Abstract

We describe the temporal evolution of the electric field of few-cycle optical pulses with arbitrary, time-varying polarization states by means of the instantaneous polarization ellipse and phase, whose physical meanings for few-cycle pulses are clarified. A physically meaningful definition of carrier–envelope phase (CEP) for arbitrarily polarized pulses is introduced. This description is used to study the changes in the temporal evolution of the electric field of a few-cycle pulsed beam. Propagation is found to result in significant changes in the polarization state, phase, and CEP. Approximate analytical formulas for these effects are provided.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. M. Nisoli, G. Sansone, S. Stagira, and S. De Silvestri, “Effects of carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra,” Phys. Rev. Lett. 91, 213905 (2003).
    [CrossRef]
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2012 (1)

2009 (2)

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

2008 (1)

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

2003 (4)

T. Brixner, N. H. Damrauer, G. Krampert, P. Niklaus, and G. Gerber, “Adaptive shaping of femtosecond polarization profiles,” J. Opt. Soc. Am. B 20, 878–881 (2003).
[CrossRef]

D. B. Milosevic, G. G. Paulus, and W. Becker, “Above-threshold ionization with few-cycle laser pulses and the relevance of the absolute phase,” Laser Physics 13, 948–958 (2003).

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

M. Nisoli, G. Sansone, S. Stagira, and S. De Silvestri, “Effects of carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra,” Phys. Rev. Lett. 91, 213905 (2003).
[CrossRef]

2002 (5)

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

D. B. Milosevic, G. G. Paulus, and W. Becker, “Phase dependent effects of a few-cycle pulse,” Phys. Rev. Lett. 89, 153001 (2002).
[CrossRef]

M. Kakehata, Y. Kobayashi, H. Takada, and K. Torizuka, “Single-shot measurement of a carrier-envelope phase by use of a time-dependent polarization pulse,” Opt. Lett. 27, 1247–1249 (2002).
[CrossRef]

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, S133–S144 (2002).
[CrossRef]

S. Stagira, E. Priori, G. Sansone, M. Nisoli, and S. De Silvestri, “Nonlinear guided propagation of few-optical-cycle laser pulses with arbitrary polarization states,” Phys. Rev. A 66, 033810 (2002).
[CrossRef]

2001 (1)

2000 (2)

1997 (1)

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

1994 (2)

Agrawal, G. P.

Baltuska, A.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Becker, W.

D. B. Milosevic, G. G. Paulus, and W. Becker, “Above-threshold ionization with few-cycle laser pulses and the relevance of the absolute phase,” Laser Physics 13, 948–958 (2003).

D. B. Milosevic, G. G. Paulus, and W. Becker, “Phase dependent effects of a few-cycle pulse,” Phys. Rev. Lett. 89, 153001 (2002).
[CrossRef]

Benedetti, E.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

Born, E.

E. Born and E. Wolf, Principles of Optics (Pergamon, 1975), pp. 23–32.

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Brixner, T.

Burnett, N. H.

Calegari, F.

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

Caumes, J. P.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

Chang, Z.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

Corkum, P. B.

Damrauer, N. H.

De Silvestri, S.

M. Nisoli, G. Sansone, S. Stagira, and S. De Silvestri, “Effects of carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra,” Phys. Rev. Lett. 91, 213905 (2003).
[CrossRef]

S. Stagira, E. Priori, G. Sansone, M. Nisoli, and S. De Silvestri, “Nonlinear guided propagation of few-optical-cycle laser pulses with arbitrary polarization states,” Phys. Rev. A 66, 033810 (2002).
[CrossRef]

Dietrich, P.

Ferrai, F.

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

Gerber, G.

Gilbertson, S.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

Gohte, Ch.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Goulielmakis, E.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Hänsch, T. W.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Hentschel, M.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Holzwarth, R.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Horvath, Z. L.

Ivanov, M. Y.

James, D. F. V.

Kakehata, M.

Khan, S. D.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

Kobayashi, Y.

Krampert, G.

T. Brixner, N. H. Damrauer, G. Krampert, P. Niklaus, and G. Gerber, “Adaptive shaping of femtosecond polarization profiles,” J. Opt. Soc. Am. B 20, 878–881 (2003).
[CrossRef]

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, S133–S144 (2002).
[CrossRef]

Krausz, F.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

P. Dietrich and F. Krausz, “Determining the absolute carrier phase of a few-cycle laser pulse,” Opt. Lett. 25, 16–18 (2000).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Li, C.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

Major, B.

Mashiko, H.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

Milosevic, D. B.

D. B. Milosevic, G. G. Paulus, and W. Becker, “Above-threshold ionization with few-cycle laser pulses and the relevance of the absolute phase,” Laser Physics 13, 948–958 (2003).

D. B. Milosevic, G. G. Paulus, and W. Becker, “Phase dependent effects of a few-cycle pulse,” Phys. Rev. Lett. 89, 153001 (2002).
[CrossRef]

Moon, E.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

Niklaus, P.

T. Brixner, N. H. Damrauer, G. Krampert, P. Niklaus, and G. Gerber, “Adaptive shaping of femtosecond polarization profiles,” J. Opt. Soc. Am. B 20, 878–881 (2003).
[CrossRef]

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, S133–S144 (2002).
[CrossRef]

Nisoli, M.

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

M. Nisoli, G. Sansone, S. Stagira, and S. De Silvestri, “Effects of carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra,” Phys. Rev. Lett. 91, 213905 (2003).
[CrossRef]

S. Stagira, E. Priori, G. Sansone, M. Nisoli, and S. De Silvestri, “Nonlinear guided propagation of few-optical-cycle laser pulses with arbitrary polarization states,” Phys. Rev. A 66, 033810 (2002).
[CrossRef]

Paulus, G. G.

D. B. Milosevic, G. G. Paulus, and W. Becker, “Above-threshold ionization with few-cycle laser pulses and the relevance of the absolute phase,” Laser Physics 13, 948–958 (2003).

D. B. Milosevic, G. G. Paulus, and W. Becker, “Phase dependent effects of a few-cycle pulse,” Phys. Rev. Lett. 89, 153001 (2002).
[CrossRef]

Poletto, L.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

Porras, M. A.

Priori, E.

S. Stagira, E. Priori, G. Sansone, M. Nisoli, and S. De Silvestri, “Nonlinear guided propagation of few-optical-cycle laser pulses with arbitrary polarization states,” Phys. Rev. A 66, 033810 (2002).
[CrossRef]

Sansone, G.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

M. Nisoli, G. Sansone, S. Stagira, and S. De Silvestri, “Effects of carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra,” Phys. Rev. Lett. 91, 213905 (2003).
[CrossRef]

S. Stagira, E. Priori, G. Sansone, M. Nisoli, and S. De Silvestri, “Nonlinear guided propagation of few-optical-cycle laser pulses with arbitrary polarization states,” Phys. Rev. A 66, 033810 (2002).
[CrossRef]

Scrinzi, A.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Shakya, M. M.

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), pp. 626–697.

Stagira, S.

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

M. Nisoli, G. Sansone, S. Stagira, and S. De Silvestri, “Effects of carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra,” Phys. Rev. Lett. 91, 213905 (2003).
[CrossRef]

S. Stagira, E. Priori, G. Sansone, M. Nisoli, and S. De Silvestri, “Nonlinear guided propagation of few-optical-cycle laser pulses with arbitrary polarization states,” Phys. Rev. A 66, 033810 (2002).
[CrossRef]

Strelkov, V.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

Takada, H.

Torizuka, K.

Udem, Th.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Uiberacker, M.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Villoresi, P.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

Vozzi, C.

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

Wolf, E.

Yakovlev, V. S.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Appl. Phys. B (1)

T. Brixner, G. Krampert, P. Niklaus, and G. Gerber, “Generation and characterization of polarization-shaped femtosecond laser pulses,” Appl. Phys. B 74, S133–S144 (2002).
[CrossRef]

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

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

J. Phys. B (1)

G. Sansone, F. Ferrai, C. Vozzi, F. Calegari, S. Stagira, and M. Nisoli, “Towards atomic unit pulse duration by polarization-controlled few-cycle pulses,” J. Phys. B 42, 134005 (2009).
[CrossRef]

Laser Physics (1)

D. B. Milosevic, G. G. Paulus, and W. Becker, “Above-threshold ionization with few-cycle laser pulses and the relevance of the absolute phase,” Laser Physics 13, 948–958 (2003).

Nature (1)

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohte, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic prosesses by intense light fields,” Nature 421, 611–615 (2003).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

S. Stagira, E. Priori, G. Sansone, M. Nisoli, and S. De Silvestri, “Nonlinear guided propagation of few-optical-cycle laser pulses with arbitrary polarization states,” Phys. Rev. A 66, 033810 (2002).
[CrossRef]

G. Sansone, E. Benedetti, J. P. Caumes, S. Stagira, C. Vozzi, M. Nisoli, L. Poletto, P. Villoresi, and V. Strelkov, “Shaping of attosecond pulses by phase-stabilized polarization gating,” Phys. Rev. A 80, 063837 (2009).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (4)

D. B. Milosevic, G. G. Paulus, and W. Becker, “Phase dependent effects of a few-cycle pulse,” Phys. Rev. Lett. 89, 153001 (2002).
[CrossRef]

H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double Optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

M. Nisoli, G. Sansone, S. Stagira, and S. De Silvestri, “Effects of carrier-envelope phase differences of few-optical-cycle light pulses in single-shot high-order-harmonic spectra,” Phys. Rev. Lett. 91, 213905 (2003).
[CrossRef]

Other (2)

A. E. Siegman, Lasers (University Science Books, 1986), pp. 626–697.

E. Born and E. Wolf, Principles of Optics (Pergamon, 1975), pp. 23–32.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Components ex(t) and ey(t) of a few-cycle pulse. Both components are Gaussian pulses of different amplitudes, centers, durations, and chirps. (b) Trajectory of the electric field (thick curve), electric field at a particular instant of time (arrow), and associated instantaneous polarization ellipse (thin curve). (c) and (d) The same as in (a) and (b), respectively, when a phase δ=π/2 is added to both components. Electric fields are in arbitrary units.

Fig. 2.
Fig. 2.

(a)–(c) Pulse shapes (thick curves), instantaneous polarization ellipses at the instant tm=1fs of maximum major semi-axis (thin curves), and electric fields at this instant for the same pulse (arrows), except that a phase shift δ is added to make the CEPs ϕ(tm) equal to (a) 0, (b) π/2, and (c) π. The Cartesian components of the pulse are given by Eq. (2) with ω0=3.2fs1, Ax(t)=exp[(t+1)2/2.52], Ay(t)=0.8exp[(t1)2/2.52], Φx(t)=0.08(t+1)2+δ, Φy(t)=0.08(t1)2+π/2+δ, i.e., 1+1/2–cycle Gaussian pulses with different amplitudes, temporal peaks, and chirps. The global phases are δ=3.262, 1.691, and 0.120 in (a)–(c), respectively. (d)–(f) Major and minor semi-axes of the instantaneous polarization ellipses as functions of time (black curves) enveloping the instantaneous strength of the electric field e(t) (gray curves) for the three pulses in (a)–(c), respectively. Electric fields are in arbitrary units.

Fig. 3.
Fig. 3.

Graphs for the propagated pulse in Fig. 2(a) at z=0 with a Gaussian transversal profile of spot size s=0.1mm. (a) Pulse shape at the point z=5LR(ω0)=267mm, r=2s(ω0)=0.721mm (thick curve), along with polarization ellipses and electric fields at the local times τ=1fs and τm(P)=0.3fs (arrows and ellipses). (b) Orientation and ellipticity of the instantaneous polarization ellipses (solid curves) compared to those of the source pulse (dashed curves). (c) Instantaneous strength of the electric field (gray curve), major and minor semi-axes of the instantaneous polarization ellipses (solid black curves) compared to those of the source pulse (dashed curves).

Fig. 4.
Fig. 4.

Graphs for the source pulse with Cartesian components given by Eq. (2) with ω0=3.2fs1, Gaussian amplitudes Ax(t)=exp[t2/3.342] (two-cycle in FWHM), Ay(t)=exp[(t1)2/1.672] (single-cycle), Φx(t)=0.045t2, and Φy(t)=π/3. (a) Pulse shape. (b) Electric field strength (gray curve) and major and minor semi-axes (black curves). The time of maximum major semi-axis is tm=0.79fs, yielding a CEP Φ(tm)=3.075rad. (c) and (d) When the source spot size is s=0.1mm, temporal shape of the pulse at z=10L(ω0)=533, r=0mm, evaluated numerically (solid curves), its zeroth-order approximation [dotted curve in (c)], and its first-order approximation [dashed curve in (d)].

Fig. 5.
Fig. 5.

For the propagated pulse in Fig. 4, (a) orientation, (b) ellipticity, and (c) electric field strength (gray curve) and major and minor semi-axes of the instantaneous polarization ellipses as functions of the local time, evaluated numerically (solid curves), and from the first-order approximation (dashed curves). In the zeroth-order approximation (dotted curves), the orientation and ellipticity are those for the source pulse, and the major and minor semi-axes those at source scaled by a0. The time of maximum major semi-axis shifts from τ=0.79fs for the source to τm(P)=0.62fs for the propagated pulse, yielding a CEP of Φ(P)(τm)=2.237rad.

Equations (36)

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Ex(t)=Ax(t)exp(iω0t),Ey(t)=Ay(t)exp(iω0t),
ex(t)=Ax(t)cos[Φx(t)ω0t],ey(t)=Ay(t)cos[Φy(t)ω0t],
ψ(t)={ψ˜(t)ifα(t)π/4,ψ˜(t)+π/2ifα(t)>π/4andψ˜(t)<0,ψ˜(t)π/2ifα(t)>π/4andψ˜(t)0,
χ(t)=12sin1{sin[2α(t)]sinΦ(t)}.
Φ(t)=Φx(t)Φy(t)
α(t)=tan1[Ay(t)Ax(t)]
Ax(t)=Ax(t)[1+tan2α(t)1+tan2χ(t)]1/2,
ex(t)=Ax(t)cos[Φx(t)ω0t],
ey(t)=sign[χ(t)]Ay(t)sin[Φx(t)ω0t],
Φx(t)=Φx(t)sign[ψ(t)χ(t)]cos1{cosψ(t)cosχ(t)cosα(t)}
ϕ(t)=Φx(t)ω0t
ϕ(tm)=Φx(tm)ω0tm
e2(t)=Ax2(t)+Ay2(t)2+Ax2(t)Ay2(t)2cos[2ϕ(t)],
Ex,y(t)=1π0dωe^x,y(ω)exp(iωt),
ex,y(t)=12πdωe^x,y(ω)exp(iωt).
Ax,y(t)=1π0dωe^x,y(ω)exp[i(ωω0)t],
Ex,y(S)(t)=1π0dωe^x,y(ω)Uexp(iωt),
Ex,y(P)(t)=1π0dωe^x,y(ω)U^(ω)exp(iωt),
U^(ω)=iL(ω)q(ω)exp[iωr22cq(ω)]exp(iωc),
U^(ω)=a(ω)exp[iφ(ω)],
a(ω)=ss(ω)exp[r2s2(ω)],
φ(ω)=ωcztan1[zL(ω)]+ωr22cR(ω),
Ex,y(P)(τ)=Ax,y(P)(τ)exp(iω0τ),
Ax,y(P)(τ)a0Ax,y(τ)[1+ia0a0dAx,y(τ)/dτAx,y(τ)]×exp[i(φ0ω0φ0)],
τ=tφ0
a0=ss(ω0)exp[r2s2(ω0)],
a0=1ω0z2z2+L2(ω0)[12r2s2(ω0)]a0,
φ0=ω0cztan1[zL(ω0)]+ω0r22cR(ω0),
φ0=zc+zω0L(ω0)s2s2(ω0)+r22cR(ω0)z2L2(ω0)z,
Ax,y(P)(τ)a0Ax,y(τ)exp[ia0a0dAx,y(τ)/dτAx,y(τ)]×exp[i(φ0ω0φ0)],
dAx,y(τ)/dτAx,y(τ)=dlnAx,y(τ)dτ+idΦx,y(τ)dτ,
Ax,y(P)(τ)a0exp[a0a0dΦx,y(τ)dτ]Ax,y(τ)×exp{i[Φx,y(τ)+a0a0dlnAx,y(τ)dτ+φ0ω0φ0]},
Ax,y(P)(τ)a0exp[a0a0dΦx,y(τ)dτ]Ax,y(τ),
Φx,y(P)(τ)Φx,y(τ)+φ0ω0φ0+a0a0dlnAx,y(τ)dτ.
Φ(P)(τ)Φ(τ)a0a0ddt{ln[tanα(τ)]},
tanα(P)(τ)exp[a0a0dΦ(τ)dτ]tanα(τ).

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