Abstract

We unveil the origin of the recently revealed polarization-state changes of polarization-shaped few-cycle pulses induced by free-space beam propagation. Simple rules are formulated to show how the orientation and ellipticity of the instantaneous polarization ellipse of the source and propagated pulses relate to each other. We demonstrate our findings with examples that clearly display the relationships found and highlight their relevance. We show, for example, that pulses often used in high-harmonic generation or attosecond pulse production rotate as a whole during free-space beam propagation or upon focusing. A pulse that may reverse its ellipticity from right-handed to left-handed during propagation is also introduced. It is shown that these effects are independent of the beam size and/or focal length. We also present how these instantaneous polarization-state changes could be noticed in classical measurements of light polarization using polarizers, phase retarders, and time-integrating detectors.

© 2014 Optical Society of America

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References

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  1. S. Cundiff and A. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4, 760–766 (2010).
    [Crossref]
  2. M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
    [Crossref]
  3. T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
    [Crossref]
  4. L. Polachek, D. Oron, and Y. Silberberg, “Full control of the spectral polarization of ultrashort pulses,” Opt. Lett. 31, 631–633 (2006).
    [Crossref]
  5. I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
    [Crossref]
  6. M. A. Porras, “Propagation-induced changes in the instantaneous polarization state, phase, and carrier–envelope phase of few-cycle pulsed beams,” J. Opt. Soc. Am. B 30, 1652–1659 (2013).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  14. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
    [Crossref]
  15. D. Oron, Y. Silberberg, N. Dudovich, and D. M. Villeneuve, “Efficient polarization gating of high-order harmonic generation by polarization-shaped ultrashort pulses,” Phys. Rev. A 72, 063816 (2005).
    [Crossref]
  16. 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).
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    [Crossref]
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    [Crossref]

2013 (1)

2012 (1)

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

2010 (1)

S. Cundiff and A. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4, 760–766 (2010).
[Crossref]

2009 (1)

2006 (2)

L. Polachek, D. Oron, and Y. Silberberg, “Full control of the spectral polarization of ultrashort pulses,” Opt. Lett. 31, 631–633 (2006).
[Crossref]

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

2005 (3)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[Crossref]

D. Oron, Y. Silberberg, N. Dudovich, and D. M. Villeneuve, “Efficient polarization gating of high-order harmonic generation by polarization-shaped ultrashort pulses,” Phys. Rev. A 72, 063816 (2005).
[Crossref]

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[Crossref]

2004 (1)

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

2003 (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

2002 (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 (1)

1989 (1)

1988 (1)

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref]

Bartels, R. A.

Baumert, T.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Benedetti, E.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

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.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Caumes, J.-P.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Constant, E.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Cundiff, S.

S. Cundiff and A. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4, 760–766 (2010).
[Crossref]

Dacic, Z.

Dudovich, N.

D. Oron, Y. Silberberg, N. Dudovich, and D. M. Villeneuve, “Efficient polarization gating of high-order harmonic generation by polarization-shaped ultrashort pulses,” Phys. Rev. A 72, 063816 (2005).
[Crossref]

Elouga, L.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Gagnon, J.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Gerber, G.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Goulielmakis, E.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Graefe, O.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Grguraš, I.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Hassan, M. T.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Helseth, L. E.

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[Crossref]

Horn, C.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

James, D. F. V.

Kakehata, M.

Kobayashi, Y.

Korotkova, O.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[Crossref]

Krampert, G.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Krausz, F.

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

Liese, D.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Luu, T. T.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Masihzadeh, O.

Mével, E.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Moulet, A.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Nisoli, M.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Oron, D.

L. Polachek, D. Oron, and Y. Silberberg, “Full control of the spectral polarization of ultrashort pulses,” Opt. Lett. 31, 631–633 (2006).
[Crossref]

D. Oron, Y. Silberberg, N. Dudovich, and D. M. Villeneuve, “Efficient polarization gating of high-order harmonic generation by polarization-shaped ultrashort pulses,” Phys. Rev. A 72, 063816 (2005).
[Crossref]

Pervak, V.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Pfeifer, T.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Polachek, L.

Poletto, L.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Porras, M. A.

Sansone, G.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Schlup, P.

Selle, R.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Sherman, G. C.

Silberberg, Y.

L. Polachek, D. Oron, and Y. Silberberg, “Full control of the spectral polarization of ultrashort pulses,” Opt. Lett. 31, 631–633 (2006).
[Crossref]

D. Oron, Y. Silberberg, N. Dudovich, and D. M. Villeneuve, “Efficient polarization gating of high-order harmonic generation by polarization-shaped ultrashort pulses,” Phys. Rev. A 72, 063816 (2005).
[Crossref]

Sola, I. J.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Stagira, S.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Strelkov, V.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Takada, H.

Torizuka, K.

Trebino, R.

Villeneuve, D. M.

D. Oron, Y. Silberberg, N. Dudovich, and D. M. Villeneuve, “Efficient polarization gating of high-order harmonic generation by polarization-shaped ultrashort pulses,” Phys. Rev. A 72, 063816 (2005).
[Crossref]

Villoresi, P.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Vozzi, C.

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Weiner, A.

S. Cundiff and A. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4, 760–766 (2010).
[Crossref]

Wirth, A.

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Wolf, E.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

Z. Dačić and E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[Crossref]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

Wollenhaupt, M.

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

Xu, L.

J. Opt. Soc. Am. A (3)

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

Nat. Photonics (1)

S. Cundiff and A. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4, 760–766 (2010).
[Crossref]

Nat. Phys. (1)

I. J. Sola, E. Mével, L. Elouga, E. Constant, V. Strelkov, L. Poletto, P. Villoresi, E. Benedetti, J.-P. Caumes, S. Stagira, C. Vozzi, G. Sansone, and M. Nisoli, “Controlling attosecond electron dynamics by phase-stabilized polarization gating,” Nat. Phys. 2, 319–322 (2006).
[Crossref]

Opt. Commun. (1)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[Crossref]

Opt. Lett. (2)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

Phys. Rev. A (1)

D. Oron, Y. Silberberg, N. Dudovich, and D. M. Villeneuve, “Efficient polarization gating of high-order harmonic generation by polarization-shaped ultrashort pulses,” Phys. Rev. A 72, 063816 (2005).
[Crossref]

Phys. Rev. E (2)

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[Crossref]

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

Phys. Rev. Lett. (3)

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

T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese, and T. Baumert, “Quantum control by ultrafast polarization shaping,” Phys. Rev. Lett. 92, 208301 (2004).
[Crossref]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref]

Rev. Sci. Instrum. (1)

M. T. Hassan, A. Wirth, I. Grguraš, A. Moulet, T. T. Luu, J. Gagnon, V. Pervak, and E. Goulielmakis, “Invited article: Attosecond photonics: synthesis and control of light transients,” Rev. Sci. Instrum. 83, 111301 (2012).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

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Figures (5)

Fig. 1.
Fig. 1.

Time evolution of the electric field of (a) PG pulse of carrier angular frequency ω0=2.355fs1 (800 nm carrier wavelength), Gaussian amplitude A(t)=exp(t2/ΔT2) of duration ΔT=3.4fs (the intensity FWHM is 1.5 carrier cycles), and delay Δt=4.67fs [see Eq. (19)]. (b) PG pulse after propagation up to the far field, or to a focus, when the pulse is emitted by the source with U(r)=exp(r2/s2). The straight dashed lines indicate the simulated direction of the polarizer, resulting in a maximum transmitted fluence for the source and propagated PG pulse. For better comparison, a global phase π/2 has been added to the propagated pulse to eliminate the effect of the Gouy phase shift π/2 (it does not alter the state of polarization) on propagation from a finite source to the far field.

Fig. 2.
Fig. 2.

(a) Time variation of the orientation (dashed black curve ψ) and ellipticity (solid gray curve χ) of the momentary polarization ellipse of the PG pulse in Fig. 1(a), and the same properties for the propagated PG pulse in Fig. 1(b) (solid black curve ψ(p) and dashed–dotted black curve χ(p)). The dotted black curve stands for the predicted polarization rotation by Eq. (13). (b) Simulated fluence after a polarizer for the PG pulse in Fig. 1(a) (dashed curve) and for the propagated pulse in Fig. 1(b) (solid curve) as a function of the transmission direction of the polarizer. (c),(d) Simulated rotation of the direction θmax of the polarizer yielding the maximum fluence behind it as a function of propagation distance z. For the collimated pulse in (c) the distance z is in units of the diffraction length ω0s2/2c, and for the focused pulse in (d) the distance z is measured from the focal point and is in units of the focal depth 2cf2/ω0s2.

Fig. 3.
Fig. 3.

(a) Rotation of the observable polarization direction of the PG pulse upon propagation up to the far field or a focus for different pulse durations. The time delay Δt is scaled proportionally with the pulse duration ΔT to retain a similar pulse shape (Δt/ΔT=2ln2); see Eq. (19). All other pulse properties are the same as in Fig. 1. (b) Ellipticity of the rotating pulse at the far field or at a focus for different pulse durations. During scaling pulse duration ΔT the product ωdΔT is kept constant to retain similar pulse shape (ωdΔT=π/2ln2); see Eq. (20). Note that this means that for small values of ΔT the ωdω0 assumption is not true. Other pulse properties are detailed later in Fig. 4.

Fig. 4.
Fig. 4.

Temporal evolution of the electric field of (a) rotating pulse with ω0=2.355fs1, ωd=0.235fs1, Gaussian amplitude A(t)=exp(t2/ΔT2) of duration ΔT=11.3fs, al=0.95, and ar=1.0 (see Eq. (20)), and (b) of the rotating pulse after propagation up to the far field when the pulse is emitted by the source with U(r)=exp(r2/s2).

Fig. 5.
Fig. 5.

(a) Temporal variation of the orientation (solid gray curve ψ) and the ellipticity (dashed black curve χ) of the instantaneous ellipse of the rotating pulse in Fig. 4(a), and the same properties for the propagated rotating pulse in Fig. 4(b) (dashed–dotted black curve ψ(p) and solid black curve χ(p)). The dotted black curve stands for the ellipticity prediction by Eq. (17). (b) Change of the ellipticity with propagation distance as it would be obtained by standard polarimetric techniques. The propagation distance z is in units of the diffraction length ω0s2/2c.

Equations (21)

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E(t)=El(t)ul+Er(t)ur,
Al(t)=12[Ax(t)iAy(t)],
Ar(t)=12[Ax(t)+iAy(t)].
ψ(t)=[Φr(t)Φl(t)]2
tanχ(t)=Ar(t)Al(t)Ar(t)+Al(t),
U(r,z,ω)=iL(ω)q(z,ω)exp[iωr22cq(z,ω)]exp[iωcz],
E(p)(t)=1π0E^(ω)U(r,z,ω)exp(iωt)
Al,r(p)(t)=1π0E^l,r(ω)U(r,z,ω)exp[i(ωω0)t]
Al,r(p)(τ)a0Al,r(τ)exp[ia0a0dlnAl,r(τ)dτ]exp[i(φ0ω0φ0)]
Al,r(p)(τ)a0exp[a0a0dΦl,r(τ)dτ]Al,r(τ)
Φl,r(p)(τ)Φl,r(τ)+φ0ω0φ0+a0a0dlnAl,r(τ)dτ,
ψ(p)(τ)ψ(τ)+12a0a0dln[Ar(τ)/Al(τ)]dτ.
ψ(p)(τ)ψ(τ)+a0a011tan2χ(τ)dtanχ(τ)dτ,
ψ(p)(τ)ψ(τ)1ω011tan2χ(τ)dtanχ(τ)dτ
Ar(p)(τ)±Al(p)(τ)a0exp[a0a0dΦM(τ)dτ]×{[Ar(τ)±Al(τ)]a0a0dψ(τ)dτ[Ar(τ)Al(τ)]},
tanχ(p)(τ)tanχ(τ)a0a0dψ(τ)dτ1a0a0dψ(τ)dτtanχ(τ),
tanχ(p)(τ)tanχ(τ)a0a0[1tan2χ(τ)]dψ(τ)dτ.
tanχ(p)(τ)tanχ(τ)1ω0[1tan2χ(τ)]dψ(τ)dτ,
El(t)=A(t+Δt/2)exp[iω0(t+Δt/2)],Er(t)=A(tΔt/2)exp[iω0(tΔt/2)]
El(t)=alA(t)exp[i(ω0+ωd)t]Er(t)=arA(t)exp(iω0t)
F(θ,ϵ)=dt{Re[E(p)(t,θ,ϵ)]}2=12dt|E(p)(t,θ,ϵ)|2

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