Abstract

A novel class of partially coherent light sources termed optical coherence grids (OCGs) are introduced that can yield stable optical grids in the far field. The optical grids, of which the light distributes in a network of straight lines crossing each other to form a series of hollow cages, can be seen as a better controlled optical lattice. Propagation properties of OCG beams in free space, including spectral density, transverse coherence, and M 2 factor, are investigated in detail. It is interesting that a periodic grid pattern is produced at a distance and remains stable on further propagation, and we stress that the structure of far-field optical grids can be flexibly tuned by modulating the correlation parameters of the source. In addition, by performing convolution of degree of coherence, we also propose perfect optical coherence grids (POCG). The far-field grid pattern of POCG is in a fully controllable fashion. This work is expected to find applications in cooling atoms, trapping microscopic particles, or assembling cells, etc.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Free-space propagation of optical coherence lattices and periodicity reciprocity

Liyuan Ma and Sergey A. Ponomarenko
Opt. Express 23(2) 1848-1856 (2015)

Parabolic-Gaussian Schell-model sources and their propagations

Adeel Abbas, Jisen Wen, Chenni Xu, and Li-Gang Wang
J. Opt. Soc. Am. A 35(8) 1283-1287 (2018)

Vector optical coherence lattices generating controllable far-field beam profiles

Chunhao Liang, Chenkun Mi, Fei Wang, Chengliang Zhao, Yangjian Cai, and Sergey A. Ponomarenko
Opt. Express 25(9) 9872-9885 (2017)

References

  • View by:
  • |
  • |
  • |

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  3. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [Crossref] [PubMed]
  4. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [Crossref] [PubMed]
  5. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  6. Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
    [Crossref] [PubMed]
  7. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell- model beam,” Phys. Rev. A 91(1), 013823 (2015).
    [Crossref]
  8. Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
    [Crossref] [PubMed]
  9. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  10. S. Cui, Z. Chen, L. Zhang, and J. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013).
    [Crossref] [PubMed]
  11. O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014).
    [Crossref] [PubMed]
  12. L. Ma and S. A. Ponomarenko, “Optical coherence gratings and lattices,” Opt. Lett. 39(23), 6656–6659 (2014).
    [Crossref] [PubMed]
  13. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
    [Crossref] [PubMed]
  14. Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
    [Crossref]
  15. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
    [Crossref] [PubMed]
  16. Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017).
    [Crossref] [PubMed]
  17. F. Wang, J. Li, G. Martinez-Piedra, and O. Korotkova, “Propagation dynamics of partially coherent crescent-like optical beams in free space and turbulent atmosphere,” Opt. Express 25(21), 26055–26066 (2017).
    [Crossref] [PubMed]
  18. M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Partially coherent sources with circular coherence,” Opt. Lett. 42(8), 1512–1515 (2017).
    [Crossref] [PubMed]
  19. Z. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
    [Crossref] [PubMed]
  20. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
    [Crossref] [PubMed]
  21. Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014).
    [Crossref] [PubMed]
  22. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
    [Crossref] [PubMed]
  23. L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
    [Crossref] [PubMed]
  24. Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016).
    [Crossref] [PubMed]
  25. C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Self-focusing of a partially coherent beam with circular coherence,” J. Opt. Soc. Am. A 34(8), 1441–1447 (2017).
    [Crossref] [PubMed]
  26. X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
    [Crossref] [PubMed]
  27. 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]
  28. M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
    [Crossref] [PubMed]
  29. J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
    [Crossref] [PubMed]
  30. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
    [Crossref]
  31. M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
    [Crossref]
  32. E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13(10), 3777–3786 (2005).
    [Crossref] [PubMed]
  33. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, and P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16(1), 106–112 (1999).
    [Crossref]
  34. O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
    [Crossref] [PubMed]
  35. M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
    [Crossref]

2017 (4)

2016 (4)

2015 (7)

2014 (7)

2013 (5)

2011 (1)

2007 (2)

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

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

2005 (2)

2001 (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[Crossref] [PubMed]

1999 (1)

Ahufinger, V.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Basu, S.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

Bloch, I.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

Borghi, R.

Cai, Y.

Chen, R.

Chen, Y.

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell- model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Chen, Z.

Cincotti, G.

Cooper, J.

Courtial, J.

Cui, S.

Damski, B.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

de Sande, J. C. G.

Ding, C.

Gbur, G.

Gori, F.

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell- model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Gu, Y.

Huang, W.

Hyde, M. W.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

Jordan, P.

Koivurova, M.

Korotkova, O.

Lajunen, H.

Lewenstein, M.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Li, J.

Li, Z.

Liang, C.

Liu, L.

Liu, X.

Ma, L.

Maluenda, D.

Mao, Y.

Martínez-Herrero, R.

Martinez-Piedra, G.

Mei, Z.

Padgett, M.

Pan, L.

Piestun, R.

Piquero, G.

Ponomarenko, S. A.

Pu, J.

Qu, J.

Saastamoinen, T.

Sanpera, A.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Santarsiero, M.

Schchepakina, E.

Schonbrun, E.

Sen, A.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Sen, U.

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Shchepakina, E.

Sun, C.

Tang, M.

Turunen, J.

Vahimaa, P.

Voelz, D.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

Wang, F.

Wang, H.

Wang, J.

Wu, G.

Wulff, K.

Xiao, X.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

Xiong, M.

Yu, J.

Yuan, Y.

Zhang, L.

Zhao, D.

Zhou, Y.

Zhu, S.

Adv. Phys. (1)

M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. 56(2), 243–379 (2007).
[Crossref]

Appl. Phys. Lett. (1)

Y. Chen, S. A. Ponomarenko, and Y. Cai, “Experimental generation of optical coherence lattices,” Appl. Phys. Lett. 109(6), 061107 (2016).
[Crossref]

J. Opt. (1)

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17(5), 055607 (2015).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Phys. (1)

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1(1), 23–30 (2005).
[Crossref]

Opt. Express (11)

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13(10), 3777–3786 (2005).
[Crossref] [PubMed]

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]

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref] [PubMed]

J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016).
[Crossref] [PubMed]

F. Wang, J. Li, G. Martinez-Piedra, and O. Korotkova, “Propagation dynamics of partially coherent crescent-like optical beams in free space and turbulent atmosphere,” Opt. Express 25(21), 26055–26066 (2017).
[Crossref] [PubMed]

L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
[Crossref] [PubMed]

Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016).
[Crossref] [PubMed]

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

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, and Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[Crossref] [PubMed]

Z. Chen, S. Cui, L. Zhang, C. Sun, M. Xiong, and J. Pu, “Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence,” Opt. Express 22(15), 18278–18283 (2014).
[Crossref] [PubMed]

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

Opt. Lett. (15)

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

Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015).
[Crossref] [PubMed]

Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
[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. Cui, Z. Chen, L. Zhang, and J. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013).
[Crossref] [PubMed]

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

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

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

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

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref] [PubMed]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017).
[Crossref] [PubMed]

M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Partially coherent sources with circular coherence,” Opt. Lett. 42(8), 1512–1515 (2017).
[Crossref] [PubMed]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

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

Phys. Rev. A (1)

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell- model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[Crossref] [PubMed]

Other (1)

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Degree of coherence of OCGs in the source plane for (a) and (d) N = M = 2, (b) and (e) N = M = 3, (c) and (f) N = M = 4, while sdy is set to zero for (d)-(f).
Fig. 2
Fig. 2 Spectral density of OCGs propagating in free space. (a) z = 0, (b) z = 10m, (c) z = 20m, (d) z = 30m, (e) z = 100m and (f) z = 1000m. Parameters are set as N = M = 6, Rx = Ry = 3, δx = δy = 3 mm, σ0 = 6 mm, L = 6 mm−1 and λ is fixed at 632.8 nm henceforth. (g) Stereograms of optical grids that corresponding to (e).
Fig. 3
Fig. 3 Degree of coherence of OCGs beams for (a) different propagation distance z and (b) different parameter L in the source plane.
Fig. 4
Fig. 4 Spectral density distribution of OCGs beam in transverse plane z = 100 m for different values of L. (a) L = 0.1, (b) L = 0.5, (c) L = 1, (d) L = 2, (e) L = 5 and (f) L = 20 with units being mm−1. Sizes of pictures (a)-(f) are identical and thus omitted.
Fig. 5
Fig. 5 Propagation factor Mx2 of OCGs beams versus the number of periodicity N for different values of δx.
Fig. 6
Fig. 6 (a) Spectral density distribution of rectangle optical grids at z = 60 m. (b) Spectral density distribution of single cage at z = 60 m.
Fig. 7
Fig. 7 Spectral density distribution in several transverse planes when δy→0. (a) z = 0, (b) z = 1 m, (c) z = 3 m, (d) z = 5 m, (e) z = 15 m and (e) z = 30 m.
Fig. 8
Fig. 8 Degree of coherence in the source plane. (a) POCG and (b) OCGs for parameter L = 50 mm−1; (c) POCG and (d) OCGs for parameter L = 1 mm−1.
Fig. 9
Fig. 9 Radiant spectral density of POCG in far field for (a) N = M = 8 and (b) N = 6, M = 12.

Equations (25)

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

W ( 0 ) ( ρ 1 , ρ 2 , ω ) = U ( ρ 1 , ω ) U ( ρ 2 , ω ) ,
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 ) ,
W ( 0 ) ( ρ 1 , ρ 2 ) = S ( ρ 1 ) S ( ρ 2 ) μ ( ρ 1 ρ 2 ) ,
p ( v x , v y ) = δ x N M L 2 exp ( v y 2 L 2 ) n x = P P [ exp ( 2 π 2 ( δ x v x + n x R x ) 2 ) + exp ( 2 π 2 ( δ x v x n x R x ) 2 ) ] , + δ y N M L 2 exp ( v x 2 L 2 ) n y = Q Q [ exp ( 2 π 2 ( δ y v y + n y R y ) 2 ) + exp ( 2 π 2 ( δ y v y n y R y ) 2 ) ]
W ( 0 ) ( ρ 1 , ρ 2 ) = 1 N M exp ( ρ 1 2 + ρ 2 2 4 σ 0 2 ) [ exp ( ( s 1 x s 2 x ) 2 2 δ x 2 ) exp ( L 2 π 2 ( s 1 y s 2 y ) 2 ) n x = P P cos ( 2 π n x R x ( s 1 x s 2 x ) δ x ) . + exp ( ( s 1 y s 2 y ) 2 2 δ y 2 ) exp ( L 2 π 2 ( s 1 x s 2 x ) 2 ) n y = Q Q cos ( 2 π n y R y ( s 1 y s 2 y ) δ y ) ]
L min = max { ( N 1 ) R x δ x 2 ln 2 , ( M 1 ) R y δ y 2 ln 2 } .
S ( r ) = ( 2 π k / r ) 2 W ˜ ( 0 ) ( k u , k u ) cos 2 θ ,
W % ( 0 ) ( f 1 , f 2 ) = ( 2 π ) 4 W ( 0 ) ( ρ 1 , ρ 2 ) exp [ i ( f 1 ρ 1 + f 2 ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 .
S ( r ) = 2 σ 0 2 k 2 cos 2 θ r 2 { 1 a x b exp ( 2 k 2 u y 2 b ) exp ( k 2 u x 2 2 a x ) × n x = P P exp ( c n x 2 2 a x ) cos h ( c n x 2 k u x a x ) + 1 a y b exp ( 2 k 2 u x 2 b ) × exp ( k 2 u y 2 2 a y ) n y = Q Q exp ( c n y 2 2 a y ) cos h ( c n y 2 k u y a y ) } ,
a j = 1 4 σ 0 2 + 1 δ j 2 ; b = 1 σ 0 2 + 8 π 2 L 2 ; c n j = 2 π n j R j δ j ,
k 2 > > 2 a j ; 2 k 2 > > b .
1 4 σ 0 2 + 1 δ j 2 < < 2 π 2 λ 2 ; 1 4 σ 0 2 +2 π 2 L 2 < < 2 π 2 λ 2 .
W ( x 1 , y 1 , x 2 , y 2 , z ) = d s 1 x d s 2 x d s 1 y d s 2 y W ( 0 ) ( ρ 1 , ρ 2 ) × exp { i k 2 z [ ( x 1 s 1 x ) 2 + ( y 1 s 1 y ) 2 - ( x 2 s 2 x ) 2 - ( y 2 s 2 y ) 2 ] }
W ( r 1 , r 2 , z ) = k 2 σ 0 2 4 z 2 N M g exp [ i k 2 z ( r 1 2 r 2 2 ) ] { 1 β x exp [ α ( x 1 x 2 ) 2 ] × exp [ η y 2 g ] × n x = P P [ exp ( ξ x + 2 β x ) + exp ( ξ x 2 β x ) ] + 1 β y exp [ α ( y 1 y 2 ) 2 ] × exp [ η x 2 g ] × n y = Q Q [ exp ( ξ y + 2 β y ) + exp ( ξ y 2 β y ) ] } ,
α = k 2 σ 0 2 2 z 2 , β j = 1 8 σ 0 2 + 1 2 δ j + α , g = 1 8 σ 0 2 + π 2 L 2 + α , η j = k 2 4 σ 0 2 z 2 ( j 1 2 + j 2 2 ) + π 2 L 2 k 2 z 2 ( j 1 j 2 ) 2 i k 3 2 z 3 ( j 1 2 j 2 2 ) , ξ j ± = ( 3 k 2 σ 0 2 4 z 2 α 2 ) ( j 1 j 2 ) + i k 4 z ( j 1 + j 2 ) ± i π n j R j δ j ; j = x , y ,
μ ( r 1 , r 2 , z ) = W ( r 1 , r 2 , z ) W ( r 1 , r 1 , z ) W ( r 2 , r 2 , z ) .
M x 2 = 4 π σ x 0 σ x ; M y 2 = 4 π σ y 0 σ y ,
σ x 0 2 = 1 I t o t a l s x 2 W ( 0 ) ( s x , s y , s x , s y ) d s x d s y ; σ y 0 2 = 1 I t o t a l s y 2 W ( 0 ) ( s x , s y , s x , s y ) d s x d s y ,
σ x 2 = 1 4 π 2 I t o t a l [ W ( 0 ) ( s 1 x , s 1 y , s 2 x , s 2 y ) s 1 x s 2 x | s 1 x = s 2 x = s x s 1 y = s 2 y = s y ] d s x d s y ; σ y 2 = 1 4 π 2 I t o t a l [ W ( 0 ) ( s 1 x , s 1 y , s 2 x , s 2 y ) s 1 y s 2 y | s 1 x = s 2 x = s x s 1 y = s 2 y = s y ] d s x d s y ,
I t o t a l = W ( 0 ) ( s x , s y , s x , s y ) d s x d s y .
M x 2 = 2 N + M { 2 σ 0 2 δ x 2 n x = P P [ 1 + 2 π 2 ( δ x 2 L 2 + 2 n x 2 R x 2 ) ] + 1 } ; M y 2 = 2 N + M { 2 σ 0 2 δ y 2 n y = Q Q [ 1 + 2 π 2 ( δ y 2 L 2 + 2 n y 2 R y 2 ) ] + 1 } .
μ POCG ( 0 ) ( ρ 1 - ρ 2 ) = { 1 N + M exp ( ( s 1 x s 2 x ) 2 2 δ x 2 ) exp ( L 2 π 2 ( s 1 y s 2 y ) 2 ) n x = P P cos ( 2 π n x R x ( s 1 x s 2 x ) δ x ) } A y si nc ( A y ( s 1 y s 2 y ) ) , + { 1 N + M exp ( ( s 1 y s 2 y ) 2 2 δ y 2 ) exp ( L 2 π 2 ( s 1 x s 2 x ) 2 ) n y = Q Q cos ( 2 π n y R y ( s 1 y s 2 y ) δ y ) } A x si nc ( A x ( s 1 x s 2 x ) )
S POCG ( r ) = 2 σ 0 2 k 2 cos 2 θ r 2 { 1 a x b exp ( 2 k 2 u y 2 b ) exp ( k 2 u x 2 2 a x ) rect ( k u y A y ) n x = P P exp ( c n x 2 2 a x ) cos h ( c n x 2 k u x a x ) + 1 a y b exp ( 2 k 2 u x 2 b ) exp ( k 2 u y 2 2 a x ) rect ( k u x A y ) n y = Q Q exp ( c n y 2 2 a y ) cos h ( c n y 2 k u y a y ) } ,

Metrics