Abstract

We report spatial self- and cross-phase modulation effects in metal-dielectric nanocomposites (MDNCs) whose nonlinear (NL) response is dominated by the quintic or septic refractive nonlinearity. The MDNCs consist of silver nanoparticles (NPs) suspended either in acetone or carbon disulfide and their effective NL susceptibilities are controlled by adjusting the volume fraction occupied by the NPs and the incident laser intensity. A theoretical treatment based on the Maxwell-Garnett model was developed to include contributions up to the seventh-order susceptibility showing a very good agreement with the experimental results.

© 2014 Optical Society of America

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2014 (2)

A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
[Crossref]

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

2013 (3)

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

2012 (6)

2011 (3)

2010 (6)

2009 (1)

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

2008 (2)

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

2007 (3)

2006 (3)

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
[Crossref]

M. A. R. C. Alencar and C. B. de Araújo, “Conical diffraction instability due to cross-phase modulation in Kerr media,” J. Opt. Soc. Am. B 23(2), 302–307 (2006).
[Crossref]

2005 (2)

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, A. Galembeck, M. M. Oliveira, and A. J. G. Zarbin, “Nonlinear susceptibility of colloids consisting of silver nanoparticles in carbon disulfide,” J. Opt. Soc. Am. B 22(11), 2444–2449 (2005).
[Crossref]

2004 (1)

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

2001 (1)

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

1992 (2)

J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
[Crossref] [PubMed]

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

1991 (1)

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[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(4), 760–769 (1990).
[Crossref]

1982 (1)

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

1981 (1)

Ahmad, M. B.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Alencar, M. A. R. C.

Alfano, R. R.

Ara, M. H. M.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Arakelian, S. M.

Barbano, E. C.

Barbosa-Silva, R.

Blau, W. J.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Boudebs, G.

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

Boyd, R.

Boyd, R. W.

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
[Crossref]

J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
[Crossref] [PubMed]

Brito-Silva, A. M.

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010).
[Crossref] [PubMed]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007).
[Crossref]

Carrasco, M. L. A.

Castillo, M. D. I.

Cerda, S. C.

Chari, R.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

Cheng, X.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Cheng, Y.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Coghlan, D.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Daneshfar, A.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Darroudi, M.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

de Araújo, C. B.

A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
[Crossref]

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010).
[Crossref] [PubMed]

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues., “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B 24(12), 2948–2956 (2007).
[Crossref]

M. A. R. C. Alencar and C. B. de Araújo, “Conical diffraction instability due to cross-phase modulation in Kerr media,” J. Opt. Soc. Am. B 23(2), 302–307 (2006).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, A. Galembeck, M. M. Oliveira, and A. J. G. Zarbin, “Nonlinear susceptibility of colloids consisting of silver nanoparticles in carbon disulfide,” J. Opt. Soc. Am. B 22(11), 2444–2449 (2005).
[Crossref]

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

de Assumpção, T. A.

De Boni, L.

Dehghani, Z.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Deng, L.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Divsar, F.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Dolgaleva, K.

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
[Crossref]

Dong, N.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Durbin, S. D.

Eichelbaum, M.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Falcão-Filho, E. L.

Galembeck, A.

Ganeev, R. A.

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Ginger, D. S.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Gomes, A. S. L.

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

Gómez, L. A.

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007).
[Crossref]

Graener, H.

Guo, S.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Guyer, S. R.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Hagan, D. J.

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(4), 760–769 (1990).
[Crossref]

He, K.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Herrmann, J.

Hickmann, J. M.

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

Hoell, A.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Hou, L.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Huang, J. P.

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
[Crossref]

Husakou, A.

Javadi, Z.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Jayabalan, J.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

J. Jayabalan, “Origin and time dependence of higher-order nonlinearities in metal nanocomposites,” J. Opt. Soc. Am. B 28(10), 2448–2455 (2011).
[Crossref]

Kassab, L. R. P.

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

L. De Boni, E. C. Barbano, T. A. de Assumpção, L. Misoguti, L. R. P. Kassab, and S. C. Zilio, “Femtosecond third-order nonlinear spectra of lead-germanium oxide glasses containing silver nanoparticles,” Opt. Express 20(6), 6844–6850 (2012).
[Crossref] [PubMed]

Khan, S.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

Kim, K.-H.

Kulkarni, A. P.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Lange, J.

Leblond, H.

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

Lee, P. C.

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

Li, C.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Liu, X.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Lu, L.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Luo, Z.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Ma, H.

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

Mahdi, M. A.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Meisel, D.

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

Midorikawa, K.

Ming, N.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Misoguti, L.

Miura, K.

Mohan, S.

Mukai, K.

Munechika, K.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Myint, T.

Nakashima, S.

Noone, K. M.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Oliveira, M. M.

Oliveira, T. R.

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

Otero, M. M. M.

Pacchioni, G.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Papazoglou, D. G.

Rademann, K.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Ramirez, E. V. G.

Reyna, A. S.

A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
[Crossref]

Rodrigues, J. J.

Ryasnyansky, A. I.

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Sadrolhosseini, A. R.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Sahraei, R.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Said, A. A.

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

Seifert, G.

Shameli, K.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[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(4), 760–769 (1990).
[Crossref]

Shen, Y. R.

Shimizu, M.

Shimotsuma, Y.

Shin, H.

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

Silva, D. M.

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

Singh, A.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

Sipe, J. E.

K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
[Crossref]

J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
[Crossref] [PubMed]

Skarka, V.

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

Sobral-Filho, R. G.

Stegeman, G.

Stepanov, A. L.

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Stößer, R.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Sugioka, K.

Tanaka, K.

Tatchev, D. M.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Tzortzakis, S.

Umran, F. A.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Usmanov, T.

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Van Stryland, E. W.

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(4), 760–769 (1990).
[Crossref]

Wang, G.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Wang, H.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Wang, J.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Wei, T.-H.

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

Weigel, W.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Xu, T.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Yu, K. W.

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
[Crossref]

Yu, L.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Zakaria, A.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Zamiri, R.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Zarbin, A. J. G.

Zhang, L.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Zhang, S.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Zhou, T.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Zilio, S. C.

Appl. Phys. B (2)

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

Appl. Phys. Lett. (2)

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (1)

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

J. Lumin. (1)

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

J. Nanomater. (1)

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

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

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

J. Opt. Soc. Am. B (5)

J. Phys. Chem. (1)

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Nano Lett. (2)

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Nanotechnology (1)

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Opt. Express (8)

S. Nakashima, K. Sugioka, K. Tanaka, M. Shimizu, Y. Shimotsuma, K. Miura, K. Midorikawa, and K. Mukai, “Plasmonically enhanced Faraday effect in metal and ferrite nanoparticles composite precipitated inside glass,” Opt. Express 20(27), 28191–28199 (2012).
[Crossref] [PubMed]

L. De Boni, E. C. Barbano, T. A. de Assumpção, L. Misoguti, L. R. P. Kassab, and S. C. Zilio, “Femtosecond third-order nonlinear spectra of lead-germanium oxide glasses containing silver nanoparticles,” Opt. Express 20(6), 6844–6850 (2012).
[Crossref] [PubMed]

K.-H. Kim, A. Husakou, and J. Herrmann, “Saturable absorption in composites doped with metal nanoparticles,” Opt. Express 18(21), 21918–21925 (2010).
[Crossref] [PubMed]

K.-H. Kim, A. Husakou, and J. Herrmann, “Slow light in dielectric composite materials of metal nanoparticles,” Opt. Express 20(23), 25790–25797 (2012).
[Crossref] [PubMed]

E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010).
[Crossref] [PubMed]

G. Stegeman, D. G. Papazoglou, R. Boyd, and S. Tzortzakis, “Nonlinear birefringence due to non-resonant, higher-order Kerr effect in isotropic media,” Opt. Express 19(7), 6387–6399 (2011).
[Crossref] [PubMed]

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[Crossref]

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Linear absorption spectra of the colloids with f = 4.0 × 10 5 and solvents: (a) sample A (cell thickness: 1 mm) and (b) sample B (cell thickness: 5 mm).
Fig. 2
Fig. 2 Normalized Z-scan profiles obtained for different NPs volume fractions f. For sample A (laser peak intensity: 5 G W / c m 2 ): (a) closed-aperture scheme; (b) open-aperture scheme. For sample B (laser peak intensity: 0.2 G W / c m 2 ): (c) closed-aperture and (d) open-aperture scheme.
Fig. 3
Fig. 3 Closed-aperture Z-scan profiles, for sample A, obtained for different laser peak intensities with two NPs volume fractions: (a) f = 1.6 × 10 5 and (b) f = 5.0 × 10 5 . The solid lines were obtained using Eq. (16).
Fig. 4
Fig. 4 Dependence of the effective third-, fifth- and seventh-order coefficients of sample B as a function of the volume fraction, f. Notice that the sample presents: septic refractive nonlinearity at f = 3.3 × 10 5 (a) and quintic refractive nonlinearity at f = 1.5 × 10 5 (b). Figures (a) and (b) were obtained with laser peak intensity of 10 8 W / c m 2 . When f = 1.2 × 10 5 and the laser peak intensity is 2.5 × 10 8 W / c m 2 , the NL absorption is due to Im χ e f f ( 7 ) (c).
Fig. 5
Fig. 5 Diffraction patterns of a Gaussian beam in a quintic refractive composite placed in the (a)–(f) focal plane and (g)–(j) around the focal plane position. Laser peak intensities: (a) I 1 = 70 G W / c m 2 ; (b) I 2 = 90 G W / c m 2 ; (c) I 3 = 100 G W / c m 2 . Experimental (black line) and theoretical (red line) intensity distribution versus the radial coordinates for (d) I 1 ; (e) I 2 ; (f) I 3 . Sample placed 0.3 mm (g) before and (h) after of the focal plane with laser peak intensity: I = 70 G W / c m 2 at the focus. Intensity distribution as function of the radial coordinates considering (i) R = 0.4 m m ; (j) R = + 0.4 m m . The red curves were obtained using the value of n 4 determined in the Z-scan experiments.
Fig. 6
Fig. 6 Pump beam profile after propagation through sample B, for f = 3.3 × 10 5 : (a) I 1 = 10 6 W / c m 2 ; (b) I 2 = 10 8 W / c m 2 . (c) Self-defocusing traces due to χ e f f ( 7 ) obtained from (a) and (b). The red and blue lines are theoretical results for the respective intensities.
Fig. 7
Fig. 7 Experimental probe beam profile due to χ e f f ( 7 ) for a distance of (a) x = 0 ; (b) x = w 0 and (c) x = 2.2 w 0 between the centers of the incident pump (white line) and probe (pink line) beams; (d)-(f) Normalized intensity distribution of the probe beam transverse profile without the pump beam (red dashed line) and with the pump beam (black line); (g)-(i) Theoretical probe beam profile calculated using Eqs. (18) and (19) with the NL coefficients determined in the Z-scan experiments for the relative distances of (a)-(c). Pump beam intensity: 10 8 W / c m 2 ; probe beam intensity: 10 7 W / c m 2 .

Equations (19)

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P = ε h L { ε h ε h L [ χ h + 3 β f 1 β f ] } E 0 = ε h L χ ( | E 0 | 2 ) E 0 ,
ε n p ( N L ) = ε h L [ 3 4 χ n p ( 3 ) | E n p | 2 + 5 8 χ n p ( 5 ) ( | E n p | 2 ) 2 + 35 64 χ n p ( 7 ) ( | E n p | 2 ) 3 ] ,
ε h ( N L ) = ε h L [ 3 4 χ h ( 3 ) | E 0 | 2 ] ,
χ ( y ) = χ ( 0 ) + ( [ χ ( y ) ] y ) y = 0 y + 1 2 ( 2 [ χ ( y ) ] y 2 ) y = 0 y 2 + 1 6 ( 3 [ χ ( y ) ] y 3 ) y = 0 y 3 ,
( [ χ ( y ) ] y ) y = 0 = 3 4 χ h ( 3 ) + 3 4 f L 2 | L | 2 χ n p ( 3 ) ,
( 2 [ χ ( y ) ] y 2 ) y = 0 = 5 4 f L 2 | L | 4 χ n p ( 5 ) 3 8 f L 3 | L | 4 ( χ n p ( 3 ) ) 2 ,
( 3 [ χ ( y ) ] y 3 ) y = 0 = 9 32 f L 4 | L | 6 ( χ n p ( 3 ) ) 3 15 8 f L 3 | L | 6 ( χ n p ( 3 ) χ n p ( 5 ) ) + 105 32 f L 2 | L | 6 χ n p ( 7 ) ,
| E n p | 2 ( | η | 2 ) z = 0 { 1 + 1 2 [ 2 ( | η | 2 ) z 2 ] z = 0 | E 0 | 2 + 1 4 [ 2 ( | η | 2 ) z 2 ] 2 z = 0 ( | E 0 | 2 ) 2 } | E 0 | 2 ,
| E n p | 2 | L | 2 { 1 1 2 | L | 2 Re [ L χ N P ( 3 ) ] | E 0 | 2 + 1 4 | L | 4 ( Re [ L χ N P ( 3 ) ] ) 2 | E 0 | 4 } | E 0 | 2 .
χ ( z 2 ) = χ ( 0 ) + ( [ χ ( z 2 ) ] [ z 2 ] ) z 2 = 0 z 2 + 1 2 ( 2 [ χ ( z 2 ) ] [ z 2 ] 2 ) z 2 = 0 ( z 2 ) 2 + 1 6 ( 3 [ χ ( z 2 ) ] [ z 2 ] 3 ) z 2 = 0 ( z 2 ) 3 ,
χ ( | E 0 | 2 ) = χ ( 0 ) + | L | 2 ( [ χ ( z 2 ) ] [ z 2 ] ) z 2 = 0 | E 0 | 2 + 1 2 | L | 4 ( 2 [ χ ( z 2 ) ] [ z 2 ] 2 [ χ ( z 2 ) ] [ z 2 ] Re [ L χ N P ( 3 ) ] ) z 2 = 0 | E 0 | 4 + 1 6 | L | 6 ( 3 [ χ ( z 2 ) ] [ z 2 ] 3 3 2 [ χ ( z 2 ) ] [ z 2 ] 2 Re [ L χ N P ( 3 ) ] + 3 2 [ χ ( z 2 ) ] [ z 2 ] ( Re [ L χ N P ( 3 ) ] ) 2 ) z 2 = 0 | E 0 | 6 .
χ e f f ( | E 0 | 2 ) = χ e f f ( 1 ) + 3 4 χ e f f ( 3 ) | E 0 | 2 + 5 8 χ e f f ( 5 ) | E 0 | 4 + 35 64 χ e f f ( 7 ) | E 0 | 6 .
χ e f f ( 3 ) = f L 2 | L | 2 χ n p ( 3 ) + χ h ( 3 ) ,
χ e f f ( 5 ) = f L 2 | L | 4 χ n p ( 5 ) 6 10 f L 3 | L | 4 ( χ n p ( 3 ) ) 2 3 10 f L | L | 6 | χ n p ( 3 ) | 2 ,
χ e f f ( 7 ) = f L 2 | L | 6 χ n p ( 7 ) + 12 35 f L 4 | L | 6 ( χ n p ( 3 ) ) 3 + 3 35 f | L | 8 [ 4 L 2 χ n p ( 3 ) + | L | 2 ( χ n p ( 3 ) ) * ] | χ n p ( 3 ) | 2 4 7 f L | L | 6 [ 2 L 2 χ n p ( 3 ) + | L | 2 ( χ n p ( 3 ) ) * ] χ n p ( 5 ) ,
T ( z , Δ Φ 0 ) 1 + N = 1 3 ( 4 N ) Δ Φ 0 ( 2 N + 1 ) z / z 0 [ ( z / z 0 ) 2 + ( 2 N + 1 ) 2 ] [ ( z / z 0 ) 2 + 1 ] N ,
I = I 0 | 0 J 0 ( k θ r ) exp [ r 2 w p 2 i ϕ ( r ) ] r d r | 2 ,
2 i k E 1 z + Δ E 1 = ω 2 c 2 [ 3 χ e f f ( 3 ) ( | E 1 | 2 + 2 | E 2 | 2 ) E 1 + 10 χ e f f ( 5 ) ( | E 1 | 4 + 6 | E 1 | 2 | E 2 | 2 + 3 | E 2 | 4 ) E 1 + 35 χ e f f ( 7 ) ( | E 1 | 6 + 18 | E 1 | 2 | E 2 | 4 + 12 | E 1 | 4 | E 2 | 2 + 4 | E 2 | 6 ) E 1 ] ,
2 i k E 2 z + Δ E 2 = ω 2 c 2 [ 3 χ e f f ( 3 ) ( 2 | E 1 | 2 + | E 2 | 2 ) E 2 + 10 χ e f f ( 5 ) ( 3 | E 1 | 4 + 6 | E 1 | 2 | E 2 | 2 + | E 2 | 4 ) E 2 + 35 χ e f f ( 7 ) ( 4 | E 1 | 6 + 12 | E 1 | 2 | E 2 | 4 + 18 | E 1 | 4 | E 2 | 2 + | E 2 | 6 ) E 2 ] ,

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