Attenuation of waveguide modes in narrow metal capillaries

P. V. Tuev; P. V. Tuev; K. V. Lotov; K. V. Lotov

doi:10.1364/JOSAA.410552

Attenuation of waveguide modes in narrow metal capillaries

P. V. Tuev and K. V. Lotov

Journal of the Optical Society of America A
Vol. 38,
Issue 1,
pp. 108-114
(2021)
•https://doi.org/10.1364/JOSAA.410552

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P. V. Tuev and K. V. Lotov, "Attenuation of waveguide modes in narrow metal capillaries," J. Opt. Soc. Am. A 38, 108-114 (2021)

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Abstract

The channeling of laser pulses in waveguides filled with a rare plasma is one of the promising techniques of laser wakefield acceleration. A solid-state capillary can precisely guide tightly focused pulses. Regardless of the material of the capillary, its walls behave like a plasma under the influence of a high-intensity laser pulse. Therefore, the waveguide modes in the capillaries have a universal structure, which depends only on the shape of the cross-section. Due to the large ratio of the capillary radius to the laser wavelength, the modes in circular capillaries differ from classical TE and TM modes. We consider the structure of capillary modes in a circular capillary, calculate the attenuation rates, discuss the mode expansion of the incident pulse using minimal simplifications, and analyze the accuracy of commonly used approximations. The attenuation length for such modes is two orders of magnitude longer than that obtained from the classical formula, and the incident pulse of the proper radius can transfer up to 98% of its initial energy to the fundamental mode. However, finding eigenmodes in capillaries of arbitrary cross-sections is a complex mathematical problem that remains to be solved.

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2018 (2)

M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza, “Diagnostics for plasma-based electron accelerators,” Rev. Mod. Phys. 90, 035002 (2018).
[Crossref]

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2017 (1)

A. Curcio, D. Giulietti, and M. Petrarca, “Tuning of betatron radiation in laser-plasma accelerators via multimodal laser propagation through capillary waveguides,” Phys. Plasmas 24, 023104 (2017).
[Crossref]

2016 (3)

S. Steinke, J. van Tilborg, C. Benedetti, C. G. R. Geddes, C. B. Schroeder, J. Daniels, K. K. Swanson, A. J. Gonsalves, K. Nakamura, N. H. Matlis, B. H. Shaw, E. Esarey, and W. P. Leemans, “Multistage coupling of independent laser-plasma accelerators,” Nature 530, 190–193 (2016).
[Crossref]

M. J. Hogan, “Electron and positron beam-driven plasma acceleration,” Rev. Accel Sci. Technol. 9, 63–83 (2016).
[Crossref]

E. Adli and P. Muggli, “Proton-beam-driven plasma acceleration,” Rev. Accel Sci. Technol. 9, 85–104 (2016).
[Crossref]

2015 (2)

A. J. Gonsalves, K. Nakamura, J. Daniels, H.-S. Mao, C. Benedetti, C. B. Schroeder, Cs. Toth, J. van Tilborg, D. E. Mittelberger, S. S. Bulanov, J.-L. Vay, C. G. R. Geddes, E. Esarey, and W. P. Leemans, “Generation and pointing stabilization of multi-GeV electron beams from a laser plasma accelerator driven in a pre-formed plasma waveguide,” Phys. Plasmas 22, 056703 (2015).
[Crossref]

K. V. Lotov, K. V. Gubin, V. E. Leshchenko, V. I. Trunov, and E. V. Pestryakov, “Guiding femtosecond high-intensity high-contrast laser pulses by copper capillaries,” Phys. Plasmas 22, 103111 (2015).
[Crossref]

2014 (4)

N. E. Andreev, S. V. Kuznetsov, and M. E. Veysman, “Laser wakefield electron acceleration in capillary waveguides under non-symmetric coupling conditions,” Nucl. Instrum. Methods A 740, 273–279 (2014).
[Crossref]

W. P. Leemans, A. J. Gonsalves, H.-S. Mao, K. Nakamura, C. Benedetti, C. B. Schroeder, Cs. Toth, J. Daniels, D. E. Mittelberger, S. S. Bulanov, J.-L. Vay, C. G. R. Geddes, and E. Esarey, “Multi-GeV electron beams from capillary-discharge-guided subpetawatt laser pulses in the self-trapping regime,” Phys. Rev. Lett. 113, 245002 (2014).
[Crossref]

M. Hansson, L. Senje, A. Persson, O. Lundh, C.-G. Wahlström, F. G. Desforges, J. Ju, T. L. Audet, B. Cros, S. D. Dufrenoy, and P. Monot, “Enhanced stability of laser wakefield acceleration using dielectric capillary tubes,” Phys. Rev. ST Accel. Beams 17, 031303 (2014).
[Crossref]

J. Ju, G. Genoud, H. E. Ferrari, O. Dadoun, B. Paradkar, K. Svensson, F. Wojda, M. Burza, A. Persson, O. Lundh, N. E. Andreev, C.-G. Wahlström, and B. Cros, “Analysis of x-ray emission and electron dynamics in a capillary-guided laser wakefield accelerator,” Phys. Rev. ST Accel. Beams 17, 051302 (2014).
[Crossref]

2013 (5)

J. Ju, K. Svensson, H. Ferrari, A. Döpp, G. Genoud, F. Wojda, M. Burza, A. Persson, O. Lundh, C.-G. Wahlström, and B. Cros, “Study of electron acceleration and x-ray radiation as a function of plasma density in capillary-guided laser wakefield accelerators,” Phys. Plasmas 20, 083106 (2013).
[Crossref]

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751 (2013).
[Crossref]

S. M. Hooker, “Developments in laser-driven plasma accelerators,” Nat. Photonics 7, 775–782 (2013).
[Crossref]

B. S. Paradkar, B. Cros, P. Mora, and G. Maynard, “Numerical modeling of multi-GeV laser wakefield electron acceleration inside a dielectric capillary tube,” Phys. Plasmas 20, 083120 (2013).
[Crossref]

N. E. Andreev, V. E. Baranov, B. Cros, G. Maynard, P. Mora, and M. E. Veysman, “Laser wakefield compression and acceleration of externally injected electron bunches in guiding structures,” J. Plasma Phys. 79, 143–152 (2013).
[Crossref]

2012 (2)

M. Veysman, N. E. Andreev, G. Maynard, and B. Cros, “Nonsymmetric laser-pulse propagation in capillary tubes with variable radius,” Phys. Rev. E 86, 066411 (2012).
[Crossref]

V. Eremin, Yu. Malkov, V. Korolikhin, A. Kiselev, S. Skobelev, A. Stepanov, and N. Andreev, “Study of the plasma wave excited by intense femtosecond laser pulses in a dielectric capillary,” Phys. Plasmas 19, 093121 (2012).
[Crossref]

2011 (3)

G. Genoud, K. Cassou, F. Wojda, H. E. Ferrari, C. Kamperidis, M. Burza, A. Persson, J. Uhlig, S. Kneip, S. P. D. Mangles, A. Lifschitz, B. Cros, and C.-G. Wahlström, “Laser-plasma electron acceleration in dielectric capillary tubes,” Appl. Phys. B 105, 309 (2011).
[Crossref]

H. Lu, M. Liu, W. Wang, C. Wang, J. Liu, A. Deng, J. Xu, C. Xia, W. Li, H. Zhang, X. Lu, C. Wang, J. Wang, X. Liang, Y. Leng, B. Shen, K. Nakajima, R. Li, and Z. Xu, “Laser wakefield acceleration of electron beams beyond 1 GeV from an ablative capillary discharge waveguide,” Appl. Phys. Lett. 99, 091502 (2011).
[Crossref]

S. Cipiccia, M. R. Islam, B. Ersfeld, R. P. Shanks, E. Brunetti, G. Vieux, X. Yang, R. C. Issac, S. M. Wiggins, G. H. Welsh, M.-P. Anania, D. Maneuski, R. Montgomery, G. Smith, M. Hoek, D. J. Hamilton, N. R. C. Lemos, D. Symes, P. P. Rajeev, V. O. Shea, J. M. Dias, and D. A. Jaroszynski, “Gamma-rays from harmonically resonant betatron oscillations in a plasma wake,” Nat. Phys. 7, 867–871 (2011).
[Crossref]

2010 (1)

M. Veysman, N. E. Andreev, K. Cassou, Y. Ayoul, G. Maynard, and B. Cros, “Theoretical and experimental study of laser beam propagation in capillary tubes for non-symmetrical coupling conditions,” J. Opt. Soc. Am. B 27, 1400–1408 (2010).
[Crossref]

2009 (4)

Y. Mori, Y. Sentoku, K. Kondo, K. Tsuji, N. Nakanii, S. Fukumochi, M. Kashihara, K. Kimura, K. Takeda, K. A. Tanaka, T. Norimatsu, T. Tanimoto, H. Nakamura, M. Tampo, R. Kodama, E. Miura, K. Mima, and Y. Kitagawa, “Autoinjection of electrons into a wake field using a capillary with attached cone,” Phys. Plasmas 16, 123103 (2009).
[Crossref]

T. Kameshima, H. Kotaki, M. Kando, I. Daito, K. Kawase, Y. Fukuda, L. M. Chen, T. Homma, S. Kondo, T. Zh. Esirkepov, N. A. Bobrova, P. V. Sasorov, and S. V. Bulanov, “Laser pulse guiding and electron acceleration in the ablative capillary discharge plasma,” Phys. Plasmas 16, 093101 (2009).
[Crossref]

W. P. Leemans, E. Esarey, C. G. R. Geddes, Cs. Toth, C. B. Schroeder, K. Nakamura, A. J. Gonsalves, D. Panasenko, E. Cormier-Michel, G. R. Plateau, C. Lin, D. L. Bruhwiler, and J. R. Cary, “Progress on laser plasma accelerator development using transversely and longitudinally shaped plasmas,” C. R. Phys. 10, 130–139 (2009).
[Crossref]

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229 (2009).
[Crossref]

2007 (2)

S. Karsch, J. Osterhoff, A. Popp, T. P. Rowlands-Rees, Zs. Major, M. Fuchs, B. Marx, R. Horlein, K. Schmid, L. Veisz, S. Becker, U. Schramm, B. Hidding, G. Pretzler, D. Habs, F. Gruner, F. Krausz, and S. M. Hooker, “GeV-scale electron acceleration in a gas-filled capillary discharge waveguide,” New J. Phys. 9, 415 (2007).
[Crossref]

S. M. Hooker, E. Brunetti, E. Esarey, J. G. Gallacher, C. G. R. Geddes, A. J. Gonsalves, D. A. Jaroszynski, C. Kamperidis, S. Kneip, K. Krushelnick, W. P. Leemans, S. P. D. Mangles, C. D. Murphy, B. Nagler, Z. Najmudin, K. Nakamura, P. A. Norreys, D. Panasenko, T. P. Rowlands-Rees, C. B. Schroeder, Cs. Toth, and R. Trines, “GeV plasma accelerators driven in waveguides,” Plasma Phys. Controlled Fusion 49, B403 (2007).
[Crossref]

2006 (2)

W. P. Leemans, B. Nagler, A. J. Gonsalves, Cs. Toth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “GeV electron beams from a centimetre-scale accelerator,” Nat. Phys. 2, 696–699 (2006).
[Crossref]

M. Veysman, B. Cros, N. E. Andreev, and G. Maynard, “Theory and simulation of short intense laser pulse propagation in capillary tubes with wall ablation,” Phys. Plasmas 13, 053114 (2006).
[Crossref]

2004 (2)

I. A. Kotelnikov, “Attenuation in waveguide,” Tech. Phys. 49, 1196–1201 (2004).
[Crossref]

Y. Kitagawa, Y. Sentoku, S. Akamatsu, W. Sakamoto, R. Kodama, K. A. Tanaka, K. Azumi, T. Norimatsu, T. Matsuoka, H. Fujita, and H. Yoshida, “Electron acceleration in an ultraintense-laser-illuminated capillary,” Phys. Rev. Lett. 92, 205002 (2004).
[Crossref]

2002 (1)

B. Cros, C. Courtois, G. Matthieussent, A. Di Bernardo, D. Batani, N. Andreev, and S. Kuznetsov, “Eigenmodes for capillary tubes with dielectric walls and ultraintense pulse guiding,” Phys. Rev. E 65, 026405 (2002).
[Crossref]

2001 (2)

K. V. Lotov, “Laser wakefield acceleration in narrow plasma-filled channels,” Laser Part. Beams 19, 219–222 (2001).
[Crossref]

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2000 (3)

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1999 (2)

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1998 (1)

M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and A. A. Offenberger, “Guiding of a 10-TW picosecond laser pulse through hollow capillary tubes,” Phys. Rev. E 57, R4899 (1998).
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1995 (1)

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E. Adli and P. Muggli, “Proton-beam-driven plasma acceleration,” Rev. Accel Sci. Technol. 9, 85–104 (2016).
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Amiranoff, F.

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Borghesi, M.

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751 (2013).
[Crossref]

M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and A. A. Offenberger, “Guiding of a 10-TW picosecond laser pulse through hollow capillary tubes,” Phys. Rev. E 57, R4899 (1998).
[Crossref]

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Cormier-Michel, E.

Courtois, C.

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M. Veysman, N. E. Andreev, G. Maynard, and B. Cros, “Nonsymmetric laser-pulse propagation in capillary tubes with variable radius,” Phys. Rev. E 86, 066411 (2012).
[Crossref]

Curcio, A.

Dadoun, O.

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J. R. Davies and J. T. Mendonca, “Basic physics of laser propagation in hollow waveguides,” Phys. Rev. E 62, 7168 (2000).
[Crossref]

Debus, A.

M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza, “Diagnostics for plasma-based electron accelerators,” Rev. Mod. Phys. 90, 035002 (2018).
[Crossref]

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M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza, “Diagnostics for plasma-based electron accelerators,” Rev. Mod. Phys. 90, 035002 (2018).
[Crossref]

Dufrenoy, S. D.

Eremin, V.

Ersfeld, B.

Esarey, E.

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229 (2009).
[Crossref]

Esirkepov, T. Zh.

Fang, M.

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Fukumochi, S.

Gaillard, R.

M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and A. A. Offenberger, “Guiding of a 10-TW picosecond laser pulse through hollow capillary tubes,” Phys. Rev. E 57, R4899 (1998).
[Crossref]

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Gubin, K. V.

Guethlein, G.

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Hamoniaux, G.

Hansson, M.

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M. J. Hogan, “Electron and positron beam-driven plasma acceleration,” Rev. Accel Sci. Technol. 9, 63–83 (2016).
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S. M. Hooker, “Developments in laser-driven plasma accelerators,” Nat. Photonics 7, 775–782 (2013).
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Islam, M. R.

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M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza, “Diagnostics for plasma-based electron accelerators,” Rev. Mod. Phys. 90, 035002 (2018).
[Crossref]

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I. A. Kotelnikov, “Attenuation in waveguide,” Tech. Phys. 49, 1196–1201 (2004).
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E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229 (2009).
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Leng, Y.

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Liang, X.

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L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8 of Electrodynamics of Continuous Media (Pergamon, 1984).

Lin, C.

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Lotov, K. V.

K. V. Lotov, “Laser wakefield acceleration in narrow plasma-filled channels,” Laser Part. Beams 19, 219–222 (2001).
[Crossref]

K. V. Lotov, “Driver channeling for laser wakefield accelerator,” Part. Accel. 63, 139–146 (1999).

Lu, H.

Lu, X.

Lundh, O.

Macchi, A.

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751 (2013).
[Crossref]

Mackinnon, A. J.

M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and A. A. Offenberger, “Guiding of a 10-TW picosecond laser pulse through hollow capillary tubes,” Phys. Rev. E 57, R4899 (1998).
[Crossref]

Major, Zs.

Malka, G.

Malkov, Yu.

Maneuski, D.

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E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

Marques, J. R.

Marx, B.

Matlis, N. H.

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Matthieussent, G.

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M. Veysman, N. E. Andreev, G. Maynard, and B. Cros, “Nonsymmetric laser-pulse propagation in capillary tubes with variable radius,” Phys. Rev. E 86, 066411 (2012).
[Crossref]

Mendonca, J. T.

J. R. Davies and J. T. Mendonca, “Basic physics of laser propagation in hollow waveguides,” Phys. Rev. E 62, 7168 (2000).
[Crossref]

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N. W. Ashcroft and N. D. Mermin, Solid State Physics (International Thomson Edition, 1976).

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Mima, K.

Miquel, J. L.

Mittelberger, D. E.

Miura, E.

Monot, P.

Montgomery, R.

Mora, P.

More, R. M.

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E. Adli and P. Muggli, “Proton-beam-driven plasma acceleration,” Rev. Accel Sci. Technol. 9, 85–104 (2016).
[Crossref]

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M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and A. A. Offenberger, “Guiding of a 10-TW picosecond laser pulse through hollow capillary tubes,” Phys. Rev. E 57, R4899 (1998).
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A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751 (2013).
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Sasorov, P. V.

Schmeltzer, R. A.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
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Schmid, K.

Schramm, U.

M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza, “Diagnostics for plasma-based electron accelerators,” Rev. Mod. Phys. 90, 035002 (2018).
[Crossref]

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E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229 (2009).
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Velikoroussov, T.

Veysman, M.

M. Veysman, N. E. Andreev, G. Maynard, and B. Cros, “Nonsymmetric laser-pulse propagation in capillary tubes with variable radius,” Phys. Rev. E 86, 066411 (2012).
[Crossref]

Veysman, M. E.

Vickers, C.

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White, W. E.

Wiggins, S. M.

Willi, O.

M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and A. A. Offenberger, “Guiding of a 10-TW picosecond laser pulse through hollow capillary tubes,” Phys. Rev. E 57, R4899 (1998).
[Crossref]

Wojda, F.

Wu, Y.

Xia, C.

Xu, J.

Xu, Z.

Yang, X.

Yoshida, H.

Yu, C.

Zgadzaj, R.

M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza, “Diagnostics for plasma-based electron accelerators,” Rev. Mod. Phys. 90, 035002 (2018).
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Zhang, H.

Zhang, Z.

Appl. Phys. B (1)

Appl. Phys. Lett. (1)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43, 1783–1809 (1964).
[Crossref]

C. R. Phys. (1)

IEEE Trans. Plasma Sci. (1)

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

J. Plasma Phys. (1)

Laser Part. Beams (2)

K. V. Lotov, “Laser wakefield acceleration in narrow plasma-filled channels,” Laser Part. Beams 19, 219–222 (2001).
[Crossref]

Nat. Photonics (1)

S. M. Hooker, “Developments in laser-driven plasma accelerators,” Nat. Photonics 7, 775–782 (2013).
[Crossref]

Nat. Phys. (2)

Nature (1)

New J. Phys. (1)

Nucl. Instrum. Methods A (1)

Part. Accel. (1)

K. V. Lotov, “Driver channeling for laser wakefield accelerator,” Part. Accel. 63, 139–146 (1999).

Phys. Plasmas (10)

Phys. Rev. E (5)

M. Borghesi, A. J. Mackinnon, R. Gaillard, O. Willi, and A. A. Offenberger, “Guiding of a 10-TW picosecond laser pulse through hollow capillary tubes,” Phys. Rev. E 57, R4899 (1998).
[Crossref]

M. Veysman, N. E. Andreev, G. Maynard, and B. Cros, “Nonsymmetric laser-pulse propagation in capillary tubes with variable radius,” Phys. Rev. E 86, 066411 (2012).
[Crossref]

J. R. Davies and J. T. Mendonca, “Basic physics of laser propagation in hollow waveguides,” Phys. Rev. E 62, 7168 (2000).
[Crossref]

Phys. Rev. Lett. (4)

Phys. Rev. ST Accel. Beams (2)

Plasma Phys. Controlled Fusion (1)

Rev. Accel Sci. Technol. (2)

M. J. Hogan, “Electron and positron beam-driven plasma acceleration,” Rev. Accel Sci. Technol. 9, 63–83 (2016).
[Crossref]

E. Adli and P. Muggli, “Proton-beam-driven plasma acceleration,” Rev. Accel Sci. Technol. 9, 85–104 (2016).
[Crossref]

Rev. Mod. Phys. (3)

M. C. Downer, R. Zgadzaj, A. Debus, U. Schramm, and M. C. Kaluza, “Diagnostics for plasma-based electron accelerators,” Rev. Mod. Phys. 90, 035002 (2018).
[Crossref]

E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229 (2009).
[Crossref]

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751 (2013).
[Crossref]

Tech. Phys. (1)

I. A. Kotelnikov, “Attenuation in waveguide,” Tech. Phys. 49, 1196–1201 (2004).
[Crossref]

Other (2)

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8 of Electrodynamics of Continuous Media (Pergamon, 1984).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (International Thomson Edition, 1976).

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Fig. 1. Attenuation rate for modes (a)

${R}_{11}$

${{\rm TM}}_{11}$

and (b)

${R}_{13}$

${{\rm TM}}_{13}$

calculated from classical formula (19) (curves, C), approximate expression (23) (A), and numerically solved Eq. (10) (N). Black dots on curve N are obtained by solving Eq. (15). Thin vertical lines mark the boundary between approximations (

$\delta {=1}$

), and dotted vertical lines show the considered parameter set.

Download Full Size | PPT Slide | PDF

Fig. 2. Attenuation rates for various modes, obtained numerically (exact) and with approximation (23) (approx.) for the baseline parameter set.

Download Full Size | PPT Slide | PDF

Fig. 3. Ratio of attenuation rates obtained approximately and numerically for various

$R$

modes at the baseline parameter set.

Download Full Size | PPT Slide | PDF

Fig. 4. Transverse electric fields for different modes, calculated by (a) the exact Eq. (10) and with approximations (b) (21) and (c) (16).

Download Full Size | PPT Slide | PDF

Fig. 5. Laser energy falling into separate waveguide modes in relation to the radius of the incident pulse. The black solid line is the total laser energy entering the capillary. The black dotted line is the energy summed over the first 15 modes.

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Equations (37)

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ε = {\begin{array}{cc} 1, & r < a, \\ ε_{w}, & r \geq a . \end{array}

E_{z} = e^{i k z - i ω t + i m ϕ} {\begin{array}{cc} E_{1} J_{m} (ϰ_{1} r), & r < a, \\ E_{2} K_{m} (ϰ_{2} r), & r \geq a, \end{array}

B_{z} = e^{i k z - i ω t + i m ϕ} {\begin{array}{cc} B_{1} J_{m} (ϰ_{1} r), & r < a, \\ B_{2} K_{m} (ϰ_{2} r), & r \geq a, \end{array}

E_{r} = \frac{{(- 1)}^{j}}{ϰ_{j}^{2}} (- i k \frac{\partial E_{z}}{\partial r} + \frac{m ω}{c r} B_{z}),

E_{ϕ} = \frac{{(- 1)}^{j}}{ϰ_{j}^{2}} (\frac{k m}{r} E_{z} + \frac{i ω}{c} \frac{\partial B_{z}}{\partial r}),

B_{r} = \frac{{(- 1)}^{j}}{ϰ_{j}^{2}} (- ik \frac{\partial B_{z}}{\partial r} - \frac{m ω ε}{c r} E_{z}),

B_{ϕ} = \frac{{(- 1)}^{j}}{ϰ_{j}^{2}} (\frac{k m}{r} B_{z} - \frac{i ω ε}{c} \frac{\partial E_{z}}{\partial r}),

ϰ_{1}^{2} = ω^{2} / c^{2} - k^{2},

ϰ_{2}^{2} = k^{2} - ε_{w} ω^{2} / c^{2},

\begin{aligned} (\frac{J_{m}^{'}}{J_{m}} + \frac{ϰ_{1}}{ϰ_{2}} \frac{K_{m}^{'}}{K_{m}}) (\frac{1}{ε_{w}} \frac{J_{m}^{'}}{J_{m}} + \frac{ϰ_{1}}{ϰ_{2}} \frac{K_{m}^{'}}{K_{m}}) \\ = \frac{1}{ε_{w}} {(\frac{m k c}{ω ϰ_{1} a})}^{2} {(1 + \frac{ϰ_{1}^{2}}{ϰ_{2}^{2}})}^{2}, \end{aligned}

ε_{w} (ω) = 1 + i \frac{ω_{p}^{2} τ}{ω (1 - i ω τ)},

E_{r} = - i m E_{ϕ}

ε_{w} \approx - 30 + 1.1 i, | ε_{w} | ≫ 1, ϰ_{2} ≫ ω / c ≫ 1 / a,

E_{ϕ} = ζ B_{z}, E_{z} = - ζ B_{ϕ},

(\frac{J_{m}^{'}}{J_{m}} - \frac{i ϰ_{1} c ζ}{ω}) (\frac{J_{m}^{'}}{J_{m}} - \frac{i ϰ_{1} c}{ω ζ}) = {(\frac{m k c}{ω ϰ_{1} a})}^{2} .

| ζ | ≪ ϰ_{1} c / ω

J_{m} (ϰ_{1} a) = 0, E_{z} \neq 0, B_{z} \equiv 0

J_{m^{'}} (ϰ_{1} a) = 0, E_{z} \equiv 0, B_{z} \neq 0.

α = \frac{ω R e (ζ)}{k a c},

α = \frac{c ϰ_{1}^{2} R e (ζ)}{ω k a} (1 + \frac{m^{2} ω^{2}}{c^{2} ϰ_{1}^{2} (a^{2} ϰ_{1}^{2} - m^{2})})

| ζ | ≫ ϰ_{1} c / ω

J_{m \pm 1} (ϰ_{1} a) = 0, \vec{B} = \pm i \vec{E}, E_{r} = \pm i E_{ϕ},

α = \frac{ϰ_{1}^{2} R e (ζ)}{2 k^{2} a | ζ |^{2}} .

ζ \approx 0.0032 - 0.18 i, δ \equiv \frac{ϰ_{1} c}{| ζ | ω} \sim 0.12.

E_{y} = E_{0} e^{- r^{2} / σ_{r}^{2}}, E_{x} = 0.

C_{m o d e} = \frac{{(\int E_{y} E_{m o d e, y} d S)}^{2}}{\int E_{y}^{2} d S \int E_{m o d e}^{2} d S},

{\vec{E}}_{⊥} = \frac{i k}{ϰ^{2}} \nabla_{⊥} E_{z} + \frac{i ω}{c ϰ^{2}} [\nabla_{⊥} B_{z} \times {\vec{e}}_{z}],

{\vec{B}}_{⊥} = \frac{i k}{ϰ^{2}} \nabla_{⊥} B_{z} - \frac{i ω}{c ϰ^{2}} [\nabla_{⊥} E_{z} \times {\vec{e}}_{z}],

ϰ^{2} = ω^{2} / c^{2} - k^{2} .

{\vec{E}}_{τ} = ζ [\vec{n} \times {\vec{B}}_{τ}],

(Δ_{⊥} + ϰ^{2}) E_{z} = 0, (Δ_{⊥} + ϰ^{2}) B_{z} = 0,

\frac{ϰ}{k ζ} E_{z} = \frac{i}{ϰ} (\frac{\partial B_{z}}{\partial τ} + \frac{ω}{k c} \frac{\partial E_{z}}{\partial n}),

\frac{ϰ ζ}{k} B_{z} = - \frac{i}{ϰ} (\frac{\partial E_{z}}{\partial τ} - \frac{ω}{k c} \frac{\partial B_{z}}{\partial n}),

\frac{\partial}{\partial τ} \sim \frac{\partial}{\partial n} \sim \frac{1}{a} \sim | ϰ | .

\frac{\partial E_{z}}{\partial τ} = \frac{\partial B_{z}}{\partial n}, \frac{\partial B_{z}}{\partial τ} = - \frac{\partial E_{z}}{\partial n} .

F = E_{z} + i B_{z},

(Δ_{⊥} + ϰ^{2}) F = 0, \frac{\partial F}{\partial n} = i \frac{\partial F}{\partial τ} .