Abstract

Delivering femtosecond laser light in the focal plane of a high numerical aperture microscope objective is still a challenge, despite significant developments in the generation of ultrashort pulses. One of the most popular techniques, used to correct phase distortions resulting from propagation through transparent media, is the multiphoton intrapulse interference phase scan (MIIPS). The accuracy of MIIPS, however, is limited when higher-order phase distortions are present. Here we introduce an improvement, called G-MIIPS, which avoids the shortcomings of MIIPS, reduces the influence of higher-order phase terms, and can be used to more efficiently compress broadband laser pulses even with a simple 4f pulse shaper setup. In this work, we present analytical formulas for MIIPS and G-MIIPS, which are valid for chirped Gaussian pulses; we show an approximated analytic expression for G-MIIPS, which is valid for arbitrary pulse shapes. Finally we demonstrate the increased accuracy of G-MIIPS via experiments and numerical simulations.

© 2014 Optical Society of America

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References

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  1. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, 2006).
  2. M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G. J. Brakenhoff, “Dispersion pre-compensation of 15 femtosecond optical pulses for high-numerical-aperture objectives,” J. Microsc. 191, 141–150 (1998).
    [CrossRef]
  3. N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
    [CrossRef]
  4. A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
    [CrossRef]
  5. D. N. Fittinghoff, J. A. Squier, C. P. J. Barty, J. N. Sweetser, R. Trebino, and M. Müller, “Collinear type II second-harmonic-generation frequency-resolved optical gating for use with high-numerical-aperture objectives,” Opt. Lett. 23, 1046–1048 (1998).
    [CrossRef]
  6. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
    [CrossRef]
  7. V. Loriot, G. Gitzinger, and N. Forget, “Self-referenced characterization of femtosecond laser pulses by chirp scan,” Opt. Express 21, 24879–24893 (2013).
    [CrossRef]
  8. B. Xu, Y. Coello, V. V. Lozovoy, D. A. Harris, and M. Dantus, “Pulse shaping of octave spanning femtosecond laser pulses,” Opt. Express 14, 10939–10944 (2006).
    [CrossRef]
  9. B. Xu, J. M. Gunn, J. M. D. Cruz, V. V. Lozovoy, and M. Dantus, “Quantitative investigation of the multiphoton intrapulse interference phase scan method for simultaneous phase measurement and compensation of femtosecond laser pulses,” J. Opt. Soc. Am. B 23, 750–759 (2006).
    [CrossRef]
  10. D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
    [CrossRef]
  11. M. Dantus and V. V. Lozovoy, “Experimental coherent laser control of physicochemical processes,” Chem. Rev. 104, 1813–1859 (2004).
    [CrossRef]
  12. K. A. Walowicz, I. Pastirk, V. V. Lozovoy, and M. Dantus, “Multiphoton intrapulse interference. 1. Control of multiphoton processes in condensed phases,” J. Phys. Chem. A 106, 9369–9373 (2002).
    [CrossRef]
  13. M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
    [CrossRef]
  14. V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” Chem. Phys. Chem. 6, 1970–2000 (2005).
    [CrossRef]
  15. V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775–777 (2004).
    [CrossRef]
  16. D. Brinks, R. Hildner, F. D. Stefani, and N. F. van Hulst, “Beating spatio-temporal coupling: implications for pulse shaping and coherent control experiments,” Opt. Express 19, 26486–26499 (2011).
    [CrossRef]
  17. J. C. Vaughan, T. Feurer, K. W. Stone, and K. A. Nelson, “Analysis of replica pulses in femtosecond pulse shaping with pixelated devices,” Opt. Express 14, 1314–1328 (2006).
    [CrossRef]

2014 (2)

D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
[CrossRef]

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

2013 (1)

2011 (2)

2006 (3)

2005 (1)

V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” Chem. Phys. Chem. 6, 1970–2000 (2005).
[CrossRef]

2004 (2)

2002 (1)

K. A. Walowicz, I. Pastirk, V. V. Lozovoy, and M. Dantus, “Multiphoton intrapulse interference. 1. Control of multiphoton processes in condensed phases,” J. Phys. Chem. A 106, 9369–9373 (2002).
[CrossRef]

2001 (1)

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

1998 (3)

Accanto, N.

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

Barty, C. P. J.

Brakenhoff, G. J.

M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G. J. Brakenhoff, “Dispersion pre-compensation of 15 femtosecond optical pulses for high-numerical-aperture objectives,” J. Microsc. 191, 141–150 (1998).
[CrossRef]

Brinks, D.

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

D. Brinks, R. Hildner, F. D. Stefani, and N. F. van Hulst, “Beating spatio-temporal coupling: implications for pulse shaping and coherent control experiments,” Opt. Express 19, 26486–26499 (2011).
[CrossRef]

Castro-Lopez, M.

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

Coello, Y.

Cruz, J. M. D.

Dantus, M.

D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
[CrossRef]

B. Xu, Y. Coello, V. V. Lozovoy, D. A. Harris, and M. Dantus, “Pulse shaping of octave spanning femtosecond laser pulses,” Opt. Express 14, 10939–10944 (2006).
[CrossRef]

B. Xu, J. M. Gunn, J. M. D. Cruz, V. V. Lozovoy, and M. Dantus, “Quantitative investigation of the multiphoton intrapulse interference phase scan method for simultaneous phase measurement and compensation of femtosecond laser pulses,” J. Opt. Soc. Am. B 23, 750–759 (2006).
[CrossRef]

V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” Chem. Phys. Chem. 6, 1970–2000 (2005).
[CrossRef]

M. Dantus and V. V. Lozovoy, “Experimental coherent laser control of physicochemical processes,” Chem. Rev. 104, 1813–1859 (2004).
[CrossRef]

V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775–777 (2004).
[CrossRef]

K. A. Walowicz, I. Pastirk, V. V. Lozovoy, and M. Dantus, “Multiphoton intrapulse interference. 1. Control of multiphoton processes in condensed phases,” J. Phys. Chem. A 106, 9369–9373 (2002).
[CrossRef]

Diels, J.-C.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, 2006).

Feurer, T.

J. C. Vaughan, T. Feurer, K. W. Stone, and K. A. Nelson, “Analysis of replica pulses in femtosecond pulse shaping with pixelated devices,” Opt. Express 14, 1314–1328 (2006).
[CrossRef]

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

Fittinghoff, D. N.

Forget, N.

Gitzinger, G.

Gunn, J. M.

Hacker, M.

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

Harris, D. A.

Hildner, R.

Iaconis, C.

Loriot, V.

Lozovoy, V. V.

D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
[CrossRef]

B. Xu, Y. Coello, V. V. Lozovoy, D. A. Harris, and M. Dantus, “Pulse shaping of octave spanning femtosecond laser pulses,” Opt. Express 14, 10939–10944 (2006).
[CrossRef]

B. Xu, J. M. Gunn, J. M. D. Cruz, V. V. Lozovoy, and M. Dantus, “Quantitative investigation of the multiphoton intrapulse interference phase scan method for simultaneous phase measurement and compensation of femtosecond laser pulses,” J. Opt. Soc. Am. B 23, 750–759 (2006).
[CrossRef]

V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” Chem. Phys. Chem. 6, 1970–2000 (2005).
[CrossRef]

M. Dantus and V. V. Lozovoy, “Experimental coherent laser control of physicochemical processes,” Chem. Rev. 104, 1813–1859 (2004).
[CrossRef]

V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775–777 (2004).
[CrossRef]

K. A. Walowicz, I. Pastirk, V. V. Lozovoy, and M. Dantus, “Multiphoton intrapulse interference. 1. Control of multiphoton processes in condensed phases,” J. Phys. Chem. A 106, 9369–9373 (2002).
[CrossRef]

Müller, M.

M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G. J. Brakenhoff, “Dispersion pre-compensation of 15 femtosecond optical pulses for high-numerical-aperture objectives,” J. Microsc. 191, 141–150 (1998).
[CrossRef]

D. N. Fittinghoff, J. A. Squier, C. P. J. Barty, J. N. Sweetser, R. Trebino, and M. Müller, “Collinear type II second-harmonic-generation frequency-resolved optical gating for use with high-numerical-aperture objectives,” Opt. Lett. 23, 1046–1048 (1998).
[CrossRef]

Nelson, K. A.

Netz, R.

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

Nieder, J. B.

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

Pastirk, I.

V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775–777 (2004).
[CrossRef]

K. A. Walowicz, I. Pastirk, V. V. Lozovoy, and M. Dantus, “Multiphoton intrapulse interference. 1. Control of multiphoton processes in condensed phases,” J. Phys. Chem. A 106, 9369–9373 (2002).
[CrossRef]

Pastorelli, F.

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

Pestov, D.

D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
[CrossRef]

Piatkowski, L.

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

Rasskazov, G.

D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
[CrossRef]

Roth, M.

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

Rudolph, W.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, 2006).

Ryabtsev, A.

D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
[CrossRef]

Sauerbrey, R.

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

Simon, U.

M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G. J. Brakenhoff, “Dispersion pre-compensation of 15 femtosecond optical pulses for high-numerical-aperture objectives,” J. Microsc. 191, 141–150 (1998).
[CrossRef]

Squier, J.

M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G. J. Brakenhoff, “Dispersion pre-compensation of 15 femtosecond optical pulses for high-numerical-aperture objectives,” J. Microsc. 191, 141–150 (1998).
[CrossRef]

Squier, J. A.

Stefani, F. D.

Stobrawa, G.

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

Stone, K. W.

Sweetser, J. N.

Trebino, R.

van Hulst, N. F.

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

D. Brinks, R. Hildner, F. D. Stefani, and N. F. van Hulst, “Beating spatio-temporal coupling: implications for pulse shaping and coherent control experiments,” Opt. Express 19, 26486–26499 (2011).
[CrossRef]

Vaughan, J. C.

Walmsley, I. A.

Walowicz, K. A.

K. A. Walowicz, I. Pastirk, V. V. Lozovoy, and M. Dantus, “Multiphoton intrapulse interference. 1. Control of multiphoton processes in condensed phases,” J. Phys. Chem. A 106, 9369–9373 (2002).
[CrossRef]

Weiner, A. M.

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
[CrossRef]

Wolleschensky, R.

M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G. J. Brakenhoff, “Dispersion pre-compensation of 15 femtosecond optical pulses for high-numerical-aperture objectives,” J. Microsc. 191, 141–150 (1998).
[CrossRef]

Xu, B.

Appl. Phys. B (1)

M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey, “Frequency doubling of phase-modulated, ultrashort laser pulses,” Appl. Phys. B 73, 273–277 (2001).
[CrossRef]

Chem. Phys. Chem. (1)

V. V. Lozovoy and M. Dantus, “Systematic control of nonlinear optical processes using optimally shaped femtosecond pulses,” Chem. Phys. Chem. 6, 1970–2000 (2005).
[CrossRef]

Chem. Rev. (1)

M. Dantus and V. V. Lozovoy, “Experimental coherent laser control of physicochemical processes,” Chem. Rev. 104, 1813–1859 (2004).
[CrossRef]

J. Microsc. (1)

M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G. J. Brakenhoff, “Dispersion pre-compensation of 15 femtosecond optical pulses for high-numerical-aperture objectives,” J. Microsc. 191, 141–150 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. A (1)

K. A. Walowicz, I. Pastirk, V. V. Lozovoy, and M. Dantus, “Multiphoton intrapulse interference. 1. Control of multiphoton processes in condensed phases,” J. Phys. Chem. A 106, 9369–9373 (2002).
[CrossRef]

Light Sci. Appl. (1)

N. Accanto, J. B. Nieder, L. Piatkowski, M. Castro-Lopez, F. Pastorelli, D. Brinks, and N. F. van Hulst, “Phase control of femtosecond pulses on the nanoscale using second harmonic nanoparticles,” Light Sci. Appl. 3, e143 (2014).
[CrossRef]

Opt. Commun. (1)

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
[CrossRef]

Opt. Eng. (1)

D. Pestov, A. Ryabtsev, G. Rasskazov, V. V. Lozovoy, and M. Dantus, “Real-time single-shot measurement and correction of pulse phase and amplitude for ultrafast lasers,” Opt. Eng. 53, 051511 (2014).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Other (1)

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, 2006).

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Figures (8)

Fig. 1.
Fig. 1.

Multiphoton intrapulse interference phase scans. (a) Simulated MIIPS trace of a femtosecond laser pulse with large cubic phase distortion. (b) Equivalent G-MIIPS scan (c) from the top: phase modulation used in the MIIPS scan; real part of the phase-amplitude modulation used in G-MIIPS; normalized SHG spectrum after phase compensation using either MIIPS (blue curve) or G-MIIPS (green curve). (d) Group delay dispersion obtained geometrically by skewing and rescaling the G-MIIPS trace.

Fig. 2.
Fig. 2.

Result of single MIIPS iterations, simulated for various phase modulation frequencies while keeping the maximum correction constant Φ 0 τ 2 = 2.5 · 10 4 fs 2 . The figure refers to a Δ t = 10 fs laser pulse after propagation through 10 cm of glass. (a) MIIPS-corrected SHG spectra for different phase-modulation frequencies; the ideal SHG spectrum (black-dashed line) is also shown for reference. (b) Residual phase after MIIPS correction using different phase-modulation frequencies.

Fig. 3.
Fig. 3.

Phase dependence of the SHG intensity at the central frequency of a 100 nm broad laser pulse centered at 800 nm. (a) Map of the SHG in function of the second- and fourth-order phase terms. The dashed line represents the points explored by a typical MIIPS trace. (b) The maximum SHG intensity along the MIIPS trajectory does not correspond to zero GDD.

Fig. 4.
Fig. 4.

Simulation of the compensation of a 10 fs laser pulse centered at 2.4 rad / fs after propagation through 10 cm of BK7 glass. Standard MIIPS (a) and G-MIIPS (b) maps, obtained with Φ 0 = 100 rad , τ = 10 fs and σ = 0.5 rad . The black-dashed lines indicate the GDD range, which can be measured with this set of parameters; the white-dashed line represents both the center of the gate and the points where the GDD is zero. (c) GDD measured by a single iteration of MIIPS (red line) and G-MIIPS (blue line), together with actual GDD value (black line). (d) Residual phase after a single iteration of MIIPS (red line) and G-MIIPS (blue line). (e) SHG signal of the ideal SHG (black line) and after compensation with a single iteration of MIIPS (red line) and G-MIIPS (blue line).

Fig. 5.
Fig. 5.

G-MIIPS for different settings of the gate width, simulated for a 10 fs laser pulse centered at 2.4 rad / fs after propagation through 10 cm of BK7 glass. For all cases, the modulation parameters were Φ 0 = 200 rad and τ = 10 fs . Standard MIIPS is also shown for comparison. (a) G-MIIPS map corresponding to σ = 1 rad . (b) G-MIIPS map corresponding to σ = 0.2 rad . The black-dashed lines indicate the GDD range, which can be measured with this set of parameters; the white-dashed line represents both the center of the gate and the points where the GDD is zero. (c) SHG after a single iteration of G-MIIPS for σ varying between 1 and 0.2 rad. (d) Residual phase after a single iteration of G-MIIPS for σ varying between 1 and 0.2 rad.

Fig. 6.
Fig. 6.

Simulation of the compensation of a 10 fs laser pulse centered at 2.4 rad / fs with significant ( 10 5 fs 4 ) fourth-order phase distortion. Standard MIIPS (a) and G-MIIPS (b) maps, obtained with Φ 0 = 20 rad , τ = 10 fs , and gate width σ = 0.5 rad . The black-dashed lines indicate the GDD range, which can be measured with this set of parameters; the white-dashed line represents both the center of the gate and the points where the GDD is zero. (c) GDD measured by a single iteration of MIIPS (red line) and G-MIIPS (blue line), together with actual GDD value (black line). (d) Residual phase after a single iteration of MIIPS (red line) and G-MIIPS (blue line). (e) SHG signal of the ideal SHG (black line) and compensated with a single iteration of MIIPS (red line) and G-MIIPS (blue line). Standard MIIPS in this case reduces SHG.

Fig. 7.
Fig. 7.

Comparison of iterative MIIPS and G-MIIPS. Panels (a) and (c) represent the case of a 10 fs pulse after propagation through 10 cm of glass. Panels (b) and (d) refer to the case of a 10 fs subject to a fourth-order spectral phase of 10 5 fs 4 . In the panels (a) and (b), it is plotted the residual phase after multiple iterations of either MIIPS (blue dots) or G-MIIPS (green diamonds). In these graphs the first point, labeled with 0, corresponds to the situation before the first iteration. The panels (c) and (d) report the SHG intensity, normalized to its theoretical limit.

Fig. 8.
Fig. 8.

Experimental comparison of MIIPS and G-MIIPS regarding the compensation of the GDD introduced by 23 mm of SF10 glass on a 15 fs pulse centered at 800 nm. (a) Standard MIIPS trace obtained with scanning parameters Φ 0 = 10 rad and τ = 25 fs . (b) G-MIIPS scan obtained with scanning parameters Φ 0 = 10 rad , τ = 25 fs , σ = 0.18 rad . (c) GDD measured by a single MIIPS (black curve) and G-MIIPS (red curve) iteration, compared with the theoretical GDD calculated using the dispersion equation. (d) SHG spectrum after a single iteration of either MIIPS (black curve) or G-MIIPS (red curve), compared with the SHG obtained after full pulse compression (green curve).

Tables (1)

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Table 1. List of Symbols Used in this Article

Equations (18)

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SHG ( 2 ω , ϕ ) = | + | E ( ω Ω ) | | E ( ω + Ω ) | · exp [ i ( ϕ ( ω Ω ) + ϕ ( ω + Ω ) ) ] d Ω | 2 ,
SHG ( 2 ω , ϕ ) = | + | E ( ω Ω ) | | E ( ω + Ω ) | exp ( i 2 ϕ ( ω ) ω 2 Ω 2 ) d Ω | 2 .
f ( ω ) = exp { i Φ 0 sin [ τ ( ω ω 0 ) ψ ] } .
ϕ ( ω Ω ) + ϕ ( ω + Ω ) + 2 Φ 0 cos ( τ Ω ) sin [ τ ( ω ω 0 ) ψ ] = 0 ,
ϕ ¨ ( ω ) = Φ 0 τ 2 sin [ τ ( ω ω 0 ) ψ m ( ω ) ] ,
| GDD min | = Φ 0 τ 2 ( 1 + ( ϵ Δ ψ ) 2 ) 1 / 2 .
D = ( 1 + ( ϵ Δ ψ ) 2 ) 1 / 2 ϵ Δ ψ ,
M ( ω ) = exp { [ τ ( ω ω 0 ) ψ σ ] 2 + i Φ 0 sin [ τ ( ω ω 0 ) ψ ] } .
G ( 2 ω , ϕ ) = SHG ( 2 ω , ϕ ) · exp { 4 [ τ ( ω ω 0 ) ψ σ ] 2 } .
G ( 2 ω , ϕ ) = | + | E ( ω Ω ) | | E ( ω + Ω ) | · exp ( 2 τ 2 Ω 2 σ 2 ) · exp { i [ ϕ ( ω Ω ) + ϕ ( ω + Ω ) + 2 Φ 0 cos ( τ Ω ) sin [ τ ( ω ω 0 ) ψ ] ] } d Ω | 2 .
E ( ω ) = exp ( ( ω ω 0 ) 2 Δ ω 2 + i ϕ ( ω ) ) .
SHG ( 2 ω , ϕ ) = exp { 4 ( ω ω 0 ) 2 Δ ω 2 } · | exp { 2 Ω 2 Δ ω 2 + i [ ϕ ( ω Ω ) + ϕ ( ω + Ω ) + 2 Φ 0 cos ( τ Ω ) sin ( τ ( ω ω 0 ) ψ ) ] } d Ω | 2 .
exp ( i z cos ( θ ) ) = n = + ( i n J n ( z ) exp ( i n θ ) ) ,
SHG ( 2 ω , ϕ ) = π Δ ω 2 Δ ω 4 ϕ ¨ ( ω ) 2 + 4 exp ( 4 ( ω ω 0 ) 2 Δ ω 2 ) · n = + exp ( Δ ω 2 τ 2 n 2 Δ ω 4 ϕ ¨ ( ω ) 2 + 4 ) · J n ( 2 Φ 0 sin ( τ ( ω ω 0 ) ψ ) ) 2 .
SHG ( 2 ω , ϕ ) = π Δ ω 2 exp ( 4 ( ω ω 0 ) 2 Δ ω 2 ) · { 4 + Δ ω 4 [ ϕ ¨ ( ω ) τ 2 Φ 0 sin ( τ ( ω ω 0 ) ψ ) ] 2 } 1 / 2 .
G ( 2 ω , ϕ ) = exp { 4 ( ω ω 0 ) 2 Δ ω 2 } · | exp { 2 Ω 2 L 2 + i [ ϕ ( ω Ω ) + ϕ ( ω + Ω ) + 2 Φ 0 cos ( τ Ω ) sin ( ( ω ω 0 ) τ ψ ) ] } d Ω | 2 .
G ( 2 ω , ϕ ) = π L 2 exp ( 4 ( ω ω 0 ) 2 Δ ω 2 ) · { 4 + L 4 [ ϕ ¨ ( ω ) τ 2 Φ 0 sin ( τ ( ω ω 0 ) ψ ) ] 2 } 1 / 2 .
G ( 2 ω , ϕ ) = π σ 2 4 { [ ϕ ¨ ( ω ) τ 2 Φ 0 sin ( τ ( ω ω 0 ) ψ ) ] 2 σ 4 + 4 τ 4 } 3 / 2 · { 4 σ 4 | E ( ω ) | 4 [ ϕ ¨ ( ω ) τ 2 Φ 0 sin ( τ ( ω ω 0 ) ψ ) ] 2 + [ σ 2 ( | E ( ω ) | 2 | E ( ω ) | ω 2 ( | E ( ω ) | ω ) 2 ) + 4 τ 2 | E ( ω ) | 2 ] 2 } .

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