Abstract

We study theoretically ultrafast modulation spectroscopy in Doppler-broadened cascading three-level and four-level systems. The pump beams consist of two frequency components that drive two-pathway excitation. The four-wave mixing (FWM) signal exhibits damping oscillation as the relative time delay τ between the two pump beams is varied, from which the resonant frequencies between energy levels can be deduced. Furthermore, Doppler-free precision in the measurement of the resonant frequencies can be achieved when the pump beams have either narrow-band or broadband linewidths. We study two-photon resonant FWM induced by time-delayed correlated fluctuating fields in a Doppler-broadened cascading three-level system. It is related intrinsically to the sum-frequency trilevel echo when the pump beams have broadband linewidths. However, unlike the corresponding incoherent-light stimulated photon echo, the maximum of the FWM signal is shifted from τ0. If we consider the fluctuations of the pump beams, the FWM signal exhibits damping oscillation when the laser frequency is off resonant from the two-photon transition.

© 2001 Optical Society of America

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  3. P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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1998

X. Mi, Z. Yu, Q. Jiang, X. Li, and P. Fu, “Four-level ultrafast modulation spectroscopy,” Opt. Commun. 152, 361–364 (1998).
[CrossRef]

1995

P. Fu, X. Mi, Z. Yu, Q. Jiang, Y. Zhang, and X. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

V. D. Kleiman, L. Zhu, J. Allen, and R. J. Gordon, “Coherent control over the photodissociation of CH3I,” J. Chem. Phys. 103, 10800–10803 (1995).
[CrossRef]

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett. 74, 3596–3599 (1995).
[CrossRef] [PubMed]

1994

P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

1993

X. Mi, Z. Yu, Q. Jiang, and P. Fu, “Time-delayed Raman-enhanced nondegenerate four-wave mixing with a broadband laser source,” Phys. Rev. A 48, 3203–3208 (1993).
[CrossRef] [PubMed]

1992

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530–1539 (1992).
[CrossRef] [PubMed]

1991

G. Vemuri, G. S. Agarwal, R. Roy, M. H. Anderson, J. Cooper, and S. J. Smith, “Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities,” Phys. Rev. A 44, 6009–6016 (1991).
[CrossRef] [PubMed]

1990

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

V. Finkelstein and P. R. Berman, “Optical transient signal induced by strong fluctuating pulses,” Phys. Rev. A 42, 3145–3128 (1990).
[CrossRef] [PubMed]

1988

P. Tchénio, A. Débarre, J.-C. Keller, and J.-L. Le Gouët, “Inhibition of inhomogeneous dephasing by a strong stochastic optical field,” Phys. Rev. A 38, 5235–5248 (1988).
[CrossRef] [PubMed]

A. G. Kofman, A. M. Levine, and Y. Prior, “Stochastic fluctuations in four-wave mixing: a unified theory,” Phys. Rev. A 37, 1248–1260 (1988).
[CrossRef] [PubMed]

1987

T. Hattori, A. Terasaki, and T. Kobayashi, “Coherent Stokes Raman scattering with incoherent light for vibrational-dephasing-time measurement,” Phys. Rev. A 35, 715–724 (1987).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[CrossRef]

1986

D. DeBeer, V. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

1985

J. E. Rothenberg and D. Grischkowsky, “Observation of a 1.9-psec polarization beat,” Opt. Lett. 10, 22–24 (1985).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467–3474 (1985).
[CrossRef] [PubMed]

1984

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

1979

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

1978

P. F. Liao, D. M. Bloom, and N. P. Econmou, “cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[CrossRef]

1976

J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
[CrossRef]

Agarwal, G. S.

G. Vemuri, G. S. Agarwal, R. Roy, M. H. Anderson, J. Cooper, and S. J. Smith, “Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities,” Phys. Rev. A 44, 6009–6016 (1991).
[CrossRef] [PubMed]

Allen, J.

V. D. Kleiman, L. Zhu, J. Allen, and R. J. Gordon, “Coherent control over the photodissociation of CH3I,” J. Chem. Phys. 103, 10800–10803 (1995).
[CrossRef]

Anderson, M. H.

G. Vemuri, G. S. Agarwal, R. Roy, M. H. Anderson, J. Cooper, and S. J. Smith, “Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities,” Phys. Rev. A 44, 6009–6016 (1991).
[CrossRef] [PubMed]

Asaka, S.

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Beach, R.

D. DeBeer, V. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467–3474 (1985).
[CrossRef] [PubMed]

Berman, P. R.

V. Finkelstein and P. R. Berman, “Optical transient signal induced by strong fluctuating pulses,” Phys. Rev. A 42, 3145–3128 (1990).
[CrossRef] [PubMed]

Bjorkholm, J. E.

J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
[CrossRef]

Bloom, D. M.

P. F. Liao, D. M. Bloom, and N. P. Econmou, “cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[CrossRef]

Buchanan, M.

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett. 74, 3596–3599 (1995).
[CrossRef] [PubMed]

Chen, C.

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

Cooper, J.

G. Vemuri, G. S. Agarwal, R. Roy, M. H. Anderson, J. Cooper, and S. J. Smith, “Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities,” Phys. Rev. A 44, 6009–6016 (1991).
[CrossRef] [PubMed]

Corkum, P. B.

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett. 74, 3596–3599 (1995).
[CrossRef] [PubMed]

Débarre, A.

P. Tchénio, A. Débarre, J.-C. Keller, and J.-L. Le Gouët, “Inhibition of inhomogeneous dephasing by a strong stochastic optical field,” Phys. Rev. A 38, 5235–5248 (1988).
[CrossRef] [PubMed]

DeBeer, D.

D. DeBeer, V. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467–3474 (1985).
[CrossRef] [PubMed]

Dupont, E.

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett. 74, 3596–3599 (1995).
[CrossRef] [PubMed]

Econmou, N. P.

P. F. Liao, D. M. Bloom, and N. P. Econmou, “cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[CrossRef]

Elliott, D. S.

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

Finkelstein, V.

V. Finkelstein and P. R. Berman, “Optical transient signal induced by strong fluctuating pulses,” Phys. Rev. A 42, 3145–3128 (1990).
[CrossRef] [PubMed]

Flusberg, A. M.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Fu, P.

X. Mi, Z. Yu, Q. Jiang, X. Li, and P. Fu, “Four-level ultrafast modulation spectroscopy,” Opt. Commun. 152, 361–364 (1998).
[CrossRef]

P. Fu, X. Mi, Z. Yu, Q. Jiang, Y. Zhang, and X. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

X. Mi, Z. Yu, Q. Jiang, and P. Fu, “Time-delayed Raman-enhanced nondegenerate four-wave mixing with a broadband laser source,” Phys. Rev. A 48, 3203–3208 (1993).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530–1539 (1992).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[CrossRef]

Fujiwara, M.

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Gordon, R. J.

V. D. Kleiman, L. Zhu, J. Allen, and R. J. Gordon, “Coherent control over the photodissociation of CH3I,” J. Chem. Phys. 103, 10800–10803 (1995).
[CrossRef]

Grischkowsky, D.

Hartmann, S. R.

D. DeBeer, V. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

R. Beach, D. DeBeer, and S. R. Hartmann, “Time-delayed four-wave mixing using intense incoherent light,” Phys. Rev. A 32, 3467–3474 (1985).
[CrossRef] [PubMed]

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Hattori, T.

T. Hattori, A. Terasaki, and T. Kobayashi, “Coherent Stokes Raman scattering with incoherent light for vibrational-dephasing-time measurement,” Phys. Rev. A 35, 715–724 (1987).
[CrossRef] [PubMed]

Jiang, Q.

X. Mi, Z. Yu, Q. Jiang, X. Li, and P. Fu, “Four-level ultrafast modulation spectroscopy,” Opt. Commun. 152, 361–364 (1998).
[CrossRef]

P. Fu, X. Mi, Z. Yu, Q. Jiang, Y. Zhang, and X. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

X. Mi, Z. Yu, Q. Jiang, and P. Fu, “Time-delayed Raman-enhanced nondegenerate four-wave mixing with a broadband laser source,” Phys. Rev. A 48, 3203–3208 (1993).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530–1539 (1992).
[CrossRef] [PubMed]

Kachru, R.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Keller, J.-C.

P. Tchénio, A. Débarre, J.-C. Keller, and J.-L. Le Gouët, “Inhibition of inhomogeneous dephasing by a strong stochastic optical field,” Phys. Rev. A 38, 5235–5248 (1988).
[CrossRef] [PubMed]

Kleiman, V. D.

V. D. Kleiman, L. Zhu, J. Allen, and R. J. Gordon, “Coherent control over the photodissociation of CH3I,” J. Chem. Phys. 103, 10800–10803 (1995).
[CrossRef]

Kobayashi, T.

T. Hattori, A. Terasaki, and T. Kobayashi, “Coherent Stokes Raman scattering with incoherent light for vibrational-dephasing-time measurement,” Phys. Rev. A 35, 715–724 (1987).
[CrossRef] [PubMed]

Kofman, A. G.

A. G. Kofman, A. M. Levine, and Y. Prior, “Stochastic fluctuations in four-wave mixing: a unified theory,” Phys. Rev. A 37, 1248–1260 (1988).
[CrossRef] [PubMed]

Le Gouët, J.-L.

P. Tchénio, A. Débarre, J.-C. Keller, and J.-L. Le Gouët, “Inhibition of inhomogeneous dephasing by a strong stochastic optical field,” Phys. Rev. A 38, 5235–5248 (1988).
[CrossRef] [PubMed]

Levine, A. M.

A. G. Kofman, A. M. Levine, and Y. Prior, “Stochastic fluctuations in four-wave mixing: a unified theory,” Phys. Rev. A 37, 1248–1260 (1988).
[CrossRef] [PubMed]

Li, X.

X. Mi, Z. Yu, Q. Jiang, X. Li, and P. Fu, “Four-level ultrafast modulation spectroscopy,” Opt. Commun. 152, 361–364 (1998).
[CrossRef]

P. Fu, X. Mi, Z. Yu, Q. Jiang, Y. Zhang, and X. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

Liao, P. F.

P. F. Liao, D. M. Bloom, and N. P. Econmou, “cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[CrossRef]

J. E. Bjorkholm and P. F. Liao, “Line shape and strength of two-photon absorption in an atomic vapor with a resonant or nearly resonant intermediate state,” Phys. Rev. A 14, 751–760 (1976).
[CrossRef]

Liu, H. C.

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett. 74, 3596–3599 (1995).
[CrossRef] [PubMed]

Matsuoka, M.

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Mi, X.

X. Mi, Z. Yu, Q. Jiang, X. Li, and P. Fu, “Four-level ultrafast modulation spectroscopy,” Opt. Commun. 152, 361–364 (1998).
[CrossRef]

P. Fu, X. Mi, Z. Yu, Q. Jiang, Y. Zhang, and X. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

X. Mi, Z. Yu, Q. Jiang, and P. Fu, “Time-delayed Raman-enhanced nondegenerate four-wave mixing with a broadband laser source,” Phys. Rev. A 48, 3203–3208 (1993).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530–1539 (1992).
[CrossRef] [PubMed]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[CrossRef]

Morita, N.

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

Mossberg, T. W.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Nakatsuka, H.

S. Asaka, H. Nakatsuka, M. Fujiwara, and M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd3+-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Prior, Y.

A. G. Kofman, A. M. Levine, and Y. Prior, “Stochastic fluctuations in four-wave mixing: a unified theory,” Phys. Rev. A 37, 1248–1260 (1988).
[CrossRef] [PubMed]

Rothenberg, J. E.

Roy, R.

G. Vemuri, G. S. Agarwal, R. Roy, M. H. Anderson, J. Cooper, and S. J. Smith, “Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities,” Phys. Rev. A 44, 6009–6016 (1991).
[CrossRef] [PubMed]

Smith, S. J.

G. Vemuri, G. S. Agarwal, R. Roy, M. H. Anderson, J. Cooper, and S. J. Smith, “Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities,” Phys. Rev. A 44, 6009–6016 (1991).
[CrossRef] [PubMed]

Tchénio, P.

P. Tchénio, A. Débarre, J.-C. Keller, and J.-L. Le Gouët, “Inhibition of inhomogeneous dephasing by a strong stochastic optical field,” Phys. Rev. A 38, 5235–5248 (1988).
[CrossRef] [PubMed]

Terasaki, A.

T. Hattori, A. Terasaki, and T. Kobayashi, “Coherent Stokes Raman scattering with incoherent light for vibrational-dephasing-time measurement,” Phys. Rev. A 35, 715–724 (1987).
[CrossRef] [PubMed]

Van Wagenen, V. G.

D. DeBeer, V. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Vemuri, G.

G. Vemuri, G. S. Agarwal, R. Roy, M. H. Anderson, J. Cooper, and S. J. Smith, “Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities,” Phys. Rev. A 44, 6009–6016 (1991).
[CrossRef] [PubMed]

Wasilewski, Z. R.

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P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
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X. Mi, Z. Yu, Q. Jiang, and P. Fu, “Time-delayed Raman-enhanced nondegenerate four-wave mixing with a broadband laser source,” Phys. Rev. A 48, 3203–3208 (1993).
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P. Fu, Z. Yu, X. Mi, Q. Jiang, and Z. Zhang, “Theoretical study of the suppression of thermal background in the Raman-enhanced nondegenerate four-wave-mixing spectrum by a time-delayed method,” Phys. Rev. A 46, 1530–1539 (1992).
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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

J. Phys. (Paris)

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function theory of laser-induced molecular reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[CrossRef]

Opt. Commun.

X. Mi, Z. Yu, Q. Jiang, X. Li, and P. Fu, “Four-level ultrafast modulation spectroscopy,” Opt. Commun. 152, 361–364 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. A

P. Fu, Z. Yu, X. Mi, X. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

P. Fu, X. Mi, Z. Yu, Q. Jiang, Y. Zhang, and X. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

X. Mi, Z. Yu, Q. Jiang, and P. Fu, “Time-delayed Raman-enhanced nondegenerate four-wave mixing with a broadband laser source,” Phys. Rev. A 48, 3203–3208 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett.

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “Phase-controlled currents in semiconductors,” Phys. Rev. Lett. 74, 3596–3599 (1995).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

(a) Cascading three-level and (b) four-level systems to be treated in UMS. (c) Schematic diagram of the geometry of UMS.

Fig. 2
Fig. 2

FWM signal intensity versus Δ2/k1u. Parameters: Γ10/k1u=Γ20/k1u=0.02 and ζ2=1.1. (a) Δ1/k1u=0, (b) Δ1/k1u=-1, (c) Δ1/k1u=-2.5. We note that the signal intensity for (a) is normalized to 1.

Fig. 3
Fig. 3

Normalized FWM signal intensity versus Γ20aτ for Δ2a=0. α2/Γ20a=20 (solid curve), 5 (dashed curve), 2 (dotted-dashed curve), 1 (dotted curve).

Fig. 4
Fig. 4

Normalized FWM signal intensity versus Γ20aτ for (a) α2/Γ20a=1 and (b) α2/Γ20a=5. Δ2a=8 (solid curve), 6 (dashed curve), 4 (dotted-dashed curve), 2 (dotted curve).

Equations (41)

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E1(r, t)=A2(r, t)exp(-iω2 t),
E2(r, t)=A2(r, t)exp(-iω2 t).
E3(r, t)=A1(r, t)exp(-iω1 t).
ρ00(0)A1̲ρ10(1)A2̲ρ20(2)(A2)*̲ρ10(3).
ρ10(3)=iμ1iμ22 exp(-iω1 t) 0 dt3 0 dt2 0 dt1×A1(r, t-t1-t2-t3)A2(r, t-t2-t3)×[A2(r, t-t3)]* exp[-(Γ10+iΔ1)(t1+t3)]×exp{-[Γ20+i(Δ1+Δ2)]t2}.
r (t)=r (t)-(t-t)v;
A1(r, t-t1-t2-t3)A2(r, t-t2-t3)[A2(r, t-t3)]*
=12(2)*u2(t-t2-t3)u*2(t-t3-τ)
×exp{i[(k1+k2-k2)·r-ω2τ]}×exp{-iv·[k1 t1+(k1+k2)t2+(k1+k2-k2)t3]}.
PT=Nμ1 -dvW(v)ρ10(3)(v).
W(v)=1πu exp[-(v/u)2].
u2(t1)u2*(t2)=exp(-α2|t1-t2|),
PT=S2(r)exp[-i(ω1 t+ω2τ)]×- dvW(v)0 dt3 0 dt2 0 dt1×exp(-α2|t2-τ|)×exp{-[Γ10+i(Δ1+k1·v)]t1}×exp(-{Γ20+i[(Δ1+Δ2)+(k1+k2)·v]}t2)×exp(-{Γ10+i[Δ1+(k1+k2-k2)·v]}t3),
S2(r)=-iNμ12μ2212(2)*×exp[i(k2-k2+k3)·r].
PT=S2(r)exp(-iω1 t)- dvW(v)L(v)Γ20-α2+i[Δ1+Δ2+(k1+k2)·v]×exp(-α2|τ|-iω2τ)-2α2 exp{-Γ20|τ|-i[Ω2+Δ1+(k1+k2)·v]τ}Γ20+α2+i[Δ1+Δ2+(k1+k2)·v],
PT=S2(r)exp(-iω1 t)- dvW(v)L(v)exp(-α2|τ|-iω2τ)Γ20+α2+i[Δ1+Δ2+(k1+k2)·v].
L(v)=1[Γ10+i(Δ1+k1·v)]{Γ10+i[Δ1+(k1+k2-k2)]·v}.
PT=S2(r)exp[-α2|τ|-i(ω1 t+ω2τ)]- dvW(v) L(v)Γ20+i[Δ1+Δ2+(k1+k2)·v].
PT1πuS2(r)exp[-α2|τ|-i(ω1 t+ω2τ)]- dv exp[-(v/u)2][Γ10+i(Δ1-k1v)]2 1Γ20+i[Δ1+Δ2-k1(1-ζ2)v],
PT-1πS2(r)exp[-α2|τ|-i(ω1 t+ω2τ)]×(1/Δ1)2g(Δ1+Δ2),
g(Δ)=- dx exp(-x2)Γ20+i[Δ-k1u(1-ζ2)x].
1πu - dv
×exp[-(v/u)2]exp{ik1[t1+t3-(ζ2-1)t2]v}
=exp(-{k1u[t1+t3-(ζ2-1)t2]}2/4)
2πk1uδ [t1+t3-(ζ2-1)t2].
PT=2πk1uS2(r)exp[-i(ω1 t+ω2τ)]×0 dt3 0 dt2 0 dt1δ[t1+t3-(ζ2-1)t2]×exp(-α2|t2-τ|)exp[-(Γ10+iΔ1)(t1+t3)]×exp{-[Γ20+i(Δ1+Δ2)]t2}.
PT=2πk1uS2(r)exp[-i(ω1 t+ω2τ)]×0 dx1 0 dx2 x1 dx3δ[x3-(ζ2-1)x2]×exp(-α2|x2-τ|)×exp[-(Γ10+iΔ1)x3]exp{-[Γ20+i(Δ1+Δ2)]x2}.
PT=2(ζ2-1)πk1uS2(r)exp(-iω1 t)×exp(-α2|τ|-iω2τ)(Γ20a-α2+iΔ2a)2-2 exp{-Γ20a|τ|-i(Ω2+ζ2Δ1)τ}(Γ20a+α2+iΔ2a)(Γ20a-α2+iΔ2a)×2α2(Γ20a+iΔ2a)(Γ20a+α2+iΔ2a)(Γ20a-α2+iΔ2a)+α2τ
PT=2(ζ2-1)πk1uS2(r)exp(-iω1 t)×exp(-α2|τ|-iω2τ)(Γ20a+α2+iΔ2a)2.
PT=2(ζ2-1)πk1uS2(r)exp(-iω1 t)×exp(-α2|τ|-iω2τ)(Γ20a+iΔ2a)2.
I(τ)exp(-2α2|τ|)[(Γ20a)2+(Δ2a)2]2.
PT=4(ζ2-1)πk1uS2(r)exp(-iω1 t) α2τα22+(Δ2a)2×exp[-Γ20a|τ|-i(Ω2+ζ2Δ1)τ],
I(τ)τ2 exp(-2Γ20a|τ|).
PT=S2(r)exp[-i(ω1 t+ω2τ)]0 dt3 0 dt2 0 dt1 - dvW(v)×exp[-iθ(v)]exp(-α2|t2-τ|)×exp[-(Γ10+iΔ1)t1]×exp{-[Γ20+i(Δ1+Δ2)]t2}×exp[-(Γ10+iΔ1)t3],
- dvW(v)exp[-iθ(v)],
P=PS+PT.
PS=2πk1u 1Γ0+1Γ1S3(r)exp(-iω1 t)×exp(-α3|τ|-iω3τ)Γ10a-α3+iΔ3a-2α3 exp[-Γ10a|τ|-i(Ω1+ζ3Δ1)τ](Γ10a-α3+iΔ3a)(Γ10a+α3+iΔ3a),
PS=2πk1u 1Γ0+1Γ1S3(r)×exp(-iω1 t) exp(-α3|τ|-iω3τ)Γ10a+α3+iΔ3a.
S3(r)=-iN(μ1/)43(3)*1×exp[i(k3-k3+k1)·r],
ρ00(0)A1̲ρ10(1)A2̲ρ30(2)(A2)*̲ρ10(3).
PT=2(ζ2-1)πk1uS2(r)exp(-iω1 t)×τ22 exp(-α2|τ|-iω2τ)+exp[-Γ20a|τ|-i(Ω2+ζ2Δ1)τ]Γ20a+α2+iΔ2a×1Γ20a+α2+iΔ2a+τ.

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