Abstract

The expression of spectral density of cosine-Gaussian-correlated Schell-model (CGSM) beams diffracted by an aperture is derived, and used to study the changes in the spectral density distribution of CGSM beams upon propagation, where the effect of aperture diffraction is emphasized. It is shown that, comparing with that of GSM beams, the spectral density distribution of CGSM beams diffracted by an aperture has dip and shows dark hollow intensity distribution when the order-parameter n is big enough. The central intensity increases with increasing truncation parameter of aperture. The comparative study of spectral density distributions of CGSM beams with aperture and that of without aperture is performed. Furthermore, the effect of order-parameter n and spatial coherence of CGSM beams on the spectral density distribution is discussed in detail. The results obtained may be useful in optical particulate manipulation.

© 2014 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, 1st ed. (Cambridge University, 2007).
  3. F. Gori, M. Santarsiero, R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
    [CrossRef] [PubMed]
  4. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996).
    [CrossRef]
  5. H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [CrossRef] [PubMed]
  6. Z. S. Tong, O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [CrossRef] [PubMed]
  7. Z. R. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [CrossRef] [PubMed]
  8. Z. R. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [CrossRef] [PubMed]
  9. Z. Mei, E. Shchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
    [CrossRef] [PubMed]
  10. Z. R. Mei, O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013).
    [CrossRef] [PubMed]
  11. S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [CrossRef] [PubMed]
  12. O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [CrossRef] [PubMed]
  13. Y. Y. Zhang, D. M. Zhao, “Scattering of multi-Gaussian Schell-model beams on a random medium,” Opt. Express 21(21), 24781–24792 (2013).
    [CrossRef] [PubMed]
  14. Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
    [CrossRef]
  15. Y. T. Zhang, L. Liu, C. L. Zhao, Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
    [CrossRef]
  16. Z. R. Mei, “Light sources generating self-focusing beams of variable focal length,” Opt. Lett. 39(2), 347–350 (2014).
    [CrossRef] [PubMed]
  17. Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
    [CrossRef]
  18. S. W. Cui, Z. Y. Chen, L. Zhang, J. X. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013).
    [CrossRef] [PubMed]
  19. C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [CrossRef] [PubMed]
  20. Y. H. Chen, F. Wang, C. L. Zhao, Y. J. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
    [CrossRef] [PubMed]
  21. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [CrossRef] [PubMed]

2014 (6)

2013 (7)

2012 (3)

2011 (1)

2008 (1)

1996 (1)

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[CrossRef] [PubMed]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996).
[CrossRef]

Cai, Y. J.

Y. T. Zhang, L. Liu, C. L. Zhao, Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. H. Chen, F. Wang, C. L. Zhao, Y. J. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[CrossRef] [PubMed]

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

Chen, Y. H.

Y. H. Chen, F. Wang, C. L. Zhao, Y. J. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[CrossRef] [PubMed]

Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

Chen, Z. Y.

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996).
[CrossRef]

Cui, S. W.

Eyyuboglu, H. T.

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

Gori, F.

Korotkova, O.

C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[CrossRef] [PubMed]

Z. R. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Z. R. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

Z. Mei, E. Shchepakina, O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[CrossRef] [PubMed]

Z. R. Mei, O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21(22), 27246–27259 (2013).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[CrossRef] [PubMed]

Z. S. Tong, O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

Lajunen, H.

Liang, C. H.

Liu, L.

Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. T. Zhang, L. Liu, C. L. Zhao, Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Liu, X. L.

C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

Mei, Z.

Mei, Z. R.

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996).
[CrossRef]

Pu, J. X.

Qu, J.

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Tong, Z. S.

Wang, F.

C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. H. Chen, F. Wang, C. L. Zhao, Y. J. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[CrossRef] [PubMed]

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

Yuan, Y. S.

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

Zhang, L.

Zhang, Y. T.

Y. T. Zhang, L. Liu, C. L. Zhao, Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Zhang, Y. Y.

Zhao, C. L.

Y. T. Zhang, L. Liu, C. L. Zhao, Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. H. Chen, F. Wang, C. L. Zhao, Y. J. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[CrossRef] [PubMed]

Zhao, D. M.

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

Opt. Commun. (2)

Y. S. Yuan, X. L. Liu, F. Wang, Y. H. Chen, Y. J. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305(0), 57–65 (2013).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1–3), 113–121 (1996).
[CrossRef]

Opt. Express (4)

Opt. Lett. (10)

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[CrossRef] [PubMed]

Z. R. Mei, “Light sources generating self-focusing beams of variable focal length,” Opt. Lett. 39(2), 347–350 (2014).
[CrossRef] [PubMed]

S. W. Cui, Z. Y. Chen, L. Zhang, J. X. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013).
[CrossRef] [PubMed]

C. H. Liang, F. Wang, X. L. Liu, Y. J. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[CrossRef] [PubMed]

Z. S. Tong, O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

Z. R. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Z. R. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[CrossRef] [PubMed]

Phys. Lett. A (1)

Y. T. Zhang, L. Liu, C. L. Zhao, Y. J. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Phys. Rev. A (1)

Y. H. Chen, F. Wang, L. Liu, C. L. Zhao, Y. J. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, 1st ed. (Cambridge University, 2007).

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of CGSM beams diffracted by an aperture.

Fig. 2
Fig. 2

Normalized spectral density distribution S(u,z)/S(0,0) of CGSM beams as a function of propagation distance z and relative coordinate u for different values of order-parameter n (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 4 and (e), (f), (g), (h) the color-coded plot corresponding to (a), (b), (c), (d) respectively. The other parameters are δ = 0.4, σ/w0 = 0.5.

Fig. 3
Fig. 3

Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of propagation distance (a) z/z0 = 0.2, (b) z/z0 = 0.3, (c) z/z0 = 0.4, (d) z/z0→∞. The other parameters are δ = 0.4, σ/w0 = 0.5, n = 2.

Fig. 4
Fig. 4

Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of truncation parameter. The other parameters are σ/w0 = 0.5, n = 2, (a) z/z0 = 0.3, (b) z/z0 = 0.4.

Fig. 5
Fig. 5

Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of order-parameter n. The other parameters are σ/w0 = 0.5, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞.

Fig. 6
Fig. 6

Normalized transverse spectral density distribution S(u,z)/S(0,0) of CGSM beams for different values of coherence parameter σ/w0. The other parameters are n = 2, δ = 0.4, (a) z/z0 = 0.3, (b) z/z0→∞.

Equations (11)

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W ( 0 ) ( x 1 , x 2 )=exp[ x 1 2 + x 2 2 w 0 2 ]cos[ n 2π ( x 2 x 1 ) σ ]exp[ ( x 2 x 1 ) 2 2 σ 2 ],
S(x,z)=W(x,x,z)= k 2πz a a a a W (0) ( x 1 , x 2 ,z=0) ×exp{ ik 2z [ ( x 1 2 x 2 2 )2x( x 1 x 2 ) ] }d x 1 d x 2 2 .
S(u,z)= i 4 π Q 2 z 0 z × δ δ exp{ 1 4 (σ/ w 0 ) 4 Q 2 [ 4π 2π nu z 0 z ( σ w 0 ) 3 4 π 2 u 2 ( z 0 z ) 2 ( σ w 0 ) 4 2π n 2 ( σ w 0 ) 2 +2i( n 2π σ w 0 +2πu z 0 z ( σ w 0 ) 2 +4πu z 0 z Q 2 ( σ w 0 ) 4 ) u +( 14 Q 1 Q 2 ( σ w 0 ) 4 ) u 2 ] } ×{ H 1 exp[ in 2π (σ/ w 0 ) 3 Q 2 u ]+ H 2 exp[ 2 2 n π 3/2 ( z 0 /z) (σ/ w 0 ) Q 2 u ] }d u ,
H 1 =[ cos( u 2π u σ/ w 0 )+isin( u 2π u σ/ w 0 ) ]×{ Erf[ i u + σ w 0 ( 2π n2πu z 0 z σ w 0 )2iδ Q 2 ( σ w 0 ) 2 2 ( σ/ w 0 ) 2 Q 2 ] Erf[ i u + σ w 0 ( 2π n2πu z 0 z σ w 0 )+2iδ Q 2 ( σ w 0 ) 2 2 ( σ/ w 0 ) 2 Q 2 ] },
H 2 =[ cos( u 2π u σ/ w 0 )isin( u 2π u σ/ w 0 ) ]×{ Erf[ i u + σ w 0 ( 2π n+2πu z 0 z σ w 0 )2iδ Q 2 ( σ w 0 ) 2 2 ( σ/ w 0 ) 2 Q 2 ] Erf[ i u + σ w 0 ( 2π n+2πu z 0 z σ w 0 )+2iδ Q 2 ( σ w 0 ) 2 2 ( σ/ w 0 ) 2 Q 2 ] },
δ= a w 0 , ( truncation parameter )
z 0 = w 0 2 λ ,
u = x w 0 ,( relative transversal coordinate at z=0 plane )
u= x w 0 , ( relative transversal coordinate at z plane )
Q 1 =1 1 2 (σ/ w 0 ) 2 iπ z 0 z ,
Q 2 =1 1 2 (σ/ w 0 ) 2 +iπ z 0 z ,

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