Abstract

Smith–Purcell radiation (SPR) is the electromagnetic wave generated by free electrons passing above a diffraction grating, and it has played an important role in free-electron light sources and particle accelerators. Orbital angular momentum (OAM) is a new degree of freedom that can significantly promote the capacity of information carried by an electro-magnetic beam. In this paper, we propose an integrable method for generating vortex Smith–Purcell radiation (VSPR), namely, SPR carrying OAM, by having free-electron bunches pass on planar holographic gratings. VSPRs generated by different electron energies, with different topological charges of the OAM, radiation angles, and frequencies are demonstrated numerically. It is also found that, for high-order radiation, the topological charge of the OAM wave will be multiplied by the radiation order. This work introduces a new way to generate SPR with OAM and provides a method to achieve an integratable and tunable free-electron OAM wave source at different frequency regions.

© 2020 Chinese Laser Press

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  9. K. J. Woods, J. E. Walsh, R. E. Stoner, H. G. Kirk, and R. C. Fernow, “Forward directed Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 74, 3808–3811 (1995).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
    [Crossref]
  27. K. Oyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
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  30. M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
    [Crossref]
  31. J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21, 20230–20239 (2013).
    [Crossref]
  32. Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
    [Crossref]
  33. E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A. Marinelli, C. Hast, and J. B. Rosenzweig, “Coherent optical vortices from relativistic electron beams,” Nat. Phys. 9, 549–553 (2013).
    [Crossref]
  34. L. Xiao, J. Chen, L. Chen, Q. Zhang, L. Guo, and M. Yang, “Electron beam excited surface plasmon polaritons carrying orbital angular momentum,” in 12th International Symposium on Antennas, Propagation and EM Theory (ISAPE) (IEEE, 2018), pp. 1–3.
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    [Crossref]
  37. F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
    [Crossref]
  38. A. Nicolas, L. Veissier, E. Giacobino, D. Maxein, and J. Laurat, “Quantum state tomography of orbital angular momentum photonic qubits via a projection-based technique,” New J. Phys. 17, 033037 (2015).
    [Crossref]
  39. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
    [Crossref]
  40. K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Prog. Electromagn. Res. 116, 441–456 (2011).
    [Crossref]

2019 (2)

Y. Ye, F. Liu, M. Wang, L. Tai, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Deep-ultraviolet Smith-Purcell radiation,” Optica 6, 592–597 (2019).
[Crossref]

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

2018 (2)

A. Massuda, C. Roques-Carmes, Y. Yang, S. E. Kooi, Y. Yang, C. Murdia, K. K. Berggren, I. Kaminer, and M. Soljačić, “Smith-Purcell radiation from low-energy electrons,” ACS Photon. 5, 3513–3518 (2018).
[Crossref]

Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, “Maximal spontaneous photon emission and energy loss from free electrons,” Nat. Phys. 14, 894–899 (2018).
[Crossref]

2017 (2)

R. Remez, N. Shapira, C. Roques-Carmes, R. Tirole, Y. Yang, Y. Lereah, M. Soljacic, I. Kaminer, and A. Arie, “Spectral and spatial shaping of Smith-Purcell radiation,” Phys. Rev. A 96, 061801 (2017).
[Crossref]

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

2016 (1)

Z. Wang, K. Yao, M. Chen, H. Chen, and Y. Liu, “Manipulating Smith-Purcell emission with Babinet metasurfaces,” Phys. Rev. Lett. 117, 157401 (2016).
[Crossref]

2015 (3)

A. Nicolas, L. Veissier, E. Giacobino, D. Maxein, and J. Laurat, “Quantum state tomography of orbital angular momentum photonic qubits via a projection-based technique,” New J. Phys. 17, 033037 (2015).
[Crossref]

Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
[Crossref]

J. K. So, F. J. García de Abajo, K. F. MacDonald, and N. I. Zheludev, “Amplification of the evanescent field of free electrons,” ACS Photon. 2, 1236–1240 (2015).
[Crossref]

2014 (1)

2013 (3)

E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A. Marinelli, C. Hast, and J. B. Rosenzweig, “Coherent optical vortices from relativistic electron beams,” Nat. Phys. 9, 549–553 (2013).
[Crossref]

J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21, 20230–20239 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

2012 (2)

K. Oyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[Crossref]

M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
[Crossref]

2011 (1)

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Prog. Electromagn. Res. 116, 441–456 (2011).
[Crossref]

2010 (1)

J. Gardelle, P. Modin, and J. T. Donohue, “Start current and gain measurements for a Smith-Purcell free-electron laser,” Phys. Rev. Lett. 105, 224801 (2010).
[Crossref]

2009 (1)

G. Adam, K. F. MacDonald, N. I. Zheludev, Y. H. Fu, C. M. Wang, D. P. Tsai, and F. J. GarciadeAbajo, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

2008 (1)

2005 (1)

S. E. Korbly, A. S. Kesar, J. R. Sirigiri, and R. J. Temkin, “Observation of frequency-locked coherent terahertz Smith-Purcell radiation,” Phys. Rev. Lett. 94, 054803 (2005).
[Crossref]

2004 (1)

H. L. Andrews and C. A. Brau, “Gain of a Smith-Purcell free-electron laser,” Phys. Rev. ST Accel. Beams 7, 070701 (2004).
[Crossref]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref]

1998 (1)

J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, “Superradiant Smith-Purcell emission,” Phys. Rev. Lett. 80, 516–519 (1998).
[Crossref]

1995 (2)

K. J. Woods, J. E. Walsh, R. E. Stoner, H. G. Kirk, and R. C. Fernow, “Forward directed Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 74, 3808–3811 (1995).
[Crossref]

K. Ishi, Y. Shibata, T. Takahashi, S. Hasebe, M. Ikezawa, K. Takami, T. Matsuyama, K. Kobayashi, and Y. Fujita, “Observation of coherent Smith-Purcell radiation from short-bunched electrons,” Phys. Rev. E 51, R5212–R5215 (1995).
[Crossref]

1992 (2)

G. Doucas, J. H. Mulvey, M. Omori, J. Walsh, and M. F. Kimmitt, “First observation of Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 69, 1761–1764 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. 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]

1984 (1)

1980 (1)

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[Crossref]

1979 (1)

J. M. Wachtel, “Free-electron lasers using the Smith-Purcell effect,” J. Appl. Phys. 50, 49–56 (1979).
[Crossref]

1975 (1)

K. Mizuno, S. Ono, and O. Shimoe, “Interaction between coherent light waves and free electrons with a reflection grating,” Nature 253, 184–185 (1975).
[Crossref]

1973 (1)

1961 (1)

K. Ishiguro and T. Tako, “An estimation of Smith-Purcell effect as the light source in the infra-red region,” Opt. Act. 8, 25–31 (1961).
[Crossref]

1960 (1)

D. Francia and G. Toraldo, “On the theory of some Čerenkovian effects,” Il Nuov. Cim. 16, 61–77 (1960).
[Crossref]

1953 (1)

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92, 1069 (1953).
[Crossref]

Adam, G.

G. Adam, K. F. MacDonald, N. I. Zheludev, Y. H. Fu, C. M. Wang, D. P. Tsai, and F. J. GarciadeAbajo, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

Ahmed, I.

K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Prog. Electromagn. Res. 116, 441–456 (2011).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Andrews, H. L.

H. L. Andrews and C. A. Brau, “Gain of a Smith-Purcell free-electron laser,” Phys. Rev. ST Accel. Beams 7, 070701 (2004).
[Crossref]

Aoki, N.

K. Oyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[Crossref]

Arie, A.

R. Remez, N. Shapira, C. Roques-Carmes, R. Tirole, Y. Yang, Y. Lereah, M. Soljacic, I. Kaminer, and A. Arie, “Spectral and spatial shaping of Smith-Purcell radiation,” Phys. Rev. A 96, 061801 (2017).
[Crossref]

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Berggren, K.

Y. Yang, C. Roques-Carmes, I. Kaminer, A. Zaidi, A. Massuda, Y. Yang, S. E. Kooi, K. Berggren, and S. Marin, “Manipulating Smith-Purcell radiation polarization with metasurfaces,” in CLEO: QELS_Fundamental Science (2018), paper FW4H.1.

Berggren, K. K.

A. Massuda, C. Roques-Carmes, Y. Yang, S. E. Kooi, Y. Yang, C. Murdia, K. K. Berggren, I. Kaminer, and M. Soljačić, “Smith-Purcell radiation from low-energy electrons,” ACS Photon. 5, 3513–3518 (2018).
[Crossref]

Brau, C. A.

H. L. Andrews and C. A. Brau, “Gain of a Smith-Purcell free-electron laser,” Phys. Rev. ST Accel. Beams 7, 070701 (2004).
[Crossref]

Chen, H.

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

Z. Wang, K. Yao, M. Chen, H. Chen, and Y. Liu, “Manipulating Smith-Purcell emission with Babinet metasurfaces,” Phys. Rev. Lett. 117, 157401 (2016).
[Crossref]

Chen, J.

M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
[Crossref]

L. Xiao, J. Chen, L. Chen, Q. Zhang, L. Guo, and M. Yang, “Electron beam excited surface plasmon polaritons carrying orbital angular momentum,” in 12th International Symposium on Antennas, Propagation and EM Theory (ISAPE) (IEEE, 2018), pp. 1–3.

Chen, L.

L. Xiao, J. Chen, L. Chen, Q. Zhang, L. Guo, and M. Yang, “Electron beam excited surface plasmon polaritons carrying orbital angular momentum,” in 12th International Symposium on Antennas, Propagation and EM Theory (ISAPE) (IEEE, 2018), pp. 1–3.

Chen, M.

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

Z. Wang, K. Yao, M. Chen, H. Chen, and Y. Liu, “Manipulating Smith-Purcell emission with Babinet metasurfaces,” Phys. Rev. Lett. 117, 157401 (2016).
[Crossref]

Christensen, T.

Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, “Maximal spontaneous photon emission and energy loss from free electrons,” Nat. Phys. 14, 894–899 (2018).
[Crossref]

Collier, R.

R. Collier, Optical Holography (Elsevier, 2013).

Cui, K.

Y. Ye, F. Liu, M. Wang, L. Tai, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Deep-ultraviolet Smith-Purcell radiation,” Optica 6, 592–597 (2019).
[Crossref]

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
[Crossref]

Donohue, J. T.

J. Gardelle, P. Modin, and J. T. Donohue, “Start current and gain measurements for a Smith-Purcell free-electron laser,” Phys. Rev. Lett. 105, 224801 (2010).
[Crossref]

Doucas, G.

G. Doucas, J. H. Mulvey, M. Omori, J. Walsh, and M. F. Kimmitt, “First observation of Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 69, 1761–1764 (1992).
[Crossref]

Dunning, M.

E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A. Marinelli, C. Hast, and J. B. Rosenzweig, “Coherent optical vortices from relativistic electron beams,” Nat. Phys. 9, 549–553 (2013).
[Crossref]

Dvorkis, P.

Feng, S.

Feng, X.

Y. Ye, F. Liu, M. Wang, L. Tai, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Deep-ultraviolet Smith-Purcell radiation,” Optica 6, 592–597 (2019).
[Crossref]

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
[Crossref]

Fernow, R. C.

K. J. Woods, J. E. Walsh, R. E. Stoner, H. G. Kirk, and R. C. Fernow, “Forward directed Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 74, 3808–3811 (1995).
[Crossref]

Flossmann, F.

Francia, D.

D. Francia and G. Toraldo, “On the theory of some Čerenkovian effects,” Il Nuov. Cim. 16, 61–77 (1960).
[Crossref]

Franke-Arnold, S.

Fu, Y. H.

G. Adam, K. F. MacDonald, N. I. Zheludev, Y. H. Fu, C. M. Wang, D. P. Tsai, and F. J. GarciadeAbajo, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

Fujita, Y.

K. Ishi, Y. Shibata, T. Takahashi, S. Hasebe, M. Ikezawa, K. Takami, T. Matsuyama, K. Kobayashi, and Y. Fujita, “Observation of coherent Smith-Purcell radiation from short-bunched electrons,” Phys. Rev. E 51, R5212–R5215 (1995).
[Crossref]

Gao, F.

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

García de Abajo, F. J.

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Y. Yang, C. Roques-Carmes, I. Kaminer, A. Zaidi, A. Massuda, Y. Yang, S. E. Kooi, K. Berggren, and S. Marin, “Manipulating Smith-Purcell radiation polarization with metasurfaces,” in CLEO: QELS_Fundamental Science (2018), paper FW4H.1.

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[Crossref]

Solanki, A.

A. Massuda, C. Roques-Carmes, A. Solanki, Y. Yang, S. E. Kooi, F. Habbal, I. Kaminer, and S. Marin, “High-order Smith-Purcell radiation in silicon nanowires,” in CLEO: QELS_Fundamental Science (2017), paper JTh5B.8.

Soljacic, M.

A. Massuda, C. Roques-Carmes, Y. Yang, S. E. Kooi, Y. Yang, C. Murdia, K. K. Berggren, I. Kaminer, and M. Soljačić, “Smith-Purcell radiation from low-energy electrons,” ACS Photon. 5, 3513–3518 (2018).
[Crossref]

Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, “Maximal spontaneous photon emission and energy loss from free electrons,” Nat. Phys. 14, 894–899 (2018).
[Crossref]

R. Remez, N. Shapira, C. Roques-Carmes, R. Tirole, Y. Yang, Y. Lereah, M. Soljacic, I. Kaminer, and A. Arie, “Spectral and spatial shaping of Smith-Purcell radiation,” Phys. Rev. A 96, 061801 (2017).
[Crossref]

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Stoner, R. E.

K. J. Woods, J. E. Walsh, R. E. Stoner, H. G. Kirk, and R. C. Fernow, “Forward directed Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 74, 3808–3811 (1995).
[Crossref]

Tai, L.

Takahashi, T.

K. Ishi, Y. Shibata, T. Takahashi, S. Hasebe, M. Ikezawa, K. Takami, T. Matsuyama, K. Kobayashi, and Y. Fujita, “Observation of coherent Smith-Purcell radiation from short-bunched electrons,” Phys. Rev. E 51, R5212–R5215 (1995).
[Crossref]

Takami, K.

K. Ishi, Y. Shibata, T. Takahashi, S. Hasebe, M. Ikezawa, K. Takami, T. Matsuyama, K. Kobayashi, and Y. Fujita, “Observation of coherent Smith-Purcell radiation from short-bunched electrons,” Phys. Rev. E 51, R5212–R5215 (1995).
[Crossref]

Tako, T.

K. Ishiguro and T. Tako, “An estimation of Smith-Purcell effect as the light source in the infra-red region,” Opt. Act. 8, 25–31 (1961).
[Crossref]

Temkin, R. J.

S. E. Korbly, A. S. Kesar, J. R. Sirigiri, and R. J. Temkin, “Observation of frequency-locked coherent terahertz Smith-Purcell radiation,” Phys. Rev. Lett. 94, 054803 (2005).
[Crossref]

Tirole, R.

R. Remez, N. Shapira, C. Roques-Carmes, R. Tirole, Y. Yang, Y. Lereah, M. Soljacic, I. Kaminer, and A. Arie, “Spectral and spatial shaping of Smith-Purcell radiation,” Phys. Rev. A 96, 061801 (2017).
[Crossref]

Toraldo, G.

D. Francia and G. Toraldo, “On the theory of some Čerenkovian effects,” Il Nuov. Cim. 16, 61–77 (1960).
[Crossref]

Tsai, D. P.

G. Adam, K. F. MacDonald, N. I. Zheludev, Y. H. Fu, C. M. Wang, D. P. Tsai, and F. J. GarciadeAbajo, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

Urata, J.

J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, “Superradiant Smith-Purcell emission,” Phys. Rev. Lett. 80, 516–519 (1998).
[Crossref]

Van den Berg, P. M.

Veissier, L.

A. Nicolas, L. Veissier, E. Giacobino, D. Maxein, and J. Laurat, “Quantum state tomography of orbital angular momentum photonic qubits via a projection-based technique,” New J. Phys. 17, 033037 (2015).
[Crossref]

Vuong, L. T.

M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
[Crossref]

Wachtel, J. M.

J. M. Wachtel, “Free-electron lasers using the Smith-Purcell effect,” J. Appl. Phys. 50, 49–56 (1979).
[Crossref]

Walsh, J.

G. Doucas, J. H. Mulvey, M. Omori, J. Walsh, and M. F. Kimmitt, “First observation of Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 69, 1761–1764 (1992).
[Crossref]

Walsh, J. E.

J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, “Superradiant Smith-Purcell emission,” Phys. Rev. Lett. 80, 516–519 (1998).
[Crossref]

K. J. Woods, J. E. Walsh, R. E. Stoner, H. G. Kirk, and R. C. Fernow, “Forward directed Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 74, 3808–3811 (1995).
[Crossref]

Wang, C. M.

G. Adam, K. F. MacDonald, N. I. Zheludev, Y. H. Fu, C. M. Wang, D. P. Tsai, and F. J. GarciadeAbajo, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

Wang, H. T.

M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
[Crossref]

Wang, M.

Y. Ye, F. Liu, M. Wang, L. Tai, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Deep-ultraviolet Smith-Purcell radiation,” Optica 6, 592–597 (2019).
[Crossref]

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

Wang, X.

Wang, Y.

Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
[Crossref]

Wang, Z.

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

Z. Wang, K. Yao, M. Chen, H. Chen, and Y. Liu, “Manipulating Smith-Purcell emission with Babinet metasurfaces,” Phys. Rev. Lett. 117, 157401 (2016).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Woods, K. J.

K. J. Woods, J. E. Walsh, R. E. Stoner, H. G. Kirk, and R. C. Fernow, “Forward directed Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 74, 3808–3811 (1995).
[Crossref]

Xiang, D.

E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A. Marinelli, C. Hast, and J. B. Rosenzweig, “Coherent optical vortices from relativistic electron beams,” Nat. Phys. 9, 549–553 (2013).
[Crossref]

Xiao, L.

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

L. Xiao, J. Chen, L. Chen, Q. Zhang, L. Guo, and M. Yang, “Electron beam excited surface plasmon polaritons carrying orbital angular momentum,” in 12th International Symposium on Antennas, Propagation and EM Theory (ISAPE) (IEEE, 2018), pp. 1–3.

Xu, S.

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

Yang, M.

L. Xiao, J. Chen, L. Chen, Q. Zhang, L. Guo, and M. Yang, “Electron beam excited surface plasmon polaritons carrying orbital angular momentum,” in 12th International Symposium on Antennas, Propagation and EM Theory (ISAPE) (IEEE, 2018), pp. 1–3.

Yang, Y.

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

A. Massuda, C. Roques-Carmes, Y. Yang, S. E. Kooi, Y. Yang, C. Murdia, K. K. Berggren, I. Kaminer, and M. Soljačić, “Smith-Purcell radiation from low-energy electrons,” ACS Photon. 5, 3513–3518 (2018).
[Crossref]

A. Massuda, C. Roques-Carmes, Y. Yang, S. E. Kooi, Y. Yang, C. Murdia, K. K. Berggren, I. Kaminer, and M. Soljačić, “Smith-Purcell radiation from low-energy electrons,” ACS Photon. 5, 3513–3518 (2018).
[Crossref]

Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, “Maximal spontaneous photon emission and energy loss from free electrons,” Nat. Phys. 14, 894–899 (2018).
[Crossref]

R. Remez, N. Shapira, C. Roques-Carmes, R. Tirole, Y. Yang, Y. Lereah, M. Soljacic, I. Kaminer, and A. Arie, “Spectral and spatial shaping of Smith-Purcell radiation,” Phys. Rev. A 96, 061801 (2017).
[Crossref]

A. Massuda, C. Roques-Carmes, A. Solanki, Y. Yang, S. E. Kooi, F. Habbal, I. Kaminer, and S. Marin, “High-order Smith-Purcell radiation in silicon nanowires,” in CLEO: QELS_Fundamental Science (2017), paper JTh5B.8.

Y. Yang, C. Roques-Carmes, I. Kaminer, A. Zaidi, A. Massuda, Y. Yang, S. E. Kooi, K. Berggren, and S. Marin, “Manipulating Smith-Purcell radiation polarization with metasurfaces,” in CLEO: QELS_Fundamental Science (2018), paper FW4H.1.

Y. Yang, C. Roques-Carmes, I. Kaminer, A. Zaidi, A. Massuda, Y. Yang, S. E. Kooi, K. Berggren, and S. Marin, “Manipulating Smith-Purcell radiation polarization with metasurfaces,” in CLEO: QELS_Fundamental Science (2018), paper FW4H.1.

Yao, K.

Z. Wang, K. Yao, M. Chen, H. Chen, and Y. Liu, “Manipulating Smith-Purcell emission with Babinet metasurfaces,” Phys. Rev. Lett. 117, 157401 (2016).
[Crossref]

Ye, J.

Ye, Y.

Y. Ye, F. Liu, M. Wang, L. Tai, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Deep-ultraviolet Smith-Purcell radiation,” Optica 6, 592–597 (2019).
[Crossref]

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

Zaidi, A.

Y. Yang, C. Roques-Carmes, I. Kaminer, A. Zaidi, A. Massuda, Y. Yang, S. E. Kooi, K. Berggren, and S. Marin, “Manipulating Smith-Purcell radiation polarization with metasurfaces,” in CLEO: QELS_Fundamental Science (2018), paper FW4H.1.

Zhang, D.

Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
[Crossref]

Zhang, Q.

L. Xiao, J. Chen, L. Chen, Q. Zhang, L. Guo, and M. Yang, “Electron beam excited surface plasmon polaritons carrying orbital angular momentum,” in 12th International Symposium on Antennas, Propagation and EM Theory (ISAPE) (IEEE, 2018), pp. 1–3.

Zhang, W.

Y. Ye, F. Liu, M. Wang, L. Tai, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Deep-ultraviolet Smith-Purcell radiation,” Optica 6, 592–597 (2019).
[Crossref]

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

Zhang, Y.

Zhao, P.

Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
[Crossref]

Zheludev, N. I.

J. K. So, F. J. García de Abajo, K. F. MacDonald, and N. I. Zheludev, “Amplification of the evanescent field of free electrons,” ACS Photon. 2, 1236–1240 (2015).
[Crossref]

G. Adam, K. F. MacDonald, N. I. Zheludev, Y. H. Fu, C. M. Wang, D. P. Tsai, and F. J. GarciadeAbajo, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

Zheng, B.

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

ACS Photon. (2)

A. Massuda, C. Roques-Carmes, Y. Yang, S. E. Kooi, Y. Yang, C. Murdia, K. K. Berggren, I. Kaminer, and M. Soljačić, “Smith-Purcell radiation from low-energy electrons,” ACS Photon. 5, 3513–3518 (2018).
[Crossref]

J. K. So, F. J. García de Abajo, K. F. MacDonald, and N. I. Zheludev, “Amplification of the evanescent field of free electrons,” ACS Photon. 2, 1236–1240 (2015).
[Crossref]

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D. Francia and G. Toraldo, “On the theory of some Čerenkovian effects,” Il Nuov. Cim. 16, 61–77 (1960).
[Crossref]

J. Appl. Phys. (1)

J. M. Wachtel, “Free-electron lasers using the Smith-Purcell effect,” J. Appl. Phys. 50, 49–56 (1979).
[Crossref]

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

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[Crossref]

Nat. Photonics (1)

F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–292 (2017).
[Crossref]

Nat. Phys. (2)

E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A. Marinelli, C. Hast, and J. B. Rosenzweig, “Coherent optical vortices from relativistic electron beams,” Nat. Phys. 9, 549–553 (2013).
[Crossref]

Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, “Maximal spontaneous photon emission and energy loss from free electrons,” Nat. Phys. 14, 894–899 (2018).
[Crossref]

Nature (2)

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[Crossref]

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[Crossref]

New J. Phys. (1)

A. Nicolas, L. Veissier, E. Giacobino, D. Maxein, and J. Laurat, “Quantum state tomography of orbital angular momentum photonic qubits via a projection-based technique,” New J. Phys. 17, 033037 (2015).
[Crossref]

Opt. Act. (1)

K. Ishiguro and T. Tako, “An estimation of Smith-Purcell effect as the light source in the infra-red region,” Opt. Act. 8, 25–31 (1961).
[Crossref]

Opt. Express (3)

Optica (1)

Phys. Rev. (1)

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92, 1069 (1953).
[Crossref]

Phys. Rev. A (3)

R. Remez, N. Shapira, C. Roques-Carmes, R. Tirole, Y. Yang, Y. Lereah, M. Soljacic, I. Kaminer, and A. Arie, “Spectral and spatial shaping of Smith-Purcell radiation,” Phys. Rev. A 96, 061801 (2017).
[Crossref]

M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. 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]

Phys. Rev. E (1)

K. Ishi, Y. Shibata, T. Takahashi, S. Hasebe, M. Ikezawa, K. Takami, T. Matsuyama, K. Kobayashi, and Y. Fujita, “Observation of coherent Smith-Purcell radiation from short-bunched electrons,” Phys. Rev. E 51, R5212–R5215 (1995).
[Crossref]

Phys. Rev. Lett. (7)

J. Urata, M. Goldstein, M. F. Kimmitt, A. Naumov, C. Platt, and J. E. Walsh, “Superradiant Smith-Purcell emission,” Phys. Rev. Lett. 80, 516–519 (1998).
[Crossref]

G. Doucas, J. H. Mulvey, M. Omori, J. Walsh, and M. F. Kimmitt, “First observation of Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 69, 1761–1764 (1992).
[Crossref]

K. J. Woods, J. E. Walsh, R. E. Stoner, H. G. Kirk, and R. C. Fernow, “Forward directed Smith-Purcell radiation from relativistic electrons,” Phys. Rev. Lett. 74, 3808–3811 (1995).
[Crossref]

Z. Wang, K. Yao, M. Chen, H. Chen, and Y. Liu, “Manipulating Smith-Purcell emission with Babinet metasurfaces,” Phys. Rev. Lett. 117, 157401 (2016).
[Crossref]

S. E. Korbly, A. S. Kesar, J. R. Sirigiri, and R. J. Temkin, “Observation of frequency-locked coherent terahertz Smith-Purcell radiation,” Phys. Rev. Lett. 94, 054803 (2005).
[Crossref]

G. Adam, K. F. MacDonald, N. I. Zheludev, Y. H. Fu, C. M. Wang, D. P. Tsai, and F. J. GarciadeAbajo, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103, 113901 (2009).
[Crossref]

J. Gardelle, P. Modin, and J. T. Donohue, “Start current and gain measurements for a Smith-Purcell free-electron laser,” Phys. Rev. Lett. 105, 224801 (2010).
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K. H. Lee, I. Ahmed, R. S. M. Goh, E. H. Khoo, E. P. Li, and T. G. G. Hung, “Implementation of the FDTD method based on Lorentz-Drude dispersive model on GPU for plasmonics applications,” Prog. Electromagn. Res. 116, 441–456 (2011).
[Crossref]

Research (1)

L. Jing, Z. Wang, X. Lin, B. Zheng, S. Xu, L. Shen, Y. Yang, F. Gao, M. Chen, and H. Chen, “Spiral field generation in Smith-Purcell radiation by helical metagratings,” Research 2019, 3806132 (2019).
[Crossref]

Sci. Rep. (1)

Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5, 10958 (2015).
[Crossref]

Science (1)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

Other (4)

L. Xiao, J. Chen, L. Chen, Q. Zhang, L. Guo, and M. Yang, “Electron beam excited surface plasmon polaritons carrying orbital angular momentum,” in 12th International Symposium on Antennas, Propagation and EM Theory (ISAPE) (IEEE, 2018), pp. 1–3.

R. Collier, Optical Holography (Elsevier, 2013).

Y. Yang, C. Roques-Carmes, I. Kaminer, A. Zaidi, A. Massuda, Y. Yang, S. E. Kooi, K. Berggren, and S. Marin, “Manipulating Smith-Purcell radiation polarization with metasurfaces,” in CLEO: QELS_Fundamental Science (2018), paper FW4H.1.

A. Massuda, C. Roques-Carmes, A. Solanki, Y. Yang, S. E. Kooi, F. Habbal, I. Kaminer, and S. Marin, “High-order Smith-Purcell radiation in silicon nanowires,” in CLEO: QELS_Fundamental Science (2017), paper JTh5B.8.

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Figures (7)

Fig. 1.
Fig. 1. Schematic diagram of generating vortex Smith–Purcell radiation (with a TC of l=1 for example) with free-electron bunches and holographic grating. The free electrons are periodically bunched in the y direction with pitch Λ much smaller than the radiating wavelength. The velocity of the electron bunches is v. The holographic grating (top view is shown in the inset) is made of metal or dielectrics depending on the frequency of VSPR and has a fork structure with two teeth. When periodic free-electron bunches pass on the holographic grating, an orbital angular momentum (OAM) wave would be emitted at an angle θ with respect to the y axis.
Fig. 2.
Fig. 2. Simulation of a THz OAM wave with TCs of (a) 0 (traditional SPR from periodic grating), (b) 1, (c) 2, (d) 3, (e) 4, and (f) 10. Left: Top view of the gratings (gray part represents PEC, white part represents vacuum), in which the teeth of the fork structures are marked. Middle: Side view of the Ey field (Ey in the yz plane, the observation planes are shown as red-dashed lines in the right colomn) of the OAM wave with different TCs. The radiation angle is about 90°, and the vortex wave propagates along the z direction. Right: Top view (the observation planes are shown as green-dashed lines in the middle column) of the Ey field above the grating, which clearly shows the vortex shape and the phase change of l×2π (l=0,1,2,3,4, and 10) with φ varying 2π.
Fig. 3.
Fig. 3. Simulation of THz VSPR with mixed mode of l=1 and l=2 (ratio of intensity: 0.5:0.5). (a) The holographic grating used in this simulation; (b) side view of the radiation field (Ey in the yz plane); (c) top view of the radiation field (Ey in the xy plane). The observation plane is 10 mm away from the holographic grating. (d) Normalized OAM spectrum of the radiated beam, in which the intensity is 0.40 for the l=1 mode and 0.41 for the l=2 mode.
Fig. 4.
Fig. 4. OAM wave at f=1  THz with l=1 generated by electrons with velocity of (a) v=c/3 (Ek=31.0  keV), (b) v=c/10 (Ek=2.57  keV), (c) v=c/20 (Ek=0.640  keV). Left: The holographic grating for different electron velocities. Middle: Side view of the radiated Ey field (the observation planes are shown as the red-dashed lines in the right colomn). The green rectangles circle the holographic gratings. Right: Top view of the radiated Ey field. The observation planes for the right figures are perpendicular to the z axis (shown as the green-dashed lines in the middle colomn). The field is normalized individually in each figure.
Fig. 5.
Fig. 5. OAM wave generated by free electrons with bunching frequency of (a) 0.68 THz, (b) 0.6 THz, and (c) 0.43 THz. Left, Ey field in the yz plane. Right, Ey field of the OAM wave in the observation plane perpendicular to the propagating direction (corresponding to the green dashed line). The corresponding radiation frequency (wavelength) is 1.36 THz (220 μm), 1.2 THz (250 μm), and 0.86 THz (350 μm), which follows the formula raised by Smith and Purcell. The parameters of the holographic grating are the same as those of Fig. 2(c), and the free-electron velocity is v=c/3.
Fig. 6.
Fig. 6. (a) Schematic of the Huygens construction for conventional SPR with a radiation angle of θ=90°. A train of free-electron bunches passes above a conventional grating. The red and black dashed arc lines represent the wave fronts of the wavelets generated by the electrons in respective color, while the gray ones show the equiphase surfaces of the wavelets corresponding to second-order SPR, which has 2π×N (N=1,2,3) phase delay compared with the wave front. The blue and orange lines show the equiphase surfaces of the SPR. Three-dimensional schematics of the equiphase surfaces of the conventional SPR are shown below. The distances between each two adjacent equiphase surfaces (namely, wavelength) are λ1 and λ1/2 for the first- and second-order SPR, respectively. (b) Schematic of the equiphase surface of the first- and second-order OAM generated by holographic grating. (c) Second-harmonic OAM wave at frequency of 2 THz. The holographic gratings shown in the inset are the same as those illustrated in the left figures of Fig. 2. The TC of second-harmonic OAM wave is doubled compared with that of fundamental mode.
Fig. 7.
Fig. 7. OAM wave generated in different frequency regions. The field plotted in each figure is normalized individually. The holographic gratings for different frequencies have similar profile to the left figure in Fig. 2(d), but the scales of the holographic gratings and material permittivity are different, which are shown in Table 1.

Tables (1)

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Table 1. Parameters of the Simulations in Different Frequency Regions