Abstract

We present a simple theory based on the method of moments for fitting experimental measurements of optical nonlinearities obtained by the Z-scan technique. Our method provides a simple and intuitive image for Gaussian beam propagation through a nonlinear optical plate: a linear transformation that consists of a displacement and a contraction of the beam waist. We study cubic nonlinearities of samples of arbitrary thickness. We also consider, for thin samples, higher-order nonlinearities such as cubic–quintic nonlinearities and the effect of nonlinear absorption.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2002

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

G. Fang, Y. Mo, Y. Song, Y. Wang, C. Li, and L. Song, “Nonlinear refractive properties of organometallic fullerene-C60 derivatives,” Opt. Commun. 205, 337–341 (2002).
[CrossRef]

2001

R. Quintero-Torres, F. Carreto-Parra, Z. Navarrete-Meza, and L. Zambrano-Valencia, “Straightforward representation of a thick nonlinear optical material, with linear and nonlinear absorption,” Mod. Phys. Lett. B 15, 796–803 (2001).
[CrossRef]

2000

V. M. Pérez-García, P. Torres, J. J. García-Ripoll, and H. Michinel, “Moment analysis of paraxial propagation in a nonlinear graded index fibre,” J. Opt. B 2, 353–358 (2000).
[CrossRef]

1999

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional z-scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043–7050 (1999).
[CrossRef]

1998

1997

Th. Busch, J. I. Cirac, V. M. Pérez-García, and P. Zoller, “Stability and collective excitations of a two-component Bose–Einstein condensed gas: a moment approach,” Phys. Rev. A 56, 2978–2983 (1997).
[CrossRef]

1995

H. Michinel, “Non-linear propagation of Gaussian beams in planar graded-index waveguides: a variational approach,” Pure Appl. Opt. 4, 701–708 (1995).
[CrossRef]

1994

1991

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

1990

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

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

1989

1988

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

1983

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 6, 3135–3145 (1983).
[CrossRef]

1979

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 6, 3135–3145 (1983).
[CrossRef]

Arntzen, P. O.

Bubner, T.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

Busch, Th.

Th. Busch, J. I. Cirac, V. M. Pérez-García, and P. Zoller, “Stability and collective excitations of a two-component Bose–Einstein condensed gas: a moment approach,” Phys. Rev. A 56, 2978–2983 (1997).
[CrossRef]

Carreto-Parra, F.

R. Quintero-Torres, F. Carreto-Parra, Z. Navarrete-Meza, and L. Zambrano-Valencia, “Straightforward representation of a thick nonlinear optical material, with linear and nonlinear absorption,” Mod. Phys. Lett. B 15, 796–803 (2001).
[CrossRef]

Chapple, P. B.

Chen, P.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional z-scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043–7050 (1999).
[CrossRef]

Cirac, J. I.

Th. Busch, J. I. Cirac, V. M. Pérez-García, and P. Zoller, “Stability and collective excitations of a two-component Bose–Einstein condensed gas: a moment approach,” Phys. Rev. A 56, 2978–2983 (1997).
[CrossRef]

Eriksson, A.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

A. Eriksson, M. Lindgren, S. Svensson, and P. O. Arntzen, “Numerical analysis of Z-scan experiments by use of a mode expansion,” J. Opt. Soc. Am. B 15, 810–816 (1998).
[CrossRef]

Fan, X. Z.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

Fang, G.

G. Fang, Y. Mo, Y. Song, Y. Wang, C. Li, and L. Song, “Nonlinear refractive properties of organometallic fullerene-C60 derivatives,” Opt. Commun. 205, 337–341 (2002).
[CrossRef]

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

García-Ripoll, J. J.

V. M. Pérez-García, P. Torres, J. J. García-Ripoll, and H. Michinel, “Moment analysis of paraxial propagation in a nonlinear graded index fibre,” J. Opt. B 2, 353–358 (2000).
[CrossRef]

Gu, B.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

Guo, S. L.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

Hagan, D. J.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317–319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

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

Hermann, J. A.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

Li, C.

G. Fang, Y. Mo, Y. Song, Y. Wang, C. Li, and L. Song, “Nonlinear refractive properties of organometallic fullerene-C60 derivatives,” Opt. Commun. 205, 337–341 (2002).
[CrossRef]

Lindgren, M.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

A. Eriksson, M. Lindgren, S. Svensson, and P. O. Arntzen, “Numerical analysis of Z-scan experiments by use of a mode expansion,” J. Opt. Soc. Am. B 15, 810–816 (1998).
[CrossRef]

McDuff, R. G.

McKay, T. J.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

Michinel, H.

V. M. Pérez-García, P. Torres, J. J. García-Ripoll, and H. Michinel, “Moment analysis of paraxial propagation in a nonlinear graded index fibre,” J. Opt. B 2, 353–358 (2000).
[CrossRef]

H. Michinel, “Non-linear propagation of Gaussian beams in planar graded-index waveguides: a variational approach,” Pure Appl. Opt. 4, 701–708 (1995).
[CrossRef]

Miller, D. A. B.

Ming, N. B.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

Mo, Y.

G. Fang, Y. Mo, Y. Song, Y. Wang, C. Li, and L. Song, “Nonlinear refractive properties of organometallic fullerene-C60 derivatives,” Opt. Commun. 205, 337–341 (2002).
[CrossRef]

Navarrete-Meza, Z.

R. Quintero-Torres, F. Carreto-Parra, Z. Navarrete-Meza, and L. Zambrano-Valencia, “Straightforward representation of a thick nonlinear optical material, with linear and nonlinear absorption,” Mod. Phys. Lett. B 15, 796–803 (2001).
[CrossRef]

Oulianov, D. A.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional z-scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043–7050 (1999).
[CrossRef]

Pérez-García, V. M.

V. M. Pérez-García, P. Torres, J. J. García-Ripoll, and H. Michinel, “Moment analysis of paraxial propagation in a nonlinear graded index fibre,” J. Opt. B 2, 353–358 (2000).
[CrossRef]

Th. Busch, J. I. Cirac, V. M. Pérez-García, and P. Zoller, “Stability and collective excitations of a two-component Bose–Einstein condensed gas: a moment approach,” Phys. Rev. A 56, 2978–2983 (1997).
[CrossRef]

Quintero-Torres, R.

R. Quintero-Torres, F. Carreto-Parra, Z. Navarrete-Meza, and L. Zambrano-Valencia, “Straightforward representation of a thick nonlinear optical material, with linear and nonlinear absorption,” Mod. Phys. Lett. B 15, 796–803 (2001).
[CrossRef]

Rentzepis, P. M.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional z-scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043–7050 (1999).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 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]

Samad, R. E.

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Sheik-Bahae, M.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317–319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 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]

Smith, S. D.

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Song, L.

G. Fang, Y. Mo, Y. Song, Y. Wang, C. Li, and L. Song, “Nonlinear refractive properties of organometallic fullerene-C60 derivatives,” Opt. Commun. 205, 337–341 (2002).
[CrossRef]

Song, Y.

G. Fang, Y. Mo, Y. Song, Y. Wang, C. Li, and L. Song, “Nonlinear refractive properties of organometallic fullerene-C60 derivatives,” Opt. Commun. 205, 337–341 (2002).
[CrossRef]

Staromlynska, J.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin and the thick sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Stryland, E. W. V.

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

Svensson, S.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

A. Eriksson, M. Lindgren, S. Svensson, and P. O. Arntzen, “Numerical analysis of Z-scan experiments by use of a mode expansion,” J. Opt. Soc. Am. B 15, 810–816 (1998).
[CrossRef]

Tomov, I. V.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional z-scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043–7050 (1999).
[CrossRef]

Torres, P.

V. M. Pérez-García, P. Torres, J. J. García-Ripoll, and H. Michinel, “Moment analysis of paraxial propagation in a nonlinear graded index fibre,” J. Opt. B 2, 353–358 (2000).
[CrossRef]

Van Stryland, E. W.

Vieira Jr., N. Dias

Wang, H. T.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

Wang, Y.

G. Fang, Y. Mo, Y. Song, Y. Wang, C. Li, and L. Song, “Nonlinear refractive properties of organometallic fullerene-C60 derivatives,” Opt. Commun. 205, 337–341 (2002).
[CrossRef]

Weaire, D.

Wei, T. H.

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

Wherrett, B. S.

Wilson, P. J.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

Wright, E. M.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Xia, T.

Xu, L.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

Yan, J.

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

Zambrano-Valencia, L.

R. Quintero-Torres, F. Carreto-Parra, Z. Navarrete-Meza, and L. Zambrano-Valencia, “Straightforward representation of a thick nonlinear optical material, with linear and nonlinear absorption,” Mod. Phys. Lett. B 15, 796–803 (2001).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Zoller, P.

Th. Busch, J. I. Cirac, V. M. Pérez-García, and P. Zoller, “Stability and collective excitations of a two-component Bose–Einstein condensed gas: a moment approach,” Phys. Rev. A 56, 2978–2983 (1997).
[CrossRef]

IEEE J. Quantum Electron.

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

J. Appl. Phys.

P. Chen, D. A. Oulianov, I. V. Tomov, and P. M. Rentzepis, “Two-dimensional z-scan for arbitrary beam shape and sample thickness,” J. Appl. Phys. 85, 7043–7050 (1999).
[CrossRef]

J. Lightwave Technol.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

J. Nonlinear Opt. Phys. Mater.

J. A. Hermann, T. Bubner, T. J. McKay, P. J. Wilson, J. Staromlynska, A. Eriksson, M. Lindgren, and S. Svensson, “Optical limiting capability of thick nonlinear absorbers,” J. Nonlinear Opt. Phys. Mater. 8, 253–275 (1999).
[CrossRef]

J. Opt. A

S. L. Guo, J. Yan, L. Xu, B. Gu, X. Z. Fan, H. T. Wang, and N. B. Ming, “Second Z-scan in materials with nonlinear refraction and nonlinear absorption,” J. Opt. A 4, 504–508 (2002).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Sketch of propagation of a focused Gaussian beam through a nonlinear plate. Top, evolution of the beam width relative to the nonlinear material. Bottom, effect of the nonlinear plate consisting of a translation (δnl) of the focus and a contraction of the beam waist. Solid curves, variation of the beam width in the linear case w(z); dashed curves, variation of the modified beam width w˜(z), with the nonlinear plate taken into account.

Fig. 2
Fig. 2

Measured Z scan of a BaF2 sample from Ref. 10, indicating self-focusing that is due to the electronic Kerr effect. Experimental data are L=2.5 mm, λ=0.532 μm, w0=13 μm, and n2R|A0|2=2.879×10-6. Solid curve, theoretical fit from expression (22). The GD fit is also shown.

Fig. 3
Fig. 3

Measured Z scan of a CS2 sample from Ref. 24. Experimental data are L=10 mm, λ=1.064 μm, w0=8.9 μm, n1=1.63, n2R=3.6×10-18 m2/W, and |A0|2=0.4 GW/m2. Solid curve, theoretical fit from Eq. (27).

Fig. 4
Fig. 4

Measured Z scan of a CS2 sample from Ref. 25. Experimental data are L=24 mm, λ=10.6 μm, w0=43 μm, n1=1.63, and n2R|A0|2=1.1×10-3. Solid curve, theoretical fit from Eq. (27).

Fig. 5
Fig. 5

Normalized Z-scan transmittance of a ZnSe sample from Ref. 9 with the aperture removed. Experimental data are L=2.7 mm, λ=0.532 μm, w0=18.44 μm, n1=2.7, n2I=1.54×10-4 cm2/GW, and |A0|2=0.21 GW/cm2. Solid curve, theoretical fit from expression (31).

Fig. 6
Fig. 6

Normalized Z-scan transmittance of a ZnSe sample from Ref. 9 with 40% aperture. Experimental data are L=2.7 mm, λ=0.532 μm, w0=18.44 μm, n1=2.7, n2I=1.54×10-4 cm2/GW, n2R=6.8×10-5 cm2/GW, and |A0|2=0.21 GW/cm2. Solid curve, theoretical fit from expression (36).

Fig. 7
Fig. 7

Z-scan curve of a C60–chromium derivative from Ref. 21. Experimental data are L=2 mm, λ=532 nm, w0=33 μm, n2R|A0|2=-2.38×10-5, and n4|A0|4=-2.35×10-3. Solid curve, theoretical fit from expression (40).

Equations (58)

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2ik0n1ψz+2ψ+2k02n1Δnψ=0,
ψ(r, z, τ)=A(z)exp-12 [β(z)r2+βττ2],
ψz=ik0n112 βr2-β+ik0Δnψ.
F(z)dxdydτ|ψ|2=|A(z)|2π3/2w2(z)wτ,
w(z)=1βR(z),
M(z)=w(z)βI(z).
Δn=n2|ψ|2-n4|ψ|4,
dFdz=-k0n2IF22π3w2wτ,
dwdz=-Mk0n1+k0n2IF28π3wwτ,
dMdz=-1k0n1w3+Fk0n2R8π3w3wτ+k0n2IFM28π3w2wτ-4k0n4F293π3w5wτ2.
d2wdz2=1k02n12w3-n2RF8π3n1w3wτ-5k02(n2I)2F232π3w3wτ2+4n4F293π3n1w5wτ2.
d2wdz2=w04zRN21w3,
1zRN2=1n12zR2-1zN2=1k02n12w04-n2R|A0|222πn1w02,
n2R|A0|2λ22π3/2n1w02.
w2(z)=w021+z2n02zR2.
w˜2(z)=w˜021+z˜2n02z˜R2,
δnl=z+L-z˜,
w˜02=w021+η2n02zR2-L2zN2/
1+ηn0zR-Ln1zRzN22,
z˜=w˜02w02η-Ln0n1zR2zN2,
wal2=w021+(D-δl)2n02zR2,
wanl2=w˜021+(D-δNL)2n02z˜R2.
Tw˜02w021+[2ηL-(n0/n1)L2](n0/n1)zN2[1+(η2/n02zR2)]-2ηL.
T1+2ηL(n0/n1)zN2[1+(η2/n02zR2)].
ΔTpv=2n1zRLzN2=k0L2π n2R|A0|20.400k0Ln2R|A0|2.
ΔTpvGD=0.405k0Ln2R|A0|2.
dT=1+2x(n0/n1)zN2[1+(x2/n02zR2)]dx.
T(z)=1+n0n1zR2zN2ln[z+L(n0/n1)]2+n02zR2z2+n02zR2.
ΔTpvthick2n0n1zR2zN2ln(L2/n12)+zR2zR2.
F(z+L)F(z)-2k0n2IF2(z)8π3w2(z)wτ L.
TOA(z)1-k0n2IF(z)2π3w2(z)wτ L,
TOA(z)1-k0n2I|A0|2n02zR22(z2+n02zR2) L.
d2wdz2=1k02n12w3-n2R|A0|2w0222n1w3-5k02(n2I)2|A0|4w0432w3,
d2wdz2=w04zRNabs21w3,
1zRNabs2=1n12zRN2-1zNabs2=1k02n12w04-n2R|A0|222πn1w02-5k02(n2I)2|A0|432π,
TNL(z)1+n0n1zR2zNabs2ln[z+L(n0/n1)]2+n02zR2z2+n02zR2.
TCA(z)TOA(z)Tnl(z).
d2wdz2=1k02n12w3-n2R|A0|2w0222n1w3+4n4|A0|4w0493n1w5.
d2wdz2=w04zRNCQ21w3,
1zRNCQ2=1n12zR2-1zNCQ2=1k02n12w04-n2R|A0|222πn1w02+4n4|A0|4n02zR293πn1w02(z2+n02zR2).
T1+2ηL(n0/n1)zNCQ2[1+(η2/n02zR2)],
F˙=dxdydτΨ˙*Ψ+H.c.,
Wx(z)dxdydτx2|Ψ|2,
Px(z)dxdydτΨ*x2Ψ.
Wx=F2βR,
Px=-F2βR+(βI)2βR.
W˙x=dxdydτx2Ψ˙*Ψ+H.c.,
P˙x=dxdydτΨ˙*x2Ψ+H.c.,
W˙x=F˙2βR-Fβ˙R2(βR)2,
P˙x=-F˙21w2+M2+Fw˙w3-MM˙.
dxdydτ|Ψ|2=F,
dxdydτx2|Ψ|2=dxdydτy2|Ψ|2=F2βR,
dxdydτx2y2|Ψ|2=F4(βR)2,
dxdydτx4|Ψ|2=3F4(βR)2,
dxdydτ|Ψ|4=F2βR(βτR)1/28π3,
dxdydτx2|Ψ|4=F2(βτR)1/248π3,
dxdydτ|Ψ|6=F3(βR)2βτR33π3,
dxdydτx2|Ψ|6=F3βRβτR183π3.

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