Optimal triplicator design applied to a geometric phase vortex grating

David Marco; María M. Sánchez-López; Aarón Cofré; Asticio Vargas; Ignacio Moreno

doi:10.1364/OE.27.014472

Optimal triplicator design applied to a geometric phase vortex grating

David Marco, María M. Sánchez-López, Aarón Cofré, Asticio Vargas, and Ignacio Moreno

Optics Express
Vol. 27,
Issue 10,
pp. 14472-14486
(2019)
•https://doi.org/10.1364/OE.27.014472

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David Marco, María M. Sánchez-López, Aarón Cofré, Asticio Vargas, and Ignacio Moreno, "Optimal triplicator design applied to a geometric phase vortex grating," Opt. Express 27, 14472-14486 (2019)

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Table of Contents Category
- Fourier Optics, Image and Signal Processing
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: February 15, 2019
- Revised Manuscript: March 15, 2019
- Manuscript Accepted: March 15, 2019
- Published: May 6, 2019
Virtual Issues
Optics Express Liquid Crystals beyond Displays (2019)

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Open Access

Abstract

In this work, a geometric phase liquid-crystal diffraction grating based on the optimal triplicator design is realized, i.e., a phase-only profile that generates three diffraction orders with equal intensity and maximum diffraction efficiency. We analyze the polarization properties of this special diffraction grating and then use embedded spiral phases to design geometric phase vortex diffraction gratings. Finally, the fabrication of a two-dimensional version of such a design using a micro-patterned half-wave retarder is demonstrated, where the phase distribution is encoded as the orientation of the fast axis of the retarder. This proof-of-concept element is made of liquid crystal on BK7 substrate where the orientation of the LC is controlled via photoalignment, using a commercially available fabrication facility. Experimental results demonstrate the parallel generation of vortex beams with different topological charge and different states of polarization.

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References

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Publication

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]
J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7(8), 1514–1518 (1990).
[Crossref]
F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157(1-6), 13–16 (1998).
[Crossref]
F. Aroca and I. Moreno, “Comparison and experimental realization of different phase-only grating designs and optimal triplicators,” Opt. Pura Apl. 49(3), 155–166 (2016).
[Crossref]
A. Cofré, P. García-Martínez, A. Vargas, and I. Moreno, “Vortex beam generation and other advanced optics experiments reproduced with a twisted-nematic liquid-crystal display with limited phase modulation,” Eur. J. Phys. 38(1), 014005 (2017).
[Crossref]
L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[Crossref]
A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76(10), 916–921 (2008).
[Crossref]
I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X.-C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[Crossref] [PubMed]
T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]
J. A. Davis, D. M. Cottrell, K. R. McCormick, J. Albero, and I. Moreno, “Arithmetic of focused vortex beams in three-dimensional optical lattice arrays,” Appl. Opt. 53(10), 2040–2050 (2014).
[Crossref] [PubMed]
C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]
I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]
C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25(21), 25697–25706 (2017).
[Crossref] [PubMed]
F. Yue, D. Wen, C. Zhang, B. D. Gerardot, W. Wang, S. Zhang, and X. Chen, “Multichannel polarization-controllable superpositions of orbital angular momentum states,” Adv. Mater. 29(15), 1603838 (2017).
[Crossref] [PubMed]
J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]
L. De Sio, D. E. Roberts, Z. Liao, S. Nersisyan, O. Uskova, L. Wickboldt, N. Tabiryan, D. M. Steeves, and B. R. Kimball, “Digital polarization holography advancing geometrical phase optics,” Opt. Express 24(16), 18297–18306 (2016).
[Crossref] [PubMed]
P. Chen, Y.-Q. Lu, and W. Hu, “Beam shaping via photopatterned liquid crystals,” Liq. Cryst. 43(13–15), 2051–2061 (2016).
[Crossref]
P. Chen, B.-Y. Wei, W. Ji, S.-J. Ge, W. Hu, F. Xu, V. Chigrinov, and Y.-Q. Lu, “Arbitrary and reconfigurable optical vortex generation: A high-efficiency technique using director-varying liquid crystal fork gratings,” Photon. Res. 3(4), 133–139 (2015).
[Crossref]
P. Chen, L.-L. Ma, W. Duan, J. Chen, S.-J. Ge, Z.-H. Zhu, M.-J. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y.-Q. Lu, “Digitalizing self-assembled chiral superstructures for optical vortex processing,” Adv. Mater. 30(10), 1705865 (2018).
[Crossref] [PubMed]
J. P. Balthasar Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref] [PubMed]
M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[Crossref]
A. Cofré, A. Vargas, F. A. Torres-Ruíz, M. M. Sánchez-López, and I. Moreno, “Geometrical-phase lens based optical system for the spin-splitting of vector beams,” Opt. Lasers Eng. 110, 401–409 (2018).
[Crossref]
https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=9098
M. Yannai, E. Maguid, A. Faerman, Q. Li, J.-H. Song, V. Kleiner, M. L. Brongersma, and E. Hasman, “Spectrally interleaved topologies using geometric phase metasurfaces,” Opt. Express 26(23), 31031–31038 (2018).
[Crossref] [PubMed]
C. Zhou and L. Liu, “Numerical study of Dammann array illuminators,” Appl. Opt. 34(26), 5961–5969 (1995).
[Crossref] [PubMed]
I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[Crossref] [PubMed]
P. Chen, S.-J. Ge, L.-L. Ma, W. Hu, V. Chigrinov, and Y.-Q. Lu, “Generation of equal-energy orbital angular momentum beams via photopatterned liquid crystals,” Phys. Rev. Appl. 5(4), 044009 (2016).
[Crossref]
Z. Xie, T. Lei, X. Weng, L. Du, S. Gao, Y. Yuan, S. Feng, Y. Zhang, and X. Yuan, “A miniaturized polymer grating for topological order detection of cylindrical vector beams,” IEEE Photonics Technol. Lett. 28(24), 2799–2802 (2016).
[Crossref]
P. Chen, S.-J. Ge, W. Duan, B.-Y. Wei, G.-X. Cui, W. Hu, and Y.-Q. Lu, “Digitalized geometric phases for parallel optical spin and orbital angular momentum encoding,” ACS Photonics 4(6), 1333–1338 (2017).
[Crossref]
L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

2018 (3)

P. Chen, L.-L. Ma, W. Duan, J. Chen, S.-J. Ge, Z.-H. Zhu, M.-J. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y.-Q. Lu, “Digitalizing self-assembled chiral superstructures for optical vortex processing,” Adv. Mater. 30(10), 1705865 (2018).
[Crossref] [PubMed]

A. Cofré, A. Vargas, F. A. Torres-Ruíz, M. M. Sánchez-López, and I. Moreno, “Geometrical-phase lens based optical system for the spin-splitting of vector beams,” Opt. Lasers Eng. 110, 401–409 (2018).
[Crossref]

M. Yannai, E. Maguid, A. Faerman, Q. Li, J.-H. Song, V. Kleiner, M. L. Brongersma, and E. Hasman, “Spectrally interleaved topologies using geometric phase metasurfaces,” Opt. Express 26(23), 31031–31038 (2018).
[Crossref] [PubMed]

2017 (6)

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25(21), 25697–25706 (2017).
[Crossref] [PubMed]

P. Chen, S.-J. Ge, W. Duan, B.-Y. Wei, G.-X. Cui, W. Hu, and Y.-Q. Lu, “Digitalized geometric phases for parallel optical spin and orbital angular momentum encoding,” ACS Photonics 4(6), 1333–1338 (2017).
[Crossref]

J. P. Balthasar Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref] [PubMed]

I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]

F. Yue, D. Wen, C. Zhang, B. D. Gerardot, W. Wang, S. Zhang, and X. Chen, “Multichannel polarization-controllable superpositions of orbital angular momentum states,” Adv. Mater. 29(15), 1603838 (2017).
[Crossref] [PubMed]

A. Cofré, P. García-Martínez, A. Vargas, and I. Moreno, “Vortex beam generation and other advanced optics experiments reproduced with a twisted-nematic liquid-crystal display with limited phase modulation,” Eur. J. Phys. 38(1), 014005 (2017).
[Crossref]

2016 (5)

F. Aroca and I. Moreno, “Comparison and experimental realization of different phase-only grating designs and optimal triplicators,” Opt. Pura Apl. 49(3), 155–166 (2016).
[Crossref]

P. Chen, Y.-Q. Lu, and W. Hu, “Beam shaping via photopatterned liquid crystals,” Liq. Cryst. 43(13–15), 2051–2061 (2016).
[Crossref]

P. Chen, S.-J. Ge, L.-L. Ma, W. Hu, V. Chigrinov, and Y.-Q. Lu, “Generation of equal-energy orbital angular momentum beams via photopatterned liquid crystals,” Phys. Rev. Appl. 5(4), 044009 (2016).
[Crossref]

Z. Xie, T. Lei, X. Weng, L. Du, S. Gao, Y. Yuan, S. Feng, Y. Zhang, and X. Yuan, “A miniaturized polymer grating for topological order detection of cylindrical vector beams,” IEEE Photonics Technol. Lett. 28(24), 2799–2802 (2016).
[Crossref]

L. De Sio, D. E. Roberts, Z. Liao, S. Nersisyan, O. Uskova, L. Wickboldt, N. Tabiryan, D. M. Steeves, and B. R. Kimball, “Digital polarization holography advancing geometrical phase optics,” Opt. Express 24(16), 18297–18306 (2016).
[Crossref] [PubMed]

2015 (3)

P. Chen, B.-Y. Wei, W. Ji, S.-J. Ge, W. Hu, F. Xu, V. Chigrinov, and Y.-Q. Lu, “Arbitrary and reconfigurable optical vortex generation: A high-efficiency technique using director-varying liquid crystal fork gratings,” Photon. Res. 3(4), 133–139 (2015).
[Crossref]

J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl. 4(3), e257 (2015).
[Crossref]

2014 (1)

J. A. Davis, D. M. Cottrell, K. R. McCormick, J. Albero, and I. Moreno, “Arithmetic of focused vortex beams in three-dimensional optical lattice arrays,” Appl. Opt. 53(10), 2040–2050 (2014).
[Crossref] [PubMed]

2011 (1)

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011).
[Crossref]

2010 (2)

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[Crossref]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X.-C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[Crossref] [PubMed]

2009 (1)

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[Crossref] [PubMed]

2008 (1)

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76(10), 916–921 (2008).
[Crossref]

2007 (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

1998 (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157(1-6), 13–16 (1998).
[Crossref]

1995 (1)

C. Zhou and L. Liu, “Numerical study of Dammann array illuminators,” Appl. Opt. 34(26), 5961–5969 (1995).
[Crossref] [PubMed]

1990 (1)

J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7(8), 1514–1518 (1990).
[Crossref]

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Albero, J.

Aroca, F.

F. Aroca and I. Moreno, “Comparison and experimental realization of different phase-only grating designs and optimal triplicators,” Opt. Pura Apl. 49(3), 155–166 (2016).
[Crossref]

Badham, K.

I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]

Balthasar Mueller, J. P.

Beresna, M.

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Bhebhe, N.

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25(21), 25697–25706 (2017).
[Crossref] [PubMed]

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157(1-6), 13–16 (1998).
[Crossref]

Brongersma, M. L.

Capasso, F.

Carpentier, A. V.

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76(10), 916–921 (2008).
[Crossref]

Chen, J.

Chen, P.

P. Chen, Y.-Q. Lu, and W. Hu, “Beam shaping via photopatterned liquid crystals,” Liq. Cryst. 43(13–15), 2051–2061 (2016).
[Crossref]

Chen, X.

Chigrinov, V.

Cincotti, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157(1-6), 13–16 (1998).
[Crossref]

Cofré, A.

Cottrell, D. M.

I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X.-C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[Crossref] [PubMed]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[Crossref] [PubMed]

Cui, G.-X.

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Davis, J. A.

I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X.-C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[Crossref] [PubMed]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[Crossref] [PubMed]

De Sio, L.

Devlin, R. C.

di Fabrizio, E.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157(1-6), 13–16 (1998).
[Crossref]

Dickey, F. M.

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[Crossref]

Du, L.

Duan, W.

Escuti, M. J.

J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]

Faerman, A.

Feng, S.

Forbes, A.

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25(21), 25697–25706 (2017).
[Crossref] [PubMed]

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Gao, S.

Gao, W.

García-Martínez, P.

Ge, S.-J.

Gecevicius, M.

Gentili, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157(1-6), 13–16 (1998).
[Crossref]

Gerardot, B. D.

Gertus, T.

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157(1-6), 13–16 (1998).
[Crossref]

Groever, B.

Hasman, E.

Holland, J. E.

I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]

Hu, W.

P. Chen, Y.-Q. Lu, and W. Hu, “Beam shaping via photopatterned liquid crystals,” Liq. Cryst. 43(13–15), 2051–2061 (2016).
[Crossref]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Ji, W.

Jia, P.

Kazansky, P. G.

Kim, J.

J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]

Kimball, B. R.

Kleiner, V.

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Kudenov, M. W.

J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]

Lei, T.

Li, Q.

Li, T.

Li, Y.

J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]

Li, Z.

Liao, Z.

Lin, J.

Liu, G. N.

Liu, L.

C. Zhou and L. Liu, “Numerical study of Dammann array illuminators,” Appl. Opt. 34(26), 5961–5969 (1995).
[Crossref] [PubMed]

Lu, Y.-Q.

P. Chen, Y.-Q. Lu, and W. Hu, “Beam shaping via photopatterned liquid crystals,” Liq. Cryst. 43(13–15), 2051–2061 (2016).
[Crossref]

Ma, L.-L.

Maguid, E.

Mait, J. N.

J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7(8), 1514–1518 (1990).
[Crossref]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

McCormick, K. R.

Michinel, H.

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76(10), 916–921 (2008).
[Crossref]

Min, C.

Miskiewicz, M. N.

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[Crossref]

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I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[Crossref] [PubMed]

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I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]

F. Aroca and I. Moreno, “Comparison and experimental realization of different phase-only grating designs and optimal triplicators,” Opt. Pura Apl. 49(3), 155–166 (2016).
[Crossref]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X.-C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[Crossref] [PubMed]

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[Crossref] [PubMed]

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J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]

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[Crossref]

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L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref]

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[Crossref]

Opt. Express (3)

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25(21), 25697–25706 (2017).
[Crossref] [PubMed]

Opt. Lasers Eng. (1)

Opt. Lett. (2)

I. Moreno, J. A. Davis, B. M. L. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009).
[Crossref] [PubMed]

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X.-C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35(10), 1536–1538 (2010).
[Crossref] [PubMed]

Opt. Pura Apl. (1)

F. Aroca and I. Moreno, “Comparison and experimental realization of different phase-only grating designs and optimal triplicators,” Opt. Pura Apl. 49(3), 155–166 (2016).
[Crossref]

Optica (1)

J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
[Crossref]

Photon. Res. (1)

Phys. Rev. Appl. (1)

Phys. Rev. Lett. (2)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

Prog. Opt. (1)

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[Crossref]

Sci. Rep. (1)

I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. 7(1), 2216 (2017).
[Crossref] [PubMed]

Other (1)

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Fig. 1 (a) Gori’s triplicator phase profile compared to the linear blaze grating profile. (b) Gray level image to generate a triplicator in a phase-only SLM. (c) Schematics of the triplication grating action.

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Fig. 2 Scheme showing the design procedure to achieve the 3x3 vortex grating design with optimal efficiency.

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Fig. 3 Scheme of the expected array of diffraction orders and the corresponding topological charges. The pair of numbers on each diffraction order denote the order (n_x,n_y).

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Fig. 4 (a) Illustration of the phase grating encoding as a geometric phase element. (b) Picture of the grating between crossed polarizers. (c) Detail of the center of the grating viewed through a polarizing microscope between crossed polarizers. (d) Fourier transform pattern, saturated on purpose in order to visualize how the energy is basically confined to the 3 × 3 target orders.

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Fig. 5 Scheme of the experimental setup. PSG: polarization state generator; PSA: polarization state analyzer; Q: quarter-wave retarder; L: linear polarizer; L₁, L₂ converging lenses.

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Fig. 6 Experimental results of the Fourier plane for six different configurations of the PSG without analyzer (left column), and with six different configurations of the PSA.

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Fig. 7 Experimental results at the Fourier plane for the 2D vortex grating illuminated with two circularly polarized vortex beams of charges (a) + 1 (b) −1. No PSA is included. The yellow arrow indicates the position of the bright spot, revealing the input charge.

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Equations (32)

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ϕ (x) = arc \tan [a \cos (\frac{2 π x}{p})],

τ_{t r i p} (x) = \exp [i ϕ (x)] = \sum_{- \infty}^{+ \infty} τ_{n} \exp (i n \frac{2 π x}{p}),

τ_{n} = \frac{1}{p} \int_{- p / 2}^{p / 2} \exp [i ϕ (x)] \exp (- i n \frac{2 π x}{p}) d x .

τ_{t r i p} (x) = \exp [i ϕ (x)] ≅ τ_{0} + τ_{+ 1} \exp (i \frac{2 π x}{p}) + τ_{- 1} \exp (- i \frac{2 π x}{p}) .

τ_{+ 1} = τ_{- 1} = i τ_{0} .

τ_{t r i p} (x) = L U T [τ_{l i n e a r} (x)],

τ_{f o r k} (x, θ) =exp [i (\frac{2 π x}{p} + l θ)] .

τ_{t r i p - f o r k} (x, θ) = L U T [τ_{f o r k} (x, θ)] = \sum_{n = - \infty}^{+ \infty} τ_{n} exp [i n (\frac{2 π x}{p} + l θ)] .

τ_{t r i p - f o r k} (x, θ) = τ_{0} {1 + i \exp [i (\frac{2 π x}{p} + l θ)] + i \exp [- i (\frac{2 π x}{p} + l θ)]} .

τ_{t r i p - f o r k} (x, θ) = τ_{0} [1 + 2 i \cos (\frac{2 π x}{p} + l θ)],

τ_{2 D t r i p - f o r k} (x, y, θ) = τ_{0}^{2} [1 + 2 i \cos (\frac{2 π x}{p} + θ)] [1 + 2 i \cos (\frac{2 π y}{p} + 3 θ)],

\begin{array}{l} \frac{τ_{2 D t r i p - f o r k} (x, y, θ)}{τ_{0}^{2}} = 1 + 2 i \cos (\frac{2 π x}{p} + θ) - 2 \cos (\frac{2 π (y - x)}{p} + 2 θ) + \\ 2 i \cos (\frac{2 π y}{p} + 3 θ) - 2 \cos (\frac{2 π (y + x)}{p} + 4 θ) \end{array}

l (n_{x}, n_{y}) = n_{x} + 3 n_{y},

M = (\begin{matrix} \cos 2 α & \sin 2 α \\ \sin 2 α & - \cos 2 α \end{matrix})

M = \frac{1}{2} (A e^{i ϕ} + A^{†} e^{- i ϕ}),

A = (\begin{matrix} 1 & - i \\ - i & - 1 \end{matrix}) .

M (τ) = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) R e (τ) + (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) Im (τ) .

\begin{matrix} M = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) \times [1 - 2 \cos (\frac{2 π (y - x)}{p} + 2 θ) - 2 \cos (\frac{2 π (y + x)}{p} + 4 θ)] + \\ (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) \times [2 \cos (\frac{2 π x}{p} + θ) + 2 \cos (\frac{2 π y}{p} + 3 θ)] \end{matrix}

τ_{0} = \frac{1}{p} \int_{- p / 2}^{p / 2} exp [i ϕ (x)] d x = \frac{1}{p} \int_{- p / 2}^{p / 2} exp [arc \tan (a \cos (\frac{2 π x}{p}))] d x .

τ_{0} = \frac{1}{2 π} \int_{- π}^{π} exp [arc \tan (a \cos δ)] d δ,

τ_{0} = \frac{1}{2 π} \int_{- π}^{π} \cos [arc \tan (a \cos δ)] d δ + \frac{i}{2 π} \int_{- π}^{π} \sin [arc \tan (a \cos δ)] d δ .

\cos (arc \tan φ) = \frac{1}{\sqrt{1 + φ^{2}}},

\sin (arc \tan φ) = \frac{φ}{\sqrt{1 + φ^{2}}},

τ_{0} = \frac{1}{2 π} \int_{- π}^{π} \frac{1}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ + \frac{i}{2 π} \int_{- π}^{π} \frac{a \cos δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ,

τ_{0} = \frac{1}{π} \int_{0}^{π} \frac{1}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ + \frac{i}{π} \int_{0}^{π} \frac{a \cos δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ,

τ_{0} = \frac{1}{π} \int_{0}^{π} \frac{1}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ = \frac{2}{π} K (- a^{2}),

τ_{\pm 1} = \frac{1}{p} \int_{- p / 2}^{p / 2} exp [arc \tan (a \cos (\frac{2 π x}{p}))] exp (\pm i \frac{2 π x}{p}) d x,

τ_{\pm 1} = \frac{1}{2 π} \int_{- π}^{π} exp [arc \tan (a \cos δ)] exp (\pm i δ) d δ .

\begin{matrix} τ_{\pm 1} = \frac{1}{2 π} \int_{- π}^{π} cos (arc \tan (a \cos δ)) \cos δ d δ + \\ \mp \frac{1}{2 π} \int_{- π}^{π} sin (arc \tan (a \cos δ)) \sin δ d δ + \\ \pm \frac{i}{2 π} \int_{- π}^{π} cos (arc \tan (a \cos δ)) \sin δ d δ + \\ + \frac{i}{2 π} \int_{- π}^{π} sin (arc \tan (a \cos δ)) \cos δ d δ, \end{matrix}

\begin{matrix} τ_{\pm 1} = \frac{1}{2 π} \int_{- π}^{π} \frac{\cos δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ \mp \frac{1}{2 π} \int_{- π}^{π} \frac{a \cos δ \sin δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ + \\ \pm \frac{i}{2 π} \int_{- π}^{π} \frac{\sin δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ + \frac{i}{2 π} \int_{- π}^{π} \frac{a \cos^{2} δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ . \end{matrix}

τ_{\pm 1} = \frac{1}{π} \int_{0}^{π} \frac{\cos δ}{\sqrt{1 + a^{2} \cos^{2} δ}} + \frac{i}{π} \int_{0}^{π} \frac{a \cos^{2} δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ .

τ_{\pm 1} = \frac{i a}{π} \int_{0}^{π} \frac{\cos^{2} δ}{\sqrt{1 + a^{2} \cos^{2} δ}} d δ = i \frac{2}{π a} [E (- a^{2}) - K (- a^{2})],

María M. Sánchez-López	https://orcid.org/0000-0002-6286-0079
Asticio Vargas	https://orcid.org/0000-0002-2364-2351
Ignacio Moreno	https://orcid.org/0000-0002-1550-0601