Abstract

The carrier-envelope phase (CEP) shift of few-cycle pulses along a focal region is evaluated without neglecting the pulse reshaping inherent to any focusing process. The CEP shift is then found to strongly depend on the chirp of the focused pulse. Based on this effect, realistic optical systems for focusing few-cycle pulses are proposed to control their focal CEP shift for their phase-sensitive interactions with matter.

© 2012 Optical Society of America

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References

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  1. F. Krausz and M. Yu. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
    [CrossRef]
  2. C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
    [CrossRef]
  3. M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
    [CrossRef]
  4. T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
    [CrossRef]
  5. G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
    [CrossRef]
  6. F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
    [CrossRef]
  7. T. Tritschler, K. D. Hof, M. W. Klein, and M. Wegener, “Variation of the carrier-envelope phase of few-cycle laser pulses owing to the Gouy phase: a solid-state-based measurement,” Opt. Lett. 30, 753–755 (2005).
    [CrossRef]
  8. M. A. Porras, “Characterization of the electric field of focused pulsed Gaussian beams for phase-sensitive interactions with matter,” Opt. Lett. 34, 1546–1548 (2009).
    [CrossRef]
  9. M. A. Porras, Z. L. Horvath, and B. Major, “On the use of lenses to focus few-cycle pulses with controlled carrier-envelope phase,” Appl. Phys. B 108, 521–531 (2012).
    [CrossRef]
  10. I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
    [CrossRef]
  11. C. J. R. Sheppard and X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
    [CrossRef]
  12. A. E. Kaplan, “Diffraction-induced transformation of near-cycle and sub-cycle pulses,” J. Opt. Soc. Am. B 15, 951–956(1998).
    [CrossRef]
  13. G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
    [CrossRef]
  14. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).
  16. See, for example, A. E. Siegman, Lasers (University Science, 1986).
  17. M. A. Porras, “Ultrashort pulsed Gaussian beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [CrossRef]

2012

M. A. Porras, Z. L. Horvath, and B. Major, “On the use of lenses to focus few-cycle pulses with controlled carrier-envelope phase,” Appl. Phys. B 108, 521–531 (2012).
[CrossRef]

2009

F. Krausz and M. Yu. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

M. A. Porras, “Characterization of the electric field of focused pulsed Gaussian beams for phase-sensitive interactions with matter,” Opt. Lett. 34, 1546–1548 (2009).
[CrossRef]

2006

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

2005

2004

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

2003

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

2002

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

2000

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

1998

M. A. Porras, “Ultrashort pulsed Gaussian beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

A. E. Kaplan, “Diffraction-induced transformation of near-cycle and sub-cycle pulses,” J. Opt. Soc. Am. B 15, 951–956(1998).
[CrossRef]

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

1997

C. J. R. Sheppard and X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

1985

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

Baltuska, A.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).

Cavalieri, A. L.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

Cheng, Z.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Chipperfield, L. E.

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

Christov, I. P.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Drescher, M.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Gan, X.

C. J. R. Sheppard and X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Goulielmakis, E.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Gu, X.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

Hambach, D.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Haworth, C. A.

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

Heinzmann, U.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Helml, W.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

Hentschel, M.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Hof, K. D.

Horvath, B.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

Horvath, Z. L.

M. A. Porras, Z. L. Horvath, and B. Major, “On the use of lenses to focus few-cycle pulses with controlled carrier-envelope phase,” Appl. Phys. B 108, 521–531 (2012).
[CrossRef]

Ivanov, M. Yu.

F. Krausz and M. Yu. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

Kaplan, A. E.

Kienberger, R.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

Klein, M. W.

Knight, P. L.

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

Krausz, F.

F. Krausz and M. Yu. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Lezius, M.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Lim, Y.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Lindner, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Major, B.

M. A. Porras, Z. L. Horvath, and B. Major, “On the use of lenses to focus few-cycle pulses with controlled carrier-envelope phase,” Appl. Phys. B 108, 521–531 (2012).
[CrossRef]

Marangos, J. P.

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

Paulus, G. G.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Porras, M. A.

M. A. Porras, Z. L. Horvath, and B. Major, “On the use of lenses to focus few-cycle pulses with controlled carrier-envelope phase,” Appl. Phys. B 108, 521–531 (2012).
[CrossRef]

M. A. Porras, “Characterization of the electric field of focused pulsed Gaussian beams for phase-sensitive interactions with matter,” Opt. Lett. 34, 1546–1548 (2009).
[CrossRef]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Robinson, J. S.

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

Schätzel, M. G.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

Schmahl, G.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Schnürer, M.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Siegman, A. E.

See, for example, A. E. Siegman, Lasers (University Science, 1986).

Tisch, J. W. G.

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

Tritschler, T.

Walther, H.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Wegener, M.

Wilhein, T.

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

Wittmann, T.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).

Appl. Phys. B

M. Schnürer, Z. Cheng, M. Hentschel, F. Krausz, T. Wilhein, D. Hambach, G. Schmahl, M. Drescher, Y. Lim, and U. Heinzmann, “Few-cycle-driven XUV laser harmonics: generation and focusing,” Appl. Phys. B 70, S227–S232 (2000).
[CrossRef]

M. A. Porras, Z. L. Horvath, and B. Major, “On the use of lenses to focus few-cycle pulses with controlled carrier-envelope phase,” Appl. Phys. B 108, 521–531 (2012).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Phys.

T. Wittmann, B. Horvath, W. Helml, M. G. Schätzel, X. Gu, A. L. Cavalieri, G. G. Paulus, and R. Kienberger, “Single-shot carrier-envelope phase measurement of few-cycle laser pulses,” Nat. Phys. 5, 357–362 (2009).
[CrossRef]

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos, and J. W. G. Tisch, “Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval,” Nat. Phys. 3, 52–57 (2006).
[CrossRef]

Opt. Commun.

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

C. J. R. Sheppard and X. Gan, “Free-space propagation of femtosecond light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. E

M. A. Porras, “Ultrashort pulsed Gaussian beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002).
[CrossRef]

Phys. Rev. Lett.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Gouy phase shift for few-cycle laser pulses,” Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Rev. Mod. Phys.

F. Krausz and M. Yu. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1987).

See, for example, A. E. Siegman, Lasers (University Science, 1986).

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Figures (4)

Fig. 1.
Fig. 1.

(a) CEP shifts from the focal point predicted by Eq. (13) in the nonreshaping approximation for pulsed Gaussian beams with spot size independent of frequency focused by a mirror ( g = 1 , γ = 0 ) (dotted curve), isodiffracting pulsed Gaussian beams focused by a mirror ( g = 0 , γ = 0 ) (dashed curve), and isodiffracting pulsed Gaussian beams focused by a lens ( g = 0 , γ = 1 ). (b) Sketch of the effect of chirp on envelope reshaping when focusing with small chromatic aberration. Bluer spectral components (dashed blue curves) placed in the trailing part of the pulse are focused at shorter distances, and redder spectral components (solid red curves) in the leading part, at larger distances. (c) CEP shifts from the focal point predicted by Eq. (21) for isodiffracting pulsed Gaussian beams ( g = 0 , γ = 1 ) with increasing chirps.

Fig. 2.
Fig. 2.

For focused Gaussian pulses with increasing chirps, CEP shifts from the focal point along the focal region of a fused silica lens, evaluated numerically (symbols) and predicted by Eq. (21) (curves). See text for details about the input pulse, the lens characteristics, and the numerical method.

Fig. 3.
Fig. 3.

Axial pulse shapes and envelopes (a) at the beginning, (b) at the middle, and (c) at the end of the focal region for the input pulse and fused silica lens described in the text. The pulse chirps are 2 C / Δ t m 2 = 0.55 (solid curves) and 2 C / Δ t m 2 = 0 (dashed curves).

Fig. 4.
Fig. 4.

For focused Gaussian pulses with different chirps, CEP shifts along the focal region of the mirror–lens system described in the text, evaluated numerically (symbols) and from Eq. (21) (curves). See the text for details about the input pulse and focusing system.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

E ( t , r , z ) = 1 π 0 d ω E ( ω , r , z ) exp ( i ω t )
E ( ω , r , z ) = p D ( ω ) f q exp [ i ( ω r 2 2 c q + ω c z ) ] ,
p D ( ω ) = p ( ω ) exp ( i ω c D n ) .
f = [ ( n 1 ) ( 1 R 1 + 1 R 2 ) ( n 1 ) 2 n D R 1 R 2 ] 1 R 2 D n ( R 1 + R 2 D ) + d
γ f 0 L R , 0 ω 0
a ( ω , z ) = f L R 1 [ 1 + ( Z / L R ) 2 ] 1 / 2 ,
φ ( ω , z ) = ω c z π 2 tan 1 ( Z L R )
E ( t , z ) = A ( t , z ) exp { i [ ω 0 t φ 0 ( z ) ] } ,
A ( τ , z ) = 1 π 0 d ω p D ( ω ) a ( ω , z ) × exp [ i 1 2 φ 0 ( z ) ( ω ω 0 ) 2 ] exp [ i τ ( ω ω 0 ) ] ,
A ( τ , z ) a 0 ( z ) A D ( τ ) + i a 0 ( z ) d A D ( τ ) d τ ,
A D ( τ ) = 1 π 0 d ω p D ( ω ) exp [ i τ ( ω ω 0 ) ]
Δ Φ ( z ) = [ ω 0 φ 0 ( z ) + φ 0 ( z ) ] [ ω 0 φ 0 ( f 0 ) + φ 0 ( f 0 ) ] .
Δ Φ ( z ) = tan 1 ζ + g ζ + γ ζ 2 1 + ζ 2 ,
g = L R , 0 L R , 0 ω 0 1 + 2 s 0 s 0 ω 0
Δ Φ total ( z ) = Δ Φ ( z ) + Δ Φ C ( z ) ,
Δ Φ C ( z ) = ω 0 [ τ p ( z ) τ p ( f 0 ) ] + ϕ ( z ) ϕ ( f 0 ) ,
A D ( τ ) = Δ t m b exp ( τ 2 b 2 ) ,
h ( z ) = f 0 f 0 ( 1 + f 0 L R , 0 ζ 1 + ζ 2 ) L R , 0 L R , 0 ( 1 ζ 2 1 + ζ 2 ) ,
| A ( τ , z ) | Δ t m | b | exp [ h 2 ( z ) Δ t m 2 ] exp ( τ ˜ 2 Δ t 2 ) a 0 ( z ) ,
Δ Φ C ( z ) = 2 C Δ t m 2 γ ζ + g ζ 2 1 + ζ 2 .
Δ Φ total ( z ) = tan 1 ζ + G ζ + Γ ζ 2 1 + ζ 2 ,
G = g 2 C Δ t m 2 γ , Γ = γ + 2 C Δ t m 2 g .
s 0 = 2 c L R , 0 ω 0 n 0 1 n 0 ω 0 , f 0 = ω 0 L R , 0 2 c s 0 ,
2 C Δ t m 2 + 0.55
γ f 0 2 f L , 0 2 n 0 ω 0 n 0 1 f L , 0 L R , 0 .
1 f L , 0 = n 0 1 n 0 ω 0 2 c ω 0 s 0 2 ,

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