Abstract

The interference between two different quantum-mechanical pathways for four-wave mixing in sodium vapor has been observed. The four-wave mixing was enhanced by two-photon resonance with the 5s state, for one pathway, and the 4d state, for the other. The phase of the interference could be varied continuously by varying laser frequencies, and constructive and destructive interference effects were studied. A model based on third-order perturbation theory gives a good description of the features observed.

© 1989 Optical Society of America

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References

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  1. V. M. Arutyunyan, E. G. Kanetsyan, and V. O. Chaltykyan, “Passage of electromagnetic radiation through a resonant medium in the presence of an intense monochromatic wave,” Zh. Eksp. Teor. Fiz. 59, 195–200 (1970) [Sov. Phys. JETP 32, 108–110 (1971)].
  2. H. Kildal and S. R. J. Brueck, “Pump depletion and saturation of two-photon resonant third-harmonic generation processes,” IEEE J. Quantum Electron. QE-16, 566–573 (1980).
    [Crossref]
  3. J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garrett, “Resonantly enhanced multiphoton ionization,” Phys. Rev. Lett. 45, 114–116 (1980).
    [Crossref]
  4. J. J. Wynne, “Polarization renormalization due to nonlinear optical generation,” Phys. Rev. Lett. 52, 751–754 (1984).
    [Crossref]
  5. M. S. Malcuit, D. T. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086–1089 (1985).
    [Crossref] [PubMed]
  6. R. W. Boyd, M. S. Malcuit, D. J. Gaulthier, and K. Rza̧żewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
    [Crossref] [PubMed]
  7. J. J. Wynne, “Nonlinear optical balance,” in Multiphoton Processes: Proceedings of the 4th International Conference on Multiphoton Processes, Boulder, Colorado, 1987, S. J. Smith and P. L. Knight, eds. (Cambridge U. Press, Cambridge, 1988), pp. 318–327.
  8. Y. Shevy, S. Hochman, and M. Rosenbluh, “Competition between stimulated three-photon scattering and parametric four-wave mixing,” Opt. Lett. 13, 215–217 (1988).
    [Crossref] [PubMed]
  9. G. Grynberg and B. Cagnac, “Doppler-free multiphotonic spectroscopy,” Rep. Prog. Phys. 40, 791–841 (1977).
    [Crossref]
  10. J. N. Elgin and G. H. C. New, “Semi-classical theory of two-photon resonant third-harmonic generation,” Opt. Commun. 16, 242–246 (1976).
    [Crossref]
  11. A. T. Georges, P. Lambropoulos, and J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300–307 (1977).
    [Crossref]
  12. P. M. Radmore and P. L. Knight, “Two-photon ionisation: interference and population trapping,” Phys. Lett. 102A, 180–185 (1984).
  13. Z. Deng, “Line narrowing and photoelectron trapping in multiphoton ionization spectroscopy,” J. Opt. Soc. Am. B 1, 874–878 (1984).
    [Crossref]
  14. Z. Deng, “Complete population trapping in two-photon ionization,” Phys. Lett. 105A, 43–44 (1984).
  15. M. C. E. Huber and R. J. Sandeman, “The measurement of oscillator strengths,” Rep. Prog. Phys. 49, 397–490 (1986).
    [Crossref]
  16. F. Biraben, B. Cagnac, E. Giacobino, and G. Grynberg, “Broadening and shift of the sodium 3S–4D and 3S–5S two-photon lines perturbed by noble gas,” J. Phys. B 10, 2369–2374 (1977).
    [Crossref]

1988 (1)

1987 (1)

R. W. Boyd, M. S. Malcuit, D. J. Gaulthier, and K. Rza̧żewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[Crossref] [PubMed]

1986 (1)

M. C. E. Huber and R. J. Sandeman, “The measurement of oscillator strengths,” Rep. Prog. Phys. 49, 397–490 (1986).
[Crossref]

1985 (1)

M. S. Malcuit, D. T. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086–1089 (1985).
[Crossref] [PubMed]

1984 (4)

J. J. Wynne, “Polarization renormalization due to nonlinear optical generation,” Phys. Rev. Lett. 52, 751–754 (1984).
[Crossref]

P. M. Radmore and P. L. Knight, “Two-photon ionisation: interference and population trapping,” Phys. Lett. 102A, 180–185 (1984).

Z. Deng, “Line narrowing and photoelectron trapping in multiphoton ionization spectroscopy,” J. Opt. Soc. Am. B 1, 874–878 (1984).
[Crossref]

Z. Deng, “Complete population trapping in two-photon ionization,” Phys. Lett. 105A, 43–44 (1984).

1980 (2)

H. Kildal and S. R. J. Brueck, “Pump depletion and saturation of two-photon resonant third-harmonic generation processes,” IEEE J. Quantum Electron. QE-16, 566–573 (1980).
[Crossref]

J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garrett, “Resonantly enhanced multiphoton ionization,” Phys. Rev. Lett. 45, 114–116 (1980).
[Crossref]

1977 (3)

G. Grynberg and B. Cagnac, “Doppler-free multiphotonic spectroscopy,” Rep. Prog. Phys. 40, 791–841 (1977).
[Crossref]

F. Biraben, B. Cagnac, E. Giacobino, and G. Grynberg, “Broadening and shift of the sodium 3S–4D and 3S–5S two-photon lines perturbed by noble gas,” J. Phys. B 10, 2369–2374 (1977).
[Crossref]

A. T. Georges, P. Lambropoulos, and J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300–307 (1977).
[Crossref]

1976 (1)

J. N. Elgin and G. H. C. New, “Semi-classical theory of two-photon resonant third-harmonic generation,” Opt. Commun. 16, 242–246 (1976).
[Crossref]

1970 (1)

V. M. Arutyunyan, E. G. Kanetsyan, and V. O. Chaltykyan, “Passage of electromagnetic radiation through a resonant medium in the presence of an intense monochromatic wave,” Zh. Eksp. Teor. Fiz. 59, 195–200 (1970) [Sov. Phys. JETP 32, 108–110 (1971)].

Arutyunyan, V. M.

V. M. Arutyunyan, E. G. Kanetsyan, and V. O. Chaltykyan, “Passage of electromagnetic radiation through a resonant medium in the presence of an intense monochromatic wave,” Zh. Eksp. Teor. Fiz. 59, 195–200 (1970) [Sov. Phys. JETP 32, 108–110 (1971)].

Biraben, F.

F. Biraben, B. Cagnac, E. Giacobino, and G. Grynberg, “Broadening and shift of the sodium 3S–4D and 3S–5S two-photon lines perturbed by noble gas,” J. Phys. B 10, 2369–2374 (1977).
[Crossref]

Boyd, R. W.

R. W. Boyd, M. S. Malcuit, D. J. Gaulthier, and K. Rza̧żewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[Crossref] [PubMed]

M. S. Malcuit, D. T. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086–1089 (1985).
[Crossref] [PubMed]

Brueck, S. R. J.

H. Kildal and S. R. J. Brueck, “Pump depletion and saturation of two-photon resonant third-harmonic generation processes,” IEEE J. Quantum Electron. QE-16, 566–573 (1980).
[Crossref]

Cagnac, B.

G. Grynberg and B. Cagnac, “Doppler-free multiphotonic spectroscopy,” Rep. Prog. Phys. 40, 791–841 (1977).
[Crossref]

F. Biraben, B. Cagnac, E. Giacobino, and G. Grynberg, “Broadening and shift of the sodium 3S–4D and 3S–5S two-photon lines perturbed by noble gas,” J. Phys. B 10, 2369–2374 (1977).
[Crossref]

Chaltykyan, V. O.

V. M. Arutyunyan, E. G. Kanetsyan, and V. O. Chaltykyan, “Passage of electromagnetic radiation through a resonant medium in the presence of an intense monochromatic wave,” Zh. Eksp. Teor. Fiz. 59, 195–200 (1970) [Sov. Phys. JETP 32, 108–110 (1971)].

Compton, R. N.

J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garrett, “Resonantly enhanced multiphoton ionization,” Phys. Rev. Lett. 45, 114–116 (1980).
[Crossref]

Deng, Z.

Z. Deng, “Complete population trapping in two-photon ionization,” Phys. Lett. 105A, 43–44 (1984).

Z. Deng, “Line narrowing and photoelectron trapping in multiphoton ionization spectroscopy,” J. Opt. Soc. Am. B 1, 874–878 (1984).
[Crossref]

Elgin, J. N.

J. N. Elgin and G. H. C. New, “Semi-classical theory of two-photon resonant third-harmonic generation,” Opt. Commun. 16, 242–246 (1976).
[Crossref]

Garrett, W. W.

J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garrett, “Resonantly enhanced multiphoton ionization,” Phys. Rev. Lett. 45, 114–116 (1980).
[Crossref]

Gaulthier, D. J.

R. W. Boyd, M. S. Malcuit, D. J. Gaulthier, and K. Rza̧żewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[Crossref] [PubMed]

Gauthier, D. T.

M. S. Malcuit, D. T. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086–1089 (1985).
[Crossref] [PubMed]

Georges, A. T.

A. T. Georges, P. Lambropoulos, and J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300–307 (1977).
[Crossref]

Giacobino, E.

F. Biraben, B. Cagnac, E. Giacobino, and G. Grynberg, “Broadening and shift of the sodium 3S–4D and 3S–5S two-photon lines perturbed by noble gas,” J. Phys. B 10, 2369–2374 (1977).
[Crossref]

Grynberg, G.

F. Biraben, B. Cagnac, E. Giacobino, and G. Grynberg, “Broadening and shift of the sodium 3S–4D and 3S–5S two-photon lines perturbed by noble gas,” J. Phys. B 10, 2369–2374 (1977).
[Crossref]

G. Grynberg and B. Cagnac, “Doppler-free multiphotonic spectroscopy,” Rep. Prog. Phys. 40, 791–841 (1977).
[Crossref]

Hochman, S.

Huber, M. C. E.

M. C. E. Huber and R. J. Sandeman, “The measurement of oscillator strengths,” Rep. Prog. Phys. 49, 397–490 (1986).
[Crossref]

Kanetsyan, E. G.

V. M. Arutyunyan, E. G. Kanetsyan, and V. O. Chaltykyan, “Passage of electromagnetic radiation through a resonant medium in the presence of an intense monochromatic wave,” Zh. Eksp. Teor. Fiz. 59, 195–200 (1970) [Sov. Phys. JETP 32, 108–110 (1971)].

Kildal, H.

H. Kildal and S. R. J. Brueck, “Pump depletion and saturation of two-photon resonant third-harmonic generation processes,” IEEE J. Quantum Electron. QE-16, 566–573 (1980).
[Crossref]

Knight, P. L.

P. M. Radmore and P. L. Knight, “Two-photon ionisation: interference and population trapping,” Phys. Lett. 102A, 180–185 (1984).

Lambropoulos, P.

A. T. Georges, P. Lambropoulos, and J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300–307 (1977).
[Crossref]

Malcuit, M. S.

R. W. Boyd, M. S. Malcuit, D. J. Gaulthier, and K. Rza̧żewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[Crossref] [PubMed]

M. S. Malcuit, D. T. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086–1089 (1985).
[Crossref] [PubMed]

Marburger, J. H.

A. T. Georges, P. Lambropoulos, and J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300–307 (1977).
[Crossref]

Miller, J. C.

J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garrett, “Resonantly enhanced multiphoton ionization,” Phys. Rev. Lett. 45, 114–116 (1980).
[Crossref]

New, G. H. C.

J. N. Elgin and G. H. C. New, “Semi-classical theory of two-photon resonant third-harmonic generation,” Opt. Commun. 16, 242–246 (1976).
[Crossref]

Payne, M. G.

J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garrett, “Resonantly enhanced multiphoton ionization,” Phys. Rev. Lett. 45, 114–116 (1980).
[Crossref]

Radmore, P. M.

P. M. Radmore and P. L. Knight, “Two-photon ionisation: interference and population trapping,” Phys. Lett. 102A, 180–185 (1984).

Rosenbluh, M.

Rza¸zewski, K.

R. W. Boyd, M. S. Malcuit, D. J. Gaulthier, and K. Rza̧żewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[Crossref] [PubMed]

Sandeman, R. J.

M. C. E. Huber and R. J. Sandeman, “The measurement of oscillator strengths,” Rep. Prog. Phys. 49, 397–490 (1986).
[Crossref]

Shevy, Y.

Wynne, J. J.

J. J. Wynne, “Polarization renormalization due to nonlinear optical generation,” Phys. Rev. Lett. 52, 751–754 (1984).
[Crossref]

J. J. Wynne, “Nonlinear optical balance,” in Multiphoton Processes: Proceedings of the 4th International Conference on Multiphoton Processes, Boulder, Colorado, 1987, S. J. Smith and P. L. Knight, eds. (Cambridge U. Press, Cambridge, 1988), pp. 318–327.

IEEE J. Quantum Electron. (1)

H. Kildal and S. R. J. Brueck, “Pump depletion and saturation of two-photon resonant third-harmonic generation processes,” IEEE J. Quantum Electron. QE-16, 566–573 (1980).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

F. Biraben, B. Cagnac, E. Giacobino, and G. Grynberg, “Broadening and shift of the sodium 3S–4D and 3S–5S two-photon lines perturbed by noble gas,” J. Phys. B 10, 2369–2374 (1977).
[Crossref]

Opt. Commun. (1)

J. N. Elgin and G. H. C. New, “Semi-classical theory of two-photon resonant third-harmonic generation,” Opt. Commun. 16, 242–246 (1976).
[Crossref]

Opt. Lett. (1)

Phys. Lett. (2)

P. M. Radmore and P. L. Knight, “Two-photon ionisation: interference and population trapping,” Phys. Lett. 102A, 180–185 (1984).

Z. Deng, “Complete population trapping in two-photon ionization,” Phys. Lett. 105A, 43–44 (1984).

Phys. Rev. A (2)

A. T. Georges, P. Lambropoulos, and J. H. Marburger, “Theory of third-harmonic generation in metal vapors under two-photon resonance conditions,” Phys. Rev. A 15, 300–307 (1977).
[Crossref]

R. W. Boyd, M. S. Malcuit, D. J. Gaulthier, and K. Rza̧żewski, “Competition between amplified spontaneous emission and the four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garrett, “Resonantly enhanced multiphoton ionization,” Phys. Rev. Lett. 45, 114–116 (1980).
[Crossref]

J. J. Wynne, “Polarization renormalization due to nonlinear optical generation,” Phys. Rev. Lett. 52, 751–754 (1984).
[Crossref]

M. S. Malcuit, D. T. Gauthier, and R. W. Boyd, “Suppression of amplified spontaneous emission by the four-wave mixing process,” Phys. Rev. Lett. 55, 1086–1089 (1985).
[Crossref] [PubMed]

Rep. Prog. Phys. (2)

G. Grynberg and B. Cagnac, “Doppler-free multiphotonic spectroscopy,” Rep. Prog. Phys. 40, 791–841 (1977).
[Crossref]

M. C. E. Huber and R. J. Sandeman, “The measurement of oscillator strengths,” Rep. Prog. Phys. 49, 397–490 (1986).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

V. M. Arutyunyan, E. G. Kanetsyan, and V. O. Chaltykyan, “Passage of electromagnetic radiation through a resonant medium in the presence of an intense monochromatic wave,” Zh. Eksp. Teor. Fiz. 59, 195–200 (1970) [Sov. Phys. JETP 32, 108–110 (1971)].

Other (1)

J. J. Wynne, “Nonlinear optical balance,” in Multiphoton Processes: Proceedings of the 4th International Conference on Multiphoton Processes, Boulder, Colorado, 1987, S. J. Smith and P. L. Knight, eds. (Cambridge U. Press, Cambridge, 1988), pp. 318–327.

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Figures (5)

Fig. 1
Fig. 1

Excitation pathways for the two-photon-resonant four-wave mixing. (a) Processes responsible for the most important terms in χ(3). (b) Processes responsible for the terms in χ(3) that have one negative resonant frequency [Eq. (1)].

Fig. 2
Fig. 2

Schematic diagram of the experiment.

Fig. 3
Fig. 3

Three-dimensional plot of the four-wave mixing signal, in arbitrary units. Δνred = 0 when ω1 = Ωbg/2, and Δνgreen = 0 when ω2 = Ωdg − Ωbg/2.

Fig. 4
Fig. 4

Comparison of results (points) with theory (solid curves) for the four-wave mixing (in arbitrary units) as a function of ω1. Each point plotted represents an average over 40 laser pulses. (a) Δνgreen = −1 GHz − 4δgreen, where δgreen ≡ 5.80 GHz; (b) Δνgreen = −1 GHz −2δgreen; (c) Δνgreen = −1 GHz − δgreen; (d) Δνgreen = −1 GHz.

Fig. 5
Fig. 5

Comparison of results (points) with theory (solid curves) for the four-wave mixing (in arbitrary units) as a function of ω2. Each point plotted represents an average over 40 laser pulses. (a) Δνred = 0 (i.e., ω1 = Ωbg/2); (b) Δνred = −δred, where δred ≡ 5.78 GHz; (c) Δνred = −2δred; (d) Δνred = −4δred.

Equations (7)

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P ( 2 ω 1 + ω 2 ) = 0 χ ( 3 ) ( - 2 ω 1 - ω 2 ; ω 1 , ω 1 , ω 2 ) E ( ω 1 ) 2 E ( ω 2 ) ,
χ ( 3 ) ( - 2 ω 1 - ω 2 ; ω 1 , ω 1 , ω 2 ) = 1 0 3 g a c N g [ b μ g c μ c b μ b a μ a g Ω b g - 2 ω 1 - i Γ b × 1 Ω a g - ω 1 ( 1 Ω c g - 2 ω 1 - ω 2 + 1 Ω c g + ω 2 ) + d μ g c μ c d μ d a μ a g Ω d g - ( ω 1 + ω 2 ) - i Γ d ( 1 Ω a g - ω 1 + 1 Ω a g - ω 2 ) × ( 1 Ω c g - 2 ω 1 - ω 2 + 1 Ω c g + ω 1 ) ] .
P ( 2 ω 1 + ω 2 ) = a 1 Ω b g - 2 ω 1 - i Γ b + a 2 Ω d g - ( ω 1 + ω 2 ) - i Γ d ,
z E ( 2 ω 1 + ω 2 ) = i μ 0 ( 2 ω 1 + ω 2 ) 2 2 k u v P ( 3 ) ( 2 ω 1 + ω 2 ) exp ( - i k u v z ) ,
S P ( 2 ω 1 + ω 2 ) 2 = | a 1 / 2 ( Ω b g / 2 ) - ω 1 - i Γ b / 2 + a 2 ( Ω d g - ω 2 ) - ω 1 - i Γ d | 2 .
P ( 2 ω 1 + ω 2 ) = a 1 π δ Ω b g - exp [ - ( Ω - Ω b g ) 2 / δ Ω b g 2 ] d Ω Ω - 2 ω 1 - i Γ b + a 2 π δ Ω d g - exp [ - ( Ω - Ω d g ) 2 / δ Ω d g 2 ] d Ω Ω - ( ω 1 + ω 2 ) - i Γ d .
δ Ω i j = Ω i j ( 2 k T / m ) 1 / 2 / c ,

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