Abstract

It has been known for some time that the spectrum of light may change on propagation, even in free space. The theory of this phenomenon was developed within the framework of scalar theory. We generalize it to electromagnetic beams, generated by planar, secondary, stochastic sources. We also derive an electromagnetic analog of the so-called scaling law. When this law is satisfied the normalized spectrum of the beam is the same throughout the far zone and across the source.

© 2006 Optical Society of America

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References

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  1. E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
    [CrossRef] [PubMed]
  2. For the review of many publications on this subject, see E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771 (1996).
  3. O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2005

2004

2003

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

1996

For the review of many publications on this subject, see E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771 (1996).

1986

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Lett. A

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Phys. Rev. Lett.

E. Wolf, Phys. Rev. Lett. 56, 1370 (1986).
[CrossRef] [PubMed]

Rep. Prog. Phys.

For the review of many publications on this subject, see E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771 (1996).

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Equations (29)

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W ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) ] = [ E i * ( ρ 1 , ω ) E j ( ρ 2 , ω ) ] ( i , j = x , y ) .
W i i ( 0 ) ( ρ 1 , ρ 2 , ω ) = S i ( 0 ) ( ( ρ 1 + ρ 2 ) 2 , ω ) μ i i ( 0 ) ( ρ 2 ρ 1 , ω )
( i = x , y ) ,
μ i i ( 0 ) ( ρ 2 ρ 1 , ω ) = E i * ( ρ 1 , ω ) E i ( ρ 2 , ω ) E i * ( ρ 1 , ω ) E i ( ρ 1 , ω ) E i * ( ρ 2 , ω ) E i ( ρ 2 , ω )
( i = x , y ) .
S ( 0 ) ( ω ) = S x ( 0 ) ( ω ) + S y ( 0 ) ( ω ) ,
S N ( 0 ) ( ω ) = S x ( 0 ) ( ω ) + S y ( 0 ) ( ω ) [ S x ( 0 ) ( ω ) + S y ( 0 ) ( ω ) ] d ω .
η ( 0 ) ( ρ 1 , ρ 2 , ω ) = Tr W ( 0 ) ( ρ 1 , ρ 2 , ω ) Tr W ( 0 ) ( ρ 1 , ρ 2 , ω ) Tr W ( 0 ) ( ρ 2 , ρ 2 , ω ) ,
( ρ 1 D , ρ 2 D ) .
η ( 0 ) ( ρ 1 , ρ 2 , ω ) = S x ( 0 ) ( ω ) μ x x ( 0 ) ( ρ 2 ρ 1 , ω ) + S y ( 0 ) ( ω ) μ y y ( 0 ) ( ρ 2 ρ 1 , ω ) S x ( 0 ) ( ω ) + S y ( 0 ) ( ω )
( ρ 1 D , ρ 2 D ) .
S ( ) ( r s , ω ) = ( 2 π k ) 2 cos 2 θ S ̃ ( 0 ) ( 0 , ω ) η ̃ ( 0 ) ( k s , ω ) r 2 .
g ̃ ( f , ω ) = 1 ( 2 π ) 2 g ( ρ , ω ) exp ( i f ρ ) d 2 ρ ,
S ( ) ( r s , ω ) = ( 2 π k ) 2 A cos 2 θ S ( 0 ) ( ω ) η ̃ ( 0 ) ( k s , ω ) r 2 ,
S N ( ) ( r s , ω ) = S ( ) ( r s , ω ) S ( ) ( r s , ω ) d ω .
S N ( ) ( r s , ω ) = ω 2 [ S x ( 0 ) ( ω ) μ ̃ x x ( 0 ) ( k s , ω ) + S y ( 0 ) ( ω ) μ ̃ y y ( 0 ) ( k s , ω ) ] ω 2 [ S x ( 0 ) ( ω ) μ ̃ x x ( 0 ) ( k s , ω ) + S y ( 0 ) ( ω ) μ ̃ y y ( 0 ) ( k s , ω ) ] d ω .
S i ( 0 ) ( ρ , ω ) = B exp [ ( ω ω ¯ ) 2 σ 2 ] ( i = x , y ) ,
μ i i ( 0 ) ( ρ 2 ρ 1 , ω ) = exp [ ( ρ 2 ρ 1 ) 2 δ i 2 2 ] ( i = x , y ) .
μ ̃ i i ( 0 ) ( f , ω ) = δ i 2 2 π exp [ δ i 2 2 f 2 ] ( i = x , y ) .
S N ( ) ( r s , ω ) = ω 2 exp [ ( ω ω ¯ ) 2 σ 2 ] { δ x 2 exp ( ω 2 K x 2 ) + δ y 2 exp ( ω 2 K y 2 ) } ω 2 exp [ ( ω ω ¯ ) 2 σ 2 ] { δ x 2 exp ( ω 2 K x 2 ) + δ y 2 exp ( ω 2 K y 2 ) } d ω ,
K i 2 = 2 c 2 sin θ δ i 2 ( i = x , y ) .
μ ̃ i i ( 0 ) ( k s , ω ) = F i i ( ω ) H ̃ ( s ) ( i = x , y ) .
S N ( ) ( r s , ω ) = k 2 [ S x ( 0 ) ( ω ) F x x ( ω ) + S y ( 0 ) ( ω ) F y y ( ω ) ] k 2 [ S x ( 0 ) ( ω ) F x x ( ω ) + S y ( 0 ) ( ω ) F y y ( ω ) ] d ω .
μ i i ( 0 ) ( ρ , ω ) = F i i ( ω ) H ̃ i i ( s , ω ) exp ( i k s ρ ) d 2 ( k s ) ,
μ i i ( 0 ) ( ρ , ω ) = k 2 F i i ( ω ) H ( k ρ ) ,
H ( k ρ ) = 1 4 π 2 H ̃ i i ( s , ω ) exp ( i k s ρ ) d 2 ( k s )
F i i ( ω ) = 1 k 2 H ( 0 ) ( i = x , y ) .
μ i i ( 0 ) ( ρ , ω ) = H ( k ρ ) H ( 0 ) ,
S N ( ) ( r s , ω ) = S ( 0 ) ( ω ) S ( 0 ) ( ω ) d ω S N ( 0 ) ( r s , ω ) .

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