Abstract

We show that using multiple shears in spectral shearing interferometry is a powerful technique for improving precision, thus enabling the measurement of more complex pulses and resolving phase ambiguities. We derive an efficient and robust optimal phase reconstruction algorithm for extracting the spectral phase from interferograms taken at an arbitrary number of different shears. We show that if the shear is easily adjustable then a multishear measurement always offers a superior precision, even when considering the loss of precision of the raw data necessitated by multiple acquisitions. We present numerical examples and demonstrate an experimental implementation of the measurement of a double pulse using two shears.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  29. C. Dorrer and I. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B 19, 1019-1029 (2002).
    [CrossRef]
  30. C. Dorrer and I. Walmsley, “Precision and consistency criteria in spectral phase interferometry for direct electric-field reconstruction,” J. Opt. Soc. Am. B 19, 1030-1038 (2002).
    [CrossRef]
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2009 (2)

2008 (6)

2007 (3)

2006 (4)

2005 (2)

2004 (1)

2003 (1)

2002 (5)

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O'Shea, A. Shreenath, R. Trebino, and R. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174-1176 (2002).
[CrossRef]

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. B 74, S225-S229 (2002).
[CrossRef]

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

C. Dorrer and I. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B 19, 1019-1029 (2002).
[CrossRef]

C. Dorrer and I. Walmsley, “Precision and consistency criteria in spectral phase interferometry for direct electric-field reconstruction,” J. Opt. Soc. Am. B 19, 1030-1038 (2002).
[CrossRef]

2000 (1)

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER,” Appl. Phys. B 70, S85-S93 (2000).
[CrossRef]

1999 (1)

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

1998 (2)

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]

S. Linden, H. Giessen, and J. Kuhl, “XFROG--a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119-124 (1998).
[CrossRef]

1993 (1)

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

1988 (1)

Anderson, M. E.

M. E. Anderson, T. Witting, and I. A. Walmsley, “Gold-SPIDER: spectral phase interferometry for direct electric field reconstruction utilizing sum-frequency generation from a gold surface,” J. Opt. Soc. Am. B 25, A13-A16 (2008).
[CrossRef]

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER,” Appl. Phys. B 70, S85-S93 (2000).
[CrossRef]

Austin, D. R.

Barthélémy, A.

Baum, P.

Bethge, J.

Birge, J. R.

Buckup, T.

Cerullo, G.

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Chambaret, J. -P.

Chériaux, G.

Chien, C. Y.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Chowdhury, I. H.

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Ultrafast double-pulse ablation of fused silica,” Appl. Phys. Lett. 86, 151110 (2005).
[CrossRef]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Comtois, D.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Dahleh, M.

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

de Araujo, L. E. E.

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER,” Appl. Phys. B 70, S85-S93 (2000).
[CrossRef]

De Silvestri, S.

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Desparois, A.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Dorrer, C.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Ell, R.

Froehly, C.

Gabolde, P.

Gallmann, L.

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Giessen, H.

S. Linden, H. Giessen, and J. Kuhl, “XFROG--a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119-124 (1998).
[CrossRef]

Gorza, S. -P.

Grebing, C.

Gu, X.

Heritage, J.

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. B 74, S225-S229 (2002).
[CrossRef]

Iaconis, C.

Jiang, Z.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Johnston, T. W.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Jones, G. A.

G. A. Jones and J. M. Jones, Elementary Number Theory (Springer, 1998).
[CrossRef]

Jones, J. M.

G. A. Jones and J. M. Jones, Elementary Number Theory (Springer, 1998).
[CrossRef]

Kang, I.

Kärtner, F. X.

Keller, U.

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Keusters, D.

Kieffer, J. -C.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Kimmel, M.

Kosik, E.

Kosik, E. M.

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER,” Appl. Phys. B 70, S85-S93 (2000).
[CrossRef]

Kuhl, J.

S. Linden, H. Giessen, and J. Kuhl, “XFROG--a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119-124 (1998).
[CrossRef]

LaFontaine, B.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Lee, D.

Lelek, M.

Linden, S.

S. Linden, H. Giessen, and J. Kuhl, “XFROG--a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119-124 (1998).
[CrossRef]

Lochbrunner, S.

Louradour, F.

Mansourian, T.

Mercier, B.

Mercure, H. P.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Möhring, J.

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. B 74, S225-S229 (2002).
[CrossRef]

Motzkus, M.

Mouradian, L.

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. B 74, S225-S229 (2002).
[CrossRef]

O'Shea, P.

Pépin, H.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Polli, D.

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Rabitz, H.

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

Radunsky, A.

Riedle, E.

Salehi, J.

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. B 74, S225-S229 (2002).
[CrossRef]

Shreenath, A.

Steinmeyer, G.

J. Bethge, C. Grebing, and G. Steinmeyer, “A fast Gabor wavelet transform for high-precision phase retrieval in spectral interferometry,” Opt. Express 15, 14313-14321 (2007).
[CrossRef] [PubMed]

G. Stibenz and G. Steinmeyer, “Optimizing spectral phase interferometry for direct electric-field reconstruction,” Rev. Sci. Instrum. 77, 073105 (2006).
[CrossRef]

A. Wyatt, I. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31, 1914-1916 (2006).
[CrossRef] [PubMed]

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Stibenz, G.

Tan, H.

Trebino, R.

Vidal, F.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

von Vacano, B.

Walmsley, I.

Walmsley, I. A.

Warren, W.

Wasylczyk, P.

Weiner, A.

Weiner, A. M.

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Ultrafast double-pulse ablation of fused silica,” Appl. Phys. Lett. 86, 151110 (2005).
[CrossRef]

Windeler, R.

Witting, T.

Wyatt, A.

Xu, L.

Xu, X.

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Ultrafast double-pulse ablation of fused silica,” Appl. Phys. Lett. 86, 151110 (2005).
[CrossRef]

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. B 74, 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. B 74, S225-S229 (2002).
[CrossRef]

Zavelani-Rossi, M.

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Zeek, E.

Appl. Opt. (1)

Appl. Phys. B (3)

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER,” Appl. Phys. B 70, S85-S93 (2000).
[CrossRef]

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. B 74, S225-S229 (2002).
[CrossRef]

M. Zavelani-Rossi, D. Polli, G. Cerullo, S. De Silvestri, L. Gallmann, G. Steinmeyer, and U. Keller, “Few-optical-cycle laser pulses by OPA: broadband chirped mirror compression and SPIDER characterization,” Appl. Phys. B 74, S245-S251 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Ultrafast double-pulse ablation of fused silica,” Appl. Phys. Lett. 86, 151110 (2005).
[CrossRef]

J. Opt. Soc. Am. B (8)

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]

M. E. Anderson, T. Witting, and I. A. Walmsley, “Gold-SPIDER: spectral phase interferometry for direct electric field reconstruction utilizing sum-frequency generation from a gold surface,” J. Opt. Soc. Am. B 25, A13-A16 (2008).
[CrossRef]

D. Keusters, H. Tan, P. O'Shea, E. Zeek, R. Trebino, and W. 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]

D. Lee, P. Gabolde, and R. Trebino, “Toward single-shot measurement of a broadband ultrafast continuum,” J. Opt. Soc. Am. B 25, A34-A40 (2008).
[CrossRef]

C. Dorrer and I. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B 19, 1019-1029 (2002).
[CrossRef]

C. Dorrer and I. Walmsley, “Precision and consistency criteria in spectral phase interferometry for direct electric-field reconstruction,” J. Opt. Soc. Am. B 19, 1030-1038 (2002).
[CrossRef]

C. Dorrer and I. Kang, “Linear self-referencing techniques for short-optical-pulse characterization,” J. Opt. Soc. Am. B 25, A1-A12 (2008).
[CrossRef]

M. Lelek, F. Louradour, A. Barthélémy, C. Froehly, T. Mansourian, L. Mouradian, J.-P. Chambaret, G. Chériaux, and B. Mercier, “Two-dimensional spectral shearing interferometry resolved in time for ultrashort optical pulse characterization,” J. Opt. Soc. Am. B 25, A17-A24 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (10)

A. Radunsky, I. Walmsley, S.-P. Gorza, and P. Wasylczyk, “Compact spectral shearing interferometer for ultrashort pulse characterization,” Opt. Lett. 32, 181-183 (2007).
[CrossRef]

E. Kosik, A. Radunsky, I. Walmsley, and C. Dorrer, “Interferometric technique for measuring broadband ultrashort pulses at the sampling limit,” Opt. Lett. 30, 326-328 (2005).
[CrossRef] [PubMed]

J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31, 2063-2065 (2006).
[CrossRef] [PubMed]

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O'Shea, A. Shreenath, R. Trebino, and R. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174-1176 (2002).
[CrossRef]

A. Wyatt, I. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31, 1914-1916 (2006).
[CrossRef] [PubMed]

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]

J. Möhring, T. Buckup, B. von Vacano, and M. Motzkus, “Parametrically amplified ultrashort pulses from a shaped photonic crystal fiber supercontinuum,” Opt. Lett. 33, 186-188 (2008).
[CrossRef] [PubMed]

A. Weiner, J. Heritage, and J. Salehi, “Encoding and decoding of femtosecond pulses,” Opt. Lett. 13, 300-302 (1988).
[CrossRef] [PubMed]

P. Baum, S. Lochbrunner, and E. Riedle, “Zero-additional-phase SPIDER: full characterization of visible and sub-20-fs ultraviolet pulses,” Opt. Lett. 29, 210-212 (2004).
[CrossRef] [PubMed]

T. Witting, D. R. Austin, and I. A. Walmsley, “Improved ancilla preparation in spectral shearing interferometry for accurate ultrafast pulse characterization,” Opt. Lett. 34, 881-883 (2009).
[CrossRef] [PubMed]

Phys. Plasmas (1)

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, H. Pépin, and H. P. Mercure, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Phys. Status Solidi B (1)

S. Linden, H. Giessen, and J. Kuhl, “XFROG--a new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206, 119-124 (1998).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Rev. Sci. Instrum. (1)

G. Stibenz and G. Steinmeyer, “Optimizing spectral phase interferometry for direct electric-field reconstruction,” Rev. Sci. Instrum. 77, 073105 (2006).
[CrossRef]

Science (1)

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

Other (1)

G. A. Jones and J. M. Jones, Elementary Number Theory (Springer, 1998).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

(a) Spectrum of a two-lobed pulse (blue solid curve) and spectrally sheared replicas with small (red dashed curve) and large (green thick curve) shears. [(b),(c)] Amplitude (blue solid curve) and phase (red dashed curve) of interference product for the (b) small and (c) large shears. The variation in the interference products with a peak SNR of 50 is shown by the shaded area.

Fig. 2
Fig. 2

Combining multiply sized shears to achieve a unit spectral displacement along the sampling points (small blue dots). (a) Shears of sizes (relative to the sampling rate) 6, 10, and 15 can be combined to form a unit displacement from the starting point (large blue dot) because the sizes are relatively prime. (b) Shears of sizes 6, 10, and 14 have a common factor of 2, and no integer combination will produce a unit displacement.

Fig. 3
Fig. 3

Schematic of a two-shear reconstruction procedure for a two-lobed spectrum where (a) one of the lobes is wider than the gap and (b) both of the lobes are narrower than the gap. The spectral amplitude (black thick lines) is sampled (black dots) at frequencies separated by the small shear Ω 1 = 1 . The arrows represent connections between different frequencies given by the shears; only a subset of these is shown. The small shear allows concatenation between adjacent samples (blue arrows, labeled “c” for concatenation). The large shear Ω 2 = 2 (red arrows, labeled “b” for bridge) bridges the gap between the lobes. In (a), the absolute phase of the large shear can be registered on one of the lobes (green arrows, labeled “r” for register) but in (b), no sufficiently wide continuous spectral region is available.

Fig. 4
Fig. 4

Modulus of the reconstruction coefficient | β k , m | [Eq. (28) in the text] for a single shear of Ω = 2 π / T (solid blue curve) and for multishear retrieval with Ω 1 = Ω (dashed red curve) and Ω 2 = 5 Ω (dotted green curve).

Fig. 5
Fig. 5

Contour plots of noise suppression factor S versus number of sampling points and shear ratio for (a) two and (b) three shears, with the third shear chosen according to the text. (c) Noise suppression factor versus shear ratio for N = 1024 for two (blue solid curve) and three (red dashed curve) shears.

Fig. 6
Fig. 6

(a) Spectral phase (red thick curve, left axis) and amplitude (blue thin curve, right axis) of the notched pulse. (b) RMS phase variation for single- (blue thin curve) and double- (red thick curve) shear reconstructions.

Fig. 7
Fig. 7

Temporal amplitude of the notched pulse (black curve) and the single- [blue (light) shade] and the double- [red (dark) shade] shear reconstructions. The latter two represent the mean ±1 standard deviation over the whole ensemble.

Fig. 8
Fig. 8

(a) Spectral phase (red thick curve, left axis) and amplitude (blue thin curve, right axis) of the pulse with a spectral phase ripple. (b) RMS phase variation for single- (blue thin curve) and double- (red thick curve) shear reconstructions.

Fig. 9
Fig. 9

SEA-SPIDER setup: test pulse TP, ancilla Anc taken from uncompressed CPA output, beam splitter BS, time delay stage TD used to adjust the shear, F1 optic focusing beams into the crystal, alignment camera C1 used for focusing into the crystal, χ ( 2 ) 30 μ m BBO cut for type II sum-frequency generation, spatial filter SF, focusing optic F2 for reimaging the upconverted beams onto the spectrometer, SP imaging spectrometer.

Fig. 10
Fig. 10

(a) Raw SEA-SPIDER trace. (b) 2D discrete Fourier transform, showing filter passband (black rectangle). (c) Amplitude of the sideband after filtering and inverse Fourier transform.

Fig. 11
Fig. 11

(a) Spectral intensity of the unknown pulse. (b) Phase difference from small (blue thick curve) and large (red thin curve) shears, averaged over the ensemble. (c) Retrieved phase, averaged over the ensemble. (d) RMS variation of the retrieved phase over the ensemble for the single- (blue thick curve) and the double- (red thin curve) shear reconstructions.

Fig. 12
Fig. 12

Temporal amplitude of the single- [blue (light) shade] and the double- [red (dark) shade] shear reconstructions, averaged over the whole ensemble. The region represents the mean ±1 standard deviation. The inset is a zoomed-in view of the shorter subpulse.

Equations (45)

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D ( ω ) = E ( ω + Ω ) E ( ω )
Γ ( ω ) = ϕ ( ω + Ω ) ϕ ( ω ) ,
D ( t ) = 1 2 π e i Ω τ E ( τ ) E ( τ t ) d τ .
σ Γ ( ω ) = σ ξ 2 | D ( ω ) | ,
ϕ n + C k ϕ n = Γ k , n ,
ω ω 0 = a 1 Ω 1 + a 2 Ω 2 + .
a 1 C 1 + a 2 C 2 + = n .
Γ k , n = Arg [ D k , n   exp ( i η k ) ] + 2 π u k , n ,
η 1 = Arg n | D 1 ( ω n ) | 2   exp [ i   Arg   D 1 ( ω n ) ] .
ϕ ̂ n = j = 0 n 1 Γ 1 , j .
Γ ̂ k , n = ϕ ̂ n + C k ϕ ̂ n = Γ 1 , n + Γ 1 , n + 1 + + Γ 1 , n + C k 1
σ arg   D k , n = σ ξ 2 | D k , n | ,
σ Γ ̂ k , n 2 = σ ξ 2 2 [ | D 1 , n | 2 + | D 1 , n + 1 | 2 + + | D 1 , n + C k 1 | 2 ] .
Γ ̂ k , n Arg   D k , n = η k     mod   2 π ,
ρ k , n 2 = σ Γ ̂ k , n 2 + σ ξ 2 2 | D k , n | 2 .
min η k n | exp ( i Γ ̂ k , n i   Arg   D k , n ) exp ( i η k ) | 2 ρ k , n 2 ,
η k = Arg n exp [ i   Arg   D k , n i Γ ̂ k , n ] ρ k , n 2 .
u k , n = round [ Γ ̂ k , n Arg   D k , n + η k 2 π ] .
G k = [ 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 ] ,
W k G k ϕ ¯ = W k Γ ¯ k ,
B ϕ ¯ = F ¯ ,
B = [ W 1 G 1 W 2 G 2 W M G M ] ,     F ¯ = [ W 1 Γ ¯ 1 W 2 Γ ¯ 2 W M Γ ¯ M ] .
P = M N k = 1 M C k ,
R = k = 1 M n = 0 N 1 | ϕ n + C k ϕ n Γ k , n | 2 Ω .
R = k = 1 M m = 0 N 1 | ϕ ̃ m ( e i Ω k t m 1 ) Γ ̃ k , m | 2 2 π / B .
ϕ ̃ m = k = 1 M Γ ̃ k , m ( e i t m Ω k 1 ) 2 k = 1 M 1 cos   Ω k t m .
ϕ ̃ m = Γ ̃ m ( e i t m Ω 1 ) ,
β k , m = ( e i t m Ω k 1 ) 2 k = 1 M 1 cos   Ω k t m ,
A 2 [ { Ω k } ] = 1 4 T m = 1 N 1 k ( e i t m Ω k 1 ) 2 [ k ( 1 cos   t m Ω k ) ] 2 .
S = A [ Ω ] A [ { Ω k = Ω } ] M α SNR .
ϵ = [ 1 J j E j E ¯ 2 ] 1 / 2 ,
σ ϕ 2 = | exp ( i ϕ ) exp ( i ϕ ) | 2 .
arg   D ( ω ) = arg [ D ¯ ( ω ) + ξ ( ω ) ] ,
= arg   D ¯ ( ω ) + I [ ξ ( ω ) D ¯ ( ω ) ] + O ( [ ξ ( ω ) D ¯ ( ω ) ] 2 ) ,
σ arg   D ( ω ) 2 = | I [ ξ ( ω ) D ¯ ( ω ) ] | 2 .
σ arg   D ( ω ) 2 = 1 2 ( | ξ ( ω ) | 2 | D ¯ ( ω ) | 2 R [ ξ ( ω ) 2 D ¯ 2 ( ω ) ] ) ,
ξ ( ω ) = 1 2 π d t ζ ̃ ( t ) F ( t ) e i ω t ,
ζ ̃ ( t ) = 1 2 π d ω ζ ( ω ) e i ω t
ξ 2 ( ω ) = 1 2 π d t d t ζ ̃ ( t ) ζ ̃ ( t ) F ( t ) F ( t ) e i ω ( t + t ) .
ξ 2 ( ω ) = 1 2 π d t d t ζ ̃ ( t ) ζ ̃ ( t ) F ( t ) F ( t ) e i ω ( t + t ) .
ζ ̃ ( t ) ζ ̃ ( t ) = | ζ ̃ | 2 δ ( t t ) .
ξ 2 ( ω ) = | ζ ̃ | 2 2 π d t F ( t ) F ( t ) .
σ arg   D ( ω ) 2 = | ξ | 2 2 | D ¯ ( ω ) | 2 ,
I ξ ( ω 1 ) D ¯ ( ω 1 ) I ξ ( ω 2 ) D ¯ ( ω 2 ) = 1 2 [ R ξ ( ω 1 ) ξ ( ω 2 ) D ¯ ( ω 1 ) D ¯ ( ω 2 ) R ξ ( ω 1 ) ξ ( ω 2 ) D ¯ ( ω 1 ) D ¯ ( ω 2 ) ] .
ξ ( ω 1 ) ξ ( ω 2 ) = | ζ ̃ | 2 2 π d t | F ( t ) | 2 e i ( ω 1 ω 2 ) t .

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