ , ν ) E n ( < ) ( i ) ( ρ , ν ) .
× E n ( > ) ( r , ν ) = i 2 π ν μ H n ( > ) ( r , ν ) , × H n ( > ) ( r , ν ) = i 2 π ν ϵ E n ( > ) ( r , ν ) , b.c. E n ( > ) ( i ) ( ρ , ν ) ,
× E n ( < ) ( r , ν ) = i 2 π ν μ H n ( < ) ( r , ν ) , × H n ( < ) ( r , ν ) = i 2 π ν ϵ E n ( < ) ( r , ν ) , b.c. E n ( < ) ( i ) ( ρ , ν ) .
E ( i ) ( ρ , ν ) = E 0 ( i ) exp ( i 2 π f ( i ) T ρ ) ,
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = [ E p ( i ) ( ρ 1 , ν ) ( E q ( i ) ( ρ 2 , ν ) ) * ] = A exp [ i 2 π f ( i ) T ( ρ 1 ρ 2 ) ] ,
A = λ 1 a 1 a 1 + + λ 2 a 2 a 2 + ,
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = λ 1 E 1 ( i ) ( ρ 1 , ν ) ( E 1 ( i ) ( ρ 2 , ν ) ) + + λ 2 E 2 ( i ) ( ρ 1 , ν ) ( E 2 ( i ) ( ρ 2 , ν ) ) + ,
E n ( i ) ( ρ , ν ) = a n exp ( i 2 π f ( i ) T ρ ) , n = 1 , 2 .
E n ( i ) ( r , ν ) = [ ( E n ( i ) ( r , ν ) ) T , E n , z ( i ) ( r , ν ) ] T ,
E n ( i ) ( r , ν ) = a n exp ( i 2 π f ̃ ( i ) T r ) , E n , z ( i ) ( r , ν ) = [ tan θ ( i ) , 0 ] E n ( i ) ( r , ν ) ,
H n ( i ) ( r , ν ) = ( ν μ ) 1 f ̃ ( i ) × E n ( i ) ( r , ν ) , n = 1 , 2 ,
E n ( r ) ( r , ν ) = [ ( E n ( r ) ( r , ν ) ) T , E n , z ( r ) ( r , ν ) ] T ,
E n ( r ) ( r , ν ) = T ( r ) a n exp ( i 2 π λ 1 d cos θ ( i ) ) exp ( i 2 π f ̃ ( r ) T r d ) ,
E n , z ( r ) ( r , ν ) = [ tan θ ( i ) , 0 ] E n ( r ) ( r , ν ) ,
T ( r ) = [ tan [ θ ( i ) θ ( t ) ] tan [ θ ( i ) + θ ( t ) ] 0 0 sin [ θ ( i ) θ ( t ) ] sin [ θ ( i ) + θ ( t ) ] ] ,
H n ( r ) ( r , ν ) = ( ν μ ) 1 f ̃ ( r ) × E n ( r ) ( r , ν ) , n = 1 , 2 ,
E n ( t ) ( r , ν ) = [ ( E n ( t ) ( r , ν ) ) T , E n , z ( t ) ( r , ν ) ] T ,
E n ( t ) ( r , ν ) = T ( t ) a n exp ( i 2 π λ 1 d cos θ ( i ) ) exp ( i 2 π f ̃ ( t ) T r d ) ,
E n , z ( t ) ( r , ν ) = [ tan θ ( t ) , 0 ] E n ( t ) ( r , ν ) ,
T ( t ) = [ sin 2 θ ( t ) sin [ θ ( i ) + θ ( t ) ] cos [ θ ( i ) θ ( t ) ] 0 0 2 cos θ ( i ) sin θ ( t ) sin [ θ ( i ) + θ ( t ) ] ] ,
H n ( t ) ( r , ν ) = ( ν μ ) 1 f ̃ ( t ) × E n ( t ) ( r , ν ) , n = 1 , 2 .
W ( e ) ( α ) ( r 1 , r 2 , ν ) = λ 1 E 1 ( α ) ( r 1 , ν ) ( E 1 ( α ) ( r 2 , ν ) ) + + λ 2 E 2 ( α ) ( r 1 , ν ) ( E 2 ( α ) ( r 2 , ν ) ) + ,
W ( h ) ( α ) ( r 1 , r 2 , ν ) = λ 1 H 1 ( α ) ( r 1 , ν ) ( H 1 ( α ) ( r 2 , ν ) ) + + λ 2 H 2 ( α ) ( r 1 , ν ) ( H 2 ( α ) ( r 2 , ν ) ) + ,
W ( m ) ( α ) ( r 1 , r 2 , ν ) = λ 1 E 1 ( α ) ( r 1 , ν ) ( H 1 ( α ) ( r 2 , ν ) ) + + λ 2 E 2 ( α ) ( r 1 , ν ) ( H 2 ( α ) ( r 2 , ν ) ) + .
R = Tr W ( e ) ( r ) ( r , r , ν ) Tr W ( e ) ( i ) ( r , r , ν ) , T = n II cos θ ( t ) cos θ ( i ) Tr W ( e ) ( t ) ( r , r , ν ) Tr W ( e ) ( i ) ( r , r , ν ) .
E 0 , TM ( i ) 2 = E 0 , x ( i ) 2 cos 2 θ ( i ) = 2 cos 2 θ ( i ) , E 0 , TE ( i ) 2 = E 0 , y ( i ) 2 = 2 .
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = E 1 ( > ) ( i ) ( ρ 1 , ν ; f 1 , f 2 ) ( E 1 ( < ) ( i ) ( ρ 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 + E 2 ( > ) ( i ) ( ρ 1 , ν ; f 1 , f 2 ) ( E 2 ( < ) ( i ) ( ρ 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 ,
E 1 ( > ) ( i ) ( ρ 1 , ν ; f 1 , f 2 ) = [ A x x ( e ) ( f 1 , f 2 , ν ) A y x ( e ) ( f 1 , f 2 , ν ) ] exp ( i 2 π f 1 T ρ 1 ) ,
E 1 ( < ) ( i ) ( ρ 2 , ν ; f 2 ) = [ 1 0 ] exp ( i 2 π f 2 T ρ 2 ) ,
E 2 ( > ) ( i ) ( ρ 1 , ν ; f 1 , f 2 ) = [ A x y ( e ) ( f 1 , f 2 , ν ) A y y ( e ) ( f 1 , f 2 , ν ) ] exp ( i 2 π f 1 T ρ 1 ) ,
E 2 ( < ) ( i ) ( ρ 2 , ν ; f 2 ) = [ 0 1 ] exp ( i 2 π f 2 T ρ 2 ) .
E n ( ) ( i ) ( ρ , ν ) = E ̃ n ( ) ( i ) exp ( i 2 π f ( i ) T ρ ) , { > , < } , n { 1 , 2 } ,
E n ( ) ( i ) ( r , ν ) = [ ( E n ( ) ( i ) ( r , ν ) ) T , E n , z ( ) ( i ) ( r , ν ) ] T ,
E n ( ) ( i ) ( r , ν ) = E ̃ n ( ) ( i ) exp ( i 2 π f ̃ ( i ) T r ) ,
E n , z ( ) ( i ) ( r , ν ) = [ cos α ( i ) cos θ ( i ) , cos β ( i ) cos θ ( i ) ] E n ( ) ( i ) ( r , ν ) ,
H n ( ) ( i ) ( r , ν ) = ( ν μ ) 1 f ̃ ( i ) × E n ( ) ( i ) ( r , ν ) , { > , < } , n { 1 , 2 } ,
E n ( ) ( r ) ( r , ν ) = [ ( E n ( ) ( r ) ( r , ν ) ) T , E n , z ( ) ( r ) ( r , ν ) ] T
E n ( ) ( r ) ( r , ν ) = T ( r ) E ̃ n ( ) ( i ) exp ( i 2 π λ 1 d cos θ ( i ) ) exp ( i 2 π f ̃ ( r ) T r d ) ,
E n , z ( ) ( r ) ( r , ν ) = [ cos α ( i ) cos θ ( i ) , cos β ( i ) cos θ ( i ) ] E n ( ) ( r ) ( r , ν ) ,
T ( r ) = 1 sin 2 θ ( i ) [ cos α ( i ) cos β ( i ) cos β ( i ) cos α ( i ) ] [ tan [ θ ( i ) θ ( t ) ] tan [ θ ( i ) θ ( t ) ] 0 0 sin [ θ ( i ) θ ( t ) ] sin [ θ ( i ) θ ( t ) ] ] [ cos α ( i ) cos β ( i ) cos β ( i ) cos α ( i ) ] ,
H n ( ) ( r ) ( r , ν ) = ( ν μ ) 1 f ̃ ( r ) × E n ( ) ( r ) ( r , ν ) , { > , < } , n { 1 , 2 } ,
E n ( ) ( t ) ( r , ν ) = [ ( E n ( ) ( t ) ( r , ν ) ) T , E n , z ( ) ( t ) ( r , ν ) ] T ,
E n ( ) ( t ) ( r , ν ) = T ( t ) E ̃ n ( ) ( i ) exp ( i 2 π λ 1 d cos θ ( i ) ) exp ( i 2 π f ̃ ( t ) T r d ) ,
E n , z ( ) ( t ) ( r , ν ) = [ cos α ( t ) cos θ ( t ) , cos β ( t ) cos θ ( t ) ] E n ( ) ( t ) ( r , ν ) ,
T ( t ) = 1 sin 2 θ ( i ) [ cos α ( i ) cos β ( i ) cos β ( i ) cos α ( i ) ] [ sin 2 θ ( t ) sin [ θ ( i ) + θ ( t ) ] cos [ θ ( i ) θ ( t ) ] 0 0 2 cos θ ( i ) sin θ ( t ) sin [ θ ( i ) + θ ( t ) ] ] [ cos α ( i ) cos β ( i ) cos β ( i ) cos α ( i ) ] ,
H n ( ) ( t ) ( r , ν ) = ( ν μ ) 1 f ̃ ( t ) × E n ( ) ( t ) ( r , ν ) , { > , < } , n { 1 , 2 } .
W ( e ) ( α ) ( r 1 , r 2 , ν ) = E 1 ( > ) ( α ) ( r 1 , ν ; f 1 , f 2 ) ( E 1 ( < ) ( α ) ( r 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 + E 2 ( > ) ( α ) ( r 1 , ν ; f 1 , f 2 ) ( E 2 ( < ) ( α ) ( r 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 ,
W ( h ) ( α ) ( r 1 , r 2 , ν ) = H 1 ( > ) ( α ) ( r 1 , ν ; f 1 , f 2 ) ( H 1 ( < ) ( α ) ( r 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 + H 2 ( > ) ( α ) ( r 1 , ν ; f 1 , f 2 ) ( H 2 ( < ) ( α ) ( r 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 ,
W ( m ) ( α ) ( r , r 2 , ν ) = E 1 ( > ) ( α ) ( r 1 , ν ; f 1 , f 2 ) ( H 1 ( < ) ( α ) ( r 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 + E 2 ( > ) ( α ) ( r 1 , ν ; f 1 , f 2 ) ( H 2 ( < ) ( α ) ( r 2 , ν ; f 2 ) ) + d 2 f 1 d 2 f 2 .
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = { ( A x x A y x ) exp ( i 2 π f ( i ) T ρ 1 ) } { ( 1 0 ) exp ( i 2 π f ( i ) T ρ 2 ) } + { ( A x y A y y ) exp ( i 2 π f ( i ) T ρ 1 ) } { ( 0 1 ) exp ( i 2 π f ( i ) T ρ 2 ) } + ,
P ( e ) ( r , ρ , ν ) [ c 1 E 1 ( i ) ( ρ , ν ) + c 2 E 2 ( i ) ( ρ , ν ) ] = c 1 P ( e ) ( r , ρ , ν ) E 1 ( i ) ( ρ , ν ) + c 2 P ( e ) ( r , ρ , ν ) E 2 ( i ) ( ρ , ν ) ,
P ( h ) ( r , ρ , ν ) [ c 1 E 1 ( i ) ( ρ , ν ) + c 2 E 2 ( i ) ( ρ , ν ) ] = c 1 P ( h ) ( r , ρ , ν ) E 1 ( i ) ( ρ , ν ) + c 2 P ( h ) ( r , ρ , ν ) E 2 ( i ) ( ρ , ν ) ,
E α ( r , ν ) = α [ P ( e ) ( r , ρ , ν ) E ( i ) ( ρ , ν ) ] = α { P ( e ) ( r , ρ , ν ) [ x E x ( i ) ( ρ , ν ) + y E y ( i ) ( ρ , ν ) ] } = α { P ( e ) ( r , ρ , ν ) [ x E x ( i ) ( ρ , ν ) ] } + α { P ( e ) ( r , ρ , ν ) [ y E y ( i ) ( ρ , ν ) ] } = P α x ( e ) ( r , ρ , ν ) E x ( i ) ( ρ , ν ) + P α y ( e ) ( r , ρ , ν ) E y ( i ) ( ρ , ν ) ,
P α β ( e ) ( r , ρ , ν ) E β ( i ) ( ρ , ν ) = α { P ( e ) ( r , ρ , ν ) [ β E β ( i ) ( ρ , ν ) ] } , β { x , y } .
P α β ( e ) ( r , ρ , ν ) [ c 1 E β , 1 ( i ) ( ρ , ν ) + c 2 E β , 2 ( i ) ( ρ , ν ) ] = c 1 P α β ( e ) ( r , ρ , ν ) E β , 1 ( i ) ( ρ , ν ) + c 2 P α β ( e ) ( r , ρ , ν ) E β , 2 ( i ) ( ρ , ν ) ,
E k ( i ) ( ρ , ν ) = n = 1 c k n ( ν ) ϕ n ( ρ ) ,
k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * E k ( i ) ( ρ 1 , ν ) ( E l ( i ) ( ρ 2 , ν ) ) *
= 1 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * [ m = 1 c k m ( ν ) ϕ m ( ρ 1 ) ] [ n = 1 c l n ( ν ) ϕ n ( ρ 2 ) ] *
= 2 k l m = 1 n = 1 c k m ( ν ) c l n * ( ν ) P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * ϕ m ( ρ 1 ) ϕ n ( ρ 2 )
= 3 k l m = 1 n = 1 c k m ( ν ) c l n * ( ν ) P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * ϕ m ( ρ 1 ) ϕ n ( ρ 2 )
= 4 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * m = 1 n = 1 c k m ( ν ) c l n * ( ν ) ϕ m ( ρ 1 ) ϕ n ( ρ 2 )
= 5 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * [ m = 1 c k m ( ν ) ϕ m ( ρ 1 ) ] [ n = 1 c l n ( ν ) ϕ n ( ρ 2 ) ] *
= 6 k l P p k ( e ) ( r 1 , ρ 1 , ν ) ( P q l ( e ) ( r 2 , ρ 2 , ν ) ) * E k ( i ) ( ρ 1 , ν ) ( E l ( i ) ( ρ 2 , ν ) ) * ,
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = [ W x x ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) W x y ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) ( W x y ( e ) ( i ) ( ρ 2 , ρ 1 , ν ) ) * W y y ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) ] = [ m = 1 n = 1 c m n ( x x ) ( ν ) ϕ m ( ρ 1 ) ϕ n ( ρ 2 ) m = 1 n = 1 c m n ( x y ) ( ν ) ϕ m ( ρ 1 ) ϕ n ( ρ 2 ) m = 1 n = 1 ( c m n ( x y ) ( ν ) ) * ϕ m ( ρ 2 ) ϕ n ( ρ 1 ) m = 1 n = 1 c m n ( y y ) ( ν ) ϕ m ( ρ 1 ) ϕ n ( ρ 2 ) ] = m = 1 n = 1 c m n ( x x ) ( ν ) ( ϕ m ( ρ 1 ) 0 ) [ ϕ n ( ρ 2 ) 0 ] + m = 1 n = 1 c m n ( x y ) ( ν ) ( ϕ m ( ρ 1 ) 0 ) [ 0 ϕ n ( ρ 2 ) ] + m = 1 n = 1 c m n ( x y ) ( ν ) * ( 0 ϕ n ( ρ 1 ) ) [ ϕ m ( ρ 2 ) 0 ] + m = 1 n = 1 c m n ( y y ) ( ν ) ( 0 ϕ m ( ρ 1 ) ) [ 0 ϕ n ( ρ 2 ) ] ,
E ( i ) ( ρ , ν ) = A ( e ) ( f , ν ) exp ( i 2 π f T ρ ) d 2 f ,
A ( e ) ( f , ν ) = E ( i ) ( ρ , ν ) exp ( i 2 π f T ρ ) d 2 ρ
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = [ A ( e ) ( f 1 , ν ) exp ( i 2 π f 1 T ρ 1 ) d 2 f 1 ] [ A ( e ) ( f 2 , ν ) exp ( i 2 π f 2 T ρ 2 ) d 2 f 2 ] + = A ( e ) ( f 1 , f 2 , ν ) exp [ i 2 π ( f 1 T ρ 1 f 2 T ρ 2 ) ] d 2 f 1 d 2 f 2 ,
A ( e ) ( f 1 , f 2 , ν ) = [ A x x ( e ) ( f 1 , f 2 , ν ) A x y ( e ) ( f 1 , f 2 , ν ) A y x ( e ) ( f 1 , f 2 , ν ) A y y ( e ) ( f 1 , f 2 , ν ) ] = ( A x x ( e ) ( f 1 , f 2 , ν ) A y x ( e ) ( f 1 , f 2 , ν ) ) ( 1 0 ) + ( A x y ( e ) ( f 1 , f 2 , ν ) A y y ( e ) ( f 1 , f 2 , ν ) ) ( 0 1 ) .
W ( e ) ( i ) ( ρ 1 , ρ 2 , ν ) = { ( A x x ( e ) ( f 1 , f 2 , ν ) A y x ( e ) ( f 1 , f 2 , ν ) ) exp ( i 2 π f 1 T ρ 1 ) } { [ 1 0 ] exp ( i 2 π f 2 T ρ 2 ) } + d 2 f 1 d 2 f 2 + { ( A x y ( e ) ( f 1 , f 2 , ν ) A y y ( e ) ( f 1 , f 2 , ν ) ) exp ( i 2 π f 1 T ρ 1 ) } { ( 0 1 ) exp ( i 2 π f 2 T ρ 2 ) } + d 2 f 1 d 2 f 2 .

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