Abstract

Using the Gaussian decomposition (GD) method, we studied the characteristics of a Z scan for a thin nonlinear medium with a large nonlinear phase shift induced by a pulsed laser. It has been verified that the GD method is still valid for analyses of Z-scan measurements with a large nonlinear phase shift and is better than some others, i.e., Fresnel–Kirchhoff diffraction and the aberration-free approximation model. By comparing the peak-to-valley configuration of Z-scan curves for a large nonlinear phase shift induced by a pulsed laser with that by a cw laser, we found that some peak-to-valley features of Z-scan curves appear as the aperture size or the light intensity increases in the case of a large nonlinear phase shift. Meanwhile, we carried out the Z-scan experiments of pure CS2 by a picosecond pulsed laser to verify the theoretical calculations in the case of a large nonlinear phase shift. The experimental results agree well with the theoretical calculations.

© 2005 Optical Society of America

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References

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  1. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  13. R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
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2004 (1)

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

2003 (2)

1999 (1)

1998 (2)

J. A. Hermann, T. McKay, and R. G. McDuff, "Z-scan with arbitrary aperture transmittance: the strongly nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

R. E. Samad and N. D. Vieira, Jr., "Analytical description of Z-scan on-axis intensity based on the Huygens Fresnel principle," J. Opt. Soc. Am. B 15, 2742-2746 (1998).
[CrossRef]

1997 (1)

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

1996 (1)

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)

1979 (1)

Baba, M.

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

Bara, S.

Byun, J. S.

Chapple, P. B.

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

Ganeev, R. A.

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

Gaskill, J. D.

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

Hagan, D. J.

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]

Hermann, J. A.

J. A. Hermann, T. McKay, and R. G. McDuff, "Z-scan with arbitrary aperture transmittance: the strongly nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

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

Hou, X.

Hwang, W.

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

Kim, J. H.

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

Kuroda, H.

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

Kwak, C. H.

C. H. Kwak, Y. L. Lee, and S. G. Kym, "Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As2S3 thin film," J. Opt. Soc. Am. B 16, 600-604 (1999).
[CrossRef]

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

Kym, S. G.

Lee, J. H.

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

Lee, S. J.

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

Lee, Y. L.

C. H. Kwak, Y. L. Lee, and S. G. Kym, "Analysis of asymmetric Z-scan measurement for large optical nonlinearities in an amorphous As2S3 thin film," J. Opt. Soc. Am. B 16, 600-604 (1999).
[CrossRef]

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

McDuff, R. G.

J. A. Hermann, T. McKay, and R. G. McDuff, "Z-scan with arbitrary aperture transmittance: the strongly nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

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

McKay, T.

J. A. Hermann, T. McKay, and R. G. McDuff, "Z-scan with arbitrary aperture transmittance: the strongly nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

McKay, T. J.

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

Michinel, H.

Miller, D. A.

Paz-Alonso, M. J.

Ren, L.

Rhee, B. K.

Ryasnyansky, A. I.

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

Said, A. A.

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] [PubMed]

Samad, R. E.

Sheik-Bahae, M.

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] [PubMed]

Smith, S. D.

Staromlynska, J.

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

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 1966).

Suzuki, M.

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

Turu, M.

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

Van Stryland, E. W.

Vieira, N. D.

Weaire, D.

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]

Wherrett, B. S.

Woo, S. Y.

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

Yao, B.

Appl. Phys. B (1)

R. A. Ganeev, M. Baba, A. I. Ryasnyansky, M. Suzuki, M. Turu, and H. Kuroda, "Nonlinear refraction in CS2," Appl. Phys. B 78, 433-438 (2004).
[CrossRef]

IEEE J. Quantum Electron. (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]

J. Nonlinear Opt. Phys. Mater. (1)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "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. (1)

J. A. Hermann, T. McKay, and R. G. McDuff, "Z-scan with arbitrary aperture transmittance: the strongly nonlinear regime," Opt. Commun. 154, 225-233 (1998).
[CrossRef]

Opt. Lett. (2)

Other (3)

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, 1966).

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

S. J. Lee, Y. L. Lee, S. Y. Woo, W. Hwang, J. H. Lee, J. H. Kim, and C. H. Kwak, "Simple method for determining Gaussian beam waist using lens Z-scan," in Conference on Lasers and Electro-Optics, Vol. 88 of 2003 OSA Technical Digest Series (Optical Society of America, 2003), p. 246.

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Figures (9)

Fig. 1
Fig. 1

Calculated Z-scan curves for a different item number of summation m. The nonlinear phase shift Δ φ 0 = 3 π .

Fig. 2
Fig. 2

Function Y m ( z = 0 , Δ φ 0 ) as a function of the item number of summation m. The nonlinear phase shift Δ φ 0 = 3 π .

Fig. 3
Fig. 3

Z-scan curves for a laser and a pulsed laser in the case of a pinhole aperture.

Fig. 4
Fig. 4

Calculated Z-scan curves by four models, where the nonlinear phase shift Δ φ 0 = 3 π .

Fig. 5
Fig. 5

Z-scan curves with a pulsed laser beam for different Δ φ 0 : 0.5 π , π , 2 π , 3 π , and 4 π . The dimensionless aperture radius Y a = 0.5 .

Fig. 6
Fig. 6

Z-scan curves with a pulsed laser beam for different Y a : 0.5, 0.8, 1, 2, and 3. The nonlinear phase shift Δ φ 0 = 3 π .

Fig. 7
Fig. 7

Radial normalized intensity distribution on the aperture plane for a sample placed at the peak, valley, and linear zone of the Z-scan curves.

Fig. 8
Fig. 8

Experimental and calculated Z-scan curves for different Y a : 0.094, 0.438, 0.656, and 1.000. The nonlinear phase shift Δ φ 0 = 1.8 π .

Fig. 9
Fig. 9

Experimental and calculated Z-scan curves for different Δ φ 0 : 0.2 π , 0.7 π , 1.4 π , and 3.0 π . The dimensionless aperture radius Y a = 0.250 .

Equations (20)

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E a ( r , z , t ) = E ( z , r = 0 , t ) exp ( α L 2 ) m = 0 [ i Δ φ 0 ( z , t ) ] m m ! w m 0 w m exp ( r 2 w m 2 i k r 2 2 R m + i θ m ) ,
w m 0 2 = w 2 ( z ) ( 2 m + 1 ) ,
d m = 1 2 k w m 0 2 ,
R m = d ( 1 g g 2 + d 2 d m 2 ) 1 ,
θ m = tan 1 ( d d m g ) ,
w m 2 = w m 0 2 ( g 2 + d 2 d m 2 ) ,
T ( r a , z ) = 0 d t 0 r a E a [ r , z , t , Δ φ 0 ( t ) ] 2 r d r 0 d t 0 r a E a ( r , z , t ) 2 r d r ,
T cw ( z , Δ φ 0 ) = P ( z , Δ φ 0 ) 2 = m = 0 C m ( z ) [ Δ φ 0 ( z ) ] m 2 ,
C m ( z ) = ( i ) m m ! w m 0 w 0 w m w 00 exp [ i ( θ m θ 0 m π 2 ) ] .
T cw ( z , Δ φ 0 ) = m = 0 K m ( z ) Y m ( z , Δ φ 0 ) ,
Y m ( z , Δ φ 0 ) = [ Δ φ 0 ( 1 + z 2 z 0 2 ) ] m m ! .
T ( x ) = 1 + 4 x Δ φ 0 ( x 2 + 1 ) ( x 2 + 9 ) ,
T cw = 1 S m = 0 n = 0 a m n [ cos ( c m n π 2 ) exp ( b m n Y a 2 ) cos ( c m n π 2 d m n Y a 2 ) ] ,
a m n = [ Δ φ 0 ( 1 + x 2 ) ] m + n [ 1 ( m + n + 1 ) m ! n ! ] ,
b m n = ( 1 + x 2 ) [ 2 m + 1 x 2 + ( 2 m + 1 ) 2 + 2 n + 1 x 2 + ( 2 n + 1 ) 2 ] ,
c m n = m n ,
d m n = 4 x ( x 2 + 1 ) ( m n ) ( m + n + 1 ) [ x 2 + ( 2 m + 1 ) 2 ] [ x 2 + ( 2 n + 1 ) 2 ] ,
S = 1 exp ( 2 Y a 2 ) ,
x = z z 0 .
T cw = 1 + 1 S m = 0 k n = 0 k m n < m 2 a m n [ cos ( c m n π 2 ) exp ( b m n y a 2 ) cos ( c m n π 2 d m n Y a 2 ) ] + 1 S m = 1 k a m n [ 1 exp ( b m n Y a 2 ) ] .

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