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]
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    [CrossRef]
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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. 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]

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]

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)

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]

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]

1997 (2)

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]

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]

1996 (2)

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]

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.

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]

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]

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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