Abstract

The propagation dynamics of radially polarized (RP) pulses in a gas-filled hollow-core fiber (HCF) is numerically studied. It is found that the stable transverse mode of RP pulse in HCF is not TM01 mode, nor any eigenmodes in terms of Bessel functions. Compared with linearly polarized (LP) pulses, the RP pulses with the same initial pulse duration and energy have higher transmission efficiency, more uniform spectral broadening, and cleaner temporal profile after highly nonlinear propagation in HCF and better focusing properties. These results suggest that energetic few-cycle RP pulses can be generated more efficiently by directly spectral broadening the RP pulses in HCF followed by temporal compression.

© 2017 Optical Society of America

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References

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2016 (3)

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

L. Qiao, D. Wang, and Y. Leng, “Spatiotemporal dynamics of an optical pulse propagating in multimode hollow-core fibers filled with prealigned molecular gases,” Phys. Rev. A 93(2), 023832 (2016).
[Crossref]

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

2014 (1)

2013 (1)

J. Andreasen and M. Kolesik, “Midinfrared femtosecond laser pulse filamentation in hollow waveguides: a comparison of simulation methods,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(5), 053303 (2013).
[Crossref] [PubMed]

2012 (2)

J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(3), 036706 (2012).
[Crossref] [PubMed]

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

2011 (2)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

2009 (2)

2007 (1)

2004 (1)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036604 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (1)

2001 (1)

C. Courtois, A. Couairon, B. Cros, J. R. Marques, and G. Matthieussent, “Propagation of intense ultrashort laser pulses in a plasma filled capillary tube: simulations and experiments,” Phys. Plasmas 8(7), 3445–3456 (2001).
[Crossref]

2000 (1)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

1996 (2)

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68(20), 2793–2795 (1996).
[Crossref]

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996).
[Crossref] [PubMed]

1993 (1)

E. G. Churin, J. Hosfeld, and T. Tschudi, “Polarization configurations with singular point formed by computer generated holograms,” Opt. Commun. 99(1–2), 13–17 (1993).
[Crossref]

1990 (1)

1966 (1)

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924 (1966).

1964 (1)

E. A. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Alahmed, Z. A.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Alisauskas, S.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Andreasen, J.

J. Andreasen and M. Kolesik, “Midinfrared femtosecond laser pulse filamentation in hollow waveguides: a comparison of simulation methods,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(5), 053303 (2013).
[Crossref] [PubMed]

J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(3), 036706 (2012).
[Crossref] [PubMed]

Andriukaitis, G.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Arpin, P.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Azzeer, A. M.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Balciunas, T.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Baltuska, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Becker, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Borot, A.

Brambilla, E.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Brown, S.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Canova, L.

Carbajo, S.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

S. Carbajo, E. Granados, D. Schimpf, A. Sell, K. H. Hong, J. Moses, and F. X. Kärtner, “Efficient generation of ultra-intense few-cycle radially polarized laser pulses,” Opt. Lett. 39(8), 2487–2490 (2014).
[Crossref] [PubMed]

Chen, M. C.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Chen, X.

Churin, E. G.

E. G. Churin, J. Hosfeld, and T. Tschudi, “Polarization configurations with singular point formed by computer generated holograms,” Opt. Commun. 99(1–2), 13–17 (1993).
[Crossref]

Corti, T.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Couairon, A.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

C. Courtois, A. Couairon, B. Cros, J. R. Marques, and G. Matthieussent, “Propagation of intense ultrashort laser pulses in a plasma filled capillary tube: simulations and experiments,” Phys. Plasmas 8(7), 3445–3456 (2001).
[Crossref]

Courtois, C.

C. Courtois, A. Couairon, B. Cros, J. R. Marques, and G. Matthieussent, “Propagation of intense ultrashort laser pulses in a plasma filled capillary tube: simulations and experiments,” Phys. Plasmas 8(7), 3445–3456 (2001).
[Crossref]

Cros, B.

C. Courtois, A. Couairon, B. Cros, J. R. Marques, and G. Matthieussent, “Propagation of intense ultrashort laser pulses in a plasma filled capillary tube: simulations and experiments,” Phys. Plasmas 8(7), 3445–3456 (2001).
[Crossref]

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

de J. Ramírez-Góngora, O.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

De Silvestri, S.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68(20), 2793–2795 (1996).
[Crossref]

Durfee, C. G.

Ford, D. H.

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Gaeta, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Gagnon, J.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Garg, M.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

Goulielmakis, E.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Granados, E.

Grguras, I.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Hassan, M. Th.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Hernández-García, C.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Hong, K. H.

Hosfeld, J.

E. G. Churin, J. Hosfeld, and T. Tschudi, “Polarization configurations with singular point formed by computer generated holograms,” Opt. Commun. 99(1–2), 13–17 (1993).
[Crossref]

Jackel, S.

Jaron-Becker, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Jullien, A.

Kapteyn, H. C.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Karpowicz, N.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

Kartner, F. X.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

Kärtner, F. X.

Keathley, P. D.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

Kimura, W. D.

Kolesik, M.

J. Andreasen and M. Kolesik, “Midinfrared femtosecond laser pulse filamentation in hollow waveguides: a comparison of simulation methods,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(5), 053303 (2013).
[Crossref] [PubMed]

J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(3), 036706 (2012).
[Crossref] [PubMed]

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036604 (2004).
[Crossref] [PubMed]

Krausz, F.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Laurent, G.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

Leger, J.

Leng, Y.

L. Qiao, D. Wang, and Y. Leng, “Spatiotemporal dynamics of an optical pulse propagating in multimode hollow-core fibers filled with prealigned molecular gases,” Phys. Rev. A 93(2), 023832 (2016).
[Crossref]

Lopez-Martens, R.

Lumer, Y.

Luu, T. T.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Machavariani, G.

Majus, D.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Malvache, A.

Marcatili, E. A.

E. A. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Marques, J. R.

C. Courtois, A. Couairon, B. Cros, J. R. Marques, and G. Matthieussent, “Propagation of intense ultrashort laser pulses in a plasma filled capillary tube: simulations and experiments,” Phys. Plasmas 8(7), 3445–3456 (2001).
[Crossref]

Matthieussent, G.

C. Courtois, A. Couairon, B. Cros, J. R. Marques, and G. Matthieussent, “Propagation of intense ultrashort laser pulses in a plasma filled capillary tube: simulations and experiments,” Phys. Plasmas 8(7), 3445–3456 (2001).
[Crossref]

Meir, A.

Miller, R. J. D.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036604 (2004).
[Crossref] [PubMed]

Moriena, G.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

Moses, J.

Moshe, I.

Moulet, A.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Mücke, O. D.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Murnane, M. M.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Nanni, E. A.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

Nisoli, M.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68(20), 2793–2795 (1996).
[Crossref]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

Pabst, S.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Perelomov, A. M.

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924 (1966).

Pervak, V.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Plaja, L.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Popmintchev, D.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Popmintchev, T.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Popov, V. S.

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924 (1966).

Pugzlys, A.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Qiao, L.

L. Qiao, D. Wang, and Y. Leng, “Spatiotemporal dynamics of an optical pulse propagating in multimode hollow-core fibers filled with prealigned molecular gases,” Phys. Rev. A 93(2), 023832 (2016).
[Crossref]

Raskazovskaya, O.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

Santra, R.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Schadt, M.

Schimpf, D.

Schmeltzer, R. A.

E. A. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Schrauth, S. E.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Sell, A.

Shim, B.

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Stalder, M.

Svelto, O.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68(20), 2793–2795 (1996).
[Crossref]

Terent’ev, M. V.

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924 (1966).

Tidwell, S. C.

Trisorio, A.

Tschudi, T.

E. G. Churin, J. Hosfeld, and T. Tschudi, “Polarization configurations with singular point formed by computer generated holograms,” Opt. Commun. 99(1–2), 13–17 (1993).
[Crossref]

Wang, D.

L. Qiao, D. Wang, and Y. Leng, “Spatiotemporal dynamics of an optical pulse propagating in multimode hollow-core fibers filled with prealigned molecular gases,” Phys. Rev. A 93(2), 023832 (2016).
[Crossref]

Wirth, A.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Wong, L. J.

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

Yakovlev, V. S.

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

Zheltikov, A. M.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

Zhokhov, P.

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322–3324 (2000).
[Crossref]

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68(20), 2793–2795 (1996).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Eur. Phys. J. Spec. Top. (1)

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Nature (1)

M. Th. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530(7588), 66–70 (2016).
[Crossref] [PubMed]

Opt. Commun. (1)

E. G. Churin, J. Hosfeld, and T. Tschudi, “Polarization configurations with singular point formed by computer generated holograms,” Opt. Commun. 99(1–2), 13–17 (1993).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Phys. Plasmas (1)

C. Courtois, A. Couairon, B. Cros, J. R. Marques, and G. Matthieussent, “Propagation of intense ultrashort laser pulses in a plasma filled capillary tube: simulations and experiments,” Phys. Plasmas 8(7), 3445–3456 (2001).
[Crossref]

Phys. Rev. A (1)

L. Qiao, D. Wang, and Y. Leng, “Spatiotemporal dynamics of an optical pulse propagating in multimode hollow-core fibers filled with prealigned molecular gases,” Phys. Rev. A 93(2), 023832 (2016).
[Crossref]

Phys. Rev. Accel. Beams (1)

S. Carbajo, E. A. Nanni, L. J. Wong, G. Moriena, P. D. Keathley, G. Laurent, R. J. D. Miller, and F. X. Kartner, “Direct longitudinal laser acceleration of electrons in free space,” Phys. Rev. Accel. Beams 19(2), 021303 (2016).
[Crossref]

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

J. Andreasen and M. Kolesik, “Midinfrared femtosecond laser pulse filamentation in hollow waveguides: a comparison of simulation methods,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(5), 053303 (2013).
[Crossref] [PubMed]

J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(3), 036706 (2012).
[Crossref] [PubMed]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036604 (2004).
[Crossref] [PubMed]

Science (2)

A. Wirth, M. Th. Hassan, I. Grguras, J. Gagnon, A. Moulet, T. T. Luu, S. Pabst, R. Santra, Z. A. Alahmed, A. M. Azzeer, V. S. Yakovlev, V. Pervak, F. Krausz, and E. Goulielmakis, “Synthesized light transients,” Science 334(6053), 195–200 (2011).
[Crossref] [PubMed]

T. Popmintchev, M. C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924 (1966).

Cited By

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Figures (7)

Fig. 1
Fig. 1

(a) Intensity profiles at the fiber inlet for optimal Gaussian distribution (solid blue line) and RP pulses with different beam sizes. (b) Coupling efficiency of RP pulses to TM01 mode versus different beam sizes.

Fig. 2
Fig. 2

(a) Transverse intensity profiles for stable linear propagation of RP input pulses in HCF with 250 μm (blue-dotted line) and 500 μm (red-square line) inner diameters, input RP pulse with κ equals 0.5 (black-circle line) and TM01 mode (green-cross line). (b) Waveguide attenuation of RP input pulses with different initial beam sizes in a 250 μm HCF versus propagation distance.

Fig. 3
Fig. 3

Propagation results of 1.5mJ LP pulse in 500 μm HCF filled with pressure-gradient gases. (a) Spatiospectral distribution of output spectrum; (b) temporal power profiles of compressed pulses versus compensated GDDs; (c) spatiotemporal distribution of compressed pulse with GDD −50fs2; (d) spectral intensity and phase profiles of the compressed pulse. The blue solid line and red dot-dash line indicate the on-axis profiles of intensity and phase; the green dashed line indicates the temporal power profile.

Fig. 4
Fig. 4

Propagation results of 1.5mJ RP pulse with κ equals 0.5 in 500 μm HCF filled with pressure-gradient gases. (a) Spatiospectral distribution of output spectrum; (b) temporal power profiles of compressed pulses versus compensated GDDs; (c) spatiotemporal distribution of compressed pulse with GDD −50fs2; (d) spectral intensity and phase profiles of the compressed pulse. The blue solid line and red dot-dash line indicate the profiles of peak intensity and phase; the green dashed line indicates the temporal power profile.

Fig. 5
Fig. 5

Comparison between RP and LP schemes. (a) Energy transmissions of LP (red-circle line) and RP (blue-dotted line) pulses along propagation in HCF; (b) output power spectra of LP (red-dashed line) and RP (blue-solid line) pulses; transverse energy distributions of (c) LP and (d) RP pulses along propagation.

Fig. 6
Fig. 6

Tight focusing characteristics of RP pulse after HCF propagation followed by 3m free diffraction and calibration. The focusing objective has an NA of 0.8. The first to third rows are intensity profiles for 700nm, 800nm and 900nm, respectively. The first to third columns are | E z | 2 , | E r | 2 and | E z | 2 + | E r | 2 , respectively. The radially polarized pulse is generated by the RP scheme.

Fig. 7
Fig. 7

Spatiotemporal intensity profiles near the focus after tight focusing with an NA = 0.8 objective. The first row indicates the intensity profiles of | E z | 2 components; the second is | E r | 2 components. The intensities are normalized to the peak values of corresponding | E z | 2 components. The radially polarized pulse is generated by the RP scheme.

Equations (5)

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z E(z,r,ω)=i( L ^ ω v g )E(z,r,ω)+ ω 2β c 2 [ iω P(z,r,ω) ε 0 j(z,r,ω) ε 0 ],
ρ t =W(I)( ρ nt ρ)+ σρI U i ,
E(z=0,t,r)= E 0 f(r)exp{ r 2 w 0 2 }exp{ t 2 2 t 2 0 },
η cp = { 0 1 r 2 exp{ r 2 κ 2 } V TM01 ( r )dr } 2 0 1 r V TM01 2 ( r )dr 0 r 3 exp{ 2 r 2 κ 2 } dr ,
E r (r,z)= 1 λ 0 θ max | P(θ) | cosθ sinθcosθ J 1 (2π r λ sinθ)exp(i2π z λ cosθ+iϕ(θ))dθ , E z (r,z)=i 1 λ 0 θ max | P(θ) | cosθ sin 2 θ J 0 (2π r λ sinθ)exp(i2π z λ cosθ+iϕ(θ))dθ ,

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