Abstract

We propose theoretically and numerically, for the first time, the generation of novel partially coherent truncated Airy beams (NPCTABs) with Airy-like distributions for both intensity and degree of coherence via Fourier phase processing. We demonstrate a clear link between the magnitude and frequency of intensity and degree of coherence distributions oscillations of generated beams, and the source coherence and the phase screen parameter. Thus, the source coherence and phase can serve as convenient parameters to control the intensity and degree of the coherence of NPCTABs. Furthermore, we discover that NPCTABs are more stable than the fully coherent truncated Airy beams (FCTABs) during their propagation in free space and can maintain their Airy-like profile for an extended propagation distance. The interesting and tunable characteristics of these novel beams may find applications in particle trapping, phase retrieval, and optical imaging.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (5)

N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019).
[Crossref]

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. P. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Z. Pang and D. Zhao, “Partially coherent dual and quad Airy beams,” Opt. Lett. 44(19), 4889–4892 (2019).
[Crossref]

C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019).
[Crossref]

P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

2018 (1)

2017 (5)

2015 (5)

2014 (3)

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]

W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

2013 (3)

S. Cui, Z. Chen, K. Hu, and J. Pu, “Investigation on partially coherent Airy beams and their propagation,” Acta. Phys. Sin. 62(9), 094205 (2013).
[Crossref]

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013).
[Crossref]

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref]

2012 (2)

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

2011 (1)

2010 (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

2009 (2)

2008 (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wave packets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

2007 (3)

2004 (1)

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69(5), 053813 (2004).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Agrawal, G. P.

Arie, A.

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Barach, G.

Barlev, O.

Baumgartl, J.

Belic, M. R.

Y. Zhang, H. Zhong, M. R. Belić, and Y. Zhang, “Guided self-accelerating Airy beams—a mini-review,” Appl. Sci. 7(4), 341 (2017).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Boudebs, G.

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69(5), 053813 (2004).
[Crossref]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Burger, L.

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013).
[Crossref]

Cai, Y.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. P. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019).
[Crossref]

P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

T. Wu, C. Liang, F. Wang, and Y. Cai, “Shaping the intensity and degree of coherence of a partially coherent beam by a 4f optical system with an amplitude filter,” J. Opt. 19(12), 124010 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

C. Liang, G. Wu, F. Wang, W. Li, Y. Cai, and S. A. Ponomarenko, “Overcoming the classical Rayleigh diffraction limit by controlling two-point correlations of partially coherent light sources,” Opt. Express 25(23), 28352–28362 (2017).
[Crossref]

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]

Cao, H.

Chen, R.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Chen, Z.

Cheng, W.

Cherukulappurath, S.

G. Boudebs and S. Cherukulappurath, “Nonlinear optical measurements using a 4f coherent imaging system with phase objects,” Phys. Rev. A 69(5), 053813 (2004).
[Crossref]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Chriki, R.

Christodoulides, D. N.

N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref]

Chu, X.

W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Cižmár, T.

Courvoisier, F.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Cui, S.

S. Cui, Z. Chen, K. Hu, and J. Pu, “Investigation on partially coherent Airy beams and their propagation,” Acta. Phys. Sin. 62(9), 094205 (2013).
[Crossref]

Davidson, N.

Dholakia, K.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Dolev, I.

Dudley, J.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Efremidis, N. K.

Forbes, A.

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013).
[Crossref]

Friesem, A. A.

Froehly, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Furfaro, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Gao, Y.

C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019).
[Crossref]

Gori, F.

Greenfield, E.

Han, W.

Hu, K.

S. Cui, Z. Chen, K. Hu, and J. Pu, “Investigation on partially coherent Airy beams and their propagation,” Acta. Phys. Sin. 62(9), 094205 (2013).
[Crossref]

Jacquot, M.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Jia, S.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

Kacerovská, B.

P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

Kaminer, I.

Khosravi, R.

P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

Kolesik, M.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref]

Konijnenberg, S.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. P. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Korotkova, O.

Lacourt, P.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Li, H.

D. Xu, X. Song, H. Li, D. Zhang, H. Wang, J. Xiong, and K. Wang, “Experimental observation of sub-Rayleigh quantum imaging with a two-photon entangled source,” Appl. Phys. Lett. 106(17), 171104 (2015).
[Crossref]

Li, W.

Liang, C.

P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019).
[Crossref]

T. Wu, C. Liang, F. Wang, and Y. Cai, “Shaping the intensity and degree of coherence of a partially coherent beam by a 4f optical system with an amplitude filter,” J. Opt. 19(12), 124010 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

C. Liang, G. Wu, F. Wang, W. Li, Y. Cai, and S. A. Ponomarenko, “Overcoming the classical Rayleigh diffraction limit by controlling two-point correlations of partially coherent light sources,” Opt. Express 25(23), 28352–28362 (2017).
[Crossref]

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]

Liang, Y.

Litvin, I.

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013).
[Crossref]

Liu, L.

C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019).
[Crossref]

Liu, X.

Lu, X.

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. P. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
[Crossref]

Lumer, Y.

Ma, L.

Ma, P.

P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

Mathis, A.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012).
[Crossref]

Mazilu, M.

Mi, C.

C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019).
[Crossref]

P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

Milonni, P. W.

Moloney, J.

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref]

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S. Cui, Z. Chen, K. Hu, and J. Pu, “Investigation on partially coherent Airy beams and their propagation,” Acta. Phys. Sin. 62(9), 094205 (2013).
[Crossref]

Adv. Photonics (1)

X. Lu, Y. Shao, C. Zhao, S. Konijnenberg, X. Zhu, Y. Tang, Y. Cai, and H. P. Urbach, “Noniterative spatially partially coherent diffractive imaging using pinhole array mask,” Adv. Photonics 1(1), 016005 (2019).
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Y. Zhang, H. Zhong, M. R. Belić, and Y. Zhang, “Guided self-accelerating Airy beams—a mini-review,” Appl. Sci. 7(4), 341 (2017).
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P. Ma, B. Kacerovská, R. Khosravi, C. Liang, J. Zeng, X. Peng, C. Mi, Y. E. Monfared, Y. Zhang, F. Wang, and Y. Cai, “Numerical approach for studying the evolution of the degrees of coherence of partially coherent beams Propagation through an ABCD Optical System,” Appl. Sci. 9(10), 2084 (2019).
[Crossref]

J. Mod. Opt. (1)

W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61(5), 379–384 (2014).
[Crossref]

J. Opt. (1)

T. Wu, C. Liang, F. Wang, and Y. Cai, “Shaping the intensity and degree of coherence of a partially coherent beam by a 4f optical system with an amplitude filter,” J. Opt. 19(12), 124010 (2017).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

C. Mi, C. Liang, F. Wang, L. Liu, Y. Gao, and Y. Cai, “Modulating the statistical properties of a vector partially coherent beam by a 4f optical system,” J. Quant. Spectrosc. Radiat. Transfer 222-223, 145–153 (2019).
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Nat. Commun. (1)

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013).
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A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
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Opt. Express (5)

Opt. Lett. (8)

Optica (2)

Phys. Rev. A (2)

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X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
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G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Modified 4f optical system which consists of the source plane, output plane and two lenses L1 and L2 with the same focal length f. A phase screen is located at the center of the spatial frequency plane.
Fig. 2.
Fig. 2. Density plots of the intensity distribution (a1-d1) and the square of the modulus of the DOC distribution (a2-d2) of the NPCTABs in the output plane with a = 350m−2/3 for different values of the initial coherence width $\delta$ of the incident GSM beam, where (a1) and (a2) $\delta = \textrm{infinity}$; (b1) and (b2) $\delta = 1\textrm{mm}$; (c1) and (c2) $\delta = 0.5\textrm{mm}$; (d1) and (d2) $\delta = 0.25\textrm{mm}$. Note that the intensity distributions are normalized to the peak intensity. The corresponding cross lines are shown as white solid curves.
Fig. 3.
Fig. 3. Density plots of the intensity distribution (a1-d1) and the square of the modulus of the DOC distribution (a2-d2) of the NPCTABs in the output plane with the initial coherence width $\delta = 1\textrm{mm}$ for different values of the phase screen parameter a, where (a1) a = 250m−2/3, (b1) a = 300m−2/3, (c1) a = 350m−2/3, (d1) a = 400m−2/3. The corresponding cross lines are shown as white solid curves.
Fig. 4.
Fig. 4. Deflection of the largest peak of the NPCTABs as a function of the propagation distance z during propagation in free space for different values of the parameter a of phase screen and the source coherence width $\delta$.
Fig. 5.
Fig. 5. Normalized intensity of the NPCTABs during propagation in free space with a = 250m−2/3 and different values of the initial coherence width $\delta$ in (a) the output plane and (b) the receiver plane at z = 60 m. Note that the beam intensity is normalized by its peak intensity.

Equations (11)

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W i ( r 1 , r 2 ) = τ ( r 1 ) τ ( r 2 ) μ i ( r 2 r 1 )   = τ ( r 1 ) τ ( r 2 ) g ~ ( r 2 r 1 ) ,
W i ( r 1 , r 2 ) = Γ ( r 1 ) × Γ ( r 2 ) ,
Γ ( r ) = τ ( r ) × F T [ g × C n ] ,
W o ( ρ 1 , ρ 2 ) = W ( r 1 , r 2 ) h ( r 1 , ρ 1 ) h ( r 2 , ρ 2 ) d 2 r 1 d 2 r 2 ,
h ( r , ρ ) = 1 λ 2 f 2 P ( ξ ) exp [ i k f ( r ρ ) ξ ] d 2 ξ = 1 λ 2 f 2 P ~ [ ( r ρ ) / ( r ρ ) λ f λ f ] ,
W o ( ρ 1 , ρ 2 ) = η ( ρ 1 ) × η ( ρ 2 ) ,
η ( ρ ) = Γ ( ρ ) h ( ρ ) ,
T ( v , z ) = ( i / i λ z λ z ) exp ( i π v 2 / i π v 2 λ z λ z ) R ( ρ ) exp ( i 2 π ρ v / i 2 π ρ v λ z λ z ) d 2 ρ   = ( i / i λ z λ z ) exp ( i π v 2 / i π v 2 λ z λ z ) R ~ ( v / v λ z λ z ) ,
g ( κ ) = 2 π δ 2 exp ( 2 π 2 δ 2 κ 2 ) ,
τ ( r ) = exp ( r 2 / r 2 ω 0 2 ω 0 2 ) ,
P ( ξ ) = exp { i k [ ( a ξ x ) 3 + ( a ξ y ) 3 ] i Φ } .

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