Abstract

Based on the simplified five-level rate-equation theory, we investigate the hybrid third-order optical nonlinear processes which combine the instantaneous nonlinearity and the one-photon absorption induced excited-state nonlinearity. We obtain the analytical third-order nonlinear absorption and refraction coefficients originating from the singlet and triplet excited-state effects. We explore the photodynamic process and give the corresponding level diagram in the nanosecond, picosecond, and femtosecond regimes. We develop the pulse-duration-dependent Z-scan theory for characterizing the simultaneous instantaneous nonlinearity and cumulative effect of the excited-state nonlinearity. We also demonstrate the validity of the presented theory to analyze the experimental results.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  7. S. V. Rao, N. K. M. N. Srinivas, and D. N. Rao, "Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques," Chem. Phys. Lett. 361, 439 (2002).
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
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    [CrossRef]
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  19. L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
    [CrossRef]
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    [CrossRef]
  23. B. Gu, W. Ji, and X. Q. Huang, "Analytical expression for femtosecond-pulsed z scans on instantaneous nonlinearity," Appl. Opt. 47, 1187 (2008).
    [CrossRef] [PubMed]
  24. B. Gu, Y. Sun, and W. Ji, "Two-photon-induced excited-state nonlinearities," Opt. Express 16, 17745 (2008).
    [CrossRef] [PubMed]
  25. H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
    [CrossRef]
  26. C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
    [CrossRef]
  27. S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
    [CrossRef]

2010

D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, "Nonlinear refraction and absorption: mechanisms and magnitudes," Adv. Opt. Photon. 2, 60 (2010).
[CrossRef]

S. C. Hayes, and C. Silva, "Analysis of the excited-state absorption spectral bandshape of oligofluorenes," J. Chem. Phys. 132, 214510 (2010).
[CrossRef] [PubMed]

T. M. Pritchett, W. F. Sun, B. Zhang, M. J. Ferry, Y. Li, J. E. Haley, D. M. Mackie, W. ShenskyIII, and A. G. Mott, "Excited-state absorption of a bipyridyl platinum(II) complex with alkynyl-benzothiazolylfluorene units," Opt. Lett. 35, 1305 (2010).
[CrossRef] [PubMed]

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

2009

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

G. Shi, C. He, Y. Li, R. Zou, X. Zhang, Y. Wang, K. Yang, Y. Song, and C. H. Wang, "Excited-state nonlinearity measurements of ZnPcBr4/DMSO," J. Opt. Soc. Am. B 26, 754 (2009).
[CrossRef]

S. Scuppa, L. Orian, D. Dini, S. Santi, and M. Meneghetti, "Nonlinear absorption properties and excited state dynamics of Ferrocene," J. Phys. Chem. A 113, 9286 (2009).
[CrossRef] [PubMed]

2008

A. Marcelli, P. Foggi, L. Moroni, C. Gellini, and P. R. Salvi, "Excited-state absorption and ultrafast relaxiation dynamics of porphyrin, diprotonated porphyrin, and tetraoxaporphyrin dication," J. Phys. Chem. A 112, 1864 (2008).

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

T. M. Pritchett, W. F. Sun, F. Guo, B. Zhang, M. J. Ferry, J. E. Rogers-Haley, W. ShenskyIII, and A. G. Mott, "Excited-state absorption in a terpyridyl platinum(II) pentynyl complex," Opt. Lett. 33, 1053 (2008).
[CrossRef] [PubMed]

B. Gu, W. Ji, and X. Q. Huang, "Analytical expression for femtosecond-pulsed z scans on instantaneous nonlinearity," Appl. Opt. 47, 1187 (2008).
[CrossRef] [PubMed]

B. Gu, Y. Sun, and W. Ji, "Two-photon-induced excited-state nonlinearities," Opt. Express 16, 17745 (2008).
[CrossRef] [PubMed]

2007

C. L. Lee, X. Yang, and N. C. Greenham, "Determination of the triplet excited-state absorption cross section in a polyfluorene by energy transfer from a phosphorescent metal complex," Phys. Rev. B 76, 245201 (2007).
[CrossRef]

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

C. P. Singh, K. S. Bindra, and S. M. Oak, "Analysis of effect of excitation pulse characteristics on numerical simulation of reverse saturable absorption," Opt. Commun. 269, 223 (2007).
[CrossRef]

2005

T. Cassano, R. Tommasi, A. P. Meacham, and M. D. Ward, "Investigation of the excited-state absorption of a Ru dioxolene complex by the Z-scan technique," J. Chem. Phys. 122, 154507 (2005).
[CrossRef] [PubMed]

2004

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

E. Kuznetsova, R. Kolesov, and O. Kocharovskaya, "Suppression of excited-state absorption: A path to ultraviolet tunable solid-state lasers," Phys. Rev. A 70, 043801 (2004).
[CrossRef]

2003

N. K. M. N. Srinivas, S. V. Rao, and D. N. Rao, "Saturable and reverse saturable absorption of Rhodamine B in methanol and water," J. Opt. Soc. Am. B 20, 2470 (2003).
[CrossRef]

A. Shukla, H. Ghosh, and S. Mazumdar, "Theory of excited-state absorption in phenylene-based π-conjugated polymers," Phys. Rev. B 67, 245203 (2003).
[CrossRef]

2002

S. V. Rao, N. K. M. N. Srinivas, and D. N. Rao, "Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques," Chem. Phys. Lett. 361, 439 (2002).
[CrossRef]

R. Lepkowicz, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, "Picosecond optical limiting in reverse saturable absorbers: a theoretical and experimental study," J. Opt. Soc. Am. B 19, 94 (2002).
[CrossRef]

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

1990

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 (1990).
[CrossRef]

1979

Adhikari, A. V.

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

Bindra, K. S.

C. P. Singh, K. S. Bindra, and S. M. Oak, "Analysis of effect of excitation pulse characteristics on numerical simulation of reverse saturable absorption," Opt. Commun. 269, 223 (2007).
[CrossRef]

Byeon, C. C.

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

Cassano, T.

T. Cassano, R. Tommasi, A. P. Meacham, and M. D. Ward, "Investigation of the excited-state absorption of a Ru dioxolene complex by the Z-scan technique," J. Chem. Phys. 122, 154507 (2005).
[CrossRef] [PubMed]

Chen, J. W.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Chen, Z.

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

Christodoulides, D. N.

Dini, D.

S. Scuppa, L. Orian, D. Dini, S. Santi, and M. Meneghetti, "Nonlinear absorption properties and excited state dynamics of Ferrocene," J. Phys. Chem. A 113, 9286 (2009).
[CrossRef] [PubMed]

Duan, W.

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

Fan, H. L.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Fan, X. Z.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

Ferry, M. J.

Foggi, P.

A. Marcelli, P. Foggi, L. Moroni, C. Gellini, and P. R. Salvi, "Excited-state absorption and ultrafast relaxiation dynamics of porphyrin, diprotonated porphyrin, and tetraoxaporphyrin dication," J. Phys. Chem. A 112, 1864 (2008).

Gellini, C.

A. Marcelli, P. Foggi, L. Moroni, C. Gellini, and P. R. Salvi, "Excited-state absorption and ultrafast relaxiation dynamics of porphyrin, diprotonated porphyrin, and tetraoxaporphyrin dication," J. Phys. Chem. A 112, 1864 (2008).

Ghosh, H.

A. Shukla, H. Ghosh, and S. Mazumdar, "Theory of excited-state absorption in phenylene-based π-conjugated polymers," Phys. Rev. B 67, 245203 (2003).
[CrossRef]

Gray, G. M.

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

Greenham, N. C.

C. L. Lee, X. Yang, and N. C. Greenham, "Determination of the triplet excited-state absorption cross section in a polyfluorene by energy transfer from a phosphorescent metal complex," Phys. Rev. B 76, 245201 (2007).
[CrossRef]

Gu, B.

Gu, J.

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

Guang, S.

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

Guo, F.

Guo, S. L.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

Hagan, D. J.

R. Lepkowicz, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, "Picosecond optical limiting in reverse saturable absorbers: a theoretical and experimental study," J. Opt. Soc. Am. B 19, 94 (2002).
[CrossRef]

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 (1990).
[CrossRef]

Haley, J. E.

Hayes, S. C.

S. C. Hayes, and C. Silva, "Analysis of the excited-state absorption spectral bandshape of oligofluorenes," J. Chem. Phys. 132, 214510 (2010).
[CrossRef] [PubMed]

He, C.

G. Shi, C. He, Y. Li, R. Zou, X. Zhang, Y. Wang, K. Yang, Y. Song, and C. H. Wang, "Excited-state nonlinearity measurements of ZnPcBr4/DMSO," J. Opt. Soc. Am. B 26, 754 (2009).
[CrossRef]

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

Hegde, P. K.

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

Huang, X. Q.

Ji, W.

Jiang, L.

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

Jiu, T.

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

Khoo, I. C.

Kobyakov, A.

Kocharovskaya, O.

E. Kuznetsova, R. Kolesov, and O. Kocharovskaya, "Suppression of excited-state absorption: A path to ultraviolet tunable solid-state lasers," Phys. Rev. A 70, 043801 (2004).
[CrossRef]

Kolesov, R.

E. Kuznetsova, R. Kolesov, and O. Kocharovskaya, "Suppression of excited-state absorption: A path to ultraviolet tunable solid-state lasers," Phys. Rev. A 70, 043801 (2004).
[CrossRef]

Kuznetsova, E.

E. Kuznetsova, R. Kolesov, and O. Kocharovskaya, "Suppression of excited-state absorption: A path to ultraviolet tunable solid-state lasers," Phys. Rev. A 70, 043801 (2004).
[CrossRef]

Lang, J.

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

Lawson, C. M.

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

Lee, C. L.

C. L. Lee, X. Yang, and N. C. Greenham, "Determination of the triplet excited-state absorption cross section in a polyfluorene by energy transfer from a phosphorescent metal complex," Phys. Rev. B 76, 245201 (2007).
[CrossRef]

Lepkowicz, R.

Li, C.

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

Li, J.

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

Li, Y.

T. M. Pritchett, W. F. Sun, B. Zhang, M. J. Ferry, Y. Li, J. E. Haley, D. M. Mackie, W. ShenskyIII, and A. G. Mott, "Excited-state absorption of a bipyridyl platinum(II) complex with alkynyl-benzothiazolylfluorene units," Opt. Lett. 35, 1305 (2010).
[CrossRef] [PubMed]

G. Shi, C. He, Y. Li, R. Zou, X. Zhang, Y. Wang, K. Yang, Y. Song, and C. H. Wang, "Excited-state nonlinearity measurements of ZnPcBr4/DMSO," J. Opt. Soc. Am. B 26, 754 (2009).
[CrossRef]

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

Liu, H.

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

Mackie, D. M.

Manjunatha, K. B.

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

Manjunatha, M. G.

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

Marcelli, A.

A. Marcelli, P. Foggi, L. Moroni, C. Gellini, and P. R. Salvi, "Excited-state absorption and ultrafast relaxiation dynamics of porphyrin, diprotonated porphyrin, and tetraoxaporphyrin dication," J. Phys. Chem. A 112, 1864 (2008).

Mazumdar, S.

A. Shukla, H. Ghosh, and S. Mazumdar, "Theory of excited-state absorption in phenylene-based π-conjugated polymers," Phys. Rev. B 67, 245203 (2003).
[CrossRef]

Mckerns, M. M.

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

Meacham, A. P.

T. Cassano, R. Tommasi, A. P. Meacham, and M. D. Ward, "Investigation of the excited-state absorption of a Ru dioxolene complex by the Z-scan technique," J. Chem. Phys. 122, 154507 (2005).
[CrossRef] [PubMed]

Meneghetti, M.

S. Scuppa, L. Orian, D. Dini, S. Santi, and M. Meneghetti, "Nonlinear absorption properties and excited state dynamics of Ferrocene," J. Phys. Chem. A 113, 9286 (2009).
[CrossRef] [PubMed]

Miller, D. A. B.

Ming, N. B.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

Moroni, L.

A. Marcelli, P. Foggi, L. Moroni, C. Gellini, and P. R. Salvi, "Excited-state absorption and ultrafast relaxiation dynamics of porphyrin, diprotonated porphyrin, and tetraoxaporphyrin dication," J. Phys. Chem. A 112, 1864 (2008).

Mott, A. G.

Nordlund, T. M.

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

Oak, S. M.

C. P. Singh, K. S. Bindra, and S. M. Oak, "Analysis of effect of excitation pulse characteristics on numerical simulation of reverse saturable absorption," Opt. Commun. 269, 223 (2007).
[CrossRef]

Orian, L.

S. Scuppa, L. Orian, D. Dini, S. Santi, and M. Meneghetti, "Nonlinear absorption properties and excited state dynamics of Ferrocene," J. Phys. Chem. A 113, 9286 (2009).
[CrossRef] [PubMed]

Poornesh, P.

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

Pritchett, T. M.

Rao, D. N.

N. K. M. N. Srinivas, S. V. Rao, and D. N. Rao, "Saturable and reverse saturable absorption of Rhodamine B in methanol and water," J. Opt. Soc. Am. B 20, 2470 (2003).
[CrossRef]

S. V. Rao, N. K. M. N. Srinivas, and D. N. Rao, "Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques," Chem. Phys. Lett. 361, 439 (2002).
[CrossRef]

Rao, S. V.

N. K. M. N. Srinivas, S. V. Rao, and D. N. Rao, "Saturable and reverse saturable absorption of Rhodamine B in methanol and water," J. Opt. Soc. Am. B 20, 2470 (2003).
[CrossRef]

S. V. Rao, N. K. M. N. Srinivas, and D. N. Rao, "Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques," Chem. Phys. Lett. 361, 439 (2002).
[CrossRef]

Ren, Q.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Rogers-Haley, J. E.

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 (1990).
[CrossRef]

Salamo, G. J.

Salvi, P. R.

A. Marcelli, P. Foggi, L. Moroni, C. Gellini, and P. R. Salvi, "Excited-state absorption and ultrafast relaxiation dynamics of porphyrin, diprotonated porphyrin, and tetraoxaporphyrin dication," J. Phys. Chem. A 112, 1864 (2008).

Santi, S.

S. Scuppa, L. Orian, D. Dini, S. Santi, and M. Meneghetti, "Nonlinear absorption properties and excited state dynamics of Ferrocene," J. Phys. Chem. A 113, 9286 (2009).
[CrossRef] [PubMed]

Scuppa, S.

S. Scuppa, L. Orian, D. Dini, S. Santi, and M. Meneghetti, "Nonlinear absorption properties and excited state dynamics of Ferrocene," J. Phys. Chem. A 113, 9286 (2009).
[CrossRef] [PubMed]

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 (1990).
[CrossRef]

Shensky, W.

Shi, G.

G. Shi, C. He, Y. Li, R. Zou, X. Zhang, Y. Wang, K. Yang, Y. Song, and C. H. Wang, "Excited-state nonlinearity measurements of ZnPcBr4/DMSO," J. Opt. Soc. Am. B 26, 754 (2009).
[CrossRef]

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

Shukla, A.

A. Shukla, H. Ghosh, and S. Mazumdar, "Theory of excited-state absorption in phenylene-based π-conjugated polymers," Phys. Rev. B 67, 245203 (2003).
[CrossRef]

Silva, C.

S. C. Hayes, and C. Silva, "Analysis of the excited-state absorption spectral bandshape of oligofluorenes," J. Chem. Phys. 132, 214510 (2010).
[CrossRef] [PubMed]

Singh, C. P.

C. P. Singh, K. S. Bindra, and S. M. Oak, "Analysis of effect of excitation pulse characteristics on numerical simulation of reverse saturable absorption," Opt. Commun. 269, 223 (2007).
[CrossRef]

Smith, S. D.

Song, W.

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

Song, Y.

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

G. Shi, C. He, Y. Li, R. Zou, X. Zhang, Y. Wang, K. Yang, Y. Song, and C. H. Wang, "Excited-state nonlinearity measurements of ZnPcBr4/DMSO," J. Opt. Soc. Am. B 26, 754 (2009).
[CrossRef]

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

Srinivas, N. K. M. N.

N. K. M. N. Srinivas, S. V. Rao, and D. N. Rao, "Saturable and reverse saturable absorption of Rhodamine B in methanol and water," J. Opt. Soc. Am. B 20, 2470 (2003).
[CrossRef]

S. V. Rao, N. K. M. N. Srinivas, and D. N. Rao, "Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques," Chem. Phys. Lett. 361, 439 (2002).
[CrossRef]

Stegeman, G. I.

Sun, W.

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

Sun, W. F.

Sun, Y.

Sun, Z. H.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Tommasi, R.

T. Cassano, R. Tommasi, A. P. Meacham, and M. D. Ward, "Investigation of the excited-state absorption of a Ru dioxolene complex by the Z-scan technique," J. Chem. Phys. 122, 154507 (2005).
[CrossRef] [PubMed]

Umesh, G.

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

Van Stryland, E. W.

Wang, C. H.

Wang, H. T.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

Wang, X. Q.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Wang, Y.

G. Shi, C. He, Y. Li, R. Zou, X. Zhang, Y. Wang, K. Yang, Y. Song, and C. H. Wang, "Excited-state nonlinearity measurements of ZnPcBr4/DMSO," J. Opt. Soc. Am. B 26, 754 (2009).
[CrossRef]

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

Ward, M. D.

T. Cassano, R. Tommasi, A. P. Meacham, and M. D. Ward, "Investigation of the excited-state absorption of a Ru dioxolene complex by the Z-scan technique," J. Chem. Phys. 122, 154507 (2005).
[CrossRef] [PubMed]

Weaire, D.

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 (1990).
[CrossRef]

Wherrett, B. S.

Wu, Y.

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

Xu, D.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Xu, L.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

Yan, J.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

Yang, J.

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

Yang, K.

Yang, X.

C. L. Lee, X. Yang, and N. C. Greenham, "Determination of the triplet excited-state absorption cross section in a polyfluorene by energy transfer from a phosphorescent metal complex," Phys. Rev. B 76, 245201 (2007).
[CrossRef]

Yu, G.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Zhang, B.

Zhang, G. H.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Zhang, W.

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

Zhang, X.

Zhao, X.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Zou, R.

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

G. Shi, C. He, Y. Li, R. Zou, X. Zhang, Y. Wang, K. Yang, Y. Song, and C. H. Wang, "Excited-state nonlinearity measurements of ZnPcBr4/DMSO," J. Opt. Soc. Am. B 26, 754 (2009).
[CrossRef]

Adv. Opt. Photon.

Appl. Opt.

Appl. Phys. B

P. Poornesh, G. Umesh, P. K. Hegde, M. G. Manjunatha, K. B. Manjunatha, and A. V. Adhikari, "Studies on third order nonlinear optical properties and reverse saturable absorption in polythiophene/poly (methylmethacrylate) composites," Appl. Phys. B 97, 117 (2009).
[CrossRef]

Appl. Phys. Lett.

C. C. Byeon, M. M. Mckerns, W. Sun, T. M. Nordlund, C. M. Lawson, and G. M. Gray, "Excited state lifetime and intersystem crossing rate of asymmetric pentaazadentate porphyrin-like metal complexes," Appl. Phys. Lett. 84, 5174 (2004).
[CrossRef]

Chem. Phys. Lett.

S. V. Rao, N. K. M. N. Srinivas, and D. N. Rao, "Nonlinear absorption and excited state dynamics in Rhodamine B studied using Z-scan and degenerate four wave mixing techniques," Chem. Phys. Lett. 361, 439 (2002).
[CrossRef]

IEEE J. Quantum Electron.

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 (1990).
[CrossRef]

J. Chem. Phys.

S. C. Hayes, and C. Silva, "Analysis of the excited-state absorption spectral bandshape of oligofluorenes," J. Chem. Phys. 132, 214510 (2010).
[CrossRef] [PubMed]

T. Cassano, R. Tommasi, A. P. Meacham, and M. D. Ward, "Investigation of the excited-state absorption of a Ru dioxolene complex by the Z-scan technique," J. Chem. Phys. 122, 154507 (2005).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, "Second Z-scan in materials with nonlinear refraction and nonlinear absorption," J. Opt. A, Pure Appl. Opt. 4, 504 (2002).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem. A

A. Marcelli, P. Foggi, L. Moroni, C. Gellini, and P. R. Salvi, "Excited-state absorption and ultrafast relaxiation dynamics of porphyrin, diprotonated porphyrin, and tetraoxaporphyrin dication," J. Phys. Chem. A 112, 1864 (2008).

S. Scuppa, L. Orian, D. Dini, S. Santi, and M. Meneghetti, "Nonlinear absorption properties and excited state dynamics of Ferrocene," J. Phys. Chem. A 113, 9286 (2009).
[CrossRef] [PubMed]

J. Phys. Chem. B

L. Jiang, T. Jiu, Y. Li, Y. Li, J. Yang, J. Li, C. Li, H. Liu, and Y. Song, "Excited-state absorption and sign tuning of nonlinear refraction in porphyrin derivatives," J. Phys. Chem. B 112, 756 (2008).
[CrossRef]

J. Yang, J. Gu, Y. Song, S. Guang, Y. Wang, W. Zhang, and J. Lang, "Excited state absorption dynamics in metal cluster polymer [WS4Cu3I(4-bpy)3]n solution," J. Phys. Chem. B 111, 7987 (2007).
[CrossRef] [PubMed]

J. Porphyrins Phthalocyanines

C. He, G. Shi, W. Duan, Y. Wu, Z. Chen, W. Song, R. Zou, and Y. Song, "Third-order nonlinear optical properties of self-assembled thin films containing tetracarboxylic copper phthalocyanine," J. Porphyrins Phthalocyanines 13, 1243 (2009).
[CrossRef]

Opt. Commun.

C. P. Singh, K. S. Bindra, and S. M. Oak, "Analysis of effect of excitation pulse characteristics on numerical simulation of reverse saturable absorption," Opt. Commun. 269, 223 (2007).
[CrossRef]

Opt. Express

Opt. Laser Technol.

H. L. Fan, X. Q. Wang, Q. Ren, X. Zhao, G. H. Zhang, J. W. Chen, D. Xu, G. Yu, and Z. H. Sun, "Investigation of the nonlinear absorption and optical limiting properties of two [Q]2[Cu(C3S5)2] compounds," Opt. Laser Technol. 42, 732 (2010).
[CrossRef]

Opt. Lett.

Phys. Rev. A

E. Kuznetsova, R. Kolesov, and O. Kocharovskaya, "Suppression of excited-state absorption: A path to ultraviolet tunable solid-state lasers," Phys. Rev. A 70, 043801 (2004).
[CrossRef]

Phys. Rev. B

A. Shukla, H. Ghosh, and S. Mazumdar, "Theory of excited-state absorption in phenylene-based π-conjugated polymers," Phys. Rev. B 67, 245203 (2003).
[CrossRef]

C. L. Lee, X. Yang, and N. C. Greenham, "Determination of the triplet excited-state absorption cross section in a polyfluorene by energy transfer from a phosphorescent metal complex," Phys. Rev. B 76, 245201 (2007).
[CrossRef]

Other

R. L. Sutherland with contributions by D. G. McLean and S. Kikpatrick, Handbook of Nonlinear Optics, 2nd ed. (Dekker, New York, 2003).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) The five-level diagram and the typical decay lifetimes of the 1PA-induced singlet and triplet ESAs as well as 2PA. Simplified level diagram for (b) 2PA, (c) 2PA and 1PA-induced singlet ESA, and (d) 1PA-induced triplet ESA.

Fig. 2
Fig. 2

Pulse-duration dependence of (a) NLA coefficient and (b) NLR index. The used parameters are listed in Table 3. The symbols show the numerical simulation results, while the solid lines give the corresponding analytical results.

Fig. 3
Fig. 3

Pulse-duration dependence of (a) OA and (b) CA (S = 0.1) traces. The used parameters are listed in Table 3. Squares, circles, uptriangles, and downtriangles are the numerical simulations by using Eqs. (6)(11) for τ = 10 ps, 100 ps, 1 ns, and 10 ns, respectively, while the corresponding solid lines are obtained by our simplified theory.

Fig. 4
Fig. 4

Pulse-duration dependence of (a) ΔTV and (b) ΔTPV (S = 0.1). The other parameters are listed in Table 3. The symbols are obtained by the numerical simulations with Eqs. (6)(11), while the corresponding solid lines are obtained by our simplified theory.

Fig. 5
Fig. 5

The OA Z-scan traces for DCu1 under the excitation of τF = 40 ps (squares) and 18 ns (circles), respectively. The symbols are the experimental data from Ref. 25, while the solid lines are the theoretical fitting results by the use of our simplified Z-scan theory.

Fig. 6
Fig. 6

The OA (squares) and CA/OA (circles) Z-scan traces of 26-bilayer CuPc(COONa)4/PDDA film under the pulsed excitation of (a) τF = 21 ps and (b) τF = 4 ns. The symbols are the experimental data from Ref. 26, while the corresponding solid lines are the theoretical fitting results by the use of our simplified Z-scan theory.

Tables (3)

Tables Icon

Table 1 The effective NLA coefficients caused by the excited singlet and triplet states for different laser pulses

Tables Icon

Table 2 The effective NLR indexes caused by the excited singlet and triplet states for different laser pulses

Tables Icon

Table 3 Typical photophysical parameters used for the numerical simulations and the analytical calculations

Equations (28)

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N S 0 t = σ 0 I h ¯ ω N S 0 β I 2 2 ħ ω + N S 1 τ S 1 + N T 1 τ T 1 ,
N S 1 t = σ 0 I ħ ω N S 0 σ S 1 I ħ ω N S 1 N S 1 τ S 1 N S 1 τ isc + N S h τ S h ,
N S h t = β I 2 2 ħ ω + σ S 1 I ħ ω N S 1 N S h τ S h ,
N T 1 t = σ T 1 I ħ ω N T 1 + N S 1 τ isc + N T h τ T h N T 1 τ T 1 ,
N T h t = σ T 1 I ħ ω N T 1 N T h τ T h ,
N = N S 0 + N S 1 + N S h + N T 1 + N T h ,
I z = ( σ 0 N S 0 + σ S N S 1 + σ T N T 1 + β I ) I ,
Δ ϕ z = k ( γ I + η S N S 1 + η T N T 1 ) ,
N S 1 ( t ) = α 0 ħ ω G ( t ) exp ( t 2 / τ 2 ) I ( t ) ,
N T 1 ( t ) = α 0 φ T I 0 ħ ω τ S t G ( t ) exp [ ( t t ) / τ T 1 ] d t ,
G ( t ) = t exp ( t 2 / τ 2 ) exp [ ( t t ) / τ S ] d t .
I ( r , z ; t ) z = α 0 I ( r , z ; t ) [ β + α S ( t ) + α T ( t ) ] I 2 ( r , z ; t ) ,
Δ ϕ ( r , z ; t ) z = k [ γ + n S ( t ) + n T ( t ) ] I ( r , z ; t ) ,
α S ( t ) = σ S α 0 ħ ω exp ( t 2 / τ 2 ) G ( t ) ,
α T ( t ) = σ T α 0 φ T ħ ω τ S exp ( t 2 / τ 2 ) t G ( t ) exp [ ( t t ) / τ T ] d t ,
n S ( t ) = η S α 0 ħ ω exp ( t 2 / τ 2 ) G ( t ) ,
n T ( t ) = η T α 0 φ T ħ ω τ S exp ( t 2 / τ 2 ) t G ( t ) exp [ ( t t ) / τ T ] d t .
E e ( r , z ; t ) = E ( r , z ; t ) exp ( α 0 L / 2 ) [ 1 + q ( r , z ; t ) ] j Δ ϕ ( r , z ; t ) / q ( r , z ; t ) 1 / 2 ,
Δ ϕ ( r , z ; t ) = k [ γ + n S ( t ) + n T ( t ) ] I ( r , z ; t ) L eff ,
q ( r , z ; t ) = [ β + α S ( t ) + α T ( t ) ] I ( r , z ; t ) L eff .
E a ( r a , z ; t ) = k j ( d z ) exp [ j k r a 2 2 ( d z ) ] 0 r E e ( r , z ; t ) exp [ j k r 2 2 ( d z ) ] J 0 ( k r r a d z ) d r ,
T ( z ) = 1 π 1 / 2 τ + ln [ 1 + p ( z , t ) exp ( t 2 / τ 2 ) ] p ( z , t ) exp ( t 2 / τ 2 ) d t ,
T ( z , S ) = exp ( α 0 L ) S ɛ i + d t 0 R a | E a ( r a , z ; t ) | 2 2 π r a d r a .
α S = σ S α 0 ħ ω 2 π τ + exp ( t 2 / τ 2 ) G ( t ) d t ,
α T = σ T α 0 φ T ħ ω τ S 2 π τ + exp ( t 2 / τ 2 ) [ t G ( t ) exp [ ( t t ) / τ T ] d t ] d t ,
n S = η S α 0 ħ ω 2 π τ + exp ( t 2 / τ 2 ) G ( t ) d t ,
n T = η T α 0 φ T ħ ω τ S 2 π τ + exp ( t 2 / τ 2 ) [ t G ( t ) exp [ ( t t ) / τ T ] d t ] d t .
T ( z ) = 1 π + ln [ 1 + ψ ( z ) exp ( ξ 2 ) ] ψ ( z ) d ξ ,

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