Abstract

In this work, we demonstrate how to generate dark and antidark beams—diffraction-free partially coherent sources—using the genuine cross-spectral density function criterion. These beams have been realized in prior work using the source’s coherent-mode representation and by transforming a J0-Bessel correlated partially coherent source using a wavefront-folding interferometer. We generalize the traditional dark and antidark beams to produce higher-order sources, which have not been realized. We simulate the generation of these beams and compare the results to the corresponding theoretical predictions. The simulated results are found to be in excellent agreement with theory, thus validating our analysis. We discuss the pros and cons of our synthesis approach vis-à-vis the prior coherent modes work. Lastly, we conclude this paper with a brief summary, and a discussion of how to physically realize these beams and potential applications.

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References

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  1. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
    [Crossref]
  2. Y. Kodama and A. Hasegawa, “Theoretical foundation of optical-soliton concept in fibers,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1992), Vol. 30, Chap. 4, pp. 205–259.
  3. S. A. Ponomarenko, W. Huang, and M. Cada, “Dark and antidark diffraction-free beams,” Opt. Lett. 32, 2508–2510 (2007).
    [Crossref]
  4. R. Borghi, F. Gori, and S. A. Ponomarenko, “On a class of electromagnetic diffraction-free beams,” J. Opt. Soc. Am. A 26, 2275–2281 (2009).
    [Crossref]
  5. 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]
  6. H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23, 28718–28727 (2015).
    [Crossref]
  7. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref]
  8. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  9. J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 54, Chap. 1, pp. 1–88.
  10. U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), Vol. 61, Chap. 4, pp. 237–281.
  11. K. Saastamoinen, A. T. Friberg, and J. Turunen, “Propagation-invariant optical beams and pulses,” in Non-Diffracting Waves, H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds. (Wiley-VCH, 2013), Chap. 14, pp. 307–326.
  12. O. Korotkova, Random Light Beams: Theory and Applications (CRC Press, 2014).
  13. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2003), Vol. 45, Chap. 3, pp. 119–204.
  14. K. R. Dhakal and V. Lakshminarayanan, “Optical tweezers: fundamentals and some biophysical applications,” in Progress in Optics, T. Visser, ed. (Elsevier, 2018), Vol. 63, Chap. 1, pp. 1–31.
  15. M. J. Padgett, J. E. Molloy, and D. McGloin, eds., Optical Tweezers: Methods and Applications (CRC Press, 2010).
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  17. X. Zhu, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Experimental realization of dark and antidark diffraction-free beams,” Opt. Lett. 44, 2260–2263 (2019).
    [Crossref]
  18. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [Crossref]
  19. F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33, 1857–1859 (2008).
    [Crossref]
  20. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
    [Crossref]
  21. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
    [Crossref]
  22. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007), p. 940.
  23. M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2017).
    [Crossref]
  24. M. W. Hyde, “Partially coherent sources generated from the incoherent sum of fields containing random-width Bessel functions,” Opt. Lett. 44, 1603–1606 (2019).
    [Crossref]
  25. F. Wang and O. Korotkova, “Random sources for beams with azimuthal intensity variation,” Opt. Lett. 41, 516–519 (2016).
    [Crossref]
  26. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
    [Crossref]
  27. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [Crossref]
  28. J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [Crossref]
  29. F. M. Dickey, ed., Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC Press, 2014).
  30. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
    [Crossref]
  31. Z. Yang, Y. Li, Q. Yu, and J. Ma, “Tunable Bessel and annular beams generated by a unimorph deformable mirror,” Opt. Eng. 57, 106107 (2018).
    [Crossref]
  32. X. Yu, A. Todi, and H. Tang, “Bessel beam generation using a segmented deformable mirror,” Appl. Opt. 57, 4677–4682 (2018).
    [Crossref]
  33. M. W. Hyde, Dark and antidark beam MATLAB R2017a simulation scripts (figshare, 2019) [retrieved 2 Mar 2019], https://doi.org/10.6084/m9.figshare.7806572 .

2019 (2)

2018 (2)

Z. Yang, Y. Li, Q. Yu, and J. Ma, “Tunable Bessel and annular beams generated by a unimorph deformable mirror,” Opt. Eng. 57, 106107 (2018).
[Crossref]

X. Yu, A. Todi, and H. Tang, “Bessel beam generation using a segmented deformable mirror,” Appl. Opt. 57, 4677–4682 (2018).
[Crossref]

2017 (1)

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2017).
[Crossref]

2016 (1)

2015 (1)

2009 (2)

2008 (1)

2007 (2)

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

2000 (1)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

1998 (1)

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[Crossref]

1991 (1)

1989 (1)

1988 (1)

1987 (3)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

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

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Borghi, R.

Bose-Pillai, S.

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2017).
[Crossref]

Cada, M.

Cai, Y.

Derevyanko, S.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), Vol. 61, Chap. 4, pp. 237–281.

Dhakal, K. R.

K. R. Dhakal and V. Lakshminarayanan, “Optical tweezers: fundamentals and some biophysical applications,” in Progress in Optics, T. Visser, ed. (Elsevier, 2018), Vol. 63, Chap. 1, pp. 1–31.

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Friberg, A. T.

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]

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref]

J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[Crossref]

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 54, Chap. 1, pp. 1–88.

K. Saastamoinen, A. T. Friberg, and J. Turunen, “Propagation-invariant optical beams and pulses,” in Non-Diffracting Waves, H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds. (Wiley-VCH, 2013), Chap. 14, pp. 307–326.

Gao, W.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2003), Vol. 45, Chap. 3, pp. 119–204.

Gori, F.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007), p. 940.

Guattari, G.

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

Hasegawa, A.

Y. Kodama and A. Hasegawa, “Theoretical foundation of optical-soliton concept in fibers,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1992), Vol. 30, Chap. 4, pp. 205–259.

Huang, W.

Hyde, M. W.

M. W. Hyde, “Partially coherent sources generated from the incoherent sum of fields containing random-width Bessel functions,” Opt. Lett. 44, 1603–1606 (2019).
[Crossref]

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2017).
[Crossref]

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[Crossref]

Kodama, Y.

Y. Kodama and A. Hasegawa, “Theoretical foundation of optical-soliton concept in fibers,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1992), Vol. 30, Chap. 4, pp. 205–259.

Korotkova, O.

Lakshminarayanan, V.

K. R. Dhakal and V. Lakshminarayanan, “Optical tweezers: fundamentals and some biophysical applications,” in Progress in Optics, T. Visser, ed. (Elsevier, 2018), Vol. 63, Chap. 1, pp. 1–31.

Levy, U.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), Vol. 61, Chap. 4, pp. 237–281.

Li, Y.

Z. Yang, Y. Li, Q. Yu, and J. Ma, “Tunable Bessel and annular beams generated by a unimorph deformable mirror,” Opt. Eng. 57, 106107 (2018).
[Crossref]

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[Crossref]

Ma, J.

Z. Yang, Y. Li, Q. Yu, and J. Ma, “Tunable Bessel and annular beams generated by a unimorph deformable mirror,” Opt. Eng. 57, 106107 (2018).
[Crossref]

Mandel, L.

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

Martínez-Herrero, R.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

Mejías, P. M.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[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]

Partanen, H.

Ponomarenko, S. A.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007), p. 940.

Saastamoinen, K.

K. Saastamoinen, A. T. Friberg, and J. Turunen, “Propagation-invariant optical beams and pulses,” in Non-Diffracting Waves, H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds. (Wiley-VCH, 2013), Chap. 14, pp. 307–326.

Santarsiero, M.

Sharmin, N.

Silberberg, Y.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), Vol. 61, Chap. 4, pp. 237–281.

Tang, H.

Tervo, J.

Todi, A.

Turunen, J.

Vasara, A.

Voelz, D. G.

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2017).
[Crossref]

Wang, F.

Wolf, E.

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

Xiao, X.

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2017).
[Crossref]

Yang, Z.

Z. Yang, Y. Li, Q. Yu, and J. Ma, “Tunable Bessel and annular beams generated by a unimorph deformable mirror,” Opt. Eng. 57, 106107 (2018).
[Crossref]

Yin, J.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2003), Vol. 45, Chap. 3, pp. 119–204.

Yu, Q.

Z. Yang, Y. Li, Q. Yu, and J. Ma, “Tunable Bessel and annular beams generated by a unimorph deformable mirror,” Opt. Eng. 57, 106107 (2018).
[Crossref]

Yu, X.

Zhao, C.

Zhu, X.

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2003), Vol. 45, Chap. 3, pp. 119–204.

Appl. Opt. (2)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

J. Opt. (1)

M. W. Hyde, S. Bose-Pillai, X. Xiao, and D. G. Voelz, “A fast and efficient method for producing partially coherent sources,” J. Opt. 19, 025601 (2017).
[Crossref]

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

Opt. Commun. (2)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

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

Opt. Eng. (1)

Z. Yang, Y. Li, Q. Yu, and J. Ma, “Tunable Bessel and annular beams generated by a unimorph deformable mirror,” Opt. Eng. 57, 106107 (2018).
[Crossref]

Opt. Express (1)

Opt. Lett. (7)

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[Crossref]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Other (12)

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 54, Chap. 1, pp. 1–88.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” in Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), Vol. 61, Chap. 4, pp. 237–281.

K. Saastamoinen, A. T. Friberg, and J. Turunen, “Propagation-invariant optical beams and pulses,” in Non-Diffracting Waves, H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds. (Wiley-VCH, 2013), Chap. 14, pp. 307–326.

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

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2003), Vol. 45, Chap. 3, pp. 119–204.

K. R. Dhakal and V. Lakshminarayanan, “Optical tweezers: fundamentals and some biophysical applications,” in Progress in Optics, T. Visser, ed. (Elsevier, 2018), Vol. 63, Chap. 1, pp. 1–31.

M. J. Padgett, J. E. Molloy, and D. McGloin, eds., Optical Tweezers: Methods and Applications (CRC Press, 2010).

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

Y. Kodama and A. Hasegawa, “Theoretical foundation of optical-soliton concept in fibers,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1992), Vol. 30, Chap. 4, pp. 205–259.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier, 2007), p. 940.

M. W. Hyde, Dark and antidark beam MATLAB R2017a simulation scripts (figshare, 2019) [retrieved 2 Mar 2019], https://doi.org/10.6084/m9.figshare.7806572 .

F. M. Dickey, ed., Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC Press, 2014).

Supplementary Material (1)

NameDescription
» Code 1       Dark and antidark beam MATLAB R2017a simulation scripts.

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Figures (3)

Fig. 1.
Fig. 1. Spectral densities S of higher-order dark beams with α = 1 and β = 1 . (a)  n = 0 , (b)  n = 2 , (c)  n = 4 , (d)  n = 6 , (e)  n = 8 , and (f)  n = 10 .
Fig. 2.
Fig. 2. Zeroth-order dark beam results with α = 1 and β = 1 . (a)  S theory [Eq. (2)], (b)  S 50 coherent modes [Eq. (24)], (c)  S genuine CSD criterion 100,000 field realizations [Eqs. (18) and (19)], (d)  W ( x 1 , 0 , x 2 , 0 ) theory [Eq. (1)], (e)  W ( x 1 , 0 , x 2 , 0 ) 50 coherent modes [Eq. (24)], (f)  W ( x 1 , 0 , x 2 , 0 ) genuine CSD criterion 100,000 field realizations [Eqs. (18) and (19)], (g) two-dimensional root-mean-square error (RMSE) and correlation coefficient ρ for the coherent modes S computed against S theory versus mode number, and (h) two-dimensional RMSE and correlation coefficient ρ for the genuine CSD criterion S computed against S theory versus trial number.
Fig. 3.
Fig. 3. Higher-order dark beam results with α = 0.5 , β = 1 , and n = 4 . (a)  S theory [Eq. (23)], (b)  S genuine CSD criterion 100,000 field realizations [Eqs. (18) and (20)], (c)  W ( x 1 , 0 , x 2 , 0 ) theory [Eq. (22)], (d)  W ( x 1 , 0 , x 2 , 0 ) genuine CSD criterion 100,000 field realizations [Eqs. (18) and (20)], and (e) two-dimensional root-mean-square error (RMSE) and correlation coefficient ρ for the genuine CSD criterion S computed against S theory versus trial number.

Equations (25)

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

W ( ρ 1 , ρ 2 ) J 0 ( β | ρ 1 ρ 2 | ) + α J 0 ( β | ρ 1 + ρ 2 | ) ,
S ( ρ ) = W ( ρ , ρ ) 1 + α J 0 ( 2 β ρ ) .
J 0 ( β | ρ 1 ρ 2 | ) = m = ( ± 1 ) m exp [ j m ( ϕ 2 ϕ 1 ) ] J m ( β ρ 1 ) J m ( β ρ 2 ) ,
W ( ρ 1 , ρ 2 ) m = [ 1 + α ( 1 ) m ] J m ( β ρ 1 ) exp ( j m ϕ 1 ) J m ( β ρ 2 ) exp ( j m ϕ 2 ) .
W ( ρ 1 , ρ 2 ) = m λ m ψ m * ( ρ 1 ) ψ m ( ρ 2 ) ,
λ m = 1 + α ( 1 ) m , ψ m ( ρ ) = J m ( β ρ ) exp ( j m ϕ ) .
W ( ρ 1 , ρ 2 ) = p ( v ) H ( ρ 1 , v ) H * ( ρ 2 , v ) d 2 v ,
H ( ρ , v ) = τ ( ρ ) [ a exp ( j β v · ρ ) + b exp ( j β v · ρ ) ] ,
W ( ρ 1 , ρ 2 ) = τ ( ρ 1 ) τ * ( ρ 2 ) [ a 2 p ˜ * ( β ρ d ) + b 2 p ˜ ( β ρ d ) ] + τ ( ρ 1 ) τ * ( ρ 2 ) a b [ p ˜ * ( β ρ a ) + p ˜ ( β ρ a ) ] ,
p ˜ ( f ) = p ( v ) exp ( j v · f ) d 2 v .
τ ( ρ ) = 1 ,
a 2 p ˜ * ( β ρ d ) + b 2 p ˜ ( β ρ d ) = J 0 ( β ρ d ) ,
a b [ p ˜ * ( β ρ a ) + p ˜ ( β ρ a ) ] = α J 0 ( β ρ a ) .
α J 0 ( β ρ a ) = 2 a b p ( v ) exp ( j β v · ρ a ) d 2 v .
α = 2 a b , p ( v ) = 1 2 π δ ( v 1 ) ,
a 2 + b 2 = 1 .
a = ± 1 2 1 ± 1 α 2 .
U ( ρ ) = 1 + 1 α 2 2 [ exp ( j β v · ρ ) + α 1 + 1 α 2 exp ( j β v · ρ ) ] ,
p ( v ) = 1 2 π δ ( v 1 ) .
p ( v ) = δ ( v 1 ) , p ( θ ) = 1 π ( 1 + δ 0 n ) cos 2 ( n θ / 2 ) ,
p ˜ ( f , ψ ) = 1 1 + δ 0 n [ J 0 ( f ) + j n cos ( n ψ ) J n ( f ) ] .
W ( ρ 1 , ρ c 2 ) = 1 1 + δ 0 n { J 0 ( β ρ d ) + j n cos ( n ϕ d ) J n ( β ρ d ) j n [ 1 ( 1 ) n ] α 2 2 ( 1 + 1 α 2 ) cos ( n ϕ d ) J n ( β ρ d ) } + α 1 + δ 0 n [ J 0 ( β ρ a ) + cos ( n π / 2 ) cos ( n ϕ a ) J n ( β ρ a ) ] ,
S ( ρ ) = 1 + α J 0 ( 2 β ρ ) + α ( 1 δ n 0 ) cos ( n π / 2 ) cos ( n ϕ ) J n ( 2 β ρ ) ,
W ( ρ 1 , ρ 2 ) m = 50 50 [ 1 + α ( 1 ) m ] J m ( β ρ 1 ) exp ( j m ϕ 1 ) J m ( β ρ 2 ) exp ( j m ϕ 2 ) ,
RMSE = 1 N 2 k = 1 N 2 ( S thy [ k ] S sim [ k ] ) 2 , ρ = k = 1 N 2 ( S sim [ k ] S ¯ sim ) ( S thy [ k ] S ¯ thy ) k = 1 N 2 ( S sim [ k ] S ¯ sim ) 2 k = 1 N 2 ( S thy [ k ] S ¯ thy ) 2 ,

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