Abstract

Generalized propagation formulas for the elements of the cross-spectral density matrix of stochastic electromagnetic beams on inverse propagation through an axially symmetrical or nonsymmetrical optical system are derived with the help of Fourier transform and inverse Fourier transform. As an example, we apply the formula to the inverse source problem of stochastic electromagnetic Gaussian Schell-model beams.

© 2009 Optical Society of America

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References

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2009 (1)

D. Kuebel, M. Lahiri, and E. Wolf, Opt. Commun. 285, 141 (2009).
[CrossRef]

2008 (2)

2002 (1)

1986 (1)

1985 (2)

Cai, Y.

Carter, W. H.

Du, X.

X. Du and D. Zhao, Opt. Commun. 281, 2711 (2008).
[CrossRef]

Hanson, S. G.

Jakobsen, M. L.

Kuebel, D.

D. Kuebel, M. Lahiri, and E. Wolf, Opt. Commun. 285, 141 (2009).
[CrossRef]

LaHaie, I. J.

Lahiri, M.

D. Kuebel, M. Lahiri, and E. Wolf, Opt. Commun. 285, 141 (2009).
[CrossRef]

Li, J.

J. Li, Laser Diffraction and Heat Effect Calculation (Science Press, 2002).

Lin, Q.

Takeda, M.

Wang, W.

Wolf, E.

D. Kuebel, M. Lahiri, and E. Wolf, Opt. Commun. 285, 141 (2009).
[CrossRef]

W. H. Carter and E. Wolf, J. Opt. Soc. Am. A 2, 1994 (1985).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

Zhao, D.

X. Du and D. Zhao, Opt. Commun. 281, 2711 (2008).
[CrossRef]

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

Opt. Commun. (2)

X. Du and D. Zhao, Opt. Commun. 281, 2711 (2008).
[CrossRef]

D. Kuebel, M. Lahiri, and E. Wolf, Opt. Commun. 285, 141 (2009).
[CrossRef]

Opt. Lett. (1)

Other (2)

J. Li, Laser Diffraction and Heat Effect Calculation (Science Press, 2002).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

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Equations (15)

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W i j ( ρ 12 , z ; ω ) = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( 0 ) ( ρ 12 , 0 ; ω ) × exp [ i k 2 ( ρ 12 T A ¯ B ¯ 1 ρ 12 2 ρ 12 T B ¯ 1 ρ 12 + ρ 12 T D ¯ B ¯ 1 ρ 12 ) ] d 4 ρ 12 ,
A ¯ = ( A 0 ̃ 0 ̃ A ) , B ¯ = ( B 0 ̃ 0 ̃ B ) , C ¯ = ( C 0 ̃ 0 ̃ C ) ,
D ¯ = ( D 0 ̃ 0 ̃ D ) ,
A = ( a x 0 0 a y ) , B = ( b x 0 0 b y ) , C = ( c x 0 0 c y ) ,
D = ( d x 0 0 d y ) ,
W i j ( ρ 12 , z ; ω ) = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( 0 ) ( A ¯ 1 u 12 , 0 ; ω ) × exp { i k 2 [ ρ 12 T ( D ¯ B ¯ 1 A ¯ 1 B ¯ 1 ) ρ 12 ] } exp { i k 2 A ¯ 1 2 B ¯ 1 2 ( u 12 ρ 12 ) 2 } 1 Det ( A ¯ ) d 4 u 12 = 1 λ 2 [ Det ( B ¯ ) ] 1 2 1 Det ( A ¯ ) W ij ( 0 ) ( A ¯ 1 ρ 12 , 0 ; ω ) exp { i k 2 [ ρ 12 T ( D ¯ B ¯ 1 A ¯ 1 B ¯ 1 ) ρ 12 ] } exp [ i k 2 A ¯ 1 2 B ¯ 1 2 ρ 12 2 ] ,
F [ W i j ( ρ 12 , z ; ω ) ] = [ Det ( B ¯ ) ] 1 2 [ Det ( A ¯ ) ] 1 [ Det ( A ¯ 1 2 B ¯ 1 2 ) ] 1 exp [ i 2 π 2 k A ¯ 1 2 B ¯ 1 2 f 12 2 ] × F { W i j ( 0 ) ( A ¯ 1 ρ 12 , 0 ; ω ) exp { i k 2 [ ρ 12 T ( D ¯ B ¯ 1 A ¯ 1 B ¯ 1 ) ρ 12 ] } } ,
W i j ( 0 ) ( A ¯ 1 ρ 12 , 0 ; ω ) = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( ρ 12 , z ; ω ) exp { i k 2 [ ρ 12 T ( D ¯ B ¯ 1 A ¯ 1 B ¯ 1 ) ρ 12 ] } exp [ i k 2 A ¯ 1 2 B ¯ 1 2 ρ 12 2 ] = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( u 12 , z ; ω ) exp { i k 2 [ u 12 T ( D ¯ B ¯ 1 A ¯ 1 B ¯ 1 ) u 12 ] } × exp [ i k 2 A ¯ 1 2 B ¯ 1 2 ( ρ 12 u 12 ) 2 ] d 4 u 12 .
W i j ( 0 ) ( ρ 12 , 0 ; ω ) = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( ρ 12 , z ; ω ) × exp [ i k 2 ( ρ 12 T D ¯ B ¯ 1 ρ 12 2 ρ 12 T B ¯ 1 ρ 12 + ρ 12 T A ¯ B ¯ 1 ρ 12 ) ] d 4 ρ 12 .
W i j ( 0 ) ( ρ 1 , ρ 2 , 0 ; ω ) = 1 ( λ b ) 2 W i j ( ρ 1 , ρ 2 , z ; ω ) × exp { i k 2 b [ d ( ρ 1 2 ρ 2 2 ) 2 ( ρ 1 ρ 1 ρ 2 ρ 2 ) + a ( ρ 1 2 ρ 2 2 ) ] } d 2 ρ 1 d 2 ρ 2 ,
W i j ( ρ 1 , ρ 2 , z ; ω ) = A i A j B i j exp ( ρ 1 2 4 σ ( z ) i 2 ) exp ( ρ 2 2 4 σ ( z ) j 2 ) exp ( ρ 2 ρ 1 2 2 δ ( z ) i j 2 ) ,
W i j ( ρ 12 , z ; ω ) = A i A j B i j exp ( i k 2 ρ 12 T M ( z ) i j 1 ρ 12 ) ,
M ( z ) i j 1 = [ ( i 2 k σ ( z ) i 2 i k δ ( z ) i j 2 ) I ( i k δ ( z ) i j 2 ) I ( i k δ ( z ) i j 2 ) I ( i 2 k σ ( z ) j 2 i k δ ( z ) i j 2 ) I ] ,
W i j ( 0 ) ( ρ 12 , 0 ; ω ) = A i A j B i j [ Det ( D ¯ + B ¯ M ( z ) i j 1 ) ] 1 2 exp [ i k 2 ρ 12 T M ( 0 ) i j 1 ρ 12 ] ,
M ( 0 ) i j 1 = ( C ¯ A ¯ M ( z ) i j 1 ) ( D ¯ + B ¯ M ( z ) i j 1 ) 1 .

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