Abstract

It was shown in the paper Opt. Lett. 40, 3739 (2015) [CrossRef]   that Laguerre–Gauss beams $({{\rm LGB}_n})$ of order $n$ and Bessel beams (BB) are asymptotically equivalent for $n \gg 1$. Here we demonstrate that an ${{\rm LGB}_n}$ and a BB are equivalent just in the inner multiring parts of the two beams. However, the outer multiring parts are completely different, and this leads us to apply a truncation on the two beams to make them indistinguishable. Since the ${{\rm LGB}_n}$ could be approximated by a BB only in the inner multiring part, we suggest another beam that could replace its outer multiring part. By considering the ${{\rm LGB}_n}$ as a sum of $n$ rings having different radii and widths, we model the ${{\rm LGB}_n}$ outer multiring part by a sum of what we call in this paper “ring shifted-Gaussian beams.” The peer-to-peer comparison of the ${{\rm LGB}_n}$ with the two cited beams allowed us to provide a new analytical description of the obstructed ${{\rm LGB}_n}$ far field. These results will be very useful to study many aspects related to the ${{\rm LGB}_n}$ diffraction by apertures and stops. As an example, we show at the end of this paper how the self-healing ability of an obstructed ${{\rm LGB}_n}$ could be studied analytically.

© 2020 Optical Society of America

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References

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2020 (1)

2019 (3)

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photon. Rev. 13, 1900140 (2019).
[Crossref]

M. A. Cox, L. Maqondo, R. Kara, G. Milione, L. Cheng, and A. Forbes, “The resilience of Hermite–and Laguerre–Gaussian modes in turbulence,” J. Lightwave Technol. 37, 3911–3917 (2019).
[Crossref]

2018 (5)

2017 (2)

2016 (4)

A. Trichili, A. B. Salem, A. Dudley, M. Zghal, and A. Forbes, “Encoding information using Laguerre Gaussian modes over free space turbulence media,” Opt. Lett. 41, 3086–3089 (2016).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Optimal phase element for generating a perfect optical vortex,” J. Opt. Soc. Am. A 33, 2376–2384 (2016).
[Crossref]

2015 (3)

2014 (3)

2013 (2)

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

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7, 839–854 (2013).
[Crossref]

2012 (1)

2010 (3)

2008 (1)

J. Sun, L. Liu, M. Yun, and L. Wan, “Series representation of the Gaussian beam far-field diffracted by annular aperture,” Optik 119, 766–768 (2008).
[Crossref]

2007 (1)

1996 (1)

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[Crossref]

1995 (2)

Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng. 34, 244–254 (1995).
[Crossref]

B. Lü, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[Crossref]

1986 (2)

1984 (2)

J. E. Harvey and J. L. Forgham, “The spot of Arago: new relevance for an old phenomenon,” Am. J. Phys. 52, 243–247 (1984).
[Crossref]

M. Simpson and A. Michette, “Imaging properties of modified Fresnel zone plates,” Optica 31, 403–413 (1984).
[Crossref]

1978 (1)

C. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
[Crossref]

1972 (1)

T. McKechnie, “The effect of condenser obstruction on the two-point resolution of a microscope,” Optica 19, 729–737 (1972).
[Crossref]

1966 (1)

Abdelhalim, B.

Agarwal, G. S.

Aguirre-Olivas, D.

Aiello, A.

Ait-Ameur, K.

Aït-Ameur, K.

Alpmann, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

C. Alpmann, C. Schöler, and C. Denz, “Elegant Gaussian beams for enhanced optical manipulation,” Appl. Phy. Lett. 106, 241102 (2015).
[Crossref]

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7, 839–854 (2013).
[Crossref]

Ameur, K. A.

Andrews, D. L.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Arrizon, V.

Arroyo-Carrasco, M. L.

Baker, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Banzer, P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Bauer, T.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Belmonte, A.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Bencheikh, A.

A. Bencheikh and A. Forbes, “The non-diffracting nature of truncated Hermite–Gaussian beams,” J. Opt. Soc. Am. A 37, C1–C6 (2020).
[Crossref]

K. Mihoubi, A. Bencheikh, and A. Manallah, “The beam propagation factor M2 of truncated standard and elegant-Hermite-Gaussian beams,” Opt. Laser Technol. 99, 191–196 (2018).
[Crossref]

A. Bencheikh, M. Fromager, and K. A. Ameur, “Generation of Laguerre–Gaussian LGp0 beams using binary phase diffractive optical elements,” Appl. Opt. 53, 4761–4767 (2014).
[Crossref]

J. Pinnell, S. Chabou, K. Mihoubi, A. Bencheikh, V. Rodriguez-Fajardo, and A. Forbes, “Revealing the modal content of obstructed beams,” arXiv preprint arXiv:2005.02331 (2020).

Berry, M. V.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Bhebhe, N.

Bigelow, N. P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Botha, R.

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng. 34, 244–254 (1995).
[Crossref]

Brunel, M.

Burger, L.

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

Cagniot, E.

Cai, Y.

Chabou, S.

J. Pinnell, S. Chabou, K. Mihoubi, A. Bencheikh, V. Rodriguez-Fajardo, and A. Forbes, “Revealing the modal content of obstructed beams,” arXiv preprint arXiv:2005.02331 (2020).

Chávez-Cerda, S.

Cheng, L.

Chu, X.

Coetmellec, S.

Cox, M. A.

de la Hoz, P.

Deng, D.

D. Deng, J. Li, and Q. Guo, “Propagation of truncated modified Laguerre–Gaussian beams,” Appl. Phys. B 98, 211–216 (2010).
[Crossref]

Dennis, M. R.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Denz, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

C. Alpmann, C. Schöler, and C. Denz, “Elegant Gaussian beams for enhanced optical manipulation,” Appl. Phy. Lett. 106, 241102 (2015).
[Crossref]

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7, 839–854 (2013).
[Crossref]

Dudley, A.

Esseling, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7, 839–854 (2013).
[Crossref]

Fickler, R.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
[Crossref]

Forbes, A.

A. Bencheikh and A. Forbes, “The non-diffracting nature of truncated Hermite–Gaussian beams,” J. Opt. Soc. Am. A 37, C1–C6 (2020).
[Crossref]

M. A. Cox, L. Maqondo, R. Kara, G. Milione, L. Cheng, and A. Forbes, “The resilience of Hermite–and Laguerre–Gaussian modes in turbulence,” J. Lightwave Technol. 37, 3911–3917 (2019).
[Crossref]

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photon. Rev. 13, 1900140 (2019).
[Crossref]

N. Bhebhe, C. Rosales-Guzman, and A. Forbes, “Classical and quantum analysis of propagation invariant vector flat-top beams,” Appl. Opt. 57, 5451–5458 (2018).
[Crossref]

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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
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M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
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K. Mihoubi, A. Bencheikh, and A. Manallah, “The beam propagation factor M2 of truncated standard and elegant-Hermite-Gaussian beams,” Opt. Laser Technol. 99, 191–196 (2018).
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J. Pinnell, S. Chabou, K. Mihoubi, A. Bencheikh, V. Rodriguez-Fajardo, and A. Forbes, “Revealing the modal content of obstructed beams,” arXiv preprint arXiv:2005.02331 (2020).

Milione, G.

Moya-Cessa, H.

Mphuthi, N.

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A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
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Neely, T. W.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013).
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Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng. 34, 244–254 (1995).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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J. Pinnell, S. Chabou, K. Mihoubi, A. Bencheikh, V. Rodriguez-Fajardo, and A. Forbes, “Revealing the modal content of obstructed beams,” arXiv preprint arXiv:2005.02331 (2020).

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Rehácek, J.

Ritsch-Marte, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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J. Pinnell, S. Chabou, K. Mihoubi, A. Bencheikh, V. Rodriguez-Fajardo, and A. Forbes, “Revealing the modal content of obstructed beams,” arXiv preprint arXiv:2005.02331 (2020).

Romero, J.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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Rosales-Guzmán, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
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[Crossref]

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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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C. Alpmann, C. Schöler, and C. Denz, “Elegant Gaussian beams for enhanced optical manipulation,” Appl. Phy. Lett. 106, 241102 (2015).
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C. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
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M. Simpson and A. Michette, “Imaging properties of modified Fresnel zone plates,” Optica 31, 403–413 (1984).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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J. Sun, L. Liu, M. Yun, and L. Wan, “Series representation of the Gaussian beam far-field diffracted by annular aperture,” Optik 119, 766–768 (2008).
[Crossref]

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Torres, J. P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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J. Sun, L. Liu, M. Yun, and L. Wan, “Series representation of the Gaussian beam far-field diffracted by annular aperture,” Optik 119, 766–768 (2008).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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White, A. G.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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Willner, A. E.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7, 839–854 (2013).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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J. Sun, L. Liu, M. Yun, and L. Wan, “Series representation of the Gaussian beam far-field diffracted by annular aperture,” Optik 119, 766–768 (2008).
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Adv. Opt. Photon. (1)

Am. J. Phys. (1)

J. E. Harvey and J. L. Forgham, “The spot of Arago: new relevance for an old phenomenon,” Am. J. Phys. 52, 243–247 (1984).
[Crossref]

Appl. Opt. (6)

Appl. Phy. Lett. (1)

C. Alpmann, C. Schöler, and C. Denz, “Elegant Gaussian beams for enhanced optical manipulation,” Appl. Phy. Lett. 106, 241102 (2015).
[Crossref]

Appl. Phys. B (1)

D. Deng, J. Li, and Q. Guo, “Propagation of truncated modified Laguerre–Gaussian beams,” Appl. Phys. B 98, 211–216 (2010).
[Crossref]

AVS Quantum Sci. (1)

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

IEE J. Microw. Opt. Acoust. (1)

C. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. (1)

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2016).
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J. Opt. Soc. Am. A (7)

Laser Photon. Rev. (2)

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. 7, 839–854 (2013).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photon. Rev. 13, 1900140 (2019).
[Crossref]

Nat. Commun. (2)

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

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[Crossref]

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[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. The ${{\rm LGB}_{10}}$ versus its equivalent BB for a beam order $n = 10$; in (a) we show the transverse profiles, while in (b) we show on-axis intensities. The two figures show that the two beams overlap in the inner multiring part, while in the outer multiring part the two beams behave differently. Insets of (a) and (b) show the truncated versions of the ${{\rm LGB}_n}$ and the BB, respectively, ensuring a perfect overlap.
Fig. 2.
Fig. 2. The similarity degree between ${{\rm LBG}_n}$ and its equivalent BB as a function of the truncation parameter $\eta$ for different beam orders $n$.
Fig. 3.
Fig. 3. The ${{\rm LGB}_n}$ versus the BB; (a) and (b) represent ${{\rm LGB}_{10}}$ and BB, respectively; (c) and (d) represent the truncated ${{\rm LGB}_{10}}$ and BB, respectively; (e) and (f) represent the remaining outer multiring parts of the truncated ${{\rm LGB}_{10}}$ and BB shown in (c) and (d).
Fig. 4.
Fig. 4. The new modeling representation of the outer multiring part of the ${{\rm LGB}_{10}}$. (a) The outer multiring part of the ${{\rm LGB}_{10}}$ and (b) a sum of rings represented by different colors located at the same ${{\rm LGB}_{10}}$ rings. The white profiles correspond to the ${{\rm LGB}_{10}}$ intensity distribution.
Fig. 5.
Fig. 5. The intensity distribution of obstructed ${{\rm LGB}_{10}}$ at the focal plane of a lens with $f = 100\,\,{\rm{mm}}$; numerical integration (in red) versus analytical expression (in blue). From the truncation on the last intensity’s zero in (a) to the truncation on the second intensity’s zero in (i), respectively.
Fig. 6.
Fig. 6. (a), (c) Density plots and (b), (d) transverse intensities of the obstructed ${{\rm LGB}_n}$ (in blue) modeled as a sum of ring shifted-Gaussian beams (in red). The white curve represents the ${{\rm LGB}_{10}}$ profile, while the five green rings are ring shifted-Gaussian beams located at the same positions as the ${{\rm LGB}_{10}}$ rings.
Fig. 7.
Fig. 7. The on-axis intensity of the obstructed ${{\rm LGB}_{10}}$ as a function of the normalized propagation distance. (a) Obstructed ${{\rm LGB}_{10}}$ (in blue), with its equivalent five-ring shifted-Gaussian beam (in red); (b) obstructed ${{\rm LGB}_{10}}$ with its equivalent two-ring shifted-Gaussian beam.
Fig. 8.
Fig. 8. The similarity degree between the obstructed ${{\rm LGB}_{10}}$ and its equivalent ring shifted-Gaussian beams for different numbers of remaining rings. The inset shows the corresponding intensity distribution of the ${{\rm LGB}_{10}}$ without its central pic.
Fig. 9.
Fig. 9. Obstructed ${{\rm LGB}_{10}}$ at the input and Fourier planes, calculated using the truncated ${{\rm LGB}_{10}}$ (blue line) and the shifted-ring Gaussian beams (red line).

Equations (18)

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( 2 r w ( z ) ) | m | L n | m | ( 2 r 2 w 2 ( z ) ) e r 2 / w 2 ( z ) Γ ( n + | m | + 1 ) n ! ( n + ( | m | + 1 ) / 2 ) | m | / 2 × J m ( 2 2 n + | m | + 1 r w ( z ) ) .
L G B n ( r , z = 0 ) = L n ( 2 r 2 w 0 2 ) exp ( r 2 w 0 2 ) ,
B B ( r , z = 0 ) = J 0 ( 2 2 n + 1 r w 0 ) .
S = 0 η L G B n ( r , z ) B B ( r , z ) r d r 0 η L G B n ( r , z ) L G B n ( r , z ) r d r 0 η B B ( r , z ) B B ( r , z ) r d r ,
0 a B B ( r , z = 0 ) J 0 ( 2 π λ r ρ r ) r d r = a 2 β J 1 ( β ) J 0 ( t ) t J 1 ( t ) J 0 ( β ) β 2 t 2 ,
L G B n ( ρ , z = f ) = a L G B n ( r , z = 0 ) J 0 ( 2 π λ r ρ r ) r d r .
L G B n ( ρ , z = f ) = 0 L G B n ( r , z = f ) J 0 ( 2 π λ r ρ r ) r d r 0 a L G B n ( r , z = f ) J 0 ( 2 π λ r ρ r ) r d r .
0 L G B n ( r , z = 0 ) J 0 ( 2 π λ f ρ r ) r d r = ( 1 ) n ( w 0 2 2 ) L n ( 2 ρ 2 ( λ f / π w 0 ) 2 ) exp ( ρ 2 ( λ f / π w 0 ) 2 ) .
0 a L G B n ( r , z = 0 ) J 0 ( 2 π λ r ρ r ) r d r a 2 β J 1 ( β ) J 0 ( t ) t J 1 ( t ) J 0 ( β ) β 2 t 2 .
L G B n ( ρ , z = f ) ( 1 ) n ( w 0 2 2 ) L n ( 2 ρ 2 ( λ f / π w 0 ) 2 ) exp ( ρ 2 ( λ f / π w 0 ) 2 ) a 2 β J 1 ( β ) J 0 ( t ) t J 1 ( t ) J 0 ( β ) β 2 t 2 .
a L G B n ( r , z = 0 ) J 0 ( 2 π λ r ρ r ) r d r 0 i n ( 1 ) i a i exp [ ( r r 0 i Δ r i ) 2 ] J 0 ( 2 π λ r ρ r ) r d r .
a L G B n ( r , z = 0 ) J 0 ( 2 π λ r r ρ ) ρ d ρ i n F T ( 1 ) i a i [ exp [ ( r Δ r i ) 2 ] δ ( r r 0 i ) ] .
F T [ exp [ ( r Δ r i ) 2 ] δ ( r r 0 i ) ] = F T [ exp [ ( r Δ r i ) 2 ] ] F T [ δ ( r r 0 i ) ] .
F T [ exp [ ( r Δ r i ) 2 ] ] F T [ δ ( r r 0 i ) ] = 1 λ f r 0 i Δ r i J 0 [ k r ρ ] exp [ ( ρ w 0 f ) 2 ] ,
a L G B n ( r , z = 0 ) J 0 ( 2 π λ r r ρ ) r d r i n ( 1 ) i a i 1 λ f r 0 i Δ r i J 0 [ k r ρ ] exp [ ( ρ w 0 f ) 2 ] .
L G B n obs ( ρ , a ) = c . L G B n ( ρ , a ) ,
c ( ρ , a ) = 1 a 2 β J 1 ( β ) J 0 ( t ) t J 1 ( t ) J 0 ( β ) β 2 t 2 ( 1 ) n ( w 0 2 2 ) L p ( 2 ρ 2 ( λ f / π w 0 ) 2 ) exp ( ρ 2 ( λ f / π w 0 ) 2 ) ,
c ( ρ , a ) = i n ( 1 ) i a i 1 λ f r 0 i Δ r i J 0 [ k r ρ ] exp [ ( ρ w 0 f ) 2 ] ( 1 ) n ( w 0 2 2 ) L p ( 2 ρ 2 ( λ f / π w 0 ) 2 ) exp ( ρ 2 ( λ f / π w 0 ) 2 ) .

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