Abstract

By employing the Gaussian decomposition method, the analytical formulas of the Gaussian-beam Z-scan traces have been derived for an optically thin material exhibiting both refractive and absorptive parts of third-order nonlinearity, with Gaussian or hyperbolic secant squared laser pulses of femtosecond duration. The formulas have been verified experimentally with femtosecond-pulsed Z scans on a carbon disulfide and acetone solution of a chalcone derivative (0.95C18H17ClO4·0.05C17H14Cl2O3). An efficient yet accurate analytical technique has been demonstrated for extracting both the nonlinear refraction coefficient and the nonlinear absorption coefficient from a single closed-aperture Z-scan trace.

© 2008 Optical Society of America

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2007 (3)

2006 (3)

S. L. Ng, I. A. Razak, H. K. Fun, P. S. Patil, S. M Dharmaprakash, and V. Shettigar, “A cocrystal of 1-(4-chlorophenyl)-3-(3,4,5-trimethoxyphenyl)prop-2-en-1-one and 3-(3-chloro-4,5-dimethoxyphenyl)-1-(4-chlorophenyl)prop-2-en-1-one (0.95:0.05),” Acta Crystallogr., Sect E: Struct. Rep. Online E62, o2611-o2613 (2006).
[CrossRef]

Y. Zhu, H. I. Elim, Y. L. Foo, T. Yu, Y. Liu, W. Ji, J. Y. Lee, Z. Shen, A. T. S. Wee, J. T. L. Thong, and C. H. Sow, “Multiwalled carbon nanotubes beaded with ZnO nanoparticles for ultrafast nonlinear optical switching,” Adv. Mater. 18, 587-592 (2006).
[CrossRef]

H. Pan, W. Z. Chen, Y. P. Feng, W. Ji, and J. Lin, “Optical limiting properties of metal nanowires,” Appl. Phys. Lett. 88, 223106 (2006).
[CrossRef]

2005 (3)

2004 (3)

2000 (1)

M. Yin, H. P. Li, S. H. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single Z-scan method, ” Appl. Phys. B 70, 587-591 (2000).
[CrossRef]

1997 (2)

P. B. Chapple, J. Staromlynska, J. A. Hermann, and T. J. Mckay, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

F. E. Hernandez, A. O. Marcano, and H. Maillotte, “Sensitivity of the total beam profile distortion Z-scan for the measurement of nonlinear refraction,” Opt. Commun. 134, 529-536 (1997).
[CrossRef]

1996 (1)

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]

Acta Crystallogr., Sect E: Struct. Rep. Online (1)

S. L. Ng, I. A. Razak, H. K. Fun, P. S. Patil, S. M Dharmaprakash, and V. Shettigar, “A cocrystal of 1-(4-chlorophenyl)-3-(3,4,5-trimethoxyphenyl)prop-2-en-1-one and 3-(3-chloro-4,5-dimethoxyphenyl)-1-(4-chlorophenyl)prop-2-en-1-one (0.95:0.05),” Acta Crystallogr., Sect E: Struct. Rep. Online E62, o2611-o2613 (2006).
[CrossRef]

Adv. Mater. (1)

Y. Zhu, H. I. Elim, Y. L. Foo, T. Yu, Y. Liu, W. Ji, J. Y. Lee, Z. Shen, A. T. S. Wee, J. T. L. Thong, and C. H. Sow, “Multiwalled carbon nanotubes beaded with ZnO nanoparticles for ultrafast nonlinear optical switching,” Adv. Mater. 18, 587-592 (2006).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (2)

M. Yin, H. P. Li, S. H. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single Z-scan method, ” Appl. Phys. B 70, 587-591 (2000).
[CrossRef]

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]

Appl. Phys. Lett. (1)

H. Pan, W. Z. Chen, Y. P. Feng, W. Ji, and J. Lin, “Optical limiting properties of metal nanowires,” Appl. Phys. Lett. 88, 223106 (2006).
[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. Nonlinear Opt. Phys. Mater. (1)

P. B. Chapple, J. Staromlynska, J. A. Hermann, and T. J. Mckay, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 251-293 (1997).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (2)

B. Gu, X. Q. Huang, S. Q. Tan, and H. T. Wang, “A precise data processing method for extracting χ(3) from Z-scan technique,” Opt. Commun. 277, 209-213 (2007).
[CrossRef]

F. E. Hernandez, A. O. Marcano, and H. Maillotte, “Sensitivity of the total beam profile distortion Z-scan for the measurement of nonlinear refraction,” Opt. Commun. 134, 529-536 (1997).
[CrossRef]

Opt. Express (2)

Other (2)

R. L. Sutherland with contributions by D. G. McLean and S. Kikpatrick, Handbook of Nonlinear Optics, 2nd ed.(Dekker, 2003).
[CrossRef]

Mathematica 4.0, Wolfram Research, Inc., 1999.

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Figures (3)

Fig. 1
Fig. 1

Closed- and open-aperture Z scans for a cw laser (solid curve), a Gaussian shape pulsed laser (dashed curve), and a hyperbolic secant squared pulsed laser (dotted curve) when q 0 = 0.2 π , Δ ϕ 0 = π , s = 0.2 , and M = 11 in Eq. (16).

Fig. 2
Fig. 2

Z-scan traces at I 0 = 68 GW / cm 2 for C S 2 . The squares and circles are the closed- and open-aperture Z scans, respectively, while the solid and dashed curves are the fits to Eqs. (16, 24), respectively.

Fig. 3
Fig. 3

Z-scan traces at I 0 = 68 GW / cm 2 for (a) acetone and (b) an acetone solution of 3TC. The squares and circles are the closed- and open-aperture Z scans, respectively, while the solid and dashed curves are the fits to Eqs. (16, 24), respectively.

Equations (28)

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E ( r , z , t ) = E 0 ( t ) ω 0 ω ( z ) exp [ r 2 ω 2 ( z ) i k r 2 2 R ( z ) ] ,
d Δ ϕ ( r , z , t ) d z = k γ I ( r , z , t ) ,
d I ( r , z , t ) d z = α I ( r , z , t ) β I 2 ( r , z , t ) ,
E e ( r , z , t ) = E ( r , z , t ) exp ( α L / 2 ) [ 1 + q ( r , z , t ) ] ( i Δ ϕ 0 / q 0 1 / 2 ) ,
E e ( r , z , t ) = E ( r , z , t ) exp ( α L / 2 ) m = 0 q ( r , z , t ) m m ! n = 1 m ( i Δ ϕ 0 q 0 1 2 n + 1 ) .
E a ( r , z , t ) = E ( r = 0 , z , t ) exp ( α L / 2 ) m = 0 f m ω m 0 ω m exp ( r 2 ω m 2 i k r 2 2 R m + i θ m ) .
g = 1 + d R ( z ) ,
ω m 0 2 = ω 2 ( z ) 2 m + 1 ,
d m = 1 2 k ω m 0 2 ,
ω m 2 = ω m 0 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 ) ,
f m = 1 m ! ( i Δ ϕ 0 h ( t ) 1 + z 2 / z 0 2 ) m n = 1 m ( 1 + i ( n 1 2 ) q 0 Δ ϕ 0 ) with f 0 = 1 .
P T ( z , t ) = c ε 0 n 0 π 0 r a | E a ( r , z , t ) | 2 r d r ,
T ( z , s ) = P T ( z , t ) d t s P i ( t ) d t ,
T ( x , s ) = m , m = 0 M g m ( x , Δ ϕ 0 , q 0 ) g m * ( x , Δ ϕ 0 , q 0 ) A m m S m m ( x , s ) ,
g m ( x , Δ ϕ 0 , q 0 ) = i m Δ ϕ 0 m ( x + i ) m ! ( x 2 + 1 ) m [ x + i ( 2 m + 1 ) ] n = 1 m [ 1 + i ( n 1 2 ) q 0 Δ ϕ 0 ] ,
A m m = + h ( t ) m + m + 1 d t + h ( t ) d t ,
S m m ( x , s ) = 1 exp [ B m m ( x ) ln ( 1 s ) ] B m m ( x ) s ,
B m m ( x ) = ( m + m + 1 ) ( 1 + x 2 ) [ x + i ( 2 m + 1 ) ] [ x i ( 2 m + 1 ) ] ,
A m m = 1 ( m + m + 1 ) 1 / 2 .
A m m = 2 m + m ( m + m ) ! ( 2 m + 2 m + 1 ) !! .
| Δ ϕ 0 M M ! n = 1 M [ 1 + i ( n 1 2 ) q 0 Δ ϕ 0 ] | 1 2 .
T ( x , s = 1 ) = m = 0 ( q 0 ) m ( x 2 + 1 ) m ( m + 1 ) 3 / 2 ,
T ( x , s = 1 ) = m = 0 ( q 0 ) m 2 m m ! ( x 2 + 1 ) m ( m + 1 ) ( 2 m + 1 ) !! .
T ( x , s 0 ) = 1 + 4 x Δ ϕ 0 ( x 2 + 3 ) q 0 ( x 2 + 1 ) ( x 2 + 9 ) + 4 Δ ϕ 0 2 ( 3 x 2 5 ) + q 0 2 ( x 4 + 17 x 2 + 40 ) 8 Δ ϕ 0 q 0 x ( x 2 + 9 ) ( x 2 + 1 ) 2 ( x 2 + 9 ) ( x 2 + 25 ) .
T ( x , s 0 ) = 1 + 1 2 4 x Δ ϕ 0 ( x 2 + 3 ) q 0 ( x 2 + 1 ) ( x 2 + 9 ) + 1 3 4 Δ ϕ 0 2 ( 3 x 2 5 ) + q 0 2 ( x 4 + 17 x 2 + 40 ) 8 Δ ϕ 0 q 0 x ( x 2 + 9 ) ( x 2 + 1 ) 2 ( x 2 + 9 ) ( x 2 + 25 ) .
T ( x , s 0 ) = 1 + 2 3 4 x Δ ϕ 0 ( x 2 + 3 ) q 0 ( x 2 + 1 ) ( x 2 + 9 ) + 8 15 4 Δ ϕ 0 2 ( 3 x 2 5 ) + q 0 2 ( x 4 + 17 x 2 + 40 ) 8 Δ ϕ 0 q 0 x ( x 2 + 9 ) ( x 2 + 1 ) 2 ( x 2 + 9 ) ( x 2 + 25 ) .

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