Abstract

Smith-Purcell radiation is emitted when an electron passes above the surface of a metallic grating. Its mechanism can be explained with Huygens’ principle by the radiation of a moving oscillating dipole, which is formed by the moving charge and its image in the metallic grating. Here, an alternative way is presented to enhance the THz Smith-Purcell radiation. By drilling a hole in the fins of a grating as an effective electron channel, the oscillation dipole happens in two dimensions here, instead of one dimension. As a result, the Smith-Purcell radiation power is ten times more than the case in which the electron passes very close to the grating surface. This method is expected to improve the efficiency of the devices which are based on the Smith-Purcell radiation.

© 2017 Optical Society of America

Full Article  |  PDF Article

Corrections

10 May 2017: A correction was made to Fig. 1.


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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2016 (2)

Y. Zhou, Y. Zhang, and S. Liu, “Electron-beam-driven enhanced terahertz coherent Smith-Purcell radiation within a cylindrical quasi-optical cavity,” IEEE Trans. THz Sci. Technol. 6(2), 262–267 (2016).

Y. Kalkal and V. Kumar, “Three-dimensional analysis of the surface mode supported in Čerenkov and Smith-Purcell free-electron lasers,” Phys. Rev. Accel. Beams 19(6), 060702 (2016).
[Crossref]

2015 (3)

M. Cao, W. Liu, Y. Wang, and K. Li, “Enhance the terahertz Smith-Purcell superradiant radiation by using dielectric loaded grating,” Phys. Plasmas 22(8), 083107 (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 Photonics 2(9), 1236–1240 (2015).
[Crossref]

L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, and M. Soljačić, “Towards graphene plasmon-based free-electron infrared to X-ray sources,” Nat. Photonics 10(1), 46–52 (2015).
[Crossref]

2014 (2)

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

2012 (3)

Y. Zhang, L. Dong, and Y. Zhou, “Enhanced coherent terahertz Smith-Purcell superradiation excited by two electron-beams,” Opt. Express 20(20), 22627–22635 (2012).
[Crossref] [PubMed]

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

2011 (1)

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

2009 (3)

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

S. Liu, M. Hu, Y. Zhang, Y. Li, and R. Zhong, “Electromagnetic diffraction radiation of a subwavelength-hole array excited by an electron beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 036602 (2009).
[Crossref] [PubMed]

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

2007 (3)

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

S. Taga, K. Inafune, and E. Sano, “Analysis of Smith-Purcell radiation in optical region,” Opt. Express 15(24), 16222–16229 (2007).
[Crossref] [PubMed]

2006 (1)

D. Li, Z. Yang, K. Imasaki, and G. S. Park, “Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9(4), 040701 (2006).
[Crossref]

2005 (2)

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(5), 054803 (2005).
[Crossref] [PubMed]

A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, “Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016501 (2005).
[Crossref] [PubMed]

2004 (1)

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

2002 (2)

S. Yamaguti, J. I. Inoue, O. Haeberlé, and K. Ohtaka, “Photonic crystals versus diffraction gratings in Smith-Purcell radiation,” Phys. Rev. B 66(19), 195202 (2002).
[Crossref]

K. Ohtaka and S. Yamaguti, “Smith-Purcell radiation from a charge running near the surface of a photonic crystal,” Opt. Quantum Electron. 34(1), 235–250 (2002).
[Crossref]

1989 (1)

L. Schächter and A. Ron, “Smith-Purcell free-electron laser,” Phys. Rev. A Gen. Phys. 40(2), 876–896 (1989).
[Crossref] [PubMed]

1984 (1)

1973 (2)

1953 (1)

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4), 1069 (1953).
[Crossref]

Adamo, G.

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

Akselrod, G. M.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Andrews, H. L.

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

Argyropoulos, C.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Arnold, R.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Biedron, S.

G. P. Gallerano and S. Biedron, “Overview of terahertz radiation sources,” in Proceedings of the 2004 FEL Conference (2004), pp. 216–221.

Blackmore, V.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Brau, C. A.

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

Cao, M.

M. Cao, W. Liu, Y. Wang, and K. Li, “Enhance the terahertz Smith-Purcell superradiant radiation by using dielectric loaded grating,” Phys. Plasmas 22(8), 083107 (2015).
[Crossref]

Chen, X.

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

Chuang, S. L.

Ciracì, C.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

de Abajo, F. J.

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

Dong, L.

Doucas, G.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Fang, C.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Fu, Y. H.

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

Gallerano, G. P.

G. P. Gallerano and S. Biedron, “Overview of terahertz radiation sources,” in Proceedings of the 2004 FEL Conference (2004), pp. 216–221.

García de Abajo, F. J.

J. K. So, F. J. García de Abajo, K. F. MacDonald, and N. I. Zheludev, “Amplification of the evanescent field of free electrons,” ACS Photonics 2(9), 1236–1240 (2015).
[Crossref]

Gong, S.

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

Haeberlé, O.

S. Yamaguti, J. I. Inoue, O. Haeberlé, and K. Ohtaka, “Photonic crystals versus diffraction gratings in Smith-Purcell radiation,” Phys. Rev. B 66(19), 195202 (2002).
[Crossref]

Hess, M.

A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, “Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016501 (2005).
[Crossref] [PubMed]

Hoang, T. B.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Hu, M.

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

S. Liu, M. Hu, Y. Zhang, Y. Li, and R. Zhong, “Electromagnetic diffraction radiation of a subwavelength-hole array excited by an electron beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 036602 (2009).
[Crossref] [PubMed]

Huang, J.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Ilic, O.

L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, and M. Soljačić, “Towards graphene plasmon-based free-electron infrared to X-ray sources,” Nat. Photonics 10(1), 46–52 (2015).
[Crossref]

Imasaki, K.

D. Li, Z. Yang, K. Imasaki, and G. S. Park, “Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9(4), 040701 (2006).
[Crossref]

Inafune, K.

Inoue, J. I.

S. Yamaguti, J. I. Inoue, O. Haeberlé, and K. Ohtaka, “Photonic crystals versus diffraction gratings in Smith-Purcell radiation,” Phys. Rev. B 66(19), 195202 (2002).
[Crossref]

Jang, K. H.

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

Joannopoulos, J. D.

L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, and M. Soljačić, “Towards graphene plasmon-based free-electron infrared to X-ray sources,” Nat. Photonics 10(1), 46–52 (2015).
[Crossref]

Kalkal, Y.

Y. Kalkal and V. Kumar, “Three-dimensional analysis of the surface mode supported in Čerenkov and Smith-Purcell free-electron lasers,” Phys. Rev. Accel. Beams 19(6), 060702 (2016).
[Crossref]

Kaminer, I.

L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, and M. Soljačić, “Towards graphene plasmon-based free-electron infrared to X-ray sources,” Nat. Photonics 10(1), 46–52 (2015).
[Crossref]

Kesar, A. S.

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(5), 054803 (2005).
[Crossref] [PubMed]

A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, “Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016501 (2005).
[Crossref] [PubMed]

Kimmitt, M. F.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Kong, J. A.

Korbly, S. E.

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(5), 054803 (2005).
[Crossref] [PubMed]

A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, “Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016501 (2005).
[Crossref] [PubMed]

Kumar, V.

Y. Kalkal and V. Kumar, “Three-dimensional analysis of the surface mode supported in Čerenkov and Smith-Purcell free-electron lasers,” Phys. Rev. Accel. Beams 19(6), 060702 (2016).
[Crossref]

Li, D.

D. Li, Z. Yang, K. Imasaki, and G. S. Park, “Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9(4), 040701 (2006).
[Crossref]

Li, K.

M. Cao, W. Liu, Y. Wang, and K. Li, “Enhance the terahertz Smith-Purcell superradiant radiation by using dielectric loaded grating,” Phys. Plasmas 22(8), 083107 (2015).
[Crossref]

Li, Y.

S. Liu, M. Hu, Y. Zhang, Y. Li, and R. Zhong, “Electromagnetic diffraction radiation of a subwavelength-hole array excited by an electron beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 036602 (2009).
[Crossref] [PubMed]

Liu, S.

Y. Zhou, Y. Zhang, and S. Liu, “Electron-beam-driven enhanced terahertz coherent Smith-Purcell radiation within a cylindrical quasi-optical cavity,” IEEE Trans. THz Sci. Technol. 6(2), 262–267 (2016).

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

S. Liu, M. Hu, Y. Zhang, Y. Li, and R. Zhong, “Electromagnetic diffraction radiation of a subwavelength-hole array excited by an electron beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 036602 (2009).
[Crossref] [PubMed]

Liu, S. G.

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

Liu, W.

M. Cao, W. Liu, Y. Wang, and K. Li, “Enhance the terahertz Smith-Purcell superradiant radiation by using dielectric loaded grating,” Phys. Plasmas 22(8), 083107 (2015).
[Crossref]

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

Liu, W. H.

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

MacDonald, K. F.

J. K. So, F. J. García de Abajo, K. F. MacDonald, and N. I. Zheludev, “Amplification of the evanescent field of free electrons,” ACS Photonics 2(9), 1236–1240 (2015).
[Crossref]

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

Mikkelsen, M. H.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Molloy, S.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Ohtaka, K.

S. Yamaguti, J. I. Inoue, O. Haeberlé, and K. Ohtaka, “Photonic crystals versus diffraction gratings in Smith-Purcell radiation,” Phys. Rev. B 66(19), 195202 (2002).
[Crossref]

K. Ohtaka and S. Yamaguti, “Smith-Purcell radiation from a charge running near the surface of a photonic crystal,” Opt. Quantum Electron. 34(1), 235–250 (2002).
[Crossref]

Ottewell, B.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Park, G. S.

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

D. Li, Z. Yang, K. Imasaki, and G. S. Park, “Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9(4), 040701 (2006).
[Crossref]

Perry, C.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Purcell, E. M.

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4), 1069 (1953).
[Crossref]

Ron, A.

L. Schächter and A. Ron, “Smith-Purcell free-electron laser,” Phys. Rev. A Gen. Phys. 40(2), 876–896 (1989).
[Crossref] [PubMed]

Sano, E.

Schächter, L.

L. Schächter and A. Ron, “Smith-Purcell free-electron laser,” Phys. Rev. A Gen. Phys. 40(2), 876–896 (1989).
[Crossref] [PubMed]

Shin, Y. M.

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

Sirigiri, J. R.

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(5), 054803 (2005).
[Crossref] [PubMed]

Smith, D. R.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

Smith, S. J.

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4), 1069 (1953).
[Crossref]

So, J. K.

J. K. So, F. J. García de Abajo, K. F. MacDonald, and N. I. Zheludev, “Amplification of the evanescent field of free electrons,” ACS Photonics 2(9), 1236–1240 (2015).
[Crossref]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

Soljacic, M.

L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, and M. Soljačić, “Towards graphene plasmon-based free-electron infrared to X-ray sources,” Nat. Photonics 10(1), 46–52 (2015).
[Crossref]

Srivastava, A.

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

Taga, S.

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(5), 054803 (2005).
[Crossref] [PubMed]

A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, “Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016501 (2005).
[Crossref] [PubMed]

Tsai, D. P.

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

Van den Berg, P. M.

Wang, C. M.

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

Wang, Y.

M. Cao, W. Liu, Y. Wang, and K. Li, “Enhance the terahertz Smith-Purcell superradiant radiation by using dielectric loaded grating,” Phys. Plasmas 22(8), 083107 (2015).
[Crossref]

Won, J. H.

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

Wong, L. J.

L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, and M. Soljačić, “Towards graphene plasmon-based free-electron infrared to X-ray sources,” Nat. Photonics 10(1), 46–52 (2015).
[Crossref]

Woods, M.

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Yamaguti, S.

S. Yamaguti, J. I. Inoue, O. Haeberlé, and K. Ohtaka, “Photonic crystals versus diffraction gratings in Smith-Purcell radiation,” Phys. Rev. B 66(19), 195202 (2002).
[Crossref]

K. Ohtaka and S. Yamaguti, “Smith-Purcell radiation from a charge running near the surface of a photonic crystal,” Opt. Quantum Electron. 34(1), 235–250 (2002).
[Crossref]

Yang, Z.

D. Li, Z. Yang, K. Imasaki, and G. S. Park, “Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9(4), 040701 (2006).
[Crossref]

Zhang, C.

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

Zhang, P.

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhou, Y. Zhang, and S. Liu, “Electron-beam-driven enhanced terahertz coherent Smith-Purcell radiation within a cylindrical quasi-optical cavity,” IEEE Trans. THz Sci. Technol. 6(2), 262–267 (2016).

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

Y. Zhang, L. Dong, and Y. Zhou, “Enhanced coherent terahertz Smith-Purcell superradiation excited by two electron-beams,” Opt. Express 20(20), 22627–22635 (2012).
[Crossref] [PubMed]

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

S. Liu, M. Hu, Y. Zhang, Y. Li, and R. Zhong, “Electromagnetic diffraction radiation of a subwavelength-hole array excited by an electron beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 036602 (2009).
[Crossref] [PubMed]

Zhao, T.

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[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 Photonics 2(9), 1236–1240 (2015).
[Crossref]

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

Zhong, R.

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

S. Liu, M. Hu, Y. Zhang, Y. Li, and R. Zhong, “Electromagnetic diffraction radiation of a subwavelength-hole array excited by an electron beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 036602 (2009).
[Crossref] [PubMed]

Zhong, R. B.

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

Zhou, J.

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

Zhou, Y.

Y. Zhou, Y. Zhang, and S. Liu, “Electron-beam-driven enhanced terahertz coherent Smith-Purcell radiation within a cylindrical quasi-optical cavity,” IEEE Trans. THz Sci. Technol. 6(2), 262–267 (2016).

Y. Zhang, L. Dong, and Y. Zhou, “Enhanced coherent terahertz Smith-Purcell superradiation excited by two electron-beams,” Opt. Express 20(20), 22627–22635 (2012).
[Crossref] [PubMed]

ACS Photonics (1)

J. K. So, F. J. García de Abajo, K. F. MacDonald, and N. I. Zheludev, “Amplification of the evanescent field of free electrons,” ACS Photonics 2(9), 1236–1240 (2015).
[Crossref]

Appl. Phys. Lett. (2)

S. Liu, C. Zhang, M. Hu, X. Chen, P. Zhang, S. Gong, T. Zhao, and R. Zhong, “Coherent and tunable terahertz radiation from graphene surface plasmon polaritons excited by an electron beam,” Appl. Phys. Lett. 104(20), 201104 (2014).
[Crossref]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Superradiant terahertz Smith-Purcell radiation from surface plasmon excited by counterstreaming electron beams,” Appl. Phys. Lett. 90(3), 031502 (2007).
[Crossref]

Chin. Phys. B (1)

P. Zhang, Y. Zhang, J. Zhou, W. H. Liu, R. B. Zhong, and S. G. Liu, “Enhancement of Smith-Purcell radiation with surface-plasmon excitation,” Chin. Phys. B 21(10), 104102 (2012).
[Crossref]

IEEE Trans. THz Sci. Technol. (1)

Y. Zhou, Y. Zhang, and S. Liu, “Electron-beam-driven enhanced terahertz coherent Smith-Purcell radiation within a cylindrical quasi-optical cavity,” IEEE Trans. THz Sci. Technol. 6(2), 262–267 (2016).

J. Opt. Soc. Am. (2)

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

J. Phys. D Appl. Phys. (1)

P. Zhang, Y. Zhang, M. Hu, W. Liu, J. Zhou, and S. Liu, “Diffraction radiation of a sub-wavelength hole array with dielectric medium loading,” J. Phys. D Appl. Phys. 45(14), 145303 (2012).
[Crossref]

Nat. Photonics (2)

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8(11), 835–840 (2014).
[Crossref]

L. J. Wong, I. Kaminer, O. Ilic, J. D. Joannopoulos, and M. Soljačić, “Towards graphene plasmon-based free-electron infrared to X-ray sources,” Nat. Photonics 10(1), 46–52 (2015).
[Crossref]

Opt. Express (2)

Opt. Quantum Electron. (1)

K. Ohtaka and S. Yamaguti, “Smith-Purcell radiation from a charge running near the surface of a photonic crystal,” Opt. Quantum Electron. 34(1), 235–250 (2002).
[Crossref]

Phys. Plasmas (1)

M. Cao, W. Liu, Y. Wang, and K. Li, “Enhance the terahertz Smith-Purcell superradiant radiation by using dielectric loaded grating,” Phys. Plasmas 22(8), 083107 (2015).
[Crossref]

Phys. Rev. (1)

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4), 1069 (1953).
[Crossref]

Phys. Rev. A Gen. Phys. (1)

L. Schächter and A. Ron, “Smith-Purcell free-electron laser,” Phys. Rev. A Gen. Phys. 40(2), 876–896 (1989).
[Crossref] [PubMed]

Phys. Rev. Accel. Beams (1)

Y. Kalkal and V. Kumar, “Three-dimensional analysis of the surface mode supported in Čerenkov and Smith-Purcell free-electron lasers,” Phys. Rev. Accel. Beams 19(6), 060702 (2016).
[Crossref]

Phys. Rev. B (1)

S. Yamaguti, J. I. Inoue, O. Haeberlé, and K. Ohtaka, “Photonic crystals versus diffraction gratings in Smith-Purcell radiation,” Phys. Rev. B 66(19), 195202 (2002).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

S. Liu, M. Hu, Y. Zhang, Y. Li, and R. Zhong, “Electromagnetic diffraction radiation of a subwavelength-hole array excited by an electron beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(3), 036602 (2009).
[Crossref] [PubMed]

A. S. Kesar, M. Hess, S. E. Korbly, and R. J. Temkin, “Time- and frequency-domain models for Smith-Purcell radiation from a two-dimensional charge moving above a finite length grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016501 (2005).
[Crossref] [PubMed]

S. Liu, M. Hu, Y. Zhang, W. Liu, P. Zhang, and J. Zhou, “Theoretical investigation of a tunable free-electron light source,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(6), 066609 (2011).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

G. Adamo, K. F. MacDonald, Y. H. Fu, C. M. Wang, D. P. Tsai, F. J. de Abajo, and N. I. Zheludev, “Light well: a tunable free-electron light source on a chip,” Phys. Rev. Lett. 103(11), 113901 (2009).
[Crossref] [PubMed]

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(5), 054803 (2005).
[Crossref] [PubMed]

Y. M. Shin, J. K. So, K. H. Jang, J. H. Won, A. Srivastava, and G. S. Park, “Evanescent tunneling of an effective surface plasmon excited by convection electrons,” Phys. Rev. Lett. 99(14), 147402 (2007).
[Crossref] [PubMed]

Phys. Rev. Spec. Top. Accel. Beams (3)

D. Li, Z. Yang, K. Imasaki, and G. S. Park, “Particle-in-cell simulation of coherent and superradiant Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9(4), 040701 (2006).
[Crossref]

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

V. Blackmore, G. Doucas, C. Perry, B. Ottewell, M. F. Kimmitt, M. Woods, S. Molloy, and R. Arnold, “First measurements of the longitudinal bunch profile of a 28.5 GeV beam using coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 12(3), 032803 (2009).
[Crossref]

Other (4)

G. P. Gallerano and S. Biedron, “Overview of terahertz radiation sources,” in Proceedings of the 2004 FEL Conference (2004), pp. 216–221.

L. Schächter, Beam-wave interaction in periodic and quasi-periodic structures (Springer Science & Business Media, 2011).

S. G. Liu, H. F. Li, W. X. Wang, and Y. L. Mo, Introduction to Microwave Electronics (National Defense Industry Press, 1985).

K. Q. Zhang and D. J. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer Science & Business Media, 2013).

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Figures (9)

Fig. 1
Fig. 1 (a) The electron is moving above the grating. From the time t0 to t1, the distances of the electron and its image charge are variable, shown in (b) and (c), and it forms an oscillation dipole. (d) The electron is moving through the hole in the structure of a hole in the fins of a grating, and there are four dipole oscillations forming, shown as (e) and (f). At the time of t1, there is no metal in the X direction and the positive Y direction, it means the location of the image charge is infinite.
Fig. 2
Fig. 2 the schematic of Hole-grating, a hole is drilled in the fins of the grating.
Fig. 3
Fig. 3 The dispersion relation of the hole-grating structure. (a) the dispersion relation v.s. hole shape (hole width), (b) the dispersion relation v.s. hole position, which denotes the distance between the lowest position of the groove and the center position of the hole, (c) The dispersion relation v.s. grating period, and (d) the dispersion relation v.s. grating groove depth.
Fig. 4
Fig. 4 The electron beam channels are compared in two structure models. (a) A traditional grating structure, with the electron beam moving above it. (b) The corresponding Ez field distributions in transverse and longitudinal views. (c) A hole is drilled in the fins of the grating as an electron channel. (d) The corresponding Ez field distributions in transverse and longitudinal views. The e-beam channels are shown in red dot square in Figs. 4(b) and 4(d).
Fig. 5
Fig. 5 The PIC simulation results. (a) The Ez field contour map for the grating structure; (b) the Ez field contour map for the hole-grating structure; and (c) the comparison of surface waves at the measurement line.
Fig. 6
Fig. 6 The comparison of surface waves in time domain at the same observation points.
Fig. 7
Fig. 7 The comparison of SP radiation. (a) The Ez field contour map for the hole-grating structure, (b) the comparison of SP radiation field between the traditional grating and the hole-grating structure, and (c) the corresponding spectrum of the radiation field.
Fig. 8
Fig. 8 (a) The schematic of the grating with hole in the fingers excited by the well bunched electron beam, and (b) the SP super-radiation.
Fig. 9
Fig. 9 The simulation results of the SP radiation excited by a well-bunched electron beam. (a) The contour map of Ez fields, (b) the time domain of Ez field for grating and hole-grating structure, and (c) the corresponding spectrums.

Equations (2)

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λ = D m ( 1 β cos θ )
a D n = ( sin c k z n a 2 ) 2 j k y n h = 1 k h cot ( k h )

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