Abstract

Analytical expressions for the normalized transmittance of a thin material with simultaneous nonlocal nonlinear change in refraction and absorption are reported. Gaussian decomposition method was used to obtain the formulas that are adequate for any magnitude of the nonlinear changes. Particular cases of no locality are compared with the local case. Experimental results are reproduced (fitted) with the founded expressions.

© 2014 Optical Society of America

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Analytical solution of on-axis beam propagation for Z-scan technique

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References

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  1. M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14(17), 955–957 (1989).
    [Crossref] [PubMed]
  2. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
    [Crossref]
  3. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4(10), 331–333 (1979).
    [Crossref] [PubMed]
  4. S. Hughes and B. S. Wherrett, “Fast Fourier Transform techniques for efficient simulation of z-scan measurements,” J. Opt. Soc. Am. B 12(10), 1888–1893 (1995).
    [Crossref]
  5. R. E. Samad and N. D. Viera, “Analytical description of z-scan on-axis intensity based on the Huygens-Fresnel principle,” J. Opt. Soc. Am. B 15(11), 2742–2747 (1998).
    [Crossref]
  6. B. Yao, L. Ren, and X. Hou, “Z-scan theory based on a diffraction model,” J. Opt. Soc. Am. B 20(6), 1290–1294 (2003).
    [Crossref]
  7. G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition,” J. Opt. Soc. Am. B 20(4), 670–676 (2003).
    [Crossref]
  8. L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
    [Crossref]
  9. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
    [Crossref]
  10. S. Q. Chen, Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, F. Song, and C. P. Zhang, “Study on Z-scan characteristics for a large nonlinear phase shift,” J. Opt. Soc. Am. B 22(9), 1911–1916 (2005).
    [Crossref]
  11. F. Q. Li, X. F. Zhang, F. Yang, N. Zong, Q. J. Peng, D. F. Cui, and Z. Y. Xu, “Analytical solution of on-axis beam propagation for z-scan technique,” J. Opt. Soc. Am. B 26(11), 2125–2130 (2009).
    [Crossref]
  12. G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan analysis for high order nonlinearities through Gaussian decomposition,” Opt. Commun. 225(4-6), 253–268 (2003).
    [Crossref]
  13. B. Gu, J. Chen, Y. Fan, J. Ding, and H. T. Wang, “Study on Z-scan characteristics for a large nonlinear phase shift,” J. Opt. Soc. Am. B 22(9), 1911–1915 (2005).
    [Crossref]
  14. B. Gu, K. Lou, J. Chen, H. T. Wang, and W. Ji, “Determination of the nonlinear refractive index in multiphoton absorbers by Z-scan measurements,” J. Opt. Soc. Am. B 27(11), 2438–2442 (2010).
    [Crossref]
  15. E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
    [Crossref]
  16. A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
    [Crossref]
  17. J. A. Herman, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1(5), 729–736 (1984).
    [Crossref]
  18. E. V. Ramirez, M. L. Carrasco, M. M. Otero, S. C. Cerda, and M. D. Castillo, “Far field intensity distributions due to spatial self phase modulation of a Gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18(21), 22067–22079 (2010).
    [Crossref] [PubMed]
  19. I. Mamedbeili and H. Nasibov, “Large third order optical nonlinearities in brilliant green solutions induced by CW He–Ne laser,” Laser Phys. 19(10), 2002–2007 (2009).
    [Crossref]

2014 (1)

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

2011 (1)

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[Crossref]

2010 (2)

2009 (3)

F. Q. Li, X. F. Zhang, F. Yang, N. Zong, Q. J. Peng, D. F. Cui, and Z. Y. Xu, “Analytical solution of on-axis beam propagation for z-scan technique,” J. Opt. Soc. Am. B 26(11), 2125–2130 (2009).
[Crossref]

L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

I. Mamedbeili and H. Nasibov, “Large third order optical nonlinearities in brilliant green solutions induced by CW He–Ne laser,” Laser Phys. 19(10), 2002–2007 (2009).
[Crossref]

2005 (2)

2003 (3)

1998 (1)

1997 (1)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
[Crossref]

1995 (1)

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]

1989 (1)

1984 (1)

1979 (1)

Almási, G.

L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

Arroyo Carrasco, M. L.

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[Crossref]

Balbuena Ortega, A.

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

Carrasco, M. L.

Castillo, M. D.

Cerda, S. C.

Chapple, P. B.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
[Crossref]

Chavez-Cerda, S.

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[Crossref]

Chen, J.

Chen, S. Q.

Cui, D. F.

Delgado Macuil, R.

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

Ding, J.

Fakis, M.

G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition,” J. Opt. Soc. Am. B 20(4), 670–676 (2003).
[Crossref]

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan analysis for high order nonlinearities through Gaussian decomposition,” Opt. Commun. 225(4-6), 253–268 (2003).
[Crossref]

Fan, Y.

Fülöp, J. A.

L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

Garcia Ramirez, E. V.

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[Crossref]

Gayou, V. L.

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

Giannetas, V.

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan analysis for high order nonlinearities through Gaussian decomposition,” Opt. Commun. 225(4-6), 253–268 (2003).
[Crossref]

G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition,” J. Opt. Soc. Am. B 20(4), 670–676 (2003).
[Crossref]

Gu, B.

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]

Hebling, J.

L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

Herman, J. A.

Hermann, J. A.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
[Crossref]

Hou, X.

Hughes, S.

Iturbe Castillo, M. D.

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[Crossref]

Ji, W.

Li, F. Q.

Liu, Z. B.

Lou, K.

Mamedbeili, I.

I. Mamedbeili and H. Nasibov, “Large third order optical nonlinearities in brilliant green solutions induced by CW He–Ne laser,” Laser Phys. 19(10), 2002–2007 (2009).
[Crossref]

Martínez Gutiérrez, H.

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

Mcduff, R. G.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
[Crossref]

Mckay, T. J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
[Crossref]

Mendez Otero, M. M.

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[Crossref]

Méndez Otero, M. M.

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

Miller, D. A. B.

Nasibov, H.

I. Mamedbeili and H. Nasibov, “Large third order optical nonlinearities in brilliant green solutions induced by CW He–Ne laser,” Laser Phys. 19(10), 2002–2007 (2009).
[Crossref]

Otero, M. M.

Pálfalvi, L.

L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

Peng, Q. J.

Persephonis, P.

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan analysis for high order nonlinearities through Gaussian decomposition,” Opt. Commun. 225(4-6), 253–268 (2003).
[Crossref]

G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition,” J. Opt. Soc. Am. B 20(4), 670–676 (2003).
[Crossref]

Polyzos, I.

G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition,” J. Opt. Soc. Am. B 20(4), 670–676 (2003).
[Crossref]

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan analysis for high order nonlinearities through Gaussian decomposition,” Opt. Commun. 225(4-6), 253–268 (2003).
[Crossref]

Ramirez, E. V.

Ren, L.

Reynoso Lara, E.

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[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]

M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14(17), 955–957 (1989).
[Crossref] [PubMed]

Samad, R. E.

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]

M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14(17), 955–957 (1989).
[Crossref] [PubMed]

Smith, S. D.

Song, F.

Staromlynska, J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
[Crossref]

Tian, J. G.

Tóth, B. C.

L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[Crossref]

Tsibouri, M.

Tsigaridas, G.

G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition,” J. Opt. Soc. Am. B 20(4), 670–676 (2003).
[Crossref]

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan analysis for high order nonlinearities through Gaussian decomposition,” Opt. Commun. 225(4-6), 253–268 (2003).
[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]

M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14(17), 955–957 (1989).
[Crossref] [PubMed]

Viera, N. D.

Wang, H. T.

Weaire, D.

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]

Wherrett, B. S.

Xu, Z. Y.

Yang, F.

Yao, B.

Zang, W. P.

Zhang, C. P.

Zhang, X. F.

Zhou, W. Y.

Zong, N.

Appl. Phys. B (1)

L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, “A general Z-scan theory,” Appl. Phys. B 97(3), 679–685 (2009).
[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]

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

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-beam z-scan: measurement techniques and analysis,” Int. J. Nonlinear Opt. Phys. 6(03), 251–293 (1997).
[Crossref]

J. Mod. Opt. (1)

A. Balbuena Ortega, M. L. Arroyo Carrasco, M. M. Méndez Otero, V. L. Gayou, R. Delgado Macuil, H. Martínez Gutiérrez, and M. D. Iturbe Castillo, “Nonlocal nonlinear refractive index of gold nanoparticles synthesized by ascorbic acid reduction: comparison of fitting models,” J. Mod. Opt. 61(sup1), S68–S72 (2014).
[Crossref]

J. Opt. (1)

E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, E. Reynoso Lara, S. Chavez-Cerda, and M. D. Iturbe Castillo, “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011).
[Crossref]

J. Opt. Soc. Am. B (9)

J. A. Herman, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1(5), 729–736 (1984).
[Crossref]

S. Hughes and B. S. Wherrett, “Fast Fourier Transform techniques for efficient simulation of z-scan measurements,” J. Opt. Soc. Am. B 12(10), 1888–1893 (1995).
[Crossref]

R. E. Samad and N. D. Viera, “Analytical description of z-scan on-axis intensity based on the Huygens-Fresnel principle,” J. Opt. Soc. Am. B 15(11), 2742–2747 (1998).
[Crossref]

G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giannetas, “Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition,” J. Opt. Soc. Am. B 20(4), 670–676 (2003).
[Crossref]

B. Yao, L. Ren, and X. Hou, “Z-scan theory based on a diffraction model,” J. Opt. Soc. Am. B 20(6), 1290–1294 (2003).
[Crossref]

S. Q. Chen, Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, F. Song, and C. P. Zhang, “Study on Z-scan characteristics for a large nonlinear phase shift,” J. Opt. Soc. Am. B 22(9), 1911–1916 (2005).
[Crossref]

B. Gu, J. Chen, Y. Fan, J. Ding, and H. T. Wang, “Study on Z-scan characteristics for a large nonlinear phase shift,” J. Opt. Soc. Am. B 22(9), 1911–1915 (2005).
[Crossref]

F. Q. Li, X. F. Zhang, F. Yang, N. Zong, Q. J. Peng, D. F. Cui, and Z. Y. Xu, “Analytical solution of on-axis beam propagation for z-scan technique,” J. Opt. Soc. Am. B 26(11), 2125–2130 (2009).
[Crossref]

B. Gu, K. Lou, J. Chen, H. T. Wang, and W. Ji, “Determination of the nonlinear refractive index in multiphoton absorbers by Z-scan measurements,” J. Opt. Soc. Am. B 27(11), 2438–2442 (2010).
[Crossref]

Laser Phys. (1)

I. Mamedbeili and H. Nasibov, “Large third order optical nonlinearities in brilliant green solutions induced by CW He–Ne laser,” Laser Phys. 19(10), 2002–2007 (2009).
[Crossref]

Opt. Commun. (1)

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan analysis for high order nonlinearities through Gaussian decomposition,” Opt. Commun. 225(4-6), 253–268 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

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Figures (8)

Fig. 1
Fig. 1 Z-scan curves for closed (a) and open (b) -aperture with ΔΦ0 = −1 rad. and ΔΨ0 of: 0.3 (red line), 0.0 (black line) and −0.3 (blue line).
Fig. 2
Fig. 2 Z-scan curves for closed (a) and open (b) – aperture with ΔΦ0 = −1 rad., ΔΨ0 = −0.3 and m of 1(red line), 2(black line) and 4(blue line).
Fig. 3
Fig. 3 Δzp-v (a) and ΔTp-v (b) as a function of the m obtained from evaluation of Eq. (16) with N = 20 and ΔΦ0 = −0.5π rad. (dotted), ΔΦ0 = -π rad. (dashed line) and ΔΦ0 = −4π rad. (solid line).
Fig. 4
Fig. 4 Closed-aperture Z-scan curves obtained from Eq. (17) with ΔΦ0 of −0.5 rad. and m parameter of: 1 (red line), 2 (black line) and 4 (blue line).
Fig. 5
Fig. 5 Results from Eq. (17) for Δzp-v (a) and ΔTp-v (b) as functions of the m parameter for ΔΦ0 = −0.5 rad. (dashed line) and ΔΦ0 = −1 rad. (solid line).
Fig. 6
Fig. 6 Closed-aperture Z-scan curves for an on-axis nonlinear phase shift of ΔΦ0 = −0.1 rad., ΔΨ0 = −0.02 and m values of: 1 (red line), 2 (black line) and 4(blue line).
Fig. 7
Fig. 7 Open-aperture Z-scan curves with ∆Ψ0 = 0.05 and m values of: 1 (red line), 2 (black line) and 4 (blue line).
Fig. 8
Fig. 8 Experimental (symbol) and numerical (lines), a) closed- and b) open-aperture Z-scan curves for incident power of: 1 mW(red), 3 mW(magenta) and 5 mW(blue).

Equations (24)

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E ( r , z ) = A 0 ω 0 ω ( z ) exp [ r 2 ω ( z ) 2 ] exp [ i k z i k r 2 2 R ( z ) + i ε ( z ) ]
n ( I ) = n 0 + γ I ,
α ( I ) = α 0 + β I ,
E o u t = E ( r , z ) exp ( α 0 L / 2 ) ( 1 + q ) ( i k γ / β 1 / 2 ) ,
I o u t ( z , r ) = I ( z , r ) exp ( α 0 L ) 1 + q ( z , r )
Δ ϕ = ( k γ / β ) ln [ 1 + q ( z , r ) ] .
I ( r , z ) = I 0 G l o c ,
G l o c = exp[- 2 r 2 / w 2 ( z ) ] ( 1 + ( z / z 0 ) 2 ) .
Δ ϕ m ( r ) = Δ Φ 0 G l o c m / 2 ,
q m ( r ) = Δ Ψ 0 G l o c m / 2 ,
E o u t = E ( r , z ) exp ( α 0 L / 2 ) [ 1 + Δ Ψ 0 G l o c m / 2 ] ( i ( Δ Φ 0 / Δ Ψ 0 ) 1 / 2 ) .
E o u t = E ( r , z ) exp ( α 0 L / 2 ) n = 0 [ ( i Δ ϕ 0 ( z ) ) n n ! × n ' = 0 n ( 1 i ( 2 n 1 ) Δ Ψ 0 Δ Φ 0 ) ] exp [ m n r 2 w 2 ( z ) ] ,
E a = E ( r , z ) n = 0 [ i Δ ϕ 0 ( z )   ] n n ! n ' = 0 n ( 1 i ( 2 n 1 ) Δ Ψ 0 Δ Φ 0 ) w n 0 w n exp ( r 2 w n 2 i k r 2 2 R n + i θ n ) ,
E a = E ( r , z ) n = 0 [ i Δ ϕ 0 ( z )   ] n n ! w n 0 w n exp ( r 2 w n 2 i k r 2 2 R n + i θ n ) .
w n 0 2 = w 2 ( z ) m n + 1
d n = k w 2 n 0 ( z ) 2
w n 2 = w n 0 2 [ g 2 + ( d d n ) 2 ]
R n = d [ 1 g g 2 + d 2 / d n 2 ] 1
θ n = tan 1 ( d / d n g ) .
T m ( z , Δ ϕ 0 ) = | E ( r = 0 , z , Δ ϕ 0 ) | 2 | E ( r = 0 , z , Δ ϕ 0 = 0 ) | 2 .
T m ( z , Δ Φ 0 ) = | n = 0 N 1 n ! [ Δ Φ 0 ( z ) ( 1 + x 2 ) m / 2 ] n i n ( x + i ) x + i ( m n + 1 ) | 2 ,
T m ( z , Δ Φ 0 ) = 1 + 2 m Δ Φ 0 x [ x 2 + ( m + 1 ) 2 ] ( x 2 + 1 ) m 2 + m 2 (3x 2 - (2m+1))Δ Φ 0 2 [ x 2 + ( m + 1 ) 2 ] [ x 2 + ( 2 m + 1 ) 2 ] ( x 2 + 1 ) m .
T ( z , Δ Φ 0 , Δ Ψ 0 ) = 1 + 2 m Δ Φ 0 + Δ Ψ 0 ( x 2 + ( m + 1 ) ) ( x 2 + ( m + 1 ) 2 ) ( x 2 + 1 ) .
T m ( z , Δ Ψ 0 ) = ln [ 1 + q 0 ( z ) ] q 0 ( z ) ,

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