Abstract

We examine paraxial propagation of recently introduced optical coherence lattices in free space and demonstrate a novel phenomenon of periodicity reciprocity between their intensity and coherence properties. The periodicity reciprocity arises because an aperiodic source intensity profile of an optical coherence lattice evolves into a lattice-like far-field profile, while the periodic spectral degree of coherence at the source becomes aperiodic on free-space propagation. We discuss how the discovered periodicity reciprocity can make optical coherence lattices attractive for robust free-space optical communications.

© 2015 Optical Society of America

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References

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  1. J. Turunen and A. T. Friberg, “Propagation-Invariant Optical Fields,” Progress in Optics,  54, 1–88 (2010).
    [Crossref]
  2. S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1–4 (2006).
    [Crossref]
  3. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508–2510 (2007).
    [Crossref] [PubMed]
  4. F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [Crossref]
  5. A. Starikov and E. Wolf, “Coherent-mode respresentation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [Crossref]
  6. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  7. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
    [Crossref]
  8. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [Crossref]
  9. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
    [Crossref]
  10. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent separable vortex beams,” Opt. Lett. 28, 878–880 (2003).
    [Crossref] [PubMed]
  11. F. Gori, “Mode propagation of the fields generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [Crossref]
  12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1997).
  13. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [Crossref]
  14. C. Palma, R. Borghi, and C. Cincotti, “Beams originated by J0-correlated Schell-model sources,” Opt. Commun. 125, 113–121 (1996).
    [Crossref]
  15. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29, 2159–2164 (2012).
    [Crossref]
  16. C. Liang, X. Liu, F. Wang, Y. Cai, and O. Korotkova, “Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39, 769–772 (2014).
    [Crossref] [PubMed]
  17. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36, 4104–4106 (2011).
    [Crossref] [PubMed]
  18. H. T. Eyyuboglu and Y. Baykal, “Transmittance pf partially coherent cosh-Gaussian, cos-Gaussian, and annular beams through turbulence,” Opt. Commun. 27817–22 (2007).
    [Crossref]
  19. G. Zhou and X. Chu, “Propagation of partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 1710529–10534 (2009).
    [Crossref] [PubMed]
  20. Z. Mei and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 2117512–17519 (2013).
    [Crossref] [PubMed]
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  22. L. Liu, “Partially coherent diffraction effect between Lau and Talbott effects,” J. Opt. Soc. Am. A 5, 1709–1716 (1988).
    [Crossref]
  23. A. W. Lohman and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt. 30, 475–479 (1983).
    [Crossref]
  24. G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta: Int. J. Opt.,  31, 4–8 (1984).
    [Crossref]
  25. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991).
    [Crossref]
  26. S. Teng, L. Liu, J. Zu, Z. Luan, and L. Liu, “Uniform theory of the Talbott effect with partially coherent light illumination,” J. Opt. Soc. Am. A 20, 1747–1754 (2003).
    [Crossref]
  27. M Santarsiro, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
    [Crossref]
  28. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. doc. ID 224490 (posted 24 October 2014, in press).
  29. S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express 19, 17086 (2011).
    [Crossref] [PubMed]
  30. S. A. Ponomarenko and E. Wolf, “Coherence properties of light in Young’s interference pattern formed with partially coherent light,” Opt. Commun. 170, 1–8 (1999).
    [Crossref]
  31. S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun.,  227, 73–74 (2003).
    [Crossref]
  32. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

2014 (1)

2013 (2)

Z. Mei and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 2117512–17519 (2013).
[Crossref] [PubMed]

M Santarsiro, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (1)

J. Turunen and A. T. Friberg, “Propagation-Invariant Optical Fields,” Progress in Optics,  54, 1–88 (2010).
[Crossref]

2009 (1)

2007 (2)

S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508–2510 (2007).
[Crossref] [PubMed]

H. T. Eyyuboglu and Y. Baykal, “Transmittance pf partially coherent cosh-Gaussian, cos-Gaussian, and annular beams through turbulence,” Opt. Commun. 27817–22 (2007).
[Crossref]

2006 (1)

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1–4 (2006).
[Crossref]

2003 (3)

2001 (2)

1999 (1)

S. A. Ponomarenko and E. Wolf, “Coherence properties of light in Young’s interference pattern formed with partially coherent light,” Opt. Commun. 170, 1–8 (1999).
[Crossref]

1996 (1)

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

1994 (1)

1993 (1)

1991 (1)

1988 (1)

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

1984 (1)

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta: Int. J. Opt.,  31, 4–8 (1984).
[Crossref]

1983 (3)

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1983).
[Crossref]

A. W. Lohman and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt. 30, 475–479 (1983).
[Crossref]

F. Gori, “Mode propagation of the fields generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

1982 (1)

1980 (1)

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1–4 (2006).
[Crossref]

Baykal, Y.

H. T. Eyyuboglu and Y. Baykal, “Transmittance pf partially coherent cosh-Gaussian, cos-Gaussian, and annular beams through turbulence,” Opt. Commun. 27817–22 (2007).
[Crossref]

Bogatyryova, G. V.

Borghi, R.

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

Cada, M.

Cai, Y.

Chu, X.

Cincotti, C.

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

de Sande, J. C. G.

M Santarsiro, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[Crossref]

Eyyuboglu, H. T.

H. T. Eyyuboglu and Y. Baykal, “Transmittance pf partially coherent cosh-Gaussian, cos-Gaussian, and annular beams through turbulence,” Opt. Commun. 27817–22 (2007).
[Crossref]

Fel’de, C. V.

Friberg, A. T.

Gori, F.

M Santarsiro, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori, “Mode propagation of the fields generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1983).
[Crossref]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

Huang, W.

Indebetouw, G.

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta: Int. J. Opt.,  31, 4–8 (1984).
[Crossref]

Korotkova, O.

Lajunen, H.

Liang, C.

Liu, L.

Liu, X.

Lohman, A. W.

A. W. Lohman and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt. 30, 475–479 (1983).
[Crossref]

Luan, Z.

Ma, L.

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. doc. ID 224490 (posted 24 October 2014, in press).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1997).

Mei, Z.

Mukunda, N.

Ojeda-Castaneda, J.

A. W. Lohman and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt. 30, 475–479 (1983).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

Palma, C.

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

Piquero, G.

M Santarsiro, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[Crossref]

Polyanskii, P. V.

Ponomarenko, S. A.

S. A. Ponomarenko, “Complex Gaussian representation of statistical pulses,” Opt. Express 19, 17086 (2011).
[Crossref] [PubMed]

S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508–2510 (2007).
[Crossref] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1–4 (2006).
[Crossref]

S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun.,  227, 73–74 (2003).
[Crossref]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent separable vortex beams,” Opt. Lett. 28, 878–880 (2003).
[Crossref] [PubMed]

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
[Crossref]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

S. A. Ponomarenko and E. Wolf, “Coherence properties of light in Young’s interference pattern formed with partially coherent light,” Opt. Commun. 170, 1–8 (1999).
[Crossref]

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. doc. ID 224490 (posted 24 October 2014, in press).

Saastamoinen, T.

Sahin, S.

Santarsiro, M

M Santarsiro, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[Crossref]

Shchepakina, E.

Simon, R.

Soskin, M. S.

Starikov, A.

Teng, S.

Tervonen, E.

Turunen, J.

Vasara, A.

Wang, F.

Wolf, E.

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent separable vortex beams,” Opt. Lett. 28, 878–880 (2003).
[Crossref] [PubMed]

S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun.,  227, 73–74 (2003).
[Crossref]

S. A. Ponomarenko and E. Wolf, “Coherence properties of light in Young’s interference pattern formed with partially coherent light,” Opt. Commun. 170, 1–8 (1999).
[Crossref]

A. Starikov and E. Wolf, “Coherent-mode respresentation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1997).

Zhou, G.

Zu, J.

J. Opt. (1)

M Santarsiro, J. C. G. de Sande, G. Piquero, and F. Gori, “Coherence-polarization properties of fields radiated from transversely periodic electromagnetic sources,” J. Opt. 15, 055701 (2013).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Acta: Int. J. Opt. (2)

A. W. Lohman and J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta: Int. J. Opt. 30, 475–479 (1983).
[Crossref]

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta: Int. J. Opt.,  31, 4–8 (1984).
[Crossref]

Opt. Commun. (9)

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1983).
[Crossref]

S. A. Ponomarenko and E. Wolf, “Coherence properties of light in Young’s interference pattern formed with partially coherent light,” Opt. Commun. 170, 1–8 (1999).
[Crossref]

S. A. Ponomarenko and E. Wolf, “The spectral degree of coherence of fully spatially coherent electromagnetic beams,” Opt. Commun.,  227, 73–74 (2003).
[Crossref]

H. T. Eyyuboglu and Y. Baykal, “Transmittance pf partially coherent cosh-Gaussian, cos-Gaussian, and annular beams through turbulence,” Opt. Commun. 27817–22 (2007).
[Crossref]

F. Gori, “Mode propagation of the fields generated by Collett-Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

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

S. A. Ponomarenko and G. P. Agrawal, “Linear optical bullets,” Opt. Commun. 261, 1–4 (2006).
[Crossref]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. E (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64, 036618 (2001).
[Crossref]

Progress in Optics (1)

J. Turunen and A. T. Friberg, “Propagation-Invariant Optical Fields,” Progress in Optics,  54, 1–88 (2010).
[Crossref]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1997).

L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. doc. ID 224490 (posted 24 October 2014, in press).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

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Figures (4)

Fig. 1
Fig. 1

Intensity profile (in arbitrary units) of a uniformly distributed OCL for several propagation distances Z. The lattice is composed of N = 10 lobes and the lattice constant is a = 1.

Fig. 2
Fig. 2

Magnitude of the spectral degree of coherence of a uniformly distributed OCL for several propagation distances Z. The lattice is composed of N = 10 lobes and the lattice constant is a = 1.

Fig. 3
Fig. 3

Intensity profile (in arbitrary units) of a symmetric, non-uniformly distributed OCL for several propagation distances Z. The the lattice constant is a = 1 and the weight distribution parameter is λ = 5.

Fig. 4
Fig. 4

Magnitude of the spectral degree of coherence of a symmetric, non-uniformly distributed OCL for several propagation distances Z. The lattice constant is a = 1 and the weight distribution parameter is λ = 5.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

W ( X 1 , Y 1 , X 2 , Y 2 , 0 ) = s = X , Y W ( s 1 , s 2 , 0 ) .
W ( s 1 , s 2 , 0 ) = d 2 α 𝒫 s ( α ) ψ α * ( s 1 , 0 ) ψ α ( s 2 , 0 ) ,
d s ψ α * ( s , 0 ) ψ α ( s , 0 ) = 1 ,
d 2 α ψ α * ( s 1 , 0 ) ψ α ( s 2 , 0 ) = δ ( s 1 s 2 ) .
𝒫 s ( α ) = n s ν n s δ ( α α n s ) , ν n s 0 .
W ( s 1 , s 2 ; 0 ) = n s ν n s ψ α n s * ( s 1 , 0 ) ψ α n s ( s 2 , 0 ) .
α n s = i π n s a s 2 ,
ψ α n s ( s , 0 ) = e ( Im α n s ) 2 π 1 / 4 exp [ ( s 2 α n s ) 2 2 ] .
( 2 i Z + 2 2 1 2 ) W ( X 1 , Y 1 , X 2 , Y 2 ; Z ) = 0 .
W ( X 1 , Y 1 , X 2 , Y 2 ; Z ) = s = X , Y W ( s 1 , s 2 ; Z ) ,
W ( s 1 , s 2 ; Z ) = n s ν n s ψ α n s * ( s 1 , Z ) ψ α n s ( s 2 , Z ) .
( 2 i Z + s 2 ) ψ α n s ( s , Z ) = 0 .
ψ α n s ( s , Z ) = e ( Im α n s ) 2 π 1 / 4 ( 1 + i Z ) 1 / 2 exp [ ( s 2 α n s ) 2 2 ( 1 + i Z ) ] .
W ( s 1 , s 2 ; Z ) = exp [ i ( s 2 2 s 1 2 ) 2 R ( Z ) ] π ( 1 + Z 2 ) n s ν n s exp { i π n s a s [ s 2 s 1 σ 2 ( Z ) ] } × exp [ ( s 1 π n s Z / a s ) 2 + ( s 2 π n s Z / a s ) 2 2 σ 2 ( Z ) ] .
R ( Z ) = Z + 1 / Z , σ ( Z ) = 1 + Z 2 .
I ( X , Y ; Z ) s = X , Y W ( s , s ; Z ) = 1 π ( 1 + Z 2 ) s = X , Y n s ν n s exp [ ( s π n s Z / a s ) 2 σ 2 ( Z ) ] .
μ ( X 1 , Y 1 , X 2 , Y 2 ; Z ) = W ( X 1 , Y 1 , X 2 , Y 2 ; Z ) I ( X 1 , Y 1 ; Z ) I ( X 2 , Y 2 ; Z ) ,
ν n X = ν n Y = ν 0 = const ; 0 n X , Y N .
I ( X , Y , 0 ) e ( X 2 + Y 2 ) ,
| μ ( X 1 , Y 1 , X 2 , Y 2 ; 0 ) | = | sin [ π N 2 a ( X 2 X 1 ) ] sin [ π N 2 a ( Y 2 Y 1 ) ] N 2 sin [ π 2 a ( X 2 X 1 ) ] sin [ π 2 a ( Y 2 Y 1 ) ] | ,
ν n s = 𝒜 λ s n s n s ! ; n s 0 ,
| μ ( X 1 , Y 1 , X 2 , Y 2 ; 0 ) | = exp { 2 s = X , Y λ s sin 2 [ π ( s 2 s 1 ) 2 a s ] } .

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