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.

© 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]
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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)

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]

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]

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]

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, 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]

2015 (3)

2014 (1)

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)

2009 (1)

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)

1990 (1)

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.

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]

Beresna, M.

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]

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.

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.

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]

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.

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]

Chen, P.

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]

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]

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]

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]

Chen, X.

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]

Chigrinov, V.

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]

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]

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.

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]

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]

Cottrell, D. M.

Cui, G.-X.

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]

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.

De Sio, L.

Devlin, R. C.

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]

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.

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]

Duan, W.

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]

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]

Escuti, M. J.

Faerman, A.

Feng, S.

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]

Forbes, A.

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.

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]

Gao, W.

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]

García-Martínez, P.

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]

Ge, S.-J.

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]

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]

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]

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]

Gecevicius, M.

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]

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.

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]

Gertus, T.

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]

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.

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]

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, 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]

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]

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]

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]

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.

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]

Kazansky, P. G.

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]

Kim, J.

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.

Lei, T.

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]

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]

Li, Q.

Li, T.

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]

Li, Y.

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. 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.

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]

Liao, Z.

Lin, J.

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]

Liu, G. N.

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]

Liu, L.

Lu, Y.-Q.

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]

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]

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]

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]

Ma, L.-L.

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]

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]

Maguid, E.

Mait, J. N.

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.

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]

Miskiewicz, M. N.

Mitry, M. J.

Moreno, I.

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]

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]

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]

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]

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]

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]

Nersisyan, S.

Niu, H.

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]

Oh, C.

Olivieri, D.

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]

Paparo, D.

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]

Pascoguin, B. M. L.

Ritsch-Marte, M.

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]

Roberts, D. E.

Romero, L. A.

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

Rosales-Guzmán, C.

Rubin, N. A.

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]

Salgueiro, J. R.

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]

Sánchez-López, M. M.

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]

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]

Santarsiero, 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]

Song, J.-H.

Steeves, D. M.

Tabiryan, N.

Tang, M.-J.

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]

Torres-Ruíz, F. A.

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]

Uskova, O.

Vargas, A.

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]

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]

Vicalvi, S.

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]

Wang, W.

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]

Wei, B.-Y.

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]

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]

Wen, D.

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]

Weng, X.

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]

Wickboldt, L.

Xie, Z.

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]

Xu, F.

Xu, R.

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]

Xu, X.

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]

Yannai, M.

Yu, C.

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]

Yuan, X.

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]

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]

Yuan, X.-C.

Yuan, Y.

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]

Yue, F.

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]

Zhang, C.

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]

Zhang, M.

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]

Zhang, N.

Zhang, S.

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]

Zhang, Y.

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]

Zhou, C.

Zhu, Z.-H.

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]

ACS Photonics (1)

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]

Adv. Mater. (2)

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]

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]

Am. J. Phys. (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]

Appl. Opt. (2)

Appl. Phys. Lett. (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]

Eur. J. Phys. (1)

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]

IEEE Photonics Technol. Lett. (1)

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]

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

Light Sci. Appl. (1)

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]

Liq. Cryst. (1)

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

New J. Phys. (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]

Opt. Acta (Lond.) (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]

Opt. Commun. (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]

Opt. Express (3)

Opt. Lasers Eng. (1)

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]

Opt. Lett. (2)

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)

Photon. Res. (1)

Phys. Rev. Appl. (1)

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]

Phys. Rev. Lett. (2)

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]

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)

https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=9098

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Figures (7)

Fig. 1
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.
Fig. 2
Fig. 2 Scheme showing the design procedure to achieve the 3x3 vortex grating design with optimal efficiency.
Fig. 3
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 (nx,ny).
Fig. 4
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.
Fig. 5
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.
Fig. 6
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.
Fig. 7
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.

Equations (32)

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ϕ(x)=arctan[ acos( 2πx p ) ],
τ trip ( x )=exp[ iϕ( x ) ]= + τ n exp( in 2πx p ) ,
τ n = 1 p p/2 p/2 exp[ iϕ( x ) ] exp( in 2πx p )dx.
τ trip ( x )=exp[ iϕ( x ) ] τ 0 + τ +1 exp( i 2πx p )+ τ 1 exp( i 2πx p ).
τ +1 = τ 1 =i τ 0 .
τ trip ( x )=LUT[ τ linear ( x ) ],
τ fork ( x,θ )=exp[ i( 2πx p +lθ ) ].
τ tripfork ( x,θ )=LUT[ τ fork ( x,θ ) ]= n= + τ n exp[ in( 2πx p +lθ ) ].
τ tripfork ( x,θ )= τ 0 { 1+iexp[ i( 2πx p +lθ ) ]+iexp[ i( 2πx p +lθ ) ] }.
τ tripfork ( x,θ )= τ 0 [ 1+2i cos( 2πx p +lθ ) ],
τ 2Dtripfork ( x,y,θ )= τ 0 2 [ 1+2i cos( 2πx p +θ ) ][ 1+2i cos( 2πy p +3θ ) ],
τ 2Dtripfork ( x,y,θ ) τ 0 2 =1+2i cos( 2πx p +θ )2cos( 2π( yx ) p +2θ )+ 2i cos( 2πy p +3θ )2cos( 2π( y+x ) p +4θ )
l( n x , n y )= n x +3 n y ,
M=( cos2α sin2α sin2α cos2α )
M= 1 2 ( A e iϕ + A e iϕ ),
A=( 1 i i 1 ).
M( τ )=( 1 0 0 1 )Re( τ )+( 0 1 1 0 )Im( τ ).
M=( 1 0 0 1 )×[ 12cos( 2π(yx) p +2θ )2cos( 2π(y+x) p +4θ ) ]+ ( 0 1 1 0 )×[ 2cos( 2πx p +θ )+2cos( 2πy p +3θ ) ]
τ 0 = 1 p p/2 p/2 exp[ iϕ( x ) ]dx= 1 p p/2 p/2 exp[ arctan( acos( 2πx p ) ) ]dx.
τ 0 = 1 2π π π exp[ arctan( acosδ ) ]dδ,
τ 0 = 1 2π π π cos[ arctan( acosδ ) ]dδ+ i 2π π π sin[ arctan( acosδ ) ]dδ.
cos( arctanφ )= 1 1+ φ 2 ,
sin( arctanφ )= φ 1+ φ 2 ,
τ 0 = 1 2π π π 1 1+ a 2 cos 2 δ dδ+ i 2π π π acosδ 1+ a 2 cos 2 δ dδ,
τ 0 = 1 π 0 π 1 1+ a 2 cos 2 δ dδ+ i π 0 π acosδ 1+ a 2 cos 2 δ dδ,
τ 0 = 1 π 0 π 1 1+ a 2 cos 2 δ dδ = 2 π K( a 2 ),
τ ±1 = 1 p p/2 p/2 exp[ arctan( acos( 2πx p ) ) ]exp( ±i 2πx p )dx,
τ ±1 = 1 2π π π exp[ arctan( acosδ ) ]exp( ±iδ )dδ.
τ ±1 = 1 2π π π cos( arctan( acosδ ) )cosδdδ+  1 2π π π sin( arctan( acosδ ) )sinδdδ+ ± i 2π π π cos( arctan( acosδ ) )sinδdδ+ + i 2π π π sin( arctan( acosδ ) )cosδdδ,
τ ±1 = 1 2π π π cosδ 1+ a 2 cos 2 δ dδ  1 2π π π acosδsinδ 1+ a 2 cos 2 δ dδ+ ±  i 2π π π sinδ 1+ a 2 cos 2 δ dδ+ i 2π π π a cos 2 δ 1+ a 2 cos 2 δ dδ.
τ ±1 =  1 π 0 π cosδ 1+ a 2 cos 2 δ + i π 0 π a cos 2 δ 1+ a 2 cos 2 δ dδ.
τ ±1 = ia π 0 π cos 2 δ 1+ a 2 cos 2 δ dδ=i 2 πa [ E( a 2 )K( a 2 ) ],

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