Abstract

Analytical expressions for Gaussian-beam Z-scan and eclipsing Z-scan signals are obtained in a unified format in the case of third-order optical nonlinearities. Considering optically thin media in the presence of both nonlinear refraction and nonlinear absorption, the Gaussian decomposition method is applied to express the normalized transmittance through an aperture or a disk in terms of a sum of heterodyne and homodyne contributions. The expressions presented are also valid for arbitrary circularly symmetric real ABCD post-sample optical systems.

© 2020 Optical Society of America

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References

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  4. R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
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    [Crossref]
  28. J. A. Hermann and P. J. Wilson, “Factors affecting optical limiting and scanning with thin nonlinear samples,” J. Nonlinear Opt. Phys. Mater. 02, 613–629 (1993).
    [Crossref]
  29. B. Gu, W. Ji, and X.-Q. Huang, “Analytical expression for femtosecond-pulsed Z-scans on instantaneous nonlinearity,” Appl. Opt. 47, 1187–1192 (2008).
    [Crossref]
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2014 (2)

2009 (1)

2008 (1)

2005 (2)

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys. 123, 054509 (2005).
[Crossref]

A. Gnoli, L. Razzari, and M. Righini, “Z-scan measurements using high repetition rate lasers: how to manage thermal effects,” Opt. Express 13, 7976–7981 (2005).
[Crossref]

2004 (3)

2001 (1)

J. A. Gardecki, G. Yu, S. Constantine, J. Peng, Y. Zhou, and L. D. Ziegler, “A unified treatment of ultrafast optical heterodyne detected and Z-scan spectroscopies,” J. Chem. Phys. 114, 3586–3597 (2001).
[Crossref]

1999 (2)

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]

D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[Crossref]

1998 (2)

1997 (2)

X. J. Zhang, W. Ji, and S. H. Tang, “Determination of optical nonlinearities and carrier lifetime in ZnO,” J. Opt. Soc. Am. B 14, 1951–1955 (1997).
[Crossref]

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659–662 (1997).
[Crossref]

1995 (1)

1994 (1)

1993 (2)

J. A. Hermann and P. J. Wilson, “Factors affecting optical limiting and scanning with thin nonlinear samples,” J. Nonlinear Opt. Phys. Mater. 02, 613–629 (1993).
[Crossref]

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

1992 (2)

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[Crossref]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46–51 (1992).
[Crossref]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. 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 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Arntzen, P.-O.

Baba, M.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Bondar, M. V.

Buckup, T.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys. 123, 054509 (2005).
[Crossref]

Burzler, J. M.

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]

Constantine, S.

J. A. Gardecki, G. Yu, S. Constantine, J. Peng, Y. Zhou, and L. D. Ziegler, “A unified treatment of ultrafast optical heterodyne detected and Z-scan spectroscopies,” J. Chem. Phys. 114, 3586–3597 (2001).
[Crossref]

Correia, R. R. B.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys. 123, 054509 (2005).
[Crossref]

Coulter, D. R.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46–51 (1992).
[Crossref]

Cui, D.-F.

Cui, Y.

B. Gu, D. Liu, J.-L. Wu, J. He, and Y. Cui, “Z-scan characterization of optical nonlinearities of an imperfect sample profits from radially polarized beams,” Appl. Phys. B 117, 1141–1147 (2014).
[Crossref]

Cunha, S. L. S.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys. 123, 054509 (2005).
[Crossref]

da Silveira, N. P.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys. 123, 054509 (2005).
[Crossref]

Ensley, T. R.

Eriksson, A.

Ferdinandus, M. R.

Fishman, D. A.

Ganeev, R. A.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Gardecki, J. A.

J. A. Gardecki, G. Yu, S. Constantine, J. Peng, Y. Zhou, and L. D. Ziegler, “A unified treatment of ultrafast optical heterodyne detected and Z-scan spectroscopies,” J. Chem. Phys. 114, 3586–3597 (2001).
[Crossref]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).

Gnoli, A.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Gu, B.

Hagan, D. J.

M. Reichert, H. Hu, M. R. Ferdinandus, M. Seidel, P. Zhao, T. R. Ensley, D. Peceli, J. M. Reed, D. A. Fishman, S. Webster, D. J. Hagan, and E. W. Van Stryland, “Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide,” Optica 1, 436–445 (2014).
[Crossref]

D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[Crossref]

O. V. Przhonska, J. H. Lim, D. J. Hagan, E. W. Van Stryland, M. V. Bondar, and Y. L. Slominsky, “Nonlinear light absorption of polymethine dyes in liquid and solid media,” J. Opt. Soc. Am. B 15, 802–809 (1998).
[Crossref]

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]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46–51 (1992).
[Crossref]

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. 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.

B. Gu, D. Liu, J.-L. Wu, J. He, and Y. Cui, “Z-scan characterization of optical nonlinearities of an imperfect sample profits from radially polarized beams,” Appl. Phys. B 117, 1141–1147 (2014).
[Crossref]

He, J.-L.

Heisler, I. A.

I. A. Heisler, R. R. B. Correia, T. Buckup, S. L. S. Cunha, and N. P. da Silveira, “Time-resolved optical Kerr-effect investigation on CS2/polystyrene mixtures,” J. Chem. Phys. 123, 054509 (2005).
[Crossref]

Hermann, J. A.

J. A. Hermann and P. J. Wilson, “Factors affecting optical limiting and scanning with thin nonlinear samples,” J. Nonlinear Opt. Phys. Mater. 02, 613–629 (1993).
[Crossref]

Hu, H.

Huang, X.-Q.

Hughes, S.

Ishizawa, N.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Ji, W.

Kovsh, D. I.

Kuroda, H.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Li, F.-Q.

Li, H.

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659–662 (1997).
[Crossref]

Lim, J. H.

Lindgren, M.

Liu, D.

B. Gu, D. Liu, J.-L. Wu, J. He, and Y. Cui, “Z-scan characterization of optical nonlinearities of an imperfect sample profits from radially polarized beams,” Appl. Phys. B 117, 1141–1147 (2014).
[Crossref]

Liu, Z.-B.

Mukamel, S.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, 1995).

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]

Palffy-Muhoray, P.

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

Peceli, D.

Peng, J.

J. A. Gardecki, G. Yu, S. Constantine, J. Peng, Y. Zhou, and L. D. Ziegler, “A unified treatment of ultrafast optical heterodyne detected and Z-scan spectroscopies,” J. Chem. Phys. 114, 3586–3597 (2001).
[Crossref]

Peng, Q.-J.

Perry, J. W.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46–51 (1992).
[Crossref]

Przhonska, O. V.

Razzari, L.

Reed, J. M.

Reichert, M.

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]

Righini, M.

Ryasnyansky, A. I.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Said, A. A.

Sakakibara, S.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Seidel, M.

Sence, M. J.

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46–51 (1992).
[Crossref]

Sheik-Bahae, M.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Slominsky, Y. L.

Song, F.

Spruce, G.

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, 2003).

Suzuki, M.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Svensson, S.

Tang, S. H.

Tian, J.-G.

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]

Turu, M.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, “Nonlinear refraction in CS2,” Appl. Phys. B 78, 433–438 (2004).
[Crossref]

Van Stryland, E. W.

M. Reichert, H. Hu, M. R. Ferdinandus, M. Seidel, P. Zhao, T. R. Ensley, D. Peceli, J. M. Reed, D. A. Fishman, S. Webster, D. J. Hagan, and E. W. Van Stryland, “Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide,” Optica 1, 436–445 (2014).
[Crossref]

D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[Crossref]

O. V. Przhonska, J. H. Lim, D. J. Hagan, E. W. Van Stryland, M. V. Bondar, and Y. L. Slominsky, “Nonlinear light absorption of polymethine dyes in liquid and solid media,” J. Opt. Soc. Am. B 15, 802–809 (1998).
[Crossref]

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]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46–51 (1992).
[Crossref]

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. 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]

Wang, H.-T.

Wang, J.

Wang, Q.

Webster, S.

Wei, T.

M. Sheik-Bahae, A. A. Said, T. 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]

Wei, T. H.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992).
[Crossref]

T. H. Wei, D. J. Hagan, M. J. Sence, E. W. Van Stryland, J. W. Perry, and D. R. Coulter, “Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46–51 (1992).
[Crossref]

Wherrett, B. S.

Wilson, P. J.

J. A. Hermann and P. J. Wilson, “Factors affecting optical limiting and scanning with thin nonlinear samples,” J. Nonlinear Opt. Phys. Mater. 02, 613–629 (1993).
[Crossref]

Wu, J.-L.

B. Gu, D. Liu, J.-L. Wu, J. He, and Y. Cui, “Z-scan characterization of optical nonlinearities of an imperfect sample profits from radially polarized beams,” Appl. Phys. B 117, 1141–1147 (2014).
[Crossref]

Xia, T.

Xu, Z.-Y.

Yan, J.

Yang, F.

Yang, S.

Young, J.

Yu, G.

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[Crossref]

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[Crossref]

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H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659–662 (1997).
[Crossref]

Zhou, W.-Y.

Zhou, Y.

J. A. Gardecki, G. Yu, S. Constantine, J. Peng, Y. Zhou, and L. D. Ziegler, “A unified treatment of ultrafast optical heterodyne detected and Z-scan spectroscopies,” J. Chem. Phys. 114, 3586–3597 (2001).
[Crossref]

Ziegler, L. D.

J. A. Gardecki, G. Yu, S. Constantine, J. Peng, Y. Zhou, and L. D. Ziegler, “A unified treatment of ultrafast optical heterodyne detected and Z-scan spectroscopies,” J. Chem. Phys. 114, 3586–3597 (2001).
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[Crossref]

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Figures (2)

Fig. 1.
Fig. 1. Z-scan simplified scheme. The exit plane of the sample and the mask plane are connected by a paraxial ABCD system.
Fig. 2.
Fig. 2. Magnitudes of first few contributions to the normalized transmittance of a CW beam through a mask in the far-field region, simulated with ${\sigma_{\rm x}}=0.1$, ${Q_0}=0.2$, and $\Delta {\Phi_0}=\pi /10$.

Tables (1)

Tables Icon

Table 1. | Δ Φ 0 | and Q 0 Upper Values for Which T a x i s w and T a w (with 0 < S a 0.5 ) Provide Z-Scan Traces with an Absolute Value of the Relative Error, Δ T a w , Less than 1%

Equations (87)

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d I d z = [ α + Δ α ( I ) ] I ,
d Δ ϕ d z = k Δ n ( I ) ,
Δ n ( t ) = n 2 , e l I ( t ) + + R ( t t ) I ( t ) d t ,
Δ n ( t ) = [ n 2 , e l + m n 2 , m ] I ( t ) = n 2 I ( t ) ,
I e x i t = I ( z = 0 ) exp ( α L ) 1 + q ,
Δ ϕ e x i t = k n 2 β ln ( 1 + q ) ,
E e x i t = E ( z = 0 ) exp ( α L / 2 ) ( 1 + q ) ( i k n 2 / β 1 / 2 ) ,
E ( r , z , t ) = E ( t ) w f w ( z ) exp [ r 2 w 2 ( z ) + i π r 2 λ R ( z ) i θ ( z ) ] ,
w ( z ) = w f ( 1 + z 2 / z f 2 ) 1 / 2 ,
R ( z ) = z ( 1 + z f 2 / z 2 ) ,
θ ( z ) = tan 1 ( z / z f )
I ( r , z , t ) = I ( t ) 1 + z 2 / z f 2 exp [ 2 r 2 w 2 ( z ) ] ,
E e x i t ( r , z , t ) = E ( r , z , t ) [ 1 + q ( r , z , t ) ] ( i k n 2 / β 1 / 2 ) = m = 0 E ( r = 0 , z , t ) f m exp [ r 2 w m 0 2 + i π r 2 λ R ( z ) ] = m = 0 E m 0 ( r , z , t ) ,
w m 0 2 = w 2 ( z ) 1 + 2 m ,
f m = i m m ! [ Δ Φ 0 f ( t ) 1 + z 2 / z f 2 ] m Π m ,
Π m = n = 1 m [ 1 + i ( 2 n 1 ) Q 0 2 Δ Φ 0 ] ,
Δ Φ 0 = k n 2 I 0 L e f f ,
Q 0 = β I 0 L e f f ,
E m a s k = 2 π i λ B 0 E e x i t exp [ i π λ B ( A r 2 + D ρ 2 ) ] × J 0 ( 2 π ρ λ B r ) r d r ,
E m = E ( r = 0 , z , t ) f m i λ B exp [ i π λ B D ρ 2 ] H m
H m = 2 π 0 exp ( a m r 2 ) J 0 ( 2 π b r ) r d r ,
a m = π i λ B [ g + i B d m ] ,
b = ρ λ B ,
g = A + B / R ( z ) ,
d m = π w m 0 2 / λ .
H m = π a m exp [ π 2 b 2 a m ] .
E m = E ( r = 0 , z , t ) f m w m 0 w m exp [ ρ 2 w m 2 + i π ρ 2 λ R m i θ m ] ,
w m = w m 0 [ g 2 + B 2 d m 2 ] 1 / 2 ,
R m = B [ D g g 2 + B 2 / d m 2 ] 1 ,
θ m = tan 1 [ B / d m g ] .
2 π + x I 0 ( ρ , t ) ρ d ρ d t = σ x + P ( t ) d t ,
S x = 1 exp ( 2 R x 2 / w 0 2 ) .
T x = + x | E m a s k | 2 ρ d ρ d t + x | E 0 | 2 ρ d ρ d t = m = 0 n m C mn x ,
C mn x = + x ( 2 δ mn ) R e { E m E n } ρ d ρ d t + x | E 0 | 2 ρ d ρ d t ,
C mn x = K mn S x + δ a x 1 [ Π mn σ mn x + Π mn ( γ mn x δ a x ) ] ,
K mn = ( 2 δ mn ) Δ Φ 0 m + n F mn m ! n ! ( 1 + m + n ) ( 1 + z 2 / z f 2 ) m + n ,
F mn = + [ f ( t ) ] m + n f ( t ) d t + f ( t ) d t ,
σ mn x = ( 1 S x ) A mn sin [ ln ( 1 S x ) B mn ] ,
γ mn x = ( 1 S x ) A mn cos [ ln ( 1 S x ) B mn ] ,
A mn = 1 2 ( 1 + Z 2 ) [ 1 + 2 m ( 1 + 2 m ) 2 + Z 2 + 1 + 2 n ( 1 + 2 n ) 2 + Z 2 ] ,
B mn = Z 2 ( 1 + Z 2 ) [ 1 ( 1 + 2 m ) 2 + Z 2 + 1 ( 1 + 2 n ) 2 + Z 2 ] ,
Z = z z f + z f ( 1 + z 2 z f 2 ) A B ,
Π mn = i m + n 1 Π m Π n ,
Π mn = exp ( α m + α n ) sin [ ( m n ) π / 2 + β m β n ] ,
Π mn = exp ( α m + α n ) cos [ ( m n ) π / 2 + β m β n ] ,
α m = n = 1 m 1 2 ln { 1 + [ ( 2 n 1 ) Q 0 2 Δ Φ 0 ] 2 } ,
β m = n = 1 m tan 1 [ ( 2 n 1 ) Q 0 2 Δ Φ 0 ] ,
R e { a m } = 1 w m 0 2 + π λ [ A I m B R e A R e B I m B R e 2 + B I m 2 ] .
F mn ( G a u s s i a n ) = 1 ( 1 + m + n ) 1 / 2 ,
F mn ( s e c h 2 ) = 2 m + n ( m + n ) ! ( 2 m + 2 n + 1 ) ! ! = [ 2 m + n ( m + n ) ! ] 2 ( 2 m + 2 n + 1 ) ! ,
Δ Φ 0 m + n F mn = + [ Δ Φ ( t ) ] m + n I ( t ) d t + I ( t ) d t = Δ Φ m + n .
T o a = m = 0 n m C mn o a ,
T o a = C 00 o a + C 01 o a + ( C 02 o a + C 11 o a ) + ( C 03 o a + C 12 o a ) + ( C 04 o a + C 13 o a + C 22 o a ) + ( C 05 o a + C 14 o a + C 23 o a ) + = μ = 0 C μ o a ,
C μ o a = m n ( μ = m + n ) C mn o a ,
T o a = m = 0 ( 1 ) m Q 0 m F m 0 ( 1 + m ) ( 1 + z 2 / z f 2 ) m ,
T a x i s = m = 0 n m K mn [ B mn Π mn + A mn Π mn ] .
T a x i s T a x i s w = 1 + 4 Z Δ Φ ( 3 + Z 2 ) Q ( 1 + z 2 / z f 2 ) ( 9 + Z 2 ) ,
T x T x w = 1 + ( 1 S x ) A sin [ ln ( 1 S x ) B ] ( S x + δ a x 1 ) ( 1 + z 2 / z f 2 ) Δ Φ + ( 1 S x ) A cos [ ln ( 1 S x ) B ] δ a x 2 ( S x + δ a x 1 ) ( 1 + z 2 / z f 2 ) Q ,
A = A 01 = 2 ( 3 + Z 2 ) / ( 9 + Z 2 ) ,
B = B 01 = 4 Z / ( 9 + Z 2 ) .
| E m a s k | 2 = m = 0 M n = 0 M E m E n
= m = 0 M n m M ( 2 δ mn ) R e { E m E n } .
| Δ Φ 0 M Π M | M ! ( 1 + 2 M ) 1 / 2 < [ 3 δ a x + ( 1 δ a x ) / 5 ] × 10 2 ,
d m = d 0 / ( 1 + 2 m ) ,
a m = a 0 + 2 m / w 2 .
h m = 1 1 + 2 m / ( a 0 w 2 ) × exp [ π 2 b 2 a 0 ( 1 1 + 2 m / ( a 0 w 2 ) 1 ) ] .
C mn x = K mn ( 1 + m + n ) σ x w 0 2 / 4 H mn x ,
H mn x = x R e { i Π mn h m h n exp ( 2 ρ 2 / w 0 2 ) } ρ d ρ .
Z = g d 0 / B = z z f + z f ( 1 + z 2 z f 2 ) A B ,
a 0 w 2 = 1 i Z ,
π 2 b 2 / a 0 = ( 1 + i Z ) ρ 2 / w 0 2 .
h m h n exp ( 2 ρ 2 / w 0 2 ) = U mn 1 + m + n exp ( 2 U mn ρ 2 / w 0 2 ) ,
H mn x = 1 1 + m + n x R e { i Π mn U mn exp ( 2 U mn ρ 2 / w 0 2 ) } ρ d ρ = 1 1 + m + n I m { Π mn U mn x exp ( 2 U mn ρ 2 / w 0 2 ) ρ d ρ } = w 0 2 / 4 1 + m + n I m { Π mn exp ( 2 U mn ρ 2 / w 0 2 ) } | x = w 0 2 / 4 1 + m + n [ Π mn σ ~ mn | x + Π mn γ ~ mn | x ] ,
σ ~ mn = exp ( 2 A mn ρ 2 / w 0 2 ) sin ( 2 B mn ρ 2 / w 0 2 ) ,
γ ~ mn = exp ( 2 A mn ρ 2 / w 0 2 ) cos ( 2 B mn ρ 2 / w 0 2 ) ,
C mn x = K mn σ x [ Π mn σ ~ mn | x + Π mn γ ~ mn | x ] .
C mn a = K mn S a [ Π mn σ mn a + Π mn ( γ mn a 1 ) ] ,
C mn d = K mn 1 S d [ Π mn σ mn d + Π mn γ mn d ] .
C mn x = K mn S x + δ a x 1 [ Π mn σ mn x + Π mn ( γ mn x δ a x ) ] ,
c n = a n + i b n = | c n | exp ( i φ n ) = ( a n 2 + b n 2 ) 1 / 2 exp [ i tan 1 ( b n / a n ) ]
Π ~ m = n = 1 m c n = [ n = 1 m | c n | ] [ n = 1 m exp ( i φ n ) ] ,
n = 1 m | c n | = exp [ n = 1 m ln | c n | ] ,
n = 1 m exp ( i φ n ) = exp [ i n = 1 m φ n ] .
α ~ m = n = 1 m 1 2 ln ( a n 2 + b n 2 ) ,
β ~ m = n = 1 m tan 1 ( b n / a n ) ,
Π m = n = 1 m [ 1 + i ( 2 n 1 ) Q 0 2 Δ Φ 0 ] = exp ( α m + i β m ) ,
Π mn = i m + n 1 Π m Π n = i exp [ i ( m n ) π / 2 ] exp ( α m i β m ) exp ( α n + i β n ) = i exp ( α m + α n ) exp { i [ ( m n ) π / 2 + β m β n ] } = exp ( α m + α n ) { sin [ ( m n ) π / 2 + β m β n ] + i cos [ ( m n ) π / 2 + β m β n ] } ,