Abstract

We present an extraction algorithm for spectral phase interferometry for direct field reconstruction (SPIDER) in the so-called X-SPIDER configuration. Our approach largely extends the measurable time windows of pulses without requiring any modification to the experimental X-SPIDER setup.

© 2013 OSA

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    [CrossRef]
  27. L. Cohen, “Wigner distribution for finite duration or band-limited signals and limiting cases,” IEEE Trans. Acoust. Speech35(6), 796–806 (1987).
    [CrossRef]
  28. F. Hlawatsch, “Interference terms in the Wigner distribution,” Digit. Signal Process.84, 363–367 (1984).
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2012 (1)

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

2011 (2)

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses,” Opt. Commun.284(15), 3785–3794 (2011).
[CrossRef]

2010 (5)

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics4(11), 760–766 (2010).
[CrossRef]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010).
[CrossRef]

D. R. Austin, T. Witting, and I. A. Walmsley, “Resolution of the relative phase ambiguity in spectral shearing interferometry of ultrashort pulses,” Opt. Lett.35(12), 1971–1973 (2010).
[CrossRef] [PubMed]

2009 (4)

D. R. Austin, T. Witting, and I. A. Walmsley, “High precision self-referenced phase retrieval of complex pulses with multiple-shearing spectral interferometry,” J. Opt. Soc. Am. B26(9), 1818–1830 (2009).
[CrossRef]

T. Witting, D. R. Austin, and I. A. Walmsley, “Ultrashort pulse characterization by spectral shearing interferometry with spatially chirped ancillae,” Opt. Express17(21), 18983–18994 (2009).
[CrossRef] [PubMed]

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett.95(5), 051101 (2009).
[CrossRef]

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1(2), 308–437 (2009).
[CrossRef]

2008 (2)

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008).
[CrossRef] [PubMed]

2007 (1)

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics1(8), 463–467 (2007).
[CrossRef]

2006 (2)

2003 (1)

P. Londero, M. E. Anderson, C. Radzewicz, C. Iaconis, and I. A. Walmsley, “Measuring ultrafast pulses in the near-ultraviolet using spectral phase interferometry for direct electric field reconstruction,” J. Mod. Opt.50, 179–184 (2003).

2002 (3)

1999 (3)

1998 (1)

1994 (1)

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical-data with an up-conversion time microscope,” Appl. Phys. Lett.65(20), 2513–2515 (1994).
[CrossRef]

1987 (1)

L. Cohen, “Wigner distribution for finite duration or band-limited signals and limiting cases,” IEEE Trans. Acoust. Speech35(6), 796–806 (1987).
[CrossRef]

1984 (1)

F. Hlawatsch, “Interference terms in the Wigner distribution,” Digit. Signal Process.84, 363–367 (1984).

1982 (1)

Anderson, M. E.

P. Londero, M. E. Anderson, C. Radzewicz, C. Iaconis, and I. A. Walmsley, “Measuring ultrafast pulses in the near-ultraviolet using spectral phase interferometry for direct electric field reconstruction,” J. Mod. Opt.50, 179–184 (2003).

Andrekson, P. A.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Austin, D. R.

Azaña, J.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Begishev, I. A.

Bennett, C. V.

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical-data with an up-conversion time microscope,” Appl. Phys. Lett.65(20), 2513–2515 (1994).
[CrossRef]

Bogris, A.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Bowlan, P.

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses,” Opt. Commun.284(15), 3785–3794 (2011).
[CrossRef]

Boyraz, O.

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett.95(5), 051101 (2009).
[CrossRef]

Bromage, J.

Cardoso, L.

Chambaret, J.-P.

Chauhan, V.

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses,” Opt. Commun.284(15), 3785–3794 (2011).
[CrossRef]

Chu, S. T.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Cohen, J.

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses,” Opt. Commun.284(15), 3785–3794 (2011).
[CrossRef]

Cohen, L.

L. Cohen, “Wigner distribution for finite duration or band-limited signals and limiting cases,” IEEE Trans. Acoust. Speech35(6), 796–806 (1987).
[CrossRef]

Cundiff, S. T.

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics4(11), 760–766 (2010).
[CrossRef]

Dasgupta, S.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

de Beauvoir, B.

Dorrer, C.

Ellis, A. D.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Essiambre, R.-J.

Figueira, G.

Fontaine, N. K.

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

Foschini, G. J.

Foster, M. A.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008).
[CrossRef] [PubMed]

Gaeta, A. L.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008).
[CrossRef] [PubMed]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Gallmann, L.

Geraghty, D. F.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008).
[CrossRef] [PubMed]

Goebel, B.

Grüner-Nielsen, L.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Heritage, J.

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

Herstrøm, S.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Hirasawa, M.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B74(9Suppl. S), s225–s229 (2002).
[CrossRef]

Hlawatsch, F.

F. Hlawatsch, “Interference terms in the Wigner distribution,” Digit. Signal Process.84, 363–367 (1984).

Huang, C.-B.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics1(8), 463–467 (2007).
[CrossRef]

Iaconis, C.

P. Londero, M. E. Anderson, C. Radzewicz, C. Iaconis, and I. A. Walmsley, “Measuring ultrafast pulses in the near-ultraviolet using spectral phase interferometry for direct electric field reconstruction,” J. Mod. Opt.50, 179–184 (2003).

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett.24(18), 1314–1316 (1999).
[CrossRef] [PubMed]

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron.35(4), 501–509 (1999).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett.23(10), 792–794 (1998).
[CrossRef] [PubMed]

Ina, H.

Jakobsen, D.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Jiang, Z.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics1(8), 463–467 (2007).
[CrossRef]

Kakande, J.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Keller, U.

Kobayashi, S.

Kolner, B. H.

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical-data with an up-conversion time microscope,” Appl. Phys. Lett.65(20), 2513–2515 (1994).
[CrossRef]

Kramer, G.

Le Blanc, C.

Leaird, D. E.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics1(8), 463–467 (2007).
[CrossRef]

Légaré, F.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

Lipson, M.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008).
[CrossRef] [PubMed]

Little, B. E.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Londero, P.

P. Londero, M. E. Anderson, C. Radzewicz, C. Iaconis, and I. A. Walmsley, “Measuring ultrafast pulses in the near-ultraviolet using spectral phase interferometry for direct electric field reconstruction,” J. Mod. Opt.50, 179–184 (2003).

Lopes, N.

Lundström, C.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Matuschek, N.

Morandotti, R.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Morita, R.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B74(9Suppl. S), s225–s229 (2002).
[CrossRef]

Moss, D. J.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Nakagawa, N.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B74(9Suppl. S), s225–s229 (2002).
[CrossRef]

O'Gorman, J.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Park, Y.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Parmigiani, F.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Pasquazi, A.

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
[CrossRef]

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Peccianti, M.

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

Petropoulos, P.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Phelan, R.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Qing, F.

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett.95(5), 051101 (2009).
[CrossRef]

Radzewicz, C.

P. Londero, M. E. Anderson, C. Radzewicz, C. Iaconis, and I. A. Walmsley, “Measuring ultrafast pulses in the near-ultraviolet using spectral phase interferometry for direct electric field reconstruction,” J. Mod. Opt.50, 179–184 (2003).

Ranc, S.

Richardson, D. J.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Rousseau, J.-P.

Rousseau, P.

Salem, R.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008).
[CrossRef] [PubMed]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Salin, F.

Sang, X. Z.

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett.95(5), 051101 (2009).
[CrossRef]

Scott, R.

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

Scott, R. P.

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical-data with an up-conversion time microscope,” Appl. Phys. Lett.65(20), 2513–2515 (1994).
[CrossRef]

Shigekawa, H.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B74(9Suppl. S), s225–s229 (2002).
[CrossRef]

Sjödin, M.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Slavík, R.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Soares, F. M.

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

Song, Q.

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett.95(5), 051101 (2009).
[CrossRef]

Steinmeyer, G.

Sutter, D. H.

Sygletos, S.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Syvridis, D.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Takeda, M.

Tien, E. K.

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett.95(5), 051101 (2009).
[CrossRef]

Trebino, R.

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses,” Opt. Commun.284(15), 3785–3794 (2011).
[CrossRef]

Turner, A. C.

Turner-Foster, A. C.

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Usechak, N. G.

Vaughan, P.

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses,” Opt. Commun.284(15), 3785–3794 (2011).
[CrossRef]

Walmsley, I. A.

D. R. Austin, T. Witting, and I. A. Walmsley, “Resolution of the relative phase ambiguity in spectral shearing interferometry of ultrashort pulses,” Opt. Lett.35(12), 1971–1973 (2010).
[CrossRef] [PubMed]

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1(2), 308–437 (2009).
[CrossRef]

T. Witting, D. R. Austin, and I. A. Walmsley, “Ultrashort pulse characterization by spectral shearing interferometry with spatially chirped ancillae,” Opt. Express17(21), 18983–18994 (2009).
[CrossRef] [PubMed]

D. R. Austin, T. Witting, and I. A. Walmsley, “High precision self-referenced phase retrieval of complex pulses with multiple-shearing spectral interferometry,” J. Opt. Soc. Am. B26(9), 1818–1830 (2009).
[CrossRef]

P. Londero, M. E. Anderson, C. Radzewicz, C. Iaconis, and I. A. Walmsley, “Measuring ultrafast pulses in the near-ultraviolet using spectral phase interferometry for direct electric field reconstruction,” J. Mod. Opt.50, 179–184 (2003).

C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B19(5), 1019–1029 (2002).
[CrossRef]

C. Dorrer and I. A. Walmsley, “Precision and consistency criteria in spectral phase interferometry for direct electric-field reconstruction,” J. Opt. Soc. Am. B19(5), 1030–1038 (2002).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron.35(4), 501–509 (1999).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett.24(18), 1314–1316 (1999).
[CrossRef] [PubMed]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett.23(10), 792–794 (1998).
[CrossRef] [PubMed]

Weerasuriya, R.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

Weiner, A.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics1(8), 463–467 (2007).
[CrossRef]

Weiner, A. M.

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics4(11), 760–766 (2010).
[CrossRef]

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Winzer, P. J.

Witting, T.

Yamamoto, K.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B74(9Suppl. S), s225–s229 (2002).
[CrossRef]

Yamashita, M.

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B74(9Suppl. S), s225–s229 (2002).
[CrossRef]

Yoo, S.

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

Zhou, L.

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

Zuegel, J. D.

Adv. Opt. Photonics (1)

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1(2), 308–437 (2009).
[CrossRef]

Appl. Phys. B (1)

M. Hirasawa, N. Nakagawa, K. Yamamoto, R. Morita, H. Shigekawa, and M. Yamashita, “Sensitivity improvement of spectral phase interferometry for direct electric-field reconstruction for the characterization of low-intensity femtosecond pulses,” Appl. Phys. B74(9Suppl. S), s225–s229 (2002).
[CrossRef]

Appl. Phys. Lett. (2)

E. K. Tien, X. Z. Sang, F. Qing, Q. Song, and O. Boyraz, “Ultrafast pulse characterization using cross phase modulation in silicon,” Appl. Phys. Lett.95(5), 051101 (2009).
[CrossRef]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical-data with an up-conversion time microscope,” Appl. Phys. Lett.65(20), 2513–2515 (1994).
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[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Pasquazi, Y. Park, S. T. Chu, B. E. Little, F. Légaré, R. Morandotti, J. Azaña, and D. J. Moss, “Time-lens measurement of subpicosecond optical pulses in CMOS compatible high-index glass waveguides,” IEEE J. Sel. Top. Quantum Electron.18(2), 629–636 (2012).
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J. Opt. Soc. Am. (1)

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

Nat. Photonics (5)

A. Pasquazi, M. Peccianti, Y. Park, B. E. Little, S. T. Chu, R. Morandotti, J. Azaña, and D. J. Moss, “Sub-picosecond phase-sensitive optical pulse characterization on a chip,” Nat. Photonics5(10), 618–623 (2011).
[CrossRef]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics4(10), 690–695 (2010).
[CrossRef]

N. K. Fontaine, R. Scott, L. Zhou, F. M. Soares, J. Heritage, and S. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4(4), 248–254 (2010).
[CrossRef]

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics1(8), 463–467 (2007).
[CrossRef]

S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics4(11), 760–766 (2010).
[CrossRef]

Nature (1)

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456(7218), 81–84 (2008).
[CrossRef] [PubMed]

Opt. Commun. (1)

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Single-shot multiple-delay crossed-beam spectral interferometry for measuring extremely complex pulses,” Opt. Commun.284(15), 3785–3794 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

J. Wemans, G. Figueira, N. Lopes, and L. Cardoso, “Self-referencing spectral phase interferometry for direct electric-field reconstruction with chirped pulses,” Opt. Lett.31(14), 2217–2219 (2006).
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R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett.33(10), 1047–1049 (2008).
[CrossRef] [PubMed]

D. R. Austin, T. Witting, and I. A. Walmsley, “Resolution of the relative phase ambiguity in spectral shearing interferometry of ultrashort pulses,” Opt. Lett.35(12), 1971–1973 (2010).
[CrossRef] [PubMed]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett.23(10), 792–794 (1998).
[CrossRef] [PubMed]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett.24(18), 1314–1316 (1999).
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Figures (8)

Fig. 1
Fig. 1

(a) Sketch of the classical SPIDER setup for amplitude and phase retrieval. Two delayed replicas of the pulse under test (PUT) interact nonlinearly with a dispersed pump via frequency mixing in a bulk crystal. The nonlinear product is composed by two replicas shifted in frequency. Their spectrum is collected with a spectrometer and elaborated to extract the complete information (amplitude and phase) of the PUT. (b) Same interaction with a highly chirped PUT: in this case the two replicas of the signal cover a significant temporal portion of the pump and their spectrum is distorted when compared to the spectrum of the PUT. As a result, the standard algorithm fails to retrieve the PUT. Conversely, the FLEA is able to correctly work on this kind of pulses, by addressing the two replicas of the nonlinear product as a Fresnel Integral of the PUT.

Fig. 2
Fig. 2

Phase extraction procedure.

Fig. 3
Fig. 3

(a) DC and (b-c) AC components as defined by Eqs. (3.1a)(3.1b) in the phase space (ωeqt), for a Gaussian PUT with first and second order chirps. The straight dashed line in red in (b-c) are the loci ωeq = ± Δt/ ϕ; where ϕ is the total first order chirp of the Fresnel integral of the PUT, encompassing both the pump and the PUT chirps. In (d) the phase space representation of the complete interferogram is reported: in this case, the separation in frequency is not guaranteed for low values of the time delay Δt. (e-h) same as (a-d), but with a larger overall total dispersion ϕ: the spectral content of the DC (e) and AC (f-g) components is localized along ωeq = 0 and ωeq = ± Δt/ ϕ, respectively. Thus, their separation in the interferogram (h) is guaranteed for a large set of delays Δt. Please notice that in (a), the pattern with hyperbolic loci around ωeq Δt/ 2 = const is due to the cosine term cos (ωeq Δt/ 2) multiplying the integral in Eq. (3.1a). When the energy is localized for frequency components |ωeq| <π/Δt, as in the case (e) the hyperbolic pattern is not visible anymore.

Fig. 4
Fig. 4

Phase jump reconstruction. Here ϕP = 10ps2, ΔτP = 250ps. (a) PSD of the PUT (black) used in the simulation. The spectral amplitude for a single replica of the nonlinear product obtained with a PUT phase jump α = 10, and the PSD reconstructed with the FLEA are displayed in red and green lines, respectively. (b) same as (a) for a PUT phase jump α = 40. (c) PUT phase for α = 10, and reconstructed phases with the classical algorithm and FLEA in black, red and green solid lines, respectively. The dashed red and green lines report the differential error for classic algorithm and FLEA respectively. (d) same as (c) for α = 40.

Fig. 5
Fig. 5

RMS error as in Eq. (3) for the classic algorithm (red plots) and FLEA (green plots) vs α (higher values are for steeper phase jumps). (a) and (b) show the error calculated by using SFG and DFWM, respectively. Light red and light green are for ϕP = 10ps2, while dark red an dark green are for ϕP = 20ps2.

Fig. 6
Fig. 6

Estimation of the accuracy of FLEA vs the standard algorithm for a PUT propagating through first order dispersive systems. (a,b) RMS error for the retrieved waveform vs pump (y axis) and PUT(x axis) total dispersions for the SFG case, and (c,d) for the DFWM case; (a,c) standard algorithm, (b,d) FLEA. The color bar reports the RMS error. The picture is saturated for RMS>0.2. In (b) the green dotted line corresponds to the limit imposed by the limitation of the pump temporal window and the cyan dotted lines report the “blind region” of the algorithm, around the opposite of the value of the pump dispersion. This graph clearly evidences that the low-error (white) zone is significantly increased when FLEA is used as compared to the standard algorithm. (c,d) same comments as (a,b) for the DFMW case.

Fig. 7
Fig. 7

Same as Fig. 6 for a PUT propagating through second order dispersive systems.

Fig. 8
Fig. 8

Same as Fig. 7 for a PUT propagating through third order dispersive systems.

Tables (1)

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Table 1 Equivalent Quantities for the Nonlinear Interaction in Different Frequency Mixing Cases

Equations (46)

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e( t )=| e( t ) |exp[ i φ e ( t ) ]= E( ω )exp( iωt ) dω 2π , E( ω )=| E( ω ) |exp[ i φ E ( ω ) ]= e( t )exp( iωt ) dt.
s( t )=[ e( t+ Δt 2 )+e( t Δt 2 ) ]
P(ω)= P o (ω)exp( i ω 2 ϕ P 2 )
p(t)= 1 2iπ ϕ P exp( i t 2 2 ϕ P )* p o (t) = 1 2iπ ϕ P exp( i t 2 2 ϕ P ) p o (τ)exp( i τ 2 2 ϕ P iτt ϕ P ) dτ
p(t) 1 2iπ ϕ P exp( i t 2 2 ϕ P ) p o (τ)exp( iτt ϕ P ) dτ = 1 2iπ ϕ P exp( i t 2 2 ϕ P ) P o ( t ϕ P )exp( i t 2 2 ϕ P )
i(t)s(t)p(t)s(t) P o ( t ϕ P )exp( i t 2 2 ϕ P )s(t)exp( i t 2 2 ϕ P )
i (t) * s(t)p (t) * s(t) P o ( t ϕ P )exp( i t 2 2 ϕ P )s(t)exp( i t 2 2 ϕ P )
i (t) * s(t) [ p (t) * ] 2 s(t) [ P o ( t ϕ P ) ] 2 exp( i t 2 ϕ P )s(t)exp( i t 2 ϕ P )
Δ τ S =Δt+Δ τ E Δ τ PN
| I(ω) | 2 | S(ω)*P(ω) | 2 | S(ω)*exp( i ω 2 ϕ P 2 ) | 2 TWMSFG
| I(ω) | 2 | S(ω)* P * (ω) | 2 | S(ω)*exp( i ω 2 ϕ P 2 ) | 2 TWMDFG
| I(ω) | 2 | S(ω)* P * (ω)* P * (ω) | 2 | S(ω)*exp( i ω 2 ϕ P 4 ) | 2 DFWM
F s (ω)=S(ω)exp( i ω 2 ϕ P 2 )
f s (t)= 1 2iπ ϕ P exp( i t 2 2 ϕ P ) s(τ)exp( i τ 2 2 ϕ P iτt ϕ P ) dτ
| I(ω) | 2 | exp( i ω 2 ϕ P 2 ) S(x) exp( i x 2 ϕ P 2 )exp( ixω ϕ P ) dx 2π | 2 = = | F s (x) exp( ixω ϕ P ) dx 2π | 2 | f s ( ω ϕ P ) | 2
t eq =ω ϕ P
F E (ω)=E(ω)exp( i ω 2 ϕ P 2 )
f e (t)= 1 2iπ ϕ P exp( i t 2 2 ϕ P ) e(τ)exp( i τ 2 2 ϕ P iτt ϕ P ) dτ
| I(ω) | 2 | f s ( t eq ) | 2 = | f e ( t eq + Δt 2 )+ f e ( t eq Δt 2 ) | 2
| I(ω) | 2 dc( t eq )+a c + ( t eq )+a c ( t eq )
dc( t eq )= | f e ( t eq + Δt 2 ) | 2 + | f e ( t eq Δt 2 ) | 2
a c ± ( t eq )=| f e ( t eq + Δt 2 ) || f e ( t eq Δt 2 ) |exp[ ±iΔθ( t eq ) ]
Δθ( t eq )= φ fe ( t eq + Δt 2 ) φ fe ( t eq Δt 2 )
f e ( t eq )e( t eq )*exp( i t eq 2 2 ϕ P )E( t eq ϕ P )exp( i t eq 2 2 ϕ P )
Δ τ E 2 <<2π ϕ P 2π Δ τ P Δ ω P
| I(ω) | 2 | E( ω+ Ω 2 )exp( +iω Δt 2 )+E( ω Ω 2 )exp( iω Δt 2 ) | 2
Ω= Δt ϕ P
dc( ω )= | E( ω+ Ω 2 ) | 2 + | E( ω Ω 2 ) | 2
a c ± ( ω )=| E( ω+ Ω 2 ) || E( ω Ω 2 ) |exp[ ±iΔθ( ω ) ]
Δθ( ω )=Δ φ E ( ω )+ωΔt= φ E ( ω+ Ω 2 ) φ E ( ω Ω 2 )+ωΔt
DC( ω eq )=cos( Δt ω eq 2 ) F e ( s+ ω eq 2 ) F e * ( s ω eq 2 ) ds 2π
A C ± ( ω eq )= 1 2 F e ( s+ ω eq 2 ) F e * ( s ω eq 2 )exp( isΔt 2 ) ds 2π
μΔθ( t eq )
μΔθ( t eq )=μ( φ fe ( t eq + Δt 2 ) φ fe ( t eq Δt 2 ) )μ d φ fe ( t eq ) d t eq Δt
μ Δθ( t eq ) d t eq Δt =μ Δθ( ω ) ϕ P dω Δt μ φ fe ( t eq )
μ Δθ( ω ) ϕ P dω Δt =μ Δ φ E ( ω )ωΔt dω Ω μ( φ E ( ω ) ω 2 ϕ P 2 )
| f e ( t eq ) |exp( i φ f e ( t eq ) )= f e ( t eq )
| f e ( t eq ) |exp( i φ f e ( t eq ) )= f e * ( t eq )
F e ( ω eq )=| E( ω eq ) |exp( i φ E ( ω eq )i ϕ P ω eq 2 2 )
F e * ( ω eq )=| E( ω eq ) |exp( i φ E ( ω eq )+i ϕ P ω eq 2 2 )
E( ω eq )=| E( ω eq ) |exp( i φ E ( ω eq ) )
E'( ω eq )=| E( ω eq ) |exp( i φ E ( ω eq )+i ϕ P ω eq 2 )
E(ω)=exp( ω 2 Δ ω E 2 )
P(ω)=exp( ω 16 Δ ω P 16 )exp( i ω 2 ϕ P 2 )
ε= [ | E R (ω)E(ω) | 2 dω 2π | E(ω) | 2 dω 2π ] 1/2
φ E (ω)= π 2 tanh( α ω Δ ω E )

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