Abstract

We develop an analytical approach to the dynamics of band populations of reverse saturable absorbers modeled by the three-level approximation of the five-level rate equations. We find high-accuracy approximate solutions to these rate equations, taking into account the temporal shape of the incident laser pulse for different regimes of excitation. The results obtained are confirmed by direct numerical integration of the rate equations and are verified by solution of the full system of the rate and the propagation equations. The validity ranges of the approximations are determined. We also prove that for input pulses that are much longer than the lifetime of the first excited state the ground-state depletion obeys the same functional dependence on the input fluence as in the case of rectangular input pulses. The dynamics of the excited states, however, explicitly depends on the pulse shape. We quantitatively estimate the effect of various parameters on the nonlinear absorption coefficient and discuss implementation of the approach by a beam propagation method to reduce the computational time.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  5. J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
    [CrossRef]
  6. J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
    [CrossRef]
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    [CrossRef]
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  13. X. Shang, Y. Liu, G. Tang, G. Zhang, and W. Chen, “Optical nonlinearities of hypocrellin A with the excitation of nanosecond pulses,” J. Opt. Soc. Am. B 15, 1502–1511 (1998).
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    [CrossRef]
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    [CrossRef]
  19. K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
    [CrossRef]
  20. S. R. Mishra, H. S. Rawat, and S. C. Mehendale, “Reverse saturable absorption and optical limiting in C60 solution in the near-infrared,” Appl. Phys. Lett. 71, 46–48 (1997).
    [CrossRef]
  21. F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
    [CrossRef]
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  24. J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
    [CrossRef]
  25. S. N. R. Swatton, K. R. Welford, R. C. Hollins, and J. R. Sambles, “A time resolved double pump–probe experimental technique to characterize excited-state parameters of organic dyes,” Appl. Phys. Lett. 71, 10–12 (1997).
    [CrossRef]
  26. J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
    [CrossRef]

1999

J. Oberle, L. Bramerie, G. Jonusauskas, and C. Rulliere, “Optical-limiting properties of a push–pull diphenyl-butadiene,” Opt. Commun. 169, 325–332 (1999).
[CrossRef]

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

P. Miles, “Bottleneck optical pulse limiters revisited,” Appl. Opt. 38, 566–570 (1999).
[CrossRef]

D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[CrossRef]

1998

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

X. Shang, Y. Liu, G. Tang, G. Zhang, and W. Chen, “Optical nonlinearities of hypocrellin A with the excitation of nanosecond pulses,” J. Opt. Soc. Am. B 15, 1502–1511 (1998).
[CrossRef]

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

1997

S. N. R. Swatton, K. R. Welford, R. C. Hollins, and J. R. Sambles, “A time resolved double pump–probe experimental technique to characterize excited-state parameters of organic dyes,” Appl. Phys. Lett. 71, 10–12 (1997).
[CrossRef]

S. R. Mishra, H. S. Rawat, and S. C. Mehendale, “Reverse saturable absorption and optical limiting in C60 solution in the near-infrared,” Appl. Phys. Lett. 71, 46–48 (1997).
[CrossRef]

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

T. Xia, D. J. Hagan, A. Dogariu, A. Said, and E. W. Van Stryland, “Optimization of optical limiting devices based on excited state absorption,” Appl. Opt. 36, 4110–4122 (1997).
[CrossRef] [PubMed]

S. Hughes, J. M. Burzler, and T. Kobayashi, “Modeling of picosecond-pulse propagation for optical limiting applications in the visible spectrum,” J. Opt. Soc. Am. B 11, 2925–2929 (1997).
[CrossRef]

1996

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

1995

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

1994

C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse saturable absorption in metallophthalocyanine,” Phys. Rev. A 49, 1149–1157 (1994).
[CrossRef] [PubMed]

P. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. 33, 6965–6979 (1994).
[CrossRef] [PubMed]

1993

D. G. Mclean, R. L. Sutherland, M. C. Brant, D. M. Brandelik, P. A. Fleitz, and T. Pottenger, “Nonlinear absorption study of a C60–toluene solution,” Opt. Lett. 18, 858–860 (1993).
[CrossRef]

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

1967

Barroso, J.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

Bartoli, F. J.

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

Bedworth, P. V.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Boggess, T. F.

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

Bovier, C.

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Boyle, M. E.

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

Bramerie, L.

J. Oberle, L. Bramerie, G. Jonusauskas, and C. Rulliere, “Optical-limiting properties of a push–pull diphenyl-butadiene,” Opt. Commun. 169, 325–332 (1999).
[CrossRef]

Brandelik, D. M.

Brant, M. C.

Brinkmann, D.

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Burzler, J. M.

S. Hughes, J. M. Burzler, and T. Kobayashi, “Modeling of picosecond-pulse propagation for optical limiting applications in the visible spectrum,” J. Opt. Soc. Am. B 11, 2925–2929 (1997).
[CrossRef]

Chen, C. T.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Chen, W.

Costela, A.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

Dogariu, A.

Dou, K.

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

Ehrenberg, B.

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

Fleitz, P. A.

Flom, S. R.

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

Garcia-Moreno, I.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

Guo, Y.

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

Hagan, D. J.

Hercher, M.

Hollins, R. C.

S. N. R. Swatton, K. R. Welford, R. C. Hollins, and J. R. Sambles, “A time resolved double pump–probe experimental technique to characterize excited-state parameters of organic dyes,” Appl. Phys. Lett. 71, 10–12 (1997).
[CrossRef]

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Honerlage, B.

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Hughes, S.

S. Hughes, J. M. Burzler, and T. Kobayashi, “Modeling of picosecond-pulse propagation for optical limiting applications in the visible spectrum,” J. Opt. Soc. Am. B 11, 2925–2929 (1997).
[CrossRef]

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Jonusauskas, G.

J. Oberle, L. Bramerie, G. Jonusauskas, and C. Rulliere, “Optical-limiting properties of a push–pull diphenyl-butadiene,” Opt. Commun. 169, 325–332 (1999).
[CrossRef]

Joucla, M.

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Knobbe, E. T.

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

Kobayashi, T.

S. Hughes, J. M. Burzler, and T. Kobayashi, “Modeling of picosecond-pulse propagation for optical limiting applications in the visible spectrum,” J. Opt. Soc. Am. B 11, 2925–2929 (1997).
[CrossRef]

Kovsh, D. I.

Lee, Y. S.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Levy, R.

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Li, C.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse saturable absorption in metallophthalocyanine,” Phys. Rev. A 49, 1149–1157 (1994).
[CrossRef] [PubMed]

Li, F.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Liu, S.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Liu, Y.

Mansour, K.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Marder, S. R.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Mclean, D. G.

Mehendale, S. C.

S. R. Mishra, H. S. Rawat, and S. C. Mehendale, “Reverse saturable absorption and optical limiting in C60 solution in the near-infrared,” Appl. Phys. Lett. 71, 46–48 (1997).
[CrossRef]

Miles, P.

P. Miles, “Bottleneck optical pulse limiters revisited,” Appl. Opt. 38, 566–570 (1999).
[CrossRef]

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

P. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. 33, 6965–6979 (1994).
[CrossRef] [PubMed]

Mishra, S. R.

S. R. Mishra, H. S. Rawat, and S. C. Mehendale, “Reverse saturable absorption and optical limiting in C60 solution in the near-infrared,” Appl. Phys. Lett. 71, 46–48 (1997).
[CrossRef]

Ng, D.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Oberle, J.

J. Oberle, L. Bramerie, G. Jonusauskas, and C. Rulliere, “Optical-limiting properties of a push–pull diphenyl-butadiene,” Opt. Commun. 169, 325–332 (1999).
[CrossRef]

Ohlmann, D.

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Parkhill, R.

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

Perry, J. W.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Pong, R. G. S.

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

Pottenger, T.

Rawat, H. S.

S. R. Mishra, H. S. Rawat, and S. C. Mehendale, “Reverse saturable absorption and optical limiting in C60 solution in the near-infrared,” Appl. Phys. Lett. 71, 46–48 (1997).
[CrossRef]

Rehspringer, J. L.

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Röder, B.

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

Rückmann, I.

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

Rulliere, C.

J. Oberle, L. Bramerie, G. Jonusauskas, and C. Rulliere, “Optical-limiting properties of a push–pull diphenyl-butadiene,” Opt. Commun. 169, 325–332 (1999).
[CrossRef]

Said, A.

Sambles, J. R.

S. N. R. Swatton, K. R. Welford, R. C. Hollins, and J. R. Sambles, “A time resolved double pump–probe experimental technique to characterize excited-state parameters of organic dyes,” Appl. Phys. Lett. 71, 10–12 (1997).
[CrossRef]

Sasabe, H.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Sastre, R.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

Schell, J.

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Serughetti, J.

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

Shang, X.

Shirk, J. S.

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

Snow, A. W.

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

Song, Y.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Spruce, G.

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Stiel, H.

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

Sun, X.

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

Sutherland, R. L.

Swatton, S. N. R.

S. N. R. Swatton, K. R. Welford, R. C. Hollins, and J. R. Sambles, “A time resolved double pump–probe experimental technique to characterize excited-state parameters of organic dyes,” Appl. Phys. Lett. 71, 10–12 (1997).
[CrossRef]

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Tang, G.

Tian, M.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Till, S. J.

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Tutt, L. W.

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

Van Stryland, E. W.

Volkmer, A.

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

Wada, T.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Wang, H.

C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse saturable absorption in metallophthalocyanine,” Phys. Rev. A 49, 1149–1157 (1994).
[CrossRef] [PubMed]

Wang, X.

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

Wang, Y.

C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse saturable absorption in metallophthalocyanine,” Phys. Rev. A 49, 1149–1157 (1994).
[CrossRef] [PubMed]

Welford, K. R.

S. N. R. Swatton, K. R. Welford, R. C. Hollins, and J. R. Sambles, “A time resolved double pump–probe experimental technique to characterize excited-state parameters of organic dyes,” Appl. Phys. Lett. 71, 10–12 (1997).
[CrossRef]

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Wherrett, B. S.

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Wu, X. L.

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Wu, Y.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Xia, T.

Yang, K.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Yang, M.

C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse saturable absorption in metallophthalocyanine,” Phys. Rev. A 49, 1149–1157 (1994).
[CrossRef] [PubMed]

Yang, S.

Yu, C.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Zeug, A.

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

Zhang, G.

Zhang, L.

C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse saturable absorption in metallophthalocyanine,” Phys. Rev. A 49, 1149–1157 (1994).
[CrossRef] [PubMed]

Zhu, P.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Zuo, X.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. R. Mishra, H. S. Rawat, and S. C. Mehendale, “Reverse saturable absorption and optical limiting in C60 solution in the near-infrared,” Appl. Phys. Lett. 71, 46–48 (1997).
[CrossRef]

S. N. R. Swatton, K. R. Welford, R. C. Hollins, and J. R. Sambles, “A time resolved double pump–probe experimental technique to characterize excited-state parameters of organic dyes,” Appl. Phys. Lett. 71, 10–12 (1997).
[CrossRef]

Chem. Phys.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

IEEE J. Quantum Electron.

K. Dou, X. Sun, X. Wang, R. Parkhill, Y. Guo, and E. T. Knobbe, “Optical limiting and nonlinear absorption of excited states in metalloporphyrin-doped sol gels,” IEEE J. Quantum Electron. 35, 1004–1014 (1999).
[CrossRef]

J. Appl. Phys.

F. Li, Y. Song, K. Yang, S. Liu, C. Li, Y. Wu, X. Zuo, C. Yu, and P. Zhu, “Determination of nonlinear absorption mechanisms using a single pulse width laser,” J. Appl. Phys. 82, 2004–2006 (1997).
[CrossRef]

J. Chem. Phys.

J. Schell, D. Brinkmann, D. Ohlmann, B. Honerlage, R. Levy, M. Joucla, J. L. Rehspringer, J. Serughetti, and C. Bovier, “Optical limiting properties and dynamics of induced absorption in C60-doped solid xerogel matrices,” J. Chem. Phys. 108, 8599–8604 (1998).
[CrossRef]

J. Schell, D. Ohlmann, D. Brinkmann, R. Levy, M. Joucla, J. L. Rehspringer, and B. Honerlage, “Reverse saturable absorption in C60-doped porous glasses studied by single- and double-pulse pump–probe experiments,” J. Chem. Phys. 111, 5929–5937 (1999).
[CrossRef]

J. Opt. Soc. Am. B

X. Shang, Y. Liu, G. Tang, G. Zhang, and W. Chen, “Optical nonlinearities of hypocrellin A with the excitation of nanosecond pulses,” J. Opt. Soc. Am. B 15, 1502–1511 (1998).
[CrossRef]

S. Hughes, J. M. Burzler, and T. Kobayashi, “Modeling of picosecond-pulse propagation for optical limiting applications in the visible spectrum,” J. Opt. Soc. Am. B 11, 2925–2929 (1997).
[CrossRef]

Mater. Res. Soc. Symp. Proc.

K. R. Welford, S. N. R. Swatton, S. Hughes, S. J. Till, G. Spruce, R. C. Hollins, and B. S. Wherrett, “Nonlinear absorption in organic dyes,” Mater. Res. Soc. Symp. Proc. 374, 239–256 (1995).
[CrossRef]

Opt. Commun.

H. Stiel, A. Volkmer, I. Rückmann, A. Zeug, B. Ehrenberg, and B. Röder, “Non-linear and transient absorption spectroscopy of magnesium(II)-tetrabenzoporphyrin in solution,” Opt. Commun. 155, 135–143 (1998).
[CrossRef]

J. Oberle, L. Bramerie, G. Jonusauskas, and C. Rulliere, “Optical-limiting properties of a push–pull diphenyl-butadiene,” Opt. Commun. 169, 325–332 (1999).
[CrossRef]

Opt. Lett.

Phys. Rev. A

C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse saturable absorption in metallophthalocyanine,” Phys. Rev. A 49, 1149–1157 (1994).
[CrossRef] [PubMed]

Prog. Quantum Electron.

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

Pure Appl. Opt.

J. S. Shirk, R. G. S. Pong, S. R. Flom, F. J. Bartoli, M. E. Boyle, and A. W. Snow, “Lead phthalocyanine reverse saturable absorption optical limiters,” Pure Appl. Opt. 5, 701–707 (1996).
[CrossRef]

Science

J. W. Perry, K. Mansour, Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear response,” Science 273, 1533–1536 (1996).
[CrossRef]

Other

E. W. Van Stryland, D. J. Hagan, T. Xia, and A. A. Said, “Application of nonlinear optics to passive optical limiting,” in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. (CRC Press, New York, 1997), pp. 840–860.

J. W. Perry, “Organic and metal-containing reverse saturable absorbers for optical limiters,” in Nonlinear Optics of Organic Molecules and Polymers, H. S. Nalwa and S. Miyata, eds. (CRC Press, New York, 1997), pp. 813–840.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

E. W. Van Stryland and M. Sheik-Bahae, “Z-scan,” in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, M. G. Kuzyk and C. W. Dirk, eds. (Marcel Dekker, New York, 1998), pp. 655–692.

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the five-level model of ESA: 0, ground state; 1, 2, excited singlet states; 3, 4, triplet states.

Fig. 2
Fig. 2

Dynamics of fractional populations excited by long hyperbolic secant f(T)=A sech2 T pulses. The results are obtained by use of analytical formulas (15), (16), and (4) (solid curves) and by numerical integration of the three-level model of Eqs. (3) (dashed curves). The exact potential P(T) given by Eq. (7) (dashed curves) and approximate potential P˜(T) given by formula (11) (solid curves) are shown at the right. Parameters: (a) w=20, A=2.5, ϕ=0.5; (b) w=5, A=2.5, ϕ=0.5; (c) w=10, A=2.5, ϕ=0.8; (d) w=10, A=2.5, ϕ=0.3. The input pulse shape is shown for reference by a dotted line in (a).

Fig. 3
Fig. 3

Dynamics of fractional populations excited by short hyperbolic secant f(T)=A sech2 T pulses. The results are obtained by use of analytical formulas (19), (21), (solid curves) and by numerical integration of the three-level model of Eqs. (3) (dashed curves). Parameters: (a) w=0.05, A=1, ϕ=0.5; (b) w=0.2, A=1, ϕ=0.5; (c) w=0.05, A=1, ϕ=0.8; (d) w=0.05, A=1, ϕ=0.3. The input pulse shape is shown for reference by a dotted curve in (a).

Fig. 4
Fig. 4

Population dynamics for a hyperbolic secant pulse f(T)=A sech2 T of intermediate length. Results obtained with exact [Eq. (7)] and approximate [Eq. (22)] potential are compared. Parameters: w=2, A=3, ϕ=0.5α=-0.207, β=-1.

Fig. 5
Fig. 5

Irradiance effective cross section σeffI given by Eq. (23) obtained with analytical formulas for (a) long, w=10, and (b) short, w=0.05, input pulses. Parameters: σ¯s=10, σ¯t=30, ϕ=0.5; the pulse shape is a hyperbolic secant.

Fig. 6
Fig. 6

Transmitted fluence F(Z)/F(0) calculated with Eq. (29) (solid curves) and with Eq. (30) (dashed curve), F(0)/Fs=5. The results obtained from the numerical integration of the full system of partial differential equations (3) and (31) are shown by circles. Parameters: (a) σ¯s=10, σ¯t=30, w=10, ϕ=0.5; (b) σ¯s=20, σ¯t=5, w=7.5, ϕ=0.3.

Fig. 7
Fig. 7

Numerical integration of the full system of Eqs. (3) and (31). The initial normalized shape of the pulse is f(T)=A sech2 T, A=2.5. Parameters correspond to those of Fig. 6.

Fig. 8
Fig. 8

Transmitted fluence F(Z)/F(0) as a function of initial fluence F(Z=0)/Fs for a fixed linear transmittance (a), (b) Z=0.1TL=0.9 and (c), (d) Z=0.3TL=0.74. Solid and dashed curves, results of integration of Eqs. (29) and (30), respectively. Circles, numerical solutions of the rate and the propagation equations (3) and (31). Parameters: (a), (c) σ¯s=10, σ¯t=30, w=10, ϕ=0.5; (b), (d) σ¯s=20, σ¯t=5, w=7.5, ϕ=0.3.

Fig. 9
Fig. 9

Transmitted fluence F(Z)/F(0) for absorption of short pulses. Solid curve, prediction of the slow- absorber approximation [Eq. (30)]. Numerical integration of Eqs. (3) and (31) is shown by circles for w=0.05 and by diamonds for w=0.2. Parameters: A=1F/Fs=2, ϕ=0.5; initial irradiance profile, f(T)=A sech2 T.

Equations (39)

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

dI(z, t)dz=-I(z, t)[σg N0(I, t)+σs N1(I, t)+σt N3(I, t)],
dN0dt=-σgI(t)ωN0+N1τ10,
dN1dt=σgI(t)ωN0-N1τ1,
dN3dt=N1τ13.
dn0dT=-f(T)n0+w(1-ϕ)n1,
dn1dT=f(T)n0-wn1,
dn3dT=wϕn1,
n0(T)+n1(T)+n3(T)=1.
u(T)=n0(T)exp½[f(T)+w]dT,
u+P(T)u=0.
P(T)=f(T)2-f2(T)4+wϕ-12f(T)-w24.
v+v2+P(T)=0.
n0(T)=C0+C1exp[-2V0(T)]dT×expV0(T)-F(T)+wT2,
n1(T)=expV0(T)-F(T)+wT2w(1-ϕ)×C1 exp[-2V0(T)]+C0+C1 exp[-2V0(T)]dT×v0(T)+f(T)-w2,
n1(T)=C2+f(T)n0(T) exp(wT)dTexp(-wT),
P(T)P˜(T)
=ϕ-12f(T)-12-ϕf(T)+w22.
ΔP(T)=P(T)-P˜(T)=(1-ϕ)f(T)+ϕ(ϕ-1)f2(T).
n0(T)exp[-ϕΔF(T)],
n1(T)f(T)wn0(T),
n0(T)exp[-ϕA(tanh T+1)],
n1(T)Aw sech2T exp[-ϕA(tanh T+1)].
P(T)P˜(T)=f(T)2-f2(T)4.
n0(T)={1+0[F(T)]} exp[-ΔF(T)-wΔT/2],
n0(T)1+w4ΔS(T)×exp[-A(tanh T+1)-w(T-T0)/2],
n1(T)=1-n0(T)-n3(T),
n3(T)w2{ΔT-1[F(T)]},
n3(T)=w4[2ΔT-ΔS¯(T)],
P˜(T)=-αf(T)-[αf(T)+β]2.
σeffI(T)=n0(T)+σ¯sn1(T)+σ¯tn3(T),
dFdZ=--tσeffI(t)I(t)dt=-0FσeffI(F)dF=-σeffF(F)F.
σeffF(F)=1F 0FσeffI(F)dF,
σeffI,long(T)=σ¯t+n0(T)1-σ¯t+(σ¯s-σ¯t) f(T)w.
σeffI,long(F)=σ¯t+1-σ¯t+σ¯s-σ¯twFs dFdTexp-ϕ FFs.
σeffF,long(F)
=σ¯t+(1-σ¯t) FsϕF×[1-exp(-ϕ F/Fs)]+2(σ¯s-σ¯t)wϕ FsϕF×2 exp(-ϕF/Fs)-1ϕF/Fs+exp(-ϕF/Fs)+1.
dYdZ=(σ¯t-1)[1-exp(-Y)]-σ¯tY-μ2 exp(-Y)-1Y+exp(-Y)+1,
dYdZ=(σ¯t-1)[1-exp(-Y)]-σ¯tY,
df(Z, T)dZ=-σeffI(T)f(Z, T),

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