Abstract

We present experimental results on the measurement of the population-relaxation times of the excited states of some metalloporphyrins such as zinc meso-tetra-phenylporphyrin (ZnmTPP), zinc meso-tetra (p-methoxyphenyl) tetrabenzporphyrin (ZnmpTBP), tetratollylporphyrin (TTP), cobalt tetratollylporphyrin (CoTTP) and nickel tetratollylporphyrin (NiTTP) with time-resolved degenerate four-wave mixing with the incoherent emission of a nanosecond pumped-dye laser. The experiments are carried out in the counterpropagating geometry (k4=k1+k2-k3), where k1 and k2 are counterpropagating to one another and k3 is at angle θ (<90° with respect to k1). Excited-state and ground-state relaxation times are calculated based on the theoretical formalism developed by Hiromi Okamoto [J. Opt. Soc. Am. B , 10, 2353, 1993]. The signals are recorded by delaying the probe beam (k3) for different fixed delays of the backward pump beam (k2). We observe strong signal at a second peak owing to the coherence of beams k2 and k3 for samples exhibiting excited-state absorption and a narrow peak corresponding to the coherence of beams k1 and k3 for samples that do not exhibit excited-state or reverse-saturable absorption. The nonlinear absorption is studied with the Z-scan technique. From the experimental data, we estimate the relaxation times of excited states.

© 1997 Optical Society of America

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  1. C. B. Harris, E. P. Ippen, G. A. Mourou, and A. H. Zewail, eds. Ultrafast Phenomena VII (Springer-Verlag, Berlin, 1990).
  2. N. Morita and T. Yajima, “Ultrahigh-time-resolved coherent transient spectroscopy with incoherentlight,” Phys. Rev. A 30, 2525–2536 (1984).
    [CrossRef]
  3. T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond–picosecondrelaxation in condensed phase,” Appl. Phys. B 47, 107–125 (1988).
    [CrossRef]
  4. H. Okamoto, “Ultrafast population relaxation by time-resolved degenerate four-wavemixing with incoherent light and analysis under breakdown of the two-levelapproximation,” J. Opt. Soc. Am. B 10, 2353–2357 (1993).
    [CrossRef]
  5. N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
    [CrossRef]
  6. D. V. G. L. N. Rao, F. J. Aranda, J. F. Roach, and D. E. Remy, “Third-order nonlinear optical interactions of some benzporphyrins,” Appl. Phys. Lett. 58, 1241–1243 (1991).
    [CrossRef]
  7. W. Blau, H. Byrne, W. H. Dennis, and J. M. Kelly, “Reverse saturable absorption in tetra phenyl porphyrins,” Opt. Commun. 56, 25–29 (1985).
    [CrossRef]
  8. D. V. G. L. N. Rao, F. J. Aranda, D. E. Remy, and J. F. Roach, “Third-order nonlinear optical interactions in tetrabenzporphyrins,” Int. J. Nonlinear Opt. Phys. 3, 511–529 (1994).
    [CrossRef]
  9. J. Bredas and R. Silbey, eds., Conjugated Polymers (Kluwer Academic, Dordrecht, The Netherlands, 1991).
  10. B. Meunier, “Metallo porphyrins as versatile catalysts for oxidation reactions andoxidative DNA cleavage,” Chem. Rev. 92, 1411–1456 (1992).
    [CrossRef]
  11. B. W. Handerson and T. J. Dougherty, PhotodynamicTherapy: Basic Principles and Clinical Applications (Marcel Dekker, New York, 1992).
  12. P. N. Prasad and D. R. Ulrich, eds., NonlinearOptical and Electroactive Polymers (Plenum, New York, 1988).
  13. A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds., “NonlinearOptical Properties of Polymers,” Mater. Res. Soc. Symp. Proc. 109 (1988).
  14. D. Narayana Rao and V. Nirmal Kumar, “Experimental demonstration of spectral modification in a Mach–Zehnderinterferometer,” J. Mod. Opt. 41, 1757–1763 (1994).
    [CrossRef]
  15. D. J. Ulness and A. C. Albrecht, “Four wave mixing in a Bloch two-level system with incoherent laserlight having a Lorentzian spectral density: analytic and diagrammatic approach,” Phys. Rev. A 53, 1081 (1996).
    [CrossRef] [PubMed]
  16. J. Liu, S. Huang, W. Qin, and J. Yu, “Investigation of picosecond relaxation processes in cresyl violet,” Opt. Commun. 91, 87 (1992).
    [CrossRef]
  17. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  18. G. L. Wood, M. J. Miller, and A. G. Mott, “Investigation of tetrabenzporphyrin by the Z-scan technique,” Opt. Lett. 20, 973–975 (1995).
    [CrossRef] [PubMed]
  19. F. Laermer, T. Elsaesser, and W. Kaiser, “Ultrashort vibronic and thermal relaxation of dye molecules after femtosecondultraviolet excitation,” Chem. Phys. Lett. 156, 381–386 (1989); A. Mokhtari, J. Chesnoy, and A. Lauberau, “Femtosecond time- and frequency-resolved fluorescence spectroscopyof a dye molecule,” Chem. Phys. Lett. 155, 593–597 (1989); G. Angel, R. Gagel, and A. Lauberau, “Femtosecond polarisation spectroscopy of liquid dye solutions,” Chem. Phys. CMPHC2 131, 129–134 (1989).
    [CrossRef]
  20. J. Rodriguez, C. Kirmaier, and D. Holten, “Optical properties of metalloporphyrin excited states,” J. Am. Chem. Soc. 111, 6500–6506 (1989); J. Rodriguez, C. Kirmaier, and D. Holten, “Time-resolved and static optical properties of vibrationally excitedporphyrins,” J. Chem. Phys. 94, 6020–6029 (1994).
    [CrossRef]

1996 (1)

D. J. Ulness and A. C. Albrecht, “Four wave mixing in a Bloch two-level system with incoherent laserlight having a Lorentzian spectral density: analytic and diagrammatic approach,” Phys. Rev. A 53, 1081 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (2)

D. Narayana Rao and V. Nirmal Kumar, “Experimental demonstration of spectral modification in a Mach–Zehnderinterferometer,” J. Mod. Opt. 41, 1757–1763 (1994).
[CrossRef]

D. V. G. L. N. Rao, F. J. Aranda, D. E. Remy, and J. F. Roach, “Third-order nonlinear optical interactions in tetrabenzporphyrins,” Int. J. Nonlinear Opt. Phys. 3, 511–529 (1994).
[CrossRef]

1993 (2)

H. Okamoto, “Ultrafast population relaxation by time-resolved degenerate four-wavemixing with incoherent light and analysis under breakdown of the two-levelapproximation,” J. Opt. Soc. Am. B 10, 2353–2357 (1993).
[CrossRef]

N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
[CrossRef]

1992 (2)

B. Meunier, “Metallo porphyrins as versatile catalysts for oxidation reactions andoxidative DNA cleavage,” Chem. Rev. 92, 1411–1456 (1992).
[CrossRef]

J. Liu, S. Huang, W. Qin, and J. Yu, “Investigation of picosecond relaxation processes in cresyl violet,” Opt. Commun. 91, 87 (1992).
[CrossRef]

1991 (1)

D. V. G. L. N. Rao, F. J. Aranda, J. F. Roach, and D. E. Remy, “Third-order nonlinear optical interactions of some benzporphyrins,” Appl. Phys. Lett. 58, 1241–1243 (1991).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

1988 (1)

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond–picosecondrelaxation in condensed phase,” Appl. Phys. B 47, 107–125 (1988).
[CrossRef]

1985 (1)

W. Blau, H. Byrne, W. H. Dennis, and J. M. Kelly, “Reverse saturable absorption in tetra phenyl porphyrins,” Opt. Commun. 56, 25–29 (1985).
[CrossRef]

1984 (1)

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

Albrecht, A. C.

D. J. Ulness and A. C. Albrecht, “Four wave mixing in a Bloch two-level system with incoherent laserlight having a Lorentzian spectral density: analytic and diagrammatic approach,” Phys. Rev. A 53, 1081 (1996).
[CrossRef] [PubMed]

Aranda, F. J.

D. V. G. L. N. Rao, F. J. Aranda, D. E. Remy, and J. F. Roach, “Third-order nonlinear optical interactions in tetrabenzporphyrins,” Int. J. Nonlinear Opt. Phys. 3, 511–529 (1994).
[CrossRef]

D. V. G. L. N. Rao, F. J. Aranda, J. F. Roach, and D. E. Remy, “Third-order nonlinear optical interactions of some benzporphyrins,” Appl. Phys. Lett. 58, 1241–1243 (1991).
[CrossRef]

Awaji, H.

N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
[CrossRef]

Blau, W.

W. Blau, H. Byrne, W. H. Dennis, and J. M. Kelly, “Reverse saturable absorption in tetra phenyl porphyrins,” Opt. Commun. 56, 25–29 (1985).
[CrossRef]

Byrne, H.

W. Blau, H. Byrne, W. H. Dennis, and J. M. Kelly, “Reverse saturable absorption in tetra phenyl porphyrins,” Opt. Commun. 56, 25–29 (1985).
[CrossRef]

Dennis, W. H.

W. Blau, H. Byrne, W. H. Dennis, and J. M. Kelly, “Reverse saturable absorption in tetra phenyl porphyrins,” Opt. Commun. 56, 25–29 (1985).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Hattori, T.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond–picosecondrelaxation in condensed phase,” Appl. Phys. B 47, 107–125 (1988).
[CrossRef]

Huang, S.

J. Liu, S. Huang, W. Qin, and J. Yu, “Investigation of picosecond relaxation processes in cresyl violet,” Opt. Commun. 91, 87 (1992).
[CrossRef]

Kelly, J. M.

W. Blau, H. Byrne, W. H. Dennis, and J. M. Kelly, “Reverse saturable absorption in tetra phenyl porphyrins,” Opt. Commun. 56, 25–29 (1985).
[CrossRef]

Kobayashi, N.

N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
[CrossRef]

Kobayashi, T.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond–picosecondrelaxation in condensed phase,” Appl. Phys. B 47, 107–125 (1988).
[CrossRef]

Kurokawa, K.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond–picosecondrelaxation in condensed phase,” Appl. Phys. B 47, 107–125 (1988).
[CrossRef]

Liu, J.

J. Liu, S. Huang, W. Qin, and J. Yu, “Investigation of picosecond relaxation processes in cresyl violet,” Opt. Commun. 91, 87 (1992).
[CrossRef]

Meunier, B.

B. Meunier, “Metallo porphyrins as versatile catalysts for oxidation reactions andoxidative DNA cleavage,” Chem. Rev. 92, 1411–1456 (1992).
[CrossRef]

Miller, M. J.

Mizunuma, S.

N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
[CrossRef]

Morita, N.

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

Mott, A. G.

Narayana Rao, D.

D. Narayana Rao and V. Nirmal Kumar, “Experimental demonstration of spectral modification in a Mach–Zehnderinterferometer,” J. Mod. Opt. 41, 1757–1763 (1994).
[CrossRef]

Nevin, W. A.

N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
[CrossRef]

Nirmal Kumar, V.

D. Narayana Rao and V. Nirmal Kumar, “Experimental demonstration of spectral modification in a Mach–Zehnderinterferometer,” J. Mod. Opt. 41, 1757–1763 (1994).
[CrossRef]

Okamoto, H.

Qin, W.

J. Liu, S. Huang, W. Qin, and J. Yu, “Investigation of picosecond relaxation processes in cresyl violet,” Opt. Commun. 91, 87 (1992).
[CrossRef]

Rao, D. V. G. L. N.

D. V. G. L. N. Rao, F. J. Aranda, D. E. Remy, and J. F. Roach, “Third-order nonlinear optical interactions in tetrabenzporphyrins,” Int. J. Nonlinear Opt. Phys. 3, 511–529 (1994).
[CrossRef]

D. V. G. L. N. Rao, F. J. Aranda, J. F. Roach, and D. E. Remy, “Third-order nonlinear optical interactions of some benzporphyrins,” Appl. Phys. Lett. 58, 1241–1243 (1991).
[CrossRef]

Remy, D. E.

D. V. G. L. N. Rao, F. J. Aranda, D. E. Remy, and J. F. Roach, “Third-order nonlinear optical interactions in tetrabenzporphyrins,” Int. J. Nonlinear Opt. Phys. 3, 511–529 (1994).
[CrossRef]

D. V. G. L. N. Rao, F. J. Aranda, J. F. Roach, and D. E. Remy, “Third-order nonlinear optical interactions of some benzporphyrins,” Appl. Phys. Lett. 58, 1241–1243 (1991).
[CrossRef]

Roach, J. F.

D. V. G. L. N. Rao, F. J. Aranda, D. E. Remy, and J. F. Roach, “Third-order nonlinear optical interactions in tetrabenzporphyrins,” Int. J. Nonlinear Opt. Phys. 3, 511–529 (1994).
[CrossRef]

D. V. G. L. N. Rao, F. J. Aranda, J. F. Roach, and D. E. Remy, “Third-order nonlinear optical interactions of some benzporphyrins,” Appl. Phys. Lett. 58, 1241–1243 (1991).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Terasaki, A.

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond–picosecondrelaxation in condensed phase,” Appl. Phys. B 47, 107–125 (1988).
[CrossRef]

Ulness, D. J.

D. J. Ulness and A. C. Albrecht, “Four wave mixing in a Bloch two-level system with incoherent laserlight having a Lorentzian spectral density: analytic and diagrammatic approach,” Phys. Rev. A 53, 1081 (1996).
[CrossRef] [PubMed]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

Wood, G. L.

Yajima, T.

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

Yamaguchi, M.

N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
[CrossRef]

Yu, J.

J. Liu, S. Huang, W. Qin, and J. Yu, “Investigation of picosecond relaxation processes in cresyl violet,” Opt. Commun. 91, 87 (1992).
[CrossRef]

Appl. Phys. B (1)

T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, “The application of incoherent light for the study of femtosecond–picosecondrelaxation in condensed phase,” Appl. Phys. B 47, 107–125 (1988).
[CrossRef]

Appl. Phys. Lett. (1)

D. V. G. L. N. Rao, F. J. Aranda, J. F. Roach, and D. E. Remy, “Third-order nonlinear optical interactions of some benzporphyrins,” Appl. Phys. Lett. 58, 1241–1243 (1991).
[CrossRef]

Chem. Phys. Lett. (1)

N. Kobayashi, W. A. Nevin, S. Mizunuma, H. Awaji, and M. Yamaguchi, “Ring-expanded porphyrins as an approach towards highly conductive molecularsemiconductors,” Chem. Phys. Lett. 205, 51–54 (1993).
[CrossRef]

Chem. Rev. (1)

B. Meunier, “Metallo porphyrins as versatile catalysts for oxidation reactions andoxidative DNA cleavage,” Chem. Rev. 92, 1411–1456 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[CrossRef]

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

D. V. G. L. N. Rao, F. J. Aranda, D. E. Remy, and J. F. Roach, “Third-order nonlinear optical interactions in tetrabenzporphyrins,” Int. J. Nonlinear Opt. Phys. 3, 511–529 (1994).
[CrossRef]

J. Mod. Opt. (1)

D. Narayana Rao and V. Nirmal Kumar, “Experimental demonstration of spectral modification in a Mach–Zehnderinterferometer,” J. Mod. Opt. 41, 1757–1763 (1994).
[CrossRef]

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

Opt. Commun. (2)

W. Blau, H. Byrne, W. H. Dennis, and J. M. Kelly, “Reverse saturable absorption in tetra phenyl porphyrins,” Opt. Commun. 56, 25–29 (1985).
[CrossRef]

J. Liu, S. Huang, W. Qin, and J. Yu, “Investigation of picosecond relaxation processes in cresyl violet,” Opt. Commun. 91, 87 (1992).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

D. J. Ulness and A. C. Albrecht, “Four wave mixing in a Bloch two-level system with incoherent laserlight having a Lorentzian spectral density: analytic and diagrammatic approach,” Phys. Rev. A 53, 1081 (1996).
[CrossRef] [PubMed]

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

Other (7)

C. B. Harris, E. P. Ippen, G. A. Mourou, and A. H. Zewail, eds. Ultrafast Phenomena VII (Springer-Verlag, Berlin, 1990).

J. Bredas and R. Silbey, eds., Conjugated Polymers (Kluwer Academic, Dordrecht, The Netherlands, 1991).

B. W. Handerson and T. J. Dougherty, PhotodynamicTherapy: Basic Principles and Clinical Applications (Marcel Dekker, New York, 1992).

P. N. Prasad and D. R. Ulrich, eds., NonlinearOptical and Electroactive Polymers (Plenum, New York, 1988).

A. J. Heeger, J. Orenstein, and D. R. Ulrich, eds., “NonlinearOptical Properties of Polymers,” Mater. Res. Soc. Symp. Proc. 109 (1988).

F. Laermer, T. Elsaesser, and W. Kaiser, “Ultrashort vibronic and thermal relaxation of dye molecules after femtosecondultraviolet excitation,” Chem. Phys. Lett. 156, 381–386 (1989); A. Mokhtari, J. Chesnoy, and A. Lauberau, “Femtosecond time- and frequency-resolved fluorescence spectroscopyof a dye molecule,” Chem. Phys. Lett. 155, 593–597 (1989); G. Angel, R. Gagel, and A. Lauberau, “Femtosecond polarisation spectroscopy of liquid dye solutions,” Chem. Phys. CMPHC2 131, 129–134 (1989).
[CrossRef]

J. Rodriguez, C. Kirmaier, and D. Holten, “Optical properties of metalloporphyrin excited states,” J. Am. Chem. Soc. 111, 6500–6506 (1989); J. Rodriguez, C. Kirmaier, and D. Holten, “Time-resolved and static optical properties of vibrationally excitedporphyrins,” J. Chem. Phys. 94, 6020–6029 (1994).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Structure of the different compounds used in the study.

Fig. 2
Fig. 2

Absorption spectra of (a) CoTTP (in chloroform), (b) ink (in distilled water), and (c) Rhodamine B (in methanol).

Fig. 3
Fig. 3

Phase-conjugate intensity as a function of delay τ for ZnmTPP: (a) δ=-4 ps (b) δ=0 ps and (c) δ=+6.5 ps.

Fig. 4
Fig. 4

Phase-conjugate intensity as a function of delay τ for (a) TTP, (b) CoTTP, and (c) NiTTP. δ=+4 ps for all samples.

Fig. 5
Fig. 5

Phase-conjugate intensity as a function of delay τ for Rhodamine B for δ=+1.5 ps. Inset: autocorrelation trace.

Fig. 6
Fig. 6

Open aperture z-scans for different samples (a) CoTTP, (b) Ink, and (c) Rhodamine B.

Fig. 7
Fig. 7

Energy-level diagram for the porphyrins. K, transition rate.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

J(τ, δ)=-dt0|P3(t0;τ, δ)|2,
E(t)=(t)exp[-iω0(t)+ik1·r]+(t-δ)×exp[-iω0(t-δ)-ik2·r]+(t-τ)×exp[-iω0(t-τ)+ik3·r]+c.c.
I0α[4/(γgg+γee)+1/γgg+1/γee](1/τc),
Iδα(1/γgg+1/γee)2(1/τc)2+(1/γgg+1/γee)(4/τc),
Ibα[4/(γgg+τee)+1/γgg+1/γee](1/τc)+8.
I0α1/(γggτc),
Iδα1/(γggτc)2,
Ibα1/(γggτc),

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