Abstract

This paper explores the far-zone behavior of partially coherent arrays. We derive an expression for the far-zone spectral density valid for any array composed of circular elements and fed by fields with Schell-model cross-spectral density functions. This expression is written as the sum of convolution integrals, making it easy to physically interpret. We discuss this expression at length and present examples. Lastly, we validate our analysis by comparing Monte Carlo averages from wave-optics simulations with theory. We conclude this paper with a brief summary of the results and potential uses of our work.

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
    [Crossref]
  3. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  4. O. Korotkova, Random Light Beams: Theory and Applications (CRC Press, 2014).
  5. G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
    [Crossref]
  6. Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
    [Crossref]
  7. P. Sprangle, B. Hafizi, A. Ting, and R. Fischer, “High-power lasers for directed-energy applications,” Appl. Opt. 54, F201–F209 (2015).
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  8. W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55, 1757–1764 (2016).
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  9. A. Brignon, ed., Coherent Laser Beam Combining (Wiley-VCH, 2013).
  10. N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
    [Crossref]
  11. B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005–2009 (2000).
    [Crossref]
  12. P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640–1647 (2009).
    [Crossref]
  13. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188–12195 (2006).
    [Crossref]
  14. M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
    [Crossref]
  15. M. Vorontsov, G. Filimonov, V. Ovchinnikov, E. Polnau, S. Lachinova, T. Weyrauch, and J. Mangano, “Comparative efficiency analysis of fiber-array and conventional beam director systems in volume turbulence,” Appl. Opt. 55, 4170–4185 (2016).
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  16. Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
    [Crossref]
  17. Y. Cai, Y. Chen, H. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
    [Crossref]
  18. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
    [Crossref]
  19. P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
    [Crossref]
  20. F. D. Kashani and M. Yousefi, “Analyzing the propagation behavior of coherence and polarization degrees of a phase-locked partially coherent radial flat-topped array laser beam in underwater turbulence,” Appl. Opt. 55, 6311–6320 (2016).
    [Crossref]
  21. M. Yousefi, F. D. Kashani, S. Golmohammady, and A. Mashal, “Scintillation and bit error rate analysis of a phase-locked partially coherent flat-topped array laser beam in oceanic turbulence,” J. Opt. Soc. Am. A 34, 2126–2137 (2017).
    [Crossref]
  22. X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
    [Crossref]
  23. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282, 1993–1997 (2009).
    [Crossref]
  24. X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42, 604–609 (2010).
    [Crossref]
  25. Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
    [Crossref]
  26. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 329–334.
  27. C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 639–643.
  28. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 314–323.
  29. C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 182–191.
  30. C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 338–346.
  31. C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 890–891.
  32. By making this statement, we have effectively assumed that the Fourier transform of a sample function drawn from a wide-sense stationary random process exists. In general, this is not the case [1,3,34]. In the context of this work, the mathematically proper way to handle this outcome is to define the cross-spectral density (CSD) function in the array plane and propagate that CSD function to the target plane [1,4,34]. With all of this in mind, the spectral density obtained in this manner is equal to our expression given in Eq. (11).
  33. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  34. J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).
  35. S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22, 31691–31707 (2014).
    [Crossref]
  36. D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40, 352–355 (2015).
    [Crossref]
  37. H. T. Yura and S. G. Hanson, “Digital simulation of an arbitrary stationary stochastic process by spectral representation,” J. Opt. Soc. Am. A 28, 675–685 (2011).
    [Crossref]
  38. C. A. Mack, “Generating random rough edges, surfaces, and volumes,” Appl. Opt. 52, 1472–1480 (2013).
    [Crossref]
  39. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).
  40. D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

2017 (2)

2016 (3)

2015 (2)

2014 (2)

2013 (2)

M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
[Crossref]

C. A. Mack, “Generating random rough edges, surfaces, and volumes,” Appl. Opt. 52, 1472–1480 (2013).
[Crossref]

2012 (2)

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
[Crossref]

2011 (1)

2010 (3)

G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42, 604–609 (2010).
[Crossref]

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
[Crossref]

2009 (3)

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640–1647 (2009).
[Crossref]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282, 1993–1997 (2009).
[Crossref]

2008 (2)

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[Crossref]

2007 (1)

Y. Cai, Y. Chen, H. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[Crossref]

2006 (1)

2000 (1)

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 329–334.

C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 639–643.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 314–323.

C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 182–191.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 338–346.

C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 890–891.

Bartell, R. J.

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Basu, S.

S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22, 31691–31707 (2014).
[Crossref]

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Baykal, Y.

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
[Crossref]

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
[Crossref]

Y. Cai, Y. Chen, H. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[Crossref]

Cai, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
[Crossref]

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
[Crossref]

Y. Cai, Y. Chen, H. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[Crossref]

Chen, J.

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
[Crossref]

Chen, Y.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Y. Cai, Y. Chen, H. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[Crossref]

Chu, X.

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640–1647 (2009).
[Crossref]

Cusumano, S. J.

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Davis, C. C.

Du, X.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282, 1993–1997 (2009).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
[Crossref]

Eyyuboglu, H.

Y. Cai, Y. Chen, H. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[Crossref]

Eyyuboglu, H. T.

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
[Crossref]

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
[Crossref]

Filimonov, G.

Fiorino, S. T.

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Fischer, R.

Gbur, G.

G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31, 2038–2045 (2014).
[Crossref]

G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

Golmohammady, S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

Hafizi, B.

Hanson, S. G.

Hyde, M. W.

Ji, X.

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42, 604–609 (2010).
[Crossref]

Kashani, F. D.

Korotkova, O.

Lachinova, S.

Li, X.

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42, 604–609 (2010).
[Crossref]

Liu, L.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Liu, X.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Liu, Z.

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640–1647 (2009).
[Crossref]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
[Crossref]

Lü, B.

Ma, H.

Ma, Y.

Mack, C. A.

Mandel, L.

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

Mangano, J.

Mashal, A.

McCrae, J. E.

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Nelson, W.

Ovchinnikov, V.

Polnau, E.

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

Shay, T. M.

Sprangle, P.

Ting, A.

Van Zandt, N. R.

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Visser, T.

G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

Voelz, D.

Voelz, D. G.

Vorontsov, M.

Vorontsov, M. A.

M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
[Crossref]

Wang, X.

Weyrauch, T.

Wolf, E.

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

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

Xiao, X.

Xu, X.

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
[Crossref]

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640–1647 (2009).
[Crossref]

Yang, F.

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42, 604–609 (2010).
[Crossref]

Yousefi, M.

Yu, J.

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Yuan, Y.

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
[Crossref]

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
[Crossref]

Yura, H. T.

Zhao, C.

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
[Crossref]

Zhao, D.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282, 1993–1997 (2009).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
[Crossref]

Zhou, P.

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
[Crossref]

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640–1647 (2009).
[Crossref]

Zhu, Y.

Adv. Opt. Technol. (1)

M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
[Crossref]

Appl. Opt. (6)

Appl. Phys. B (2)

Y. Cai, Y. Chen, H. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467–475 (2007).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93, 901–905 (2008).
[Crossref]

J. Mod. Opt. (1)

Y. Yuan, Y. Cai, C. Zhao, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of laser array beams in turbulent atmosphere,” J. Mod. Opt. 57, 621–631 (2010).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282, 1993–1997 (2009).
[Crossref]

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640–1647 (2009).
[Crossref]

Opt. Eng. (1)

N. R. Van Zandt, S. J. Cusumano, R. J. Bartell, S. Basu, J. E. McCrae, and S. T. Fiorino, “Comparison of coherent and incoherent laser beam combination for tactical engagements,” Opt. Eng. 51, 104301 (2012).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

X. Li, X. Ji, and F. Yang, “Beam quality of radial Gaussian Schell-model array beams,” Opt. Laser Technol. 42, 604–609 (2010).
[Crossref]

Opt. Lasers Eng. (1)

Y. Yuan, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50, 752–759 (2012).
[Crossref]

Opt. Lett. (1)

Prog. Opt. (2)

G. Gbur and T. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010).
[Crossref]

Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017).
[Crossref]

Other (15)

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

A. Brignon, ed., Coherent Laser Beam Combining (Wiley-VCH, 2013).

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

O. Korotkova, Random Light Beams: Theory and Applications (CRC Press, 2014).

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 329–334.

C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 639–643.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 314–323.

C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 182–191.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), pp. 338–346.

C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. (Wiley, 2016), pp. 890–891.

By making this statement, we have effectively assumed that the Fourier transform of a sample function drawn from a wide-sense stationary random process exists. In general, this is not the case [1,3,34]. In the context of this work, the mathematically proper way to handle this outcome is to define the cross-spectral density (CSD) function in the array plane and propagate that CSD function to the target plane [1,4,34]. With all of this in mind, the spectral density obtained in this manner is equal to our expression given in Eq. (11).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial (SPIE, 2011).

Supplementary Material (4)

NameDescription
» Visualization 1       Figure 1(d)
» Visualization 2       Figure 2(d)
» Visualization 3       Figure 3(b)
» Visualization 4       Figure 3(f)

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Figures (3)

Fig. 1.
Fig. 1. Scenario 1 results: (a)  S thy and (b)  S sim with δ x x = δ y y = 31    cm ; (c)  S thy and (d)  S sim with δ x x = δ y y = 7.75    cm (see Visualization 1); (e)  S thy and (f)  S sim with δ x x = δ y y = 1.94    cm ; (g)  y = 0 slices (a)–(f); and (h)  x = 0 slices (a)–(f).
Fig. 2.
Fig. 2. Scenario 2 results: (a)  S thy and (b)  S sim with δ x x = δ y y = 31    cm ; (c)  S thy and (d)  S sim with δ x x = δ y y = 7.75    cm (see Visualization 2); (e)  S thy and (f)  S sim with δ x x = δ y y = 1.94    cm ; (g)  y = 0 slices (a)–(f); and (h)  x = 0 slices (a)–(f).
Fig. 3.
Fig. 3. Scenario 3 results: Scenario 3A—(a)  S thy and (b)  S sim with δ x x = δ y y = 31    cm (see Visualization 3); (c)  y = 0 slices (a) and (b); and (d)  x = 0 slices (a) and (b); Scenario 3B—(e)  S thy and (f)  S sim with δ x x = δ y y = 31    cm (see Visualization 4); (g)  y = 0 slices (e) and (f); (h)  x = 0 slices (e) and (f).

Equations (18)

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E ( r , ω ) = θ ^ j k 4 π r e j k r [ L x ( r , ω ) sin ϕ + L y ( r , ω ) cos ϕ ] + ϕ ^ j k 4 π r e j k r [ L x ( r , ω ) cos θ cos ϕ L y ( r , ω ) cos θ sin ϕ ] ,
L x , y ( r , ω ) = 2 S E y , x src ( ρ , ω ) e j k r ^ · ρ d 2 ρ ,
S ( r , ω ) E ( r , ω ) · E * ( r , ω ) = k 2 ( 4 π r ) 2 ( cos 2 θ cos 2 ϕ + sin 2 ϕ ) L x ( r , ω ) L x * ( r , ω ) + k 2 ( 4 π r ) 2 ( cos 2 θ sin 2 ϕ + cos 2 ϕ ) L y ( r , ω ) L y * ( r , ω ) + k 2 ( 4 π r ) 2 sin 2 θ sin ( 2 ϕ ) Re [ L x ( r , ω ) L y * ( r , ω ) ] ,
S ( ρ , z , ω ) k 2 ( 4 π z ) 2 [ L x ( ρ , z , ω ) L x * ( ρ , z , ω ) + L y ( ρ , z , ω ) L y * ( ρ , z , ω ) ] ,
L x , y ( ρ , z , ω ) 2 S E y , x src ( ρ , ω ) exp ( j k z ρ · ρ ) d 2 ρ .
E src ( ρ , ω ) = m = 1 N circ ( | ρ ρ m | d / 2 ) × [ x ^ C x , m ( ω ) t x , m ( ρ , ω ) + y ^ C y , m ( ω ) t y , m ( ρ , ω ) ] ,
E α src ( ρ 1 ) E α src * ( ρ 2 ) = m = 1 N n = 1 N C α , m C α , n * P m ( ρ 1 ) P n ( ρ 2 ) t α , m ( ρ 1 ) t α , n * ( ρ 2 ) = m = 1 N n = 1 N C α , m C α , n * P m ( ρ 1 ) P n ( ρ 2 ) μ α α , m n ( ρ 1 ρ 2 ) ,
S ( ρ , z ) = 1 ( λ z ) 2 α = x , y m = 1 N n = 1 N C α , m C α , n * μ α α , m n ( t ) × [ P m ( s ) P n ( s t ) d 2 s ] exp ( j k z t · ρ ) d 2 t = 1 ( λ z ) 2 α = x , y m = 1 N n = 1 N C α , m C α , n * × μ α α , m n ( t ) Λ m n ( t ) exp ( j 2 π t · ρ λ z ) d 2 t ,
S ( ρ , z ) = 1 ( λ z ) 2 α = x , y m = 1 N n = 1 N C α , m C α , n * × μ ˜ α α , m n ( f ) Λ ˜ m n ( f ρ λ z ) d 2 f ,
S ( ρ , z ) = α = x , y m = 1 N n = 1 N C α , m C α , n * [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( 2 π d 2 ρ ) × exp [ j 2 π ( x ρ m ρ n ) · ρ ] μ ˜ α α , m n ( ρ ) | ρ = ρ λ z ,
S ( ρ , z ) = S incoh ( ρ , z ) + S coh ( ρ , z ) , S incoh ( ρ , z ) = α = x , y m = 1 N | C α , m | 2 [ π ( d / 2 ) 2 λ z ] 2 × jinc 2 ( 2 π d 2 ρ ) μ ˜ α α , m m ( ρ ) | ρ = ρ λ z , S coh ( ρ , z ) = α = x , y m = 1 N n m N C α , m C α , n * [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( 2 π d 2 ρ ) × exp [ j 2 π ( ρ m ρ n ) · ρ ] μ ˜ α α , m n ( ρ ) | ρ = ρ λ z .
S incoh ( ρ , z ) = α = x , y | C α | 2 N [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( 2 π d 2 ρ ) μ ˜ α α ( ρ ) | ρ = ρ λ z , S coh ( ρ , z ) = α = x , y | C α | 2 m = 1 N n m N [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( 2 π d 2 ρ ) × exp [ j 2 π ( ρ m ρ n ) · ρ ] μ ˜ α α ( ρ ) | ρ = ρ λ z , S ( ρ , z ) = α = x , y | C α | 2 { N 2 [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( 2 π d 2 ρ ) × | 1 N m = 1 N exp ( j 2 π ρ m · ρ ) | 2 } μ ˜ α α ( ρ ) | ρ = ρ λ z .
S incoh ( ρ , z ) = S ( ρ , z ) = α = x , y | C α | 2 { N [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( 2 π d 2 ρ ) } μ ˜ α α ( ρ ) | ρ = ρ λ z .
S incoh ( ρ , z ) = α = x , y m = 1 N | C α , m | 2 [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( k z d 2 ρ ) , S coh ( ρ , z ) = α = x , y m = 1 N n m N C α , m C α , n * [ π ( d / 2 ) 2 λ z ] 2 × jinc 2 ( k z d 2 ρ ) exp [ j k z ( ρ m ρ n ) · ρ ] .
S incoh ( ρ , z ) = S ( ρ , z ) = 2 [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( k z d 2 ρ ) ,
S incoh ( ρ , z ) = 2 [ π ( d / 2 ) 2 λ z ] 2 jinc 2 ( k z d 2 ρ ) , S coh ( ρ , z ) = S incoh ( ρ , z ) 1 2 cos [ k z ( ρ 1 ρ 2 ) · ρ ] ,
μ α α , m n ( ρ ) = Γ α α , m n exp ( ρ 2 2 δ α α 2 ) ,
t α , m [ i , j ] = k l r α , m [ k , l ] μ ˜ α α , m m [ k , l ] 2 × exp ( j 2 π N i k ) exp ( j 2 π N j l ) 1 N Δ ,

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