Smith-Purcell radiation from periodic beams

D. Y. Sergeeva; A. P. Potylitsyn; A. A. Tishchenko; M. N. Strikhanov

doi:10.1364/OE.25.026310

Smith-Purcell radiation from periodic beams

D. Y. Sergeeva, A. P. Potylitsyn, A. A. Tishchenko, and M. N. Strikhanov

Optics Express
Vol. 25,
Issue 21,
pp. 26310-26328
(2017)
•https://doi.org/10.1364/OE.25.026310

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D. Y. Sergeeva, A. P. Potylitsyn, A. A. Tishchenko, and M. N. Strikhanov, "Smith-Purcell radiation from periodic beams," Opt. Express 25, 26310-26328 (2017)

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Related Topics
Table of Contents Category
- Diffraction and Gratings
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Lasers, frequency modulated (140.3518)
- Diffraction gratings (230.1950)
- Sources (230.6080)
- Traveling-wave devices (230.7020)
- Radiation (350.5610)

About this Article
History
- Original Manuscript: September 1, 2017
- Revised Manuscript: October 10, 2017
- Manuscript Accepted: October 11, 2017
- Published: October 13, 2017

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Open Access

Abstract

Smith-Purcell effect is well known as a source of monochromatic electromagnetic radiation. In this paper we present the generalized theory of Smith-Purcell radiation from periodic beams. The form-factors describing both coherent and incoherent regimes of radiation are calculated. The radiation characteristics are investigated in two practically important frequency ranges, THz and X-ray, for two modulation profiles, most frequently used in practice – a train of microbunches and a Gaussian-shaped one, characterized by sinusoidal modulation with an arbitrary modulation depth. On the base of the theory developed we show that a modulated electron beam consisting of a set of bunches makes it possible to improve significantly the spectral line monochromaticity of the light emitted, reaching values better than 1% for short gratings. We demonstrate as well that Smith-Purcell radiation can be used for non-destructive diagnostics of the depth of modulation for partially modulated beams. These findings not only open up a new way to manipulate the light emission using Smith-Purcell effect but also promise a profound impact for other radiation sources based on charged particle beams, such as undulator radiation in FELs, next-generation X-ray radiation source based on inverse Compton scattering, in a wide range from THz to X-rays.

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References

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D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Conical diffraction effect in optical and x-ray Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 18, 052801 (2015).
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, 032803 (2009).
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D. Li, M. Hangyo, Y. Tsunawaki, Z. Yang, Y. Wei, S. Miyamoto, M. Asakawa, and K. Imasakia, “Super-radiant Smith-Purcell radiation from periodic line charges”, Nucl. Instrum. Methods Phys. Res., Sect. A 674, 20–23 (2012).
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A. Gover, “Superradiant and stimulated-superradiant emission in prebunched electron-beam radiators. I. Formulation,” Phys. Rev. Spec. Top. Accel. Beams 8, 030701 (2005).
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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).
H. L. Andrews, C. H. Boulware, C. A. Brau, and J. D. Jarvis, “Dispersion and attenuation in a Smith-Purcell free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 8, 050703 (2005).
D. Li, Y. Wang, M. Nakajima, M. Tani, M. Hashida, M. R. Asakawa, Y. Wei, and S. Miyamoto, “Coherent radiation at the fundamental frequency by a Smith-Purcell free-electron laser with dielectric substrate,” Appl. Phys. Lett. 110, 151108 (2017).
Y. E. Sun, P. Piot, A. Johnson, A. H. Lumpkin, T. J. Maxwell, J. Ruan, and R. Thurman-Keup, “Tunable subpicosecond electron-bunch-train generation using a transverse-to-longitudinal phase-space exchange technique,” Phys. Rev. Lett. 105(23), 234801 (2010).
[PubMed]
W. S. Graves, F. X. Kärtner, D. E. Moncton, and P. Piot, “Intense Superradiant X Rays from a Compact Source Using a Nanocathode Array and Emittance Exchange,” Phys. Rev. Lett. 108(26), 263904 (2012).
[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).
[PubMed]
A. S. Kesar, R. A. Marsh, and R. J. Temkin, “Power measurement of frequency-locked Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9, 022801 (2006).
G. A. Naumenko, A. P. Potylitsyn, D. Yu. Sergeeva, A. A. Tishchenko, M. N. Strikhanov, and V. V. Bleko, “First experimental observation of the conical effect in Smith-Purcell radiation,” JETP Lett. 105, 553–560 (2017).
O. Haeberle, P. Rullhusen, J.-M. Salome, and N. Maene, “Smith-Purcell radiation from electrons moving parallel to a gratingat oblique incidence to the rulings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55, 4675 (1997).
D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Microscopic theory of Smith-Purcell radiation from 2D photonic crystal,” Nucl. Instrum. Methods Phys. Res. B 402, 206–211 (2017).
H. Zhang, I. V. Konoplev, A. J. Lancaster, H. Harrison, G. Doucas, A. Aryshev, M. Shevelev, N. Terunuma, and J. Urakawa, “Non-destructive measurement and monitoring of separation of charged particle micro-bunches,” Appl. Phys. Lett. 111, 043505 (2017).
P. Piot, Y.-E. Sun, T. J. Maxwell, J. Ruan, A. H. Lumpkin, M. M. Rihaoui, and R. Thurman-Keup, “Observation of coherently enhanced tunable narrow-band terahertz transition radiation from a relativistic sub-picosecond electron bunch train,” Appl. Phys. Lett. 98, 261501 (2011).
Y. Zhang and L. Dong, “Enhanced coherent terahertz Smith-Purcell superradiation excited by two electron-beams,” Opt. Express 20(20), 22627–22635 (2012).
[PubMed]
P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).
L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).
A. P. Potylitsyn, M. I. Ryazanov, M. N. Strikhanov, and A. A. Tishchenko, Diffraction Radiation from Relativistic Particles (Springer, 2011).
D. Yu. Sergeeva and A. A. Tishchenko, “X-ray Smith-Purcell radiation from a beam skimming a grating surface,” in Proceedings of the 36th International Free Electron Laser Conference, J. Chrin, ed. (JACoW, 2015), pp. 378–383.
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G. Doucas, M. F. Kimmitt, A. Doria, G. P. Gallerano, E. Giovenale, G. Messina, H. L. Andrews, and J. H. Brownell, “Determination of longitudinal bunch shape by means of coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 5, 072802 (2002).
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[PubMed]
P. Musumeci, C. Pellegrini, and J. B. Rosenzweig, “Higher harmonic inverse free-electron laser interaction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016501 (2005).
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J. C. T. Thangaraj, R. Thurman-Keup, J. Ruan, A. S. Johnson, A. H. Lumpkin, and J. Santucci, “Experimental studies on coherent synchrotron radiation at an emittance exchange beam line,” Phys. Rev. Spec. Top. Accel. Beams 15, 110702 (2012).
B. Marchetti, A. Bacci, E. Chiadroni, A. Cianchi, M. Ferrario, A. Mostacci, R. Pompili, C. Ronsivalle, B. Spataro, and I. Zagorodnov, “Novel schemes for the optimization of the SPARC narrow band THz source,” Rev. Sci. Instrum. 86(7), 073301 (2015).
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[PubMed]
H. Backe, W. Lauth, H. Mannweiler, H. Rochholz, K. Aulenbacher, R. Barday, H. Euteneuer, K.-H. Kaiser, G. Kube, F. Schwellnus, and V. Tioukine, “Investigation of Far-Infrared Smith-Purcell Radiation at the 3.41 MeV Electron Injector LINAC of the Mainz Microtron MAMI,” in Advanced Radiation Sources and Applications, H. Wiedemann, ed. (Springer, 2006).
X. Z. Meng, M. H. Wang, L. M. Zhang, and Z. M. Ren, “Characteristic analysis of a Smith–Purcell terahertz source,” Photonics Res. 4, 162–167 (2016).
A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B 1, 723–728 (1984).
A. P. Kazantsev and G. I. Surdutovich, “Radiation of a charged particle flying by metal screen,” Sov. Phys. Dokl. 7, 990 (1963).
G. Kube, H. Backe, H. Euteneuer, A. Grendel, F. Hagenbuck, H. Hartmann, K. H. Kaiser, W. Lauth, H. Schöpe, G. Wagner, T. Walcher, and M. Kretzschmar, “Observation of optical Smith-Purcell radiation at an electron beam energy of 855 MeV,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056501 (2002).
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[PubMed]
D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Influence of beam divergence on form-factor in X-ray diffraction radiation,” Nucl. Instrum. Methods Phys. Res. Sect. B 355, 175–179 (2015).
A. P. Potylitsyn, P. V. Karataev, and G. A. Naumenko, “Resonant diffraction radiation from an ultrarelativistic particle moving close to a tilted grating,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(6 Pt B), 7039–7045 (2000).
[PubMed]
E. Chiadroni, M. Bellaveglia, P. Calvani, M. Castellano, L. Catani, A. Cianchi, G. Di Pirro, M. Ferrario, G. Gatti, O. Limaj, S. Lupi, B. Marchetti, A. Mostacci, E. Pace, L. Palumbo, C. Ronsivalle, R. Pompili, and C. Vaccarezza, “Characterization of the THz radiation source at the Frascati linear accelerator,” Rev. Sci. Instrum. 84(2), 022703 (2013).
[PubMed]

2017 (5)

D. Li, Y. Wang, M. Nakajima, M. Tani, M. Hashida, M. R. Asakawa, Y. Wei, and S. Miyamoto, “Coherent radiation at the fundamental frequency by a Smith-Purcell free-electron laser with dielectric substrate,” Appl. Phys. Lett. 110, 151108 (2017).

G. A. Naumenko, A. P. Potylitsyn, D. Yu. Sergeeva, A. A. Tishchenko, M. N. Strikhanov, and V. V. Bleko, “First experimental observation of the conical effect in Smith-Purcell radiation,” JETP Lett. 105, 553–560 (2017).

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Microscopic theory of Smith-Purcell radiation from 2D photonic crystal,” Nucl. Instrum. Methods Phys. Res. B 402, 206–211 (2017).

H. Zhang, I. V. Konoplev, A. J. Lancaster, H. Harrison, G. Doucas, A. Aryshev, M. Shevelev, N. Terunuma, and J. Urakawa, “Non-destructive measurement and monitoring of separation of charged particle micro-bunches,” Appl. Phys. Lett. 111, 043505 (2017).

L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).

2016 (3)

P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).

J. L. Steinmann, E. Blomley, M. Brosi, E. Bründermann, M. Caselle, J. L. Hesler, N. Hiller, B. Kehrer, Y.-L. Mathis, M. J. Nasse, J. Raasch, M. Schedler, P. Schönfeldt, M. Schuh, M. Schwarz, M. Siegel, N. Smale, M. Weber, and A.-S. Müller, “Frequency-comb spectrum of periodic-patterned signals,” Phys. Rev. Lett. 117(17), 174802 (2016).
[PubMed]

X. Z. Meng, M. H. Wang, L. M. Zhang, and Z. M. Ren, “Characteristic analysis of a Smith–Purcell terahertz source,” Photonics Res. 4, 162–167 (2016).

2015 (4)

B. Marchetti, A. Bacci, E. Chiadroni, A. Cianchi, M. Ferrario, A. Mostacci, R. Pompili, C. Ronsivalle, B. Spataro, and I. Zagorodnov, “Novel schemes for the optimization of the SPARC narrow band THz source,” Rev. Sci. Instrum. 86(7), 073301 (2015).
[PubMed]

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Influence of beam divergence on form-factor in X-ray diffraction radiation,” Nucl. Instrum. Methods Phys. Res. Sect. B 355, 175–179 (2015).

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Conical diffraction effect in optical and x-ray Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 18, 052801 (2015).

P. Zhang, L. K. Ang, and A. Gover, “Enhancement of coherent Smith-Purcell radiation at terahertz frequency by optimized grating, prebunched beams, and open cavity,” Phys. Rev. 18, 020702 (2015).

2014 (1)

H. L. Andrews, F. B. Taheri, J. Barros, R. Bartolini, L. Cassinari, C. Clarke, S. Le Corre, N. Delerue, G. Doucas, N. Fuster-Martinez, I. Konoplev, M. Labat, C. Perry, A. Reichold, S. Stevenson, and M. V. Grosjean, “Longitudinal profile monitors using Coherent Smith–Purcell radiation,” Nucl. Instrum. Methods Phys. Res., Sect. A 740, 212–215 (2014).

2013 (2)

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “X-ray Diffraction and Transition Radiation from Charged Particles Bunches,” Nucl. Instrum. Methods Phys. Res., Sect. B 309, 189–193 (2013).

E. Chiadroni, M. Bellaveglia, P. Calvani, M. Castellano, L. Catani, A. Cianchi, G. Di Pirro, M. Ferrario, G. Gatti, O. Limaj, S. Lupi, B. Marchetti, A. Mostacci, E. Pace, L. Palumbo, C. Ronsivalle, R. Pompili, and C. Vaccarezza, “Characterization of the THz radiation source at the Frascati linear accelerator,” Rev. Sci. Instrum. 84(2), 022703 (2013).
[PubMed]

2012 (4)

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

J. C. T. Thangaraj, R. Thurman-Keup, J. Ruan, A. S. Johnson, A. H. Lumpkin, and J. Santucci, “Experimental studies on coherent synchrotron radiation at an emittance exchange beam line,” Phys. Rev. Spec. Top. Accel. Beams 15, 110702 (2012).

D. Li, M. Hangyo, Y. Tsunawaki, Z. Yang, Y. Wei, S. Miyamoto, M. Asakawa, and K. Imasakia, “Super-radiant Smith-Purcell radiation from periodic line charges”, Nucl. Instrum. Methods Phys. Res., Sect. A 674, 20–23 (2012).

W. S. Graves, F. X. Kärtner, D. E. Moncton, and P. Piot, “Intense Superradiant X Rays from a Compact Source Using a Nanocathode Array and Emittance Exchange,” Phys. Rev. Lett. 108(26), 263904 (2012).
[PubMed]

2011 (1)

P. Piot, Y.-E. Sun, T. J. Maxwell, J. Ruan, A. H. Lumpkin, M. M. Rihaoui, and R. Thurman-Keup, “Observation of coherently enhanced tunable narrow-band terahertz transition radiation from a relativistic sub-picosecond electron bunch train,” Appl. Phys. Lett. 98, 261501 (2011).

2010 (1)

Y. E. Sun, P. Piot, A. Johnson, A. H. Lumpkin, T. J. Maxwell, J. Ruan, and R. Thurman-Keup, “Tunable subpicosecond electron-bunch-train generation using a transverse-to-longitudinal phase-space exchange technique,” Phys. Rev. Lett. 105(23), 234801 (2010).
[PubMed]

2009 (1)

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, 032803 (2009).

2008 (2)

Y. Li, Y.-E. Sun, and K.-J. Kim, “High-power beam-based coherently enhanced THz radiation source,” Phys. Rev. Spec. Top. Accel. Beams 11, 080701 (2008).

P. Musumeci, J. T. Moody, R. J. England, J. B. Rosenzweig, and T. Tran, “Experimental generation and Characterization of Uniformly Filled Ellipsoidal Electron-Beam Distributions,” Phys. Rev. Lett. 100(24), 244801 (2008).
[PubMed]

2007 (1)

M. Boscolo, M. Ferrario, C. Vaccarezza, I. Boscolo, F. Castelli, and S. Cialdi, “A train of micro-bunches for PWFA experiments produced by RF photoinjectors,” Int. J. Mod. Phys. B 21, 415–421 (2007).

2006 (1)

A. S. Kesar, R. A. Marsh, and R. J. Temkin, “Power measurement of frequency-locked Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9, 022801 (2006).

2005 (4)

P. Musumeci, C. Pellegrini, and J. B. Rosenzweig, “Higher harmonic inverse free-electron laser interaction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016501 (2005).
[PubMed]

A. Gover, “Superradiant and stimulated-superradiant emission in prebunched electron-beam radiators. I. Formulation,” Phys. Rev. Spec. Top. Accel. Beams 8, 030701 (2005).

H. L. Andrews, C. H. Boulware, C. A. Brau, and J. D. Jarvis, “Dispersion and attenuation in a Smith-Purcell free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 8, 050703 (2005).

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).
[PubMed]

2002 (3)

A. Tremaine, X. J. Wang, M. Babzien, I. Ben-Zvi, M. Cornacchia, A. Murokh, H.-D. Nuhn, R. Malone, C. Pellegrini, S. Reiche, J. Rosenzweig, J. Skaritka, and V. Yakimenko, “Fundamental and harmonic microbunching in a high-gain self-amplified spontaneous-emission free-electron laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(3 Pt 2B), 036503 (2002).
[PubMed]

G. Doucas, M. F. Kimmitt, A. Doria, G. P. Gallerano, E. Giovenale, G. Messina, H. L. Andrews, and J. H. Brownell, “Determination of longitudinal bunch shape by means of coherent Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 5, 072802 (2002).

G. Kube, H. Backe, H. Euteneuer, A. Grendel, F. Hagenbuck, H. Hartmann, K. H. Kaiser, W. Lauth, H. Schöpe, G. Wagner, T. Walcher, and M. Kretzschmar, “Observation of optical Smith-Purcell radiation at an electron beam energy of 855 MeV,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056501 (2002).
[PubMed]

2000 (1)

A. P. Potylitsyn, P. V. Karataev, and G. A. Naumenko, “Resonant diffraction radiation from an ultrarelativistic particle moving close to a tilted grating,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61(6 Pt B), 7039–7045 (2000).
[PubMed]

1998 (2)

J. H. Brownell, J. Walsh, and G. Doucas, “Spontaneous Smith-Purcell radiation described through induced surface currents,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57, 1075 (1998).

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).

1997 (1)

O. Haeberle, P. Rullhusen, J.-M. Salome, and N. Maene, “Smith-Purcell radiation from electrons moving parallel to a gratingat oblique incidence to the rulings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55, 4675 (1997).

1994 (1)

Y. Shibata, T. Takahashi, T. Kanai, K. Ishi, M. Ikezawa, J. Ohkuma, S. Okuda, and T. Okada, “Diagnostics of an electron beam of a linear accelerator using coherent transition radiation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(2), 1479–1484 (1994).
[PubMed]

1984 (1)

A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B 1, 723–728 (1984).

1963 (1)

A. P. Kazantsev and G. I. Surdutovich, “Radiation of a charged particle flying by metal screen,” Sov. Phys. Dokl. 7, 990 (1963).

1953 (1)

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92, 1069–1070 (1953).

Andrews, H. L.

H. L. Andrews, C. H. Boulware, C. A. Brau, and J. D. Jarvis, “Dispersion and attenuation in a Smith-Purcell free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 8, 050703 (2005).

Ang, L. K.

P. Zhang, L. K. Ang, and A. Gover, “Enhancement of coherent Smith-Purcell radiation at terahertz frequency by optimized grating, prebunched beams, and open cavity,” Phys. Rev. 18, 020702 (2015).

Araki, S.

A. Aryshev, S. Araki, M. Fukuda, K. Lekomtsev, M. Shevelev, J. Urakawa, A. Potylitsyn, and K. Sakaue, “Development of advanced THz generation schemes at KEK LUCX facility,” in Proceedings of the 10 Annual Meeting of Particle Accelerator Society of Japan (2013), pp. 873–876.

Arnold, R.

Aryshev, A.

Asakawa, M.

Asakawa, M. R.

Babzien, M.

Bacci, A.

Backe, H.

Barros, J.

Bartolini, R.

Bellaveglia, M.

Ben-Zvi, I.

Bhasin, L.

P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).

Blackmore, V.

Bleko, V. V.

Blomley, E.

Boscolo, I.

Boscolo, M.

Boulware, C. H.

H. L. Andrews, C. H. Boulware, C. A. Brau, and J. D. Jarvis, “Dispersion and attenuation in a Smith-Purcell free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 8, 050703 (2005).

Brau, C. A.

H. L. Andrews, C. H. Boulware, C. A. Brau, and J. D. Jarvis, “Dispersion and attenuation in a Smith-Purcell free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 8, 050703 (2005).

Brosi, M.

Brownell, J. H.

Bründermann, E.

Calvani, P.

Caselle, M.

Cassinari, L.

Castellano, M.

Castelli, F.

Catani, L.

Chiadroni, E.

Cialdi, S.

Cianchi, A.

Clarke, C.

Coleman, J.

J. Coleman, C. Ekdahl, and J. Oertel, Intense X-ray and EUV light source, patent US9686844B2 (2017).

Cornacchia, M.

Delerue, N.

Di Pirro, G.

Dong, L.

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

Doria, A.

Doucas, G.

Dvorkis, P.

A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B 1, 723–728 (1984).

Ekdahl, C.

J. Coleman, C. Ekdahl, and J. Oertel, Intense X-ray and EUV light source, patent US9686844B2 (2017).

Elisha, U.

A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B 1, 723–728 (1984).

England, R. J.

Euteneuer, H.

Ferrario, M.

Fukuda, M.

Fuster-Martinez, N.

Gallerano, G. P.

Gatti, G.

Giovenale, E.

Goldstein, M.

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).

Gover, A.

P. Zhang, L. K. Ang, and A. Gover, “Enhancement of coherent Smith-Purcell radiation at terahertz frequency by optimized grating, prebunched beams, and open cavity,” Phys. Rev. 18, 020702 (2015).

A. Gover, “Superradiant and stimulated-superradiant emission in prebunched electron-beam radiators. I. Formulation,” Phys. Rev. Spec. Top. Accel. Beams 8, 030701 (2005).

A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B 1, 723–728 (1984).

Graves, W. S.

Grendel, A.

Grosjean, M. V.

Haeberle, O.

Hagenbuck, F.

Hangyo, M.

Harrison, H.

Hartmann, H.

Hashida, M.

Hesler, J. L.

Hiller, N.

Ikezawa, M.

Imasakia, K.

Ishi, K.

Jarvis, J. D.

H. L. Andrews, C. H. Boulware, C. A. Brau, and J. D. Jarvis, “Dispersion and attenuation in a Smith-Purcell free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 8, 050703 (2005).

Jia, O.

L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).

Johnson, A.

Johnson, A. S.

Kaiser, K. H.

Kanai, T.

Karataev, P. V.

Kärtner, F. X.

Kazantsev, A. P.

A. P. Kazantsev and G. I. Surdutovich, “Radiation of a charged particle flying by metal screen,” Sov. Phys. Dokl. 7, 990 (1963).

Kehrer, B.

Kesar, A. S.

A. S. Kesar, R. A. Marsh, and R. J. Temkin, “Power measurement of frequency-locked Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9, 022801 (2006).

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).
[PubMed]

Kim, K.-J.

Y. Li, Y.-E. Sun, and K.-J. Kim, “High-power beam-based coherently enhanced THz radiation source,” Phys. Rev. Spec. Top. Accel. Beams 11, 080701 (2008).

Kimmitt, M. F.

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).

Konoplev, I.

Konoplev, I. V.

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).
[PubMed]

Kretzschmar, M.

Kube, G.

Kumar, A.

P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).

Kumar, M.

P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).

Kumar, P.

P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).

Labat, M.

Lancaster, A. J.

Lauth, W.

Le Corre, S.

Lekomtsev, K.

Li, D.

Li, Y.

Y. Li, Y.-E. Sun, and K.-J. Kim, “High-power beam-based coherently enhanced THz radiation source,” Phys. Rev. Spec. Top. Accel. Beams 11, 080701 (2008).

Liang, L.

L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).

Limaj, O.

Liu, W.

L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).

Lu, Y.

L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).

Lumpkin, A. H.

Lupi, S.

Maene, N.

Malone, R.

Marchetti, B.

Marsh, R. A.

A. S. Kesar, R. A. Marsh, and R. J. Temkin, “Power measurement of frequency-locked Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9, 022801 (2006).

Mathis, Y.-L.

Maxwell, T. J.

Meng, X. Z.

X. Z. Meng, M. H. Wang, L. M. Zhang, and Z. M. Ren, “Characteristic analysis of a Smith–Purcell terahertz source,” Photonics Res. 4, 162–167 (2016).

Messina, G.

Miyamoto, S.

Molloy, S.

Moncton, D. E.

Moody, J. T.

Mostacci, A.

Müller, A.-S.

Murokh, A.

Musumeci, P.

P. Musumeci, C. Pellegrini, and J. B. Rosenzweig, “Higher harmonic inverse free-electron laser interaction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016501 (2005).
[PubMed]

Nakajima, M.

Nasse, M. J.

Naumenko, G. A.

Naumov, A.

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).

Nuhn, H.-D.

Oertel, J.

J. Coleman, C. Ekdahl, and J. Oertel, Intense X-ray and EUV light source, patent US9686844B2 (2017).

Ohkuma, J.

Okada, T.

Okuda, S.

Ottewell, B.

Pace, E.

Palumbo, L.

Pellegrini, C.

P. Musumeci, C. Pellegrini, and J. B. Rosenzweig, “Higher harmonic inverse free-electron laser interaction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016501 (2005).
[PubMed]

Perry, C.

Piot, P.

Platt, C.

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).

Pompili, R.

Potylitsyn, A.

Potylitsyn, A. P.

Purcell, E. M.

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92, 1069–1070 (1953).

Raasch, J.

Reiche, S.

Reichold, A.

Ren, Z. M.

X. Z. Meng, M. H. Wang, L. M. Zhang, and Z. M. Ren, “Characteristic analysis of a Smith–Purcell terahertz source,” Photonics Res. 4, 162–167 (2016).

Rihaoui, M. M.

Ronsivalle, C.

Rosenzweig, J.

Rosenzweig, J. B.

P. Musumeci, C. Pellegrini, and J. B. Rosenzweig, “Higher harmonic inverse free-electron laser interaction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016501 (2005).
[PubMed]

Ruan, J.

Rullhusen, P.

Sakaue, K.

Salome, J.-M.

Santucci, J.

Schedler, M.

Schönfeldt, P.

Schöpe, H.

Schuh, M.

Schwarz, M.

Sergeeva, D. Yu.

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Microscopic theory of Smith-Purcell radiation from 2D photonic crystal,” Nucl. Instrum. Methods Phys. Res. B 402, 206–211 (2017).

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Conical diffraction effect in optical and x-ray Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 18, 052801 (2015).

Shevelev, M.

Shibata, Y.

Siegel, M.

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).
[PubMed]

Skaritka, J.

Smale, N.

Smith, S. J.

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92, 1069–1070 (1953).

Spataro, B.

Steinmann, J. L.

Stevenson, S.

Strikhanov, M. N.

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Microscopic theory of Smith-Purcell radiation from 2D photonic crystal,” Nucl. Instrum. Methods Phys. Res. B 402, 206–211 (2017).

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Conical diffraction effect in optical and x-ray Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 18, 052801 (2015).

Sun, Y. E.

Sun, Y.-E.

Y. Li, Y.-E. Sun, and K.-J. Kim, “High-power beam-based coherently enhanced THz radiation source,” Phys. Rev. Spec. Top. Accel. Beams 11, 080701 (2008).

Surdutovich, G. I.

A. P. Kazantsev and G. I. Surdutovich, “Radiation of a charged particle flying by metal screen,” Sov. Phys. Dokl. 7, 990 (1963).

Taheri, F. B.

Takahashi, T.

Tani, M.

Temkin, R. J.

A. S. Kesar, R. A. Marsh, and R. J. Temkin, “Power measurement of frequency-locked Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9, 022801 (2006).

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).
[PubMed]

Terunuma, N.

Thangaraj, J. C. T.

Thurman-Keup, R.

Tishchenko, A. A.

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Microscopic theory of Smith-Purcell radiation from 2D photonic crystal,” Nucl. Instrum. Methods Phys. Res. B 402, 206–211 (2017).

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Conical diffraction effect in optical and x-ray Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 18, 052801 (2015).

Tran, T.

Tremaine, A.

Tripathi, V. K.

P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).

Tsunawaki, Y.

Urakawa, J.

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).

Vaccarezza, C.

Wagner, G.

Walcher, T.

Walsh, J.

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).

Wang, L.

L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).

Wang, M. H.

X. Z. Meng, M. H. Wang, L. M. Zhang, and Z. M. Ren, “Characteristic analysis of a Smith–Purcell terahertz source,” Photonics Res. 4, 162–167 (2016).

Wang, X. J.

Wang, Y.

Weber, M.

Wei, Y.

Woods, M.

Yakimenko, V.

Yang, Z.

Zagorodnov, I.

Zhang, H.

Zhang, L. M.

X. Z. Meng, M. H. Wang, L. M. Zhang, and Z. M. Ren, “Characteristic analysis of a Smith–Purcell terahertz source,” Photonics Res. 4, 162–167 (2016).

Zhang, P.

P. Zhang, L. K. Ang, and A. Gover, “Enhancement of coherent Smith-Purcell radiation at terahertz frequency by optimized grating, prebunched beams, and open cavity,” Phys. Rev. 18, 020702 (2015).

Zhang, Y.

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

Appl. Phys. Lett. (3)

Int. J. Mod. Phys. B (1)

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

A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation pattern of Smith-Purcell radiation,” J. Opt. Soc. Am. B 1, 723–728 (1984).

JETP Lett. (1)

Nucl. Instrum. Methods Phys. Res. B (1)

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Microscopic theory of Smith-Purcell radiation from 2D photonic crystal,” Nucl. Instrum. Methods Phys. Res. B 402, 206–211 (2017).

Nucl. Instrum. Methods Phys. Res. Sect. B (1)

Nucl. Instrum. Methods Phys. Res., Sect. A (2)

Nucl. Instrum. Methods Phys. Res., Sect. B (1)

Opt. Express (2)

L. Liang, W. Liu, O. Jia, L. Wang, and Y. Lu, “High-harmonic terahertz Smith-Purcell free-electron-laser with two tandem cylindrical-gratings,” Opt. Express 25, 2960 (2017).

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

Photonics Res. (1)

X. Z. Meng, M. H. Wang, L. M. Zhang, and Z. M. Ren, “Characteristic analysis of a Smith–Purcell terahertz source,” Photonics Res. 4, 162–167 (2016).

Phys. Plasmas (1)

P. Kumar, L. Bhasin, V. K. Tripathi, A. Kumar, and M. Kumar, “Smith–Purcell terahertz radiation from laser modulated electron beam over a metallic grating,” Phys. Plasmas 23, 093301 (2016).

Phys. Rev. (2)

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92, 1069–1070 (1953).

P. Zhang, L. K. Ang, and A. Gover, “Enhancement of coherent Smith-Purcell radiation at terahertz frequency by optimized grating, prebunched beams, and open cavity,” Phys. Rev. 18, 020702 (2015).

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

P. Musumeci, C. Pellegrini, and J. B. Rosenzweig, “Higher harmonic inverse free-electron laser interaction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1 Pt 2), 016501 (2005).
[PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (4)

Phys. Rev. Lett. (6)

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).

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).
[PubMed]

Phys. Rev. Spec. Top. Accel. Beams (8)

A. S. Kesar, R. A. Marsh, and R. J. Temkin, “Power measurement of frequency-locked Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 9, 022801 (2006).

H. L. Andrews, C. H. Boulware, C. A. Brau, and J. D. Jarvis, “Dispersion and attenuation in a Smith-Purcell free electron laser,” Phys. Rev. Spec. Top. Accel. Beams 8, 050703 (2005).

D. Yu. Sergeeva, A. A. Tishchenko, and M. N. Strikhanov, “Conical diffraction effect in optical and x-ray Smith-Purcell radiation,” Phys. Rev. Spec. Top. Accel. Beams 18, 052801 (2015).

Y. Li, Y.-E. Sun, and K.-J. Kim, “High-power beam-based coherently enhanced THz radiation source,” Phys. Rev. Spec. Top. Accel. Beams 11, 080701 (2008).

A. Gover, “Superradiant and stimulated-superradiant emission in prebunched electron-beam radiators. I. Formulation,” Phys. Rev. Spec. Top. Accel. Beams 8, 030701 (2005).

Rev. Sci. Instrum. (2)

Sov. Phys. Dokl. (1)

A. P. Kazantsev and G. I. Surdutovich, “Radiation of a charged particle flying by metal screen,” Sov. Phys. Dokl. 7, 990 (1963).

Other (8)

J. Coleman, C. Ekdahl, and J. Oertel, Intense X-ray and EUV light source, patent US9686844B2 (2017).

H. Backe, W. Lauth, H. Mannweiler, H. Rochholz, K. Aulenbacher, R. Barday, H. Euteneuer, K.-H. Kaiser, G. Kube, F. Schwellnus, and V. Tioukine, “Investigation of Far-Infrared Smith-Purcell Radiation at the 3.41 MeV Electron Injector LINAC of the Mainz Microtron MAMI,” in Advanced Radiation Sources and Applications, H. Wiedemann, ed. (Springer, 2006).

C. Brau, Free-Electron Lasers (Academic, 1990).

O. Luiten, S. van der Geer, M. de Loos, F. Kiewiet, and M. van der Wiel, “Ideal waterbag electron bunches from an RF photogun,” in Proceedings of 9th European Particle Accelerator Conference 2004, L. Rivkin, ed. (CERN, 2004), pp. 725–727.

A. P. Potylitsyn, M. I. Ryazanov, M. N. Strikhanov, and A. A. Tishchenko, Diffraction Radiation from Relativistic Particles (Springer, 2011).

D. Yu. Sergeeva and A. A. Tishchenko, “X-ray Smith-Purcell radiation from a beam skimming a grating surface,” in Proceedings of the 36th International Free Electron Laser Conference, J. Chrin, ed. (JACoW, 2015), pp. 378–383.

A. J. Lancaster, G. Doucas, H. Harrison, and I. V. Konoplev, “Novel grating designs for a single-shot Smith-Purcell bunch profile monitor,” in Proceedings of the 5th International Beam Instrumentation Conference, I.Costa, ed. (JACoW, 2017), pp. 213–216.

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Fig. 1 Layout for generation of Smith-Purcell radiation.

Download Full Size | PPT Slide | PDF

Fig. 2 The longitudinal profile of the beam

f_{l} (x)

described by Eq. (7) for different parameters.

Download Full Size | PPT Slide | PDF

Fig. 3 The longitudinal form-factor

F_{l}^{f}

described by Eq. (12) for

μ = 0

and

γ = 2 \cdot 10^{4}

Download Full Size | PPT Slide | PDF

Fig. 4 The longitudinal profile of the beam described by

g_{l} (x)

from Eq. (16).

Download Full Size | PPT Slide | PDF

Fig. 5 The longitudinal form-factor of the beam

F_{l}^{g}

described by Eq. (19) at

γ = 16

(a) for

N_{b} = 16

, (b) for

{σ^{'}}_{x} = λ_{0} / 7

Download Full Size | PPT Slide | PDF

Fig. 6 The comparison between

F_{i n c}

(red dashed curve) and

N F_{c o h}^{g}

(black solid curve) for the beam of size

{σ^{'}}_{x} = 100 μ m

σ_{y, z} = 250 μ m

λ_{0} = 0.8 m m

ϕ = 0

N_{b} = 4

N / N_{b} = 10^{8}

γ = 16

Download Full Size | PPT Slide | PDF

Fig. 7 Spectral-angular distribution of SPR at THz frequencies for the trains of microbunches with population of each microbunch

N / N_{b} = 10^{8}

. Black solid curve is plotted for

λ = β^{- 1} λ_{0} \approx 0.8 m m

(see Eq. (21) for

s = 1

), red dashed curve is plotted for

λ = β^{- 1} λ_{0} / 2 \approx 0.4 m m

(see Eq. (21) for

s = 2

). For all curves

λ_{0} = 0.8 m m

{σ^{'}}_{x} = 100 μ m

σ_{y, z} = 250 μ m

ϕ = 0

N_{b} = 4

N_{s t} = 7

γ = 16

h = 0.7 m m

d = 2 m m

a = 1 m m

Download Full Size | PPT Slide | PDF

Fig. 8 Spectral-angular distribution of SPR at THz frequencies for the trains of microbunches with population of each microbunch

N / N_{b} = 10^{8}

depending on the wavelength. Black solid curve is plotted for

N_{b} = 2

, red curve

N_{b} = 4

, blue curve

N_{b} = 16

. For all the curves

λ_{0} = 0.8 m m

{σ^{'}}_{x} = 100 μ m

σ_{y, z} = 250 μ m

ϕ = 0

N_{s t} = 7

γ = 16

h = 0.7 m m

d = 2 m m,

a = 1 m m

θ = 0.2 r a d

Download Full Size | PPT Slide | PDF

Fig. 9 Spectral-angular distribution of SPR at THz frequencies (a) for

N_{b} = 2

N_{s t} = 9

d = 11 m m

; (b) for

N_{b} = 16

N_{s t} = 9

d = 11 m m

; (c) for

N_{b} = 16

N_{s t} = 9

d = 2.2 m m

. For all the curves

{σ^{'}}_{x} = 100 μ m

σ_{y, z} = 250 μ m

λ_{0} = 0.8 m m

ϕ = 0

θ = 50^{0}

γ = 16

h = 0.7 m m

a = d / 2

N = 10^{9}

. The blue curve is

2 \cdot 10^{- 2} G

with

G

from Eq. (5), the red curve is the longitudinal form-factor multiplied by 10, i.e.

10 F_{l}^{g}

, with

F_{l}^{g}

from Eq. (19).

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Fig. 10 Dependence of

R = \frac{d W (m = 3, s = 1)}{d W (m = 3, s = 2)}

on the depth of the beam modulation for

θ = 1.77 ​ r a d

. The other parameters are as in Fig. 7.

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Fig. 11 Spectral-angular distribution of SPR for the Gaussian-shaped beam with sinusoidal modulation at X-ray frequencies, according to Eqs. (2) and (28). (a) Black solid curve: radiation from fully modulated beam

μ = 0

; red dashed curve: partially modulated beam

μ = 0.2

; blue dotted curve for the single not modulated bunch. For all the curves

σ_{x} = 20 μ m

λ_{0} = 10 n m

N_{b} \approx 2000

h = 0.05 m m

d = 6.5 μ m

λ = 10.005 n m

. (b) Black solid curve:

d = 6.4 μ m

, which corresponds to Eq. (32),

m = 3

; blue dashed curve:

d = 6.6 μ m

. For all curves:

μ = 0

σ_{x} / λ_{0} = 500

λ_{0} = 3 n m

θ = 53 ​ m r a d

h = 10 μ m

. For both figures:

γ = 2 \cdot 10^{3}

σ_{y, z} = 10 μ m

ϕ = 0

a = d / 2

N = 10^{10}

ℏ ω_{p} = 26.1 e V

(beryllium),

N_{s t} = 7

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Fig. 12 Dependence of the spectral-angular distribution of the radiation on the depth of beam modulation

μ

. The black curve corresponds to the first SPR peak

m = 1

θ = 55.5 m r a d

, the blue dashed curve corresponds to the fifth SPR peak

m = 5

θ = 124.1 m r a d

. Other parameters as in Fig. 11(a).

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Equations (42)

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λ m = d (β^{- 1} - \cos θ), m = 1, 2, ...

\frac{d^{2} W (n, ω)}{d ω d Ω} = \frac{d^{2} W_{1} (n, ω)}{d ω d Ω} G F,

F = N F_{i n c} + N (N - 1) F_{c o h} .

F_{i n c} = \int_{V} d^{3} r {| e^{- i q (r - r_{0})} |}^{2} P (r), F_{c o h} = {| \int_{V} d^{3} r e^{- i q (r - r_{0})} P (r) |}^{2},

G = \frac{\sin^{2} (N_{s t} d φ / 2)}{\sin^{2} (d φ / 2)},

f (x) = C \exp [- x^{2} / σ_{x}^{2}] (1 + \sum_{n = 1}^{\infty} b_{n} \cos (n k_{0} x)),

f_{l} (x) = \frac{2}{\sqrt{π} σ_{x}} \frac{μ + \sin^{2} (π x / λ_{0})}{1 + 2 μ - \exp [- π^{2} σ_{x}^{2} / λ_{0}^{2}]} \exp [- \frac{x^{2}}{σ_{x}^{2}}],

f_{t r} (y, z) = {\begin{cases} \frac{\exp [- y^{2} / σ_{y}^{2}]}{\sqrt{π} σ_{y} σ_{z}}, h - \frac{σ_{z}}{2} \leq z \leq h + \frac{σ_{z}}{2}, \\ 0, z < h - σ_{z} / 2, z > h + σ_{z} / 2 . \end{cases}

F_{i n c}^{f} = \frac{\sin h (ρ σ_{z})}{ρ σ_{z}},

F_{c o h}^{f} = F_{t r}^{f} F_{l}^{f},

F_{t r}^{f} = \frac{\sinh^{2} (ρ σ_{z} / 2)}{{(ρ σ_{z} / 2)}^{2}} \exp [- \frac{σ_{y}^{2} k_{y}^{2}}{2}],

F_{l}^{f} = \exp [- \frac{σ_{x}^{2} ξ^{2}}{2}] {(\frac{1 + 2 μ - \exp [- π^{2} σ_{x}^{2} / λ_{0}^{2}] \cos h (π ξ σ_{x}^{2} / λ_{0})}{1 + 2 μ - \exp [- π^{2} σ_{x}^{2} / λ_{0}^{2}]})}^{2},

k_{y} = \frac{ω}{c} \sin θ \sin ϕ, ξ = ω / v, ρ = \sqrt{ξ^{2} + k_{y}^{2} - ω^{2} / c^{2}}

λ = β^{- 1} λ_{0} .

Δ λ = \frac{2 λ_{0}}{β} {(\frac{π σ_{x}}{λ_{0}} - \frac{λ_{0}}{π σ_{x}})}^{- 1} \approx \frac{8 λ_{0}}{β π N_{b}} .

g_{l} (x) = \frac{1}{N_{b} \sqrt{π} {σ^{'}}_{x}} \sum_{s = 0}^{N_{b} - 1} \exp [- \frac{{(x - s λ_{0})}^{2}}{{σ^{'}}_{x}^{2}}] .

F_{c o h}^{g} = F_{l}^{g} F_{t r}^{g},

F_{t r}^{g} = \frac{\sin h^{2} (ρ σ_{z} / 2)}{{(ρ σ_{z} / 2)}^{2}} \exp [- \frac{σ_{y}^{2} k_{y}^{2}}{2}],

F_{l}^{g} = \exp [- \frac{{σ^{'}}_{x}^{2} ξ^{2}}{2}] \frac{1}{N_{b}^{2}} \frac{\sin^{2} (N_{b} λ_{0} ξ / 2)}{\sin^{2} (λ_{0} ξ / 2)},

F_{l}^{g} = {| \sum_{p = 0}^{N_{b} - 1} σ_{p} e^{- \frac{ω}{c β} (\frac{σ_{p}^{2}}{4} \frac{ω}{c β} + i p λ_{p})} / \sum_{p = 0}^{N_{b} - 1} σ_{p} |}^{2},

λ = β^{- 1} \frac{λ_{0}}{s}, s = 1, 2, ...,

Δ λ = \frac{2}{(N_{b}^{2} s^{2} - 1)} \frac{N_{b} λ_{0}}{β} .

\frac{{σ^{'}}_{x}}{λ_{0}} \approx \frac{1}{π} \frac{N_{b}}{N_{b}^{2} s^{2} - 1} .

n = (\cos θ, \sin θ \sin ϕ, \sin θ \cos ϕ) .

\frac{d^{2} W_{1} (n, ω)}{d ℏ ω d Ω} = \frac{e^{2}}{c ℏ} \frac{\exp [- 2 ρ h]}{π^{2} β^{3} ρ^{2} φ^{2}} \frac{ω^{4}}{c^{4}} \sin^{2} (\frac{a φ}{2}) [\frac{(1 - n_{y}^{2}) + 2 n_{y}^{2} γ^{2}}{γ^{2} \sqrt{1 - n_{y}^{2}}} (1 - β n_{x}) + \frac{β (1 - n_{y}^{2}) - n_{x}}{γ^{2}}] .

λ_{0} = d (1 - β n_{x}) \frac{s}{m}, s, m = 1, 2, ...

\cos θ = β^{- 1} - \frac{λ_{0}}{d β} \frac{m}{s}, s, m = 1, 2...

\frac{d^{2} W_{1} (n, ω)}{d ℏ ω d Ω} = \frac{e^{2}}{c ℏ} e^{- 2 ρ h} {(\frac{ε (ω) - 1}{2 π β φ ρ})}^{2} \frac{ω^{4}}{c^{4}} \sin^{2} (\frac{a φ}{2}) \frac{{| [n^{'} \times n^{'} \times (\frac{ω}{β c γ^{2}} e_{x} + k_{y} e_{y} - i ρ e_{z})] |}^{2}}{{| ρ - i (ω / c) \sqrt{| ε (ω) - 1 + n_{z}^{2} |} |}^{2}},

ε (ω) = 1 - ω_{p}^{2} / ω^{2}, ω > > ω_{p},

n^{'} = \frac{1}{\sqrt{ε (ω)}} (n_{x}, n_{y}, ε (ω) - 1 + n_{z}^{2}) .

λ_{0} m = d (1 - β n_{x}), m = 1, 2, ...

\cos θ = β^{- 1} - \frac{λ_{0}}{d β} m, m = 1, 2, ...,

2 π σ_{z} < < λ γ β,

λ_{t} β m = d (1 - θ_{t}^{2} / 2 - β n_{x}), ​ ​ ​ ​ m = 1, 2...

\frac{Δ λ}{λ_{t}} = \frac{θ_{t}^{2} / 2}{β^{- 1} (1 - θ_{t}^{2} / 2) - \cos θ} \approx \frac{θ_{t}^{2}}{4 \sin^{2} (θ / 2)}

D_{i n c} = {(1 + \frac{h ω γ}{c β} σ_{α}^{2})}^{- 1 / 2}, D_{c o h} = {(1 + \frac{h ω γ}{2 c β} σ_{α}^{2})}^{- 1},

F = N F_{i n c} D_{i n c} + N (N - 1) F_{c o h} D_{c o h} .

\frac{h ω γ}{c β} σ_{α}^{2} < < 1,

σ_{α} < γ^{- 1} .

\frac{Δ λ}{λ} = (\frac{\partial λ}{\partial γ}) \frac{Δ γ}{λ}

{(\frac{Δ λ}{λ})}_{S P} = - \frac{Δ γ}{γ^{3}}

Δ λ_{S P} = \frac{2 d N_{s t} (β^{- 1} - n_{x})}{(N_{s t}^{2} m^{2} - 1)} .