Abstract

An inherent relationship between an invariant far-field beam intensity pattern and the convolution of any two legitimate degrees of coherence in the source plane is established. Two classes of random sources are introduced by modeling the source degree of coherence with the help of the convolution operation of the Gaussian Schell-model correlation function and the multi-sinc Schell-model correlation function in the polar and Cartesian symmetries. The established relationships are used to explore the far-field intensity features produced by the new sources. It is shown that the far-field intensity patterns of the novel sources have multi-sinc Schell-model transverse distributions with a Gaussian envelope, looking like the multi-sinc Schell-model beams filtered by a soft-edge Gaussian aperture. The results demonstrate the potential of coherence modulation of the source fields for far-field beam shaping applications.

© 2015 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. E. Collett and E. Wolf, “New equivalence theorems for planar sources that generate the same distributions of radiant intensity,” J. Opt. Soc. Am. 69(7), 942–950 (1979).
    [Crossref]
  3. G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28(18), 1627–1629 (2003).
    [Crossref] [PubMed]
  4. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26(1), 147–155 (2009).
    [Crossref] [PubMed]
  5. E. E. García-Guerrero, E. R. Méndez, Z.-H. Gu, T. A. Leskova, and A. A. Maradudin, “Interference of a pair of symmetric partially coherent beams,” Opt. Express 18(5), 4816–4828 (2010).
    [Crossref] [PubMed]
  6. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
    [Crossref] [PubMed]
  7. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  8. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  9. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  10. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  11. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [Crossref] [PubMed]
  12. O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
    [Crossref] [PubMed]
  13. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
    [Crossref] [PubMed]
  14. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [Crossref] [PubMed]
  15. S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
    [Crossref] [PubMed]
  16. F. Wang, C. Liang, Y. Yuan, and Y. Cai, “Generalized multi-Gaussian correlated Schell-model beam: from theory to experiment,” Opt. Express 22(19), 23456–23464 (2014).
    [Crossref] [PubMed]
  17. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [Crossref] [PubMed]
  18. X. Liu and D. Zhao, “Electromagnetic random source for circular optical frame and its statistical properties,” Opt. Express 23(13), 16702–16714 (2015).
    [Crossref] [PubMed]
  19. C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014).
    [Crossref] [PubMed]
  20. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [Crossref] [PubMed]
  21. M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
    [Crossref] [PubMed]
  22. Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014).
    [Crossref] [PubMed]
  23. Z. Mei, O. Korotkova, and Y. Mao, “Powers of the degree of coherence,” Opt. Express 23(7), 8519–8531 (2015).
    [Crossref] [PubMed]
  24. Z. Mei and O. Korotkova, “Alternating series of cross-spectral densities,” Opt. Lett. 40(11), 2473–2476 (2015).
    [Crossref] [PubMed]
  25. O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
    [Crossref] [PubMed]

2015 (4)

2014 (7)

2013 (3)

2012 (4)

2011 (1)

2010 (1)

2009 (1)

2007 (1)

2003 (1)

1979 (1)

Betancur, R.

Cai, Y.

Castañeda, R.

Collett, E.

de Sande, J. C. G.

Ding, C.

García-Guerrero, E. E.

Gbur, G.

Gori, F.

Gu, Z.-H.

Korotkova, O.

Z. Mei, O. Korotkova, and Y. Mao, “Powers of the degree of coherence,” Opt. Express 23(7), 8519–8531 (2015).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Alternating series of cross-spectral densities,” Opt. Lett. 40(11), 2473–2476 (2015).
[Crossref] [PubMed]

O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
[Crossref] [PubMed]

Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014).
[Crossref] [PubMed]

C. Ding, O. Korotkova, Y. Zhang, and L. Pan, “Cosine-Gaussian correlated Schell-model pulsed beams,” Opt. Express 22(1), 931–942 (2014).
[Crossref] [PubMed]

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

O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
[Crossref] [PubMed]

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

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

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

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

Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
[Crossref] [PubMed]

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

Kumar, A.

Lajunen, H.

Leskova, T. A.

Liang, C.

Liu, X.

Ma, L.

Mao, Y.

Maradudin, A. A.

Mei, Z.

Méndez, E. R.

Pan, L.

Piquero, G.

Ponomarenko, S. A.

Prabhakar, S.

Reddy, S. G.

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

Shchepakina, E.

Singh, R. P.

Tong, Z.

Visser, T. D.

Wang, F.

Wolf, E.

Yuan, Y.

Zhang, Y.

Zhao, D.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Express (6)

Opt. Lett. (14)

G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus?” Opt. Lett. 28(18), 1627–1629 (2003).
[Crossref] [PubMed]

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

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

S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38(21), 4441–4444 (2013).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

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

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

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

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

Z. Mei and O. Korotkova, “Alternating series of cross-spectral densities,” Opt. Lett. 40(11), 2473–2476 (2015).
[Crossref] [PubMed]

O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
[Crossref] [PubMed]

M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014).
[Crossref] [PubMed]

Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014).
[Crossref] [PubMed]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Figures (8)

Fig. 1
Fig. 1 The degree of coherence μ c derived from the convolution of μ g with δ g = 0.04 mm and μ s with δ s = 0.1 mm , m = 1 , N = 10 (upper row) and N = 11 (lower row).
Fig. 2
Fig. 2 The same as Fig. 1 but for .
Fig. 3
Fig. 3 Far field spectral densities generated by the sources with degrees of coherence μ g , μ s and μ c shown in Fig. 1.
Fig. 4
Fig. 4 Far field spectral densities generated by the sources with degrees of coherence μ g , μ s and μ c shown in Fig. 2.
Fig. 5
Fig. 5 Far field spectral density S c for the circular symmetry MSSM beams with different N (upper row N = 10 and low row N = 11 ) are filtered by Gaussian beams with different coherence length μ g . (a) and (d) μ g = 0.02 mm ; (b) and (e) μ g = 0.06 mm ; (c) and (f) μ g = 0.15 mm .
Fig. 6
Fig. 6 The degree of coherence μ c derived from the convolution of Gaussian degree of coherence μ g with δ g x = δ g y = 0.04 mm and Cartesian symmetry MSSM degree of coherence μ s with δ s x = δ s y = 0.1 mm , (a) N = 10 , m = 1 ; (b) N = 11 , m = 1 ; (c) N = 10 , m = 1.5 ; (d) N = 11 , m = 1.5 .
Fig. 7
Fig. 7 Far-field spectral density S c generated by the new sources with degrees of coherence μ c shown in Fig. 6.
Fig. 8
Fig. 8 Far field spectral density S c for the Cartesian symmetry MSSM beams with m = 1.5 and different N (upper row N = 10 and low row N = 11 ) are filtered by Gaussian beams with different coherence length μ g , (a) and (d) δ g x = δ g y = 0.02 mm ; (b) and (e) δ g x = 0.02 mm , δ g y = 0.06 mm ; (c) and (f) δ g x = δ g y = 0.15 mm .

Equations (26)

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W ( 0 ) ( ρ 1 , ρ 2 ) = S ( 0 ) ( ρ 1 ) S ( 0 ) ( ρ 2 ) μ ( ρ 1 ρ 2 ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = p ( v ) H 0 ( ρ 1 , v ) H 0 ( ρ 2 , v ) d 2 v ,
H 0 ( ρ , v ) = τ ( ρ ) exp ( 2 π i v ρ ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = τ * ( ρ 1 ) τ ( ρ 2 ) p ˜ ( ρ 1 ρ 2 ) ,
μ ( ρ 1 ρ 2 ) = p ˜ ( ρ 1 ρ 2 ) .
μ ( ρ 1 ρ 2 ) = A μ 1 ( ρ 1 ρ 2 ) μ 2 ( ρ 1 ρ 2 2 ) ,
p ( v ) = p 1 ( v ) p 2 ( v ) ,
S ( ρ , z ) = p ( v ) | H ( ρ , z , v ) | 2 d 2 v ,
| H ( ρ , z , v ) | 2 = k 2 4 π 2 z 2 H 0 ( ρ 1 , v ) H 0 ( ρ 2 , v ) × exp [ i k ( ρ ρ 1 ) 2 ( ρ ρ 2 ) 2 2 z ] d 2 ρ 1 d 2 ρ 2 .
S ( ρ , z ) = p 1 ( v ) p 2 ( v ) | H ( ρ , z , v ) | 2 d 2 v .
μ 1 ( ρ 1 ρ 2 ) = μ g ( ρ 1 ρ 2 ) = exp [ ( ρ 1 ρ 2 ) 2 2 δ g 2 ] ,
μ 2 ( ρ 1 ρ 2 ) = μ s ( ρ 1 ρ 2 ) = 1 B n = 1 N ( 1 ) n 1 C n sin c ( ρ 1 ρ 2 C n δ s ) ,
μ c ( ρ 1 ρ ) = A μ g ( ρ 1 ρ ) μ s ( ρ 1 ρ 2 ) = A exp [ ( ρ 1 ρ ) 2 2 δ g 2 ] 1 B n = 1 N ( 1 ) n 1 C n sin c ( ρ 1 ρ 2 C n δ s ) .
p c ( v ) = p g ( v ) p s ( v ) = 2 π δ g 2 exp ( 2 π 2 δ g 2 v 2 ) δ s B n = 1 N ( 1 ) n 1 r e c t ( C n δ s v ) .
τ ( ρ ) = exp [ ρ 2 / ( 2 σ 2 ) ] ,
| H ( ρ , z , v ) | 2 = σ 2 w 2 ( z ) exp [ ( ρ + 2 π z v / k ) 2 / w 2 ( z ) ] ,
w 2 ( z ) = σ 2 + z 2 / ( k 2 σ 2 ) .
W ( 0 ) ( ρ 1 , ρ 2 ) = t = x , y W ( t 1 , t 2 ) = t = x , y p ( v t ) H 0 ( t 1 , v t ) H 0 ( t 2 , v t ) d v t .
W ( 0 ) ( ρ 1 , ρ 2 ) = τ * ( ρ 1 ) τ ( ρ 2 ) t = x , y p ˜ t ( t 1 t 2 ) .
μ ( ρ 1 ρ 2 ) = t = x , y p ˜ t ( t 1 t 2 ) .
μ ( ρ 1 ρ 2 ) = A t = x , y μ 1 ( t 1 t 2 ) μ 2 ( t 1 t 2 ) ,
p ( v ) = t = x , y p ( v t ) = t = x , y p 1 ( v t ) p 2 ( v t ) .
S ( ρ , z ) = t = x , y p ( v t ) | H t ( t , z , v t ) | 2 d v t = t = x , y p 1 ( v t ) p 2 ( v t ) | H t ( t , z , v t ) | 2 d v t ,
| H t ( t , z , v t ) | 2 = σ w ( z ) exp [ ( t + 2 π z v t / k ) 2 / w 2 ( z ) ] .
μ c ( ρ 1 ρ 2 ) = A t = x , y μ g t ( t 1 t 2 ) μ s t ( t 1 t 2 ) = A t = x , y exp [ ( t 1 t 2 ) 2 2 δ g t 2 ] 1 B n = 1 N ( 1 ) n 1 C n sin c ( t 1 t 2 C n δ s t ) .
p c ( v ) = t = x , y p 1 ( v t ) p 2 ( v t ) = t = x , y 2 π δ g t 2 exp ( 2 π 2 δ g t 2 v t 2 ) δ s t B n = 1 N ( 1 ) n 1 r e c t ( C n δ s t v t ) .

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