Abstract

We consider in detail the technique of tomographic ultrafast retrieval of transverse light E fields (TURTLE) for measuring the evolution of an arbitrary, potentially complex, and ultrashort laser pulse’s intensity-and-phase and polarization-state evolutions in time. TURTLE involves making three ultrashort-pulse measurements using established single-polarization pulse-measurement techniques. Two of the measurements are of the pulse’s orthogonal linear polarizations (e.g., horizontal and vertical) and the third occurs for an arbitrary additional polarization angle (e.g., 45°). If the field projections are measured using second-harmonic-generation frequency-resolved optical gating, we demonstrate that a simple optimization can accurately and reliably retrieve the time-dependent polarization state, even for very complex polarization-shaped pulses.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (2)

2007 (1)

2006 (1)

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, “Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer,” Appl. Opt. 45, 8345-8359 (2006).
[CrossRef]

2004 (2)

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] [PubMed]

T. Suzuki, S. Minemoto, T. Kanai, and H. Sakai, “Optimal control of multiphoton ionization process in aligned I2 molecules with time-dependent polarization pulses,” Phys. Rev. Lett. 92, 133005 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (1)

E. Zeek, A. P. Shreenath, P. O'Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal in frequency-resolved optical gating,” Appl. Phys. B. 74, 265-271 (2002).
[CrossRef]

2001 (2)

2000 (1)

M. Kakehata, R. Ueda, H. Takada, K. Torizuka, and M. Obara, “Combination of high-intensity femtosecond laser pulses for generation of time-dependent polarization pulses and ionization of atomic gas,” Appl. Phys. B 70, s207-s213 (2000).
[CrossRef]

1999 (1)

1998 (2)

M. M. Wefers, H. Kawashima, and K. A. Nelson, “Optical control over two-dimensional lattice vibrational trajectories in crystalline quartz,” J. Chem. Phys. 108, 10248-10255 (1998).
[CrossRef]

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

1997 (2)

E. Constant, V. D. Taranukhin, A. Stolow, and P. B. Corkum, “Methods for the measurement of the duration of high-harmonic pulses,” Phys. Rev. A 56, 3870-3878 (1997).
[CrossRef]

W. J. Walecki, D. N. Fittingho, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett. 22, 81-83 (1997).
[CrossRef] [PubMed]

1996 (2)

A. E. Paul, J. A. Bolger, A. L. Smirl, and J. G. Pellegrino, “Time-resolved measurements of the polarization state of four-wave mixing signals from GaAs multiple quantum wells,” J. Opt. Soc. Am. B 13, 1016-1025 (1996).
[CrossRef]

J. A. Bolger, A. E. Paul, and A. L. Smirl, “Ultrafast ellipsometry of coherent processes and exciton-exciton interactions in quantum wells at negative delays,” Phys. Rev. B 54, 11666-11671 (1996).
[CrossRef]

1995 (1)

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B. 12, 1955-1967 (1995).
[CrossRef]

1994 (1)

1993 (1)

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581-1589 (1993).
[CrossRef] [PubMed]

1980 (1)

Arnaud, J. A.

Bartel, R. A.

Bartels, R. A.

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Complete polarization state control of ultrafast laser pulses with a single linear spatial light modulator,” Opt. Express 15, 18025-18032 (2007).
[CrossRef] [PubMed]

D. Kupka, P. Schlup, and R. A. Bartels, “Simplified polarization pulse shaper using a birefringent prism,” Rev. Sci. Instrum. (to be published).
[PubMed]

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] [PubMed]

Bolger, J. A.

J. A. Bolger, A. E. Paul, and A. L. Smirl, “Ultrafast ellipsometry of coherent processes and exciton-exciton interactions in quantum wells at negative delays,” Phys. Rev. B 54, 11666-11671 (1996).
[CrossRef]

A. E. Paul, J. A. Bolger, A. L. Smirl, and J. G. Pellegrino, “Time-resolved measurements of the polarization state of four-wave mixing signals from GaAs multiple quantum wells,” J. Opt. Soc. Am. B 13, 1016-1025 (1996).
[CrossRef]

Brixner, G. T.

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] [PubMed]

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]

Colombeau, B.

Constant, E.

E. Constant, V. D. Taranukhin, A. Stolow, and P. B. Corkum, “Methods for the measurement of the duration of high-harmonic pulses,” Phys. Rev. A 56, 3870-3878 (1997).
[CrossRef]

Corkum, P. B.

E. Constant, V. D. Taranukhin, A. Stolow, and P. B. Corkum, “Methods for the measurement of the duration of high-harmonic pulses,” Phys. Rev. A 56, 3870-3878 (1997).
[CrossRef]

Dahleh, M.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581-1589 (1993).
[CrossRef] [PubMed]

Damrauer, N. H.

Delong, K. W.

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B. 12, 1955-1967 (1995).
[CrossRef]

K. W. Delong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206-2215 (1994).
[CrossRef]

Fittingho, D. N.

Fittinghoff, D. N.

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B. 12, 1955-1967 (1995).
[CrossRef]

Froehly, C.

Fuente, R. D. L.

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] [PubMed]

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]

G. T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. 26, 557-559 (2001).
[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] [PubMed]

Hass, B.

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] [PubMed]

Hunter, J.

José Ferreiro, J.

Kakehata, M.

M. Kakehata, R. Ueda, H. Takada, K. Torizuka, and M. Obara, “Combination of high-intensity femtosecond laser pulses for generation of time-dependent polarization pulses and ionization of atomic gas,” Appl. Phys. B 70, s207-s213 (2000).
[CrossRef]

Kanai, T.

T. Suzuki, S. Minemoto, T. Kanai, and H. Sakai, “Optimal control of multiphoton ionization process in aligned I2 molecules with time-dependent polarization pulses,” Phys. Rev. Lett. 92, 133005 (2004).
[CrossRef] [PubMed]

Kawashima, H.

M. M. Wefers, H. Kawashima, and K. A. Nelson, “Optical control over two-dimensional lattice vibrational trajectories in crystalline quartz,” J. Chem. Phys. 108, 10248-10255 (1998).
[CrossRef]

Keusters, D.

Kimmel, M.

E. Zeek, A. P. Shreenath, P. O'Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal in frequency-resolved optical gating,” Appl. Phys. B. 74, 265-271 (2002).
[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] [PubMed]

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]

Kupka, D.

D. Kupka, P. Schlup, and R. A. Bartels, “Simplified polarization pulse shaper using a birefringent prism,” Rev. Sci. Instrum. (to be published).
[PubMed]

Ladera, C. L.

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B. 12, 1955-1967 (1995).
[CrossRef]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Lago, E.

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] [PubMed]

Lindinger, A.

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, “Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer,” Appl. Opt. 45, 8345-8359 (2006).
[CrossRef]

Masihzadeh, O.

Minemoto, S.

T. Suzuki, S. Minemoto, T. Kanai, and H. Sakai, “Optimal control of multiphoton ionization process in aligned I2 molecules with time-dependent polarization pulses,” Phys. Rev. Lett. 92, 133005 (2004).
[CrossRef] [PubMed]

Nelson, K. A.

M. M. Wefers, H. Kawashima, and K. A. Nelson, “Optical control over two-dimensional lattice vibrational trajectories in crystalline quartz,” J. Chem. Phys. 108, 10248-10255 (1998).
[CrossRef]

Niklaus, P.

Obara, M.

M. Kakehata, R. Ueda, H. Takada, K. Torizuka, and M. Obara, “Combination of high-intensity femtosecond laser pulses for generation of time-dependent polarization pulses and ionization of atomic gas,” Appl. Phys. B 70, s207-s213 (2000).
[CrossRef]

O'Shea, P.

D. Keusters, H.-S. Tan, P. O'Shea, E. Zeek, R. Trebino, and W. S. Warren, “Relative-phase ambiguities in measurements of ultrashort pulses with well-separated multiple frequency components,” J. Opt. Soc. Am. B 20, 2226-2237 (2003).
[CrossRef]

E. Zeek, A. P. Shreenath, P. O'Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal in frequency-resolved optical gating,” Appl. Phys. B. 74, 265-271 (2002).
[CrossRef]

Paul, A. E.

A. E. Paul, J. A. Bolger, A. L. Smirl, and J. G. Pellegrino, “Time-resolved measurements of the polarization state of four-wave mixing signals from GaAs multiple quantum wells,” J. Opt. Soc. Am. B 13, 1016-1025 (1996).
[CrossRef]

J. A. Bolger, A. E. Paul, and A. L. Smirl, “Ultrafast ellipsometry of coherent processes and exciton-exciton interactions in quantum wells at negative delays,” Phys. Rev. B 54, 11666-11671 (1996).
[CrossRef]

Pellegrino, J. G.

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] [PubMed]

Piasecki, J.

Plewicki, M.

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, “Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer,” Appl. Opt. 45, 8345-8359 (2006).
[CrossRef]

Rabitz, H.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581-1589 (1993).
[CrossRef] [PubMed]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Sakai, H.

T. Suzuki, S. Minemoto, T. Kanai, and H. Sakai, “Optimal control of multiphoton ionization process in aligned I2 molecules with time-dependent polarization pulses,” Phys. Rev. Lett. 92, 133005 (2004).
[CrossRef] [PubMed]

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] [PubMed]

Shreenath, A. P.

E. Zeek, A. P. Shreenath, P. O'Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal in frequency-resolved optical gating,” Appl. Phys. B. 74, 265-271 (2002).
[CrossRef]

Smirl, A. L.

Stolow, A.

E. Constant, V. D. Taranukhin, A. Stolow, and P. B. Corkum, “Methods for the measurement of the duration of high-harmonic pulses,” Phys. Rev. A 56, 3870-3878 (1997).
[CrossRef]

Suzuki, T.

T. Suzuki, S. Minemoto, T. Kanai, and H. Sakai, “Optimal control of multiphoton ionization process in aligned I2 molecules with time-dependent polarization pulses,” Phys. Rev. Lett. 92, 133005 (2004).
[CrossRef] [PubMed]

Takada, H.

M. Kakehata, R. Ueda, H. Takada, K. Torizuka, and M. Obara, “Combination of high-intensity femtosecond laser pulses for generation of time-dependent polarization pulses and ionization of atomic gas,” Appl. Phys. B 70, s207-s213 (2000).
[CrossRef]

Tan, H. -S.

Taranukhin, V. D.

E. Constant, V. D. Taranukhin, A. Stolow, and P. B. Corkum, “Methods for the measurement of the duration of high-harmonic pulses,” Phys. Rev. A 56, 3870-3878 (1997).
[CrossRef]

Torizuka, K.

M. Kakehata, R. Ueda, H. Takada, K. Torizuka, and M. Obara, “Combination of high-intensity femtosecond laser pulses for generation of time-dependent polarization pulses and ionization of atomic gas,” Appl. Phys. B 70, s207-s213 (2000).
[CrossRef]

Trebino, R.

P. Schlup, O. Masihzadeh, L. Xu, R. Trebino, and R. A. Bartel, “Tomographic retrieval of the polarization state of an ultrafast laser pulse,” Opt. Lett. 33, 267-269 (2008).
[CrossRef] [PubMed]

L. Xu, E. Zeek, and R. Trebino, “Simulations of frequency-resolved optical gating for measuring very complex pulses,” J. Opt. Soc. Am. B 25, A70-A80 (2008).
[CrossRef]

D. Keusters, H.-S. Tan, P. O'Shea, E. Zeek, R. Trebino, and W. S. Warren, “Relative-phase ambiguities in measurements of ultrashort pulses with well-separated multiple frequency components,” J. Opt. Soc. Am. B 20, 2226-2237 (2003).
[CrossRef]

E. Zeek, A. P. Shreenath, P. O'Shea, M. Kimmel, and R. Trebino, “Simultaneous automatic calibration and direction-of-time removal in frequency-resolved optical gating,” Appl. Phys. B. 74, 265-271 (2002).
[CrossRef]

W. J. Walecki, D. N. Fittingho, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett. 22, 81-83 (1997).
[CrossRef] [PubMed]

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B. 12, 1955-1967 (1995).
[CrossRef]

K. W. Delong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206-2215 (1994).
[CrossRef]

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, 2002).
[CrossRef]

Ueda, R.

M. Kakehata, R. Ueda, H. Takada, K. Torizuka, and M. Obara, “Combination of high-intensity femtosecond laser pulses for generation of time-dependent polarization pulses and ionization of atomic gas,” Appl. Phys. B 70, s207-s213 (2000).
[CrossRef]

Vampouille, M.

Walecki, W. J.

Warren, W. S.

Weber, S. M.

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, “Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer,” Appl. Opt. 45, 8345-8359 (2006).
[CrossRef]

Wefers, M. M.

M. M. Wefers, H. Kawashima, and K. A. Nelson, “Optical control over two-dimensional lattice vibrational trajectories in crystalline quartz,” J. Chem. Phys. 108, 10248-10255 (1998).
[CrossRef]

Weise, F.

M. Plewicki, F. Weise, S. M. Weber, and A. Lindinger, “Phase, amplitude, and polarization shaping with a pulse shaper in a Mach-Zehnder interferometer,” Appl. Opt. 45, 8345-8359 (2006).
[CrossRef]

White, W. E.

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] [PubMed]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Xu, L.

Zeek, E.

Appl. Opt. (2)

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[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic visualization of the TURTLE principle. The time-evolving electric field vector E ( t ) (not shown) is characterized by measuring linear projections E ̃ x ( Ω ) , E ̃ y ( Ω ) , and E η ( Ω ) in the frequency domain using an existing ultrashort-pulse characterization technique. The algorithm establishes the relative amplitude r, delay τ, and phase θ between the projections to retrieve the full vector field.

Fig. 2
Fig. 2

TURTLE retrieval steps for a vector field consisting of two transform-limited Gaussian components separated by τ = 170   fs . (a),(b),(c) Simulations of measured SHG FROG traces for E ̃ x ( Ω ) , E ̃ y ( Ω ) , and E ̃ η ( Ω ) , respectively, with η = 45 ° . (d),(e) Pulse fields E x ( t ) , E y ( t ) obtained using the standard reconstruction algorithm (dots), compared with the generating fields (solid curve). (f) The error surface; the two minima indicate the π   rad phase ambiguity arising from the SHG FROG trace of two pulses well separated in time. (g) Sketch of the full vector field E ( t ) .

Fig. 3
Fig. 3

(a) Simulation of measured SHG FROG traces; in this case all three projections yield the same trace. (b) The error surface; the two minima indicate an ambiguity in the chirality of the vector field E ( t ) , which is shown in (c).

Fig. 4
Fig. 4

Establishing the chirality of a circularly polarized field by adding a known chirp; cf. Fig. 3. (a) Simulation of measured SHG FROG trace for E ̃ x ( Ω ) ; an identical trace is recorded for E ̃ y ( Ω ) . (b),(c) Pulse fields E x ( t ) , E y ( t ) obtained using the standard reconstruction algorithm (dots), compared with the generating fields (solid curve). (d) SHG FROG trace for the η-projected component. (e) The error surface that shows only a single minimum at θ = + π / 2 . (f) Sketch of the full vector field E ( t ) .

Fig. 5
Fig. 5

TURTLE retrieval steps for a randomly generated vector field. (a),(b),(c) Simulations of measured SHG FROG traces for E ̃ x ( Ω ) , E ̃ y ( Ω ) , and E ̃ η ( Ω ) , respectively. (d),(e) Pulse fields E x ( t ) , E y ( t ) obtained using the standard reconstruction algorithm (dots), compared with the generating fields (solid curve). (f) The error surface and (g) sketch of the full vector field E ( t ) .

Fig. 6
Fig. 6

TURTLE retrieval steps for a randomly generated very complex vector field. (a),(b),(c) Simulations of measured SHG FROG traces for E ̃ x ( Ω ) , E ̃ y ( Ω ) , and E ̃ η ( Ω ) , respectively. (d),(e) Pulse fields E x ( t ) , E y ( t ) obtained using the standard reconstruction algorithm (dots), compared with the generating fields (solid curve). (f) The error surface and (g) sketch of the full vector field E ( t ) .

Tables (1)

Tables Icon

Table 1 Different Pulses with Their Reconstructed Relative Delays And Relative Phases

Equations (7)

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E ̃ ( Ω ) = E ̃ x ( Ω ) x ̂ + r E ̃ y ( Ω ) e i ( Ω τ + θ ) y ̂ ,
P | r E ̃ ( Ω ) | 2 d Ω = r 2 | E ̃ ( Ω ) | 2 d Ω ,
E ̃ η ( Ω ) = cos   η E ̃ x ( Ω ) + r   sin   η E ̃ y ( Ω ) e i ( Ω τ + θ ) .
I FROG ( ω , T ) = | E ( t ) E ( t T ) e i ω t d t | 2 .
e = i , j = 1 N [ I η meas ( i , j ) I η calc ( i , j ) ] 2 i , j = 1 N [ I η meas ( i , j ) ] 2 ,
I FROG ( η ¯ ) ( ω i , τ j ) = I FROG ( ω i , τ j ) + η i j α / η ¯ ,
I FROG ( ω , T ) = | ( 1 + e i θ ) 2 | 2 | E x ( t ) E x ( t T ) e i ω T d t | 2 .

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