Abstract

Unlike in established methods, the phase of an optical pulse is calculated from a single time transient in combination with a single spectrum of its polarization state: For dynamically changing elliptical polarizations the vectorial nature of the electromagnetic radiation results in two independent constraints that uniquely determine the chirp and the spectral phase. The practical feasibility of the method is demonstrated, a phase of a four-wave-mixing signal from a semiconductor single quantum well with a pulse energy of only 0.5  fJ is determined, and the accuracy of the setup is analyzed by a numerical simulation.

© 1999 Optical Society of America

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References

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  1. X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, Phys. Rev. B 56, 9738 (1997).
    [Crossref]
  2. O. Buccafusca, X. Chen, W. J. Walecki, and A. L. Smirl, J. Opt. Soc. Am. B 15, 1218 (1998).
    [Crossref]
  3. D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
    [Crossref]
  4. K. W. DeLong, R. Trebino, and W. E. White, J. Opt. Soc. Am. B 12, 2463 (1995).
    [Crossref]
  5. D. N. Fittinghoff, J. Bowie, J. Sweetser, R. Jennings, M. Krumbügel, K. DeLong, and R. Trebino, Opt. Lett. 21, 884 (1996).
    [Crossref] [PubMed]
  6. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
    [Crossref]
  7. S. Linden, H. Giessen, and J. Kuhl, Phys. Status Solidi B 206, 119 (1998).
    [Crossref]
  8. W. J. Walecki, D. N. Fittinghoff, and A. L. Smirl, Opt. Lett. 22, 81 (1997).
    [Crossref] [PubMed]
  9. E. J. Akutowicz, Trans. Am. Math. Soc. 83, 179 (1956).
  10. A. E. Paul, J. A. Bolger, A. L. Smirl, and J. G. Pellegrino, J. Opt. Soc. Am. B 13, 1016 (1996).
    [Crossref]
  11. G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
    [Crossref]
  12. B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

1998 (3)

O. Buccafusca, X. Chen, W. J. Walecki, and A. L. Smirl, J. Opt. Soc. Am. B 15, 1218 (1998).
[Crossref]

S. Linden, H. Giessen, and J. Kuhl, Phys. Status Solidi B 206, 119 (1998).
[Crossref]

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

1997 (3)

W. J. Walecki, D. N. Fittinghoff, and A. L. Smirl, Opt. Lett. 22, 81 (1997).
[Crossref] [PubMed]

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, Phys. Rev. B 56, 9738 (1997).
[Crossref]

1996 (2)

1995 (1)

1994 (1)

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[Crossref]

1956 (1)

E. J. Akutowicz, Trans. Am. Math. Soc. 83, 179 (1956).

Akutowicz, E. J.

E. J. Akutowicz, Trans. Am. Math. Soc. 83, 179 (1956).

Axt, V. M.

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Bartels, G.

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Bigot, J.-Y.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[Crossref]

Bolger, J. A.

Bowie, J.

Buccafusca, O.

O. Buccafusca, X. Chen, W. J. Walecki, and A. L. Smirl, J. Opt. Soc. Am. B 15, 1218 (1998).
[Crossref]

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, Phys. Rev. B 56, 9738 (1997).
[Crossref]

Chemla, D. S.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[Crossref]

Chen, X.

O. Buccafusca, X. Chen, W. J. Walecki, and A. L. Smirl, J. Opt. Soc. Am. B 15, 1218 (1998).
[Crossref]

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, Phys. Rev. B 56, 9738 (1997).
[Crossref]

DeLong, K.

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

K. W. DeLong, R. Trebino, and W. E. White, J. Opt. Soc. Am. B 12, 2463 (1995).
[Crossref]

Faschinger, W.

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Fittinghoff, D. N.

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, Phys. Rev. B 56, 9738 (1997).
[Crossref]

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

W. J. Walecki, D. N. Fittinghoff, and A. L. Smirl, Opt. Lett. 22, 81 (1997).
[Crossref] [PubMed]

D. N. Fittinghoff, J. Bowie, J. Sweetser, R. Jennings, M. Krumbügel, K. DeLong, and R. Trebino, Opt. Lett. 21, 884 (1996).
[Crossref] [PubMed]

Giessen, H.

S. Linden, H. Giessen, and J. Kuhl, Phys. Status Solidi B 206, 119 (1998).
[Crossref]

Gutowski, J.

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Haase, B.

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Jennings, R.

Krumbügel, M.

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Kuhl, J.

S. Linden, H. Giessen, and J. Kuhl, Phys. Status Solidi B 206, 119 (1998).
[Crossref]

Linden, S.

S. Linden, H. Giessen, and J. Kuhl, Phys. Status Solidi B 206, 119 (1998).
[Crossref]

Mycek, M.-A.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[Crossref]

Neukirch, U.

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Nürnberger, J.

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Paul, A. E.

Pellegrino, J. G.

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Schäfer, W.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[Crossref]

Smirl, A. L.

Stahl, A.

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

Sweetser, J.

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Trebino, R.

Walecki, W. J.

Weiss, S.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[Crossref]

White, W. E.

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

Opt. Lett. (2)

Phys. Rev. B (2)

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, Phys. Rev. B 56, 9738 (1997).
[Crossref]

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, and W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[Crossref]

Phys. Rev. Lett. (1)

G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, Phys. Rev. Lett. 81, 5880 (1998).
[Crossref]

Phys. Status Solidi B (1)

S. Linden, H. Giessen, and J. Kuhl, Phys. Status Solidi B 206, 119 (1998).
[Crossref]

Rev. Sci. Instrum. (1)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, Rev. Sci. Instrum. 68, 3277 (1997).
[Crossref]

Trans. Am. Math. Soc. (1)

E. J. Akutowicz, Trans. Am. Math. Soc. 83, 179 (1956).

Other (1)

B. Haase, U. Neukirch, J. Gutowski, G. Bartels, A. Stahl, V. M. Axt, J. Nürnberger, and W. Faschinger, “Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave mixing experiments,” Phys. Rev. B Rapid Commun. (to be published).

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Figures (3)

Fig. 1
Fig. 1

Iteration scheme ϕxnϕxn+1 used to determine the phase function.

Fig. 2
Fig. 2

Measured amplitudes (solid curves) and retrieved phases (dashed curves) of the self-diffracted electric fields Ex and Ey in the quantum-well plane in the time and the frequency domains. The dotted curves are calculated amplitudes, determined by a FT (inverse FT) using measured amplitudes and retrieved phases. An overall phase constant has been added such that the spectral phase ϕ¯x vanishes at the center of the heavy-hole exciton line ω0=2.825 eV. The depicted phase functions ϕ^x/yt represent only the slowly varying component of the phase ϕx/yt=ϕ^x/yt-ω0t.

Fig. 3
Fig. 3

Simulation of the phase-retrieval method with numerical data, assuming unlimited (1–fs) as well as limited (100- and 200-fs) time resolution of the upconversion setup. The energy scale is relative to the heavy-role exciton line.

Equations (2)

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expiωtAxtexpiϕxtdt=A¯xωexpiϕ¯xω.
expiωtAytexpiϕxt+iϕxytdt=A¯yωexpiϕ¯xω+iϕ¯xyω,

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