Abstract

We employ a setup, based on a phase spatial light modulator (SLM), for generation of arbitrary optical fields. The process is based on two sequential phase modulations of a laser beam at two different zones of the SLM. The input beam is transformed in the first contact with the SLM by a phase modulation that includes a diffractive phase element (DPE), which encodes the desired complex field, and a Fourier transforming lens. The Fourier transform of the DPE is projected, using a mirror, to the second SLM modulation area, whose transmittance includes a second Fourier transforming lens and a phase spatial filter, used to eliminate non-desired parts of the DPE Fourier spectrum.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  13. V. Arrizón, D. Sánchez-de-La-Llave, and G. Méndez, “Holographic generation of a class of nondiffracting fields with optimum efficiency,” Opt. Lett. 37(11), 2154–2156 (2012).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  20. L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
    [Crossref]
  21. P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
    [Crossref]
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    [Crossref]
  23. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5 th ed. (Harcourt/Academic Press, 2001), p. 689.
  24. We employed the electrically addressable phase SLM, model “Pluto”(HOLOEYE Photonics AG), with format of 1080 × 1920 pixels and pixel size of 8 microns.

2016 (2)

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24(6), 6249–6264 (2016).
[Crossref]

2015 (3)

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

2014 (1)

2012 (1)

2011 (3)

V. Arrizón, D. Sánchez-de-La-Llave, G. Méndez, and U. Ruiz, “Efficient generation of periodic and quasi-periodic non-diffractive optical fields with phase holograms,” Opt. Express 19(11), 10553–10562 (2011).
[Crossref]

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011).
[Crossref]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

2010 (1)

2009 (3)

2008 (1)

2007 (1)

2000 (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[Crossref]

1999 (1)

1997 (1)

1994 (1)

1971 (1)

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ando, T.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5 th ed. (Harcourt/Academic Press, 2001), p. 689.

Arnold, A. S.

Arrizón, V.

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bernet, S.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011).
[Crossref]

A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16(7), 4479–4486 (2008).
[Crossref]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Burger, L.

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

Campos, J.

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Carrada, R.

Clark, T. W.

Cohn, R. W.

Coppola, G.

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

Cottrell, D. M.

Davis, J. A.

Di Caprio, G.

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

Dudley, A.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Ferraro, P.

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

Forbes, A.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

C. Rosales-Guzmán and A. Forbes, How to shape light with spatial light modulators (SPIE Press, 2017).

Franke-Arnold, S.

Fukuchi, N.

González, L. A.

Goto, H.

Hernández-Hernández, R. J.

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Inoue, T.

Itoh, K.

Jesacher, A.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011).
[Crossref]

A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16(7), 4479–4486 (2008).
[Crossref]

Jones, A. L.

Juskaitis, R.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[Crossref]

Kettunen, V.

Kirk, J. P.

Konishi, T.

Lancis, J.

Lavery, M. P. J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Li, L.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Liang, M.

Litvin, I.

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

Matsumoto, N.

Maurer, C.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011).
[Crossref]

A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16(7), 4479–4486 (2008).
[Crossref]

McLaren, M.

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

Memmolo, P.

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

Méndez, G.

Mendoza-Yero, O.

Miccio, L.

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

Mínguez-Vega, G.

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Moreno, I.

Neil, M. A. A.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[Crossref]

Netti, P. A.

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

Ngcobo, S.

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

Noponen, E.

Offer, R. F.

Ohtake, Y.

Padgett, M.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Paturzo, M.

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

Radwell, N.

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ren, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ricardez-Vargas, I.

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011).
[Crossref]

A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16(7), 4479–4486 (2008).
[Crossref]

Rosales-Guzmán, C.

C. Rosales-Guzmán and A. Forbes, How to shape light with spatial light modulators (SPIE Press, 2017).

Ruiz, U.

Sánchez-de-La-Llave, D.

Schwaighofer, A.

Terborg, R. A.

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Turunen, J.

Vahimaa, P.

Volke-Sepúlveda, K.

Wang, J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5 th ed. (Harcourt/Academic Press, 2001), p. 689.

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Wilson, T.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[Crossref]

Xie, G.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Yzuel, M. J.

Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Adv. Opt. Photonics (3)

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8(2), 200–227 (2016).
[Crossref]

P. Memmolo, L. Miccio, M. Paturzo, G. Di Caprio, G. Coppola, P. A. Netti, and P. Ferraro, “Recent advances in holographic 3D particle tracking,” Adv. Opt. Photonics 7(4), 713–755 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Appl. Opt. (3)

J. Microsc. (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000).
[Crossref]

J. Opt. (1)

L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

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

Laser Photonics Rev. (1)

C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “What spatial light modulators can do for optical microscopy,” Laser Photonics Rev. 5(1), 81–101 (2011).
[Crossref]

Nat. Photonics (1)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Other (3)

C. Rosales-Guzmán and A. Forbes, How to shape light with spatial light modulators (SPIE Press, 2017).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5 th ed. (Harcourt/Academic Press, 2001), p. 689.

We employed the electrically addressable phase SLM, model “Pluto”(HOLOEYE Photonics AG), with format of 1080 × 1920 pixels and pixel size of 8 microns.

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Figures (12)

Fig. 1.
Fig. 1. Optical setup for generation of arbitrary optical fields. The laser (L) illuminates the first DPE at the SLM (A) whose phase modulation includes the beam kinoform and the first Fourier transforming lens. The Fourier transform of the kinoform is projected (through the path A-M-B) to the second SLM modulation zone (B), which includes a spatial filter and the second Fourier transforming lens. This lens projects the generated beam to its focal plane, along the output axis.
Fig. 2.
Fig. 2. (a) Amplitude and (b) phase of a first order BG beam with radial period p = w0/3.5.
Fig. 3.
Fig. 3. Fourier spectra of (a) the first order BG beam under discussion and (b) its corresponding KG field. The amplitude of the approximate BG beam obtained from the bright ring in the KG field Fourier spectrum is displayed in (c).
Fig. 4.
Fig. 4. Amplitudes of PQP fields with Gaussian envelope, with parameters (a) (Q = 5, t = 0) and (b) (Q = 6, t = 1).
Fig. 5.
Fig. 5. Fourier spectra of (a) the PQP field with parameters (Q = 6, t = 1) and (b) its corresponding KG field. The amplitude of the approximate PQP field obtained from the bright spots in the KG field Fourier spectrum is displayed in (c).
Fig. 6.
Fig. 6. (a) Amplitude of the input Gaussian used in the simulations, and (b) phase modulation of the DPE displayed in the first modulation zone of the SLM, for generation of the first order BG beam under discussion.
Fig. 7.
Fig. 7. (a) Field propagated to the second SLM modulation zone, obtained by propagation of the Gaussian beam modulated by the DPE in Fig. 6(b), and (b) DPE phase modulation in the second SLM modulation zone, within the square area in (a).
Fig. 8.
Fig. 8. Numerical (top) and experimental (bottom) BG beams of orders n = 0 (left), n = 1 (center), and n = 2 (right), generated in the experimental setup of Fig. 1.
Fig. 9.
Fig. 9. Numerical (top) and experimental (bottom) PQP fields with parameters (Q = 5, t = 0) (left), (Q = 6, t = 0) (center), and (Q = 6, t = 1) (right) generated in the experimental setup of Fig. 1.
Fig. 10.
Fig. 10. Transverse amplitudes of the approximated BG beams generated by the beams’ kinoforms in the optical setup of Fig. 1(red trace). The period of kinoforms is p = w0/3.5 where w0 is the waist radius of the input Gaussian beam. The best fitting exact BG beams are displayed in blue trace. The beam orders are (a) n = 0, (b) n = 1, and (c) n = 2.
Fig. 11.
Fig. 11. Considering that the first order BG beam under discussion appears at the order M = 15 in the series for the KG field, we obtain: (a) the computed coefficients Aq [Eq. (7)], (b) the radial modulation G(ρ) of the KG field Fourier spectrum, and (c) the radial modulations for the Fourier spectra of the terms in the series with orders q = 14, 15, and 16.
Fig. 12.
Fig. 12. (a) Transverse amplitude of the KG field Fourier spectrum modified by the spatial filter, and (b) transverse amplitude of the approximated first order BG beam obtained by the Fourier transform of the filtered spectrum. For comparison, the transverse amplitude of the initially desired BG beam is displayed in (c).

Equations (17)

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b n ( r , θ ) = J n ( 2 π ρ 0 r ) e x p ( i n θ ) e x p ( r 2 / w 0 2 ) ,
k n ( r , θ ) = p h [ J n ( 2 π ρ 0 r ) ] e x p ( i n θ ) ,
k n ( r , θ ) = q = 1 A q J n ( 2 π β q r ) e x p ( i n θ ) ,
A q = 2 L 2 J n + 1 2 ( λ q ) 0 L r p h [ J n ( 2 π ρ 0 r ) ] J n ( 2 π β q r ) d r .
f n ( r , θ ) = q = 1 A q J n ( 2 π β q r ) e x p ( i n θ ) e x p ( r 2 / w 0 2 ) .
f n ( r , θ ) = q = 1 A q J n [ 2 π ( λ q / λ M ) ρ 0 r ] e x p ( i n θ ) e x p ( r 2 / w 0 2 )
A q = 2 L 2 J n + 1 2 ( λ q ) 0 L r p h [ J n ( 2 π ρ 0 r ) ] J n [ 2 π ( λ q / λ M ) ρ 0 r ] d r ,
F n ( ρ , ϕ ) = { q = 1 A q δ [ ρ ( λ q / λ M ) ρ 0 ] e x p ( i n ϕ ) } e x p ( π 2 w 0 2 ρ 2 ) ,
f ( r , θ ) = C q = 0 Q 1 e x p [ i t q ( 2 π / Q ) ] e x p { i 2 π ρ 0 r cos [ θ q ( 2 π / Q ) ] } ,
f G ( r , θ ) = f ( r , θ ) e x p ( r 2 / w 0 2 )
k G ( r , θ ) = e x p [ i ϕ ( r , θ ) ] e x p ( r 2 / w 0 2 ) .
f n ( r , θ ) = g ( r ) e x p ( i n θ )
f n q ( r , θ ) = g q ( r ) e x p ( i n θ ) ,
g ( r ) = p h [ J n ( 2 π ρ 0 r ) ] e x p ( r 2 / w 2 )
g q ( r ) = A q J n ( 2 π β q r ) e x p ( r 2 / w 2 )
F n ( ρ , ϕ ) = G ( ρ ) e x p ( i n ϕ )
F n q ( ρ , ϕ ) = G q ( ρ ) e x p ( i n ϕ )

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