Abstract

We study the dependence of optical nonlinearities in As2Se3 glass on the incident pulse-width using the Z-scan technique. In this work, 11.5-ns optical pulses from a Nd:YAG laser operating at 1.064 µm were compressed up to 1.6 ns by using stimulated Brillouin scattering in heavy fluorocarbon liquid, where the duration of the compressed pulses depends on the pump energy. Moreover picosecond optical pulses with a 1.053-µm wavelength were utilized for the Z-scan experiments. To investigate the pulse-width dependence of the optical nonlinearities leads to the separation of an ultrafast Kerr nonlinearity and a slow (cumulative) nonlinearity such as a thermal nonlinearity because the magnitude of the cumulative nonlinearity should be proportional to the pulse duration. The experimental results clearly show that both the nonlinear refractive index and the nonlinear absorption coefficient increase linearly with pulse width. The origin of such pulse-width dependence is presumably attributed to photostructural changes inherent in chalcogenide glasses, which are induced not by nonlinear absorption such as two-photon absorption or two-step absorption, but by linear absorption in the weak-absorption region.

© 2008 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]

2008 (1)

2007 (1)

2006 (1)

2005 (3)

S.-Y. Kim, M.-J. Kang, and S.-Y. Choi, “Nonlinear optical properties of As2Se3 thin films,” Thin Solid Films 493, 207–211 (2005).
[Crossref]

K. S. Abedin, “Observation of strong stimulated Brillouin scattering in single-mode As2Se3 chalcogenide fiber,” Opt. Express 13, 10266–10271 (2005).
[Crossref] [PubMed]

K. Tanaka, “Optical nonlinearity in photonic glasses,” J. Mater. Sci.: Mater. Electron. 16, 633–643 (2005).
[Crossref]

2004 (4)

2003 (3)

A. Zakery, Y. Ruan, A. V. Rode, M. Samoc, and B. Luther-Davies, “Low-loss waveguides in ultrafast laserdeposited As2S3 chalcogenide films,” J. Opt. Soc. Am. 20, 1844–1852 (2003).
[Crossref]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[Crossref]

2002 (2)

J. M. Harbold, F. Ö Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal, “Highly nonlinear As-S-Se glasses for all-optical switching,” Opt. Lett. 27, 119–121 (2002).
[Crossref]

K. Tanaka, “Wrong bond in glasses: A comparative study on oxides and chalcogenides,” J. Optoelectron. Adv. Mater. 4, 505–512 (2002).

2000 (2)

F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[Crossref]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spälter, R. E. Slusher, S. -W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
[Crossref]

1999 (1)

1997 (1)

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

1996 (1)

T. Wagner and S. O. Kasap, “Glass transformation, heat capacity and structure of AsxSe1-x glasses studied by modulated temperature differential scanning calorimetry experiments,” Philos. Mag. B 74, 667–680 (1996).
[Crossref]

1995 (1)

K. Shimakawa, A. Kolobov, and S. R. Elliott, “Photoinduced effects and metastability in amorphous semiconductors and insulators,” Adv. Phys. 44, 475–588 (1995).
[Crossref]

1992 (1)

1991 (1)

G. Pfeiffer, M. A. Paesler, and S. C. Agarwal, “Reversible photodarkening of amorphous arsenic chalcogens,” J. Non-Cryst. Solids 130, 111–143 (1991).
[Crossref]

1990 (1)

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

1980 (1)

K. Tanaka, “Reversible photostructural change: Mechanisms, properties and applications,” J. Non-Cryst. Solids 35&36, 1023–1034 (1980).
[Crossref]

Abedin, K. S.

Agarwal, S. C.

G. Pfeiffer, M. A. Paesler, and S. C. Agarwal, “Reversible photodarkening of amorphous arsenic chalcogens,” J. Non-Cryst. Solids 130, 111–143 (1991).
[Crossref]

Aggarwal, I. D.

Baker, N. J.

Barthélémy, A.

F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[Crossref]

Belykh, A. V.

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

Boudebs, G.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

Cheong, S. -W.

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

Choi, D. Y.

Choi, S.-Y.

S.-Y. Kim, M.-J. Kang, and S.-Y. Choi, “Nonlinear optical properties of As2Se3 thin films,” Thin Solid Films 493, 207–211 (2005).
[Crossref]

Couderc, V.

F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[Crossref]

Efimov, O. M.

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

Eggleton, B. J.

Elliott, S. R.

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[Crossref]

K. Shimakawa, A. Kolobov, and S. R. Elliott, “Photoinduced effects and metastability in amorphous semiconductors and insulators,” Adv. Phys. 44, 475–588 (1995).
[Crossref]

Finsterbusch, K.

Fu, L.

Glebov, L. B.

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

Hagan, D. J.

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. 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]

Harbold, J. M.

Hodelin, J.

Hwang, H. Y.

Ilday, F. Ö

Kang, M.-J.

S.-Y. Kim, M.-J. Kang, and S.-Y. Choi, “Nonlinear optical properties of As2Se3 thin films,” Thin Solid Films 493, 207–211 (2005).
[Crossref]

Kasap, S. O.

T. Wagner and S. O. Kasap, “Glass transformation, heat capacity and structure of AsxSe1-x glasses studied by modulated temperature differential scanning calorimetry experiments,” Philos. Mag. B 74, 667–680 (1996).
[Crossref]

Katsufuji, T.

Kim, S. G.

Kim, S.-Y.

S.-Y. Kim, M.-J. Kang, and S.-Y. Choi, “Nonlinear optical properties of As2Se3 thin films,” Thin Solid Films 493, 207–211 (2005).
[Crossref]

Kitao, M.

Kolobov, A.

K. Shimakawa, A. Kolobov, and S. R. Elliott, “Photoinduced effects and metastability in amorphous semiconductors and insulators,” Adv. Phys. 44, 475–588 (1995).
[Crossref]

Kwak, C. H.

Lamont, M. R. E.

Leblond, H.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

Lee, Y. L.

Lenz, G.

Li, H.

Lines, M. E.

Luther-Davies, B.

Madden, S.

Maeda, S.

Matveev, Yu. A.

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

Mekryukov, A. M.

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

Mikhailov, M. D.

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

Minakata, M.

Moss, D. J.

Nguyen, H. C.

Nguyen, V. Q.

Nishio, H.

Ogusu, K.

Paesler, M. A.

G. Pfeiffer, M. A. Paesler, and S. C. Agarwal, “Reversible photodarkening of amorphous arsenic chalcogens,” J. Non-Cryst. Solids 130, 111–143 (1991).
[Crossref]

Pfeiffer, G.

G. Pfeiffer, M. A. Paesler, and S. C. Agarwal, “Reversible photodarkening of amorphous arsenic chalcogens,” J. Non-Cryst. Solids 130, 111–143 (1991).
[Crossref]

Quemard, C.

F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[Crossref]

Richardson, K.

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

Rode, A. V.

A. Zakery, Y. Ruan, A. V. Rode, M. Samoc, and B. Luther-Davies, “Low-loss waveguides in ultrafast laserdeposited As2S3 chalcogenide films,” J. Opt. Soc. Am. 20, 1844–1852 (2003).
[Crossref]

Ruan, Y.

A. Zakery, Y. Ruan, A. V. Rode, M. Samoc, and B. Luther-Davies, “Low-loss waveguides in ultrafast laserdeposited As2S3 chalcogenide films,” J. Opt. Soc. Am. 20, 1844–1852 (2003).
[Crossref]

Said, A. A.

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. 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]

Samoc, M.

A. Zakery, Y. Ruan, A. V. Rode, M. Samoc, and B. Luther-Davies, “Low-loss waveguides in ultrafast laserdeposited As2S3 chalcogenide films,” J. Opt. Soc. Am. 20, 1844–1852 (2003).
[Crossref]

Sanchez, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

Sanghera, J.

Sanghera, J. S.

Shaw, L. B.

Sheik-Bahae, M.

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. 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]

Shimakawa, K.

K. Shimakawa, A. Kolobov, and S. R. Elliott, “Photoinduced effects and metastability in amorphous semiconductors and insulators,” Adv. Phys. 44, 475–588 (1995).
[Crossref]

Shinkawa, K.

Slusher, R. E.

Smektala, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[Crossref]

Spälter, S.

Suzuki, K.

Ta’eed, V. G.

Tanaka, K.

K. Tanaka, “Optical nonlinearity in photonic glasses,” J. Mater. Sci.: Mater. Electron. 16, 633–643 (2005).
[Crossref]

K. Tanaka, “Midgap photon effects in As2S3 glass,” Philos. Mag. Lett. 84, 601–606 (2004).
[Crossref]

K. Tanaka, “Wrong bond in glasses: A comparative study on oxides and chalcogenides,” J. Optoelectron. Adv. Mater. 4, 505–512 (2002).

K. Tanaka, “Reversible photostructural change: Mechanisms, properties and applications,” J. Non-Cryst. Solids 35&36, 1023–1034 (1980).
[Crossref]

Troles, J.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

Van Stryland, E. W.

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. 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]

Wagner, T.

T. Wagner and S. O. Kasap, “Glass transformation, heat capacity and structure of AsxSe1-x glasses studied by modulated temperature differential scanning calorimetry experiments,” Philos. Mag. B 74, 667–680 (1996).
[Crossref]

Wang, J.

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]

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]

Wise, F. W.

Yamasaki, J.

Young, J.

Zakery, A.

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[Crossref]

A. Zakery, Y. Ruan, A. V. Rode, M. Samoc, and B. Luther-Davies, “Low-loss waveguides in ultrafast laserdeposited As2S3 chalcogenide films,” J. Opt. Soc. Am. 20, 1844–1852 (2003).
[Crossref]

Zimmermann, J.

Adv. Phys. (1)

K. Shimakawa, A. Kolobov, and S. R. Elliott, “Photoinduced effects and metastability in amorphous semiconductors and insulators,” Adv. Phys. 44, 475–588 (1995).
[Crossref]

IEEE J. Quantum Electron. (1)

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

J. Mater. Sci.: Mater. Electron. (1)

K. Tanaka, “Optical nonlinearity in photonic glasses,” J. Mater. Sci.: Mater. Electron. 16, 633–643 (2005).
[Crossref]

J. Non-Cryst. Solids (5)

F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[Crossref]

K. Tanaka, “Reversible photostructural change: Mechanisms, properties and applications,” J. Non-Cryst. Solids 35&36, 1023–1034 (1980).
[Crossref]

G. Pfeiffer, M. A. Paesler, and S. C. Agarwal, “Reversible photodarkening of amorphous arsenic chalcogens,” J. Non-Cryst. Solids 130, 111–143 (1991).
[Crossref]

A. V. Belykh, O. M. Efimov, L. B. Glebov, Yu. A. Matveev, A. M. Mekryukov, M. D. Mikhailov, and K. Richardson, “Photo-structural transformation of chalcogenide glasses under non-linear absorption of laser radiation,” J. Non-Cryst. Solids 213&214, 330–335 (1997).
[Crossref]

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[Crossref]

J. Opt. Soc. Am. (1)

A. Zakery, Y. Ruan, A. V. Rode, M. Samoc, and B. Luther-Davies, “Low-loss waveguides in ultrafast laserdeposited As2S3 chalcogenide films,” J. Opt. Soc. Am. 20, 1844–1852 (2003).
[Crossref]

J. Opt. Soc. Am. B (4)

J. Optoelectron. Adv. Mater. (1)

K. Tanaka, “Wrong bond in glasses: A comparative study on oxides and chalcogenides,” J. Optoelectron. Adv. Mater. 4, 505–512 (2002).

Opt. Commun. (1)

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Philos. Mag. B (1)

T. Wagner and S. O. Kasap, “Glass transformation, heat capacity and structure of AsxSe1-x glasses studied by modulated temperature differential scanning calorimetry experiments,” Philos. Mag. B 74, 667–680 (1996).
[Crossref]

Philos. Mag. Lett. (1)

K. Tanaka, “Midgap photon effects in As2S3 glass,” Philos. Mag. Lett. 84, 601–606 (2004).
[Crossref]

Thin Solid Films (1)

S.-Y. Kim, M.-J. Kang, and S.-Y. Choi, “Nonlinear optical properties of As2Se3 thin films,” Thin Solid Films 493, 207–211 (2005).
[Crossref]

Other (1)

VITRON Spezialwerkstoffe, Data sheet of Infrared Chalcogenide Glasses, http://www.vitron.de/english/optical_glasses.php.

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Figures (5)

Fig. 1.
Fig. 1.

Typical normalized open- and closed-aperture Z-scan data of As2Se3 glass at λ=1.064 µm with an incident energy of 15 µJ. The incident pulse-width is (a) t FWHM=11.5 ns and (b) t FWHM=1.6 ns. The solid lines are the best fits based on Eqs. (15) and (12) in [12].

Fig. 2.
Fig. 2.

Dependence of (a) the effective nonlinear refractive index n 2eff and (b) the effective nonlinear absorption coefficient β eff on the incident pulse energy for the pulse width t FWHM=11.5 ns at 1.064 µm. A pulse energy of 10 µJ corresponds to the on-axis peak intensity (inside the sample) I0 =26.7 MW/cm2.

Fig. 3.
Fig. 3.

Dependence of (a) the effective nonlinear refractive index n 2eff and (b) the effective nonlinear absorption coefficient β eff on the incident pulse energy for the pulse width t FWHM=1.6 ns at 1.064 µm. A pulse energy of 10 µJ corresponds to the on-axis peak intensity (inside the sample) I0 =192 MW/cm2.

Fig. 4.
Fig. 4.

Dependence of (a) the effective nonlinear refractive index n 2eff and (b) the effective nonlinear absorption coefficient β eff on the incident optical intensity for the pulse width t FWHM=60 ps at 1.052 µm. The results at t FWHM=300 ps in [8] are also reproduced.

Fig. 5.
Fig. 5.

Dependence of (a) the effective nonlinear refractive index n 2eff and (b) the effective nonlinear absorption coefficient β eff on the incident pulse-width t FWHM. The data for t FWHM=1.6 -11.5 ns were obtained at an incident pulse energy of 15 µJ. The data at 60 ps are shown with a white circle.

Equations (12)

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d Δ T d t + Δ T τ = α I ( t ) ρ C ,
I ( t ) = I 0 exp ( t 2 t 0 2 ) ,
Δ T ( t ) = α ρ C π t 0 I 0 2 [ 1 + erf ( t t 0 ) ] ,
Δ n T ( t ) = d n d T Δ T ( t ) ,
Δ Φ 0 = k 0 Δ n ( t ) + Δ n T ( t ) L eff = k 0 ( γ I 0 2 + d n d T α 2 ρ C π t 0 I 0 ) L eff k 0 n 2 eff I 0 2 L eff ,
n 2 e ff = γ + d n d T α ρ C π t 0 2 ,
n 2 eff = γ + d n d T α 2 ρ C π t FWHM 2 ln 2 ,
d Δ ϕ d z = k 0 Δ n = k 0 ( γ I + σ r N ) ,
d I d z = ( α + β I + σ ab N ) I ,
N ( t ) = t α I ( t ) ω dt ,
n 2 eff = γ + σ r ( α 2 ω π t FWHM 2 ln 2 ) ,
β eff = β + σ ab ( σ 2 ω π t FWHM 2 ln 2 ) ,

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