Abstract

We investigate nonlinear optical absorption of a series of polymethine dyes in liquid solution and in a solid poly(urethane acrylate) elastopolymer host. Depending on the ratio between the absorption cross section from the ground to the first excited state (σ01) and from the first to the higher excited states (σ12) and on the pump wavelength, they demonstrate absorption that either increases or decreases with increasing incident energy density. Our interest is directed at studying the optical limiting behavior of the dyes for sensor protection applications, which requires that σ12 be greater than σ01. We performed a detailed investigation of the dynamics of these processes, using Z-scan and pump–probe techniques at the single wavelength of 532 nm. We obtained σ12/σ0180 for one of the dyes, which is significantly higher than any previously reported value of which we are aware. The linear absorption remained significant, a requirement for good optical limiting. However, at high incident energy density for most of the dyes the increasing absorption converted to a bleaching effect with repeated irradiation. This effect was reversed after several seconds in liquid solutions and several minutes in the polymer host, possibly because of replenishment of the dye by diffusion in both media.

© 1998 Optical Society of America

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  3. D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
    [CrossRef]
  4. L. W. Tutt and T. F. Boggess, “A revew of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
    [CrossRef]
  5. P. A. Miles, “Bottleneck optical limiters: the optimal use of excited-state absorbers,” Appl. Opt. 33, 6965–6979 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  23. 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).
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  26. T.-H. Wei, T.-H. Huang, H.-D. Lin, and S.-H. Lin, “Lifetime determination for high-lying states using Z-scan,” Appl. Phys. Lett. 67, 2266–2268 (1995).
    [CrossRef]
  27. S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
    [CrossRef]
  28. O. Przhonska, Yu. Slominsky, U. Stahl, and S. Daehne, “Excited-state relaxation processes in polymethine dye molecules in polymeric media,” J. Lumin. 69, 105–113 (1996).
    [CrossRef]
  29. E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

1997 (1)

1996 (2)

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

O. Przhonska, Yu. Slominsky, U. Stahl, and S. Daehne, “Excited-state relaxation processes in polymethine dye molecules in polymeric media,” J. Lumin. 69, 105–113 (1996).
[CrossRef]

1995 (4)

T. H. Wei, W. P. Zang, and G. Zang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik (Stuttgart) 98, 143–146 (1995).

T.-H. Wei, T.-H. Huang, H.-D. Lin, and S.-H. Lin, “Lifetime determination for high-lying states using Z-scan,” Appl. Phys. Lett. 67, 2266–2268 (1995).
[CrossRef]

G. S. He, J. D. Bhawalkar, C. F. Zhao, and P. N. Prasad, “Optical limiting effect in a two-photon absorption dye doped solid matrix,” Appl. Phys. Lett. 67, 2433–2435 (1995).
[CrossRef]

G. L. Wood, M. J. Miller, and A. G. Mott, “Investigation of tetrabenzporphyrin by the Z-scan technique,” Opt. Lett. 20, 973–975 (1995).
[CrossRef] [PubMed]

1994 (3)

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

K. Mansour, P. Fuqua, S. R. Marder, B. Dunn, and J. W. Perry, “Solid state optical limiting materials based on phthalocyanine containing polymers and organically-modified sol-gels,” in Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 239–250 (1994).
[CrossRef]

O. V. Przhonska, M. V. Bondar, and Y. A. Tikhonov, “Nonlinear light absorption by liquid and solid solutions of organic dyes,” Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 289–297 (1994).
[CrossRef]

1993 (4)

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
[CrossRef]

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

S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
[CrossRef]

A. Kost, L. Tutt, M. B. Klein, T. K. Dougherty, and W. E. Elias, “Optical limiting with C60 in poly(methyl methacrylate),” Opt. Lett. 18, 334–336 (1993).
[CrossRef] [PubMed]

1992 (2)

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[CrossRef]

M. V. Bondar, O. V. Przhonska, and Y. A. Tikhonov, “Inhonogeneous broadening of organic dyes in polymeric media: nonlinear transmission spectra and photochemical kinetics,” J. Phys. Chem. 96, 10831–10837 (1992).
[CrossRef]

1990 (1)

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

1989 (3)

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity, single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

V. I. Bezrodnyi, M. V. Bondar, G. Y. Kozak, O. V. Przhonska, and Y. A. Tikhonov, “Dye-activated polymeric media for frequency tunable dye lasers,” J. Appl. Spectrosc. 50, 441–454 (1989).
[CrossRef]

1985 (1)

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

1984 (1)

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

1981 (1)

V. I. Bezrodnyi, O. V. Przhonska, E. A. Tikhonov, and M. T. Shpak, “Nonlinear absorption and thermal defocussing of light in dye solutions,” Sov. Phys. JETP 53, 259–270 (1981).

1979 (1)

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

1970 (1)

J. A. Fleck, “Ultrashort pulse generation by Q-switched lasers,” Phys. Rev. B 1, 84–100 (1970).
[CrossRef]

1967 (1)

C. R. Guliano and L. D. Hess, “Nonlinear absorption of light: optical saturation of electronic transitions in organic molecules with high intensity laser radiation,” IEEE J. Quantum Electron. QE-3, 358–367 (1967).
[CrossRef]

Alperovich, M.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Bedworth, P. V.

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

Bezrodnyi, V. I.

V. I. Bezrodnyi, M. V. Bondar, G. Y. Kozak, O. V. Przhonska, and Y. A. Tikhonov, “Dye-activated polymeric media for frequency tunable dye lasers,” J. Appl. Spectrosc. 50, 441–454 (1989).
[CrossRef]

V. I. Bezrodnyi, O. V. Przhonska, E. A. Tikhonov, and M. T. Shpak, “Nonlinear absorption and thermal defocussing of light in dye solutions,” Sov. Phys. JETP 53, 259–270 (1981).

Bhawalkar, J. D.

G. S. He, J. D. Bhawalkar, C. F. Zhao, and P. N. Prasad, “Optical limiting effect in a two-photon absorption dye doped solid matrix,” Appl. Phys. Lett. 67, 2433–2435 (1995).
[CrossRef]

Boggess, T. F.

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

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

Bondar, M. V.

O. V. Przhonska, M. V. Bondar, and Y. A. Tikhonov, “Nonlinear light absorption by liquid and solid solutions of organic dyes,” Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 289–297 (1994).
[CrossRef]

M. V. Bondar, O. V. Przhonska, and Y. A. Tikhonov, “Inhonogeneous broadening of organic dyes in polymeric media: nonlinear transmission spectra and photochemical kinetics,” J. Phys. Chem. 96, 10831–10837 (1992).
[CrossRef]

V. I. Bezrodnyi, M. V. Bondar, G. Y. Kozak, O. V. Przhonska, and Y. A. Tikhonov, “Dye-activated polymeric media for frequency tunable dye lasers,” J. Appl. Spectrosc. 50, 441–454 (1989).
[CrossRef]

Chen, C.-T.

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

Coulter, D. R.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[CrossRef]

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

Daehne, S.

O. Przhonska, Yu. Slominsky, U. Stahl, and S. Daehne, “Excited-state relaxation processes in polymethine dye molecules in polymeric media,” J. Lumin. 69, 105–113 (1996).
[CrossRef]

Dogariu, A.

Dougherty, T. K.

Dunn, B.

K. Mansour, P. Fuqua, S. R. Marder, B. Dunn, and J. W. Perry, “Solid state optical limiting materials based on phthalocyanine containing polymers and organically-modified sol-gels,” in Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 239–250 (1994).
[CrossRef]

Dyadyusha, G.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Elias, W. E.

Fleck, J. A.

J. A. Fleck, “Ultrashort pulse generation by Q-switched lasers,” Phys. Rev. B 1, 84–100 (1970).
[CrossRef]

Fuqua, P.

K. Mansour, P. Fuqua, S. R. Marder, B. Dunn, and J. W. Perry, “Solid state optical limiting materials based on phthalocyanine containing polymers and organically-modified sol-gels,” in Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 239–250 (1994).
[CrossRef]

Guha, S.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

Guliano, C. R.

C. R. Guliano and L. D. Hess, “Nonlinear absorption of light: optical saturation of electronic transitions in organic molecules with high intensity laser radiation,” IEEE J. Quantum Electron. QE-3, 358–367 (1967).
[CrossRef]

Hagan, D. J.

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
[CrossRef]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[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, 760–769 (1990).
[CrossRef]

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

He, G. S.

G. S. He, J. D. Bhawalkar, C. F. Zhao, and P. N. Prasad, “Optical limiting effect in a two-photon absorption dye doped solid matrix,” Appl. Phys. Lett. 67, 2433–2435 (1995).
[CrossRef]

Hess, L. D.

C. R. Guliano and L. D. Hess, “Nonlinear absorption of light: optical saturation of electronic transitions in organic molecules with high intensity laser radiation,” IEEE J. Quantum Electron. QE-3, 358–367 (1967).
[CrossRef]

Huang, T.-H.

T.-H. Wei, T.-H. Huang, H.-D. Lin, and S.-H. Lin, “Lifetime determination for high-lying states using Z-scan,” Appl. Phys. Lett. 67, 2266–2268 (1995).
[CrossRef]

Hughes, S.

S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
[CrossRef]

Klein, M. B.

Kost, A.

Kozak, G. Y.

V. I. Bezrodnyi, M. V. Bondar, G. Y. Kozak, O. V. Przhonska, and Y. A. Tikhonov, “Dye-activated polymeric media for frequency tunable dye lasers,” J. Appl. Spectrosc. 50, 441–454 (1989).
[CrossRef]

Lee, I.-Y. S.

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

Lin, H.-D.

T.-H. Wei, T.-H. Huang, H.-D. Lin, and S.-H. Lin, “Lifetime determination for high-lying states using Z-scan,” Appl. Phys. Lett. 67, 2266–2268 (1995).
[CrossRef]

Lin, S.-H.

T.-H. Wei, T.-H. Huang, H.-D. Lin, and S.-H. Lin, “Lifetime determination for high-lying states using Z-scan,” Appl. Phys. Lett. 67, 2266–2268 (1995).
[CrossRef]

Lloyd, A. D.

S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
[CrossRef]

Mansour, K.

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

K. Mansour, P. Fuqua, S. R. Marder, B. Dunn, and J. W. Perry, “Solid state optical limiting materials based on phthalocyanine containing polymers and organically-modified sol-gels,” in Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 239–250 (1994).
[CrossRef]

Marder, S. R.

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

K. Mansour, P. Fuqua, S. R. Marder, B. Dunn, and J. W. Perry, “Solid state optical limiting materials based on phthalocyanine containing polymers and organically-modified sol-gels,” in Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 239–250 (1994).
[CrossRef]

Miles, P.

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

Miles, P. A.

Miller, M. J.

Miskowski, V. M.

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

Mott, A. G.

Ng, D.

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

Perry, J. W.

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

K. Mansour, P. Fuqua, S. R. Marder, B. Dunn, and J. W. Perry, “Solid state optical limiting materials based on phthalocyanine containing polymers and organically-modified sol-gels,” in Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 239–250 (1994).
[CrossRef]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[CrossRef]

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

Prasad, P. N.

G. S. He, J. D. Bhawalkar, C. F. Zhao, and P. N. Prasad, “Optical limiting effect in a two-photon absorption dye doped solid matrix,” Appl. Phys. Lett. 67, 2433–2435 (1995).
[CrossRef]

Przhonska, O.

O. Przhonska, Yu. Slominsky, U. Stahl, and S. Daehne, “Excited-state relaxation processes in polymethine dye molecules in polymeric media,” J. Lumin. 69, 105–113 (1996).
[CrossRef]

Przhonska, O. V.

O. V. Przhonska, M. V. Bondar, and Y. A. Tikhonov, “Nonlinear light absorption by liquid and solid solutions of organic dyes,” Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 289–297 (1994).
[CrossRef]

M. V. Bondar, O. V. Przhonska, and Y. A. Tikhonov, “Inhonogeneous broadening of organic dyes in polymeric media: nonlinear transmission spectra and photochemical kinetics,” J. Phys. Chem. 96, 10831–10837 (1992).
[CrossRef]

V. I. Bezrodnyi, M. V. Bondar, G. Y. Kozak, O. V. Przhonska, and Y. A. Tikhonov, “Dye-activated polymeric media for frequency tunable dye lasers,” J. Appl. Spectrosc. 50, 441–454 (1989).
[CrossRef]

V. I. Bezrodnyi, O. V. Przhonska, E. A. Tikhonov, and M. T. Shpak, “Nonlinear absorption and thermal defocussing of light in dye solutions,” Sov. Phys. JETP 53, 259–270 (1981).

Przhonskaya, O.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Said, A. A.

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
[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, 760–769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity, single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sasabe, H.

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

Sence, M. J.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[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]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity, single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Shpak, M.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Shpak, M. T.

V. I. Bezrodnyi, O. V. Przhonska, E. A. Tikhonov, and M. T. Shpak, “Nonlinear absorption and thermal defocussing of light in dye solutions,” Sov. Phys. JETP 53, 259–270 (1981).

Slominsky, Yu.

O. Przhonska, Yu. Slominsky, U. Stahl, and S. Daehne, “Excited-state relaxation processes in polymethine dye molecules in polymeric media,” J. Lumin. 69, 105–113 (1996).
[CrossRef]

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Smirl, A. L.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

Smirnova, T.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Soileau, M. J.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Spruce, G.

S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
[CrossRef]

Stahl, U.

O. Przhonska, Yu. Slominsky, U. Stahl, and S. Daehne, “Excited-state relaxation processes in polymethine dye molecules in polymeric media,” J. Lumin. 69, 105–113 (1996).
[CrossRef]

Tian, M.

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

Tikhonov, E.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Tikhonov, E. A.

V. I. Bezrodnyi, O. V. Przhonska, E. A. Tikhonov, and M. T. Shpak, “Nonlinear absorption and thermal defocussing of light in dye solutions,” Sov. Phys. JETP 53, 259–270 (1981).

Tikhonov, Y. A.

O. V. Przhonska, M. V. Bondar, and Y. A. Tikhonov, “Nonlinear light absorption by liquid and solid solutions of organic dyes,” Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 289–297 (1994).
[CrossRef]

M. V. Bondar, O. V. Przhonska, and Y. A. Tikhonov, “Inhonogeneous broadening of organic dyes in polymeric media: nonlinear transmission spectra and photochemical kinetics,” J. Phys. Chem. 96, 10831–10837 (1992).
[CrossRef]

V. I. Bezrodnyi, M. V. Bondar, G. Y. Kozak, O. V. Przhonska, and Y. A. Tikhonov, “Dye-activated polymeric media for frequency tunable dye lasers,” J. Appl. Spectrosc. 50, 441–454 (1989).
[CrossRef]

Tjurin, V.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Tolmachev, A.

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Tutt, L.

Tutt, L. W.

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

Van Stryland, E. W.

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
[CrossRef]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[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, 760–769 (1990).
[CrossRef]

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High sensitivity, single beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Vanherzeele, H.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

Wada, T.

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

Wei, T. H.

T. H. Wei, W. P. Zang, and G. Zang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik (Stuttgart) 98, 143–146 (1995).

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
[CrossRef]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[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, 760–769 (1990).
[CrossRef]

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

Wei, T.-H.

T.-H. Wei, T.-H. Huang, H.-D. Lin, and S.-H. Lin, “Lifetime determination for high-lying states using Z-scan,” Appl. Phys. Lett. 67, 2266–2268 (1995).
[CrossRef]

Welford, K. R.

S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
[CrossRef]

Wherrett, B. S.

S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
[CrossRef]

Williams, W. E.

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

Wood, G. L.

Woodall, M. A.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

Wu, X.-L.

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

Xia, T.

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
[CrossRef]

Zang, G.

T. H. Wei, W. P. Zang, and G. Zang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik (Stuttgart) 98, 143–146 (1995).

Zang, W. P.

T. H. Wei, W. P. Zang, and G. Zang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik (Stuttgart) 98, 143–146 (1995).

Zhao, C. F.

G. S. He, J. D. Bhawalkar, C. F. Zhao, and P. N. Prasad, “Optical limiting effect in a two-photon absorption dye doped solid matrix,” Appl. Phys. Lett. 67, 2433–2435 (1995).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 45–51 (1992).
[CrossRef]

Appl. Phys. Lett. (2)

T.-H. Wei, T.-H. Huang, H.-D. Lin, and S.-H. Lin, “Lifetime determination for high-lying states using Z-scan,” Appl. Phys. Lett. 67, 2266–2268 (1995).
[CrossRef]

G. S. He, J. D. Bhawalkar, C. F. Zhao, and P. N. Prasad, “Optical limiting effect in a two-photon absorption dye doped solid matrix,” Appl. Phys. Lett. 67, 2433–2435 (1995).
[CrossRef]

IEEE J. Quantum Electron. (2)

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]

C. R. Guliano and L. D. Hess, “Nonlinear absorption of light: optical saturation of electronic transitions in organic molecules with high intensity laser radiation,” IEEE J. Quantum Electron. QE-3, 358–367 (1967).
[CrossRef]

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

D. J. Hagan, T. Xia, A. A. Said, T. H. Wei, and E. W. Van Stryland, “High dynamic range passive optical limiters,” Int. J. Nonlinear Opt. Phys. 2, 483–501 (1993).
[CrossRef]

J. Appl. Spectrosc. (1)

V. I. Bezrodnyi, M. V. Bondar, G. Y. Kozak, O. V. Przhonska, and Y. A. Tikhonov, “Dye-activated polymeric media for frequency tunable dye lasers,” J. Appl. Spectrosc. 50, 441–454 (1989).
[CrossRef]

J. Lumin. (1)

O. Przhonska, Yu. Slominsky, U. Stahl, and S. Daehne, “Excited-state relaxation processes in polymethine dye molecules in polymeric media,” J. Lumin. 69, 105–113 (1996).
[CrossRef]

J. Phys. Chem. (1)

M. V. Bondar, O. V. Przhonska, and Y. A. Tikhonov, “Inhonogeneous broadening of organic dyes in polymeric media: nonlinear transmission spectra and photochemical kinetics,” J. Phys. Chem. 96, 10831–10837 (1992).
[CrossRef]

Kvant. Elektron. (Moscow) (1)

M. Alperovich, G. Dyadyusha, O. Przhonskaya, T. Smirnova, Yu. Slominsky, E. Tikhonov, A. Tolmachev, V. Tjurin, and M. Shpak, “Active media for near IR region utilizing polymethine dyes,” Kvant. Elektron. (Moscow) 6, 1231–1236 (1979).

Opt. Commun. (2)

W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2,” Opt. Commun. 50, 256–260 (1984).
[CrossRef]

S. Hughes, G. Spruce, B. S. Wherrett, K. R. Welford, and A. D. Lloyd, “The saturation limit to picosecond, induced absorption in dyes,” Opt. Commun. 100, 113–117 (1993).
[CrossRef]

Opt. Eng. (Bellingham) (1)

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two-photon absorption, nonlinear refraction and optical limiting in semiconductors,” Opt. Eng. (Bellingham) 24, 613–623 (1985).

Opt. Lett. (3)

Optik (Stuttgart) (1)

T. H. Wei, W. P. Zang, and G. Zang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik (Stuttgart) 98, 143–146 (1995).

Phys. Rev. B (1)

J. A. Fleck, “Ultrashort pulse generation by Q-switched lasers,” Phys. Rev. B 1, 84–100 (1970).
[CrossRef]

Proc. SPIE (3)

K. Mansour, P. Fuqua, S. R. Marder, B. Dunn, and J. W. Perry, “Solid state optical limiting materials based on phthalocyanine containing polymers and organically-modified sol-gels,” in Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 239–250 (1994).
[CrossRef]

O. V. Przhonska, M. V. Bondar, and Y. A. Tikhonov, “Nonlinear light absorption by liquid and solid solutions of organic dyes,” Organic, Metallo-Organic, and Nonlinear Materials for Nonlinear Optical Applications, S. R. Marder and J. W. Perry, eds., Proc. SPIE 2143, 289–297 (1994).
[CrossRef]

D. R. Coulter, V. M. Miskowski, J. W. Perry, T. H. Wei, E. W. Van Stryland, and D. J. Hagan, in Materials for Optical Switches, Isolators and Limiters, M. J. Soileau, ed., Proc. SPIE 1105, 42–51 (1989).
[CrossRef]

Prog. Quantum Electron. (1)

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

Science (1)

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

Sov. Phys. JETP (1)

V. I. Bezrodnyi, O. V. Przhonska, E. A. Tikhonov, and M. T. Shpak, “Nonlinear absorption and thermal defocussing of light in dye solutions,” Sov. Phys. JETP 53, 259–270 (1981).

Other (3)

R. H. Pantel and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, New York, 1969).

M. Brunel, M. Canva, A. Brun, F. Chaput, and J.-P. Boilot, “Dynamics and tuning ranges of reverse saturation absorption of organically doped xerogels,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 278–279.

E. P. Ippen and C. V. Shank, in Ultrashort Light Pulses, S. L. Shapiro, ed., Vol. 18 of Topics in Applied Physics (Springer-Verlag, Berlin, 1977), p. 83.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Three-level model for organic dyes. S0, S1, and S2 are singlet states, σ01 and σ12 are absorption cross sections, and 1/τ1 and 1/τ2 are rate constants. The horizontal lines in each singlet state are vibrational energy levels. (b) Calculated dependencies of transmittance versus fluence, F: curve 1, σ01 >σ12; curve 2, σ01=σ12; curve 3, σ01<σ12.

Fig. 2
Fig. 2

Chemical structures of the polymethine dyes.

Fig. 3
Fig. 3

Normalized absorption spectra of PD 1–PD 5 in ethanol solutions. The dashed curve shows absorption for PD 3 in PUA.

Fig. 4
Fig. 4

Z-scan data for PD 1 in (a) ethanol and in (b) PUA matrix and for PD 2 in (c) ethanol and in (d) PUA matrix. (a) Energy E=0.1 (curve 1), 0.5 μJ (curve 2); (b) E=0.03 (1), 0.3 μJ (2); (c) E=0.1 (1), 0.5 μJ (2); (d) E=0.1 (1), 0.5 μJ (2). The solid curves are numerical fits as described in Section 4.

Fig. 5
Fig. 5

(a) Z-scan data for PD 3 in ethanol; energy E=0.05 (curve 1), 1.0 μJ (curve 2). The solid curves are numerical fits as described in Section 4. (b) Z scan at E=4.0 μJ for PD 3 in ethanol for two repetition rates to 10 Hz (curve 1) and 0.5 Hz (curve 2).

Fig. 6
Fig. 6

Z-scan data for PD 4 in (a) ethanol solution and in (b) PUA matrix. (a) E=0.05 (curve 1), 0.5 (curve 2), 2.0 μJ (curve 3); (b) E=0.5 (curve 1), 2.0 μJ (curve 2).

Fig. 7
Fig. 7

Results of pump–probe experiments for PD 2 in 1, PUA and 2, ethanol solution at a pumping fluence F=0.015 J/cm2. The solid curves are single exponential fits.

Tables (1)

Tables Icon

Table 1 Parameters of Polymethine Dyes in Ethanol and PUA

Equations (5)

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

σ01=-ln(TL)NL,
dFdz=N ωσ01 (σ01-σ12)[exp(-Fσ01/ω)-1]-σ12NF,
dFdz=-NFσ011+(σ12-σ01) F2ω.
F(L)=F(0)exp(-αL)1+(σ12-σ01)F(0)[1-exp(-αL)]/(2ω),
T=-F(L, t)dt-F(0, t)dt.

Metrics