Abstract

Using the asymptotic formula of the Hermite polynomials for higher-orders $n \gg 1$, an elegant mathematical expression that makes Hermite–Gaussian beams and cosine beams equivalent is obtained. Two factors of merit, the similarity and the power content ratio, are used to quantify the degree of equivalence between the two beams. These results yield a new nondiffracting Hermite–Gaussian beam in one dimension (1D) and that is easily extended to 2D.

© 2020 Optical Society of America

Full Article  |  PDF Article

Corrections

23 June 2020: A typographical correction was made to the Fig. 6 caption.

References

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2019 (4)

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]

N. Mphuthi, L. Gailele, I. Litvin, A. Dudley, R. Botha, and A. Forbes, “Free-space optical communication link with shape-invariant orbital angular momentum Bessel beams,” Appl. Opt. 58, 4258–4264 (2019).
[Crossref]

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Structured light beams constituted of incoming and outgoing waves,” Phys. Rev. A 100, 053847 (2019).
[Crossref]

2018 (5)

2017 (2)

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
[Crossref]

Q. Jiang, Y. Su, H. Nie, Z. Ma, and Y. Li, “Propagation and interaction of cos-Gaussian beams in photorefractive crystals,” Phys. Lett. A 381, 2246–2249 (2017).
[Crossref]

2016 (2)

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

2015 (1)

2014 (4)

B. Boubaha, A. Bencheikh, and K. At-Ameur, “Spatial properties of rectified cosine Gaussian beams,” J. Opt. 16, 025701 (2014).
[Crossref]

W. Cheng and P. Polynkin, “Micromachining of borosilicate glass surfaces using femtosecond higher-order Bessel beams,” J. Opt. Soc. Am. B 31, C48–C52 (2014).
[Crossref]

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).
[Crossref]

X. Chu and W. Wen, “Quantitative description of the self-healing ability of a beam,” Opt. Express 22, 6899–6904 (2014).
[Crossref]

2013 (2)

C. Snoeyink, “Imaging performance of Bessel beam microscopy,” Opt. Lett. 38, 2550–2553 (2013).
[Crossref]

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24(6), 22–29 (2013).
[Crossref]

2012 (2)

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

2010 (2)

C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. 35, 1215–1217 (2010).
[Crossref]

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: ‘non-diffracting’ beams,” Laser Photon. Rev. 4, 529–547 (2010).
[Crossref]

2007 (1)

C. López-Mariscal and J. C. Gutiérrez-Vega, “The generation of nondiffracting beams using inexpensive computer-generated holograms,” Am. J. Phys. 75, 36–42 (2007).
[Crossref]

2006 (1)

C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

2005 (2)

2004 (2)

2003 (1)

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[Crossref]

1998 (1)

1997 (1)

1995 (1)

S. Wang, Q. Lin, and X. Lu, “Realization of super-diffraction limiting propagation by cos beams,” Optik 100, 8–10 (1995).

1992 (1)

M. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

1987 (2)

J. Durnin, J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[Crossref]

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

Aggarwal, M.

H. Kumar, M. Aggarwal, D. Sharma, S. Chandok, and T. S. Gill, “Significant enhancement in the propagation of cosh-Gaussian laser beam in a relativistic–ponderomotive plasma using ramp density profile,” Laser Part. Beams 36, 179–185 (2018).
[Crossref]

Ahmed, N.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Almaiman, A.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Arnold, C. B.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

Arroyo-Carrasco, M. L.

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Structured light beams constituted of incoming and outgoing waves,” Phys. Rev. A 100, 053847 (2019).
[Crossref]

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre–Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40, 3739–3742 (2015).
[Crossref]

Ashrafi, S.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

At-Ameur, K.

B. Boubaha, A. Bencheikh, and K. At-Ameur, “Spatial properties of rectified cosine Gaussian beams,” J. Opt. 16, 025701 (2014).
[Crossref]

Bandres, M. A.

Baykal, Y.

Bencheikh, A.

B. Boubaha, A. Bencheikh, and K. At-Ameur, “Spatial properties of rectified cosine Gaussian beams,” J. Opt. 16, 025701 (2014).
[Crossref]

Bergner, K.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

Botha, R.

Boubaha, B.

B. Boubaha, A. Bencheikh, and K. At-Ameur, “Spatial properties of rectified cosine Gaussian beams,” J. Opt. 16, 025701 (2014).
[Crossref]

Capasso, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

Cardano, F.

Casperson, L. W.

Chandok, S.

H. Kumar, M. Aggarwal, D. Sharma, S. Chandok, and T. S. Gill, “Significant enhancement in the propagation of cosh-Gaussian laser beam in a relativistic–ponderomotive plasma using ramp density profile,” Laser Part. Beams 36, 179–185 (2018).
[Crossref]

Chávez-Cerda, S.

Cheng, W.

Chu, X.

Cluzel, B.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

de Fornel, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

Dellinger, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

Denz, C.

Dholakia, K.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: ‘non-diffracting’ beams,” Laser Photon. Rev. 4, 529–547 (2010).
[Crossref]

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

Dong, D.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Dudley, A.

Duocastella, M.

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

Durnin, J.

J. Durnin, J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (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.

J. Durnin, J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[Crossref]

Eyyuboglu, H. T.

Flamm, D.

Fonseca, E. J.

D. G. Pires, A. F. Sonsin, A. J. Jesus-Silva, and E. J. Fonseca, “Three-dimensional speckle light self-healing-based imaging system,” Sci. Rep. 8, 1 (2018).
[Crossref]

Forbes, A.

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]

N. Mphuthi, L. Gailele, I. Litvin, A. Dudley, R. Botha, and A. Forbes, “Free-space optical communication link with shape-invariant orbital angular momentum Bessel beams,” Appl. Opt. 58, 4258–4264 (2019).
[Crossref]

I. Nape, E. Otte, A. Vallés, C. Rosales-Guzmán, F. Cardano, C. Denz, and A. Forbes, “Self-healing high-dimensional quantum key distribution using hybrid spin-orbit Bessel states,” Opt. Express 26, 26946–26960 (2018).
[Crossref]

N. Mphuthi, R. Botha, and A. Forbes, “Are Bessel beams resilient to aberrations and turbulence?” J. Opt. Soc. Am. A 35, 1021–1027 (2018).
[Crossref]

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).
[Crossref]

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24(6), 22–29 (2013).
[Crossref]

A. Forbes, Laser beam Propagation: Generation and Propagation of Customized Light (CRC Press, 2014).

Gailele, L.

Genevet, P.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

Gill, T. S.

H. Kumar, M. Aggarwal, D. Sharma, S. Chandok, and T. S. Gill, “Significant enhancement in the propagation of cosh-Gaussian laser beam in a relativistic–ponderomotive plasma using ramp density profile,” Laser Part. Beams 36, 179–185 (2018).
[Crossref]

Gong, Q.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Gross, H.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

Grossmann, D.

Gunn-Moore, F.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: ‘non-diffracting’ beams,” Laser Photon. Rev. 4, 529–547 (2010).
[Crossref]

Gutierrez-Vega, J. C.

C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

Gutiérrez-Vega, J. C.

Hellstern, J.

Helmerson, K.

Huang, H.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Iturbe-Castillo, M. D.

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Structured light beams constituted of incoming and outgoing waves,” Phys. Rev. A 100, 053847 (2019).
[Crossref]

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre–Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40, 3739–3742 (2015).
[Crossref]

Jenne, M.

Jesus-Silva, A. J.

D. G. Pires, A. F. Sonsin, A. J. Jesus-Silva, and E. J. Fonseca, “Three-dimensional speckle light self-healing-based imaging system,” Sci. Rep. 8, 1 (2018).
[Crossref]

Ji, Z.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Jiang, Q.

Q. Jiang, Y. Su, H. Nie, Z. Ma, and Y. Li, “Propagation and interaction of cos-Gaussian beams in photorefractive crystals,” Phys. Lett. A 381, 2246–2249 (2017).
[Crossref]

Jiang, Z.

Kaiser, M.

Kleiner, J.

Koschig, M.

Kumar, H.

H. Kumar, M. Aggarwal, D. Sharma, S. Chandok, and T. S. Gill, “Significant enhancement in the propagation of cosh-Gaussian laser beam in a relativistic–ponderomotive plasma using ramp density profile,” Laser Part. Beams 36, 179–185 (2018).
[Crossref]

Kumkar, M.

Lapointe, M.

M. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

Lavery, M.

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24(6), 22–29 (2013).
[Crossref]

Lavery, M. P.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Lebedev, N.

N. Lebedev, Special Functions and Their Applications (Courier, 1972).

Li, L.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Li, S.

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
[Crossref]

Li, Y.

Q. Jiang, Y. Su, H. Nie, Z. Ma, and Y. Li, “Propagation and interaction of cos-Gaussian beams in photorefractive crystals,” Phys. Lett. A 381, 2246–2249 (2017).
[Crossref]

Liao, P.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Lin, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

Lin, Q.

S. Wang, Q. Lin, and X. Lu, “Realization of super-diffraction limiting propagation by cos beams,” Optik 100, 8–10 (1995).

Litvin, I.

Lopez-Mariscal, C.

C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

López-Mariscal, C.

Lu, X.

S. Wang, Q. Lin, and X. Lu, “Realization of super-diffraction limiting propagation by cos beams,” Optik 100, 8–10 (1995).

Ma, Z.

Q. Jiang, Y. Su, H. Nie, Z. Ma, and Y. Li, “Propagation and interaction of cos-Gaussian beams in photorefractive crystals,” Phys. Lett. A 381, 2246–2249 (2017).
[Crossref]

Mao, H.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[Crossref]

Mazilu, M.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: ‘non-diffracting’ beams,” Laser Photon. Rev. 4, 529–547 (2010).
[Crossref]

McGloin, D.

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

McLaren, M.

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).
[Crossref]

Mendoza-Hernández, J.

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Structured light beams constituted of incoming and outgoing waves,” Phys. Rev. A 100, 053847 (2019).
[Crossref]

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre–Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40, 3739–3742 (2015).
[Crossref]

Mhlanga, T.

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).
[Crossref]

Miceli, J.

J. Durnin, J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[Crossref]

Molisch, A. F.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Mphuthi, N.

Nape, I.

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]

I. Nape, E. Otte, A. Vallés, C. Rosales-Guzmán, F. Cardano, C. Denz, and A. Forbes, “Self-healing high-dimensional quantum key distribution using hybrid spin-orbit Bessel states,” Opt. Express 26, 26946–26960 (2018).
[Crossref]

Nie, H.

Q. Jiang, Y. Su, H. Nie, Z. Ma, and Y. Li, “Propagation and interaction of cos-Gaussian beams in photorefractive crystals,” Phys. Lett. A 381, 2246–2249 (2017).
[Crossref]

Nolte, S.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

M. Jenne, D. Flamm, T. Ouaj, J. Hellstern, J. Kleiner, D. Grossmann, M. Koschig, M. Kaiser, M. Kumkar, and S. Nolte, “High-quality tailored-edge cleaving using aberration-corrected Bessel-like beams,” Opt. Lett. 43, 3164–3167 (2018).
[Crossref]

Ornigotti, M.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

Otte, E.

Ouaj, T.

Padgett, M.

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24(6), 22–29 (2013).
[Crossref]

Padgett, M. J.

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).
[Crossref]

Pires, D. G.

D. G. Pires, A. F. Sonsin, A. J. Jesus-Silva, and E. J. Fonseca, “Three-dimensional speckle light self-healing-based imaging system,” Sci. Rep. 8, 1 (2018).
[Crossref]

Polynkin, P.

Qiaug, L.

W. Shaomin, L. Qiaug, and L. Xuanhui, “Super-diffraction limiting propagation of cos beams,” Appl. Laser2 (1994).

Ren, Y.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Rosales-Guzmán, C.

Roux, F. S.

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).
[Crossref]

Shaomin, W.

W. Shaomin, L. Qiaug, and L. Xuanhui, “Super-diffraction limiting propagation of cos beams,” Appl. Laser2 (1994).

Sharma, D.

H. Kumar, M. Aggarwal, D. Sharma, S. Chandok, and T. S. Gill, “Significant enhancement in the propagation of cosh-Gaussian laser beam in a relativistic–ponderomotive plasma using ramp density profile,” Laser Part. Beams 36, 179–185 (2018).
[Crossref]

Shi, K.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Snoeyink, C.

Sonsin, A. F.

D. G. Pires, A. F. Sonsin, A. J. Jesus-Silva, and E. J. Fonseca, “Three-dimensional speckle light self-healing-based imaging system,” Sci. Rep. 8, 1 (2018).
[Crossref]

Steinkopf, R.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

Stevenson, D. J.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: ‘non-diffracting’ beams,” Laser Photon. Rev. 4, 529–547 (2010).
[Crossref]

Su, Y.

Q. Jiang, Y. Su, H. Nie, Z. Ma, and Y. Li, “Propagation and interaction of cos-Gaussian beams in photorefractive crystals,” Phys. Lett. A 381, 2246–2249 (2017).
[Crossref]

Szameit, A.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

Tovar, A. A.

Tur, M.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Vallés, A.

Vetter, C.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

Wang, J.

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
[Crossref]

Wang, S.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[Crossref]

S. Wang, Q. Lin, and X. Lu, “Realization of super-diffraction limiting propagation by cos beams,” Optik 100, 8–10 (1995).

Wang, Z.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Wen, W.

Willner, A. E.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Willner, A. J.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Xi, P.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Xiao, Y.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Xie, G.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Xuanhui, L.

W. Shaomin, L. Qiaug, and L. Xuanhui, “Super-diffraction limiting propagation of cos beams,” Appl. Laser2 (1994).

Yan, Y.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Yang, X.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Yu, W.

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

Zhang, W.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[Crossref]

Zhao, D.

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[Crossref]

Zhao, Z.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Am. J. Phys. (1)

C. López-Mariscal and J. C. Gutiérrez-Vega, “The generation of nondiffracting beams using inexpensive computer-generated holograms,” Am. J. Phys. 75, 36–42 (2007).
[Crossref]

Appl. Opt. (1)

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]

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)

B. Boubaha, A. Bencheikh, and K. At-Ameur, “Spatial properties of rectified cosine Gaussian beams,” J. Opt. 16, 025701 (2014).
[Crossref]

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

J. Opt. Soc. Am. B (1)

Laser Part. Beams (1)

H. Kumar, M. Aggarwal, D. Sharma, S. Chandok, and T. S. Gill, “Significant enhancement in the propagation of cosh-Gaussian laser beam in a relativistic–ponderomotive plasma using ramp density profile,” Laser Part. Beams 36, 179–185 (2018).
[Crossref]

Laser Photon. Rev. (4)

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: ‘non-diffracting’ beams,” Laser Photon. Rev. 4, 529–547 (2010).
[Crossref]

M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. 6, 607–621 (2012).
[Crossref]

W. Yu, Z. Ji, D. Dong, X. Yang, Y. Xiao, Q. Gong, P. Xi, and K. Shi, “Super-resolution deep imaging with hollow Bessel beam STED microscopy,” Laser Photon. Rev. 10, 147–152 (2016).
[Crossref]

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photon. Rev. 13, 1900103 (2019).
[Crossref]

Nat. Commun. (1)

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).
[Crossref]

Opt. Commun. (1)

D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite–cosine–Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[Crossref]

Opt. Eng. (1)

C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

M. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

Opt. Lett. (5)

Opt. Photon. News (1)

A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photon. News 24(6), 22–29 (2013).
[Crossref]

Optik (1)

S. Wang, Q. Lin, and X. Lu, “Realization of super-diffraction limiting propagation by cos beams,” Optik 100, 8–10 (1995).

Phys. Lett. A (1)

Q. Jiang, Y. Su, H. Nie, Z. Ma, and Y. Li, “Propagation and interaction of cos-Gaussian beams in photorefractive crystals,” Phys. Lett. A 381, 2246–2249 (2017).
[Crossref]

Phys. Rev. A (1)

J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Structured light beams constituted of incoming and outgoing waves,” Phys. Rev. A 100, 053847 (2019).
[Crossref]

Phys. Rev. Lett. (2)

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

J. Durnin, J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[Crossref]

Sci. Rep. (3)

D. G. Pires, A. F. Sonsin, A. J. Jesus-Silva, and E. J. Fonseca, “Three-dimensional speckle light self-healing-based imaging system,” Sci. Rep. 8, 1 (2018).
[Crossref]

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
[Crossref]

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimetre-wave communication links,” Sci. Rep. 6, 22082 (2016).
[Crossref]

Other (3)

A. Forbes, Laser beam Propagation: Generation and Propagation of Customized Light (CRC Press, 2014).

W. Shaomin, L. Qiaug, and L. Xuanhui, “Super-diffraction limiting propagation of cos beams,” Appl. Laser2 (1994).

N. Lebedev, Special Functions and Their Applications (Courier, 1972).

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Figures (9)

Fig. 1.
Fig. 1. (a) Intensity pattern of an infinite cosine beam and (b) its equivalent ${{\rm HG}_{10}}$ beam. The cross-sections of these intensities profiles are shown in (c).
Fig. 2.
Fig. 2. Two FoMs as a function of the truncation parameter $\beta$ for a cosine and equivalent HG beam both of order $n = 10$.
Fig. 3.
Fig. 3. FoMs as function of $\beta$ for $n = 10$ and $n = 40$ showing (a) the similarity factor and (b) the power content ratio factor.
Fig. 4.
Fig. 4. Intensity profiles for even $(n = 10)$ and odd $(n = 21)$ beams: (a) truncated $H{G_{10}}$ and its equivalent cosine beam with $\beta = 0.32$, (b) same beams with $\beta = 0.55$, and similarly in (c) and (d) for $n = 21$. For a truncation parameter $\beta = 1$, the degree of similarity is low ($S \lt 0.25$), resulting poor visual similarity, shown in (e) and (f) for $n = 10$ and $n = 21$, respectively.
Fig. 5.
Fig. 5. Propagation behavior on the $y^\prime - z$ plane of truncated HG beams (left) and their equivalent cosine beams (right) for $\beta = 0.32$ with (a) and (b) $n = 10$, (c) and (d) $n = 21$, and (e) and (f) $n = 40$. Skewed dashed lines show the formation of nondiffracting structure due to the interference of two oblique plane waves, the transverse dashed lines show the range of resulting nondiffracting beams.
Fig. 6.
Fig. 6. Propagation behavior on the $y^\prime - z$ plane of (a) the truncated CG beam and (b) its equivalent HG beam for $\beta = 1$.
Fig. 7.
Fig. 7. Density plot and profiles for ${{\rm HG}_{10}}$ beam and its equivalent cos beam for a given truncation parameter $\beta = 0.55$. Propagation in the $y^\prime - z$ plane at (a) and (b), respectively. (c) and (d) Their corresponding transverse patterns at the $z = {z_R}$ plane. (e) and (f) Their intensity profiles at two planes, $z = 0.2{z_R}$ and $z = {z_R}$, respectively.
Fig. 8.
Fig. 8. On-axis intensity of the truncated cosine beam and ${{\rm HG}_{10}}$ beams for (a) $\beta = 0.32$ and (b) $\beta = 0.55$.
Fig. 9.
Fig. 9. 2D non-diffracting truncated HG beams with (a) ${{\rm HG}_{10,10}}$ and (b) its equivalent cosine beam.

Equations (7)

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

E H G ( y , z ) = A 0 w 0 w y ( z ) H m ( 2 y w y ) × exp ( y 2 w y 2 i k y 2 2 R y i k z + i ϕ ( z ) ) ,
E cos ( y , z ) = exp ( i k z z ) cos ( k y y ) ,
H n ( y ) 2 ( n + 1 ) / 2 n n / 2 e n / 2 e y 2 / 2 cos ( 2 n + 1 y n π 2 ) .
H G n ( y , z ) = H n ( 2 y w ( z ) ) exp ( y 2 w ( z ) 2 ) .
E o u t ( y , z ) = 1 i λ z a + a E i n ( y , z = 0 ) exp [ i π λ z ( y y ) 2 ] d y ,
S = β + β | E cos E H G | d y β + β | E cos | 2 d y β + β | E H G | 2 d y ,
η = β + β | E cos | 2 d y β + β | E H G | 2 d y .

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