Abstract

The Z-scan technique is used to measure the nonlinear refractive index n2 and the nonlinear absorption coefficient β of ionic magnetic colloids as a function of the nanoparticles’ diameter (Drx) in the femto- and millisecond time-scale regimes. A change from a convergent-lens type (at a femtosecond time scale) to a divergent-lens type behavior (at a millisecond time scale) is observed. From the values of n2 and β measured in the femtosecond time-scale regime, the third-order susceptibility χ(3) is calculated and is shown to depend on Drx: the smaller the Drx, the bigger the χ(3). A possible origin of the high values of χ(3) found may be confining effects related to the physical nanoscale of the magnetic particles.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Boyd, Nonlinear Optics (Academic, 1992).
  2. 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]
  3. R. E. Rosensweig, Ferrohydrodynamics (Cambridge U. Press, 1985).
  4. R. Massart, "Preparation of aqueous magnetic liquids in alkaline and acid media," IEEE Trans. Magn. 17, 1247-1248 (1981).
    [CrossRef]
  5. A. M. Figueiredo Neto and S. R. A. Salinas, The Physics of Lyotropic Liquid Crystals: Phase Transitions and Structural Properties (Oxford U. Press, 2005).
  6. J. W. M. Bulte, "Magnetic nanoparticles as markers for cellular MR imaging," J. Magn. Magn. Mater. 289, 423-427 (2005).
    [CrossRef]
  7. T. Hashimoto, T. Yoko, and S. Sakka, "Third-order nonlinear optical susceptibility of α-iron (III) oxide thin film prepared by the sol-gel method," J. Ceram. Soc. Jpn. 101, 64-68 (1993).
    [CrossRef]
  8. T. Hashimoto, T. Yamada, and T. Yoko, "Third-order nonlinear optical properties of sol-gel derived α-Fe2O3, gamma-Fe2O3, and Fe3O4 thin films," J. Appl. Phys. 80, 3184-3190 (1996).
    [CrossRef]
  9. C. C. Wang, "Empirical relation between the linear and the third-order nonlinear optical susceptibility," Phys. Rev. B 2, 2045-2048 (1970).
    [CrossRef]
  10. B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
    [CrossRef]
  11. H. J. Yuan, L. Li, and P. Palffy-Muhoray, "Nonlinear birefringence of nematic liquid crystals," Proc. SPIE 1307, 363-373 (1990).
    [CrossRef]
  12. S. Alves, A. Bourdon, and A. M. Figueiredo Neto, "Generalization of the thermal lens model formalism to account for thermodiffusion in a single-beam Z-scanexperiment: determination of the Soret coefficient," J. Opt. Soc. Am. B 20, 713-718 (2003).
    [CrossRef]
  13. S. Alves, F. L. S. Cuppo, and A. M. Figueiredo Neto, "Determination of the nonlinear refractive index of lyotropic mixtures with and without the ferrofluid doping: a time-resolved Z-scan experiment in millisecond time scales," J. Opt. Soc. Am. B 23, 67-74 (2006).
    [CrossRef]
  14. S. Alves, A. Bourdon, and A. M. Figueiredo Neto, "Investigation of the Soret coefficient in magnetic fluids using the Z-scan technique," J. Magn. Magn. Mater. 289, 285-288 (2005).
    [CrossRef]
  15. M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
    [CrossRef]
  16. F. A. Tourinho, R. Franck, and R. Massart, "Aqueous ferrofluids based on manganese and cobalt ferrites," J. Mater. Sci. 25, 3249-3254 (1990).
    [CrossRef]
  17. X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
    [CrossRef] [PubMed]
  18. F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, "Observation of three-photon enhanced four-photon absorption," Phys. Rev. Lett. 91, 0639021 (2003).
    [CrossRef]

2006 (2)

S. Alves, F. L. S. Cuppo, and A. M. Figueiredo Neto, "Determination of the nonlinear refractive index of lyotropic mixtures with and without the ferrofluid doping: a time-resolved Z-scan experiment in millisecond time scales," J. Opt. Soc. Am. B 23, 67-74 (2006).
[CrossRef]

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

2005 (2)

S. Alves, A. Bourdon, and A. M. Figueiredo Neto, "Investigation of the Soret coefficient in magnetic fluids using the Z-scan technique," J. Magn. Magn. Mater. 289, 285-288 (2005).
[CrossRef]

J. W. M. Bulte, "Magnetic nanoparticles as markers for cellular MR imaging," J. Magn. Magn. Mater. 289, 423-427 (2005).
[CrossRef]

2003 (2)

2001 (1)

M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
[CrossRef]

1997 (1)

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

1996 (1)

T. Hashimoto, T. Yamada, and T. Yoko, "Third-order nonlinear optical properties of sol-gel derived α-Fe2O3, gamma-Fe2O3, and Fe3O4 thin films," J. Appl. Phys. 80, 3184-3190 (1996).
[CrossRef]

1993 (1)

T. Hashimoto, T. Yoko, and S. Sakka, "Third-order nonlinear optical susceptibility of α-iron (III) oxide thin film prepared by the sol-gel method," J. Ceram. Soc. Jpn. 101, 64-68 (1993).
[CrossRef]

1990 (2)

H. J. Yuan, L. Li, and P. Palffy-Muhoray, "Nonlinear birefringence of nematic liquid crystals," Proc. SPIE 1307, 363-373 (1990).
[CrossRef]

F. A. Tourinho, R. Franck, and R. Massart, "Aqueous ferrofluids based on manganese and cobalt ferrites," J. Mater. Sci. 25, 3249-3254 (1990).
[CrossRef]

1989 (1)

1981 (1)

R. Massart, "Preparation of aqueous magnetic liquids in alkaline and acid media," IEEE Trans. Magn. 17, 1247-1248 (1981).
[CrossRef]

1970 (1)

C. C. Wang, "Empirical relation between the linear and the third-order nonlinear optical susceptibility," Phys. Rev. B 2, 2045-2048 (1970).
[CrossRef]

Alves, S.

Bourdon, A.

S. Alves, A. Bourdon, and A. M. Figueiredo Neto, "Investigation of the Soret coefficient in magnetic fluids using the Z-scan technique," J. Magn. Magn. Mater. 289, 285-288 (2005).
[CrossRef]

S. Alves, A. Bourdon, and A. M. Figueiredo Neto, "Generalization of the thermal lens model formalism to account for thermodiffusion in a single-beam Z-scanexperiment: determination of the Soret coefficient," J. Opt. Soc. Am. B 20, 713-718 (2003).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Bulte, J. W. M.

J. W. M. Bulte, "Magnetic nanoparticles as markers for cellular MR imaging," J. Magn. Magn. Mater. 289, 423-427 (2005).
[CrossRef]

Chen, W.

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

Cuppo, F. L. S.

da Silva, G. J.

M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
[CrossRef]

Depeyrot, J.

M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
[CrossRef]

Ding, S.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Du, Y.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Figueiredo Neto, A. M.

Franck, R.

F. A. Tourinho, R. Franck, and R. Massart, "Aqueous ferrofluids based on manganese and cobalt ferrites," J. Mater. Sci. 25, 3249-3254 (1990).
[CrossRef]

Gan, F.

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

Hashimoto, T.

T. Hashimoto, T. Yamada, and T. Yoko, "Third-order nonlinear optical properties of sol-gel derived α-Fe2O3, gamma-Fe2O3, and Fe3O4 thin films," J. Appl. Phys. 80, 3184-3190 (1996).
[CrossRef]

T. Hashimoto, T. Yoko, and S. Sakka, "Third-order nonlinear optical susceptibility of α-iron (III) oxide thin film prepared by the sol-gel method," J. Ceram. Soc. Jpn. 101, 64-68 (1993).
[CrossRef]

Lara, M. C. F. L.

M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
[CrossRef]

Li, L.

H. J. Yuan, L. Li, and P. Palffy-Muhoray, "Nonlinear birefringence of nematic liquid crystals," Proc. SPIE 1307, 363-373 (1990).
[CrossRef]

Liu, M.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, "Observation of three-photon enhanced four-photon absorption," Phys. Rev. Lett. 91, 0639021 (2003).
[CrossRef]

Massart, R.

F. A. Tourinho, R. Franck, and R. Massart, "Aqueous ferrofluids based on manganese and cobalt ferrites," J. Mater. Sci. 25, 3249-3254 (1990).
[CrossRef]

R. Massart, "Preparation of aqueous magnetic liquids in alkaline and acid media," IEEE Trans. Magn. 17, 1247-1248 (1981).
[CrossRef]

Palffy-Muhoray, P.

H. J. Yuan, L. Li, and P. Palffy-Muhoray, "Nonlinear birefringence of nematic liquid crystals," Proc. SPIE 1307, 363-373 (1990).
[CrossRef]

Pang, D.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Polyakov, S.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, "Observation of three-photon enhanced four-photon absorption," Phys. Rev. Lett. 91, 0639021 (2003).
[CrossRef]

Rosensweig, R. E.

R. E. Rosensweig, Ferrohydrodynamics (Cambridge U. Press, 1985).

Said, A. A.

Sakka, S.

T. Hashimoto, T. Yoko, and S. Sakka, "Third-order nonlinear optical susceptibility of α-iron (III) oxide thin film prepared by the sol-gel method," J. Ceram. Soc. Jpn. 101, 64-68 (1993).
[CrossRef]

Salinas, S. R. A.

A. M. Figueiredo Neto and S. R. A. Salinas, The Physics of Lyotropic Liquid Crystals: Phase Transitions and Structural Properties (Oxford U. Press, 2005).

Sheik-Bahae, M.

Shen, X.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Sousa, M. H.

M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
[CrossRef]

Stegeman, G.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, "Observation of three-photon enhanced four-photon absorption," Phys. Rev. Lett. 91, 0639021 (2003).
[CrossRef]

Tang, G.

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

Tourinho, F. A.

M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
[CrossRef]

F. A. Tourinho, R. Franck, and R. Massart, "Aqueous ferrofluids based on manganese and cobalt ferrites," J. Mater. Sci. 25, 3249-3254 (1990).
[CrossRef]

Van Stryland, E. W.

Wang, C. C.

C. C. Wang, "Empirical relation between the linear and the third-order nonlinear optical susceptibility," Phys. Rev. B 2, 2045-2048 (1970).
[CrossRef]

Wang, Q.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Wang, X.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Wu, X.

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

Xie, M.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Xiong, G.

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

Yamada, T.

T. Hashimoto, T. Yamada, and T. Yoko, "Third-order nonlinear optical properties of sol-gel derived α-Fe2O3, gamma-Fe2O3, and Fe3O4 thin films," J. Appl. Phys. 80, 3184-3190 (1996).
[CrossRef]

Yoko, T.

T. Hashimoto, T. Yamada, and T. Yoko, "Third-order nonlinear optical properties of sol-gel derived α-Fe2O3, gamma-Fe2O3, and Fe3O4 thin films," J. Appl. Phys. 80, 3184-3190 (1996).
[CrossRef]

T. Hashimoto, T. Yoko, and S. Sakka, "Third-order nonlinear optical susceptibility of α-iron (III) oxide thin film prepared by the sol-gel method," J. Ceram. Soc. Jpn. 101, 64-68 (1993).
[CrossRef]

Yoshino, F.

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, "Observation of three-photon enhanced four-photon absorption," Phys. Rev. Lett. 91, 0639021 (2003).
[CrossRef]

Yu, B.

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

Yuan, H. J.

H. J. Yuan, L. Li, and P. Palffy-Muhoray, "Nonlinear birefringence of nematic liquid crystals," Proc. SPIE 1307, 363-373 (1990).
[CrossRef]

Zhang, G.

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

Zhu, C.

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

IEEE Trans. Magn. (1)

R. Massart, "Preparation of aqueous magnetic liquids in alkaline and acid media," IEEE Trans. Magn. 17, 1247-1248 (1981).
[CrossRef]

J. Appl. Phys. (1)

T. Hashimoto, T. Yamada, and T. Yoko, "Third-order nonlinear optical properties of sol-gel derived α-Fe2O3, gamma-Fe2O3, and Fe3O4 thin films," J. Appl. Phys. 80, 3184-3190 (1996).
[CrossRef]

J. Ceram. Soc. Jpn. (1)

T. Hashimoto, T. Yoko, and S. Sakka, "Third-order nonlinear optical susceptibility of α-iron (III) oxide thin film prepared by the sol-gel method," J. Ceram. Soc. Jpn. 101, 64-68 (1993).
[CrossRef]

J. Magn. Magn. Mater. (2)

J. W. M. Bulte, "Magnetic nanoparticles as markers for cellular MR imaging," J. Magn. Magn. Mater. 289, 423-427 (2005).
[CrossRef]

S. Alves, A. Bourdon, and A. M. Figueiredo Neto, "Investigation of the Soret coefficient in magnetic fluids using the Z-scan technique," J. Magn. Magn. Mater. 289, 285-288 (2005).
[CrossRef]

J. Mater. Sci. (1)

F. A. Tourinho, R. Franck, and R. Massart, "Aqueous ferrofluids based on manganese and cobalt ferrites," J. Mater. Sci. 25, 3249-3254 (1990).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Chem. B (2)

X. Wang, Y. Du, S. Ding, Q. Wang, G. Xiong, M. Xie, X. Shen, and D. Pang, "Preparation of third-order optical nonlinearities of self-assembled chitosan/CdSe-ZnS core-shell quantum dots multilayer films," J. Phys. Chem. B 110, 1566-1570 (2006).
[CrossRef] [PubMed]

M. H. Sousa, J. Depeyrot, G. J. da Silva, M. C. F. L. Lara, and F. A. Tourinho, "New electric double-layered magnetic fluids based on copper, nickel, and zinc ferrite nanostructures," J. Phys. Chem. B 105, 1168-1175 (2001).
[CrossRef]

Opt. Lett. (1)

Opt. Mater. (1)

B. Yu, C. Zhu, F. Gan, X. Wu, G. Zhang, G. Tang, and W. Chen, "Optical nonlinearities of Fe2O3 nanoparticles investigated by Z-scan technique," Opt. Mater. 8, 249-254 (1997).
[CrossRef]

Phys. Rev. B (1)

C. C. Wang, "Empirical relation between the linear and the third-order nonlinear optical susceptibility," Phys. Rev. B 2, 2045-2048 (1970).
[CrossRef]

Phys. Rev. Lett. (1)

F. Yoshino, S. Polyakov, M. Liu, and G. Stegeman, "Observation of three-photon enhanced four-photon absorption," Phys. Rev. Lett. 91, 0639021 (2003).
[CrossRef]

Proc. SPIE (1)

H. J. Yuan, L. Li, and P. Palffy-Muhoray, "Nonlinear birefringence of nematic liquid crystals," Proc. SPIE 1307, 363-373 (1990).
[CrossRef]

Other (3)

R. W. Boyd, Nonlinear Optics (Academic, 1992).

R. E. Rosensweig, Ferrohydrodynamics (Cambridge U. Press, 1985).

A. M. Figueiredo Neto and S. R. A. Salinas, The Physics of Lyotropic Liquid Crystals: Phase Transitions and Structural Properties (Oxford U. Press, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Typical ZS curves of the normalized transmittances in the femtosecond experiment, as a function of z. Water-based ionic sample IF2: (a) on-axis transmittance. The solid curve represents the best fit of Eq. (3) to the experimental data (b) Total transmittance. The solid curve represents the best fit of Eq. (5) to the experimental data.

Fig. 2
Fig. 2

Nonlinear optical absorption β as a function of the incident (peak) laser power I o . Water-based ionic sample IF2.

Fig. 3
Fig. 3

Modulus of χ ( 3 ) as a function of the magnetic particle’s diameter. Water-based ionic sample IF2.

Fig. 4
Fig. 4

Typical ZS curve of the normalized transmittances in the millisecond experiment, as a function of z. Water-based ionic sample IF2.

Fig. 5
Fig. 5

Nonlinear refraction index n 2 as a function of the magnetic particle’s diameter D r x .

Tables (2)

Tables Icon

Table 1 Femtosecond ZS Experiment Results a

Tables Icon

Table 2 Millisecond ZS Experiment Results a

Equations (9)

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

Re χ ( 3 ) = 2 n o 2 ε o c n 2 [ m 2 V 2 ] ,
Im χ ( 3 ) = n o 2 ε o c 2 β w [ m 2 V 2 ] ,
Γ N ( z ) 1 Δ Φ o 4 x ( 1 + x 2 ) ( 9 + x 2 ) Θ 2 ( x 2 + 3 ) ( 1 + x 2 ) ( 9 + x 2 ) ,
Δ Φ o = 2 π n 2 L eff I o λ ,
Γ N ( z ) 1 1 2 Θ 1 + x 2 .
Γ ( z , t ) = Γ 1 2 x [ C N ( 1 + x 2 ) 2 + C T 1 + x 2 t t + 2 t c ] + ( 1 + x 2 ) [ C N ( 1 + x 2 ) 2 + C T 1 + x 2 t t + 2 t c ] 2 ,
Γ N ( z , t 0 + ) = Γ ( z , t 0 + ) Γ = Γ N ( z ) ,
Δ Γ pv theo = Γ N ( z = z p ) Γ N ( z = z v ) ,
Γ N ( z ) = 1 1 2 x [ C N ( 1 + x 2 ) 2 ] + ( 1 + x 2 ) [ C N ( 1 + x 2 ) 2 ] .

Metrics