Abstract

We investigated nonlinear optical properties of Phenoxy-phthalocyanine (Pc1) and Phenoxy-phthalocyanine-Zinc(II) (Pc2) at a wavelength of 800 nm with 100 fs pulses .The nonlinear absorption coefficient (α) and nonlinear refractive index (n 2) are measured using standard Z-scan technique. Open aperture Z-scan indicates strong three-photon absorption in both phthalocyanines. With good solubility and excellent nonlinear optical coefficient,the samples are expected to be a potential candidate for optical applications.

©2010 Optical Society of America

1. Introduction

Phthalocyanines and metallophthalocyanines have been studied in great deal for many years. Recently, they have also found applications in many fields, especially in nonlinear optical (NLO) devices, electrochromic devices and gas sensors among others [1, 2]. Phthalocyanines are versatile because of their aromatic 18-π electron system and cability of containing more than 70 kinds of metallic and non-metallic ions in the ring cavity [3, 4]. A disadvantage of phthalocyanines and metallophthalocyanines is their low solubility. The solubility can be increased, however, by introducing long chain groups, such as alkyl or phenoxy into the peripheral positions of the phthalocyanine framework [5].

In this paper, phthalocyanines with four peripheral phenoxy substituents were prepared and their complexes with Zn (II) metal salts were investigated. In addition, we investigated the nonlinear optical properties of Phenoxy-phthalocyanine and Phenoxy-phthalocyanine- Zinc(II) using Z-scan technique with 800 nm, 100 fs pulses laser. Our studies in the femtosecond domain provide sufficient evidence that these molecules possess superior nonlinear optical performance for their potential applications.

2. Experiment

Phenoxy-phthalocyanine (IR (KBr) ν max/cm−1: 1230(C-O-C), 3100, 1615, 1530, 1336, 1187, 1130 (Pc skeletal). 1H NMR (DMSO 400MHz): δ, ppm 7.2, 7.31-7.35, 7.5(5H, m, Ph-H), 7.36-7.39, 7.8, 8.1(3H, m, Pc-H)) and Phenoxy-phthalocyanine-Zinc(II) (IR (KBr) ν max/cm−1: 1251(C-O-C), 3100, 1608, 1565, 1336, 1180, 1130 (Pc skeletal). 1H NMR (DMSO 400MHz): δ, ppm 7.2, 7.32-7.36, 7.5(5H, m, Ph-H), 7.37-7.41, 7.8, 8.1(3H, m, Pc-H)) were synthesized [6] and their molecular weight are 883u and 946u. All the experiments were performed with samples dissolved in N, N-dimethylformamide (DMF) and the solubility of Pc1 and Pc2 were estimated to be 108g/L and 19g/L, respectively. Figure 1 shows the molecular structure of the phthalocyanines.

 figure: Fig. 1

Fig. 1 Molecular structures of the phthalocyanines.

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A very convenient and fast experimental method to determine the nonlinear optical properties (NLO) of materials is the Z-scan experiment. Z-Scan experiment, based on the beam-distortion effect in a nonlinear sample, is a technique for measuring the value of the effective nonlinear absorption coefficient [710]. Standard Z-scan experimental setup is shown as Fig. 2 . A commercial Ti-sapphire laser was used as the light source with a repetition rate of 10 Hz, pulse width of 100 fs and wavelength of 800 nm. A lens with focal length of 300 mm was used to focus the laser pulses into 1mm thick quartz cuvette, which contained the sample solution at a concentration of 4.53 × 10−4 mol/L.

 figure: Fig. 2

Fig. 2 Z-scan experimental setup

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3. Results and discussion

The UV absorption spectrum of Pc1 and Pc2 is shown in Fig. 3 . Both the Q-band centered at about 700 nm and the B-band centered at around 300 nm can be observed clearly. The Q-band occurring at about 700 nm originate from the typical absorption which is correlated to π-π* transitions from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO) [11].

 figure: Fig. 3

Fig. 3 UV absorption spectrum of Pc1and Pc2

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Assuming Gaussian profile for laser pulses and using the open aperture Z-scan theory [12], the general equation for open aperture normalized transmittance is given by:

TnPA=1[1+(n1)αnLeff(I0/(1+(z/z0)2))n1]1/(n1)
α n is the effective multi-photon absorption coefficient(n = 2 for two-photon absorption; n = 3 for three-photon absorption, and so on), I 0 is the intensity of laser beam, Leff=(1exp((n1)α0L))/((n1)α0)is the effective length with α 0 the linear absorption coefficient and L is the thickness of the sample, z 0 is the diffraction length of the beam.

Figure 4 shows open aperture Z-scan of Pc1 and Pc2 at input intensities of 3.92 × 109 W/cm2. Open circles represent experimental data, the solid line represents theoretical fit with n = 3 (three-photon absorption) and dashed line represents the fit obtained with n = 2(two-photon absorption). The best fit was obtained with the transmission equation for n = 3.

 figure: Fig. 4

Fig. 4 Normalized open-aperture Z-scan transmittance of Pc1 and Pc2 while the solid line represents theoretical fit with three-photon absorption and dashed line represents the fit obtained with two-photon absorption.

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In order to distinguish between multi-photon absorption processes, we performed intensity dependent absorption studies in the open aperture Z-scan. The values of α 2 (α 3) for Pc1 (Pc2) were obtained with the theoretical fits with Eq. (1) for five different intensities in the range of 4.0 ~11.0 × 109W/cm2.

The difference between the intensity dependence of α 2 and α 3 for the samples are depicted in Fig. 5 . We find that two-photon absorption coefficient (α 2) of samples increases with the energy and three-photon absorption coefficient (α 3) remain constant. Obviously, the nonlinear absorption process involved is certainly three-photon absorption [13, 14].

 figure: Fig. 5

Fig. 5 The comparison of intensity dependence of α2 and α3 for (a) Pc1 and (b) Pc2

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The value of the α 3 was evaluated from the fits to the experimental data obtained using Eq. (1). The values estimated for Pc1 and Pc2 were 2.84 × 10−18cm3/W2 and 7.48 × 10−18cm3/W2, respectively. The nonlinear absorption coefficient of Pc1 (free base) is smaller than the Pc2, which results from the substitution of H2 by metal Zn, the substitution changes the symmetry of the molecular. We have evaluated the three-photon absorption cross-section (σ 3) using the relationσ3=α3(ω)2/N, where ω is the frequency of the laser radiation, N = NAC is the number of molecules per milliliter, NA is the Avogadro’s number, and C is the concentration in mol/L. The values for Pc1 and Pc2 were 0.64 × 10−72cm6s2 and 1.69 × 10−72 cm6s2, respectively.

The nonlinear refractive property of the samples was assessed from a division of the normalized close aperture data. The measured transmittance as a function of sample position along the Z-axis is shown in Fig. 6 at input intensities of 3.92 × 109 W/cm2 for the two samples, respectively. The close-aperture curve indicates a self-focusing effect and a positive refractive index.

 figure: Fig. 6

Fig. 6 Closed aperture Z-scan curves for (a) Pc1 and (b) Pc2 (Solid lines are the fits to the standard closed aperture equations)

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The third-order nonlinear refractive index (n 2) value can be obtained from ΔTpνusing the Eq. (2):

ΔTpv=0.406(1S)0.25|Δϕ0|Δϕ0=kn2I0(t)Leff
where ΔTpν is the measured peak-valley transmittance difference, Δϕ0is the on-axis phase-shift and I 0(t) is the peak intensity at focus and S is the linear transmittance of the aperture given byS=1exp(2ra2/wa2) where r a is the radius of the aperture and w a is the radius of the laser spot before the aperture. The nonlinear refractive index n 2 evaluated using Eq. (2) was 5.7 × 10−15 cm2 /W for the Pc1 and 8.4 × 10−15 cm2 /W for the Pc2 .

The nonlinear refractive index mainly comes from population redistribution and is better described by excited state refractive cross-section σr than by n2 in such cumulative nonlinearity [15]. Such a σr is related to the on-axis phase distortion at focusΔϕ0by [16]

Δϕ0=α0hωσrI0(t)Leff

The σr values of 6.1 × 10−18 cm2 for Pc1 and 3.4 × 10−18 cm2 for Pc2 were obtained. It is obvious that the introduction of Zn, which changes the excited singlet state absorption cross section, also has much influence on the nonlinear refractive index of phthalocyanines [17, 18].

4. Conclusions

The nonlinear optical properties of two phenoxy-phthalocyanines in DMF have been investigated by Z-scan technique using 100 fs laser pulses at 800 nm. The data indicates strong three-photon absorption and nonlinear absorption coefficients evaluated were 5.32 × 10−18cm3/W2 for the free-base phthalocyanine and 16.38 × 10−18cm3/W2 for the metallic (Zn) phthalocyanine. The values of the nonlinear refractive indices for Pc1 and Pc2 were estimated to be 0.57 × 10−17 cm2 /W and 0.84 × 10−17 cm2 /W. With good solubility and excellent third order NLO coefficient, the samples are expected to be a potential candidate for optical applications.

Acknowledgements

The research is supported by the National Natural Science Foundation of China under Grant Nos. 60478014 and 60878006.

References and Links

1. D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007). [CrossRef]  

2. M. Durmuş and T. Nyokong, “Synthesis, photophysical and photochemical studies of new water-soluble indium(III) phthalocyanines,” Photochem. Photobiol. Sci. 6(6), 659–668 (2007). [CrossRef]   [PubMed]  

3. R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007). [CrossRef]  

4. M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007). [CrossRef]  

5. H. Isago, K. Miura, and Y. Oyama, “Synthesis and properties of a highly soluble dihydoxo(tetra-tert-butylphthalocyaninato)antimony(V) complex as a precursor toward water-soluble phthalocyanines,” J. Inorg. Biochem. 102(3), 380–387 (2008). [CrossRef]   [PubMed]  

6. D. K. Modibane and T. Nyokong, “Synthesis and photophysical properties of lead phthalocyanines,” Polyhedron 27(3), 1102–1110 (2008). [CrossRef]  

7. S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]  

8. M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003). [CrossRef]  

9. G. S. Maciel, A. G. Bezerra-Jr, N. Rakov, C. B. de Araujo, A. S. L. Gomes, and W. M. de Azevedo, “Third-order nonlinear optical properties of undoped polyaniline solutions and films probed at 532 nm,” J. Opt. Soc. Am. B 18(8), 1099–1103 (2001). [CrossRef]  

10. Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009). [CrossRef]  

11. C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007). [CrossRef]  

12. R. L. Sutherland, Handbook of Nonlinear Optics. Second Edition (Marcel Dekker, Inc. New York, 2003), Chap. 9.

13. F. E. Hernández, K. D. Belfield, and I. Cohanoschi, “Three-photon absorption enhancement in a symmetrical charge transfer fluorene derivative,” Chem. Phys. Lett. 391(1-3), 22–26 (2004). [CrossRef]  

14. T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006). [CrossRef]  

15. C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007). [CrossRef]  

16. G. L. Wood, M. J. Miller, and A. G. Mott, “Investigation of tetrabenzporphyrin by the Z-scan technique,” Opt. Lett. 20(9), 973–975 (1995). [CrossRef]   [PubMed]  

17. S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009). [CrossRef]  

18. L. Howe and J. Z. Zhang, “Ultrafast Studies of Excited-State Dynamics of Phthalocyanine and Zinc Phthalocyanine Tetrasulfonate in Solution,” J. Phys. Chem. A 101(18), 3207–3213 (1997). [CrossRef]  

References

  • View by:

  1. D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
    [Crossref]
  2. M. Durmuş and T. Nyokong, “Synthesis, photophysical and photochemical studies of new water-soluble indium(III) phthalocyanines,” Photochem. Photobiol. Sci. 6(6), 659–668 (2007).
    [Crossref] [PubMed]
  3. R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007).
    [Crossref]
  4. M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
    [Crossref]
  5. H. Isago, K. Miura, and Y. Oyama, “Synthesis and properties of a highly soluble dihydoxo(tetra-tert-butylphthalocyaninato)antimony(V) complex as a precursor toward water-soluble phthalocyanines,” J. Inorg. Biochem. 102(3), 380–387 (2008).
    [Crossref] [PubMed]
  6. D. K. Modibane and T. Nyokong, “Synthesis and photophysical properties of lead phthalocyanines,” Polyhedron 27(3), 1102–1110 (2008).
    [Crossref]
  7. S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
    [Crossref]
  8. M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
    [Crossref]
  9. G. S. Maciel, A. G. Bezerra-Jr, N. Rakov, C. B. de Araujo, A. S. L. Gomes, and W. M. de Azevedo, “Third-order nonlinear optical properties of undoped polyaniline solutions and films probed at 532 nm,” J. Opt. Soc. Am. B 18(8), 1099–1103 (2001).
    [Crossref]
  10. Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
    [Crossref]
  11. C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
    [Crossref]
  12. R. L. Sutherland, Handbook of Nonlinear Optics. Second Edition (Marcel Dekker, Inc. New York, 2003), Chap. 9.
  13. F. E. Hernández, K. D. Belfield, and I. Cohanoschi, “Three-photon absorption enhancement in a symmetrical charge transfer fluorene derivative,” Chem. Phys. Lett. 391(1-3), 22–26 (2004).
    [Crossref]
  14. T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006).
    [Crossref]
  15. C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007).
    [Crossref]
  16. G. L. Wood, M. J. Miller, and A. G. Mott, “Investigation of tetrabenzporphyrin by the Z-scan technique,” Opt. Lett. 20(9), 973–975 (1995).
    [Crossref] [PubMed]
  17. S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009).
    [Crossref]
  18. L. Howe and J. Z. Zhang, “Ultrafast Studies of Excited-State Dynamics of Phthalocyanine and Zinc Phthalocyanine Tetrasulfonate in Solution,” J. Phys. Chem. A 101(18), 3207–3213 (1997).
    [Crossref]

2009 (2)

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009).
[Crossref]

2008 (2)

D. K. Modibane and T. Nyokong, “Synthesis and photophysical properties of lead phthalocyanines,” Polyhedron 27(3), 1102–1110 (2008).
[Crossref]

H. Isago, K. Miura, and Y. Oyama, “Synthesis and properties of a highly soluble dihydoxo(tetra-tert-butylphthalocyaninato)antimony(V) complex as a precursor toward water-soluble phthalocyanines,” J. Inorg. Biochem. 102(3), 380–387 (2008).
[Crossref] [PubMed]

2007 (6)

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

M. Durmuş and T. Nyokong, “Synthesis, photophysical and photochemical studies of new water-soluble indium(III) phthalocyanines,” Photochem. Photobiol. Sci. 6(6), 659–668 (2007).
[Crossref] [PubMed]

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007).
[Crossref]

R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007).
[Crossref]

2006 (1)

T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006).
[Crossref]

2004 (1)

F. E. Hernández, K. D. Belfield, and I. Cohanoschi, “Three-photon absorption enhancement in a symmetrical charge transfer fluorene derivative,” Chem. Phys. Lett. 391(1-3), 22–26 (2004).
[Crossref]

2003 (1)

M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
[Crossref]

2001 (1)

1997 (1)

L. Howe and J. Z. Zhang, “Ultrafast Studies of Excited-State Dynamics of Phthalocyanine and Zinc Phthalocyanine Tetrasulfonate in Solution,” J. Phys. Chem. A 101(18), 3207–3213 (1997).
[Crossref]

1995 (1)

1990 (1)

S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Alonzo, M.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Altindal, A.

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

Bekaroglu, Ö.

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

Belardini, A.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Belfield, K. D.

F. E. Hernández, K. D. Belfield, and I. Cohanoschi, “Three-photon absorption enhancement in a symmetrical charge transfer fluorene derivative,” Chem. Phys. Lett. 391(1-3), 22–26 (2004).
[Crossref]

Bertolotti, M.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Bezerra-Jr, A. G.

Bulut, M.

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

Chen, D.

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Chen, Y.

C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007).
[Crossref]

Cohanoschi, I.

F. E. Hernández, K. D. Belfield, and I. Cohanoschi, “Three-photon absorption enhancement in a symmetrical charge transfer fluorene derivative,” Chem. Phys. Lett. 391(1-3), 22–26 (2004).
[Crossref]

Davies, B. L.

M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
[Crossref]

de Araujo, C. B.

de Azevedo, W. M.

Dong, Z.

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Duan, W. B.

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

Durmus, M.

M. Durmuş and T. Nyokong, “Synthesis, photophysical and photochemical studies of new water-soluble indium(III) phthalocyanines,” Photochem. Photobiol. Sci. 6(6), 659–668 (2007).
[Crossref] [PubMed]

Fan, R. W.

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Fazio, E.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Giribabu, L.

S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009).
[Crossref]

R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007).
[Crossref]

Gomes, A. S. L.

Hagan, D. J.

S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

He, C. J.

C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007).
[Crossref]

He, C. Y.

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

He, G. S.

T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006).
[Crossref]

Hernández, F. E.

F. E. Hernández, K. D. Belfield, and I. Cohanoschi, “Three-photon absorption enhancement in a symmetrical charge transfer fluorene derivative,” Chem. Phys. Lett. 391(1-3), 22–26 (2004).
[Crossref]

Howe, L.

L. Howe and J. Z. Zhang, “Ultrafast Studies of Excited-State Dynamics of Phthalocyanine and Zinc Phthalocyanine Tetrasulfonate in Solution,” J. Phys. Chem. A 101(18), 3207–3213 (1997).
[Crossref]

Humphrey, M. G.

M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
[Crossref]

Isago, H.

H. Isago, K. Miura, and Y. Oyama, “Synthesis and properties of a highly soluble dihydoxo(tetra-tert-butylphthalocyaninato)antimony(V) complex as a precursor toward water-soluble phthalocyanines,” J. Inorg. Biochem. 102(3), 380–387 (2008).
[Crossref] [PubMed]

Jiang, Y. G.

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Kulac, D.

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

Kumar, R. S. S.

R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007).
[Crossref]

Larciprete, M. C.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Leahu, G.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Lin, T. C.

T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006).
[Crossref]

Maciel, G. S.

Mansoor, S. B.

S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Miller, M. J.

Miura, K.

H. Isago, K. Miura, and Y. Oyama, “Synthesis and properties of a highly soluble dihydoxo(tetra-tert-butylphthalocyaninato)antimony(V) complex as a precursor toward water-soluble phthalocyanines,” J. Inorg. Biochem. 102(3), 380–387 (2008).
[Crossref] [PubMed]

Modibane, D. K.

D. K. Modibane and T. Nyokong, “Synthesis and photophysical properties of lead phthalocyanines,” Polyhedron 27(3), 1102–1110 (2008).
[Crossref]

Mott, A. G.

Nie, Y. X.

C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007).
[Crossref]

Nyokong, T.

D. K. Modibane and T. Nyokong, “Synthesis and photophysical properties of lead phthalocyanines,” Polyhedron 27(3), 1102–1110 (2008).
[Crossref]

M. Durmuş and T. Nyokong, “Synthesis, photophysical and photochemical studies of new water-soluble indium(III) phthalocyanines,” Photochem. Photobiol. Sci. 6(6), 659–668 (2007).
[Crossref] [PubMed]

Ostuni, R.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Oyama, Y.

H. Isago, K. Miura, and Y. Oyama, “Synthesis and properties of a highly soluble dihydoxo(tetra-tert-butylphthalocyaninato)antimony(V) complex as a precursor toward water-soluble phthalocyanines,” J. Inorg. Biochem. 102(3), 380–387 (2008).
[Crossref] [PubMed]

Özkaya, A. R.

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

Prasad, P. N.

T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006).
[Crossref]

Rakov, N.

Rao, D. N.

S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009).
[Crossref]

R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007).
[Crossref]

Rao, S. V.

S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009).
[Crossref]

R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007).
[Crossref]

Said, A. A.

S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Salih, B.

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

Samoc, A.

M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
[Crossref]

Samoc, M.

M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
[Crossref]

Shi, G.

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

Sibilia, C.

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Song, W.

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

Song, Y. L.

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

Stryland, E. W. V.

S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Umesh, G.

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Venkatram, N.

S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009).
[Crossref]

Wang, D. Y.

C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007).
[Crossref]

Wei, T. H.

S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Wong, M. S.

M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
[Crossref]

Wood, G. L.

Wu, Y. Q.

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

Xia, Y. Q.

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Zhang, J. Z.

L. Howe and J. Z. Zhang, “Ultrafast Studies of Excited-State Dynamics of Phthalocyanine and Zinc Phthalocyanine Tetrasulfonate in Solution,” J. Phys. Chem. A 101(18), 3207–3213 (1997).
[Crossref]

Zhao, W.

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Zheng, Q. D.

T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006).
[Crossref]

Chem. Phys. Lett. (2)

R. S. S. Kumar, S. V. Rao, L. Giribabu, and D. N. Rao, “Femtosecond and nanosecond nonlinear optical properties of alkyl phthalocyanines studied using Z-scan technique,” Chem. Phys. Lett. 447(4-6), 274–278 (2007).
[Crossref]

F. E. Hernández, K. D. Belfield, and I. Cohanoschi, “Three-photon absorption enhancement in a symmetrical charge transfer fluorene derivative,” Chem. Phys. Lett. 391(1-3), 22–26 (2004).
[Crossref]

IEEE J. Quantum Electron. (1)

S. B. Mansoor, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

J. Appl. Phys. (1)

S. V. Rao, N. Venkatram, L. Giribabu, and D. N. Rao, “Ultrafast nonlinear optical properties of alkyl-phthalocyanine nanoparticles investigated using Z-scan technique,” J. Appl. Phys. 105(5), 053109 (2009).
[Crossref]

J. Inorg. Biochem. (1)

H. Isago, K. Miura, and Y. Oyama, “Synthesis and properties of a highly soluble dihydoxo(tetra-tert-butylphthalocyaninato)antimony(V) complex as a precursor toward water-soluble phthalocyanines,” J. Inorg. Biochem. 102(3), 380–387 (2008).
[Crossref] [PubMed]

J. Mater. Chem. (1)

T. C. Lin, G. S. He, Q. D. Zheng, and P. N. Prasad, “Degenerate two-/three-photon absorption and optical power-limiting properties in femtosecond regime of a multi-branched chromophore,” J. Mater. Chem. 16(25), 2490–2498 (2006).
[Crossref]

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

J. Phys. Chem. A (1)

L. Howe and J. Z. Zhang, “Ultrafast Studies of Excited-State Dynamics of Phthalocyanine and Zinc Phthalocyanine Tetrasulfonate in Solution,” J. Phys. Chem. A 101(18), 3207–3213 (1997).
[Crossref]

Opt. Commun. (1)

C. J. He, Y. Chen, Y. X. Nie, and D. Y. Wang, “Third order optical nonlinearities of eight-β-octa-octyloxy-phthalocyanines,” Opt. Commun. 271(1), 253–256 (2007).
[Crossref]

Opt. Laser Technol. (1)

Y. Q. Xia, Y. G. Jiang, R. W. Fan, Z. Dong, W. Zhao, D. Chen, and G. Umesh, “Ultrafast nonlinear optical properties of dye-doped PMMA discs irradiated by 40 fs laser pulses,” Opt. Laser Technol. 41(6), 700–704 (2009).
[Crossref]

Opt. Lett. (1)

Opt. Mater. (1)

M. Samoc, A. Samoc, B. L. Davies, M. G. Humphrey, and M. S. Wong, “Third-order optical nonlinearities of oligomers, dendrimers and polymers derived from solution Z-scan studies,” Opt. Mater. 21(1-3), 485–488 (2003).
[Crossref]

Org. Electron. (1)

C. Y. He, Y. Q. Wu, G. Shi, W. B. Duan, W. Song, and Y. L. Song, “Large third-order optical nonlinearities of ultrathin films containing octacarboxylic copper phthalocyanine,” Org. Electron. 8(2-3), 198–205 (2007).
[Crossref]

Photochem. Photobiol. Sci. (1)

M. Durmuş and T. Nyokong, “Synthesis, photophysical and photochemical studies of new water-soluble indium(III) phthalocyanines,” Photochem. Photobiol. Sci. 6(6), 659–668 (2007).
[Crossref] [PubMed]

Photon. Nanostructures (1)

M. C. Larciprete, R. Ostuni, A. Belardini, M. Alonzo, G. Leahu, E. Fazio, C. Sibilia, and M. Bertolotti, “Nonlinear optical absorption of zinc-phthalocyanines in polymeric matrix,” Photon. Nanostructures 5(2-3), 73–78 (2007).
[Crossref]

Polyhedron (2)

D. Kulac, M. Bulut, A. Altındal, A. R. Özkaya, B. Salih, and Ö. Bekaroğlu, “Synthesis and characterization of novel 4-nitro-2-(octyloxy)phenoxy substituted symmetrical and unsymmetrical Zn(II), Co(II) and Lu(III) phthalocyanines,” Polyhedron 26(18), 5432–5440 (2007).
[Crossref]

D. K. Modibane and T. Nyokong, “Synthesis and photophysical properties of lead phthalocyanines,” Polyhedron 27(3), 1102–1110 (2008).
[Crossref]

Other (1)

R. L. Sutherland, Handbook of Nonlinear Optics. Second Edition (Marcel Dekker, Inc. New York, 2003), Chap. 9.

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Figures (6)

Fig. 1
Fig. 1 Molecular structures of the phthalocyanines.
Fig. 2
Fig. 2 Z-scan experimental setup
Fig. 3
Fig. 3 UV absorption spectrum of Pc1and Pc2
Fig. 4
Fig. 4 Normalized open-aperture Z-scan transmittance of Pc1 and Pc2 while the solid line represents theoretical fit with three-photon absorption and dashed line represents the fit obtained with two-photon absorption.
Fig. 5
Fig. 5 The comparison of intensity dependence of α2 and α3 for (a) Pc1 and (b) Pc2
Fig. 6
Fig. 6 Closed aperture Z-scan curves for (a) Pc1 and (b) Pc2 (Solid lines are the fits to the standard closed aperture equations)

Equations (3)

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

T n P A = 1 [ 1 + ( n 1 ) α n L e f f ( I 0 / ( 1 + ( z / z 0 ) 2 ) ) n 1 ] 1 / ( n 1 )
Δ T p v = 0.406 ( 1 S ) 0.25 | Δ ϕ 0 | Δ ϕ 0 = k n 2 I 0 ( t ) L e f f
Δ ϕ 0 = α 0 h ω σ r I 0 ( t ) L e f f

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