Abstract

With the recent development of the metasurface, generating an optical vortex in optical far or near fields is realized in various ways. However, to generate vortices in both the near and far fields simultaneously is still a challenge, although it has great potential in the future compact and versatile photonic system. Here, a bi-channel optical vortex generator in both the near and far fields is proposed and demonstrated within a single metasurface, where the surface plasmon vortex and the far-field optical vortex can be simultaneously generated under circularly polarized light. The ability of generating vortices with arbitrary topological charges is experimentally demonstrated, which agrees well with simulations. This approach provides great freedom to integrate different vortex generators in a single device and offers new opportunities for integrated optical communications, trapping, and other related fields.

© 2020 Chinese Laser Press

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2018 (2)

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

J. Liu, S. M. Li, L. Zhu, A. D. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S. Y. Yu, X. L. Cai, and J. Wang, “Direct fiber vector eigenmode multiplexing transmission seeded by integrated optical vortex emitters,” Light Sci. Appl. 7, 17148 (2018).
[Crossref]

2017 (7)

S. Pidishety, S. Pachava, P. Gregg, S. Ramachandran, G. Brambilla, and B. Srinivasan, “Orbital angular momentum beam excitation using an all-fiber weakly fused mode selective coupler,” Opt. Lett. 42, 4347–4350 (2017).
[Crossref]

Y. J. Bao, S. Zu, W. Liu, L. Zhou, X. Zhu, and Z. Y. Fang, “Revealing the spin optics in conic-shaped metasurfaces,” Phys. Rev. B 95, 081406 (2017).
[Crossref]

A. Arbabi, E. Arbabi, Y. Horie, S. M. Kamali, and A. Faraon, “Planar metasurface retroreflector,” Nat. Photonics 11, 415–420 (2017).
[Crossref]

B. H. Chen, P. C. Wu, V. C. Su, Y. C. Lai, C. H. Chu, I. C. Lee, J. W. Chen, Y. H. Chen, Y. C. Lan, C. H. Kuan, and D. P. Tsai, “GaN metalens for pixel-level full-color routing at visible light,” Nano Lett. 17, 6345–6352 (2017).
[Crossref]

Y. Bao, Q. Jiang, Y. Kang, X. Zhu, and Z. Fang, “Enhanced optical performance of multifocal metalens with conic shapes,” Light Sci. Appl. 6, e17071 (2017).
[Crossref]

S. Wang, P. C. Wu, V. C. Su, Y. C. Lai, C. Hung Chu, J. W. Chen, S. H. Lu, J. Chen, B. Xu, C. H. Kuan, T. Li, S. Zhu, and D. P. Tsai, “Broadband achromatic optical metasurface devices,” Nat. Commun. 8, 187 (2017).
[Crossref]

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, 1603838 (2017).
[Crossref]

2016 (6)

M. Q. Mehmood, S. Mei, S. Hussain, K. Huang, S. Y. Siew, L. Zhang, T. Zhang, X. Ling, H. Liu, J. Teng, A. Danner, S. Zhang, and C. W. Qiu, “Visible-frequency metasurface for structuring and spatially multiplexing optical vortices,” Adv. Mater. 28, 2533–2539 (2016).
[Crossref]

F. Qin, L. Ding, L. Zhang, F. Monticone, C. C. Chum, J. Deng, S. Mei, Y. Li, J. Teng, M. Hong, S. Zhang, A. Alu, and C. W. Qiu, “Hybrid bilayer plasmonic metasurface efficiently manipulates visible light,” Sci. Adv. 2, e1501168 (2016).
[Crossref]

X. Li, L. Chen, Y. Li, X. Zhang, M. Pu, Z. Zhao, X. Ma, Y. Wang, M. Hong, and X. Luo, “Multicolor 3D meta-holography by broadband plasmonic modulation,” Sci. Adv. 2, e1601102 (2016).
[Crossref]

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

P. Vayalamkuzhi, S. Bhattacharya, U. Eigenthaler, K. Keskinbora, C. T. Samlan, M. Hirscher, J. P. Spatz, and N. K. Viswanathan, “Direct patterning of vortex generators on a fiber tip using a focused ion beam,” Opt. Lett. 41, 2133–2136 (2016).
[Crossref]

J. Jin, J. Luo, X. Zhang, H. Gao, X. Li, M. Pu, P. Gao, Z. Zhao, and X. Luo, “Generation and detection of orbital angular momentum via metasurface,” Sci. Rep. 6, 24286 (2016).
[Crossref]

2015 (4)

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. F. Li, P. W. Wong, K. W. Cheah, E. Y. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun. 6, 8241 (2015).
[Crossref]

B. Desiatov, N. Mazurski, Y. Fainman, and U. Levy, “Polarization selective beam shaping using nanoscale dielectric metasurfaces,” Opt. Express 23, 22611–22618 (2015).
[Crossref]

C. F. Chen, C. T. Ku, Y. H. Tai, P. K. Wei, H. N. Lin, and C. B. Huang, “Creating optical near-field orbital angular momentum in a gold metasurface,” Nano Lett. 15, 2746–2750 (2015).
[Crossref]

X. Ma, M. Pu, X. Li, C. Huang, Y. Wang, W. Pan, B. Zhao, J. Cui, C. Wang, Z. Zhao, and X. Luo, “A planar chiral meta-surface for optical vortex generation and focusing,” Sci. Rep. 5, 10365 (2015).
[Crossref]

2014 (6)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref]

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

W. Y. Tsai, J. S. Huang, and C. B. Huang, “Selective trapping or rotation of isotropic dielectric microparticles by optical near field in a plasmonic Archimedes spiral,” Nano Lett. 14, 547–552 (2014).
[Crossref]

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8, 234–238 (2014).
[Crossref]

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
[Crossref]

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref]

2013 (4)

J. Lin, J. P. Mueller, Q. Wang, G. Yuan, N. Antoniou, X. C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref]

X. J. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2, e72 (2013).
[Crossref]

L. L. Huang, X. Z. Chen, H. Muhlenbernd, H. Zhang, S. M. Chen, B. F. Bai, Q. F. Tan, G. F. Jin, K. W. Cheah, C. W. Qiu, J. S. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

2012 (5)

X. Chen, L. Huang, H. Muhlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun. 3, 1198 (2012).
[Crossref]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37, 2595–2597 (2012).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

S. Pidishety, V. Kumar, and N. K. Viswanathan, “Plasmon-mediated vectorial topological dipole: formation and annihilation,” Opt. Lett. 37, 4233–4235 (2012).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref]

2011 (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

2010 (1)

H. Kim, J. Park, S. W. Cho, S. Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

2008 (1)

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

2006 (2)

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

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

2003 (1)

V. Garces-Chavez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[Crossref]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

2001 (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Ahmed, N.

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
[Crossref]

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. X. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Alu, A.

F. Qin, L. Ding, L. Zhang, F. Monticone, C. C. Chum, J. Deng, S. Mei, Y. Li, J. Teng, M. Hong, S. Zhang, A. Alu, and C. W. Qiu, “Hybrid bilayer plasmonic metasurface efficiently manipulates visible light,” Sci. Adv. 2, e1501168 (2016).
[Crossref]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100, 013602 (2008).
[Crossref]

Antoniou, N.

J. Lin, J. P. Mueller, Q. Wang, G. Yuan, N. Antoniou, X. C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref]

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

Arbabi, A.

A. Arbabi, E. Arbabi, Y. Horie, S. M. Kamali, and A. Faraon, “Planar metasurface retroreflector,” Nat. Photonics 11, 415–420 (2017).
[Crossref]

Arbabi, E.

A. Arbabi, E. Arbabi, Y. Horie, S. M. Kamali, and A. Faraon, “Planar metasurface retroreflector,” Nat. Photonics 11, 415–420 (2017).
[Crossref]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
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X. J. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2, e72 (2013).
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Padgett, M. J.

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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, 163905 (2006).
[Crossref]

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H. Kim, J. Park, S. W. Cho, S. Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref]

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Pu, M.

J. Jin, J. Luo, X. Zhang, H. Gao, X. Li, M. Pu, P. Gao, Z. Zhao, and X. Luo, “Generation and detection of orbital angular momentum via metasurface,” Sci. Rep. 6, 24286 (2016).
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X. Li, L. Chen, Y. Li, X. Zhang, M. Pu, Z. Zhao, X. Ma, Y. Wang, M. Hong, and X. Luo, “Multicolor 3D meta-holography by broadband plasmonic modulation,” Sci. Adv. 2, e1601102 (2016).
[Crossref]

X. Ma, M. Pu, X. Li, C. Huang, Y. Wang, W. Pan, B. Zhao, J. Cui, C. Wang, Z. Zhao, and X. Luo, “A planar chiral meta-surface for optical vortex generation and focusing,” Sci. Rep. 5, 10365 (2015).
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D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. F. Li, P. W. Wong, K. W. Cheah, E. Y. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun. 6, 8241 (2015).
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E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
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[Crossref]

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[Crossref]

X. Chen, L. Huang, H. Muhlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C. W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun. 3, 1198 (2012).
[Crossref]

Ramachandran, S.

Ren, Y.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. Bi-channel optical vortex generated in both the near and far fields. (a) Schematic of NFVG that is based on a plasmonic metasurface which consists of a nanoslit array. (b) The nanoslit on an Au film works as the building block of NFVG with width W=80  nm, length L=250  nm, and depth T=120  nm. The depth T is also the thickness of the Au film. (c) Schematic diagram that illustrates the design principle of the bi-channel metasurface using geometric phase of the nanoslit in both the near and far fields. Under illumination of CP light with a specific spin state (LCP/RCP), the proposed metasurface can simultaneously generate focused SP vortex at the center (point O) of the metasurface and focused far-field vortex (cross-polarization, RCP/LCP) of scattered light at point F with a designed focal length f.
Fig. 2.
Fig. 2. NFVG design and the simulation results. (a) The nanoslit array of a designed NFVG that can generate SP vortex and far-field vortex with topological charge of l1=3 and l2=2 under illumination of LCP light at 671 nm, respectively. The inner radius R1 and outer radius R2 are set to be 3.5 and 9 μm, and the distance between two adjacent nanoslits is d=450  nm. (b) Simulated optical far-field intensity of cross-polarized light (i.e., RCP light) at the xz plane. The inset shows the intensity distribution at the focal xy plane (z=20  μm). (c) Simulated |Ez|2 intensity distribution of SPPs in the near field on the surface of Au film. Simulated results of (d) instantaneous vortex field distribution and (e) spatial phase profile at z=20  μm in the far field, respectively. Simulated results of (f) instantaneous Ez intensity distribution and (g) spatial phase profile for SPPs in the near field, respectively.
Fig. 3.
Fig. 3. Experimental results of optical performances for the NFVG in the near and far fields with LCP incidence. (a) Scanning electron microscopy (SEM) images of the fabricated NFVG sample. The magnified SEM image shows the appearance of a single nanoslit. The scale bar is 5 μm. (b) Experimentally measured far-field intensity of cross-polarized light (RCP light) at the xz plane. The inset shows the intensity distribution at the focal xy plane (z=20  μm) marked with a white dashed line in xz profile. (c) Measured x-component of transmitted light at z=30  μm. (d) SNOM measured optical near-field intensity distribution. Inset shows the zoom-in image of SPPs distribution at the center of the metasurface. (e) Comparison of the intensity profiles between measurement and simulation. The black dashed line corresponds to the simulation result. The blue and green solid lines show the field intensity along the blue and green dashed lines (i.e., x and y directions) in the SNOM result (d), respectively.
Fig. 4.
Fig. 4. Flexibility of the proposed NFVG for different topological charge configurations. Results of the sample designed for near- and far-field topological charges of (a) l1=2, l2=3 (Sample A), (b) l1=2, l2=2 (Sample B), and (c) l1=3, l2=2 (Sample C). Experimentally measured optical far-field intensity distributions on the transmission side of the metasurface for (a1), (b1), (c1) focal xy plane (z=20  μm); (a3), (b3), (c3) cross-sectional planes; and (a2), (b2), (c2) x-component at z=30  μm. The left of (a3)–(c3) shows the SEM images of these three samples. (a4), (b4), (c4) SNOM measured optical near-field intensity distributions for Samples A, B, and C. (a5), (b5), (c5) Simulated instantaneous Ez intensity distributions of SPPs in the near field for Samples A, B, and C. For all simulations and experimental measurements in (a)–(c), the incident beam is LCP light of 671 nm, and the results in (a1)–(c1) and (a3)–(c3) have filtered out the component of initial polarization to measure the cross-polarized light.
Fig. 5.
Fig. 5. Schematic of the experimental setup for far-field measurement. The incident CP light is generated by a polarizer and a QWP. With an actuator, the far-field intensity distribution at different distances can be collected by the CMOS camera. In this work, the incidence is LCP light, and the cross-polarized light (RCP) is measured in the far field.
Fig. 6.
Fig. 6. SEM images and far-field simulated results of Samples A and B. SEM images of fabricated (a) Sample A and (b) Sample B with FIB milling process. (c)–(h) Simulated far-field intensity distribution of RCP at the xz plane and corresponding intensity and phase distribution at the xy plane (z=20  μm) for Sample A [(c), (e), and (f)] and Sample B [(d), (g), and (h)].
Fig. 7.
Fig. 7. Simulated near-field results of Samples A and B. Simulated near-field intensity distributions of (a), (b) Sample A and (d), (e) Sample B. The simulated phase distribution of near-field SPPs of (c) Sample A and (f) Sample B. The similarity of the near-field intensity and phase distribution of these two samples confirmed the generating of near-field vortex with the same topological charge of l1=2 as designed.
Fig. 8.
Fig. 8. Comparison of experimental results and simulations. (a) The intensity profile of experimental results (along the white dashed line in the inset) and simulations (along the x direction of intensity distribution shown in Fig. 7) of Sample A (l1=2, l2=3). (b) The intensity profile of experimental results (along the white dashed line in the inset) and simulations of Sample B (l1=2, l2=2). Blue dashed lines represent simulated results and the orange ones represent experimentally measured results. The measured profiles are 0.596 μm (Sample A) and 0.656 μm (Sample B), and are close to the simulation results 0.640 μm, which confirms that the generated near-field vortex has a topological charge of 2 as designed.
Fig. 9.
Fig. 9. Simulated results of the bi-channel metalens under LCP incidence. (a) Designed patterns of the bi-channel metalens. (b) Simulated near-field intensity distribution. (c) Simulated far-field intensity distribution of RCP at the xz plane. (d) Simulated far-field intensity distribution at the xy plane [green dashed line in (c)].
Fig. 10.
Fig. 10. Simulated near- and far-field distributions of NFVG with different configurations of l1 and l2 under LCP incidence. (a) Designed patterns with different topological charges l1 (0 to 6) and l2 (6 to 0). (b) Simulated intensity (upper row) and instantaneous field (lower row) distributions of near-field vortices with different topological charges. (c) Simulated intensity (upper row) and instantaneous field (lower row) distributions of far-field vortices (RCP) with different topological charges.

Equations (3)

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σθ+ksppr+Φσ(P,O)=Φ1+2πm+l1φ,
2σθ+kr2+f2=Φ2+2πn+l2φ,
2ksppr+2Φσ(P,O)kr2+f2=2(Φ1+l1φ)(Φ2+l2φ)+2πM.

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