Abstract

Maxwell’s equations for the apparently complicated generation and propagation of femtosecond four-wave-mixing signals in optically thick samples can be solved by triple Fourier transformation into the three-dimensional (3D) frequency domain. Given the linear absorption and refractive-index spectra, the propagation problem can be solved in three dimensions under the assumption that nonlinear distortions of the excitation pulses can be neglected. A propagation function exactly incorporates the linear evolution of the excitation pulses, the nonlinear generation of the signal, and the linear propagation of the signal. A quantitative treatment of the directional filtering of the 3D susceptibility that arises from excitation with noncollinear pulses and selective interference detection of signal in one phase-matched direction is developed. This 3D treatment is used to examine the influence of phase-matching bandwidth, directional filtering, and sample absorption on femtosecond four-wave-mixing signals in the rectangular and square boxcars phase-matching geometries.

© 2005 Optical Society of America

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2004 (1)

2003 (7)

J. A. Gruetzmacher and N. F. Scherer, "Finite-difference time-domain simulation of ultrashort pulse propagation incorporating quantum mechanical response functions," Opt. Lett. 28, 573-575 (2003).
[CrossRef] [PubMed]

D. M. Jonas, "Two-dimensional femtosecond spectroscopy," Annu. Rev. Phys. Chem. 54, 425-463 (2003).
[CrossRef] [PubMed]

D. M. Jonas, "Optical analogs of 2D NMR," Science 300, 1515-1517 (2003).
[CrossRef] [PubMed]

M. Khalil, N. Demirdöven, and A. Tokmakoff, "Obtaining absorptive lineshapes in two-dimensional infrared vibrational correlation spectra," Phys. Rev. Lett. 90, 047401 (2003).
[CrossRef]

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

S. Yeremenko, M. S. Pshenichnikov, and D. A. Wiersma, "Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo," Chem. Phys. Lett. 367, 107-113 (2003).
[CrossRef]

D. Keusters and W. S. Warren, "Effect of pulse propagation on the two-dimensional photon echo spectrum of multilevel systems," J. Chem. Phys. 119, 4478-4489 (2003).
[CrossRef]

2002 (2)

S. Mukamel and A. Tortschanoff, "Multiple quantum coherences in liquid state NMR and nonlinear optics: collective vs. local origin," Chem. Phys. Lett. 357, 327-335 (2002).
[CrossRef]

N. Belabas and M. Joffre, "Visible-infrared two-dimensional Fourier-transform spectroscopy," Opt. Lett. 27, 2043-2045 (2002).
[CrossRef]

2001 (1)

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, "Two dimensional Fourier transform electronic spectroscopy," J. Chem. Phys. 115, 6606-6622 (2001).
[CrossRef]

2000 (4)

D. E. Thompson and J. C. Wright, "Model for spectral artifacts in two-dimensional four-wave mixing spectra from absorption and refractive index dispersion at infrared resonances," J. Phys. Chem. A 104, 11282-11289 (2000).
[CrossRef]

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-592 (2000).
[CrossRef]

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, "Spectral resolution and sampling issues in Fourier-transform spectral interferometry," J. Opt. Soc. Am. B 17, 1795-1802 (2000).
[CrossRef]

1999 (3)

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, 10934-10956 (1999).
[CrossRef]

A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, "Second-harmonic generation frequency resolved optical gating in the single-cycle regime," IEEE J. Quantum Electron. 35, 459-478 (1999).
[CrossRef]

A. Baltuska, M. F. Emde, M. S. Pshenichnikov, and D. A. Wiersma, "Early-time dynamics of the photoexcited hydrated electron," J. Phys. Chem. A 103, 10065-10082 (1999).
[CrossRef]

1998 (2)

1997 (2)

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, "Transient grating frequency-resolved optical gating," Opt. Lett. 22, 519-521 (1997).
[CrossRef] [PubMed]

1996 (5)

L. Lepetit and M. Joffre, "Two-dimensional nonlinear optics using Fourier-transform spectral interferometry," Opt. Lett. 21, 564-566 (1996).
[CrossRef] [PubMed]

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, and I. A. Walmsley, "Measurement of the intensity and phase of ultraweak, ultrashort laser pulses," Opt. Lett. 21, 884-886 (1996).
[CrossRef] [PubMed]

L. Lepetit, G. Chériaux, and M. Joffre, "Two-dimensional nonlinear optics spectroscopy: simulations and experimental demonstration," J. Nonlinear Opt. Phys. Mater. 5, 465-476 (1996).
[CrossRef]

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, "Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical grating," IEEE J. Quantum Electron. 32, 1253-1264 (1996).
[CrossRef]

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

1995 (3)

O. Kinrot and Y. Prior, "Nonlinear interaction of propagating short pulses in optically dense media," Phys. Rev. A 51, 4996-5007 (1995).
[CrossRef] [PubMed]

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

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]

1993 (3)

R. W. Ziolkowski and J. B. Judkins, "Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time," J. Opt. Soc. Am. B 10, 186-198 (1993).
[CrossRef]

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460-5468 (1993).
[CrossRef]

M. N. Belov, E. A. Manykin, and M. A. Selifanov, "Self-consistent theory of time-resolved four-wave mixing," Opt. Commun. 99, 101-104 (1993).
[CrossRef]

1991 (1)

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

1990 (2)

S. Mukamel, "Femtosecond optical spectroscopy: a direct look at elementary chemical events," Annu. Rev. Phys. Chem. 41, 647-681 (1990).
[CrossRef]

F. C. Spano and W. S. Warren, "Photon echo decays in optically dense media," J. Chem. Phys. 93, 1546-1556 (1990).
[CrossRef]

1982 (1)

R. W. Olson, H. W. H. Lee, F. G. Patterson, and M. D. Fayer, "Optical density effects in photon echo experiments," J. Chem. Phys. 76, 31-39 (1982).
[CrossRef]

1980 (1)

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

1977 (1)

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
[CrossRef]

1962 (1)

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962). In addition to the typographical errors noted on p. xx of Bloembergen's book [N. Bloembergen, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992)], the right hand side of Eq. (6.8) should be multiplied by exp (iphis ) and the sign of EM′ should be positive on the right-hand side of Eq. (6.16).
[CrossRef]

1932 (1)

S. A. Korff and G. Breit, "Optical dispersion," Rev. Mod. Phys. 4, 471-503 (1932).
[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, 10934-10956 (1999).
[CrossRef]

S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, "Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals," J. Opt. Soc. Am. B 15, 2338-2345 (1998).
[CrossRef]

Albrecht Ferro, A.

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, "Two dimensional Fourier transform electronic spectroscopy," J. Chem. Phys. 115, 6606-6622 (2001).
[CrossRef]

Asbury, J. B.

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

Bakker, H. J.

M. Bonn, S. Woutersen, and H. J. Bakker, "Coherent picosecond vibron polaritons as probes of vibrational lifetimes," Opt. Commun. 147, 138-142 (1998).
[CrossRef]

Baltuska, A.

A. Baltuska, M. F. Emde, M. S. Pshenichnikov, and D. A. Wiersma, "Early-time dynamics of the photoexcited hydrated electron," J. Phys. Chem. A 103, 10065-10082 (1999).
[CrossRef]

A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, "Second-harmonic generation frequency resolved optical gating in the single-cycle regime," IEEE J. Quantum Electron. 35, 459-478 (1999).
[CrossRef]

Bardeen, C. J.

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

Belabas , N.

Belabas, N.

Belov, M. N.

M. N. Belov, E. A. Manykin, and M. A. Selifanov, "Self-consistent theory of time-resolved four-wave mixing," Opt. Commun. 99, 101-104 (1993).
[CrossRef]

Bigot, J.-Y.

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

Blank, D. A.

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

Bloembergen , N.

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962). In addition to the typographical errors noted on p. xx of Bloembergen's book [N. Bloembergen, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992)], the right hand side of Eq. (6.8) should be multiplied by exp (iphis ) and the sign of EM′ should be positive on the right-hand side of Eq. (6.16).
[CrossRef]

Bonn, M.

M. Bonn, S. Woutersen, and H. J. Bakker, "Coherent picosecond vibron polaritons as probes of vibrational lifetimes," Opt. Commun. 147, 138-142 (1998).
[CrossRef]

Bowie, J. L.

Brabec , T.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-592 (2000).
[CrossRef]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Breit, G.

S. A. Korff and G. Breit, "Optical dispersion," Rev. Mod. Phys. 4, 471-503 (1932).
[CrossRef]

Broser, I.

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

Butenhoff , T. J.

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460-5468 (1993).
[CrossRef]

Chériaux, G.

L. Lepetit, G. Chériaux, and M. Joffre, "Two-dimensional nonlinear optics spectroscopy: simulations and experimental demonstration," J. Nonlinear Opt. Phys. Mater. 5, 465-476 (1996).
[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]

Cho, M.

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

DeLong, K. W.

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbügel, K. W. DeLong, R. Trebino, and I. A. Walmsley, "Measurement of the intensity and phase of ultraweak, ultrashort laser pulses," Opt. Lett. 21, 884-886 (1996).
[CrossRef] [PubMed]

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, "Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical grating," IEEE J. Quantum Electron. 32, 1253-1264 (1996).
[CrossRef]

Demirdöven, N.

M. Khalil, N. Demirdöven, and A. Tokmakoff, "Obtaining absorptive lineshapes in two-dimensional infrared vibrational correlation spectra," Phys. Rev. Lett. 90, 047401 (2003).
[CrossRef]

Dorrer, C.

Emde, M. F.

A. Baltuska, M. F. Emde, M. S. Pshenichnikov, and D. A. Wiersma, "Early-time dynamics of the photoexcited hydrated electron," J. Phys. Chem. A 103, 10065-10082 (1999).
[CrossRef]

Fayer, M. D.

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

R. W. Olson, H. W. H. Lee, F. G. Patterson, and M. D. Fayer, "Optical density effects in photon echo experiments," J. Chem. Phys. 76, 31-39 (1982).
[CrossRef]

Fittinghoff, D. N.

Fleming, G. R.

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

Gaffney, K. J.

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

Gallagher, S. M.

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, 10934-10956 (1999).
[CrossRef]

Gauvin, S.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

Goun, A.

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

Gruetzmacher , J. A.

Hahn, S.

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

Heitz, R.

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

Hierle, R.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

Hoffman, A.

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

Hulin, D.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

Hybl, J. D.

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, "Two dimensional Fourier transform electronic spectroscopy," J. Chem. Phys. 115, 6606-6622 (2001).
[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, 10934-10956 (1999).
[CrossRef]

S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, "Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals," J. Opt. Soc. Am. B 15, 2338-2345 (1998).
[CrossRef]

Jennings, R. T.

Joffre, M.

Jonas, D. M.

N. Belabas and D. M. Jonas, "Fourier algorithm for four-wave mixing signals from optically dense systems with memory," Opt. Lett. 29, 1811-1813 (2004).
[CrossRef] [PubMed]

D. M. Jonas, "Two-dimensional femtosecond spectroscopy," Annu. Rev. Phys. Chem. 54, 425-463 (2003).
[CrossRef] [PubMed]

D. M. Jonas, "Optical analogs of 2D NMR," Science 300, 1515-1517 (2003).
[CrossRef] [PubMed]

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, "Two dimensional Fourier transform electronic spectroscopy," J. Chem. Phys. 115, 6606-6622 (2001).
[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, 10934-10956 (1999).
[CrossRef]

S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, "Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals," J. Opt. Soc. Am. B 15, 2338-2345 (1998).
[CrossRef]

Judkins, J. B.

Keusters , D.

D. Keusters and W. S. Warren, "Effect of pulse propagation on the two-dimensional photon echo spectrum of multilevel systems," J. Chem. Phys. 119, 4478-4489 (2003).
[CrossRef]

Khalil, M.

M. Khalil, N. Demirdöven, and A. Tokmakoff, "Obtaining absorptive lineshapes in two-dimensional infrared vibrational correlation spectra," Phys. Rev. Lett. 90, 047401 (2003).
[CrossRef]

Kinrot , O.

O. Kinrot and Y. Prior, "Nonlinear interaction of propagating short pulses in optically dense media," Phys. Rev. A 51, 4996-5007 (1995).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
[CrossRef]

Korff , S. A.

S. A. Korff and G. Breit, "Optical dispersion," Rev. Mod. Phys. 4, 471-503 (1932).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-592 (2000).
[CrossRef]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Krumbügel, M. A.

Landin, B. L.

Lee, H. W. H.

R. W. Olson, H. W. H. Lee, F. G. Patterson, and M. D. Fayer, "Optical density effects in photon echo experiments," J. Chem. Phys. 76, 31-39 (1982).
[CrossRef]

Lepetit , L.

Lepetit, L.

L. Lepetit, G. Chériaux, and M. Joffre, "Two-dimensional nonlinear optics spectroscopy: simulations and experimental demonstration," J. Nonlinear Opt. Phys. Mater. 5, 465-476 (1996).
[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]

Likforman, J.-P.

Lummer, B.

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

Manykin, E. A.

M. N. Belov, E. A. Manykin, and M. A. Selifanov, "Self-consistent theory of time-resolved four-wave mixing," Opt. Commun. 99, 101-104 (1993).
[CrossRef]

Migus, A.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

Mukamel , S.

S. Mukamel and A. Tortschanoff, "Multiple quantum coherences in liquid state NMR and nonlinear optics: collective vs. local origin," Chem. Phys. Lett. 357, 327-335 (2002).
[CrossRef]

Mukamel, S.

S. Mukamel, "Femtosecond optical spectroscopy: a direct look at elementary chemical events," Annu. Rev. Phys. Chem. 41, 647-681 (1990).
[CrossRef]

Olson, R. W.

R. W. Olson, H. W. H. Lee, F. G. Patterson, and M. D. Fayer, "Optical density effects in photon echo experiments," J. Chem. Phys. 76, 31-39 (1982).
[CrossRef]

Park, K.

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

Patterson, F. G.

R. W. Olson, H. W. H. Lee, F. G. Patterson, and M. D. Fayer, "Optical density effects in photon echo experiments," J. Chem. Phys. 76, 31-39 (1982).
[CrossRef]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962). In addition to the typographical errors noted on p. xx of Bloembergen's book [N. Bloembergen, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992)], the right hand side of Eq. (6.8) should be multiplied by exp (iphis ) and the sign of EM′ should be positive on the right-hand side of Eq. (6.16).
[CrossRef]

Piletic, I. R.

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

Portella, M. T.

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

Prior, Y.

O. Kinrot and Y. Prior, "Nonlinear interaction of propagating short pulses in optically dense media," Phys. Rev. A 51, 4996-5007 (1995).
[CrossRef] [PubMed]

Pshenichnikov, M. S.

S. Yeremenko, M. S. Pshenichnikov, and D. A. Wiersma, "Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo," Chem. Phys. Lett. 367, 107-113 (2003).
[CrossRef]

A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, "Second-harmonic generation frequency resolved optical gating in the single-cycle regime," IEEE J. Quantum Electron. 35, 459-478 (1999).
[CrossRef]

A. Baltuska, M. F. Emde, M. S. Pshenichnikov, and D. A. Wiersma, "Early-time dynamics of the photoexcited hydrated electron," J. Phys. Chem. A 103, 10065-10082 (1999).
[CrossRef]

Rajaram, B.

Rohlfing, E. A.

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460-5468 (1993).
[CrossRef]

Scherer, N. F.

Schoenlein, R. W.

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

Selifanov, M. A.

M. N. Belov, E. A. Manykin, and M. A. Selifanov, "Self-consistent theory of time-resolved four-wave mixing," Opt. Commun. 99, 101-104 (1993).
[CrossRef]

Shank, C. V.

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

Sheng , S.-C.

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

Siegman, A. E.

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

A. E. Siegman, "Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating," J. Opt. Soc. Am. 67, 545-550 (1977).
[CrossRef]

Spano , F. C.

F. C. Spano and W. S. Warren, "Photon echo decays in optically dense media," J. Chem. Phys. 93, 1546-1556 (1990).
[CrossRef]

Steinel, T.

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

Stromberg, C.

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

Sung, J.

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

Sweetser, J. N.

Thompson , D. E.

D. E. Thompson and J. C. Wright, "Model for spectral artifacts in two-dimensional four-wave mixing spectra from absorption and refractive index dispersion at infrared resonances," J. Phys. Chem. A 104, 11282-11289 (2000).
[CrossRef]

Tokmakoff, A.

M. Khalil, N. Demirdöven, and A. Tokmakoff, "Obtaining absorptive lineshapes in two-dimensional infrared vibrational correlation spectra," Phys. Rev. Lett. 90, 047401 (2003).
[CrossRef]

Tortschanoff, A.

S. Mukamel and A. Tortschanoff, "Multiple quantum coherences in liquid state NMR and nonlinear optics: collective vs. local origin," Chem. Phys. Lett. 357, 327-335 (2002).
[CrossRef]

Toussaere, E.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

Trebino, R.

Wagner, J.-M.

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

Walmsley, I. A.

Warren, W. S.

D. Keusters and W. S. Warren, "Effect of pulse propagation on the two-dimensional photon echo spectrum of multilevel systems," J. Chem. Phys. 119, 4478-4489 (2003).
[CrossRef]

F. C. Spano and W. S. Warren, "Photon echo decays in optically dense media," J. Chem. Phys. 93, 1546-1556 (1990).
[CrossRef]

White, J. O.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

Wiersma, D. A.

S. Yeremenko, M. S. Pshenichnikov, and D. A. Wiersma, "Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo," Chem. Phys. Lett. 367, 107-113 (2003).
[CrossRef]

A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, "Second-harmonic generation frequency resolved optical gating in the single-cycle regime," IEEE J. Quantum Electron. 35, 459-478 (1999).
[CrossRef]

A. Baltuska, M. F. Emde, M. S. Pshenichnikov, and D. A. Wiersma, "Early-time dynamics of the photoexcited hydrated electron," J. Phys. Chem. A 103, 10065-10082 (1999).
[CrossRef]

Woutersen, S.

M. Bonn, S. Woutersen, and H. J. Bakker, "Coherent picosecond vibron polaritons as probes of vibrational lifetimes," Opt. Commun. 147, 138-142 (1998).
[CrossRef]

Wright, J. C.

D. E. Thompson and J. C. Wright, "Model for spectral artifacts in two-dimensional four-wave mixing spectra from absorption and refractive index dispersion at infrared resonances," J. Phys. Chem. A 104, 11282-11289 (2000).
[CrossRef]

Yeremenko, S.

S. Yeremenko, M. S. Pshenichnikov, and D. A. Wiersma, "Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo," Chem. Phys. Lett. 367, 107-113 (2003).
[CrossRef]

Zimmerman, R.

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

Ziolkowski , R. W.

Zyss, J.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

Annu. Rev. Phys. Chem. (2)

D. M. Jonas, "Two-dimensional femtosecond spectroscopy," Annu. Rev. Phys. Chem. 54, 425-463 (2003).
[CrossRef] [PubMed]

S. Mukamel, "Femtosecond optical spectroscopy: a direct look at elementary chemical events," Annu. Rev. Phys. Chem. 41, 647-681 (1990).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
[CrossRef]

Chem. Phys. Lett. (2)

S. Yeremenko, M. S. Pshenichnikov, and D. A. Wiersma, "Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo," Chem. Phys. Lett. 367, 107-113 (2003).
[CrossRef]

S. Mukamel and A. Tortschanoff, "Multiple quantum coherences in liquid state NMR and nonlinear optics: collective vs. local origin," Chem. Phys. Lett. 357, 327-335 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, "Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical grating," IEEE J. Quantum Electron. 32, 1253-1264 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, "Second-harmonic generation frequency resolved optical gating in the single-cycle regime," IEEE J. Quantum Electron. 35, 459-478 (1999).
[CrossRef]

J. Chem. Phys. (7)

J. D. Hybl, A. Albrecht Ferro, and D. M. Jonas, "Two dimensional Fourier transform electronic spectroscopy," J. Chem. Phys. 115, 6606-6622 (2001).
[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, 10934-10956 (1999).
[CrossRef]

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460-5468 (1993).
[CrossRef]

D. Keusters and W. S. Warren, "Effect of pulse propagation on the two-dimensional photon echo spectrum of multilevel systems," J. Chem. Phys. 119, 4478-4489 (2003).
[CrossRef]

R. W. Olson, H. W. H. Lee, F. G. Patterson, and M. D. Fayer, "Optical density effects in photon echo experiments," J. Chem. Phys. 76, 31-39 (1982).
[CrossRef]

F. C. Spano and W. S. Warren, "Photon echo decays in optically dense media," J. Chem. Phys. 93, 1546-1556 (1990).
[CrossRef]

M. Cho, D. A. Blank, J. Sung, K. Park, S. Hahn, and G. R. Fleming, "Intrinsic cascading contributions to the fifth- and seventh-order electronically off-resonant spectroscopies," J. Chem. Phys. 112, 2082-2094 (2000).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

L. Lepetit, G. Chériaux, and M. Joffre, "Two-dimensional nonlinear optics spectroscopy: simulations and experimental demonstration," J. Nonlinear Opt. Phys. Mater. 5, 465-476 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

J. Phys. Chem. A (2)

D. E. Thompson and J. C. Wright, "Model for spectral artifacts in two-dimensional four-wave mixing spectra from absorption and refractive index dispersion at infrared resonances," J. Phys. Chem. A 104, 11282-11289 (2000).
[CrossRef]

A. Baltuska, M. F. Emde, M. S. Pshenichnikov, and D. A. Wiersma, "Early-time dynamics of the photoexcited hydrated electron," J. Phys. Chem. A 103, 10065-10082 (1999).
[CrossRef]

Nonlinear Opt. (1)

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, and J. Zyss, "Femtosecond ultrabroad-band frequency mixing in MNA and KDP thin crystals," Nonlinear Opt. 11, 5-12 (1995).

Opt. Commun. (2)

M. N. Belov, E. A. Manykin, and M. A. Selifanov, "Self-consistent theory of time-resolved four-wave mixing," Opt. Commun. 99, 101-104 (1993).
[CrossRef]

M. Bonn, S. Woutersen, and H. J. Bakker, "Coherent picosecond vibron polaritons as probes of vibrational lifetimes," Opt. Commun. 147, 138-142 (1998).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. (1)

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606-622 (1962). In addition to the typographical errors noted on p. xx of Bloembergen's book [N. Bloembergen, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992)], the right hand side of Eq. (6.8) should be multiplied by exp (iphis ) and the sign of EM′ should be positive on the right-hand side of Eq. (6.16).
[CrossRef]

Phys. Rev. A (2)

O. Kinrot and Y. Prior, "Nonlinear interaction of propagating short pulses in optically dense media," Phys. Rev. A 51, 4996-5007 (1995).
[CrossRef] [PubMed]

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

Phys. Rev. B (1)

B. Lummer, J.-M. Wagner, R. Heitz, A. Hoffman, I. Broser, and R. Zimmerman, "Pulse-propagation-induced higher orders of diffraction in transient four-wave mixing with semiconductors," Phys. Rev. B 54, 16727-16732 (1996).
[CrossRef]

Phys. Rev. Lett. (4)

J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, C. J. Bardeen, A. Migus, and C. V. Shank, "Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes," Phys. Rev. Lett. 66, 1138-1141 (1991).
[CrossRef] [PubMed]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

M. Khalil, N. Demirdöven, and A. Tokmakoff, "Obtaining absorptive lineshapes in two-dimensional infrared vibrational correlation spectra," Phys. Rev. Lett. 90, 047401 (2003).
[CrossRef]

J. B. Asbury, T. Steinel, C. Stromberg, K. J. Gaffney, I. R. Piletic, A. Goun, and M. D. Fayer, "Hydrogen bond dynamics probed with ultrafast infrared heterodyne detected multidimensional vibrational stimulated echoes," Phys. Rev. Lett. 91, 237402 (2003).
[CrossRef]

Rev. Mod. Phys. (2)

T. Brabec and F. Krausz, "Intense few-cycle laser fields: Frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-592 (2000).
[CrossRef]

S. A. Korff and G. Breit, "Optical dispersion," Rev. Mod. Phys. 4, 471-503 (1932).
[CrossRef]

Science (1)

D. M. Jonas, "Optical analogs of 2D NMR," Science 300, 1515-1517 (2003).
[CrossRef] [PubMed]

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V. S. Letokhov and V. P. Chebotayev, Nonlinear Laser Spectroscopy (Springer-Verlag, New York, 1977).

P. H. Vaccaro, "Degenerate four-wave mixing spectroscopy," in Nonlinear Spectroscopy for Molecular Structure Determination , R. W. Field, E. Hirota, J. P. Maier, and S. Tsuchiya, eds. (Blackwell, Oxford, UK, 1997), pp. 75-126.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford U. Press, New York, 1995).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

J. C. Slater, Microwave Transmission , 1st ed. (McGraw-Hill, New York, 1942).

J. R. Reitz, F. J. Milford, and R. W. Christy, Foundations of Electromagnetic Theory , 3rd ed. (Addison-Wesley, Reading, Mass., 1980).

D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves (Prentice-Hall, Englewood Cliffs, N.J., 1994).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1991).

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-×(×E)-1c2 2Et2=10c2 2Pt2,
E(r, t)=12π -Eˆ(r, ω)exp(-iωt)dω.
-×(×Eˆ)+(ω2/c2)Eˆ=-(ω2/0c2)Pˆ.
Pˆ(r, ω)=Pˆ(1)(r, ω)+Pˆ(NL)(r, ω).
Pˆ(1)(r, ω)=[ˆ(ω)-0]Eˆ(r, ω),
-×(×Eˆ)+ω2c2 ˆ(ω)0 Eˆ=-ω20c2 Pˆ(NL).
2Eˆ+ω2c2 ˆ(ω)0 Eˆ=-ω20c2 Pˆ(NL)-[·Pˆ(NL)/ˆ(ω)],
Pˆ(3)(r, (ωt=-ωa-ωb-ωc), ωa, ωb, ωc)=Xˆ(3)(-ωa,-ωb,-ωc)Eˆa(r,-ωa)Eˆb(r,-ωb)Eˆc(r,-ωc),
Eˆγ(r, ωγ)=1(2π)3 Eˆγ[kˆγ(ωγ), ωγ]×exp[ikˆγ(ωγ)·r]d3kˆγ,
kˆ(ω)=-kˆ(-ω)*
Pˆ(3)(r, (ωt=-ωa-ωb-ωc), ωa, ωb, ωc)=1(2π)9 Xˆ(3)(-ωa,-ωb,-ωc) EˆaEˆbEˆc×exp[i(kˆa(-ωa)+kˆb(-ωb)+kˆc(-ωc))·r]×d3kˆad3kˆbd3kˆc,
Pˆ(3)(r, ω)=Pˆ(3)(r, ω)exp[ikˆp·r],
Pˆ(3)(kˆγ; r, (ωt=-ωa-ωb-ωc), ωa, ωb, ωc)=Xˆ(3)(-ωa,-ωb,-ωc)
Eˆa[kˆa(-ωa),-ωa]Eˆb[kˆb(-ωb),-ωb]Eˆc[kˆc(-ωc),-ωc],
kˆp(-ωa,-ωb,-ωc)=kˆc(-ωc)+kˆb(-ωb)+kˆa(-ωa).
(·Pˆ(3))=-(kˆp·Pˆ(3))kˆp  exp[ikˆp·r]=-(kˆp·Pˆ(3))kˆp,
2Eˆ+(kˆs·kˆs)Eˆ=-[(kˆs·kˆs)Pˆ(3)-(kˆp·Pˆ(3))kˆp]/ˆ,
Eˆp(r, ωt)=Eˆp(ωt)exp[ikˆp·r],
Eˆp(ωt)=-1ˆS(ωt) (kˆs·kˆs)Pˆ(3)-(kˆp·Pˆ(3))kˆp(kˆs·kˆs)-(kˆp·kˆp).
Eˆf(r, ωt)=Eˆs(ωt)exp[ikˆs·r]+Eˆp(ωt)exp[ikˆp·r].
Hˆf(r, ωt)=1μ0ωt kˆs×Eˆs(ωt)exp[ikˆs·r]+1μ0ωt kˆp×Eˆp(ωt)exp[ikˆp·r].
Eˆr1(r, ωt)·ex, y|z=0=Eˆf(r, ωt)·ex, y|z=0,
ˆW(ωt)Eˆr1(r, ωt)·ez|z=0=(ˆS(ωt)Eˆf(r, ωt)
+Pˆ(3)(r, ωt))·ez|z=0,
Hˆr1(r, ωt)·ex, y,z|z=0=Hˆf(r, ωt)·ex, y,z|z=0,
kˆp·ex=kˆr1·ex=kˆs·ex=kˆNL·ex,
kˆp·ey=kˆr1·ey=kˆs·ey=kˆNL·ey,
kˆr1·ez=-kˆt·ez=-[(ωt2/c2)(ˆW(ωt)/0)-(kˆp ·ex)2-(kˆp·ey)2]1/2,
kˆs·ez=[(ωt2/c2)(ˆS(ωt)/0)-(kˆp·ex)2-(kˆp·ey)2]1/2,
kˆt·ez=[(ωt2/c2)(air(ωt)/0)-(kˆp·ex)2-(kˆp·ey)2]1/2.
Eˆt(kˆγ; d, ωt, ωa, ωb, ωc)
=iωt20c Π˜(3)(kˆa, kˆb, kˆc, d)·Pˆ(3)(kˆγ; ωt, ωa, ωb, ωc).
Eˆt(kˆγ; r, ωt, ωa, ωb, ωc)=Eˆt(kˆγ; d, ωt, ωa, ωb, ωc)×exp[ikˆt·r].
Π˜(3)(kˆa, kˆb, kˆc, d)=[I-[kˆskˆsT]/(kˆs·kˆs)]×ωt(kˆs·ez)c exp[iΔkˆ(ωt, ωa, ωb, ωc)·ezd]-1iΔkˆ(ωt, ωa, ωb, ωc)·ez,
Δkˆ(ωt, ωa, ωb, ωc)=kˆp(-ωa,-ωb,-ωc)-kˆs(ωt),
Π(3)[kˆa, kˆb, kˆc, d]=Π˜(3)[kˆa, kˆb, kˆc, d]/exp[i Re[kˆs(ωt)-kt(ωt)]·ezd].
Π(3)[kˆa, kˆb, kˆc, d]=[I-[kˆskˆsT]/(kˆs·kˆs)]×ωt(kˆs·ez)c exp[iΔkˆ(ωt, ωa, ωb, ωc)·ezd]-1iΔkˆ(ωt, ωa, ωb, ωc)·ez×exp[-Im(kˆs(ωt)·ezd)],
[kˆt-kd0]·ex, y=[kˆc(-ωc)+kˆb(-ωb)+kˆa(-ωa)-kd0(ωt)]·ex, y
Sdet(kˆγ; ωt, ωa, ωb, ωc)=|Eˆt(kˆγ; r, ωt)+Eˆd(r, ωt)|2D(r, ωt)d3r,
Sdet(kˆγ; ωt, ωa, ωb, ωc)=D0  1(2π)3 |Eˆt[kˆγ; kd, ωt]+Eˆd[kd, ωt]|2d3kd,
Eˆd[kd, ωt]=Eˆd(r, ωt)exp[-ikd·r]d3r.
Eˆt[kˆγ; k(ωt), ωt]=Eˆt(kˆγ; r, ωt, ωa, ωb, ωc)×exp[-ik·r]×(2π)3δ(k-kt).
Sˆint(ωt, ωa, ωb, ωc, kˆa, kˆb, kˆc)=1(2π)3 Eˆt[kˆγ; kd(ωt), ωt]·Eˆd*[kd(ωt), ωt]d3kd.
Sˆ(ωt, ωa, ωb, ωc)=1(2π)9 Sˆint(ωt, ωa, ωb, ωc, kˆa, kˆb, kˆc)×d3kˆad3kˆbd3kˆc.
Sˆ((ωt=-ωa-ωb-ωc), ωa, ωb, ωc)=iωt20c 1(2π)12 (3)[kˆa, kˆb, kˆc, d]Xˆ(3)(-ωa,-ωb,-ωc)Eˆa[kˆa(-ωa),-ωa]Eˆb[kˆb(-ωb),-ωb]Eˆc[kˆc(-ωc),-ωc] ·Eˆd*[kd(ωt), ωt](2π)3δ(kd-kt)d3kˆad3kˆbd3kˆcd3kd.
S(t, ta, tb, tc)=Sˆ(ωa, ωb, ωc)×exp[-iωa(ta-t)]×exp[-iωb(tb-t)]×exp[-iωc(tc-t)]dωadωbdωc.
Π(3)[kˆa, kˆb, kˆc, d]Π(3)[kˆa0, kˆb0, kˆc0, d],
Eˆγ[kˆγ, ωγ]Eˆγ(kˆγ0; ωγ)Fˆγ[kγ(ωγ)],
Sˆ(ωt, ωa, ωb, ωc)iωt20c Π(3)[kˆa0, kˆb0, kˆc0, d](Xˆ(3)(-ωa,-ωb,-ωc)Eˆa(kˆa0;-ωa)Eˆb(kˆb0;-ωb)Eˆc(kˆc0;-ωc))·Eˆd*(kd0; ωt)Φ(3)(ωt, ωa, ωb, ωc),
Φ(3)(ωt, ωa, ωb, ωc)1(2π)12 Fˆa[ka(-ωa)]Fˆb[kb(-ωb)]×Fˆc[kc(-ωc)]Fˆd*[kd(ωt)](2π)3δ(kd-kt)×d3kad3kbd3kcd3kd.
Φ(3)(ωt, ωa, ωb, ωc)=1(2π)9 Fˆa[ka(-ωa)]Fˆb[kb(-ωb)]×Fˆc[kc(-ωc)]Fˆd*[kt(ωt)]d3kad3kbd3kc.
Sˆpm(3)(ud; ωa, ωb, ωc)=Xˆ(3)(ωa, ωb, ωc)×Φideal(3)(ud; ωa, ωb, ωc),
Φideal(3)(ud; ωa, ωb, ωc)
=[θ(saωa)θ(sbωb)θ(scωc)+θ(-saωa)θ(-sbωb)θ(-scωc)].
ua={+sin(α)ex+sin(β)ey+[1-sin2(α)-sin2(β)]1/2ez},
ub={+sin(α)ex-sin(β)ey+[1-sin2(α)-sin2(β)]1/2ez},
uc={-sin(α)ex+sin(β)ey+[1-sin2(α)-sin2(β)]1/2ez},
ud={-sin(α)ex-sin(β)ey+[1-sin2(α)-sin2(β)]1/2ez}.
kˆa0(ωa)=ωac[+sin(α)ex+sin(β)ey+nˆz(ωa)ez],
kˆb0(ωb)=ωbc[+sin(α)ex-sin(β)ey+nˆz(ωb)ez],
kˆc0(ωc)=ωcc[-sin(α)ex+sin(β)ey+nˆz(ωc)ez],
nˆz(ω)[nˆ2(ω)-sin2(α)-sin2(β)]1/2.
nˆz(ω)=[n2(ω)-sin2(α)-sin2(β)]1/2+iκ(ω)n(ω)/[n2(ω)-sin2(α)-sin2(β)]1/2nz(ω)+iκz(ω),
l=[n(ω)/nz(ω)]d,
kˆp0=-ωcc+ωbc-ωacsin(α)ex++ωcc-ωbc-ωacsin(β)ey+ωcnˆz(ωc)c+ωbnˆz(ωb)c-ωanˆz(-ωa)cez.
kˆs0=-ωcc+ωbc-ωacsin(α)ex++ωcc-ωbc-ωacsin(β)ey+[ωtnˆzs(ωt, ωa, ωb, ωc)/c]ez,
nˆzs(ωt, ωa, ωb, ωc){nˆ2(ωt)-(c/ωt)2[(kˆs·ex)2+(kˆs·ey)2]}1/2,
nˆzs(ωt, ωa, ωb, ωc)=nzs(ωt, ωa, ωb, ωc)+iκzs(ωt, ωa, ωb, ωc),
nzs(ωt, ωa, ωb, ωc){n2(ωt)-(c/ωt)2[(kˆs·ex)2+(kˆs·ey)2]}1/2
κzs(ωt, ωa, ωb, ωc)=κ(ωt)n(ωt)/nzs(ωt, ωa, ωb, ωc).
ΔFˆ0=ωcnˆz(ωc)c+ωbnˆz(ωb)c-ωanˆz(-ωa)c-ωtnˆzscez.
ΔFˆ0=ωcnz(ωc)c+ωbnz(ωb)c-ωanz(ωa)c-ωtnzscez+iωcκz(ωc)c+ωbκz(ωb)c+ωaκz(ωa)c-ωtκzscez.
kt0=-ωcc+ωbc-ωacsin(α)ex++ωcc-ωbc-ωacsin(β)ey+ωtnzt(ωt, ωa, ωb, ωc)c ez,
nzt(ωt, ωa, ωb, ωc){1-(c/ωt)2[(kt·ex)2+(kt·ey)2]}1/2.
Eˆ(ω, x, y, z=0)=Eˆ(ω, z=0)exp[-(x2+y2)/2w02],
Eˆ[ω, kx, ky, kz]=Eˆ(ω, z=0)2πw02  exp[-(kx2+ky2)w02/2]2πδ(kz-kzω),
F[k(ω)]=exp[-((k·e1)2+(k·e2)2)w02(ω)/2],
e1a=[sβex-sαey]/[s2α+s2β]1/2
e2a=[sα(1-s2α-s2β)1/2ex+sβ(1-s2α-s2β)1/2ey-(s2α+s2β)ez]/[s2α+s2β]1/2,
ka=ωac[axex+ayey+(1-ax2-ay2)1/2ez].
kt=ωcccx+ωbcbx-ωacaxex+ωcccy+ωbcby-ωacayey+ωtnzt(ωt, ωa, ωb, ωc)c ez,
Φ(3)(ωt, ωa, ωb, ωc)= ×exp[-((ka·e1a)2+(ka·e2a)2)w0a2(ωa)/2]×exp[-((kb·e1b)2+(kb·e2b)2)w0b2(ωb)/2]×exp[-((kc·e1c)2+(kc·e2c)2)w0c2(ωc)/2]×exp[-((kt·e1d)2+(kt·e2d)2)w0d2(ωt)/2]daxdaydbxdbydcxdcy,
k·uk(ω)-[(k0·e1)2+(k0·e2)2]/2k(ω)
Φ(3)(ωt, ωa, ωb, ωc)=exp[-(ωt-ωa)2w02s2(α)[1-2s2(α)]/4c2]×exp[-(ωb-ωc)2w02s2(α)[1-2s2(α)] / [4c2[1-s2(α)]]]
Φ(3)(ωt, ωa, ωb, ωc)=exp[-(ωb-ωa)2w02  sin2(α)/2c2]×exp[-(ωc-ωa)2w02  sin2(β)/2c2]×δ(-ωt-ωa+ωb+ωc),
(δ; δ, δ, δ),(0; 0, δ,-δ),(4δ;-2δ, δ, δ),
ΔΦδ(1;1,1,1)=,ΔΦδ(0;0,1,-1)=2 c(ln 2)1/2w0  sin(α),ΔΦδ(4;-2,1,1)=2 c(ln 2)1/23w0  sin(α).
exp[-(t-ta)2c2/w02  sin2(α)]×exp[-(tb-tc)2c2/w02  sin2(α)].
(3)(ω, ω, ω, d)=1nˆzs 1-exp[-2ωκ(ω)l/c]2ωκ(ω)/c×exp[-ωκ(ω)l/c],
Φ(3)(ω, ω, ω, ω)=1,
ΔFˆ0·ezd=[ωtnˆz(ωt)-ωtnˆzs]d/c+i2ωaκ(ωa)l/c,
(3)(ωa, ωa, ωt, d)=1nˆzs 1-exp[-2ωaκ(ωa)l/c]2ωaκ(ωa)/c×exp[-ωtκ(ωt)l/c],
Φ(3)(ωt, ωa, ωa, ωt)=1
Spp=  Eˆc(r, ωt)·Eˆt(r,-ωt)+Eˆc(r,-ωt)·Eˆt(r, ωt)z=ddxdy=2ωt    Im[Sˆpm(3)(uc;-ωa, ωa, ωt)] Eˆa(-ωa)Eˆb(ωa)Eˆc(ωt)·Eˆc(-ωt)×1-exp[-2ωaκ(ωa)l/c]2ωaκ(ωa)/c×exp[-2ωtκ(ωt)l/c].
(3)(ωa, ωa, ωt, d)=(1/nzs)sinc[ωt(nz(ωt)-nzs)d/(2c)]×exp[iωt(nz(ωt)-nzs)d/(2c)],
Φ(3)(ωt, ωa, ωa, ωt)=exp[-(ωt-ωa)2w02  sin2(β)/(2c2)],
(3)(ωa, ωa, ωt, d)(1/nsz)sinc[(ωa-ωt)nl  sin2(βI)/c]×exp[i(ωa-ωt)nl  sin2(βI)/c],
Φ(3)(ωt, ωa, ωa, ωt)=exp[-(ωt-ωa)2w02n2  sin2(βI)/(2c2)].
Re(ΔFˆ0·ezd)2ωc(ωa-ωb)+2ωb(ωa-ωc)ωc+ωb-ωa  d sin2 αnc.
Δδ(1;1,1,1)=,Δδ(0;0,1,-1)=ω0x1/2n(ω0)cd sin2(α)1/2,
Δδ(4;-2,1,1)=ω0-x1/2n(ω0)c3d sin2(α)2+ω0  x1/2n(ω0)c3d sin2(α)1/2,
2i(kˆs·)Eˆf=-(1/ˆ)[(kˆs·kˆs)Pˆ(3)-(kˆp·Pˆ(3))kˆp]exp[i(kˆp-kˆs)·r]
2Eˆ(r, ω)+ω2c2 ˆ(ω)0 Eˆ(r, ω)=0
Eˆ(r, ω)=Eˆ(ω)exp[ikˆ(ω)·r],
Eˆ(ω)·kˆ(ω)=0
[kˆ(ω)·kˆ(ω)]=(ω2/c2)(ˆ(ω)/0)(ω2/c2)nˆ2(ω).
Hˆ(r, ω)=kˆ(ω)×Eˆ(r, ω)/(ωμ0),
Eˆ(r, ω)=-kˆ(ω)×Hˆ(r, ω)/(ωˆ(ω)).
kˆ(ω)=(ω/c)p(ω, u·e)u+i(ω/c)q(ω, u·e)ek(ω)+ik(ω),
p(ω)n(ω),q(ω)κ(ω)/(u·e).
P(3)(r, t, ta, tb, tc)=000R(3)(τa, τb, τc) EaEbEcdτadτbdτc,
Pˆ(3)(r, t, ωa, ωb, ωc)=000R(3)(τa, τb, τc) Eˆa(r,-ωa)Eˆb(r,-ωb)Eˆc(r,-ωc)×exp[iωa(t-τa)]exp[iωb(t-τb)]×exp[iωc(t-τc)]dτadτbdτc.
Xˆ(3)(-ωa,-ωb,-ωc)=000R(3)(τa, τb, τc)×exp[-iωaτa]exp[-iωbτb]×exp[-iωcτc]dτadτbdτc,
Pˆ(3)(r, t, ωa, ωb, ωc)=Xˆ(3)(-ωa,-ωb,-ωc) Eˆa(r,-ωa)Eˆb(r,-ωb)Eˆc(r,-ωc)×exp[i(ωa+ωb+ωc)t].
Pˆ(3)(r, ωt, ωa, ωb, ωc)=Xˆ(3)(-ωa,-ωb,-ωc) Eˆa(r,-ωa)Eˆb(r,-ωb)Eˆc(r,-ωc)×2πδ(ωt+ωa+ωb+ωc),
Eˆγ(r,-ωγ)=Eˆγ(r,-ωγ)exp[ikˆγ(-ωγ)·r],
Pˆ(3)(r, ωt, ωa, ωb, ωc)=Xˆ(3)(-ωa,-ωb,-ωc) Eˆa(r,-ωa)Eˆb(r,-ωb)Eˆc(r,-ωc)×exp[ikˆa(-ωa)·r]exp[ikˆb(-ωb)·r]×exp[ikˆc(-ωc)·r]2πδ(-ωt-ωa-ωb-ωc).
Pˆ(3)(r,+ωt,+ωa,-ωb,-ωc)=Xˆ(3)(-ωa,+ωb,+ωc) Eˆa(r,-ωa)Eˆb(r,+ωb)Eˆc(r,+ωc)×exp[ikˆa(-ωa)·r]exp[ikˆb(ωb)·r]×exp[ikˆc(ωc)·r]2πδ(-ωt-ωa+ωb+ωc),
kˆp(-ωa, ωb, ωc)=kˆc(ωc)+kˆb(ωb)+kˆa(-ωa).
(Eˆr2(r, ωt)+Eˆf(r, ωt))·ex, y|z=d=Eˆt(r, ωt)·ex, y|z=d,
(ˆS(ωt)Eˆr2(r, ωt)+ˆS(ωt)Eˆf(r, ωt)+Pˆ(3)(r, ωt))·ez|z=d=ˆW(ωt)Eˆt(r, ωt)·ez|z=d,
(Hˆr2(r, ωt)+Hˆf(r, ωt))·ex, y,z|z=d=Hˆt(r, ωt)·ex, y,z|z=d,
EˆpTE(r, ωt)=(Eˆp(r, ωt)·en)·en,
EˆpTM(r, ωt)=Eˆp(r, ωt)-EˆpTE(r, ωt).
EˆsTE(r, ωt)=-1+Δkˆ·ez2kˆs·ez(Eˆp(ωt)·en)exp[ikˆs·r]en,
Δkˆ(ωt, ωa, ωb, ωc)=kˆp(-ωa,-ωb,-ωc)-kˆs(ωt)=(kˆp·ez-kˆs·ez)ez,
EˆtTE(r, ωt)=1+Δkˆ·ez2kˆs·ez(Eˆp·en)×[exp[iΔkˆ·ezd]-1]exp[ikˆt·r]en,
EˆtTM(r, ωt)=12 kˆs·ezkˆs·kˆs[(kˆp·ez)(Eˆp·etg)-(kˆs·etg)(Eˆp·ez)]+(Eˆp·etg)×[exp[iΔkˆ·ezd]-1]×exp[ikˆt·r]etg-kˆs·etgkˆs·ez ez,
Π˜(3)[kˆa, kˆb, kˆc, d]=[I-[kˆskˆsT]/(kˆs·kˆs)]×ωt(kˆs·ez)c exp[iΔkˆ(ωt, ωa, ωb, ωc)·ezd]-1iΔkˆ(ωt, ωa, ωb, ωc)·ez.
kˆs·kˆskˆs·ez[I-[kˆskˆsT]/(kˆs·kˆs)]=kˆs·kˆskˆs·ez000kˆs·ez-kˆs·etg0-kˆs·etg(kˆs·etg)2kˆs·ez.
Eˆt(ωt)·e¯n=kˆpz+kˆszkˆtz+kˆszpˆ exp[ikˆpzd]-2kˆsz(kˆpz-kˆrz)(kˆsz-kˆrz)(kˆtz+kˆsz)pˆ exp[ikˆszd]×exp[-ikˆtzd].
Eˆt(ωt)·etg=(KP)+KˆsPˆKˆt+Kˆs exp[ikˆpzd]-2Kˆs((KP)-KˆrPˆ)(Kˆs-Kˆr)(Kˆt+Kˆs) exp[ikˆszd]×exp[-ikˆtzd],
Eˆt(ωt)·ez=-kˆt·etgkˆt·ez Eˆt(ωt)·etg,
Eˆr(ωt)·etg=-(KP)-KˆsPˆKˆs-Kˆr-2Kˆs[(KP)-KˆsPˆ](Kˆt+Kˆs)(Kˆs-Kˆr) exp[i(kˆpz+kˆsz)d]-(Kˆt-Kˆs)[(KP)+KˆsPˆ](Kˆt+Kˆs)(Kˆs-Kˆr) exp[i2kˆszd]×n=0(rˆstTMrˆsrTM  exp[i2kˆszd])n.
Eˆt(ωt)·etg=+(KP)+KˆsPˆKˆt+Kˆs  exp[i(kˆpz-kˆtz)d]-2Kˆs[(KP)-KˆrPˆ](Kˆt+Kˆs)(Kˆs-Kˆr) exp[i(kˆsz-kˆtz)d]+(Kˆr+Kˆs)[(KP)-KˆsPˆ](Kˆt+Kˆs)(Kˆs-Kˆr)    ×exp[i(2kˆsz+kˆpz-kˆtz)d]×n=0(rˆstTMrˆsrTM  exp[i2kˆszd])n.
EˆγFW(r, ω)=EˆγFW(ω)exp[ikˆγ·r],
EˆγFW(ω)=Eˆγ(ω)n=0(rˆstrˆsr  exp[i2kˆγzd])n,
EˆγBW(r, ω)=EˆγBW(ω)exp[ikˆγBW·r],
EˆγBW(ω)=Eˆγ(ω)rˆst  exp[i2kˆγzd]n=0(rˆsrrˆst  exp[i2kˆγzd])n
kˆγBW=(kˆγ·etg)etg-(kˆγ·ez)ez.

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