Abstract

By treating laser-induced optical Kerr nonlinearity as a noninstantaneous decaying process, we present the pulse-duration-dependent Z-scan analytical expressions for an arbitrary aperture and an arbitrary nonlinear magnitude. This theory has the capacity to characterize the third-order nonlinear refraction induced by a laser pulse with its temporal duration being much longer than or comparable to the recovery time of the nonlinear effect. Through Z-scan measurements at different pulse durations, the nonlinear refractive coefficient and the recovery time could be determined unambiguously and simultaneously. Furthermore, the theory can be utilized to confirm whether the measured optical Kerr nonlinearity is instantaneous or noninstantaneous with respect to the given pulse duration.

© 2009 Optical Society of America

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  1. I. Kang, T. Krauss, and F. Wise, Opt. Lett. 22, 1077 (1997).
    [CrossRef] [PubMed]
  2. S. Smolorz, F. Wise, and N. F. Borrelli, Opt. Lett. 24, 1103 (1999).
    [CrossRef]
  3. R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, 2003), Sec. 6.
    [CrossRef]
  4. R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
    [CrossRef]
  5. A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, J. Opt. Soc. Am. B 14, 796 (1997).
    [CrossRef]
  6. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
    [CrossRef]
  7. B. Gu, J. Chen, Y. Fan, J. Ding, and H. Wang, J. Opt. Soc. Am. B 22, 2651 (2005).
    [CrossRef]
  8. B. Gu, W. Ji, and X. Huang, Appl. Opt. 47, 1187 (2008).
    [CrossRef] [PubMed]
  9. L. Palfalvi and J. Hebling, Appl. Phys. B 78, 775 (2004).
    [CrossRef]
  10. Q. Y. Chen, E. H. Sargent, N. Leclerc, and A. J. Attias, Appl. Opt. 42, 7235 (2003).
    [CrossRef]
  11. B. Gu, W. Ji, X. Q. Huang, P. S. Patil, and S. M. Dharmaprakash, Opt. Express 17, 1126 (2009).
    [CrossRef] [PubMed]
  12. K. Ogusu and K. Shinkawa, Opt. Express 16, 14780 (2008).
    [CrossRef] [PubMed]
  13. K. Shinkawa and K. Ogusu, Opt. Express 16, 18230 (2008).
    [CrossRef] [PubMed]

2009

2008

2005

2004

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

L. Palfalvi and J. Hebling, Appl. Phys. B 78, 775 (2004).
[CrossRef]

2003

Q. Y. Chen, E. H. Sargent, N. Leclerc, and A. J. Attias, Appl. Opt. 42, 7235 (2003).
[CrossRef]

R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, 2003), Sec. 6.
[CrossRef]

1999

1997

1990

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Attias, A. J.

Baba, M.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Bloembergen, N.

Borrelli, N. F.

Chen, J.

Chen, Q. Y.

Dharmaprakash, S. M.

Ding, J.

Dogariu, A.

Fan, Y.

Ganeev, R. A.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Gu, B.

Hagan, D. J.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, J. Opt. Soc. Am. B 14, 796 (1997).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Hebling, J.

L. Palfalvi and J. Hebling, Appl. Phys. B 78, 775 (2004).
[CrossRef]

Huang, X.

Huang, X. Q.

Ishizawa, N.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Ji, W.

Kang, I.

Krauss, T.

Kuroda, H.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Leclerc, N.

Ogusu, K.

Palfalvi, L.

L. Palfalvi and J. Hebling, Appl. Phys. B 78, 775 (2004).
[CrossRef]

Patil, P. S.

Ryasnyansky, A. I.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Said, A. A.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, J. Opt. Soc. Am. B 14, 796 (1997).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Sakakibara, S.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Sargent, E. H.

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Shinkawa, K.

Smolorz, S.

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics, 2nd ed. (Marcel Dekker, 2003), Sec. 6.
[CrossRef]

Suzuki, M.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Turu, M.

R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B 78, 433 (2004).
[CrossRef]

Van Stryland, E. W.

A. Dogariu, T. Xia, D. J. Hagan, A. A. Said, E. W. Van Stryland, and N. Bloembergen, J. Opt. Soc. Am. B 14, 796 (1997).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Wang, H.

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Wise, F.

Xia, T.

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Figures (2)

Fig. 1
Fig. 1

Pulse-duration dependence of the refractive index variation when taking τ R = 2 ps . Δ n ( 0 , 0 ; t ) and I ( 0 , 0 ; t ) are normalized to γ I 0 and I 0 , respectively.

Fig. 2
Fig. 2

Pulse-duration dependence of the Z-scan trace for Φ 0 = 2 and s = 0.2 when taking τ R = 2 ps . The solid curve is the Z-scan trace induced by an instantaneous nonlinearity. Inset, Δ T PV as a function of τ τ R . In the inset, the circles are numerical results, while the solid curve is obtained by Eq. (13).

Equations (13)

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E ( r , z ; t ) = E 0 ω 0 ω ( z ) exp [ r 2 ω 2 ( z ) + i k r 2 2 R ( z ) ] exp [ t 2 2 τ 2 ] ,
d Δ φ ( r , z ; t ) d z = k Δ n ( r , z ; t ) ,
d I ( r , z ; t ) d z = α 0 I ( r , z ; t ) .
τ R d Δ n ( r , z ; t ) d t + Δ n ( r , z ; t ) = γ I ( r , z ; t ) .
Δ n ( r , z ; t ) = γ I ( r , z ) f ( t ) ,
f ( t ) = 1 τ R t exp ( t 2 τ 2 ) exp ( t t τ R ) d t .
E e ( r , z ; t ) = E ( r , z ; t ) e α 0 L 2 exp [ i Δ φ ( r , z ; t ) ] ,
Δ φ ( r , z ; t ) = Φ 0 1 + z 2 z 0 2 exp [ 2 r 2 ω 2 ( z ) ] f ( t ) .
T ( x ) = 1 s [ 1 u , v = 0 M Φ 0 u + v A u v ( 1 s ) λ u v cos ψ u v u ! v ! ( u + v + 1 ) ( x 2 + 1 ) u + v ] ,
λ u v = ( u + v + 1 ) ( x 2 + 1 ) [ x 2 + ( 2 u + 1 ) ( 2 v + 1 ) ] [ x 2 + ( 2 u + 1 ) 2 ] [ x 2 + ( 2 v + 1 ) 2 ] ,
ψ u v = ( u v ) { π 2 2 ( u + v + 1 ) x ( x 2 + 1 ) ln ( 1 s ) [ x 2 + ( 2 u + 1 ) 2 ] [ x 2 + ( 2 v + 1 ) 2 ] } ,
A u v = 1 π 1 2 τ + f ( t ) u + v exp ( t 2 τ 2 ) d t .
Δ T PV = 0.406 ( 1 s ) 0.25 | Φ 0 | 2 0.76 ( τ τ R ) 3 1 + 0.76 ( τ τ R ) 3 .

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