Abstract

Compensating attosecond chirp (atto-chirp) of broadband high-order harmonic pulses in the water window region (282 to 533 eV) is a major challenge, due to the lack of natural materials that exhibit negative group velocity dispersion and low loss. Analysis shows that the amount of dispersion of fully ionized hydrogen plasma with suitable density-length product is sufficient to compensate the chirp of attosecond pulses with center photon energy above 300 eV. This is confirmed by numerical simulations based on the Strong Field Approximation.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  5. B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65(1), 011804 (2001).
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2018 (1)

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

2017 (1)

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

2016 (2)

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

A. E. Dyson, C. Thornton, and S. M. Hooker, “A compact, low cost Marx bank for generating capillary discharge plasmas,” Rev. Sci. Instrum. 87(9), 093302 (2016).
[Crossref] [PubMed]

2014 (1)

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

2013 (1)

2012 (1)

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

2009 (1)

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

2008 (1)

2005 (1)

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Phys. Rev. Lett. 94(3), 033901 (2005).
[Crossref] [PubMed]

2004 (2)

K. T. Kim, C. M. Kim, M.-G. Baik, G. Umesh, and C. H. Nam, “Single sub-50- attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A 69(5), 051805 (2004).
[Crossref]

S. Kazamias and P. Balcou, “Intrinsic chirp of attosecond pulses: Single-atom model versus experiment,” Phys. Rev. A 69(6), 063416 (2004).
[Crossref]

2003 (1)

2001 (4)

B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65(1), 011804 (2001).
[Crossref]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

D. J. Spence and S. M. Hooker, “Investigation of a hydrogen plasma waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 015401 (2001).
[PubMed]

1994 (1)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

1993 (2)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref] [PubMed]

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[Crossref] [PubMed]

Agostini, P.

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Alisauskas, S.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Andriukaitis, G.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Antipov, S.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Arpin, P.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Augé, F.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Baik, M.-G.

K. T. Kim, C. M. Kim, M.-G. Baik, G. Umesh, and C. H. Nam, “Single sub-50- attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A 69(5), 051805 (2004).
[Crossref]

Balciunas, T.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Balcou, P.

S. Kazamias and P. Balcou, “Intrinsic chirp of attosecond pulses: Single-atom model versus experiment,” Phys. Rev. A 69(6), 063416 (2004).
[Crossref]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Baltuska, A.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Becker, A.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Bobrova, N.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Brabec, T.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Breger, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Breidbach, J.

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Phys. Rev. Lett. 94(3), 033901 (2005).
[Crossref] [PubMed]

Brown, S.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Bulanov, S.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Butler, A.

Butler, J.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Cederbaum, L. S.

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Phys. Rev. Lett. 94(3), 033901 (2005).
[Crossref] [PubMed]

Chang, Z.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65(1), 011804 (2001).
[Crossref]

Chen, M.-C.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Cheng, Y.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Chew, A.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Chini, M.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

Chirla, R.

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

Corkum, P.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Corkum, P. B.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref] [PubMed]

Cunningham, E.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Daniels, J.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

DiMauro, L. F.

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[Crossref] [PubMed]

Doumy, G.

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

Drescher, M.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Dyson, A. E.

A. E. Dyson, C. Thornton, and S. M. Hooker, “A compact, low cost Marx bank for generating capillary discharge plasmas,” Rev. Sci. Instrum. 87(9), 093302 (2016).
[Crossref] [PubMed]

Frassetto, F.

Gaeta, A.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Gonsalves, A.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Guggenmos, A.

Gullikson, E. M.

Heinzmann, U.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Hentschel, M.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Hernandez-Garcia, C.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Hertrich, S.

Hofstetter, M.

Hooker, S. M.

A. E. Dyson, C. Thornton, and S. M. Hooker, “A compact, low cost Marx bank for generating capillary discharge plasmas,” Rev. Sci. Instrum. 87(9), 093302 (2016).
[Crossref] [PubMed]

D. Spence, A. Butler, and S. M. Hooker, “Gas-filled capillary discharge waveguides,” J. Opt. Soc. Am. B 20(1), 138–151 (2003).
[Crossref]

D. J. Spence and S. M. Hooker, “Investigation of a hydrogen plasma waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 015401 (2001).
[PubMed]

Hu, S.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Ivanov, M. Y.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Jaron-Becker, A.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Kapteyn, H. C.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Kazamias, S.

S. Kazamias and P. Balcou, “Intrinsic chirp of attosecond pulses: Single-atom model versus experiment,” Phys. Rev. A 69(6), 063416 (2004).
[Crossref]

Kienberger, R.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Kim, C. M.

K. T. Kim, C. M. Kim, M.-G. Baik, G. Umesh, and C. H. Nam, “Single sub-50- attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A 69(5), 051805 (2004).
[Crossref]

Kim, K. T.

K. T. Kim, C. M. Kim, M.-G. Baik, G. Umesh, and C. H. Nam, “Single sub-50- attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A 69(5), 051805 (2004).
[Crossref]

Kleineberg, U.

Krausz, F.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Kulander, K. C.

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[Crossref] [PubMed]

L’Huillier, A.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Leemans, W. P.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Lewenstein, M.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

Li, J.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Liu, F.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Milosevic, N.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Mittelberger, D. E.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Mucke, O. D.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Muller, H. G.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Mullot, G.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Murnane, M. M.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Nam, C. H.

K. T. Kim, C. M. Kim, M.-G. Baik, G. Umesh, and C. H. Nam, “Single sub-50- attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A 69(5), 051805 (2004).
[Crossref]

Nickel, B.

Paul, P. M.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Pieronek, C.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Plaja, L.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Poletto, L.

Popmintchev, D.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Popmintchev, T.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Pugzlys, A.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Rauhut, R.

Reider, G. A.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Ren, X.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Roedig, C.

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

Sasorov, P.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Schafer, K. J.

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[Crossref] [PubMed]

Schmidt, J.

Schnick, W.

Schrauth, S. E.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Seibald, M.

Shan, B.

B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65(1), 011804 (2001).
[Crossref]

Shim, B.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, 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, 1287–1291 (2012).

Spence, D.

Spence, D. J.

D. J. Spence and S. M. Hooker, “Investigation of a hydrogen plasma waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 015401 (2001).
[PubMed]

Spielmann, C.

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Thornton, C.

A. E. Dyson, C. Thornton, and S. M. Hooker, “A compact, low cost Marx bank for generating capillary discharge plasmas,” Rev. Sci. Instrum. 87(9), 093302 (2016).
[Crossref] [PubMed]

Toma, E. S.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292(5522), 1689–1692 (2001).
[Crossref] [PubMed]

Umesh, G.

K. T. Kim, C. M. Kim, M.-G. Baik, G. Umesh, and C. H. Nam, “Single sub-50- attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A 69(5), 051805 (2004).
[Crossref]

Villoresi, P.

Waldron, W.

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

Wang, Y.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Wheeler, J.

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

Wu, Y.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Yang, B.

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[Crossref] [PubMed]

Yin, Y.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Zhao, K.

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

J. Appl. Phys. (1)

A. Gonsalves, F. Liu, N. Bobrova, P. Sasorov, C. Pieronek, J. Daniels, S. Antipov, J. Butler, S. Bulanov, W. Waldron, D. E. Mittelberger, and W. P. Leemans, “Demonstration of a high repetition rate capillary discharge waveguide,” J. Appl. Phys. 119(3), 033302 (2016).
[Crossref]

J. Opt. (1)

X. Ren, J. Li, Y. Yin, K. Zhao, A. Chew, Y. Wang, S. Hu, Y. Cheng, E. Cunningham, Y. Wu, M. Chini, and Z. Chang, “Attosecond light sources in the water window,” J. Opt. 20(2), 023001 (2018).
[Crossref]

J. Opt. Soc. Am. B (2)

Nat. Commun. (1)

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
[Crossref] [PubMed]

Nat. Photonics (1)

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

Nature (1)

M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001).
[Crossref] [PubMed]

Opt. Express (1)

Phys. Rev. A (4)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994).
[Crossref] [PubMed]

K. T. Kim, C. M. Kim, M.-G. Baik, G. Umesh, and C. H. Nam, “Single sub-50- attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A 69(5), 051805 (2004).
[Crossref]

B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65(1), 011804 (2001).
[Crossref]

S. Kazamias and P. Balcou, “Intrinsic chirp of attosecond pulses: Single-atom model versus experiment,” Phys. Rev. A 69(6), 063416 (2004).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

D. J. Spence and S. M. Hooker, “Investigation of a hydrogen plasma waveguide,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 015401 (2001).
[PubMed]

Phys. Rev. Lett. (4)

G. Doumy, J. Wheeler, C. Roedig, R. Chirla, P. Agostini, and L. F. DiMauro, “Attosecond synchronization of high-order harmonics from midinfrared drivers,” Phys. Rev. Lett. 102(9), 093002 (2009).
[Crossref] [PubMed]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref] [PubMed]

K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, “Above threshold ionization beyond the high harmonic cutoff,” Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[Crossref] [PubMed]

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Phys. Rev. Lett. 94(3), 033901 (2005).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

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Figures (2)

Fig. 1
Fig. 1 (a) Driving laser field. (b) Power spectrum of the X-ray pulse. (c) Phases of X-rays with and without chirp compensation by plasma dispersion.
Fig. 2
Fig. 2 Electric fields of the X-ray pulses. (a) Without chirp compensation. (b) With chirp compensation. (c) Transform-limited. The pulse durations, τ p , are the FWHM of the intensity profiles.

Equations (8)

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ω c = I p +3× 10 13 I 0 λ 0 2 ,
Chirp=1.63× 10 18 1 I 0 λ 0 .
n( ω x )= 1 ω p 2 ω x 2 ,
ω p = e 2 N e ε 0 m e ,
n( ω x )1 1 2 ω p 2 ω x 2 .
v g ( ω x )= d ω k d k x =n( ω x )c,
GVD( ω x ) ω p 2 c ω x 3 = e 2 ε 0 m e c N e ω x 3 ,
GDD( ω x )= e 2 3 ε 0 m e c N e L p ( ω x ) 3 .