Abstract

An approach is proposed to analyze the two- and three-photon absorption process using the open-aperture Z-scan technique. The normalized transmittance equations of one pulse energy presented here can be used to investigate the effect of different temporal profiles of excitation laser pulses on the nonlinear absorption coefficients (NACs). The numerical simulations show that our method is well in accordance with the usual Z-scan in determination of two- and three-photon absorption coefficients in 0.6–1.0 ranges of nonlinear energy transmittance for the typical pulse models, such as Gaussian, sech2, Lorentz, and asymmetric sech2 models. Our results suggest that the introduced parameter in relation to the pulse envelope in the approach may become a candidate for the parameters taking account of the NAC deviation from the pulse profiles of the probe laser.

© 2013 Optical Society of America

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2012

2010

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

2009

2008

M. Balu, L. A. Padilha, D. J. Hagan, E. W. Van Stryland, S. Yao, K. Belfield, S. Zheng, S. Barlow, and S. Marder, “Broadband Z-scan characterization using a high-spectral-irradiance, high-quality supercontinuum,” J. Opt. Soc. Am. B 25, 159–165 (2008).
[CrossRef]

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

2007

Z.-B. Liu, X.-Q. Yan, J.-G. Tian, W.-Y. Zhou, and W.-P. Zang, “Nonlinear ellipse rotation modified Z-scan measurements of third-order nonlinear susceptibility tensor,” Opt. Express 15, 13351–13359 (2007).
[CrossRef]

D. S. Correa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonca, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277, 440–445 (2007).
[CrossRef]

2006

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

2005

2004

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

1996

1995

1993

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

1990

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef]

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]

1986

T. F. Boggess, K. M. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22, 360–368 (1986).
[CrossRef]

1976

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[CrossRef]

Abdollahi, S.

Baba, M.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

Balu, M.

Bao, W.

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

Barlow, S.

Belfield, K.

Bhawalkar, J. D.

Boggess, T. F.

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

T. F. Boggess, K. M. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22, 360–368 (1986).
[CrossRef]

Bohnert, K. M.

T. F. Boggess, K. M. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22, 360–368 (1986).
[CrossRef]

Boyd, I. W.

T. F. Boggess, K. M. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22, 360–368 (1986).
[CrossRef]

Chen, J.

Chen, Q.

Chen, X.

Cohanoschi, I.

D. S. Correa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonca, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277, 440–445 (2007).
[CrossRef]

Correa, D. S.

D. S. Correa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonca, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277, 440–445 (2007).
[CrossRef]

Cui, Y.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

De Boni, L.

D. S. Correa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonca, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277, 440–445 (2007).
[CrossRef]

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef]

Ding, J.

Fan, Y.-X.

Fang, J.

Ganeev, R. A.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

Gibson, A. F.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[CrossRef]

Gu, B.

Hagan, D.

Hagan, D. J.

M. Balu, L. A. Padilha, D. J. Hagan, E. W. Van Stryland, S. Yao, K. Belfield, S. Zheng, S. Barlow, and S. Marder, “Broadband Z-scan characterization using a high-spectral-irradiance, high-quality supercontinuum,” J. Opt. Soc. Am. B 25, 159–165 (2008).
[CrossRef]

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]

Hales, J.

Hatch, C. B.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[CrossRef]

He, G. S.

Hernandez, F. E.

D. S. Correa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonca, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277, 440–445 (2007).
[CrossRef]

Hou, Y.

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

Hu, G.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Huang, W.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Ishizawa, N.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

Jiang, L.

Kuroda, H.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

Li, X.

Liu, M.

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

Liu, Y.

Liu, Z.-B.

Lu, C.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Lu, Y.

Lu, Z.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Luan, J.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Ma, Y.

Maggs, P. N. D.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[CrossRef]

Mansour, K.

T. F. Boggess, K. M. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22, 360–368 (1986).
[CrossRef]

Marder, S.

Mendonca, C. R.

D. S. Correa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonca, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277, 440–445 (2007).
[CrossRef]

Misoguti, L.

D. S. Correa, L. De Boni, L. Misoguti, I. Cohanoschi, F. E. Hernandez, and C. R. Mendonca, “Z-scan theoretical analysis for three-, four- and five-photon absorption,” Opt. Commun. 277, 440–445 (2007).
[CrossRef]

Moravvej-Farshi, M. K.

Moss, S. C.

T. F. Boggess, K. M. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22, 360–368 (1986).
[CrossRef]

Padilha, L. A.

Prasad, P. N.

Pu, J.

Qi, H.

Qian, L.

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

Qian, Y.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Quan, S.

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

Ryasnyansky, A. I.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (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]

Sakakibara, S.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

Sheik-Bahae, M.

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

Smirl, A. L.

T. F. Boggess, K. M. Bohnert, K. Mansour, S. C. Moss, I. W. Boyd, and A. L. Smirl, “Simultaneous measurement of the two-photon coefficient and free-carrier cross section above the bandgap of crystalline silicon,” IEEE J. Quantum Electron. 22, 360–368 (1986).
[CrossRef]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef]

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 1996), pp. 549–590.

Suzuki, M.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

Teng, F.

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

Tian, J.-G.

Tilley, D. R.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[CrossRef]

Turu, M.

R. A. Ganeev, A. I. Ryasnyansky, N. Ishizawa, M. Baba, M. Suzuki, M. Turu, S. Sakakibara, and H. Kuroda, “Two- and three-photon absorption in CS2,” Opt. Commun. 231, 431–436 (2004).
[CrossRef]

Tutt, L. W.

L. W. Tutt and T. F. Boggess, “A review of optical limiting mechanisms and devices using organics, fullerenes, semiconductors and other materials,” Prog. Quantum Electron. 17, 299–338 (1993).
[CrossRef]

Van Stryland, E.

Van Stryland, E. W.

M. Balu, L. A. Padilha, D. J. Hagan, E. W. Van Stryland, S. Yao, K. Belfield, S. Zheng, S. Barlow, and S. Marder, “Broadband Z-scan characterization using a high-spectral-irradiance, high-quality supercontinuum,” J. Opt. Soc. Am. B 25, 159–165 (2008).
[CrossRef]

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]

Walker, A. C.

A. F. Gibson, C. B. Hatch, P. N. D. Maggs, D. R. Tilley, and A. C. Walker, “Two-photon absorption in indium antimonide and germanium,” J. Phys. C 9, 3259–3275 (1976).
[CrossRef]

Wang, D.

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

Wang, H.-T.

Wang, J.

Wang, Y.

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

Wang, Z.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Webb, W. W.

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]

Xu, C.

Xu, X.

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

Xu, Z.

S. Quan, F. Teng, Z. Xu, L. Qian, Y. Hou, Y. Wang, and X. Xu, “Solvent and concentration effects on fluorescence emission in MEH-PPV solution” Eur. Polym. J. 42, 228–233(2006).
[CrossRef]

Yan, X.-Q.

Yao, S.

Yu, Q.

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

Yun, B.

C. Lu, W. Huang, J. Luan, Z. Lu, Y. Qian, B. Yun, G. Hu, Z. Wang, and Y. Cui, “Study on the three-photon properties of a triphenylamine derivative with three conjugated branches,” Opt. Commun 281, 4038–4041 (2008).
[CrossRef]

Zang, W.-P.

Zhang, Y.

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

Zhang, Z.

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

Zhao, P.

P. Zhao, Z. Zhang, W. Bao, Q. Yu, Y. Zhang, Z. Zhang, D. Wang, and M. Liu, “Two-photon absorption coefficient in relation to the typical pulse models of laser,” Opt. Commun. 283, 1924–1928 (2010).
[CrossRef]

Zhao, Y.

Zheng, S.

Zheng, Y.

Zhou, W.-Y.

Appl. Opt.

Eur. Polym. J.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Change curves of Eq. (1) (where β 0 L I 0 = 0.3255 ) and Eq. (15). (b) Fitting result. All for the Gaussian pulse model.

Fig. 2.
Fig. 2.

Differences between Eq. (1) and Eq. (17) as a function of the depth of the transmission minimum for the Gaussian pulse (a) and the asymmetric sech 2 pulse (b).

Fig. 3.
Fig. 3.

Difference curves of Eqs. (19), (18), and (22) for rectangular, Gaussian, sech 2 , Lorentz, and asymmetric sech 2 pulse models.

Fig. 4.
Fig. 4.

Comparing Eq. (31) with Eq. (15).

Fig. 5.
Fig. 5.

Differences between Eq. (19) and Eq. (18) as a function of the depth of the transmission minimum for the Gaussian pulse (a) and the asymmetric sech 2 pulse (b).

Fig. 6.
Fig. 6.

Rationality of the correction formulas (32) and (33) for the Gaussian pulse (a), (c) and the asymmetric sech 2 pulse (b), (d) based on Eqs. (17) and (18), where β i and γ i are substituted by β corr i and γ corr i , respectively.

Tables (2)

Tables Icon

Table 1. Pulse-Shape Factors of the Typical Pulse Models and the Relative Differences of β and γ Based on the Factors in the 0.9–1.0 Ranges of Energy Transmittance in the Fully Open-Aperture Z-scan

Tables Icon

Table 2. Fitted Parameters of Eqs. (32) and (33) for the Four Pulse Models

Equations (34)

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T = ln [ 1 + β L I 0 g ( t ) / ( 1 + z 2 / z 0 2 ) ] d t β L I 0 / ( 1 + z 2 / z 0 2 ) g ( t ) d t ,
g ( t ) = exp ( 2 t 2 / τ G 2 ) .
d I / d z = α I β I 2 γ I 3 ,
I = I ( r , z , z , t ) = I 0 f ( r , z , z ) g ( t ) ,
d d z [ I 0 f ( r , z , z ) g ( t ) ] = α I 0 f ( r , z , z ) g ( t ) β [ I 0 f ( r , z , z ) g ( t ) ] 2 γ [ I 0 f ( r , z , z ) g ( t ) ] 3 .
g 2 PA = g 2 ( t ) d t g ( t ) d t = g ( t )
g 3 PA = g 3 ( t ) d t g ( t ) d t = g 2 ( t ) .
d d z [ I 0 f ( r , z , z ) ] = α [ I 0 f ( r , z , z ) ] g 2 PA β [ I 0 f ( r , z , z ) ] 2 g 3 PA γ [ I 0 f ( r , z , z ) ] 3 .
I 0 f G ( r , z , 0 ) = I 0 [ 1 + z 2 / z 0 2 ] 1 exp { 2 r 2 w 0 2 ( 1 + z 2 / z 0 2 ) } .
E L = 0 I ( r , z , L , t ) d t 2 π r d r .
E 0 = 0 I ( r , z , 0 , t ) d t 2 π r d r .
T 2 PA = ln [ 1 + g 2 PA β L / ( 1 + z 2 / z 0 2 ) ] g 2 PA β L I 0 / ( 1 + z 2 / z 0 2 ) .
T 3 PA = ln { [ 2 L γ g 3 PA I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 + [ 1 + 2 L γ g 3 PA I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 } [ 2 L γ g 3 PA I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 .
g ( t ) = exp ( 2 t 2 / τ G 2 ) .
T 2 PA G = ln { 1 + 2 β L I 0 / [ 2 ( 1 + z 2 / z 0 2 ) ] } 2 β L I 0 / [ 2 ( 1 + z 2 / z 0 2 ) ]
T 3 PA G = ln { { 2 L γ 3 I 0 2 / [ 3 ( 1 + z 2 / z 0 2 ) 2 ] } 1 2 + { 1 + 2 L γ 3 I 0 2 / [ 3 ( 1 + z 2 / z 0 2 ) 2 ] } 1 2 } { 2 L γ 3 I 0 2 / [ 3 ( 1 + z 2 / z 0 2 ) 2 ] } 1 2 .
T 2 PA i = ln [ 1 + g 2 PA i β L I 0 / ( 1 + z 2 / z 0 2 ) ] g 2 PA i β L I 0 / ( 1 + z 2 / z 0 2 )
T 3 PA i = ln { [ g 3 PA i 2 L γ I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 + [ 1 + g 3 PA i 2 L γ I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 } [ g 3 PA i 2 L γ I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 ,
T 3 PA = ln { [ 2 L γ I 0 2 g 2 ( t ) / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 + [ 1 + 2 L γ I 0 2 g 2 ( t ) / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 } d t [ 2 L γ I 0 2 g 2 ( t ) / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 d t .
δ β = β 0 β β
δ γ = γ 0 γ γ .
δ β , max G = β 0 G β G β G = 1.53 %
T 3 PA = ln { [ 2 L γ I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 + [ 1 + 2 L γ I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 } [ 2 L γ I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 ,
T 2 PA data = T 2 PA G = ln [ 1 + g 2 PA G β G I 0 / ( 1 + z 2 / z 0 2 ) ] g 2 PA G β G I 0 / ( 1 + z 2 / z 0 2 ) = T 2 PA L = ln [ 1 + g 2 PA L β L L I 0 / ( 1 + z 2 / z 0 2 ) ] g 2 PA L β L L I 0 / ( 1 + z 2 / z 0 2 )
T 3 PA data = T 3 PA G = ln { [ g 3 PA G 2 L γ G I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 + [ 1 + g 3 PA G 2 L γ G I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 } [ g 3 PA G 2 L γ G I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 = T 3 PA L = ln { [ g 3 PA L 2 L γ L I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 + [ 1 + g 3 PA L 2 L γ L I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 } [ g 3 PA L 2 L γ L I 0 2 / ( 1 + z 2 / z 0 2 ) 2 ] 1 2 .
g 2 PA G β G = g 2 PA L β L
g 3 PA G γ G = g 3 PA L γ L .
δ β L G = β L β G β G = g 2 PA G g 2 PA L 1 13.1 %
δ γ L G = γ L γ G γ G = g 3 PA G g 3 PA L 1 17.3 % .
δ β i j = β i β j β j = g 2 PA j g 2 PA i 1 , i , j = G , s , L , a s
δ γ i j = γ i γ j γ j = g 3 PA j g 3 PA i 1 , i , j = G , s , L , a s .
T 2 PA ( z ) = 1 π q 0 ( z , 0 ) ln [ 1 + q 0 ( z , 0 ) exp ( τ 2 ) ] d τ = m = 0 [ q 0 ( z , 0 ) ] m ( m + 1 ) 3 / 2 ,
β corr i = ( 1 + a 2 PA i + b 2 PA i T 2 PA , min ) β i , i = G , s , L , a s
γ corr i = ( 1 + a 3 PA i + b 3 PA i T 3 PA , min ) γ i , i = G , s , L , a s ,

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