Abstract

We introduce a generalized numerical method to calculate short-pulsed laser propagation in a wide class of multiphoton absorbing materials. The method has no restrictions on the input pulse widths varying from nanosecond to femtosecond, and its numerical solution is both radially and temporarily dependent, enabling us to check numerically the validity of assuming radially constant solutions, which ensures that the true peak intensity falls below the damage causing level. A new feature of our technique enables us to determine quantitatively the contributions to the total absorption due to every electronic energy level. We found excellent agreement between our calculations and experiments using sample materials ranging from reverse saturable absorbers, two-photon absorbers with excited-state absorption to three-photon absorbers. We applied our technique to a two-photon absorber with excited-state absorption and found approximately 1 order of magnitude increase in the absorption when femtosecond pulses were used in place of nanosecond pulses.

© 2006 Optical Society of America

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2005 (1)

2004 (7)

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, "Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores," J. Phys. Chem. 120, 5275-5284 (2004).
[CrossRef]

R. Kannan, G. S. He, T.-C. Lin, P. N. Prasad, R. A. Vaia, and L.-S. Tan, "Toward highly active two-photon absorbing liquids. Synthesis and characterization of 1,3,5-triazine-based octupolar molecules," Chem. Mater. 16, 185-194 (2004).
[CrossRef]

I. C. Khoo, A. Diaz, and J. Ding, "Nonlinear-absorbing fiber array for large-dynamic-range optical limiting application against intense short laser pulses," J. Opt. Soc. Am. B 21, 1234-1240 (2004).
[CrossRef]

E. S. Marmur, C. D. Schmults, and D. J. Goldberg, "A review of laser and photodynamic therapy for the treatment of nonmelanoma skin cancer," Dermatol. Surg. 30, 264-271 (2004).
[CrossRef] [PubMed]

J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, L.-S. Tan, and P. A. Fleitz, "Understanding the one-photon photophysical properties of a two-photon absorbing chromophore," J. Phys. Chem. A 108, 5514-5520 (2004).
[CrossRef]

W. Jia, E. P. Douglas, F. Guo, and W. Sun, "Optical limiting of semiconductor nanoparticles for nanosecond laser pulses," Appl. Phys. Lett. 85, 6326-6328 (2004).
[CrossRef]

P. N. Prasad, "Emerging opportunities at the interface of photonics, nanotechnology and biotechnology," Mol. Cryst. Liq. Cryst. 415, 1-10 (2004).
[CrossRef]

2003 (1)

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

2002 (2)

D.-Y. Wang, C.-L. Zhan, Y. Chen, Y.-J. Li, Z.-Z. Lu, and Y.-Z. Nie, "Large optical power limiting induced by three-photon absorption of two stibazolium-like dyes," Chem. Phys. Lett. 369, 621-626 (2002).
[CrossRef]

A. Baev, F. Gel'mukhanov, P. Macak, Y. Luo, and H. Ågren, "General theory for pulse propagation in two-photon active media," J. Phys. Chem. 117, 6214-6220 (2002).
[CrossRef]

2001 (3)

I.-C. Khoo, A. Diaz, M. V. Wood, and P. H. Chen, "Passive optical limiting of picosecond-nanosecond laser pulses using highly nonlinear organic liquid cored fiber array," IEEE J. Sel. Top. Quantum Electron. 7, 760-768 (2001).
[CrossRef]

D. A. Oulianov, I. V. Tomov, A. S. Dvornikow, and R. M. Rentzepis, "Observations on the measurements of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

S. M. Kirkpatrick, R. R. Naik, and M. O. Stone, "Nonlinear saturation and determination of the two-photon absorption cross section of green fluorescent protein," J. Phys. Chem. B 105, 2867-2873 (2001).
[CrossRef]

2000 (2)

M. J. Potasek, S. Kim, and D. McLaughlin, "All optical power limiting," J. Nonlinear Opt. Phys. Mater. 9, 343-365 (2000).
[CrossRef]

A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, "Analytical approach to dynamics of reverse saturable absorbers," J. Opt. Soc. Am. B 17, 1884-1894 (2000).
[CrossRef]

1999 (4)

I. C. Khoo, P. H. Chen, M. V. Wood, and M.-Y. Shih, "Molecular photonics of a highly nonlinear organic fiber core liquid for picosecond-nanosecond optical limiting application," Chem. Phys. 245, 517-531 (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]

B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

H. E. Pudavar, M. P. Joshi, P. N. Prasad, and B. A. Reinhardt, "High-density three-dimensional optical data storage in a stacked compact disk format with two-photon writing and single photon readout," Appl. Phys. Lett. 74, 1338-1340 (1999).
[CrossRef]

1998 (4)

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, and P. N. Prasad, "Highly active two-photon dyes: design, synthesis, and characterization toward application," Chem. Mater. 10, 1863-1874 (1998).
[CrossRef]

G. Witzgall, R. Vrijen, and E. Yablonovitch, "Single-shot two-photon exposure of commercial photoresist for the production of three-dimensional structures," Opt. Lett. 23, 1745-1748 (1998).
[CrossRef]

S. Maruo and S. J. Kawata, "Two-photon-absorbed near-infrared photopolymerization for three-dimensional microfabridation," J. Microelectromech. Syst. 7, 411-415 (1998).
[CrossRef]

1997 (1)

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]

1995 (1)

C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, "Excited-state nonlinear absorption in multi-energy-level molecular systems," Phys. Rev. A 51, 569-575 (1995).
[CrossRef] [PubMed]

1994 (1)

C. W. Gardiner and A. S. Parkins, "Driving atoms with light of arbitrary statistics," Phys. Rev. A 50, 1792-1806 (1994).
[CrossRef] [PubMed]

1993 (2)

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] [PubMed]

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, 299-305 (1993).
[CrossRef]

1992 (1)

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

1986 (1)

A. Barchielli, "Measurement theory and stochastic differential equations in quantum mechanics," Phys. Rev. A 34, 1642-1649 (1986).
[CrossRef] [PubMed]

1985 (1)

C. W. Gardiner and M. J. Collet, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

1982 (1)

N. Allard and J. Kielkopf, "The effect of neutral nonresonant collisions on atomic spectral lines," Rev. Mod. Phys. 54, 1103-1182 (1982).
[CrossRef]

1974 (1)

J. Kleinschmidt, S. Rentsch, W. Tottleben, and B. Wilhelmi, "Measurement of strong nonlinear absorption in stilbene-chloroform solution, explained by the superposition of two-photon absorption and one-photon absorption from the excited state," Chem. Phys. Lett. 24, 133-135 (1974).
[CrossRef]

1966 (1)

M. Lax, "Quantum noise IV. Quantum theory of noise sources," Phys. Rev. 145, 110-129 (1966).
[CrossRef]

Ågren, H.

A. Baev, F. Gel'mukhanov, P. Macak, Y. Luo, and H. Ågren, "General theory for pulse propagation in two-photon active media," J. Phys. Chem. 117, 6214-6220 (2002).
[CrossRef]

Albota, M.

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Allard, N.

N. Allard and J. Kielkopf, "The effect of neutral nonresonant collisions on atomic spectral lines," Rev. Mod. Phys. 54, 1103-1182 (1982).
[CrossRef]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Plenum, 1975).

Allen, R.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Ananthavel, S.

B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

Aranda, F. J.

Bacskai, B. J.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Baev, A.

A. Baev, F. Gel'mukhanov, P. Macak, Y. Luo, and H. Ågren, "General theory for pulse propagation in two-photon active media," J. Phys. Chem. 117, 6214-6220 (2002).
[CrossRef]

Barchielli, A.

A. Barchielli, "Measurement theory and stochastic differential equations in quantum mechanics," Phys. Rev. A 34, 1642-1649 (1986).
[CrossRef] [PubMed]

Barlow, S.

B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

Basharov, A. M.

A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves (Kluwer Academic, 1999).

Beljonne, D.

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Bhatt, J. C.

B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, and P. N. Prasad, "Highly active two-photon dyes: design, synthesis, and characterization toward application," Chem. Mater. 10, 1863-1874 (1998).
[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, 299-305 (1993).
[CrossRef]

Brandelik, D. M.

Brant, M. C.

Bredas, J. L.

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Brott, L. L.

B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, and P. N. Prasad, "Highly active two-photon dyes: design, synthesis, and characterization toward application," Chem. Mater. 10, 1863-1874 (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]

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990).

Chen, P. H.

I.-C. Khoo, A. Diaz, M. V. Wood, and P. H. Chen, "Passive optical limiting of picosecond-nanosecond laser pulses using highly nonlinear organic liquid cored fiber array," IEEE J. Sel. Top. Quantum Electron. 7, 760-768 (2001).
[CrossRef]

I. C. Khoo, P. H. Chen, M. V. Wood, and M.-Y. Shih, "Molecular photonics of a highly nonlinear organic fiber core liquid for picosecond-nanosecond optical limiting application," Chem. Phys. 245, 517-531 (1999).
[CrossRef]

Chen, Y.

D.-Y. Wang, C.-L. Zhan, Y. Chen, Y.-J. Li, Z.-Z. Lu, and Y.-Z. Nie, "Large optical power limiting induced by three-photon absorption of two stibazolium-like dyes," Chem. Phys. Lett. 369, 621-626 (2002).
[CrossRef]

Clarson, S. J.

B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, and P. N. Prasad, "Highly active two-photon dyes: design, synthesis, and characterization toward application," Chem. Mater. 10, 1863-1874 (1998).
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B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

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

Reinhardt, B. A.

H. E. Pudavar, M. P. Joshi, P. N. Prasad, and B. A. Reinhardt, "High-density three-dimensional optical data storage in a stacked compact disk format with two-photon writing and single photon readout," Appl. Phys. Lett. 74, 1338-1340 (1999).
[CrossRef]

B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, and P. N. Prasad, "Highly active two-photon dyes: design, synthesis, and characterization toward application," Chem. Mater. 10, 1863-1874 (1998).
[CrossRef]

Remy, D. E.

Rentsch, S.

J. Kleinschmidt, S. Rentsch, W. Tottleben, and B. Wilhelmi, "Measurement of strong nonlinear absorption in stilbene-chloroform solution, explained by the superposition of two-photon absorption and one-photon absorption from the excited state," Chem. Phys. Lett. 24, 133-135 (1974).
[CrossRef]

Rentzepis, R. M.

D. A. Oulianov, I. V. Tomov, A. S. Dvornikow, and R. M. Rentzepis, "Observations on the measurements of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

Roach, J. F.

Rochel, H.

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Rockel, H.

B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

Rogers, J. E.

R. L. Sutherland, M. C. Brant, J. Heinrichs, J. E. Rogers, J. E. Slagle, D. G. McLean, and P. A. Fleitz, "Excited-state characterization and effective three-photon absorption model of two-photon-induced excited state absorption in organic push-pull charge-transfer chromophores," J. Opt. Soc. Am. B 22, 1939-1948 (2005).
[CrossRef]

J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, L.-S. Tan, and P. A. Fleitz, "Understanding the one-photon photophysical properties of a two-photon absorbing chromophore," J. Phys. Chem. A 108, 5514-5520 (2004).
[CrossRef]

Rumi, M.

B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Sankaran, B.

J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, L.-S. Tan, and P. A. Fleitz, "Understanding the one-photon photophysical properties of a two-photon absorbing chromophore," J. Phys. Chem. A 108, 5514-5520 (2004).
[CrossRef]

Schmults, C. D.

E. S. Marmur, C. D. Schmults, and D. J. Goldberg, "A review of laser and photodynamic therapy for the treatment of nonmelanoma skin cancer," Dermatol. Surg. 30, 264-271 (2004).
[CrossRef] [PubMed]

Shen, Y. R.

Y. R. Shen, The Principle of Nonlinear Optics (Wiley, 1984).

Shih, M.-Y.

I. C. Khoo, P. H. Chen, M. V. Wood, and M.-Y. Shih, "Molecular photonics of a highly nonlinear organic fiber core liquid for picosecond-nanosecond optical limiting application," Chem. Phys. 245, 517-531 (1999).
[CrossRef]

Si, J.

C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, "Excited-state nonlinear absorption in multi-energy-level molecular systems," Phys. Rev. A 51, 569-575 (1995).
[CrossRef] [PubMed]

Siegner, U.

U. Siegner and U. Keller, "Nonlinear optical processes for ultrashort pulse generation," in Handbook of Optics, M.Bass, J.M.Enoch, E.W.Van Stryland, and W.Wolfe, eds. (McGraw-Hill, 2001), Vol. 4, pp. 25-31.

Skoch, J.

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Slagle, J. E.

R. L. Sutherland, M. C. Brant, J. Heinrichs, J. E. Rogers, J. E. Slagle, D. G. McLean, and P. A. Fleitz, "Excited-state characterization and effective three-photon absorption model of two-photon-induced excited state absorption in organic push-pull charge-transfer chromophores," J. Opt. Soc. Am. B 22, 1939-1948 (2005).
[CrossRef]

J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, L.-S. Tan, and P. A. Fleitz, "Understanding the one-photon photophysical properties of a two-photon absorbing chromophore," J. Phys. Chem. A 108, 5514-5520 (2004).
[CrossRef]

Stone, M. O.

S. M. Kirkpatrick, R. R. Naik, and M. O. Stone, "Nonlinear saturation and determination of the two-photon absorption cross section of green fluorescent protein," J. Phys. Chem. B 105, 2867-2873 (2001).
[CrossRef]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Subramaniam, G.

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Sun, W.

W. Jia, E. P. Douglas, F. Guo, and W. Sun, "Optical limiting of semiconductor nanoparticles for nanosecond laser pulses," Appl. Phys. Lett. 85, 6326-6328 (2004).
[CrossRef]

Sutherland, R. L.

Tan, L.-S.

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, "Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores," J. Phys. Chem. 120, 5275-5284 (2004).
[CrossRef]

R. Kannan, G. S. He, T.-C. Lin, P. N. Prasad, R. A. Vaia, and L.-S. Tan, "Toward highly active two-photon absorbing liquids. Synthesis and characterization of 1,3,5-triazine-based octupolar molecules," Chem. Mater. 16, 185-194 (2004).
[CrossRef]

J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, L.-S. Tan, and P. A. Fleitz, "Understanding the one-photon photophysical properties of a two-photon absorbing chromophore," J. Phys. Chem. A 108, 5514-5520 (2004).
[CrossRef]

Tomov, I. V.

D. A. Oulianov, I. V. Tomov, A. S. Dvornikow, and R. M. Rentzepis, "Observations on the measurements of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

Tottleben, W.

J. Kleinschmidt, S. Rentsch, W. Tottleben, and B. Wilhelmi, "Measurement of strong nonlinear absorption in stilbene-chloroform solution, explained by the superposition of two-photon absorption and one-photon absorption from the excited state," Chem. Phys. Lett. 24, 133-135 (1974).
[CrossRef]

Turro, N. J.

N. J. Turro, Modern Molecular Photochemistry (Benjamin, 1978).

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, 299-305 (1993).
[CrossRef]

Vaia, R. A.

R. Kannan, G. S. He, T.-C. Lin, P. N. Prasad, R. A. Vaia, and L.-S. Tan, "Toward highly active two-photon absorbing liquids. Synthesis and characterization of 1,3,5-triazine-based octupolar molecules," Chem. Mater. 16, 185-194 (2004).
[CrossRef]

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, "Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores," J. Phys. Chem. 120, 5275-5284 (2004).
[CrossRef]

Van Stryland, E. W.

Vrijen, R.

Wang, D.-Y.

D.-Y. Wang, C.-L. Zhan, Y. Chen, Y.-J. Li, Z.-Z. Lu, and Y.-Z. Nie, "Large optical power limiting induced by three-photon absorption of two stibazolium-like dyes," Chem. Phys. Lett. 369, 621-626 (2002).
[CrossRef]

Wang, R.

C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, "Excited-state nonlinear absorption in multi-energy-level molecular systems," Phys. Rev. A 51, 569-575 (1995).
[CrossRef] [PubMed]

Webb, W. W.

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Wilhelmi, B.

J. Kleinschmidt, S. Rentsch, W. Tottleben, and B. Wilhelmi, "Measurement of strong nonlinear absorption in stilbene-chloroform solution, explained by the superposition of two-photon absorption and one-photon absorption from the excited state," Chem. Phys. Lett. 24, 133-135 (1974).
[CrossRef]

Witzgall, G.

Wood, M. V.

I.-C. Khoo, A. Diaz, M. V. Wood, and P. H. Chen, "Passive optical limiting of picosecond-nanosecond laser pulses using highly nonlinear organic liquid cored fiber array," IEEE J. Sel. Top. Quantum Electron. 7, 760-768 (2001).
[CrossRef]

I. C. Khoo, P. H. Chen, M. V. Wood, and M.-Y. Shih, "Molecular photonics of a highly nonlinear organic fiber core liquid for picosecond-nanosecond optical limiting application," Chem. Phys. 245, 517-531 (1999).
[CrossRef]

Wu, X. L.

B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Xu, C.

M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Yablonovitch, E.

Yang, M.

C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, "Excited-state nonlinear absorption in multi-energy-level molecular systems," Phys. Rev. A 51, 569-575 (1995).
[CrossRef] [PubMed]

Yang, S.

Yuan, L.

B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, and P. N. Prasad, "Highly active two-photon dyes: design, synthesis, and characterization toward application," Chem. Mater. 10, 1863-1874 (1998).
[CrossRef]

Zhan, C.-L.

D.-Y. Wang, C.-L. Zhan, Y. Chen, Y.-J. Li, Z.-Z. Lu, and Y.-Z. Nie, "Large optical power limiting induced by three-photon absorption of two stibazolium-like dyes," Chem. Phys. Lett. 369, 621-626 (2002).
[CrossRef]

Zhang, L.

C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, "Excited-state nonlinear absorption in multi-energy-level molecular systems," Phys. Rev. A 51, 569-575 (1995).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

W. Jia, E. P. Douglas, F. Guo, and W. Sun, "Optical limiting of semiconductor nanoparticles for nanosecond laser pulses," Appl. Phys. Lett. 85, 6326-6328 (2004).
[CrossRef]

H. E. Pudavar, M. P. Joshi, P. N. Prasad, and B. A. Reinhardt, "High-density three-dimensional optical data storage in a stacked compact disk format with two-photon writing and single photon readout," Appl. Phys. Lett. 74, 1338-1340 (1999).
[CrossRef]

Chem. Mater. (2)

R. Kannan, G. S. He, T.-C. Lin, P. N. Prasad, R. A. Vaia, and L.-S. Tan, "Toward highly active two-photon absorbing liquids. Synthesis and characterization of 1,3,5-triazine-based octupolar molecules," Chem. Mater. 16, 185-194 (2004).
[CrossRef]

B. A. Reinhardt, L. L. Brott, S. J. Clarson, A. G. Dillard, J. C. Bhatt, R. Kannan, L. Yuan, G. S. He, and P. N. Prasad, "Highly active two-photon dyes: design, synthesis, and characterization toward application," Chem. Mater. 10, 1863-1874 (1998).
[CrossRef]

Chem. Phys. (1)

I. C. Khoo, P. H. Chen, M. V. Wood, and M.-Y. Shih, "Molecular photonics of a highly nonlinear organic fiber core liquid for picosecond-nanosecond optical limiting application," Chem. Phys. 245, 517-531 (1999).
[CrossRef]

Chem. Phys. Lett. (2)

J. Kleinschmidt, S. Rentsch, W. Tottleben, and B. Wilhelmi, "Measurement of strong nonlinear absorption in stilbene-chloroform solution, explained by the superposition of two-photon absorption and one-photon absorption from the excited state," Chem. Phys. Lett. 24, 133-135 (1974).
[CrossRef]

D.-Y. Wang, C.-L. Zhan, Y. Chen, Y.-J. Li, Z.-Z. Lu, and Y.-Z. Nie, "Large optical power limiting induced by three-photon absorption of two stibazolium-like dyes," Chem. Phys. Lett. 369, 621-626 (2002).
[CrossRef]

Dermatol. Surg. (1)

E. S. Marmur, C. D. Schmults, and D. J. Goldberg, "A review of laser and photodynamic therapy for the treatment of nonmelanoma skin cancer," Dermatol. Surg. 30, 264-271 (2004).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

I.-C. Khoo, A. Diaz, M. V. Wood, and P. H. Chen, "Passive optical limiting of picosecond-nanosecond laser pulses using highly nonlinear organic liquid cored fiber array," IEEE J. Sel. Top. Quantum Electron. 7, 760-768 (2001).
[CrossRef]

J. Biomed. Opt. (1)

B. J. Bacskai, J. Skoch, G. A. Hickey, R. Allen, and B. T. Hyman, "Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques," J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

J. Microelectromech. Syst. (1)

S. Maruo and S. J. Kawata, "Two-photon-absorbed near-infrared photopolymerization for three-dimensional microfabridation," J. Microelectromech. Syst. 7, 411-415 (1998).
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J. Nonlinear Opt. Phys. Mater. (1)

M. J. Potasek, S. Kim, and D. McLaughlin, "All optical power limiting," J. Nonlinear Opt. Phys. Mater. 9, 343-365 (2000).
[CrossRef]

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

J. Phys. Chem. (2)

A. Baev, F. Gel'mukhanov, P. Macak, Y. Luo, and H. Ågren, "General theory for pulse propagation in two-photon active media," J. Phys. Chem. 117, 6214-6220 (2002).
[CrossRef]

G. S. He, T.-C. Lin, J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L.-S. Tan, "Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores," J. Phys. Chem. 120, 5275-5284 (2004).
[CrossRef]

J. Phys. Chem. A (1)

J. E. Rogers, J. E. Slagle, D. G. McLean, R. L. Sutherland, B. Sankaran, R. Kannan, L.-S. Tan, and P. A. Fleitz, "Understanding the one-photon photophysical properties of a two-photon absorbing chromophore," J. Phys. Chem. A 108, 5514-5520 (2004).
[CrossRef]

J. Phys. Chem. B (1)

S. M. Kirkpatrick, R. R. Naik, and M. O. Stone, "Nonlinear saturation and determination of the two-photon absorption cross section of green fluorescent protein," J. Phys. Chem. B 105, 2867-2873 (2001).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

P. N. Prasad, "Emerging opportunities at the interface of photonics, nanotechnology and biotechnology," Mol. Cryst. Liq. Cryst. 415, 1-10 (2004).
[CrossRef]

Nature (1)

B. H. Cumpston, S. Ananthavel, S. Barlow, D. L. Dyer, J. E. Ehrlich, L. L. Erskine, A. A. Heikal, S. M. Kuebler, I. Y. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X. L. Wu, S. R. Marder, and J. W. Perry, "Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication," Nature 398, 51-54 (1999).
[CrossRef]

Opt. Commun. (1)

D. A. Oulianov, I. V. Tomov, A. S. Dvornikow, and R. M. Rentzepis, "Observations on the measurements of two-photon absorption cross-section," Opt. Commun. 191, 235-243 (2001).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (1)

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Prog. Quantum Electron. (1)

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, 299-305 (1993).
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M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. F. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Rochel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu, "Design of organic molecules with large two-photon absorption cross sections," Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Other (11)

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, 1997), pp. 813-839.

N. J. Turro, Modern Molecular Photochemistry (Benjamin, 1978).

U. Siegner and U. Keller, "Nonlinear optical processes for ultrashort pulse generation," in Handbook of Optics, M.Bass, J.M.Enoch, E.W.Van Stryland, and W.Wolfe, eds. (McGraw-Hill, 2001), Vol. 4, pp. 25-31.

Y. R. Shen, The Principle of Nonlinear Optics (Wiley, 1984).

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Figures (9)

Fig. 1
Fig. 1

Set of five basics absorption diagrams, B 0 B 4 , which are the building blocks used to describe the generic materials for our numerical method. Electronic states are labeled N 0 N 4 , and the σ s are absorption cross sections. The upward arrows correspond to photoexcitation transitions and the downward arrows correspond to intersystem electron decay (for simplicity, we omit all other possible singlet–singlet and triplet–triplet states electron decays).

Fig. 2
Fig. 2

Energy transmittance T E , defined by Eq. (21), as a function of input energy. The measured data is shown by the ∙ (dot) symbol, the solid curve is our numerical calculation, the dashed curve is the original solution. (a) Comparing with the numerical solution of McLean et al. (Ref. [42]) for C 60 nanosecond; (b) comparing with the analytical calculation of Sutherland et al. (Ref. [21]) AF455, nanosecond; (c) comparing with the analytical calculation of Wang et al. (Ref. [24]) PPAI, picosecond. All the parameters are given in Table 1.

Fig. 3
Fig. 3

C 60 nanosecond regime. Evolution of population densities (left column) and absolute contributions (right column) due to different active electronic levels superimposed with the total absorption as defined by Eq. (28), at the entrance of the slab, η = 0 , and at ρ = 0 for different incident fluence values: Φ in = { 0.51 , 2.05 , 14.1 } given in J cm 2 (from top to bottom, correspondingly).

Fig. 4
Fig. 4

AF455 nanosecond regime. Evolution of population densities (left column) and absolute contributions (right column) due to different active electronic levels superimposed with the total absorption as defined by Eq. (28), at the entrance of the slab, η = 0 , and at ρ = 0 for different incident energy values: E in = 17 μ J , 93 μ J , 0.33 mJ (from top to bottom, correspondingly).

Fig. 5
Fig. 5

PPAI picosecond regime. Evolution of the population densities at the entrance of the slab, η = 0 , and at ρ = 0 for two incident intensity values: I in = { 16.9 , 204.5 } given in GW cm 2 .

Fig. 6
Fig. 6

Numerical solution of evolution of the pulse intensity as a function of radius at τ = 0 and at different depths η = 0.00 , 0.25 , 0.50 , 0.75 , 1.00 for different MPA samples: (a) C 60 , nanosecond, Φ in = 2.05 J cm 2 ; (b) AF455, nanosecond, E in = 131 μ J ; (c) PPAI, picosecond, I in = 204.5 GW cm 2 ; (d) AF455, femtosecond, E in = 6.6 μ J , R 0 = 7.07 μ m , T 0 = 102.0 femtosecond.

Fig. 7
Fig. 7

Transmission as a function of input energy for AF455 in a 0.82 mm slab for fs pulses; (solid curves) using integrated values [as in Eq. (22)], (dashed curve) using peak values, as in Eq. (21).

Fig. 8
Fig. 8

AF455 fs regime. (a) Evolution of population densities at the entrance of the 0.824 mm slab for 102.0 fs , E in = 6.6 μ J , pulse at ρ = 0 ; (b) contributions of active electron levels to the absorption superimposed with total intensity absorption.

Fig. 9
Fig. 9

Temporal partitions used for the subsampling calculation. The dots show where the intensity values are calculated, the solid lines show where the density values are calculated, and the dashed line shows an example for the subsample times.

Tables (3)

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Table 1 Parameters for Multiphoton Absorbing Materials

Tables Icon

Table 2 Individual Contributions to the Total Absorption of Nanosecond Pulses to the Electronic States of C 60

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Table 3 Individual Contributions to the Total Absorption of Nanosecond Pulses to the Electronic States of AF455

Equations (69)

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2 E ( z , r , t ) 1 c 2 2 t 2 E ( z , r , t ) = 1 ε 0 c 2 2 t 2 P ( z , r , t ) ,
E ( z , r , t ) = 1 2 E ̃ ( z , r , t ) exp [ i ( ω 0 t k 0 z ) ] + c.c. ,
P ( z , r , t ) = P ̃ ( z , r , t ) exp [ i ( ω 0 t k 0 z ) ] + c.c. ,
( z + 1 c t i 2 k 0 2 ) E ̃ ( z , r , t ) = i k 0 ε 0 P ̃ ( z , r , t ) ,
g ̂ t = i [ H ̂ , g ̂ ] ,
g s 1 s 2 t = i s 3 ( H s 1 s 3 g s 3 s 2 g s 2 s 3 H s 3 s 1 ) ,
P ̃ = n a e ̂ ψ ψ * R d R = n a s 1 s 2 g s 1 s 2 d s 2 s 1 = n a Tr ( d g ) ,
g s 1 s 2 t = ( γ s 1 s 2 + i ω s 1 s 2 ) g s 1 s 2 + i s 3 ( H s 1 s 3 int g s 3 s 2 g s 2 s 3 H s 3 s 1 int ) .
d N ̃ ( z , r , t ) d t = [ D ̂ 0 + α = 1 N A D ̂ α α ω 0 I ̃ α ( z , r , t ) ] N ̃ ( z , r , t ) ,
d I ̃ ( z , r , t ) d z = β = 1 N B [ σ β N ̃ ( z , r , t ) ] I ̃ β ( z , r , t ) ,
d N ( η , ρ , τ ) d τ = T 0 [ D ̂ 0 + α = 1 N A D ̂ α I 0 α α ω 0 I α ( η , ρ , τ ) ] N ( η , ρ , τ ) ,
d I ( η , ρ , τ ) d η = L d f N β = 1 N B [ σ β N ( η , ρ , τ ) ] I 0 ( β 1 ) I β ( η , ρ , τ ) ,
Ω = { Ω ( ρ j ) , ρ j = j Δ ρ } ,
Ω ( ρ j ) = [ Ω N ̃ ( j ) , Ω I ̃ ( j ) ] ,
Ω N ̃ ( j ) = { ( η n + 1 2 , ρ j , τ i + 1 2 ) , η n + 1 2 = ( η 0 + Δ η 2 ) + n Δ η , τ i + 1 2 = ( τ 0 + Δ τ 2 ) + i Δ τ } ,
Ω I ̃ ( j ) = { ( η n , ρ j , τ i ) , η n = η 0 + n Δ η , τ i = τ 0 + i Δ τ }
N n + 1 2 , j , i + 1 2 ( k ) exp ( Δ τ T 0 D ̂ 0 + Δ τ T 0 D ̂ 1 I 0 ω 0 1 2 { I n , j , i + I n + 1 , j , i ( k ) } + Δ τ T 0 D ̂ 2 I 0 2 2 ω 0 1 2 { I n , j , i 2 + I n + 1 , j , i ( k ) 2 } + Δ τ T 0 D ̂ 3 I 0 3 3 ω 0 1 2 { I n , j , i 3 + I n + 1 , j , i ( k ) 3 } ) N n + 1 2 , j , i 1 2 ( k ) ,
I n + 1 , j , i ( k + 1 ) exp ( L d f N Δ η { σ 1 N n + 1 2 , j , i 1 2 ( k ) + N n + 1 2 , j , i + 1 2 ( k ) 2 } L d f N Δ η I 0 { σ 2 N n + 1 2 , j , i 1 2 ( k ) + N n + 1 2 , j , i + 1 2 ( k ) 2 } 1 2 { I n , j , i + I n + 1 , j , i ( k ) } L d f N Δ η I 0 2 { σ 3 N n + 1 2 , j , i 1 2 ( k ) + N n + 1 2 , j , i + 1 2 ( k ) 2 } 1 2 { I n , j , i 2 + I n + 1 , j , i ( k ) 2 } ) I n , i .
I n , j ( τ ̂ m ) = I n , j [ τ i 1 2 + ( m + 1 2 ) Δ τ ] = λ m I n , j ( τ i 1 2 ) + μ m I n , j ( τ i + 1 2 ) ,
I n , j α ( τ i 1 2 ) ( I α ) n , j , i ̂ 1 2 ( I n , j , i 1 α + I n , j , i α ) .
N n + 1 2 , j , i + 1 2 ( k ) m = 0 M 1 exp { Δ τ T 0 M [ D ̂ 0 + α = 1 N A D ̂ α I 0 α α ω 0 ( λ m 1 2 { ( I α ) n , j , i ̂ + ( I α ) n + 1 , j , i ̂ ( k ) } + μ m 1 2 { ( I α ) n , j , i ̂ + 1 + ( I α ) n + 1 , j , i ̂ + 1 ( k ) } ) ] } N n + 1 2 , j , i 1 2 ( k ) ,
M M ( n , j , i ) = min s 1 , s 2 { M , Δ τ T 0 M D ̂ 0 [ s 1 , s 2 ] + I 0 ω 0 1 2 { I n , j , i + I n + 1 , j , i ( k ) } D ̂ 1 [ s 1 , s 2 ] + I 0 2 2 ω 0 1 2 { ( I 2 ) n , j , i + ( I 2 ) n + 1 , j , i ( k ) } D ̂ 2 [ s 1 , s 2 ] + I 0 3 3 ω 0 1 2 { ( I 3 ) n , j , i + ( I 3 ) n + 1 , j , i ( k ) } D ̂ 3 [ s 1 , s 2 ] < ε } .
T δ = δ E out ( ρ * , τ * ) δ E in ( ρ * , τ * ) ,
( ρ * , τ * ) = arg max ρ , τ δ E out ( ρ , τ ) ,
T E = E out E in = T F 0 + d ρ 2 π ρ + d τ I ( η max , ρ , τ ) 2 π 0 + d ρ ρ + d τ I 0 e ( τ ) 2 e ( ρ ) 2 ,
Q ω = s S σ Q s ; ω ,
S σ = { s σ 1 [ s ] + σ 2 [ s ] + σ 3 [ s ] > 0 } .
Q s ; ω Q s ; n + 1 2 , j , i 1 2 = exp ( L d f N Δ η β = 1 N B I 0 β 1 { σ β [ s ] N s ; n + 1 2 , j , i 1 2 ( K ) + N s ; n + 1 2 , j , i + 1 2 ( K ) 2 } 1 2 { I n , j , i β 1 + I n + 1 , j , i ( K ) β 1 } ) .
q ω = 1 Q ω ,
p s ; ω = 1 Q s ; ω .
p ̂ s ; ω = p s ; ω s S σ p s ; ω .
q s ; { ω } = p ̂ s ; { ω } q { ω } .
I ( η = 0 , ρ , τ ) = I 0 exp ( τ 2 ) exp ( ρ 2 ) .
p ̂ s [ τ 0 , τ 1 ] = i , τ i [ τ 0 , τ 1 ] p ̂ s ; { n , j , i } .
σ 1 = [ σ 01 , σ 12 , 0 , σ 34 , 0 ] , σ 2 = σ 3 = 0 ,
D ̂ 0 = [ 0 k 10 0 k 30 0 0 ( k 13 + k 10 ) k 21 0 0 0 0 k 21 0 0 0 k 13 0 k 30 k 43 0 0 0 0 k 43 ] , D ̂ 1 = [ σ 01 0 0 0 0 σ 01 σ 12 0 0 0 0 σ 12 0 0 0 0 0 0 σ 34 0 0 0 0 σ 34 0 ] ,
D ̂ 2 = D ̂ 3 = ( 0 ) 5 × 5 ,
σ 1 = [ 0 , σ S , σ T , 0 , 0 ] , σ 2 = [ σ TPA , 0 , 0 , 0 , 0 ] , σ 3 = 0 ,
D ̂ 1 = [ 0 0 0 0 0 0 σ S 0 0 0 0 0 σ T 0 0 0 0 σ T 0 0 0 σ S 0 0 0 ] , D ̂ 2 = [ σ TPA 0 0 0 0 σ TPA 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] , D ̂ 3 = ( 0 ) 5 × 5 ,
σ 3 = [ σ 3 PA , 0 ] , D ̂ 3 = [ σ 3 PA 0 σ 3 PA 0 ] ,
g s 1 s 2 t = ( Γ s 1 s 2 + i ω s 1 s 2 ) g s 1 s 2 + i E ̃ s 3 ( d s 1 s 3 g s 3 s 2 g s 2 s 3 d s 3 s 1 ) ,
σ s 1 s 2 = ω Γ s 1 s 2 d s 1 s 2 2 n c ε 0 [ Γ s 1 s 2 2 + ( ω s 1 s 2 ω ) 2 ] ,
g 00 t = σ 01 ϕ ̃ ( g 11 g 00 ) + k 10 g 11 + k 30 g 33 ,
g 11 t = σ 12 ϕ ̃ ( g 22 g 11 ) σ 01 ϕ ̃ ( g 11 g 00 ) + k 21 g 22 ( k 13 + k 10 g 11 ) ,
g 22 t = σ 12 ϕ ̃ ( g 22 g 11 ) k 21 g 22 ,
g 33 t = σ 34 ϕ ̃ ( g 44 g 33 ) + k 43 g 44 k 30 g 33 ,
g 44 t = σ 34 ϕ ̃ ( g 44 g 33 ) k 43 g 44 ,
N ̃ 0 t = σ 01 ϕ ̃ N 0 + k 10 N 1 + k 30 N ,
N ̃ 1 t = σ 01 ϕ ̃ N 0 ( σ 12 ϕ ̃ + k 13 + k 10 ) N 1 + k 21 N 2 ,
N ̃ 2 t = σ 12 ϕ ̃ N 1 k 21 N 2 ,
N ̃ 3 t = ( σ 34 ϕ ̃ + k 30 ) N 3 + k 43 N 4 ,
N ̃ 4 t = σ 34 ϕ ̃ N 3 k 43 N 4 .
P ̃ ( z , r , t ) = i n c ε 0 ω 0 [ σ 10 N ̃ 1 ( z , r , t ) + σ 12 N ̃ 2 ( z , r , t ) + σ 34 N ̃ 3 ( z , r , t ) ] I ̃ ( z , r , t ) ,
( z + 1 c t ) I ̃ ( z , r , t ) = [ σ 01 N ̃ 0 ( z , r , t ) + σ 12 N ̃ 1 ( z , r , t ) + σ 34 N ̃ 3 ( z , r , t ) ] I ̃ ( z , r , t ) .
ln N ( η , ρ , τ ) τ = T 0 [ D ̂ 0 + α = 1 , , N A D ̂ α I 0 α α ω 0 I α ( η , ρ , τ ) ] .
N ( η , ρ , τ + Δ τ ) = N ( η , ρ , τ ) exp { τ τ + Δ τ T 0 [ D ̂ 0 + α = 1 , , N A D ̂ α I 0 α α ω 0 I α ( η , ρ , τ ) ] d τ }
N ( η , ρ , τ ) exp { Δ τ T 0 D ̂ 0 + Δ τ T 0 α = 1 , , N A D ̂ α I 0 α α ω 0 I α ( η , ρ , τ ) 1 2 [ I α ( η Δ η 2 , ρ , τ + Δ τ 2 ) + I α ( η + Δ η 2 , ρ , τ + Δ τ 2 ) ] } .
ln I ( η , ρ , τ ) η = L d f N β = 1 N B [ σ β N ̃ ( η , ρ , τ ) ] I 0 ( β 1 ) I β 1 ( η , ρ , τ ) .
I ( η + Δ η , ρ , τ ) = I ( η , ρ , τ ) exp { L d f N β = 1 N B [ σ β η η + Δ η I 0 ( β 1 ) I β 1 ( η , ρ , τ ) N ( η , ρ , τ ) d η ] }
I ( η , ρ , τ ) exp ( L d f N Δ η β = 1 N B { [ σ β N ( η + Δ η 2 , ρ , τ Δ τ 2 ) + N ( η + Δ η 2 , ρ , τ + Δ τ 2 ) 2 ] × I 0 β 1 2 [ I β 1 ( η , ρ , τ ) + I β 1 ( η + Δ η , ρ , τ ) ] } ) .
D 0 = Δ τ T 0 D ̂ 0 , D α = Δ τ T 0 D ̂ α I 0 α α ω 0 for α > 0 .
N ( η , ρ , τ + Δ τ ) = N ( η , ρ , τ ) exp [ 1 Δ τ τ τ + Δ τ D 0 + α = 1 N A D α I α ( η , ρ , τ ) d τ ] .
C 2 ( for α = 2 ) = exp [ 1 Δ τ τ τ + Δ τ D 2 I 2 ( η , ρ , τ ) d τ ] = exp [ 1 Δ τ m = 0 M 1 τ m τ m + 1 D 2 I 2 ( η , ρ , τ ) d τ ] ,
C 2 ( for α = 2 ) = m exp [ 1 Δ τ D 2 τ m τ m + 1 I 2 ( η , ρ , τ ) d τ ] ,
C 2 ( for α = 2 ) m exp { 1 Δ τ D 2 Δ τ M 1 2 [ I 2 ( η Δ η 2 , ρ , τ m + δ τ 2 ) + I 2 ( η + Δ η 2 , ρ , τ m + δ τ 2 ) ] } ,
C 2 ( for α = 2 ) m exp ( 1 M D 2 { λ m 1 2 [ I 2 ( η Δ η 2 , ρ , τ ) + I 2 ( η + Δ η 2 , ρ , τ ) ] + μ m 1 2 [ I 2 ( η Δ η 2 , ρ , τ + Δ τ ) + I 2 ( η + Δ η 2 , ρ , τ + Δ τ ) ] } ) ,
C 2 ( for α = 2 ) m exp [ 1 M D 2 ( λ m 1 2 { ( I ) 2 [ η Δ η 2 , ρ , τ ] + ( I ) 2 [ η + Δ η 2 , ρ , τ ] } + μ m 1 2 { ( I ) 2 [ η Δ η 2 , ρ , τ + Δ τ ] + ( I ) 2 [ η + Δ η 2 , ρ , τ + Δ τ ] } ) ] .
( I ) 2 [ , , τ ] 1 2 [ I 2 ( , , τ Δ τ 2 ) + I 2 ( , , τ + Δ τ 2 ) ] ,
( I ) 2 [ , , τ + Δ τ ] 1 2 [ I 2 ( , , τ + Δ τ 2 ) + I 2 ( , , τ + 3 Δ τ 2 ) ] .

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