Abstract

The interaction of light with spinning objects can lead to a frequency shift as a result of the transferred angular momentum. Subwavelength plasmonic nanostructures on an optical surface provide an efficient platform for local light manipulation and thus can be utilized for such momentum exchanges. Here we demonstrate reflective-type plasmonic metasurface q-plates based on the geometric Pancharatnam-Berry phase that are capable of changing the spin and orbital angular momentum of light. When these metasurfaces rotate at a constant angular speed, we observe a rotational Doppler frequency shift of the reflected light, which depends on the total angular momentum transfer. The flexibility in the design of the metasurfaces even enables complex reflective phase masks that can transfer different orbital angular momenta, at the same time, to the beam. Our experiments show that such complex metasurface phase masks can be used to obtain spatially variant Doppler shift in the reflected beam profile.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2017 (2)

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

2016 (1)

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

2015 (4)

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

D. Hakobyan and E. Brasselet, “Optical torque reversal and spin-orbit rotational Doppler shift experiments,” Opt. Express 23, 31230–31239 (2015).
[Crossref]

2014 (2)

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]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

2013 (4)

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. 13, 829–834 (2013).
[Crossref]

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Broadband plasmonic half-wave plates in reflection,” Opt. Lett. 38, 513–515 (2013).
[Crossref]

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

2012 (4)

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (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 (2011).
[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, 163905 (2006).
[Crossref]

2002 (1)

1998 (2)

A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
[Crossref]

J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

1996 (1)

G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
[Crossref]

1984 (1)

1979 (1)

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

1966 (1)

P. Allen, “A radiation torque experiment,” Am. J. Phys. 34, 1185–1192 (1966).
[Crossref]

1961 (1)

D. L. Franklin, W. Schlegel, and R. F. Rushmer, “Blood flow measured by Doppler frequency shift of back-scattered ultrasound,” Science 134, 564–565 (1961).
[Crossref]

1909 (1)

J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London A 82, 560–567 (1909).
[Crossref]

Aieta, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

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 (2011).
[Crossref]

Allen, L.

J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

Allen, P.

P. Allen, “A radiation torque experiment,” Am. J. Phys. 34, 1185–1192 (1966).
[Crossref]

Antoniou, N.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

Arnold, S.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

Bai, B.

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Biener, G.

Blanchard, R.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Boltasseva, A.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Bomzon, Z.

Boyd, R. W.

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]

Bozhevolnyi, S. I.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Broadband plasmonic half-wave plates in reflection,” Opt. Lett. 38, 513–515 (2013).
[Crossref]

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. 13, 829–834 (2013).
[Crossref]

Brasselet, E.

Briggs, D. P.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Cafarelli, N. J.

N. J. Cafarelli, “Doppler frequency position fixing method,” U.S. patent2,968,034 (January10, 1961).

Cai, X.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Capasso, F.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

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 (2011).
[Crossref]

Cardano, F.

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

Cheah, K. W.

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Chen, D.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Chen, S.

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Chen, X.

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Courtial, J.

J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

De Leon, I.

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]

Dholakia, K.

J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

Djurišic, A. B.

Dodge, M. J.

Dong, J.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Doppler, C.

C. Doppler, Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels: Versuch einer das Bradley’sche Aberrations-Theorem als integrirenden Theil in sich schliessenden allgemeineren Theorie (Commission bei Borrosch & André, 1842).

Eilenberger, F.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Elazar, J. M.

Emani, N. K.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Eriksen, R. L.

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. 13, 829–834 (2013).
[Crossref]

Franklin, D. L.

D. L. Franklin, W. Schlegel, and R. F. Rushmer, “Blood flow measured by Doppler frequency shift of back-scattered ultrasound,” Science 134, 564–565 (1961).
[Crossref]

Fu, D.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Gaburro, Z.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

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 (2011).
[Crossref]

Garetz, B. A.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

Genevet, P.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

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 (2011).
[Crossref]

Gerardot, B. D.

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

Hakobyan, D.

Hasman, E.

Helgert, C.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Huang, L.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Jasper, S. C.

S. C. Jasper, “Method of Doppler searching in a digital GPS receiver,” U.S. patent4,701,934 (October20, 1987).

Jin, G.

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Kang, M.

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Karimi, E.

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]

Kats, M. A.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

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 (2011).
[Crossref]

Kenney, M.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Kildishev, A. V.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Kleiner, V.

Kley, E.-B.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Kravchenko, I. I.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Lederer, F.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Li, F.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Li, G.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Li, J.

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Li, T.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

Li, X.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

Lin, J.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Liu, J.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

Majewski, M. L.

Manzo, C.

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]

Marrucci, L.

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[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]

Moitra, P.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Mühlenbernd, H.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Ni, X.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Nielsen, M. G.

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. 13, 829–834 (2013).
[Crossref]

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Broadband plasmonic half-wave plates in reflection,” Opt. Lett. 38, 513–515 (2013).
[Crossref]

Nienhuis, G.

G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
[Crossref]

Nori, F.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Padgett, M.

J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[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, 163905 (2006).
[Crossref]

Pertsch, T.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Pors, A.

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. 13, 829–834 (2013).
[Crossref]

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Broadband plasmonic half-wave plates in reflection,” Opt. Lett. 38, 513–515 (2013).
[Crossref]

Poynting, J.

J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London A 82, 560–567 (1909).
[Crossref]

Pun, E. Y.-B.

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Qassim, H.

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]

Rakic, A. D.

Reineke, B.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

Robertson, D.

J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

Rockstuhl, C.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Rodrguez-Fortuño, F.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Rushmer, R. F.

D. L. Franklin, W. Schlegel, and R. F. Rushmer, “Blood flow measured by Doppler frequency shift of back-scattered ultrasound,” Science 134, 564–565 (1961).
[Crossref]

Schlegel, W.

D. L. Franklin, W. Schlegel, and R. F. Rushmer, “Blood flow measured by Doppler frequency shift of back-scattered ultrasound,” Science 134, 564–565 (1961).
[Crossref]

Schulz, S. A.

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]

Scully, M. O.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Setzpfandt, F.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Shalaev, V. M.

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Song, X.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

Tan, Q.

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Tetienne, J.-P.

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 (2011).
[Crossref]

Tünnermann, A.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Upham, J.

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]

Valentine, J.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Walther, B.

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Wang, W.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Wang, Y.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

Wen, D.

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

Xin, J.

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

Yang, Y.

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

Yu, N.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

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 (2011).
[Crossref]

Yue, F.

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

Zayats, A. V.

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Zentgraf, T.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Zhang, P.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Zhang, S.

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

Zhang, X.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Zheng, G.

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Zhou, H.

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

ACS Photon. (2)

F. Yue, D. Wen, J. Xin, B. D. Gerardot, J. Li, and X. Chen, “Vector vortex beam generation with a single plasmonic metasurface,” ACS Photon. 3, 1558–1563 (2016).
[Crossref]

L. Huang, X. Song, B. Reineke, T. Li, X. Li, J. Liu, S. Zhang, Y. Wang, and T. Zentgraf, “Volumetric generation of optical vortices with metasurfaces,” ACS Photon. 4, 338–346 (2017).
[Crossref]

Adv. Mater. (1)

B. Walther, C. Helgert, C. Rockstuhl, F. Setzpfandt, F. Eilenberger, E.-B. Kley, F. Lederer, A. Tünnermann, and T. Pertsch, “Spatial and spectral light shaping with metamaterials,” Adv. Mater. 24, 6300–6304 (2012).
[Crossref]

Am. J. Phys. (1)

P. Allen, “A radiation torque experiment,” Am. J. Phys. 34, 1185–1192 (1966).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 013101 (2012).
[Crossref]

Light Sci. Appl. (2)

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]

H. Zhou, D. Fu, J. Dong, P. Zhang, D. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on rotational Doppler effect,” Light Sci. Appl. 6, e162512017.
[Crossref]

Nano Lett. (5)

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband focusing flat mirrors based on plasmonic gradient metasurfaces,” Nano Lett. 13, 829–834 (2013).
[Crossref]

L. Huang, X. Chen, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Dispersionless phase discontinuities for controlling light propagation,” Nano Lett. 12, 5750–5755 (2012).
[Crossref]

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14, 1394–1399 (2014).
[Crossref]

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[Crossref]

Nat. Nanotechnol. (1)

G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10, 308–312 (2015).
[Crossref]

Nat. Photonics (2)

K. Y. Bliokh, F. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

Opt. Commun. (2)

G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132, 8–14 (1996).
[Crossref]

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,” Opt. Commun. 31, 1–3 (1979).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

J. Courtial, D. Robertson, K. Dholakia, L. Allen, and M. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[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]

Proc. R. Soc. London A (1)

J. Poynting, “The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light,” Proc. R. Soc. London A 82, 560–567 (1909).
[Crossref]

Science (3)

D. L. Franklin, W. Schlegel, and R. F. Rushmer, “Blood flow measured by Doppler frequency shift of back-scattered ultrasound,” Science 134, 564–565 (1961).
[Crossref]

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 (2011).
[Crossref]

X. Ni, N. K. Emani, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Broadband light bending with plasmonic nanoantennas,” Science 335, 427 (2012).
[Crossref]

Other (3)

C. Doppler, Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels: Versuch einer das Bradley’sche Aberrations-Theorem als integrirenden Theil in sich schliessenden allgemeineren Theorie (Commission bei Borrosch & André, 1842).

N. J. Cafarelli, “Doppler frequency position fixing method,” U.S. patent2,968,034 (January10, 1961).

S. C. Jasper, “Method of Doppler searching in a digital GPS receiver,” U.S. patent4,701,934 (October20, 1987).

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Figures (6)

Fig. 1.
Fig. 1. Rotational Doppler effect for photons reflected from a spinning object. A photon with the total angular momentum quantum number j1 is reflected on a spinning surface with angular frequency Ω and moment of inertia I. During the interaction with the surface, the photon exchanges momentum, resulting in a new quantum number j2 after reflection, which leads to a Doppler shifted frequency.
Fig. 2.
Fig. 2. Metasurface design. (a) Local structure is a 30-nm-thick gold antenna with a 200-nm length and an 80-nm width. It is placed on a 130-nm-thick gold mirror with a 90-nm-thick MgF2 Spacer. This system reflects circularly polarized light into the cross polarization for a wavelength of 780 nm. Therefore, the reflection phase corresponds to two times the antenna orientation angle ϕ. (b) A q-plate is constructed by rotating the antennas q times along one ring. An incoming circularly polarized Gaussian beam (σ1=±1,l1=0) obtains an additional orbital angular momentum after being reflected (σ2=σ1,l2=2qσ1). (c) Exemplary SEM-image of the q-plate with q=2.
Fig. 3.
Fig. 3. Experimental setup. The linearly polarized Gaussian source beam is a superposition of right and left circularly polarized light. After passing the polarizing beam splitter, it is focused with a lens (f1=150  nm) on the spinning metasurface, which rotates at an angular frequency Ω. Since both polarizations are Doppler shifted with opposite signs, a local intensity beating occurs after the reflection on the polarizing beam splitter. This intensity beating corresponds to a rotation of the resulting 4q-fold beam profile. It is measured by an Si-detector placed off-axis after the beam is widened by a second lens (f2=30  nm).
Fig. 4.
Fig. 4. Experimental observations for different topological charges. The illustrated q-plate patterns create the expected 4q-fold beam patterns in the experiment. The Fourier transformed signal of an off-axis placed detector contains peaks at whole number multiples of the motor frequency F=Ω/2π, which is set around 13–14 Hz. For small topological charges q1, the expected peak at 4·(q1)·F is dominant, while for higher topological charges, the experimental implementation loses informative value. Note, higher q values lead to a stronger beam divergence and a larger spatial intensity distribution. Therefore, we increased the beam intensity for q>3 to obtain a better signal-to-noise ratio at the detector. As a consequence, the peak height of the Fourier amplitudes cannot be directly compared.
Fig. 5.
Fig. 5. Influence of the topological charge on the Doppler shift. The ratio of the measured beating frequency (Doppler shift) and the metasurface rotation frequency versus the encoded topological charge q on the metasurface. The measured values follow a linear trend, confirming the expected linear relationship [Eq. (6)] of the Doppler shift to the total transferred momentum.
Fig. 6.
Fig. 6. Mixed q-plate with q=2 and q=2. (a) Encoded phase mask on the metasurface with a diameter of 400 μm. (b) Observed static beam profile for different metasurface orientation. Between each image the sample has been rotated by 60°. (c) Rotating beam profile can be understood as a superposition of the static images. (d) Fourier transformed signal has been measured for the three rings that appear in the rotating beam profile for a motor frequency of F=11  Hz. In addition to the single q peaks at 4F and 12F, there are also frequencies based on E-field superposition that appear at 4F and 8F. Due to the different orbital scattering for q=2 and q=2, the ratio of the three peaks varies for the three rings.

Equations (8)

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j1+IΩ1=j2+IΩ2,
ω1+IΩ122=ω2+IΩ222,
Δω=ω2ω1=I2(Ω12Ω22)IΩ1(Ω1Ω2)=Ω1(j2+j1).
Δj=j2j1=2·(q1)·σ1.
Δω=ω2ω1=Ω·Δj=2Ω·(q1)·σ1.
Δωintensity=4  Ω(q1).
I(r,ϕ)=I(r)·|exp(i2qϕ)+exp(i2qϕ)|2=I(r)·2·(cos(4qϕ)+1),
I(t)=I0·(cos(Ωt)+cos(6Ωt))2=12  I0(2+3·cos(4Ωt)+2·cos(8Ωt)+cos(12Ωt)).

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