Abstract

We present a detailed theoretical investigation on the Gaussian beam Z scan for arbitrary aperture and arbitrary nonlinear refraction phase shifts, based on the Gaussian decomposition method, including cases when the medium exhibits the single (2n+1)th-order nonlinear refraction effect and the simultaneous third- and fifth-order nonlinear refraction effects. We find the optimum sum upper limit, which is of great importance to fit the Z-scan traces and extract the nonlinear refraction coefficients related to the third- and fifth-order effects. This method has not only a high accuracy but is also time saving. We also discuss the influence of two-photon absorption on the Z-scan traces when materials possess the simultaneous third- and fifth-order nonlinear refraction effects associated with the two-photon absorption using the fast Fourier transform.

© 2005 Optical Society of America

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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, 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, 760-769 (1990).
    [CrossRef]
  3. L. C. Oliveira, T. Catunda, and S. C. Zilio, "Saturation effects in Z-scan measurements," Jpn. J. Appl. Phys., Part 1 35, 2649-2652 (1996).
    [CrossRef]
  4. B. L. Yao, L. Y. Ren, and X. Hou, "Z-scan theory based on a diffraction model," J. Opt. Soc. Am. B 20, 1290-1294 (2003).
    [CrossRef]
  5. S. Hughes, J. M. Burzler, G. Spruce, and B. S. Wherrett, "Fast Fourier transform techniques for efficient simulation of Z-scan measurements," J. Opt. Soc. Am. B 12, 1888-1893 (1995).
    [CrossRef]
  6. W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
    [CrossRef]
  7. G. Tsigaridas, M. Fakis, I. Polyzos, M. Tsibouri, P. Persephonis, and V. Giznnetad, "Z-scan analysis for near-Gaussian beams through Hermite-Gaussian decomposition," J. Opt. Soc. Am. B 20, 670-676 (2003).
    [CrossRef]
  8. 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, 331-333 (1979).
    [CrossRef] [PubMed]
  9. W. P. Zang, J. G. Tian, Z. B. Liu, W. Y. Zhou, F. Song, C. P. Zhang, and J. J. Xu, "Study on Z-scan characteristics of cascaded nonlinear media," J. Opt. Soc. Am. B 21, 349-356 (2004).
    [CrossRef]
  10. G. Tsigaridas, M. Fakis, I. Plyzos, P. Persephonis, and V. Giannetas, "Z-scan analysis for high order nonlinearities through Gaussian decomposition," Opt. Commun. 225, 253-268 (2003).
    [CrossRef]
  11. C. H. Kwak, Y. L. Lee, and S. G. Kim, "Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As2S3 thin film," J. Opt. Soc. Am. B 16, 600-604 (1999).
    [CrossRef]
  12. C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
    [CrossRef]
  13. B. Gu, X. C. Peng, T. Jia, J. P. Ding, J. L. He, and H. T. Wang, "Determinations of third- and fifth-order nonlinearities by the use of the top-hat-beam Z scan: theory and experiment," J. Opt. Soc. Am. B 22, 446-452 (2005).
    [CrossRef]
  14. Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
    [CrossRef]
  15. P. B. Chapple, J. Skaromlynska, J. A. Hermann, T. J. Mckay, and R G. Mcduff, "Single-beam Z-scan: measurement techniques and analysis," Int. J. Nonlinear Opt. Phys. 6, 251-293 (1997).
    [CrossRef]
  16. J. A. Hermann, T. Mckay, and R. G. Mcuff, "Z-scan with arbitrary aperture transmittance: the strong nonlinear regime," Opt. Commun. 154, 225-233 (1998).
    [CrossRef]
  17. K. S. Bindra and A. K. Kar, "Role of femtosecond pulses in distinguishing third- and fifth-order nonlinearity for semiconductor-doped glasses," Appl. Phys. Lett. 79, 3761-3763 (2001).
    [CrossRef]
  18. R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
    [CrossRef]

2005

2004

W. P. Zang, J. G. Tian, Z. B. Liu, W. Y. Zhou, F. Song, C. P. Zhang, and J. J. Xu, "Study on Z-scan characteristics of cascaded nonlinear media," J. Opt. Soc. Am. B 21, 349-356 (2004).
[CrossRef]

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
[CrossRef]

2003

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

B. L. Yao, L. Y. Ren, and X. Hou, "Z-scan theory based on a diffraction model," J. Opt. Soc. Am. B 20, 1290-1294 (2003).
[CrossRef]

G. Tsigaridas, M. Fakis, I. Plyzos, P. Persephonis, and V. Giannetas, "Z-scan analysis for high order nonlinearities through Gaussian decomposition," Opt. Commun. 225, 253-268 (2003).
[CrossRef]

Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
[CrossRef]

2001

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

K. S. Bindra and A. K. Kar, "Role of femtosecond pulses in distinguishing third- and fifth-order nonlinearity for semiconductor-doped glasses," Appl. Phys. Lett. 79, 3761-3763 (2001).
[CrossRef]

1999

1998

J. A. Hermann, T. Mckay, and R. G. Mcuff, "Z-scan with arbitrary aperture transmittance: the strong nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

1997

P. B. Chapple, J. Skaromlynska, J. A. Hermann, T. J. Mckay, and R G. Mcduff, "Single-beam Z-scan: measurement techniques and analysis," Int. J. Nonlinear Opt. Phys. 6, 251-293 (1997).
[CrossRef]

1996

L. C. Oliveira, T. Catunda, and S. C. Zilio, "Saturation effects in Z-scan measurements," Jpn. J. Appl. Phys., Part 1 35, 2649-2652 (1996).
[CrossRef]

1995

1993

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

1990

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

1989

1979

Baba, M.

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
[CrossRef]

Bindra, K. S.

K. S. Bindra and A. K. Kar, "Role of femtosecond pulses in distinguishing third- and fifth-order nonlinearity for semiconductor-doped glasses," Appl. Phys. Lett. 79, 3761-3763 (2001).
[CrossRef]

Burzler, J. M.

Catunda, T.

L. C. Oliveira, T. Catunda, and S. C. Zilio, "Saturation effects in Z-scan measurements," Jpn. J. Appl. Phys., Part 1 35, 2649-2652 (1996).
[CrossRef]

Chapple, P. B.

P. B. Chapple, J. Skaromlynska, J. A. Hermann, T. J. Mckay, and R G. Mcduff, "Single-beam Z-scan: measurement techniques and analysis," Int. J. Nonlinear Opt. Phys. 6, 251-293 (1997).
[CrossRef]

Ding, J. P.

Fakis, M.

G. Tsigaridas, M. Fakis, I. Plyzos, P. Persephonis, and V. Giannetas, "Z-scan analysis for high order nonlinearities through Gaussian decomposition," Opt. Commun. 225, 253-268 (2003).
[CrossRef]

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

Ganeev, R. A.

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
[CrossRef]

Giannetas, V.

G. Tsigaridas, M. Fakis, I. Plyzos, P. Persephonis, and V. Giannetas, "Z-scan analysis for high order nonlinearities through Gaussian decomposition," Opt. Commun. 225, 253-268 (2003).
[CrossRef]

Giznnetad, V.

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, 760-769 (1990).
[CrossRef]

He, J. L.

Hermann, J. A.

J. A. Hermann, T. Mckay, and R. G. Mcuff, "Z-scan with arbitrary aperture transmittance: the strong nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

P. B. Chapple, J. Skaromlynska, J. A. Hermann, T. J. Mckay, and R G. Mcduff, "Single-beam Z-scan: measurement techniques and analysis," Int. J. Nonlinear Opt. Phys. 6, 251-293 (1997).
[CrossRef]

Hou, X.

Hughes, S.

Jia, T.

Kar, A. K.

K. S. Bindra and A. K. Kar, "Role of femtosecond pulses in distinguishing third- and fifth-order nonlinearity for semiconductor-doped glasses," Appl. Phys. Lett. 79, 3761-3763 (2001).
[CrossRef]

Kim, S. G.

Kuroda, H.

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
[CrossRef]

Kwak, C. H.

Lee, Y. L.

Li, D. H.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Li, Y. J.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Liu, Z. B.

W. P. Zang, J. G. Tian, Z. B. Liu, W. Y. Zhou, F. Song, C. P. Zhang, and J. J. Xu, "Study on Z-scan characteristics of cascaded nonlinear media," J. Opt. Soc. Am. B 21, 349-356 (2004).
[CrossRef]

Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
[CrossRef]

Lu, Z. Z.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Mcduff, R G.

P. B. Chapple, J. Skaromlynska, J. A. Hermann, T. J. Mckay, and R G. Mcduff, "Single-beam Z-scan: measurement techniques and analysis," Int. J. Nonlinear Opt. Phys. 6, 251-293 (1997).
[CrossRef]

Mckay, T.

J. A. Hermann, T. Mckay, and R. G. Mcuff, "Z-scan with arbitrary aperture transmittance: the strong nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

Mckay, T. J.

P. B. Chapple, J. Skaromlynska, J. A. Hermann, T. J. Mckay, and R G. Mcduff, "Single-beam Z-scan: measurement techniques and analysis," Int. J. Nonlinear Opt. Phys. 6, 251-293 (1997).
[CrossRef]

Mcuff, R. G.

J. A. Hermann, T. Mckay, and R. G. Mcuff, "Z-scan with arbitrary aperture transmittance: the strong nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

Miller, D. A.

Morita, M.

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
[CrossRef]

Nie, Y. X.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Oliveira, L. C.

L. C. Oliveira, T. Catunda, and S. C. Zilio, "Saturation effects in Z-scan measurements," Jpn. J. Appl. Phys., Part 1 35, 2649-2652 (1996).
[CrossRef]

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

Peng, X. C.

Persephonis, P.

G. Tsigaridas, M. Fakis, I. Plyzos, P. Persephonis, and V. Giannetas, "Z-scan analysis for high order nonlinearities through Gaussian decomposition," Opt. Commun. 225, 253-268 (2003).
[CrossRef]

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

Plyzos, I.

G. Tsigaridas, M. Fakis, I. Plyzos, P. Persephonis, and V. Giannetas, "Z-scan analysis for high order nonlinearities through Gaussian decomposition," Opt. Commun. 225, 253-268 (2003).
[CrossRef]

Polyzos, I.

Ren, L. Y.

Ryasnyansky, A. I.

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
[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, 760-769 (1990).
[CrossRef]

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

Sheik-Bahae, M.

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

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

Skaromlynska, J.

P. B. Chapple, J. Skaromlynska, J. A. Hermann, T. J. Mckay, and R G. Mcduff, "Single-beam Z-scan: measurement techniques and analysis," Int. J. Nonlinear Opt. Phys. 6, 251-293 (1997).
[CrossRef]

Smith, S. D.

Song, F.

Spruce, G.

Suzuki, M.

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (2004).
[CrossRef]

Tian, J. G.

W. P. Zang, J. G. Tian, Z. B. Liu, W. Y. Zhou, F. Song, C. P. Zhang, and J. J. Xu, "Study on Z-scan characteristics of cascaded nonlinear media," J. Opt. Soc. Am. B 21, 349-356 (2004).
[CrossRef]

Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
[CrossRef]

Tsibouri, M.

Tsigaridas, G.

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

G. Tsigaridas, M. Fakis, I. Plyzos, P. Persephonis, and V. Giannetas, "Z-scan analysis for high order nonlinearities through Gaussian decomposition," Opt. Commun. 225, 253-268 (2003).
[CrossRef]

Turu, M.

R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Fifth-order optical nonlinearity of pseudoisocyanine solution at 529 nm," J. Opt. A, Pure Appl. Opt. 6, 282-287 (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, 760-769 (1990).
[CrossRef]

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

Wang, D. Y.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

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, 760-769 (1990).
[CrossRef]

Wherrett, B. S.

Xu, J. J.

Xu, W.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Yao, B. L.

Zang, W. P.

W. P. Zang, J. G. Tian, Z. B. Liu, W. Y. Zhou, F. Song, C. P. Zhang, and J. J. Xu, "Study on Z-scan characteristics of cascaded nonlinear media," J. Opt. Soc. Am. B 21, 349-356 (2004).
[CrossRef]

Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
[CrossRef]

Zhan, C. L.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Zhang, C. P.

W. P. Zang, J. G. Tian, Z. B. Liu, W. Y. Zhou, F. Song, C. P. Zhang, and J. J. Xu, "Study on Z-scan characteristics of cascaded nonlinear media," J. Opt. Soc. Am. B 21, 349-356 (2004).
[CrossRef]

Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
[CrossRef]

Zhang, D. Q.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Zhang, G. Y.

Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
[CrossRef]

Zhao, L. Z.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Zhao, W.

W. Zhao and P. Palffy-Muhoray, "Z-scan technique using top-hat beams," Appl. Phys. Lett. 63, 1613-1615 (1993).
[CrossRef]

Zhou, W. Y.

W. P. Zang, J. G. Tian, Z. B. Liu, W. Y. Zhou, F. Song, C. P. Zhang, and J. J. Xu, "Study on Z-scan characteristics of cascaded nonlinear media," J. Opt. Soc. Am. B 21, 349-356 (2004).
[CrossRef]

Z. B. Liu, W. P. Zang, J. G. Tian, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, "Analysis of Z-scan of thick media with high-order nonlinearity by variational approach," Opt. Commun. 219, 411-419 (2003).
[CrossRef]

Zhu, D. B.

C. L. Zhan, D. H. Li, D. Y. Wang, D. Q. Zhang, Y. J. Li, W. Xu, Z. Z. Lu, L. Z. Zhao, Y. X. Nie, and D. B. Zhu, "The high fifth-order nonlinearity in a new stilbazolium derivative: trans-1-[p-(p-dimethylaminobenzyl-azo)-benzyl]-2-(N-methyl-4-pyridinium)-ethene iodide," Chem. Phys. Lett. 347, 410-414 (2001).
[CrossRef]

Zilio, S. C.

L. C. Oliveira, T. Catunda, and S. C. Zilio, "Saturation effects in Z-scan measurements," Jpn. J. Appl. Phys., Part 1 35, 2649-2652 (1996).
[CrossRef]

Appl. Phys. Lett.

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Figures (7)

Fig. 1
Fig. 1

Dependences of the Z-scan trace profiles on the sum upper limit M for a nonlinear phase shift of 2 π . The Z-scan traces simulated by the FFT method are also present. (a) The pure third-order case and (b) the pure fifth-order case.

Fig. 2
Fig. 2

Dependence of the optimum sum upper limit M opt on Φ 0 n .

Fig. 3
Fig. 3

Dependence of the optimum sum upper limit N opt on both Φ 01 and Φ 02 .

Fig. 4
Fig. 4

Comparison between the GD method with the optimum sum upper limit and the FFT technique for the case of the small-aperture Z scan when Φ 01 = π . Φ 02 = + 2 π (open circles) and Φ 02 = 2 π (open squares). The curves correspond to the respective Z-scan traces by the FFT technique.

Fig. 5
Fig. 5

Small-aperture Z-scan traces with the material coexisting with the third- and fifth-order nonlinear refraction for Φ 01 = 0.4 π and for different Φ 02 . (a) Φ 02 > 0 and (b) the case of Φ 02 < 0 .

Fig. 6
Fig. 6

Influence of the linear transmittance s of the aperture on the Z-scan traces, where Φ 01 = 0.4 π . (a) Φ 02 = 0.8 π and (b) Φ 02 = 0.8 π .

Fig. 7
Fig. 7

Influence of two-photon absorption on the Z-scan traces for the simultaneous third- and fifth-order nonlinear refraction when s = 0.01 . (a) Φ 01 = 0.2 π and Φ 02 = 0.4 π and (b) Φ 01 = 0.2 π and Φ 02 = 0.4 π .

Equations (51)

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E ( r , z ) = E 0 ω 0 ω ( z ) exp [ r 2 ω 2 ( z ) i k r 2 2 R ( z ) ] exp [ i ϕ ( z ) ] ,
I ( r , z ) = I 0 1 1 + ( z z 0 ) 2 exp [ 2 r 2 ω 2 ( z ) ] ,
d Δ ϕ ( r , z ) d z = k [ γ 1 I ( r , z ) + γ 2 I 2 ( r , z ) ] ,
d I ( r , z ) d z = α I ( r , z ) β I 2 ( r , z ) ,
E e ( r , z ) = E ( r , z ) exp ( α L 2 ) [ 1 + ψ ( r , z ) ] i k γ 1 β i k α γ 2 β 2 1 2 exp { i k α γ 2 β 2 [ 1 + ψ ( r , z ) α L eff ( 1 ) ] ψ ( r , z ) 1 + ψ ( r , z ) } ,
E e ( r , z ) = E ( r , z ) exp ( α L 2 ) exp [ i k γ 1 I ( r , z ) L eff ( 1 ) + i k γ 2 I 2 ( r , z ) L eff ( 2 ) ] .
E e ( r , z ) = E ( r , z ) exp ( α L 2 ) exp [ i Δ ϕ 1 ( r , z ) + i Δ ϕ 2 ( r , z ) ] ,
Δ ϕ n ( r , z ) = k γ n I n ( r , z ) L eff ( n ) = ϕ n ( z ) exp [ 2 n r 2 ω 2 ( z ) ] ,
ϕ n ( z ) = Φ 0 n [ 1 + ( z z 0 ) 2 ] n .
E e ( r , z ) = E ( r , z ) exp ( α L 2 ) exp [ i Δ ϕ n ( r , z ) ] .
E e ( r , z ) = E ( r , 0 ) exp ( α L 2 ) m = 0 [ i ϕ n ( z ) ] m m ! exp [ ( 2 m n + 1 ) r 2 ω 2 ( z ) ] exp [ i k r 2 2 R ( z ) ] .
E a ( r ) = E ( 0 , z ) exp ( α L 2 ) m = 0 [ i ϕ n ( z ) ] m m ! ω 0 m ω m exp ( r 2 ω m 2 i k r 2 2 R m + i θ m ) .
g = 1 + d R ( z ) ,
ω 0 m 2 = ω 2 ( z ) 2 m n + 1 ,
d m = 1 2 k ω 0 m 2 ,
ω m 2 = ω 0 m 2 ( g 2 + d 2 d m 2 ) ,
R m = d ( 1 g g 2 + d 2 d m 2 ) 1 ,
θ m = tan 1 ( d d m g ) .
T n ( z , s , Φ 0 n ) = m , m = 0 i ( m m ) m ! m ! Φ 0 n m + m P m m ( z ) S m m ( z , s ) ,
P m m ( z ) = 1 [ 1 + ( z z 0 ) 2 ] m n + m n g 2 + d 2 d 0 2 ( g + i d d m ) ( g i d d m ) ,
S m m ( z , s ) = 1 exp [ A m m ( z ) ln ( 1 s ) ] A m m ( z ) s ,
A m m ( z ) = ( m n + m n + 1 ) [ 1 + ( z z 0 ) 2 ] [ z z 0 + i ( 2 m n + 1 ) ] [ z z 0 ( 2 m n + 1 ) ] .
E e ( r , z ) = E ( r , 0 ) exp ( α L 2 ) m = 0 f m ν exp [ ( 2 B m ν + 1 ) r 2 2 R ( z ) ] .
f m ν = ν = 0 m ( i Φ 01 ) ν ( i Φ 02 ) m ν ν ! ( m ν ) ! 1 [ 1 + ( z z 0 ) 2 ] B m ν ,
B m ν = 2 m ν .
E a ( r ) = E ( 0 , z ) exp ( α L 2 ) m = 0 f m ν ω 0 m ν ω m ν exp ( r 2 ω m ν 2 i k r 2 2 R m ν + i θ m ν ) ,
ω 0 m ν 2 = ω 2 ( z ) 2 B m ν + 1 ,
d m ν = 1 2 k ω 0 m ν 2 ,
ω m ν 2 = ω 0 m ν 2 ( g 2 + d 2 d m ν 2 ) ,
R m ν = d ( 1 g g 2 + d 2 d m ν 2 ) 1 ,
θ m ν = tan 1 ( d d m ν g ) .
T ( z , s , Φ 01 , Φ 02 ) = m , m = 0 f m ν f m ν * ( g 2 + d 2 d 00 2 ) ( g + i d d m ν ) ( g i d d m ν ) S m m ( z , s ) ,
S m m ( z , s ) = 1 exp [ A m m ( z ) ln ( 1 s ) ] A m m ( z ) s ,
A m m ( z ) = ( B m ν + B m ν + 1 ) [ 1 + ( z z 0 ) 2 ] [ z z 0 + i ( 2 B m ν + 1 ) ] [ z z 0 i ( 2 B m ν + 1 ) ] .
T n ( x , Φ 0 n ) = m , m = 0 i ( m m ) m ! m ! Φ 0 n m + m x 2 + 1 ( x 2 + 1 ) m n + m n [ x + i ( 2 m n + 1 ) ] [ x i ( 2 m n + 1 ) ] .
T n ( x , Φ 0 n ) = 1 + 4 n x Φ 0 n ( x 2 + 1 ) n [ x 2 + ( 2 n + 1 ) 2 ] .
T n ( x , Φ 0 n ) = 1 + 4 n x Φ 0 n ( x 2 + 1 ) n [ x 2 + ( 2 n + 1 ) 2 ] + 4 n 2 [ 3 x 2 ( 4 n + 1 ) ] Φ 0 n 2 ( x 2 + 1 ) 2 n [ x 2 + ( 2 n + 1 ) 2 ] [ x 2 + ( 4 n + 1 ) 2 ] .
T 1 ( x , Φ 01 ) = 1 + 4 x Φ 01 ( x 2 + 1 ) ( x 2 + 9 ) + 4 ( 3 x 2 5 ) Φ 01 2 ( x 2 + 1 ) 2 ( x 2 + 9 ) ( x 2 + 25 ) .
T 2 ( x , Φ 02 ) = 1 + 8 x Φ 02 ( x 2 + 1 ) 2 ( x 2 + 25 ) + 48 ( x 2 3 ) Φ 02 2 ( x 2 + 1 ) 4 ( x 2 + 25 ) ( x 2 + 81 ) .
T n ( x , Φ 0 n ) = lim M m = 0 M 1 m ! [ Φ 0 n ( 1 + x 2 ) n ] m i m ( x + i ) x + i ( 2 m n + 1 ) 2 .
Φ 0 n M opt M opt ! = 0.5 .
T n ( x , s , Φ 0 n ) = 1 s { 1 m , m = 0 Φ 0 n m + m m ! m ! ( m n + m n + 1 ) ( x 2 + 1 ) m n + m n ( 1 s ) λ m m cos ψ m m } ,
λ m m = ( m n + m n + 1 ) ( x 2 + 1 ) [ x 2 + ( 2 m n + 1 ) ( 2 m n + 1 ) ] [ x 2 + ( 2 m n + 1 ) 2 ] [ x 2 + ( 2 m n + 1 ) 2 ] ,
ψ m m = ( m m ) { π 2 2 n ( m n + m n + 1 ) x ( x 2 + 1 ) ln ( 1 s ) [ x 2 + ( 2 m n + 1 ) 2 ] [ x 2 + ( 2 m n + 1 ) 2 ] } .
T ( x , Φ 01 , Φ 02 ) = m , m = 0 f m ν f m ν * x 2 + 1 [ x + i ( 2 B m ν + 1 ) ] [ x i ( 2 B m ν + 1 ) ] .
T ( x , Φ 01 , Φ 02 ) = 1 + T 1 ( x , Φ 01 ) + T 2 ( x , Φ 02 ) + 48 Φ 01 Φ 02 ( x 4 + 14 x 2 35 ) ( x 2 + 1 ) 3 ( x 2 + 9 ) ( x 2 + 25 ) ( x 2 + 49 ) ,
ν = 0 N opt 1 ν ! ( N opt ν ) ! Φ 01 ν Φ 02 N opt ν = 0.5 .
T ( x , s , Φ 01 , Φ 02 ) = 1 s { 1 m , m = 0 v , v = 0 m , m F m v m v ( 1 s ) λ m v m v cos ψ m v m v } ,
F m v m v = Φ 01 v + v Φ 02 m + m v v v ! v ! ( m v ) ! ( m v ) ! ( B m v + B m v + 1 ) ( x 2 + 1 ) B m v + B m v ,
λ m v m v = ( B m v + B m v + 1 ) ( x 2 + 1 ) [ x 2 + ( 2 B m v + 1 ) ( 2 B m v + 1 ) ] [ x 2 + ( 2 B m v + 1 ) 2 ] [ ( x 2 + ( 2 B m v + 1 ) 2 ] ,
ψ m v m v = π 2 ( m m ) 2 ( B m v B m v ) ( B m v + B m v + 1 ) x ( x 2 + 1 ) ln ( 1 s ) [ x 2 + ( 2 B m v + 1 ) 2 ] [ ( x 2 + ( 2 B m v + 1 ) 2 ] .

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