Abstract

Bessel beams are becoming a very useful tool in many areas of optics and photonics, because of the invariance of their intensity profile over an extended propagation range. Finite-Difference-Time-Domain (FDTD) approach is widely used for the modeling of the beam interaction with nanostructures. However, the generation of the Bessel beam in this approach is a computationally challenging problem. In this work, we report an approach for the generation of the infinite Bessel beams in three-dimensional FDTD. It is based on the injection of the Bessel solutions of Maxwell’s equations from a cylindrical hollow annulus. This configuration is compatible with Particle In Cell simulations of laser plasma interactions. This configuration allows using a smaller computation box and is therefore computationally more efficient than the creation of a Bessel-Gauss beam from a wall and models more precisely the analytical infinite Bessel beam. Zeroth and higher-order Bessel beams with different cone angles are successfully produced. We investigate the effects of the injector parameters on the error with respect to the analytical solution. In all cases, the relative deviation is in the range of 0.01-7.0 percent.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

2018 (4)

2017 (1)

S. Kumar, A. Parola, P. Di Trapani, and O. Jedrkiewicz, “Laser plasma wakefield acceleration gain enhancement by means of accelerating Bessel pulses,” Appl. Phys. B 123(6), 185 (2017).
[Crossref]

2016 (3)

G. A. Hine, A. J. Goers, L. Feder, J. A. Elle, S. J. Yoon, and H. M. Milchberg, “Generation of axially modulated plasma waveguides using a spatial light modulator,” Opt. Lett. 41(15), 3427–3430 (2016).
[Crossref]

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

J. J. Wang, T. Wriedt, J. A. Lock, and L. Mädler, “General description of circularly symmetric Bessel beams of arbitrary order,” J. Quant. Spectrosc. Radiat. Transf. 184, 218–232 (2016).
[Crossref]

2015 (2)

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

2013 (1)

2012 (3)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

2011 (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

2010 (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

2008 (2)

E. Mcleod and C. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref]

Y. Yu and W.-B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. Electromagn. Res. Lett. 5, 57–71 (2008).
[Crossref]

2007 (1)

2006 (1)

2005 (2)

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by a slits-truncated Bessel beam,” Appl. Phys. Lett. 86(3), 031110 (2005).
[Crossref]

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2004 (1)

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

2000 (2)

V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[Crossref]

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000).
[Crossref]

1998 (1)

J. B. Schneider, C. L. Wagner, and O. M. Ramahi, “Implementation of transparent sources in FDTD simulations,” IEEE Trans. Antennas Propag. 46(8), 1159–1168 (1998).
[Crossref]

1997 (1)

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55(3), 3539–3545 (1997).
[Crossref]

1993 (1)

C. G. Durfee and H. M. Milchberg, “Light pipe for high intensity laser pulses,” Phys. Rev. Lett. 71(15), 2409–2412 (1993).
[Crossref]

1991 (1)

S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85(2-3), 159–161 (1991).
[Crossref]

1987 (2)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

1983 (1)

J. M. Dawson, “Particle simulation of plasmas,” Rev. Mod. Phys. 55(2), 403–447 (1983).
[Crossref]

1954 (1)

Arber, T. D.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Arnold, C.

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

E. Mcleod and C. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref]

Bell, A. R.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Bennett, K.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Bergner, K.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

K. Bergner, M. Müller, R. Klas, J. Limpert, S. Nolte, and A. Tünnerman, “Scaling ultrashort laser pulse induced glass modifications for cleaving applications,” Appl. Opt. 57(21), 5941–5947 (2018).
[Crossref]

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Bhuyan, M.

R. Stoian, M. Bhuyan, G. Zhang, G. Cheng, R. Meyer, and F. Courvoisier, “Ultrafast Bessel beams: advanced tools for laser materials processing,” Adv. Opt. Technol. 7(3), 165–174 (2018).
[Crossref]

Birdsall, C.

C. Birdsall and A. Langdon, Plasma Physics via Computer Simulation (Taylor and Francis, 2004)351–385.

Brady, C. S.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Brown, C.

Chen, A.

Cheng, G.

R. Stoian, M. Bhuyan, G. Zhang, G. Cheng, R. Meyer, and F. Courvoisier, “Ultrafast Bessel beams: advanced tools for laser materials processing,” Adv. Opt. Technol. 7(3), 165–174 (2018).
[Crossref]

Couairon, A.

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Courvoisier, F.

R. Stoian, M. Bhuyan, G. Zhang, G. Cheng, R. Meyer, and F. Courvoisier, “Ultrafast Bessel beams: advanced tools for laser materials processing,” Adv. Opt. Technol. 7(3), 165–174 (2018).
[Crossref]

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Cui, Z.

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Dawson, J. M.

J. M. Dawson, “Particle simulation of plasmas,” Rev. Mod. Phys. 55(2), 403–447 (1983).
[Crossref]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

H. Little, C. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

Di Trapani, P.

S. Kumar, A. Parola, P. Di Trapani, and O. Jedrkiewicz, “Laser plasma wakefield acceleration gain enhancement by means of accelerating Bessel pulses,” Appl. Phys. B 123(6), 185 (2017).
[Crossref]

Dou, W.-B.

Y. Yu and W.-B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. Electromagn. Res. Lett. 5, 57–71 (2008).
[Crossref]

Dubietis, A.

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Dudley, J. M.

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Duocastella, M.

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

Durfee, C. G.

C. G. Durfee and H. M. Milchberg, “Light pipe for high intensity laser pulses,” Phys. Rev. Lett. 71(15), 2409–2412 (1993).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Elle, J. A.

Esarey, E.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55(3), 3539–3545 (1997).
[Crossref]

Evans, R. G.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Faccio, D.

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Feder, L.

Gaizauskas, E.

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Garcés-Chávez, V.

H. Little, C. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

Gedney, S. D.

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000).
[Crossref]

Gillies, P.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Giust, R.

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Goers, A. J.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Gross, H.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Hafizi, B.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55(3), 3539–3545 (1997).
[Crossref]

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method (Artech House, 2005)169–327.

Han, L.

Han, Y.

Hine, G. A.

Imasaki, K.

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by a slits-truncated Bessel beam,” Appl. Phys. Lett. 86(3), 031110 (2005).
[Crossref]

Itina, T.

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Jarutis, V.

E. Gaizauskas, E. Vanagas, V. Jarutis, S. Juodkazis, V. Mizeikis, and H. Misawa, “Discrete damage traces from filamentation of Gauss-Bessel pulses,” Opt. Lett. 31(1), 80–82 (2006).
[Crossref]

V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[Crossref]

Jedrkiewicz, O.

S. Kumar, A. Parola, P. Di Trapani, and O. Jedrkiewicz, “Laser plasma wakefield acceleration gain enhancement by means of accelerating Bessel pulses,” Appl. Phys. B 123(6), 185 (2017).
[Crossref]

Jukna, V.

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Juodkazis, S.

Kauranen, M.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Klas, R.

Kolesik, M.

Kumar, S.

S. Kumar, A. Parola, P. Di Trapani, and O. Jedrkiewicz, “Laser plasma wakefield acceleration gain enhancement by means of accelerating Bessel pulses,” Appl. Phys. B 123(6), 185 (2017).
[Crossref]

Langdon, A.

C. Birdsall and A. Langdon, Plasma Physics via Computer Simulation (Taylor and Francis, 2004)351–385.

Lawrence-Douglas, A.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Li, D.

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by a slits-truncated Bessel beam,” Appl. Phys. Lett. 86(3), 031110 (2005).
[Crossref]

Limpert, J.

Little, H.

Lock, J. A.

J. J. Wang, T. Wriedt, J. A. Lock, and L. Mädler, “General description of circularly symmetric Bessel beams of arbitrary order,” J. Quant. Spectrosc. Radiat. Transf. 184, 218–232 (2016).
[Crossref]

Lotti, A.

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Mädler, L.

J. J. Wang, T. Wriedt, J. A. Lock, and L. Mädler, “General description of circularly symmetric Bessel beams of arbitrary order,” J. Quant. Spectrosc. Radiat. Transf. 184, 218–232 (2016).
[Crossref]

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

Mcleod, E.

E. Mcleod and C. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref]

McLeod, J. H.

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

Meyer, R.

R. Stoian, M. Bhuyan, G. Zhang, G. Cheng, R. Meyer, and F. Courvoisier, “Ultrafast Bessel beams: advanced tools for laser materials processing,” Adv. Opt. Technol. 7(3), 165–174 (2018).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Milchberg, H. M.

Milián, C.

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Misawa, H.

Mishra, S.

S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85(2-3), 159–161 (1991).
[Crossref]

Mizeikis, V.

Moloney, J.

Müller, M.

Nolte, S.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

K. Bergner, M. Müller, R. Klas, J. Limpert, S. Nolte, and A. Tünnerman, “Scaling ultrashort laser pulse induced glass modifications for cleaving applications,” Appl. Opt. 57(21), 5941–5947 (2018).
[Crossref]

Ornigotti, M.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

Ouadghiri-Idrissi, I.

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Papazoglou, D. G.

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Parola, A.

S. Kumar, A. Parola, P. Di Trapani, and O. Jedrkiewicz, “Laser plasma wakefield acceleration gain enhancement by means of accelerating Bessel pulses,” Appl. Phys. B 123(6), 185 (2017).
[Crossref]

Paškauskas, R.

V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[Crossref]

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Ramahi, O. M.

J. B. Schneider, C. L. Wagner, and O. M. Ramahi, “Implementation of transparent sources in FDTD simulations,” IEEE Trans. Antennas Propag. 46(8), 1159–1168 (1998).
[Crossref]

Ramsay, M. G.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Ridgers, C. P.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Roden, J. A.

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000).
[Crossref]

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Roskey, D.

Rubino, E.

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Schmitz, H.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Schneider, J. B.

J. B. Schneider, C. L. Wagner, and O. M. Ramahi, “Implementation of transparent sources in FDTD simulations,” IEEE Trans. Antennas Propag. 46(8), 1159–1168 (1998).
[Crossref]

Sibbett, W.

H. Little, C. Brown, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical guiding of microscopic particles in femtosecond and continuous wave Bessel light beams,” Opt. Express 12(11), 2560–2565 (2004).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Sircombe, N. J.

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Sprangle, P.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55(3), 3539–3545 (1997).
[Crossref]

Stabinis, A.

V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[Crossref]

Steinkopf, R.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

Stoian, R.

R. Stoian, M. Bhuyan, G. Zhang, G. Cheng, R. Meyer, and F. Courvoisier, “Ultrafast Bessel beams: advanced tools for laser materials processing,” Adv. Opt. Technol. 7(3), 165–174 (2018).
[Crossref]

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, 2007)349–387.

Szameit, A.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method (Artech House, 2005)169–327.

Tamošauskas, G.

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Tünnerman, A.

Tzortzakis, S.

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Vanagas, E.

Vetter, C.

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

Wagner, C. L.

J. B. Schneider, C. L. Wagner, and O. M. Ramahi, “Implementation of transparent sources in FDTD simulations,” IEEE Trans. Antennas Propag. 46(8), 1159–1168 (1998).
[Crossref]

Wang, J.

Wang, J. J.

J. J. Wang, T. Wriedt, J. A. Lock, and L. Mädler, “General description of circularly symmetric Bessel beams of arbitrary order,” J. Quant. Spectrosc. Radiat. Transf. 184, 218–232 (2016).
[Crossref]

Wriedt, T.

J. J. Wang, T. Wriedt, J. A. Lock, and L. Mädler, “General description of circularly symmetric Bessel beams of arbitrary order,” J. Quant. Spectrosc. Radiat. Transf. 184, 218–232 (2016).
[Crossref]

Wright, E.

Wu, Z.

Xie, C.

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Yoon, S. J.

Yu, M.

Yu, Y.

Y. Yu and W.-B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. Electromagn. Res. Lett. 5, 57–71 (2008).
[Crossref]

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Zhang, G.

R. Stoian, M. Bhuyan, G. Zhang, G. Cheng, R. Meyer, and F. Courvoisier, “Ultrafast Bessel beams: advanced tools for laser materials processing,” Adv. Opt. Technol. 7(3), 165–174 (2018).
[Crossref]

Adv. Opt. Technol. (1)

R. Stoian, M. Bhuyan, G. Zhang, G. Cheng, R. Meyer, and F. Courvoisier, “Ultrafast Bessel beams: advanced tools for laser materials processing,” Adv. Opt. Technol. 7(3), 165–174 (2018).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

S. Kumar, A. Parola, P. Di Trapani, and O. Jedrkiewicz, “Laser plasma wakefield acceleration gain enhancement by means of accelerating Bessel pulses,” Appl. Phys. B 123(6), 185 (2017).
[Crossref]

Appl. Phys. Lett. (1)

D. Li and K. Imasaki, “Vacuum laser-driven acceleration by a slits-truncated Bessel beam,” Appl. Phys. Lett. 86(3), 031110 (2005).
[Crossref]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

IEEE Trans. Antennas Propag. (1)

J. B. Schneider, C. L. Wagner, and O. M. Ramahi, “Implementation of transparent sources in FDTD simulations,” IEEE Trans. Antennas Propag. 46(8), 1159–1168 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Quant. Spectrosc. Radiat. Transf. (1)

J. J. Wang, T. Wriedt, J. A. Lock, and L. Mädler, “General description of circularly symmetric Bessel beams of arbitrary order,” J. Quant. Spectrosc. Radiat. Transf. 184, 218–232 (2016).
[Crossref]

Laser Photonics Rev. (2)

C. Vetter, R. Steinkopf, K. Bergner, M. Ornigotti, S. Nolte, H. Gross, and A. Szameit, “Realization of free-space long-distance self-healing Bessel beams,” Laser Photonics Rev. 13(10), 1900103 (2019).
[Crossref]

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

Microw. Opt. Technol. Lett. (1)

J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000).
[Crossref]

Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2011).
[Crossref]

Nat. Nanotechnol. (1)

E. Mcleod and C. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref]

Nat. Photonics (2)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012).
[Crossref]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref]

Opt. Commun. (3)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184(1-4), 105–112 (2000).
[Crossref]

S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85(2-3), 159–161 (1991).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (1)

D. Faccio, E. Rubino, A. Lotti, A. Couairon, A. Dubietis, G. Tamošauskas, D. G. Papazoglou, and S. Tzortzakis, “Nonlinear light-matter interaction with femtosecond high-angle Bessel beams,” Phys. Rev. A 85(3), 033829 (2012).
[Crossref]

Phys. Rev. E (1)

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55(3), 3539–3545 (1997).
[Crossref]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

C. G. Durfee and H. M. Milchberg, “Light pipe for high intensity laser pulses,” Phys. Rev. Lett. 71(15), 2409–2412 (1993).
[Crossref]

Plasma Phys. Controlled Fusion (1)

T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, “Contemporary particle-in-cell approach to laser-plasma modelling,” Plasma Phys. Controlled Fusion 57(11), 113001 (2015).
[Crossref]

Prog. Electromagn. Res. Lett. (1)

Y. Yu and W.-B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. Electromagn. Res. Lett. 5, 57–71 (2008).
[Crossref]

Rev. Mod. Phys. (1)

J. M. Dawson, “Particle simulation of plasmas,” Rev. Mod. Phys. 55(2), 403–447 (1983).
[Crossref]

Sci. Rep. (1)

C. Xie, V. Jukna, C. Milián, R. Giust, I. Ouadghiri-Idrissi, T. Itina, J. M. Dudley, A. Couairon, and F. Courvoisier, “Tubular filamentation for laser material processing,” Sci. Rep. 5(1), 8914 (2015).
[Crossref]

Other (3)

C. Birdsall and A. Langdon, Plasma Physics via Computer Simulation (Taylor and Francis, 2004)351–385.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method (Artech House, 2005)169–327.

J. Stratton, Electromagnetic Theory (McGraw-Hill, 2007)349–387.

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Figures (9)
Fig. 1.
Fig. 1. A Bessel-Gauss beam from FDTD simulation. Fluence distribution in $zx$ plane at $y=0$ (left), and $yx$ plane at location of the maximum for $I(0, 0, z)$ (right). We aim to create a Bessel beam in the white box with a computational box of $4\lambda _\textrm {xy}\times 4\lambda _\textrm {xy}\times 2\lambda _\textrm {z}$ which is smaller than Bessel-Gauss one by a factor of 64.

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Fig. 2.
Fig. 2. Bessel pulse from Fourier spectrum integration. Equation (19) integration is calculated for cone angle of $\theta =17 ^{\circ }$ (left panel), $\theta =25 ^{\circ }$ (middle panel), and $\theta =30 ^{\circ }$ (right panel). In all cases, $m=0$, $T=60\,\textrm {fs}$, $T_0=180\,\textrm {fs}$, and $\omega _0=2.4\,\textrm {{PHz}}$. The red dashed lines show $\pm c/\sin \theta$ velocities. We show the amplitude of $E_\textrm {y}$ field in log-scale to enhance the X-shape of the pulse propagation.

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Fig. 3.
Fig. 3. Concept of the Bessel antenna in the FDTD simulation box. The yellow annulus with a thickness of $\delta s$ shows the cylindrical antenna which has an outer radius $R_\textrm {B}=N_{\perp }\lambda _\textrm {xy}$, and a length $2N_{\parallel }\lambda _\textrm {z}$. There are $N_\textrm {PML}$ grid cells in each transverse direction where the CPML boundary condition is applied.

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Fig. 4.
Fig. 4. Different-order Bessel beams. The amplitude of the $y$ component electric field and the intensity of generated Bessel beams are plotted on $xy$ plane at $z=0$: Run A (top row), Run B (middle row), and Run C (bottom row). The first three columns show the $y$ component of the electric field at different times, $t=26.7\, \textrm {fs}$, $106.8\, \textrm {fs}$, and $213.5\, \textrm {fs}$ from left to right. The rightmost column shows the intensity distribution at $t=213.5\, \textrm {fs}$, the corresponding time of the peak intensity ($T_{0}+R_\textrm {B}/c\sin \theta$). The white circle in the right column shows the inner radius of the Bessel antenna .

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Fig. 5.
Fig. 5. Different-order Bessel beams. The intensity map of generated Bessel beams in $xz$ plane at $y=0$: Run A (left), Run B (middle), and Run C (right). All snapshots are taken at $213.5\, \textrm {fs}$.

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Fig. 6.
Fig. 6. Different-order Bessel beams. Comparisons between the $y$ component electric field from the Bessel antenna (blue solid line) and from the Eq. (19) (red dashed line). Run A (left column), Run B (middle column), and Run C (right column). The bottom row shows the relative deviation.

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Fig. 7.
Fig. 7. Bessel beam with different cone angles. The amplitude of the $y$ component electric field and the intensity of generated Bessel beams are plotted on $xy$ plane at $z=0$: Run E (top row), Run D (middle row), and Run A (bottom row). The first three columns show the time evolution of the $y$ component electric field at three different times from left to right. As the simulation box is different for each row, the corresponding time for each column is different. The rightmost column shows the intensity distribution. The white circle in the right column shows the inner radius of the Bessel antenna.

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Fig. 8.
Fig. 8. Bessel beam with different cone angles. Comparisons between the $y$ component electric field from the Bessel antenna (blue solid line) and from the Eq. (19) (red dashed line). Run E (left column), Run D (middle column), and Run A (right column). The bottom row shows the relative deviation.

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Fig. 9.
Fig. 9. Effects of the antenna thickness and radius on the Bessel beam. The top row shows the relative deviation for different antenna thicknesses: Run D (top left), Run F (top middle), and Run G (top right). The bottom row shows the relative deviation for different antenna radii: Run I (bottom left), Run H (bottom middle), and Run D (bottom right).

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Tables (1)

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Table 1. Sets of simulation parameters for Bessel antenna.

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Equations (23)

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∇ 2 Π − μ ϵ ∂ t 2 Π = 0
( E 1 , E 2 ) = ( ∇ × ∇ × Π , i ω μ ∇ × Π )
( H 1 , H 2 ) = ( − i ω ϵ ∇ × Π , ∇ × ∇ × Π )
( Q , R ) = ( ∇ × Π , ∇ × ∇ × Π )
( E 1 , E 2 ) = ( R , i k 0 Q )
( η H 1 , η H 2 ) = ( − i k 0 ϵ r Q , R )
Π {x} = x ^ J m ( K r ) e i m ϕ e i ( ω t − β z )
Π {y} = y ^ J m ( K r ) e i m ϕ e i ( ω t − β z )
Π {z} = z ^ J m ( K r ) e i m ϕ e i ( ω t − β z )
Q = e i ( ω t − β z ) e i m ϕ x ^ 0 y ^ i β J m ( K r ) z ^ K 2 i [ J {m+1} ( K r ) e i ϕ + J {m-1} ( K r ) e − i ϕ ]
R = e i ( ω t − β z ) e i m ϕ x ^ k 2 + β 2 2 J m ( K r ) + K 2 4 [ J {m+2} ( K r ) e 2 i ϕ + J {m-2} ( K r ) e − 2 i ϕ ] y ^ i K 2 4 [ J {m-2} ( K r ) e − 2 i ϕ − J {m+2} ( K r ) e 2 i ϕ ] z ^ i β K 2 [ J {m-1} ( K r ) e − i ϕ − J {m+1} ( K r ) e i ϕ ]
Q = e i ( ω t − β z ) e i m ϕ − x ^ i β J m ( K r ) y ^ 0 − z ^ K 2 [ J {m+1} ( K r ) e i ϕ − J {m-1} ( K r ) e − i ϕ ]
R = e i ( ω t − β z ) e i m ϕ x ^ i K 2 4 [ J {m-2} ( K r ) e − 2 i ϕ − J {m+2} ( K r ) e 2 i ϕ ] y ^ k 2 + β 2 2 J m ( K r ) − K 2 4 [ J {m+2} ( K r ) e 2 i ϕ + J {m-2} ( K r ) e − 2 i ϕ ] − z ^ β K 2 [ J {m+1} ( K r ) e i ϕ + J {m-1} ( K r ) e − i ϕ ]
Q = e i ( ω t − β z ) e i m ϕ x ^ i K 2 [ J {m+1} ( K r ) e i ϕ + J {m-1} ( K r ) e − i ϕ ] y ^ K 2 [ J {m+1} ( K r ) e i ϕ − J {m-1} ( K r ) e − i ϕ ] z ^ 0
R = e i ( ω t − β z ) e i m ϕ − x ^ i β K 2 [ J {m+1} ( K r ) e i ϕ − J {m-1} ( K r ) e − i ϕ ] − y ^ β K 2 [ J {m+1} ( K r ) e i ϕ + J {m-1} ( K r ) e − i ϕ ] z ^ K 2 J m ( K r )
( E , H ) = k 0 − 2 ( E 0 k 0 Q , H 0 R ) e i ( ω t − β z ) e i m ϕ
( E , H ) = k 0 − 2 ( E 0 R , H 0 k 0 Q ) e i ( ω t − β z ) e i m ϕ
F ( ω ) = T 2 π e − T 2 ( ω − ω 0 ) 2 4
E y ( x , t ) = − cos ⁡ θ e i m ϕ ∫ − ∞ ∞ d ω F ( ω ) J m ( ω sin ⁡ θ c r ) e i ω ( t − T 0 − z cos ⁡ θ / c )
E B = k 0 − 2 ℜ { E 0 f ( t ) k 0 Q e i [ ω ( t − T 0 ) − β z ] e i m ϕ }
H B = k 0 − 2 ℜ { H 0 f ( t ) R e i [ ω ( t − T 0 ) − β z ] e i m ϕ }
E i t + δ t ( i + δ i / 2 , j , k ) = E i t + δ t ( i + δ i / 2 , j , k ) + E Bi t + δ t ( i + δ i / 2 , j , k )
H i t + δ t ( i , j + δ j / 2 , k + δ k / 2 ) = H i t + δ t ( i , j + δ j / 2 , k + δ k / 2 ) + H Bi t + δ t ( i , j + δ j / 2 , k + δ k / 2 )

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