Abstract

The carrier-envelope phase of a laser pulse has recently become an important quantity in extreme nonlinear optics. Because of the topological Gouy phase, it changes while the pulse propagates through the focus of a lens. This variation is measured by a simple solid-state-based approach. The experimental results are analyzed by comparison with simple analytical model calculations.

© 2005 Optical Society of America

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  1. C. R. Gouy, Acad. Sci. Paris C. R. 110, 1251 (1890).
  2. C. R. Gouy, Ann. Chim. Phys. 24, 145 (1891).
  3. A. E. Siegmann, Lasers (University Science, Mill Valley, Calif., 1986); note that the sign convention used for the Gouy phase in this textbook is different from the one used in this Letter.
  4. D. Subbarao, Opt. Lett. 20, 2162 (1995).
    [CrossRef]
  5. M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
    [CrossRef]
  6. S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).
    [CrossRef]
  7. M. Wegener, Extreme Nonlinear Optics (Springer-Verlag, Berlin, 2004).
  8. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
    [CrossRef]
  9. E. Budiarto, N.-W. Pu, S. Jeong, and J. Bokor, Opt. Lett. 23, 213 (1998).
    [CrossRef]
  10. F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
    [CrossRef]
  11. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 411 (1999).
    [CrossRef]
  12. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Opt. Lett. 27, 2127 (2002).
    [CrossRef]
  13. D. J. Jones, T. M. Fortier, and S. T. Cundiff, J. Opt. Soc. Am. B 21, 1098 (2004).
    [CrossRef]
  14. X. W. Sun and H. S. Kwok, J. Appl. Phys. 86, 408 (1999).
    [CrossRef]
  15. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Phys. Rev. Lett. 89, 127401 (2002).
    [CrossRef]
  16. S. T. Cundiff, W. H. Knox, E. P. Ippen, and H. A. Haus, Opt. Lett. 21, 662 (1996).
    [CrossRef] [PubMed]
  17. T. M. Fortier, D. J. Jones, and S. T. Cundiff, Opt. Lett. 28, 2198 (2003).
    [CrossRef] [PubMed]
  18. Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde, Appl. Phys. B: Lasers Opt. 63, 481 (1996).
    [CrossRef]
  19. S. Feng, H. G. Winful, and R. W. Hellwarth, Phys. Rev. E 59, 4630 (1999).
    [CrossRef]
  20. A. E. Kaplan, J. Opt. Soc. Am. B 15, 951 (1998).
    [CrossRef]

2004 (2)

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

D. J. Jones, T. M. Fortier, and S. T. Cundiff, J. Opt. Soc. Am. B 21, 1098 (2004).
[CrossRef]

2003 (1)

2002 (2)

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Phys. Rev. Lett. 89, 127401 (2002).
[CrossRef]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Opt. Lett. 27, 2127 (2002).
[CrossRef]

2001 (1)

1999 (4)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 411 (1999).
[CrossRef]

X. W. Sun and H. S. Kwok, J. Appl. Phys. 86, 408 (1999).
[CrossRef]

S. Feng, H. G. Winful, and R. W. Hellwarth, Phys. Rev. E 59, 4630 (1999).
[CrossRef]

1998 (2)

1996 (2)

Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde, Appl. Phys. B: Lasers Opt. 63, 481 (1996).
[CrossRef]

S. T. Cundiff, W. H. Knox, E. P. Ippen, and H. A. Haus, Opt. Lett. 21, 662 (1996).
[CrossRef] [PubMed]

1995 (1)

1984 (1)

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

1891 (1)

C. R. Gouy, Ann. Chim. Phys. 24, 145 (1891).

1890 (1)

C. R. Gouy, Acad. Sci. Paris C. R. 110, 1251 (1890).

Angelow, G.

Baltuška, A.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Berry, M. V.

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

Bokor, J.

Bor, Zs.

Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde, Appl. Phys. B: Lasers Opt. 63, 481 (1996).
[CrossRef]

Budiarto, E.

Chen, Y.

Cho, S. H.

Cundiff, S. T.

Feng, S.

S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

S. Feng, H. G. Winful, and R. W. Hellwarth, Phys. Rev. E 59, 4630 (1999).
[CrossRef]

Fortier, T. M.

Fujimoto, J. G.

Goulielmakis, E.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Gouy, C. R.

C. R. Gouy, Ann. Chim. Phys. 24, 145 (1891).

C. R. Gouy, Acad. Sci. Paris C. R. 110, 1251 (1890).

Haus, H. A.

Hellwarth, R. W.

S. Feng, H. G. Winful, and R. W. Hellwarth, Phys. Rev. E 59, 4630 (1999).
[CrossRef]

Horváth, Z. L.

Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde, Appl. Phys. B: Lasers Opt. 63, 481 (1996).
[CrossRef]

Ippen, E. P.

Jeong, S.

Jones, D. J.

Kaplan, A. E.

Kärtner, F. X.

Knox, W. H.

Krausz, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Kwok, H. S.

X. W. Sun and H. S. Kwok, J. Appl. Phys. 86, 408 (1999).
[CrossRef]

Lezius, M.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Lindner, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Morgner, U.

Mücke, O. D.

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Opt. Lett. 27, 2127 (2002).
[CrossRef]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Phys. Rev. Lett. 89, 127401 (2002).
[CrossRef]

Paulus, G. G.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Pu, N.-W.

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Scheuer, V.

Siegmann, A. E.

A. E. Siegmann, Lasers (University Science, Mill Valley, Calif., 1986); note that the sign convention used for the Gouy phase in this textbook is different from the one used in this Letter.

Subbarao, D.

Sun, X. W.

X. W. Sun and H. S. Kwok, J. Appl. Phys. 86, 408 (1999).
[CrossRef]

Tritschler, T.

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Phys. Rev. Lett. 89, 127401 (2002).
[CrossRef]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Opt. Lett. 27, 2127 (2002).
[CrossRef]

Tschudi, T.

Vinkó, J.

Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde, Appl. Phys. B: Lasers Opt. 63, 481 (1996).
[CrossRef]

von der Linde, D.

Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde, Appl. Phys. B: Lasers Opt. 63, 481 (1996).
[CrossRef]

Walther, H.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Wegener, M.

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Opt. Lett. 27, 2127 (2002).
[CrossRef]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Phys. Rev. Lett. 89, 127401 (2002).
[CrossRef]

M. Wegener, Extreme Nonlinear Optics (Springer-Verlag, Berlin, 2004).

Whitaker, J. F.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Winful, H. G.

S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

S. Feng, H. G. Winful, and R. W. Hellwarth, Phys. Rev. E 59, 4630 (1999).
[CrossRef]

Acad. Sci. Paris C. R. (1)

C. R. Gouy, Acad. Sci. Paris C. R. 110, 1251 (1890).

Ann. Chim. Phys. (1)

C. R. Gouy, Ann. Chim. Phys. 24, 145 (1891).

Appl. Phys. B: Lasers Opt. (1)

Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde, Appl. Phys. B: Lasers Opt. 63, 481 (1996).
[CrossRef]

J. Appl. Phys. (1)

X. W. Sun and H. S. Kwok, J. Appl. Phys. 86, 408 (1999).
[CrossRef]

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

Opt. Lett. (7)

Phys. Rev. E (1)

S. Feng, H. G. Winful, and R. W. Hellwarth, Phys. Rev. E 59, 4630 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Phys. Rev. Lett. 89, 127401 (2002).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

Other (2)

A. E. Siegmann, Lasers (University Science, Mill Valley, Calif., 1986); note that the sign convention used for the Gouy phase in this textbook is different from the one used in this Letter.

M. Wegener, Extreme Nonlinear Optics (Springer-Verlag, Berlin, 2004).

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Figures (2)

Fig. 1
Fig. 1

Scheme of the experimental setup. The output of a 5 - fs mode-locked laser oscillator is split into two balanced arms. In each arm the light is focused with a Cassegrain lens (L) onto a ZnO crystal, followed by a set of optical filters (F), and a detector (D). One arm is used to stabilize the carrier-envelope frequency of the laser and additionally serves as the reference (REF) input for a lock-in amplifier. The other arm delivers the signal (SIG) input for the lock-in amplifier. The amplitude and phase of the output of the lock-in amplifier are recorded while scanning the ZnO crystal of the signal arm through the focus.

Fig. 2
Fig. 2

(a) Gaussian beam width w ( z , ω 0 ) versus propagation coordinate z in units of the Rayleigh length z R ( ω 0 ) = 2.8 μ m at laser carrier frequency ω 0 . The open circles are the result of a knife-edge measurement. The corresponding raw data are shown by the gray-scale plot in the background. Here 50% of the maximum is set to black and 0% and 100% are set to white. (b) Lock-in phase (open circles), which is identical to the shift of carrier-envelope phase ϕ ( z ) . The solid curves correspond to the theoretical expectation for cases (i)-(iii). Case (ii) is identical to the Gouy phase. Amplitude A of the lock-in signal is shown by the gray area at the bottom.

Equations (7)

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w 2 ( z , ω ) = w 0 2 ( ω ) { 1 + [ z z R ( ω ) ] 2 } .
z R ( ω ) = w 0 2 ( ω ) 2 c 0 ω
φ G ( z , ω ) = arctan [ z z R ( ω ) ] .
E ( t ) = E ̃ ( t ) cos [ ω 0 t + ϕ ( z ) ] ,
ϕ ( z ) = ω 0 c 0 2 z [ v phase ( z ) v group ( z ) ] d z = φ G ( z , ω 0 ) ω 0 φ G ω ( z , ω 0 ) .
ϕ ( z ) = arctan ( Z ) + F Z 1 + Z 2 ,
z R ( ω ) ω F .

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