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Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation

Xinyue Du and Daomu Zhao

Open Access Open Access

Abstract

The cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources that are completely unpolarized or completely polarized are derived. We find that both the completely unpolarized stochastic electromagnetic Gaussian Schell-model beam and the completely polarized stochastic electromagnetic Gaussian Schell-model beam will keep their spectral degree of polarization or become partially polarized under different constraint conditions during their propagation in free space or through turbulent atmosphere. We give necessary theoretical explanation to the physical phenomena. They are considered as coherence-induced polarization changes and spectral density-induced polarization changes.

©2008 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. Illustrating the notation.

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Fig. 2.
Fig. 2. Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely unpolarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax =Ay =1, Bxy =0, σx σy ,=1cm, δxx =2mm, and δyy is shown in the figure.

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Fig.3. .
Fig.3. . s Fig. 2, but passing through the turbulent atmosphere with C 2 n =10-12 m-2/3.

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Fig. 4.
Fig. 4. Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely polarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax =2, Ay =1, Bxy =exp(iπ/3), σx=1cm, δxx =δyy =δxy =2mm, and σy is shown in the figure.

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Fig. 5.
Fig. 5. As Fig. 4, but passing through turbulent atmosphere with C 2 n =10-12 m-2/3.

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Equations (35)

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W ↔ ( r 1 , r 2 , ω ) ≡ [ W ij ( r 1 , r 2 , ω ) ] = [ 〈 E i * ( r 1 , ω ) E j ( r 2 , ω ) 〉 ] , ( i = x , y ; j = x , y ) ,
W ↔ ( ρ 1 , ρ 2 , z , ω ) = ∫ ∫ W ↔ ( 0 ) ( ρ ′ 1 , ρ ′ 2 , ω ) K ( ρ 1 − ρ ′ 1 , ρ 2 − ρ ′ 2 , z , ω ) d 2 ρ ′ 1 d 2 ρ ′ 2 ,
K ( ρ 1 − ρ ′ 1 , ρ 2 − ρ ′ 2 , z , ω ) = G * ( ρ 1 − ρ ′ 1 , z , ω ) G ( ρ 2 − ρ ′ 2 , z , ω ) ,
K rm ( ρ 1 − ρ ′ 1 , ρ 2 − ρ ′ 2 , z , ω ) = 〈 G m * ( ρ 1 − ρ ′ 1 , z , ω ) G m ( ρ 2 − ρ ′ 2 , z , ω ) 〉 rm ,
P ( ρ , ρ , z , ω ) = 1 − 4 Det W ↔ ( ρ , ρ , z , ω ) [ T r W ↔ ( ρ , ρ , z , ω ) ] 2 ,
W ↔ ( 0 ) ( ρ ′ 1 , ρ ′ 2 , ω ) = a ( ρ ′ 1 , ρ ′ 2 , ω ) [ 1 0 0 1 ] .
W ↔ ( 0 ) ( ρ ′ 1 , ρ ′ 2 , ω ) = [ e x * ( ρ ′ 1 , ω ) e x ( ρ ′ 2 , ω ) e x * ( ρ ′ 1 , ω ) e y ( ρ ′ 2 , ω ) e y * ( ρ ′ 1 , ω ) e x ( ρ ′ 2 , ω ) e y * ( ρ ′ 1 , ω ) e y ( ρ ′ 2 , ω ) ] ,
W ij ( 0 ) ( ρ ′ 1 , ρ ′ 2 , ω ) = A i A j B ij exp ( − ρ ′ 1 2 4 σ i 2 − ρ ′ 2 2 4 σ j 2 ) exp ( − ∣ ρ ′ 2 − ρ ′ 1 ∣ 2 2 δ ij 2 ) ,
W ij ( 0 ) ( ρ ′ , ρ ′ , ω ) = A i A j B ij exp ( − ρ ′ 2 4 σ i 2 − ρ ′ 2 4 σ j 2 ) .
W ↔ ( 0 ) ( ρ ′ , ρ ′ , ω ) = A 2 exp ( − ρ ′ 2 2 σ 2 ) [ 1 0 0 1 ] .
max { δ xx , δ yy } ≤ δ xy ≤ min { δ xx ∣ B xy ∣ , δ yy ∣ B xy ∣ } ,
W ↔ ( 0 ) ( ρ ′ , ρ ′ , ω ) = [ A x A x B xx exp ( − ρ ′ 2 4 σ x 2 − ρ ′ 2 4 σ x 2 ) A x A y B xy exp ( − ρ ′ 2 4 σ x 2 − ρ ′ 2 4 σ y 2 ) A y A x B yx exp ( − ρ ′ 2 4 σ y 2 − ρ ′ 2 4 σ x 2 ) A y A y B yy exp ( − ρ ′ 2 4 σ y 2 − ρ ′ 2 4 σ y 2 ) ] .
W ij ( ρ 12 , z , ω ) = A i A j B ij [ Det ( I ¯ + BP ¯ + B ¯ M ′ ij − 1 ) ] − 1 2 exp { − ik 2 ρ 12 T [ ( B ¯ − 1 + P ¯ ) ,
− ( B ¯ − 1 − 1 2 P ¯ ) T ( B ¯ − 1 + P ¯ + M ′ ij − 1 ) − 1 ( B ¯ − 1 − 1 2 P ¯ ) ] ρ 12 }
M ′ ij − 1 = [ − i 2 k σ i − 2 − i k δ ij − 2 0 i k δ ij − 2 0 0 − i 2 k σ i − 2 − i k δ ij − 2 0 i k δ ij − 2 i k δ ij − 2 0 − i 2 k σ j − 2 − i k δ ij − 2 0 0 i k δ ij − 2 0 − i 2 k σ j − 2 − i k δ ij − 2 ] .
B ¯ = [ z I 0 0 − z I ] , P ¯ = 2 ik ρ 0 2 [ I − I − I I ] ,
W ij ( ρ 12 , z , ω ) = A i A j B ij [ Det ( I ¯ + B ¯ M ′ ij − 1 ) ] − 1 2
× exp { − ik 2 ρ 12 T [ B ¯ − 1 − B ¯ − 1 T ( B ¯ − 1 + M ′ ij − 1 ) − 1 B ¯ − 1 ] ρ 12 } ,
F ij ( ρ 12 , z , ω ) = [ Det ( I ¯ + BP ¯ + B ¯ M ′ ij − 1 ) ] − 1 2 exp { − ik 2 ρ 12 T [ ( B ¯ − 1 + P ¯ ) ,
− ( B ¯ − 1 − 1 2 P ¯ ) T ( B ¯ − 1 + P ¯ + M ′ ij − 1 ) − 1 ( B ¯ − 1 − 1 2 P ¯ ) ] ρ 12 }
W xx ( ρ 12 , z , ω ) = W yy ( ρ 12 , z , ω ) = A 2 F ( ρ 12 , z , ω ) ,
W xy ( ρ 12 , z , ω ) = W yx ( ρ 12 , z , ω ) = 0 .
P ( ρ , ρ , z , ω ) = 1 − 4 Det W ↔ ( ρ , ρ , z , ω ) [ Tr W ↔ ( ρ , ρ , z , ω ) ] 2
= ∣ W xx ( ρ 12 , z , ω ) − W yy ( ρ 12 , z , ω ) ∣ W xx ( ρ 12 , z , ω ) + W yy ( ρ 12 , z , ω ) ,
= 0
S i ( 0 ) ( ρ ' , ω ) = A i 2 exp ( − ρ ' 2 2 σ i 2 ) , ( i = x , y ) .
W xx ( ρ 12 , z , ω ) = A x A x B xx F ( ρ 12 , z , ω ) ,
W xy ( ρ 12 , z , ω ) = A x A y B xy F ( ρ 12 , z , ω ) ,
W yx ( ρ 12 , z , ω ) = A y A x B yx F ( ρ 12 , z , ω ) ,
W yy ( ρ 12 , z , ω ) = A y A y B yy F ( ρ 12 , z , ω ) .
Det W ↔ ( ρ , ρ , z , ω ) = W xx ( ρ 12 , z , ω ) W yy ( ρ 12 , z , ω ) − W xy ( ρ 12 , z , ω ) W yx ( ρ 12 , z , ω )
= A x 2 A y 2 ( B xx B yy − B xy B yx ) F 2 ( ρ 12 , z , ω )
= 0
P ( ρ , ρ , z , ω ) = 1 − 0 [ Tr W ↔ ( ρ , ρ , z , ω ) ] 2 .
= 1
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