Abstract

Using the Z-scan technique, we studied the nonlinear absorption and refraction behaviors of a dilute toluene solution of a silicon naphthalocyanine (Si(OSi(n-hexyl)3)2, SiNc) at 532 nanometer with both a 2.8-nanosecond pulse and a 21-nanosecond (HW1/eM) pulse train containing 11 18-picosecond pulses 7 nanosecond apart. A thermal acoustic model and its steady-state approximation account for the heat generated by the nonradiative relaxations subsequent to the absorption. We found that when the steady-state approximation satisfactorily explained the results obtained with a 21-nanosecond pulse train, only the thermal-acoustic model fit the 2.8-nanosecond experimental results, which supports the approximation criterion established by Kovsh et al.

© 2007 Optical Society of America

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  1. J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.
  2. J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
    [Crossref]
  3. T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
    [Crossref]
  4. A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).
  5. J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
    [Crossref]
  6. S. R. Mishra, H. S. Rawat, M. P. Joshi, and S. C. Mehendale, “The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27,157–163, (1994).
    [Crossref]
  7. T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
    [Crossref] [PubMed]
  8. J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
    [Crossref]
  9. 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]
  10. P. Brochard and V. Grolier-Mazza, “Thermal nonlinear refraction in dye solution: a study of the transient regime,” J. Opt. Soc. Am. B 14,405–414 (1997).
    [Crossref]
  11. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W.Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron 26,760–769 (1990).
    [Crossref]
  12. 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]
  13. T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).
  14. T. H. Wei and T. H. Huang, “A study of photophysics using the Z-scan technique: lifetime determination for high-lying excited states,” Opt. and Quantum Electron 281495–1508 (1996).
    [Crossref]
  15. A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed.,W. KaiseerSpringer-Verlag, Berlin, 1993), pp. 305.
  16. J. H. Brannon and D. Madge, “Picosecond laser Photophysics. group 3A phthalocyanines,” J. Am. Chem. Soc 102,62–65 (1980).
    [Crossref]
  17. C. JensenHigh Power Dye Lasers, F.J Durate, eds. (Springer-Verlag, Berlin, 1991), pp. 48.
  18. C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, “Dynamic and steady-state behaviors of reverse satura absorption in metallophthalocyanines,” Phys. Rev. A 49,1149–1157 (1994).
    [Crossref] [PubMed]
  19. T. H. Wei, T. H. Huang, and M. S. Lin, “Signs of nonlinear refraction in chloroaluminum phthalocyanine solution,” Appl. Phys. Lett 72,2505–2507 (1998).
    [Crossref]
  20. D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed.,D. R. Lideet al, eds. (CRC Press, Boca Raton, 1996), pp. 6-128.
  21. D. I. Kovsh, D. J. Hagan, and E. W. Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Express 4,315–327 (1999).
    [Crossref] [PubMed]
  22. D. Landau and E. M. Lifshitz, in Course of theoretical physics (Pergamon Press), Vol. 6.
  23. J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm 44,267–272 (1983).
    [Crossref]
  24. C. W. Chang and M. S. thesis, National Chung Cheng University, pp. 28, (1999).

2005 (1)

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

1999 (3)

D. I. Kovsh, D. J. Hagan, and E. W. Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Express 4,315–327 (1999).
[Crossref] [PubMed]

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]

C. W. Chang and M. S. thesis, National Chung Cheng University, pp. 28, (1999).

1998 (1)

T. H. Wei, T. H. Huang, and M. S. Lin, “Signs of nonlinear refraction in chloroaluminum phthalocyanine solution,” Appl. Phys. Lett 72,2505–2507 (1998).
[Crossref]

1997 (1)

1996 (3)

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

T. H. Wei and T. H. Huang, “A study of photophysics using the Z-scan technique: lifetime determination for high-lying excited states,” Opt. and Quantum Electron 281495–1508 (1996).
[Crossref]

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

1995 (1)

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

1994 (2)

S. R. Mishra, H. S. Rawat, M. P. Joshi, and S. C. Mehendale, “The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27,157–163, (1994).
[Crossref]

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

1993 (1)

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]

1992 (1)

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

1991 (1)

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

1990 (1)

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

1989 (1)

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

1983 (1)

J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm 44,267–272 (1983).
[Crossref]

1980 (1)

J. H. Brannon and D. Madge, “Picosecond laser Photophysics. group 3A phthalocyanines,” J. Am. Chem. Soc 102,62–65 (1980).
[Crossref]

Alvarez, D.

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

Asai, N.

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

Bartoli, F. J.

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

Bedwhorth, P.V.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Brandelik, D. M.

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]

Brannon, J. H.

J. H. Brannon and D. Madge, “Picosecond laser Photophysics. group 3A phthalocyanines,” J. Am. Chem. Soc 102,62–65 (1980).
[Crossref]

Brant, M. C.

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]

Brochard, P.

Chang, C. W.

C. W. Chang and M. S. thesis, National Chung Cheng University, pp. 28, (1999).

Chen, C. T.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Chen, C. W.

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

Coulter, D. L.

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

Coulter, D. R.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

Fleitz, P. A.

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]

Grolier-Mazza, V.

Habiro, M.

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

Hagan, D. J.

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]

D. I. Kovsh, D. J. Hagan, and E. W. Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Express 4,315–327 (1999).
[Crossref] [PubMed]

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

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

Heritier, J. -M.

J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm 44,267–272 (1983).
[Crossref]

Hoffman, C. A.

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

Hu, J. K.

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

Huang, T. H.

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

T. H. Wei, T. H. Huang, and M. S. Lin, “Signs of nonlinear refraction in chloroaluminum phthalocyanine solution,” Appl. Phys. Lett 72,2505–2507 (1998).
[Crossref]

T. H. Wei and T. H. Huang, “A study of photophysics using the Z-scan technique: lifetime determination for high-lying excited states,” Opt. and Quantum Electron 281495–1508 (1996).
[Crossref]

Jensen, C.

C. JensenHigh Power Dye Lasers, F.J Durate, eds. (Springer-Verlag, Berlin, 1991), pp. 48.

Joshi, M. P.

S. R. Mishra, H. S. Rawat, M. P. Joshi, and S. C. Mehendale, “The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27,157–163, (1994).
[Crossref]

Kafafi, A. H.

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

Kaiser, W.

A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed.,W. KaiseerSpringer-Verlag, Berlin, 1993), pp. 305.

Khundkar, L. R.

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

Kijima, Y.

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

Kishii, N.

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

Kovsh, D.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

Kovsh, D. I.

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]

D. I. Kovsh, D. J. Hagan, and E. W. Stryland, “Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Express 4,315–327 (1999).
[Crossref] [PubMed]

Kuwano, A.

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

Landau, D.

D. Landau and E. M. Lifshitz, in Course of theoretical physics (Pergamon Press), Vol. 6.

Lee, J. Y. S.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Li, C.

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

Lide, D. R.

D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed.,D. R. Lideet al, eds. (CRC Press, Boca Raton, 1996), pp. 6-128.

Lifshitz, E. M.

D. Landau and E. M. Lifshitz, in Course of theoretical physics (Pergamon Press), Vol. 6.

Lin, M. S.

T. H. Wei, T. H. Huang, and M. S. Lin, “Signs of nonlinear refraction in chloroaluminum phthalocyanine solution,” Appl. Phys. Lett 72,2505–2507 (1998).
[Crossref]

Lindle, J. R.

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

Liu, D.

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

Madge, D.

J. H. Brannon and D. Madge, “Picosecond laser Photophysics. group 3A phthalocyanines,” J. Am. Chem. Soc 102,62–65 (1980).
[Crossref]

Mansour, K.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Marder, S. R.

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Matsuzawa, N.

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

Mclean, D. G.

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]

Mehendale, S. C.

S. R. Mishra, H. S. Rawat, M. P. Joshi, and S. C. Mehendale, “The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27,157–163, (1994).
[Crossref]

Miles, P.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Mishra, S. R.

S. R. Mishra, H. S. Rawat, M. P. Joshi, and S. C. Mehendale, “The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27,157–163, (1994).
[Crossref]

Ng, D.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Perry, J. W.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Pottenger, T.

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]

Rawat, H. S.

S. R. Mishra, H. S. Rawat, M. P. Joshi, and S. C. Mehendale, “The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27,157–163, (1994).
[Crossref]

Said, A. A.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

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

Sasabe, H.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Seilmeier, A.

A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed.,W. KaiseerSpringer-Verlag, Berlin, 1993), pp. 305.

Sence, M. J.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

Seto, J.

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

Sheik-Bahae, M.

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

Shirk, J. S.

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

Snow, A. W.

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

Stryland, E. W.

Stryland, E. W. V.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

Stryland, E. W.Van

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]

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

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

Sutherland, R. L.

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]

Takane, N.

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

Tamura, S.

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

thesis, M. S.

C. W. Chang and M. S. thesis, National Chung Cheng University, pp. 28, (1999).

Tian, M.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Tomiyama, T.

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

Wada, T.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Wajsgrus, A.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

Wang, H.

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

Wang, Y.

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

Watanabe, I.

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

Wei, T. H.

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

T. H. Wei, T. H. Huang, and M. S. Lin, “Signs of nonlinear refraction in chloroaluminum phthalocyanine solution,” Appl. Phys. Lett 72,2505–2507 (1998).
[Crossref]

T. H. Wei and T. H. Huang, “A study of photophysics using the Z-scan technique: lifetime determination for high-lying excited states,” Opt. and Quantum Electron 281495–1508 (1996).
[Crossref]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

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

Xia, T.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

Xu, X. L.

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

Yamada, M.

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

Yang, M.

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

Yang, S.

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

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]

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

Zhang, L.

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

Appl. Opt (2)

T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, “Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt 34,8201–8208 (1995).
[Crossref] [PubMed]

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]

Appl. Phys. B (1)

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54,46–51 (1992).
[Crossref]

Appl. Phys. Lett (2)

J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, “Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett 55,1287–1288 (1989).
[Crossref]

T. H. Wei, T. H. Huang, and M. S. Lin, “Signs of nonlinear refraction in chloroaluminum phthalocyanine solution,” Appl. Phys. Lett 72,2505–2507 (1998).
[Crossref]

IEEE J. Quantum Electron (1)

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

J. Am. Chem. Soc (1)

J. H. Brannon and D. Madge, “Picosecond laser Photophysics. group 3A phthalocyanines,” J. Am. Chem. Soc 102,62–65 (1980).
[Crossref]

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

J. Phys. B:At. Mol. Phys. (1)

S. R. Mishra, H. S. Rawat, M. P. Joshi, and S. C. Mehendale, “The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27,157–163, (1994).
[Crossref]

Mol. Phys (1)

T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu, and C. W. Chen, “Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys 103,1847–1857 (2005).

Opt. and Quantum Electron (1)

T. H. Wei and T. H. Huang, “A study of photophysics using the Z-scan technique: lifetime determination for high-lying excited states,” Opt. and Quantum Electron 281495–1508 (1996).
[Crossref]

Opt. Comm (1)

J. -M. Heritier, “Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm 44,267–272 (1983).
[Crossref]

Opt. Express (1)

Opt. Lett (1)

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]

Phys. Rev. A (1)

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

Pure and Appl. Chem (1)

J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, “Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem 68,1429–1434 (1996).
[Crossref]

Other (8)

J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, S. R. Marder, T. H. Wei, M. J. Sence, E. W.Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991), Vol. 194 pp.369–382.

A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus, S. Yang, D. Kovsh, and E. W.Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE- 2853, (1996).

J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science23,1533–1536 (1996).
[Crossref]

C. JensenHigh Power Dye Lasers, F.J Durate, eds. (Springer-Verlag, Berlin, 1991), pp. 48.

A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed.,W. KaiseerSpringer-Verlag, Berlin, 1993), pp. 305.

D. Landau and E. M. Lifshitz, in Course of theoretical physics (Pergamon Press), Vol. 6.

D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed.,D. R. Lideet al, eds. (CRC Press, Boca Raton, 1996), pp. 6-128.

C. W. Chang and M. S. thesis, National Chung Cheng University, pp. 28, (1999).

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Figures (7)

Fig. 1.
Fig. 1.

The Z-scan experimental setup. D4, D5, and D6 are photodetectors. BS1 and BS2 are beam splitters. A sample placed on a motion control stage can be moved from -z to +z.

Fig. 2.
Fig. 2.

The temporal profile of the full pulse envelope. The numbers above spikes mark their order.

Fig. 3.
Fig. 3.

A five-energy-band model for SiNc-toluene: upward-pointing arrows, wiggly lines, and downward-pointing arrows indicate optical excitation, non-radiative relaxation and radiative relaxation, respectively. ∣ ν) refers to vibrational eigenstate. τ denotes lifetime (ISC ≡intersystem crossing, IC ≡internal conversion, and f ≡fluorescence).

Fig. 4.
Fig. 4.

The Z-scan curves for 21-ns pulse trains with an energy level of 0.8 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT a : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NT d : dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

Fig. 5.
Fig. 5.

The Z-scan curve for 21-ns pulse trains with an energy level of 1.4 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT a : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NTd: dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

Fig. 6.
Fig. 6.

The Z-scan curve for 2.8-ns pulses with an energy level of 1.4 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT a : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NT d : dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

Fig. 7.
Fig. 7.

The Z-scan curve for 2.8-ns pulses with an energy level of 2.5 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT a : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NT d : dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

Equations (19)

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

I 0 = I z r , t = I 00 ( n ) × [ w 0 2 w 2 ( z ) ] × exp [ 2 r 2 w 2 ( z ) ] × exp [ ( t n × 7 ns τ ) 2 ]
ϕ 0 = ϕ ( z , r , t ) = kr 2 2 R ( z ) .
I 00 ( n ) = 2 ε ( n ) π 3 2 w 0 2 τ .
ε ( n ) = ε ( 0 ) × exp [ ( n × 7 ns τ env ) 2 ] .
ε ( n ) = ε t × exp [ ( n × 7 ns τ env ) 2 ] m = 5 5 exp [ ( m × 7 ns τ e n v ) ] .
I 0 = I z r , t = I 00 × [ w 0 2 w 2 ( z ) ] × exp [ 2 r 2 w 2 ( z ) ] × exp [ ( t τ ) 2 ]
dI dz ' = [ ( σa ) S 0 N S 0 + ( σa ) S 1 N S 1 + ( σa ) T 1 ] I βN S 0 I 2
dz ' = [ ( σr ) S 0 N S 0 + ( σr ) S 1 N S 1 + ( σr ) T 1 N T 1 ] I + γ N S 0 I + kn 2 I + k Δ n therm ,
dN S 0 dt = ( σa ) S 0 N S 0 I ħω βN S 0 I 2 2 ħω + N S 1 τ f + N T 1 τ T 1 ,
dN S 1 dt = ( σa ) S 0 N S 0 I ħω + βN S 0 I 2 2 ħω N S 1 τ f N S 1 τ ISC ,
dN T 1 dt = N S 1 τ ISC N T 1 τ T 1 .
dQ dt = ( σa ) S 0 N S 0 I ħω × ħ ( ω ω S 1 ) + ( σa ) S 1 N S 1 I + βN S 0 I 2 2 ħω × ħ ( 2 ω ω S 1 ) + ( σa ) T 1 N T 1 I ,
Δ θ = 1 ρC p t dQ dt ' dt ' .
Δθ n = 1 ρC p dQ n dt ' dt ' ,
Δ θ = 5 n 1 Δ θ n .
2 ( Δρ ) t 2 v s 2 2 ( Δ ρ ) = v s 2 2 ( Δθ ) γ e 2 nc 2 I ,
Δn therm = ( n θ ) ρ Δ θ + ( n ρ ) θ Δ ρ .
Δ n therm γ e 2 Δ ρ .
Δ n therm e 2 n Δ θ + γ e 2 4 n 2 cv s 2 I .

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