Abstract

An analytical formula for the cross-spectral density matrix of the electric field of anisotropic electromagnetic Gaussian–Schell model beams propagating in free space is derived by using a tensor method. The effects of coherence on those beams are studied. It is shown that two anisotropic stochastic electromagnetic beams that propagate from the source plane z=0 into the half-space z>0 may have different beam shapes (i.e., spectral density) and states of polarization in the half-space, even though they have the same beam shape and states of polarization in the source plane. This fact is due to a difference in the coherence properties of the field in the source plane.

© 2007 Optical Society of America

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References

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Figures (1)

Fig. 1
Fig. 1

Evolution of the normalized spectral density [black-to-white (gray-scale) background] and the state of polarization of two anisotropic Gaussian–Schell model beams propagating in free space. The ellipses superimposed on the background are the spectral polarization ellipses of the fully polarized portion of the field. Their centers are chosen to be located along the two mutually orthogonal directions at points where the spectral density normalized by its axial value takes on the values 1, 0.7, 0.35, and 0.1. The ellipses (major axes) are scaled by the degree of polarization. The parameters are taken as λ = 632.8 nm , σ i = σ j = σ , δ i j = δ , A x = A y = 1 , B x y = B y x * = 0.16 exp [ ( i Π ) 4 ] , σ x x = 2 cm , σ y y = 1 cm , σ x y = + , δ x y ( a ) = δ x y ( b ) = δ x x ( a ) = δ x x ( b ) = 1 mm , δ y y ( a ) = 1 mm , δ y y ( b ) = 0.4 mm , and z = 50 m . Image scale: a, 8 cm × 8 cm ; b, 15 cm × 15 cm .

Equations (10)

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W = [ W i j ( r 1 , r 2 ; ω ) ] = [ E i * ( r 1 ; ω ) E j ( r 2 ; ω ) ] ,
W i j ( r 1 , r 2 ; ω ) = A i A j B i j exp { 1 4 [ r 1 T σ i r 1 + r 2 T σ j r 2 ] } × exp [ 1 2 ( r 1 r 2 ) T δ i j ( r 1 r 2 ) ] ,
σ i = [ σ i 11 2 σ i 12 2 σ i 21 2 σ i 22 2 ] ,
σ j = [ σ j 11 2 σ j 12 2 σ j 21 2 σ j 22 2 ] ,
δ i j = [ δ i j 11 2 δ i j 12 2 δ i j 21 2 δ i j 22 2 ] .
W i j ( ρ 1 , ρ 2 , z ; ω ) = ( k 2 π z ) 2 z = 0 W i j ( 0 ) ( r 1 , r 2 ; ω ) exp [ i k ( ρ 1 r 1 ) 2 ( ρ 2 r 2 ) 2 2 z ] d r 1 d r 2 ,
W i j ( ρ 1 , ρ 2 , z ; ω ) = k 2 A i A j B i j 4 π 2 [ det ( F ̃ ) ] 1 2 exp { i k 2 [ r ̃ T ( F ̃ 1 + M ̃ i j 1 ) r ̃ 2 r ̃ T F ̃ 1 ρ ̃ + ρ ̃ T F ̃ 1 ρ ̃ ] } d r ̃ ,
M ̃ i j 1 = [ i 2 k σ i i k δ i j i k δ i j i k δ i j i 2 k σ j i k δ i j ] ,
F ̃ = [ z I 0 0 z I ] , I
W i j ( ρ 1 , ρ 2 , z ; ω ) = A i A j B i j [ det ( I ̃ + F ̃ M ̃ i j 1 ) ] 1 2 exp { i k 2 [ ρ ̃ T ( F ̃ + M ̃ i j ) 1 ρ ̃ ] } ,

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