Abstract

Ultrashort-pulse characterization techniques, such as the numerous variants of frequency-resolved optical gating (FROG) and spectral phase interferometry for direct electric-field reconstruction, fail to fully determine the relative phases of well-separated frequency components. If well-separated frequency components are also well separated in time, the cross-correlation variants (e.g., XFROG) succeed, but only if short, well-characterized gate pulses are used.

© 2003 Optical Society of America

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    [Crossref]
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  17. Mode-locked pulse trains have this structure, but we are not considering mode-locked pulse trains here; the short range of relative delays used in most measurements allows for the measurement of only one pulse in such a train, removing the modes from the pulse spectrum.
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    [Crossref]
  26. An example of a pulse with well-separated frequency components whose peaks in the spectrogram do overlap is a three-component pulse, described by E(ω)=|A˜A(ω-ωA)|exp[iϕ˜A(ω-ωA)]exp(iϕA)+|A˜B(ω-ωB)|exp[iϕ˜B(ω-ωB)]exp(iϕB)+|A˜C(ω-ωC)|exp[iϕ˜C(ω-ωC)]exp(iϕC). The second-harmonic FROG spectrogram of this pulse will have peaks, among others, centered at ωA+ωCand 2ωBwith phases ϕA+ϕCand 2ϕB,respectively. When ωA+ωC=2ωB,these two contributions to the peak will beat against each other with fringes dependent on 2ϕB-(ϕA+ϕC).If (ϕB-ϕC)is known, then (ϕB-ϕA)can be deduced and vice versa. This will also be true if components Aand Chave a similar delay that is different from that of component B.
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    [Crossref]
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    [Crossref]
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    [Crossref]

2002 (4)

X. Gu, L. Xu, M. Kimmel, P. O’Shea, A. P. Shreenath, R. Trebino, and R. S. 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. 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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (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]

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002).
[Crossref]

2001 (3)

2000 (2)

1999 (6)

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10,934–10,956 (1999).
[Crossref]

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[Crossref]

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[Crossref]

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. 35, 501–508 (1999).
[Crossref]

I. A. Walmsley, “Measuring ultrafast optical pulses using spectral interferometry,” Opt. Photonics News 10(4), 28–33 (1999).
[Crossref]

K. Sundermann and R. de Vivie-Riedle, “Extensions to quantum optimal control algorithms and applications to special problems in state selective molecular dynamics,” J. Chem. Phys. 110, 1896–1904 (1999).
[Crossref]

1998 (2)

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Heterodyne-detected stimulated photon echo: applications to optical dynamics in solution,” Chem. Phys. Lett. 233, 287–309 (1998).

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]

1997 (2)

J. X. Tull, M. A. Dugan, and W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[Crossref]

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

1996 (1)

1995 (2)

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Short-time solvation dynamics probed by phase-locked heterodyne detected pump–probe,” Chem. Phys. Lett. 247, 264–270 (1995).
[Crossref]

L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995).
[Crossref]

1994 (2)

1992 (2)

J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992).
[Crossref]

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[Crossref]

1991 (2)

J. L. A. Chilla and O. E. Martı́nez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16, 39–41 (1991).
[Crossref] [PubMed]

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

1990 (1)

1971 (1)

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

Albrecht, A. W.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10,934–10,956 (1999).
[Crossref]

Alphonse, G. A.

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[Crossref]

Boyd, R. W.

Carlson, R. J.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (2002).
[Crossref]

Chériaux, G.

Chilla, J. L. A.

Cina, J. A.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Connolly, J. C.

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[Crossref]

Davis, J.

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[Crossref]

Davis, J. C.

de Boeij, W. P.

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Heterodyne-detected stimulated photon echo: applications to optical dynamics in solution,” Chem. Phys. Lett. 233, 287–309 (1998).

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Short-time solvation dynamics probed by phase-locked heterodyne detected pump–probe,” Chem. Phys. Lett. 247, 264–270 (1995).
[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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (2002).
[Crossref]

de Vivie-Riedle, R.

K. Sundermann and R. de Vivie-Riedle, “Extensions to quantum optimal control algorithms and applications to special problems in state selective molecular dynamics,” J. Chem. Phys. 110, 1896–1904 (1999).
[Crossref]

Delfyett, P. J.

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[Crossref]

DeLong, K. W.

Du, M.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Dugan, M. A.

J. X. Tull, M. A. Dugan, and W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[Crossref]

Fainman, Y.

Fetterman, M.

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[Crossref]

Fetterman, M. R.

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Fleming, G. R.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Foing, J.-P.

J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992).
[Crossref]

Fuji, T.

A. Yabushita, T. Fuji, and T. Kobayashi, “SHG FROG and XFROG methods for phase/intensity characterization of pulses propagated through an absorptive optical medium,” Opt. Commun. 198, 227–232 (2001).
[Crossref]

Gallagher Faeder, S. M.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10,934–10,956 (1999).
[Crossref]

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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (2002).
[Crossref]

Gee, S.

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[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]

Goswami, D.

M. R. Fetterman, J. C. Davis, H.-S. Tan, W. Yang, D. Goswami, J.-K. Rhee, and W. S. Warren, “Fast-frequency-hopping modulation and detection demonstration,” J. Opt. Soc. Am. B 18, 1372–1376 (2001).
[Crossref]

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[Crossref]

Gu, X.

He, G. S.

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002).
[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. B 74, S225–S229 (2002).
[Crossref]

Huang, F.

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[Crossref]

Hunter, J.

Hybl, J. D.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10,934–10,956 (1999).
[Crossref]

Iaconis, C.

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. 35, 501–508 (1999).
[Crossref]

Joffre, M.

L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995).
[Crossref]

J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992).
[Crossref]

Jonas, D. M.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10,934–10,956 (1999).
[Crossref]

Kane, D. J.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

K. W. DeLong, R. Trebino, and D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B 11, 1595–1608 (1994).
[Crossref]

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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (2002).
[Crossref]

Keusters, D.

D. Keusters, H.-S. Tan, and W. S. Warren, “The failure of nonlinear pulse shape detection,” in Nonlinear Optics, Vol. 72 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002), pp. PD3-1–PD3-2.

Kimmel, M.

Kobayashi, T.

A. Yabushita, T. Fuji, and T. Kobayashi, “SHG FROG and XFROG methods for phase/intensity characterization of pulses propagated through an absorptive optical medium,” Opt. Commun. 198, 227–232 (2001).
[Crossref]

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[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]

Leaird, D. E.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[Crossref]

Lepetit, L.

Likforman, J.-P.

J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992).
[Crossref]

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]

Malcuit, M. S.

Marom, D.

Marti´nez, O. E.

Matro, A.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Migus, A.

J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992).
[Crossref]

Miller, E. 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.

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]

Nitta, I.

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[Crossref]

Norris, T. B.

O’Shea, P.

Paek, E. G.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[Crossref]

Panasenko, D.

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (2002).
[Crossref]

Proch, D.

Pshenichnikov, M. S.

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Heterodyne-detected stimulated photon echo: applications to optical dynamics in solution,” Chem. Phys. Lett. 233, 287–309 (1998).

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Short-time solvation dynamics probed by phase-locked heterodyne detected pump–probe,” Chem. Phys. Lett. 247, 264–270 (1995).
[Crossref]

Reitze, D. H.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[Crossref]

Rhee, J.-K.

Rice, S. A.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Rokitski, R.

Romerorochin, V.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Ruggiero, A. J.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Scherer, N. F.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Shi, H.

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[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. B 74, S225–S229 (2002).
[Crossref]

Shkunov, V. V.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

Shreenath, A. P.

Sosnowski, T. S.

Steinmeyer, 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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (2002).
[Crossref]

Sun, P.-C.

Sundermann, K.

K. Sundermann and R. de Vivie-Riedle, “Extensions to quantum optimal control algorithms and applications to special problems in state selective molecular dynamics,” J. Chem. Phys. 110, 1896–1904 (1999).
[Crossref]

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Tan, H.-S.

M. R. Fetterman, J. C. Davis, H.-S. Tan, W. Yang, D. Goswami, J.-K. Rhee, and W. S. Warren, “Fast-frequency-hopping modulation and detection demonstration,” J. Opt. Soc. Am. B 18, 1372–1376 (2001).
[Crossref]

D. Keusters, H.-S. Tan, and W. S. Warren, “The failure of nonlinear pulse shape detection,” in Nonlinear Optics, Vol. 72 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002), pp. PD3-1–PD3-2.

Tien, A.-C.

Treacy, E. B.

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

Trebino, R.

Tull, J. X.

J. X. Tull, M. A. Dugan, and W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[Crossref]

Walmsley, I. A.

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. 35, 501–508 (1999).
[Crossref]

I. A. Walmsley, “Measuring ultrafast optical pulses using spectral interferometry,” Opt. Photonics News 10(4), 28–33 (1999).
[Crossref]

Warren, W. S.

M. R. Fetterman, J. C. Davis, H.-S. Tan, W. Yang, D. Goswami, J.-K. Rhee, and W. S. Warren, “Fast-frequency-hopping modulation and detection demonstration,” J. Opt. Soc. Am. B 18, 1372–1376 (2001).
[Crossref]

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[Crossref]

J. X. Tull, M. A. Dugan, and W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[Crossref]

D. Keusters, H.-S. Tan, and W. S. Warren, “The failure of nonlinear pulse shape detection,” in Nonlinear Optics, Vol. 72 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002), pp. PD3-1–PD3-2.

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[Crossref]

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[Crossref]

White, W. E.

Wiersma, D. A.

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Heterodyne-detected stimulated photon echo: applications to optical dynamics in solution,” Chem. Phys. Lett. 233, 287–309 (1998).

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Short-time solvation dynamics probed by phase-locked heterodyne detected pump–probe,” Chem. Phys. Lett. 247, 264–270 (1995).
[Crossref]

Windeler, R. S.

Witte, T.

Xu, L.

Yabushita, A.

A. Yabushita, T. Fuji, and T. Kobayashi, “SHG FROG and XFROG methods for phase/intensity characterization of pulses propagated through an absorptive optical medium,” Opt. Commun. 198, 227–232 (2001).
[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]

Yang, W.

M. R. Fetterman, J. C. Davis, H.-S. Tan, W. Yang, D. Goswami, J.-K. Rhee, and W. S. Warren, “Fast-frequency-hopping modulation and detection demonstration,” J. Opt. Soc. Am. B 18, 1372–1376 (2001).
[Crossref]

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (2002).
[Crossref]

Zeidler, D.

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

Adv. Magn. Opt. Reson. (1)

J. X. Tull, M. A. Dugan, and W. S. Warren, “High-resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. 20, 1–65 (1997).
[Crossref]

Appl. Phys. B (2)

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 compres-sions and SPIDER characterization,” Appl. Phys. B 74, S245–S251 (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]

Chem. Phys. Lett. (2)

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Short-time solvation dynamics probed by phase-locked heterodyne detected pump–probe,” Chem. Phys. Lett. 247, 264–270 (1995).
[Crossref]

W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Heterodyne-detected stimulated photon echo: applications to optical dynamics in solution,” Chem. Phys. Lett. 233, 287–309 (1998).

IEEE J. Quantum Electron. (4)

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[Crossref]

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. 35, 501–508 (1999).
[Crossref]

J.-P. Foing, J.-P. Likforman, M. Joffre, and A. Migus, “Femtosecond pulse phase measurement by spectrally resolved up-conversion: application to continuum compression,” IEEE J. Quantum Electron. 28, 2285–2290 (1992).
[Crossref]

P. J. Delfyett, H. Shi, S. Gee, I. Nitta, J. C. Connolly, and G. A. Alphonse, “Joint time-frequency measurements of mode-locked semiconductor diode lasers and dynamics using frequency resolved optical gating,” IEEE J. Quantum Electron. 35, 487–500 (1999).
[Crossref]

IEEE Photonics Technol. Lett. (1)

W. Yang, F. Huang, M. Fetterman, J. Davis, D. Goswami, and W. S. Warren, “Real time adaptive amplitude feedback in an AOM–based ultrafast optical pulse shaping system,” IEEE Photonics Technol. Lett. 11, 1665–1667 (1999).
[Crossref]

J. Appl. Phys. (1)

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

J. Chem. Phys. (3)

K. Sundermann and R. de Vivie-Riedle, “Extensions to quantum optimal control algorithms and applications to special problems in state selective molecular dynamics,” J. Chem. Phys. 110, 1896–1904 (1999).
[Crossref]

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romerorochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry—time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

A. W. Albrecht, J. D. Hybl, S. M. Gallagher Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10,934–10,956 (1999).
[Crossref]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

A. Yabushita, T. Fuji, and T. Kobayashi, “SHG FROG and XFROG methods for phase/intensity characterization of pulses propagated through an absorptive optical medium,” Opt. Commun. 198, 227–232 (2001).
[Crossref]

Opt. Lett. (5)

Opt. Photonics News (1)

I. A. Walmsley, “Measuring ultrafast optical pulses using spectral interferometry,” Opt. Photonics News 10(4), 28–33 (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]

Prog. Quantum Electron. (1)

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quantum Electron. 26, 131–191 (2002).
[Crossref]

Rev. Sci. Instrum. (2)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929–1960 (2000).
[Crossref]

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Other (5)

R. Trebino, Frequency-Resolved Optical Gating: the Measurement of Ultrashort Laser Pulses (Kluwer Scientific, Boston, Mass., 2002).

D. Keusters, H.-S. Tan, and W. S. Warren, “The failure of nonlinear pulse shape detection,” in Nonlinear Optics, Vol. 72 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002), pp. PD3-1–PD3-2.

Mode-locked pulse trains have this structure, but we are not considering mode-locked pulse trains here; the short range of relative delays used in most measurements allows for the measurement of only one pulse in such a train, removing the modes from the pulse spectrum.

An example of a pulse with well-separated frequency components whose peaks in the spectrogram do overlap is a three-component pulse, described by E(ω)=|A˜A(ω-ωA)|exp[iϕ˜A(ω-ωA)]exp(iϕA)+|A˜B(ω-ωB)|exp[iϕ˜B(ω-ωB)]exp(iϕB)+|A˜C(ω-ωC)|exp[iϕ˜C(ω-ωC)]exp(iϕC). The second-harmonic FROG spectrogram of this pulse will have peaks, among others, centered at ωA+ωCand 2ωBwith phases ϕA+ϕCand 2ϕB,respectively. When ωA+ωC=2ωB,these two contributions to the peak will beat against each other with fringes dependent on 2ϕB-(ϕA+ϕC).If (ϕB-ϕC)is known, then (ϕB-ϕA)can be deduced and vice versa. This will also be true if components Aand Chave a similar delay that is different from that of component B.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).

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Figures (7)

Fig. 1
Fig. 1

Comparison of the spectrum, the intensity profile, and autocorrelation of pulses described by Eq. (4) for ϕA-ϕB=0 (solid curves) and ϕA-ϕB=0.6π (dashed curves). (a) Components of a pulse are well separated in both the time and the frequency domains (Tδt and Δωδω). (b) Components overlap in the time domain but are well separated in the frequency domain (T=0 and Δωδω). (c) Components are well separated in the time domain but overlap in the frequency domain (Tδt and Δω=0). If the components do not overlap in the frequency domain [(a) and (b)], then the spectrum and the autocorrelation are identical for every ϕA-ϕB. Only if the components of the pulse overlap in the frequency domain [(c)] can the phase relation between the components be determined from the spectrum.

Fig. 2
Fig. 2

Comparison of SHG FROG spectrograms for pulses described by Eq. (4) with ϕA-ϕB=0 (left) and ϕA-ϕB=0.6π (right). (a) SHG FROG spectrogram for a pulse with components well separated in the time domain and in the frequency domain. (b) Spectrogram for a pulse with components well separated in the frequency domain but overlapping in the time domain. (c) Spectrogram for a pulse with components well separated in the time domain but overlapping in the frequency domain. The spectrograms for pulses that are well separated in the frequency domain [(a) and (b)] are identical for all phase differences ϕA-ϕB. Only if the pulses overlap in the frequency domain [(c)] can the SHG FROG determine phase difference ϕA-ϕB.

Fig. 3
Fig. 3

Comparison of PG FROG spectrograms for pulses described by Eq. (4). (a) PG FROG spectrogram for a pulse with components separated both in the time domain and in the frequency domain. (b) Spectrogram for a pulse with components well separated in the frequency domain but arriving at the same time. (c) Spectrogram for a pulse with components at the same frequency but well separated in the time domain, with ϕA-ϕB=0. (d) Same as (c) but with ϕA-ϕB=0.6π. The spectrograms for pulses well separated in the frequency domain [(a) and (b)] are identical for all phase differences ϕA-ϕB. Only if the pulses overlap in the frequency domain can the PG FROG determine the phase difference ϕA-ϕB.

Fig. 4
Fig. 4

Comparison of SFG XFROG spectrogram for pulses described by Eq. (4). (a) SFG XFROG spectrogram for pulse with components well separated in the frequency domain but overlapping in time, with ϕA-ϕB=0. (b) Same as (a) but with ϕA-ϕB=0.6π. Here a reference pulse is used that is just short enough to cause the two peaks in the spectrogram to overlap such that interference fringes can be observed. The position of the fringes permits determination of ϕA-ϕB. If a longer reference is used, the two peaks in the spectrogram do not overlap, and ϕA-ϕB cannot be determined. (c) The same reference as for (a) and (b) was used but the components of the pulse were well separated both in the time domain and in the frequency domain. Now the spectrogram is identical for all phase differences ϕA-ϕB.

Fig. 5
Fig. 5

SFG XFROG diagram for the pulses depicted in Fig. 4(c) (separated both in frequency and in time), with a chirped reference pulse. If the amount of chirp imposed on the reference is (a) not enough or (c) too much, no interference is observed, and the phase difference between the components cannot be determined. Only if the pulse is chirped by the correct amount (b) can an interference pattern be observed that permits determination of ϕA-ϕB.

Fig. 6
Fig. 6

XSPIDER spectrum for small spectral shear (left) and for large spectral shear (right), for pulses with ϕA-ϕB=0 and ϕA-ϕB=0.6π. The components are well separated both in time and in frequency. For small spectral shear, the two peaks in the SPIDER spectrum correspond to the SPIDER spectra of the individual components, and the spectrum is identical for all ϕA-ϕB. If the large spectral shear is chosen exactly correctly, the central peak in the spectrogram gives information about ϕA-ϕB (inset), although now information about the phase within each component is not available.

Fig. 7
Fig. 7

Spectrogram of SPC SFG FROG for a pulse consisting of three components, all arriving at different times and with different center frequencies. In the spectrogram at the left, the phase difference among all components is zero; in the spectrogram at the right the phase difference between the first and the second components is 0.3π and between the first and third components is 0.6π. From the shift in the fringes, these phase differences can be recovered.

Tables (1)

Tables Icon

Table 1 Summary of Capability of Different Pulse Characterization Methods to Determine the Phase Difference Between Well-Separated Components of an Ultrafast Laser Pulse

Equations (35)

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E(ω)=A˜(ω)exp[iφ˜(ω)]+c.c.,
E(t)=AA(t)exp(-iωAt)exp(iϕA)+AB(t-T)exp(-iωBt)exp(iϕB),
E(ω)=|A˜A(ω-ωA)|exp[iϕ˜A(ω-ωA)]exp(iϕA)+|A˜B(ω-ωB)|exp[iϕ˜B(ω-ωB)]×exp(iϕB),
E(t)=exp(-α2t2)exp(-iωAt)exp(iϕA)+exp[-α2(t-T)2]exp(-iωBt)exp(iϕB).
ISHG(Ω, τ)=E(t)E(t-τ)exp(iΩt)dt2.
ISHG(Ω, τ)=exp(iωAτ)exp(2iϕA)×AA(t)AA(t-τ)exp[i(Ω-2ωA)t]dt
+exp(iωBτ)exp[i(ϕA+ϕB)]×AA(t)AB(t-T-τ)×exp[i(Ω-ωA-ωB)t]dt
+exp(iωAτ)exp[i(ϕA+ϕB)]×AB(t-T)AA(t-τ)×exp[i(Ω-ωA-ωB)t]dt
+exp(iωBτ)exp(2iϕB)×AB(t-T)AB(t-T-τ)×exp[i(Ω-2ωB)t]dt2.
ISHG(Ω, τ)=E(ω)E(Ω-ω)exp(iωτ)dω2,
IPG(Ω, τ)=E(t)|E(t-τ)|2exp(iΩt)dt.
IPG(Ω, τ)=exp(iϕA)AA(t)|AA(t-τ)|2exp[i(Ω-ωA)t]dt+exp(iϕA)AA(t)|AB(t-T-τ)|2exp[i(Ω-ωA)t]dt+exp[i(ωA-ωB)τ]exp(iϕA)AB(t-T)AA(t-τ)AB*(t-T-τ)exp[i(Ω-ωA)t]dt+exp(iϕB)AB(t-T)|AB(t-T-τ)|2exp[i(Ω-ωB)t]dt+exp(iϕB)AB(t-T)|AA(t-τ)|2exp[i(Ω-ωB)t]dt+exp[i(ωB-ωA)τ]exp(iϕB)AA(t)AB(t-T-τ)AA*(t-τ)exp[i(Ω-ωB)t]dt+exp[i(ωA-ωB)τ]exp[i(2ϕA-ϕB)]AA(t)AA(t-τ)AB*(t-T-τ)exp[i{Ω-(2ωA-ωB)}t]dt+exp[i(ωB-ωA)τ]exp[i(2ϕB-ϕA)]AB(t-T)AB(t-T-τ)×AA*(t-τ)exp{i[Ω-(2ωB-ωA]t}dt2.
ISPIDER(ω)
=|E(ω-ω0)E(ω0)+E(ω-ω0-Ω)E(ω0+Ω)×exp(-iϕS)exp(iωτ)|2
=|E(ω-ω0)E(ω0)|2+|E(ω-ω0-Ω)E(ω0+Ω)|2+2 Re[E(ω-ω0)×E*(ω-ω0-Ω)E(ω0)E*(ω0+Ω)exp(iϕS)exp(-iωτ)].
2 Re({A˜A(ω-ωA-ω0)exp[iϕ˜A(ω-ωA-ω0)]exp(iϕA)+A˜B(ω-ωB-ω0)exp[iϕ˜B(ω-ωB-ω0)]exp(iϕB)}×{A˜A(ω-ωA-ω0-Ω)exp[-iϕ˜A(ω-ωA-ω0-Ω)]exp(-iϕA)+A˜B(ω-ωB-ω0-Ω)exp[-iϕ˜B(ω-ωB-ω0-Ω)]exp(-iϕB)}×{A˜A(ω0-ωA)exp[iϕ˜A(ω0-ωA)]exp(iϕA)+A˜B(ω0-ωB)exp[iϕ˜B(ω0-ωB)]exp(iϕB)}×{A˜A(ω0+Ω-ωA)exp[-iϕ˜A(ω0+Ω-ωA)]exp(-iϕA)+A˜B(ω0+Ω-ωB)×exp[-iϕ˜B(ω0+Ω-ωB)exp(-iϕB)}exp(iϕS)exp(-iωτ)).
2 Re({A˜A(ω-ωA-ω0)exp[iϕ˜A(ω-ωA-ω0)]exp(iϕA)A˜A(ω-ωA-ω0-Ω)exp[-iϕ˜A(ω-ωA-ω0-Ω)]×exp(-iϕA)+A˜A(ω-ωA-ω0)exp[iϕ˜A(ω-ωA-ω0)]exp(iϕA)A˜B(ω-ωB-ω0-Ω)×exp[-iϕ˜B(ω-ωB-ω0-Ω)]exp(-iϕB)+A˜B(ω-ωB-ω0)exp[iϕ˜B(ω-ωB-ω0)]×exp(iϕB)A˜A(ω-ωA-ω0-Ω)exp[-iϕ˜A(ω-ωA-ω0-Ω)]exp(-iϕA)+A˜B(ω-ωB-ω0)×exp[iϕ˜B(ω-ωB-ω0)]exp(iϕB)+A˜B(ω-ωB-ω0-Ω)exp[-iϕ˜B(ω-ωB-ω0-Ω)]exp(-iϕB)}×A˜A(ω0-ωA)exp[iϕ˜A(ω0-ωA)]exp(iϕA)A˜A(ω0+Ω-ωA)exp[-iϕ˜A(ω0+Ω-ωA)]exp(-iϕA)×exp(iϕS)exp(-iωτ)).
IXSFG(Ω, τ)=E(t)EX(t-τ)exp(iΩt)dt2,
IXSFG(Ω, τ)=exp(iω0τ)exp(iϕA)AA(t)AX(t-τ)×exp[i(Ω-ωA-ω0)t]dt+exp(iω0τ)exp(iϕB)AB(t-T)×AX(t-τ)exp[i(Ω-ωB-ω0)t]dt2,
IXSPIDER(ω)=|E(ω-ω0)EX(ω0)|2+|E(ω-ω0-Ω)EX(ω0+Ω)|2+2 Re[E(ω-ω0)E*(ω-ω0-Ω)EX(ω0)EX*(ω0+Ω)exp(iϕS)exp(-iωτ)].
2 Re({A˜A(ω-ωA-ω0)exp[iϕ˜A(ω-ωA-ω0)]exp(iϕA)A˜A(ω-ωA-ω0-Ω)×exp[-iϕ˜A(ω-ωA-0-Ω)]exp(-iϕA)+A˜B(ω-ωB-ω0)exp[iϕ˜B(ω-ωB-ω0)]×exp(iϕB)A˜B(ω-ωB-ω0-Ω)exp[-iϕ˜B(ω-ωB-ω0-Ω)]exp(-iϕB)}×A˜X(ω0)exp[iφ˜X(ω0)]A˜X(ω0+Ω)exp[-iφ˜X(ω0+Ω)]exp(iϕS)exp(-iωτ)).
2 Re{A˜A(ω-ωA-ω0)exp[iϕ˜A(ω-ωA-ω0)]exp(iϕA)A˜B(ω-ωB-ω0-Ω)×exp[-iϕ˜B(ω-ωB-ω0-Ω)]exp(-iϕB)A˜X(ω0)exp[iφ˜X(ω0)]A˜X(ω0+Ω)×exp[-iφ˜X(ω0+Ω)]exp(-iϕS)exp(-iωτ)=|A˜A(ω-ωA-ω0)A˜B(ω-ωB-ω0-Ω)A˜X(ω0)A˜X(ω0+Ω)|×cos[ϕ˜A(ω-ωA-ω0)-ϕ˜B(ω-ωB-ω0-Ω)+ϕA-ϕB+φ˜X(ω0)-φ˜X(ω0+Ω)+ϕS-ωτ].
ISF-SPC(Ω, τ)=E(t)ESPC(t-τ)exp(iΩt)dt2.
ESPC(ω)=E*(ω),
ESPC(t)=ASPC(t)exp(-iωt)=A*(-t)exp(-iωt).
ESPC(t)=AA*(-t)exp(-iωAt)exp(-iϕA)+AB*(-t-T)exp(-iωBt)exp(-iϕB).
ISF-SPC(Ω, τ)=exp(iωAτ)AA(t)AA*(-t-τ)exp[i(Ω-2ωA)t]dt+exp(iωBτ)exp[i(ϕA-ϕB)]AA(t)AB*(-t-T-τ)exp[i(Ω-ωA-ωB)t]dt+exp(iωAτ)exp[i(-ϕA+ϕB]AB(t-T)AA*(-t-τ)exp[i(Ω-ωA-ωB)t]dt+exp(iωBτ)AB(t-T)AB*(-t-T-τ)exp[i(Ω-2ωB)t]dt2.
E(t)=A(t)exp(-iωAt)exp(iϕA)+B(t-T)×exp(-iωB)(t-T)exp(iϕB);
E˜(ω)=˜A(ω-ωA)exp(iϕA)+˜B(ω-ωB)×exp[i(ω-ωB)T]exp(iϕB),
-ωAt+ϕA+ω0t,
-ωB(t-T)+ϕB+ω0t.
E(t)=A(t)exp(-iωAt)exp(iϕA)+B(t-T)×exp[i(ωB-ω0)T]exp(-iωBt)exp(iϕB).
AA(t)=A(t),
AB(t-T)=B(t-T)exp[-i(ωB-ω0)T].
E(ω)=|A˜A(ω-ωA)|exp[iϕ˜A(ω-ωA)]exp(iϕA)+|A˜B(ω-ωB)|exp[iϕ˜B(ω-ωB)]exp(iϕB)+|A˜C(ω-ωC)|exp[iϕ˜C(ω-ωC)]exp(iϕC).

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