Abstract

The generalized Stokes parameters of 2D stochastic electromagnetic beams are developed to the 3D case, which can be addressed as certain linear combinations of the 3 × 3 cross-spectral density matrix in terms of the nine Gell-Mann matrices. Using the electromagnetic Gaussian Shell-model source as an example, we investigate their precise propagation laws of coherence properties and polarization properties with the help of the 3D generalized Stokes parameters. Some numerical examples and detailed comparisons of the obtained results with the 2D case are made. It is shown that 3D generalized Stokes parameters are required for the exact description of stochastic electromagnetic beams.

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References

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  1. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica (Utrecht) 5(8), 785–795 (1938).
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    [CrossRef] [PubMed]
  4. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24(4), 1063–1068 (2007).
    [CrossRef]
  5. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
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    [CrossRef] [PubMed]
  7. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
    [CrossRef] [PubMed]
  8. T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88(12), 123902 (2002).
    [CrossRef] [PubMed]
  9. A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253(1-3), 10–14 (2005).
    [CrossRef]
  10. T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34(21), 3394–3396 (2009).
    [CrossRef] [PubMed]
  11. A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71(6), 063815 (2005).
    [CrossRef]
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    [CrossRef]

2009 (1)

2007 (2)

2005 (3)

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253(1-3), 10–14 (2005).
[CrossRef]

A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71(6), 063815 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

2002 (1)

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88(12), 123902 (2002).
[CrossRef] [PubMed]

1976 (1)

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica (Utrecht) 5(8), 785–795 (1938).
[CrossRef]

Du, X.

Duan, K.

Friberg, A. T.

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88(12), 123902 (2002).
[CrossRef] [PubMed]

Korotkova, O.

Lindfors, K.

Lü, B.

Luis, A.

A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24(4), 1063–1068 (2007).
[CrossRef]

A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71(6), 063815 (2005).
[CrossRef]

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253(1-3), 10–14 (2005).
[CrossRef]

Mandel, L.

Setälä, T.

Tervo, J.

Wolf, E.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica (Utrecht) 5(8), 785–795 (1938).
[CrossRef]

Zhao, D.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253(1-3), 10–14 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71(6), 063815 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88(12), 123902 (2002).
[CrossRef] [PubMed]

Physica (Utrecht) (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica (Utrecht) 5(8), 785–795 (1938).
[CrossRef]

Other (1)

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

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Figures (6)

Fig. 1
Fig. 1

The spectral density of a stochastic electromagnetic beam in the plane z = 15zR , The source is assumed to be a Gaussian Shell-model source with Ax = 1.5, Ay = 1, Bxy = 0.3exp(/6), Byx = 0.3exp(-/6), (a) f 1 = f 2 = 0.005, fxx = fyy = fxy = fyx = 0.015, (b) f 1 = f 2 = 0.2, fxx = fyy = fxy = fyx = 0.45.

Fig. 2
Fig. 2

The contour graphs of the spectral densityof a stochastic electromagnetic beam in the plane z = 15zR . The source parameters are the same as in Fig. 1(b). (a) S 0, 2D , (b) S 0, 3D .

Fig. 3
Fig. 3

The changes in the spectral degree of coherence along z-axis direction of a stochastic electromagnetic Gaussian Schell-model beam. The source parameters are the same as in Fig. 1 except for fxy = 0.6fxx . Pairs of field point ρ 2 = ρ 1 = ( 0.2 m m ,   0.2 m m ) .

Fig. 4
Fig. 4

(a) and (b) are the changes in the spectral degree P of polarization along z-axis direction of a stochastic electromagnetic Gaussian Schell-model beam, (c) and (d) are the transverse distribution of P in the plane z = 15zR . The source parameters are the same as in Fig. 1 except for fαβ , fxy = fyx showed in figures, (a) and (c) fxx = fyy = 0.015, (b) and (d) fxx = fyy = 0.45.

Fig. 5
Fig. 5

There-dimensional distributions of spectral degree of polarization P 2D and corresponding contour graphs of a stochastic electromagnetic beam in the plane z = 15zR . fxy = 0.6fxx and the other source parameters are the same as in Fig. 4(d).

Fig. 6
Fig. 6

As Fig. 5 but for P 3D, (a) Ax = 1.5, Ay = 1; (b) Ax = 1, Ay = 1; (c) Ax = 1, Ay = 1.5.

Equations (23)

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S i , 2 D ( r 1 , r 2 , ω ) = tr [ W 2 D ( r 1 , r 2 , ω ) σ i ] ,
S j , 3 D ( r 1 , r 2 , ω ) = tr [ W 3 D ( r 1 , r 2 , ω ) λ j ] ,
S 0 , 3 D ( r 1 , r 2 , ω ) = W x x , 3 D ( r 1 , r 2 , ω ) + W y y , 3 D ( r 1 , r 2 , ω ) + W z z , 3 D ( r 1 , r 2 , ω ) ,
S 1 , 3 D ( r 1 , r 2 , ω ) = W x y , 3 D ( r 1 , r 2 , ω ) + W y x , 3 D ( r 1 , r 2 , ω ) ,
S 2 , 3 D ( r 1 , r 2 , ω ) = i [ W y x , 3 D ( r 1 , r 2 , ω ) W x y , 3 D ( r 1 , r 2 , ω ) ] ,
S 3 , 3 D ( r 1 , r 2 , ω ) = W x x , 3 D ( r 1 , r 2 , ω ) W y y , 3 D ( r 1 , r 2 , ω ) ,
S 4 , 3 D ( r 1 , r 2 , ω ) = W x z , 3 D ( r 1 , r 2 , ω ) + W z x , 3 D ( r 1 , r 2 , ω ) ,
S 5 , 3 D ( r 1 , r 2 , ω ) = i [ W x z , 3 D ( r 1 , r 2 , ω ) W z x , 3 D ( r 1 , r 2 , ω ) ] ,
S 6 , 3 D ( r 1 , r 2 , ω ) = W y z , 3 D ( r 1 , r 2 , ω ) + W z y , 3 D ( r 1 , r 2 , ω ) ,
S 7 , 3 D ( r 1 , r 2 , ω ) = i [ W y z , 3 D ( r 1 , r 2 , ω ) W z y , 3 D ( r 1 , r 2 , ω ) ] ,
S 8 , 3 D ( r 1 , r 2 , ω ) = 3 [ W x x , 3 D ( r 1 , r 2 , ω ) + W y y , 3 D ( r 1 , r 2 , ω ) 2 W z z , 3 D ( r 1 , r 2 , ω ) ] / 3.
μ 3 D ( r 1 , r 2 , ω ) = S 0 , 3 D ( r 1 , r 2 , ω ) / ( S 0 , 3 D ( r 1 , r 1 , ω ) S 0 , 3 D ( r 2 , r 2 , ω ) ) .
P 3 D ( r , ω ) = 3 2 ( j = 1 8 S j , 3 D 2 ( r , ω ) ) 1 / 2 / S 0 , 3 D ( r , ω ) .
W α β ( 0 ) ( ρ 10 , ρ 20 , ω ) = A α A β B α β exp ( ρ 10 2 4 σ α 2 ) exp ( ρ 20 2 4 σ β 2 ) exp ( | ρ 20 ρ 10 | 2 2 δ α β 2 ) ,
W α β ( r 1 , r 2 , ω ) = ( z λ ) 2 W α β ( 0 ) ( ρ 10 , ρ 20 , ω ) { exp [ i k ( R 2 R 1 ) ] } / R 2 2 R 1 2 d 2 ρ 10 d 2 ρ 20 ,     ( α , β = x , y ) ,
W α z ( r 1 , r 2 , ω ) = z λ 2 [ W α x ( 0 ) ( ρ 10 , ρ 20 , ω ) ( x 2 x 20 ) + W α y ( 0 ) ( ρ 10 , ρ 20 , ω ) ( y 2 y 20 ) ]                          × { exp [ i k ( R 2 R 1 ) ] } / R 2 2 R 1 2 d 2 ρ 10 d 2 ρ 20                  ( α = x , y ) ,
W z z ( r 1 , r 2 , ω ) = 1 λ 2 [ W x x ( 0 ) ( ρ 10 , ρ 20 , 0 ) ( x 1 x 10 ) ( x 2 x 20 ) + 2 W x y ( 0 ) ( ρ 10 , ρ 20 , 0 ) ( x 1 x 10 )               × ( y 2 y 20 ) + W y y ( 0 ) ( ρ 10 , ρ 20 , 0 ) ( y 1 y 10 ) ( y 2 y 20 ) ] { exp [ i k ( R 2 R 1 ) ] } / R 2 2 R 1 2 d 2 ρ 10 d 2 ρ 20 .
W α β ( r 1 , r , 2 ω ) = A α A β B α β z 2 { exp [ i k ( r 2 r 1 ) ] / r 1 2 r 2 2 } exp ( F α β ) / P α β ,           ( α , β = x , y ) ,
W α z ( r 1 , r , 2 ω ) = A α z exp [ i k ( r 2 r 1 ) ] r 1 2 r 2 2 { A x B α x J α x P α x exp ( F α x ) A y B α y J α y P α y exp ( F α y ) }                               ( α = x , y ) ,
W z z ( r 1 , r , 2 ω ) = exp [ i k ( r 2 r 1 ) ] / ( r 1 2 r 2 2 ) { A x 2 Q x x exp ( F x x ) / P x x   + A y 2 Q y y exp ( F y y ) / P y y                          + A x A y ( B x y + B x y ) Q x y exp ( F x y ) / P x y } ,
F α β = k [ C 1 α β ρ 2 2 / r 2 2 + C 2 α β ρ 1 2 / r 1 2 k f α β 2 ( x 1 x 2 + y 1 y 2 ) / ( r 1 r 2 ) ] / P α β ,
Q x y = ( 1 i / ( 2 C 1 x y r ) ) x 1 y 2 + i k f x y 2 y 2 G x y ( x ) / P x y + H x y ( x ) G x y ( y ) 2 k f x y 2 C 1 x y G x y ( x ) G x y ( y ) / P x y 2 ,
Q α α = ( 1 i 2 C 1 α α r 1 ) α 1 α 2 ( i f α α 2 k α 2 P α α + H α α ( α ) ) G α α ( α ) 2 C 1 α α k f α α 2 P α α 2 ( G α α ( α ) ) 2 + f α α 2 P α α ,

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