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Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere

Xinyue Du, Daomu Zhao, and Olga Korotkova

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Abstract

We report analytic formulas for the elements of the 2×2 cross-spectral density matrix of a stochastic electromagnetic anisotropic beam propagating through the turbulent atmosphere with the help of vector integration. From these formulas the changes in the spectral density (spectrum), in the spectral degree of polarization, and in the spectral degree of coherence of such a beam on propagation are determined. As an example, these quantities are calculated for a so-called anisotropic electromagnetic Gaussian Schell-model beam propagating in the isotropic and homogeneous atmosphere. In particular, it is shown numerically that for a beam of this class, unlike for an isotropic electromagnetic Gaussian Schell-model beam, its spectral degree of polarization does not return to its value in the source plane after propagating at sufficiently large distances in the atmosphere. It is also shown that the spectral degree of coherence of such a beam tends to zero with increasing distance of propagation through the turbulent atmosphere, in agreement with results previously reported for isotropic beams.

©2007 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. Illustrating the notation relating to propagation of a stochastic electromagnetic beam through the turbulent atmosphere.

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Fig. 2.
Fig. 2. Changes in the normalized spectral density S/S0 along the z-axis of anisotropic electromagnetic beams through the turbulent atmosphere with different C2 n . The source is assumed to be electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax =2, Ay =1, Bxy =0.2exp(iπ/3), σx =1cm, σy =2cm, δxx =δyy =2mm, δxy =3mm. The unit of C2 n is m-2 3.

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Fig. 3.
Fig. 3. Changes in the spectral degree of coherence µ along the z-axis of anisotropic electromagnetic Gaussian Schell-model beams propagating through the turbulent atmosphere with different C2 n . The source parameters are the same as Fig. 2. Pairs of field points: (a) ρ T 12=(1mm, 0, -1mm, 0), (b) ρ T 12=(3 mm, 0, -3 mm, 0) The unit of C2 n is m-2/3.

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Fig. 4.
Fig. 4. Changes in the spectral degree of polarization P along the z-axis of anisotropic electromagnetic Gaussian Schell-model beams propagating through the turbulent atmosphere with different C2 n . The source parameters are the same as in Fig. 2, but (a) σx =σy 1 cm, (b) σx =1 cm, σy =2 cm. The unit of C2 n is m-2/3.

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Equations (19)

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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 ) .
E i ( ρ , z , ω ) = − ik 2 π z exp ( ikz ) ∬ E i ( 0 ) ( ρ ′ , ω ) exp [ ik 2 z ( ρ − ρ ′ ) 2 ] exp [ ψ ( ρ , ρ ′ , z , ω ) ] d 2 ρ ′ ,
W ij ( ρ 1 , ρ 2 , z , ω ) = k 2 4 π 2 z 2 ∬ ∬ W ij ( 0 ) ( ρ 1 ′ , ρ 2 ′ , ω ) exp [ − ik 2 z ( ρ 1 − ρ 1 ′ ) 2 + ik 2 z ( ρ 2 − ρ 2 ′ ) 2 ]
× 〈 exp [ ψ * ( ρ 1 , ρ 1 ′ , z , ω ) + ψ ( ρ 2 , ρ 2 ′ , z , ω ) ] 〉 m d 2 ρ 1 ′ d 2 ρ 2 ′ ,
〈 exp [ ψ * ( ρ 1 , ρ 1 ′ , z , ω ) + ψ ( ρ 2 , ρ 2 ′ , z , ω ) ] 〉 m = exp [ − ( 1 ⁄ 2 ) D ψ ( ρ d ′ , ρ d ) ]
≅ exp [ − ( 1 ⁄ ρ 0 2 ) ( ρ d ′ 2 + ρ d ′ · ρ d + ρ d 2 ) ] ,
W ij ( ρ 12 , z , ω ) = k 2 4 π 2 [ Det ( B ¯ ) ] − 1 ⁄ 2 ∬ ∬ W ij ( 0 ) ( ρ 12 ′ , ω )
× exp [ − ik 2 ( ρ 12 ′ T B ¯ − 1 ρ 12 ′ − 2 ρ 12 ′ T B ¯ − 1 ρ 12 + ρ 12 T B ¯ − 1 ρ 12 ) ] ,
× exp [ − ik 2 ( ρ 12 ′ T P ¯ ρ 12 ′ + ρ 12 ′ T P ¯ ρ 12 + ρ 12 T P ¯ ρ 12 ) ] d 4 ρ 12 ′
B ¯ = [ z I 0 0 − z I ] , P ¯ = 2 ik ρ 0 2 [ I − I − I I ] ,
W ij ( 0 ) ( ρ 1 ′ , ρ 2 ′ , ω ) = S i ( 0 ) ( ρ 1 ′ , ω ) S j ( 0 ) ( ρ 2 ′ , ω ) η ij ( 0 ) ( ρ 2 ′ − ρ 1 ′ , 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 12 ′ , ω ) = A i A j B ij exp ( − ik 2 ρ 12 ′ T M ij ′ − 1 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 ]
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 }
S ( ρ , z , ω ) = Tr W ↔ ( ρ , z , ω ) ,
μ ( ρ 12 , z , ω ) = Tr W ↔ ( ρ 12 , z , ω ) Tr W ↔ ( ρ 11 , z , ω ) Tr W ↔ ( ρ 22 , z , ω ) ,
P ( ρ , z , ω ) = 1 − 4 Det W ↔ ( ρ , z , ω ) [ Tr W ↔ ( ρ , z , ω ) ] 2 .
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