Abstract

Based on the complex Gaussian expansion method for two dimensional rectangular hard-edged aperture, the analytical formulas for the generalized Stokes parameters of random electromagnetic beams through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. With the help of the analytical formulae, the changes in statistical properties of rectangular hard-edge diffracted random electromagnetic beams, such as in the spectral density, in the spectral degree of coherence, in the polarization properties and so on, can be determined. Numerical examples of such changes are presented and discussed.

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References

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  1. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
  2. J. Eillis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 2205 (2004).
  3. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005).
    [CrossRef] [PubMed]
  4. Y. Zhu and D. Zhao, “Generalized Stokes parameters of a stochastic electromagnetic beam propagation through a paraxial ABCD optical system,” J. Opt. Soc. Am. A 25(8), 1944 (2008).
    [CrossRef]
  5. J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34(20), 3074–3076 (2009).
    [CrossRef] [PubMed]
  6. B. Kanseri, S. Rath, and H. C. Kandpal, “Direct determination of the generalized Stokes parameters from the usual Stokes parameters,” Opt. Lett. 34(6), 719–721 (2009).
    [CrossRef] [PubMed]
  7. S. Sahin, “Generalized Stokes parameters in phase space,” Opt. Lett. 35(10), 1704–1706 (2010).
    [CrossRef] [PubMed]
  8. Z. Mei, “Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams,” Opt. Express 18(22), 22826 (2010).
    [CrossRef] [PubMed]
  9. Z. Mei and J. Gu, “Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams,” Appl. Phys. B 99(3), 571–578 (2010).
    [CrossRef]
  10. J. J. Wen and M. A. Breazeale, “A diffracton beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752 (1988).
    [CrossRef]
  11. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
    [CrossRef] [PubMed]
  12. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics, Cambridge Univ. Press, England, 1995.

2010 (3)

2009 (2)

2008 (1)

2007 (1)

2005 (1)

2004 (1)

J. Eillis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 2205 (2004).

1988 (1)

J. J. Wen and M. A. Breazeale, “A diffracton beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752 (1988).
[CrossRef]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffracton beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752 (1988).
[CrossRef]

Dogariu, A.

J. Eillis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 2205 (2004).

Du, X.

Eillis, J.

J. Eillis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 2205 (2004).

Friberg, A. T.

Gu, J.

Z. Mei and J. Gu, “Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams,” Appl. Phys. B 99(3), 571–578 (2010).
[CrossRef]

Kandpal, H. C.

Kanseri, B.

Korotkova, O.

Mei, Z.

Z. Mei and J. Gu, “Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams,” Appl. Phys. B 99(3), 571–578 (2010).
[CrossRef]

Z. Mei, “Generalized Stokes parameters of three-dimensional stochastic electromagnetic beams,” Opt. Express 18(22), 22826 (2010).
[CrossRef] [PubMed]

Rath, S.

Réfrégier, P.

Roueff, A.

Sahin, S.

Setälä, T.

Tervo, J.

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffracton beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752 (1988).
[CrossRef]

Wolf, E.

Zhao, D.

Zhu, Y.

Appl. Phys. B (1)

Z. Mei and J. Gu, “Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams,” Appl. Phys. B 99(3), 571–578 (2010).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, “A diffracton beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (2)

Opt. Lett. (5)

Other (2)

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

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics, Cambridge Univ. Press, England, 1995.

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Figures (6)

Fig. 1
Fig. 1

Changes in the Stokes parameters along z-axis direction of a rectangular hard-edge diffracted random electromagnetic beam on propagation in free space. The source is assumed to be a Gaussian Schell-model source with λ = 632.8nm, Ax = 1.5, Ay = 1.0, Bxy = 0.3exp(iπ/3), Byx = 0.3exp(-iπ/3), σx = σy = 10mm, δxx = 0.15mm, δxy = δyx = 0.25mm, δyy = 0.2mm.

Fig. 2
Fig. 2

Three-dimensional spectral density distributions and corresponding contour graphs of a hard-edged aperture (a = b = 5mm) diffracted random electromagnetic beam in the transverse plane z = 10m (a1-a2) and z = 100m (b1-b2). The source parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Changes in the spectral degree of polarization along z-axis direction of a rectangular hard-edged aperture diffracted random electromagnetic beams on propagation in free space. All parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

There-dimensional spectral degree of polarization distributions and corresponding contour graphs of a rectangular hard-edged aperture (a = b = 5mm) diffracted random electromagnetic beam in the transverse plane z = 10m (a1-a2) and z = 100m (b1-b2). The source parameters are the same as in Fig. 1.

Fig. 5
Fig. 5

Changes in the spectral degree of coherence along z-axis direction of a rectangular hard-edge diffracted random electromagnetic beam on propagation in free space. Pairs of field point ρ 2 = ρ 1 = (0.05mm, 0.05mm, 0.05mm,0.05mm), the source parameters are the same as in Fig. 1.

Fig. 6
Fig. 6

There-dimensional spectral degree of coherence distributions and corresponding contour graphs of different rectangular hard-edged diffracted random electromagnetic Gaussian Schell-model beam in the transverse plane z = 10m when the pair of field points are located symmetrically with respect to the z-axis, i. e. ρ 2 = -ρ 1. (a1-a2) a = b = 5mm; (b1-b2) a = b = 50mm.

Equations (28)

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S 0 ( r 1 , r 2 , ω ) = E x * ( r 1 , ω ) E x ( r 2 , ω ) + E y * ( r 1 , ω ) E y ( r 2 , ω ) ,
S 1 ( r 1 , r 2 , ω ) = E x * ( r 1 , ω ) E x ( r 2 , ω ) E y * ( r 1 , ω ) E y ( r 2 , ω ) ,
S 2 ( r 1 , r 2 , ω ) = E x * ( r 1 , ω ) E y ( r 2 , ω ) + E y * ( r 1 , ω ) E x ( r 2 , ω ) ,
S 3 ( r 1 , r 2 , ω ) = i [ E y * ( r 1 , ω ) E x ( r 2 , ω ) E x * ( r 1 , ω ) E y ( r 2 , ω ) ] ,
E i ( ρ , z , ω ) = i λ B a a b b E i ( ρ 0 , 0 , ω ) exp [ i k 2 B [ A ρ 0 2 2 ρ ρ 0 + D ρ 2 ] ] d 2 ρ 0 ,
F ( ρ ) = { 1 ,        | x | a ,   | y | b 0 ,       | x | > a ,   | y | > b ,
E i ( ρ , z , ω ) = i λ B F ( ρ 0 ) E i ( ρ 0 , 0 , ω ) exp [ i k 2 B [ A ρ 0 2 2 ρ ρ 0 + D ρ 2 ] ] d 2 ρ 0 .
F ( ρ ) = h = 1 N g = 1 N A h A g exp [ B h x 2 / a 2 B g y 2 / b 2 ] ,
E i * ( ρ 1 , ρ 2 , z , ω ) E j ( ρ 1 , ρ 2 , z , ω )         = 1 ( λ B ) 2 F ( ρ 10 , ρ 20 ) E i * ( ρ 10 , ρ 20 , 0 , ω ) E j ( ρ 10 , ρ 20 , 0 , ω )           × exp { i k [ A ( ρ 10 2 ρ 10 2 ) 2 ( ρ 10 ρ 1 ρ 20 ρ 2 ) + D ( ρ 10 2 ρ 10 2 ) ] / 2 B } d 2 ρ 10 d 2 ρ 20  ,
F ( ρ 10 , ρ 20 ) = h 1 = 1 N g 1 = 1 N h 2 = 1 N g 2 = 1 N A h 1 * A g 1 * A h 2 A g 2 exp [ B h 1 * a 2 x 10 2 B g 1 * b 2 y 10 2 B h 2 a 2 x 20 2 B g 2 b 2 y 20 2 ] ,
S α ( ρ 1 , ρ 2 , z , ω ) = 1 ( λ B ) 2 F ( ρ 10 , ρ 20 ) S α ( 0 ) ( ρ 10 , ρ 20 , ω ) exp { i k 2 B [ A ( ρ 10 2 ρ 10 2 )                             2 ( ρ 10 ρ 1 ρ 20 ρ 2 ) + D ( ρ 10 2 ρ 10 2 ) ] } d 2 ρ 10 d 2 ρ 20             ( α = 0 , 1 , 2 , 3 ) .
S 0 ( 0 ) ( ρ 10 , ρ 20 , ω ) = W x x ( 0 ) ( ρ 10 , ρ 20 , ω ) + W y y ( 0 ) ( ρ 10 , ρ 20 , ω ) ,
S 1 ( 0 ) ( ρ 10 , ρ 20 , ω ) = W x x ( 0 ) ( ρ 10 , ρ 20 , ω ) W y y ( 0 ) ( ρ 10 , ρ 20 , ω ) ,
S 2 ( 0 ) ( ρ 10 , ρ 20 , ω ) = W x y ( 0 ) ( ρ 10 , ρ 20 , ω ) + W y x ( 0 ) ( ρ 10 , ρ 20 , ω ) ,
S 3 ( 0 ) ( ρ 10 , ρ 20 , ω ) = i [ W y x ( 0 ) ( ρ 10 , ρ 20 , ω ) W x y ( 0 ) ( ρ 10 , ρ 20 , ω ) ] ,
W i j ( 0 ) ( ρ 10 , ρ 20 , ω ) = S i ( 0 ) ( ρ 10 , ω ) S j ( 0 ) ( ρ 20 , ω ) η i j ( 0 ) ( ρ 20 ρ 10 , ω ) ,
W i j ( 0 ) ( ρ 10 , ρ 20 , ω ) = A i A j B i j exp ( ρ 10 2 4 σ i 2 ) exp ( ρ 20 2 4 σ j 2 ) exp ( | ρ 20 ρ 10 | 2 2 δ i j 2 ) ,
S 0 ( ρ 1 , ρ 2 , z , ω ) = exp [ i k D ( ρ 1 2 ρ 2 2 ) / ( 2 B ) ] h 1 N h 2 N g 1 N g 2 N A h 1 A h 2 A g 1 A g 2                             × [ A x 2 exp ( F x x ) / P x x ( h ) P x x ( g ) + A y 2 exp ( F y y ) / P y y ( h ) P y y ( g ) ] ,
S 1 ( ρ 1 , ρ 2 , z , ω ) = exp [ i k D ( ρ 1 2 ρ 2 2 ) / ( 2 B ) ] h 1 N h 2 N g 1 N g 2 N A h 1 A h 2 A g 1 A g 2                             × [ A x 2 exp ( F x x ) / P x x ( h ) P x x ( g ) A y 2 exp ( F y y ) / P y y ( h ) P y y ( g ) ] ,
S 2 ( ρ 1 , ρ 2 , z , ω ) = exp [ i k D ( ρ 1 2 ρ 2 2 ) / ( 2 B ) ] h 1 N h 2 N g 1 N g 2 N A h 1 A h 2 A g 1 A g 2 [ A x A y B x y                               × exp ( F x y ) / P x x ( h ) P x x ( g ) + A y A x B y x exp ( F y x ) / P y y ( h ) P y y ( g ) ] ,
S 3 ( ρ 1 , ρ 2 , z , ω ) = i exp [ i k D ( ρ 1 2 ρ 2 2 ) / ( 2 B ) ] h 1 N h 2 N g 1 N g 2 N A h 1 A h 2 A g 1 A g 2 [ A y A x B y x                             × exp ( F y x ) / P x x ( h ) P x x ( g ) A x A y B x y exp ( F x y ) / P y y ( h ) P y y ( g ) ] ,
F i j = exp [ k 2 B ( x 1 2 C h 1 + y 1 2 C g 1 ) + 1 P i j ( h ) ( k C h 1 x 2 2 B x 1 2 δ i j 2 ) 2 + 1 P i j ( g ) ( k C g 1 y 2 2 B y 1 2 δ i j 2 ) 2 ] ,
P i j ( γ ) = C γ 1 C γ 2 B 2 / ( δ i j 4 k 2 ) ,  
C h γ = B / ( 2 k σ i 2 ) + B / ( k δ i j 2 ) + ( 1 ) γ + 1 i A + 2 B B h γ / ( k a 2 ) ,  
C g γ = B / ( 2 k σ i 2 ) + B / ( k δ i j 2 ) + ( 1 ) γ + 1 i A + 2 B B g γ / ( k b 2 ) ,  
  ( γ = 1 ,   2 )
P 2 ( ρ , z , ω ) = i = 1 3 S i 2 ( ρ , z , ω ) / S 0 2 ( ρ , z , ω ) .
μ ( r 1 , r 2 , ω ) = S 0 ( r 1 , r 2 , ω ) / ( S 0 ( r 1 , r 1 , ω ) S 0 ( r 2 , r 2 , ω ) ) .

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