Abstract

Using a Gaussian decomposition method and a distributed lens model, we have analyzed the Z-scan measurements of thick optical nonlinear media with nonlinear refraction and nonlinear absorption. With the introduction of correction functions that include saturation of nonlinear refraction and coupling between nonlinear refraction and nonlinear absorption, analytic solutions to closed-aperture and open-aperture Z-scan characteristics were obtained. Detailed comparisons of the analytic solutions with numerical solutions were made, and the two kinds of solution coincided well, no matter how much nonlinear refraction or nonlinear absorption there was.

© 2004 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 measurement,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  2. 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]
  3. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
    [CrossRef]
  4. J.-G. Tian, W.-P. Zang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media by the variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).
  5. J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993).
    [CrossRef]
  6. J.-G. Tian, W.-P. Zang, C.-Z. Zhang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media,” Appl. Opt. 34, 4331–4336 (1995).
    [CrossRef] [PubMed]
  7. J. A. Hermann, “Nonlinear optical absorption in thick media,” J. Opt. Soc. Am. B 14, 814–823 (1997).
    [CrossRef]
  8. P. B. Chapple, J. A. Hermann, and R. G. McDuff, “Power saturation effects in thick single-element optical limiters,” Opt. Quantum Electron. 31, 555–569 (1999).
    [CrossRef]
  9. P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and the thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
    [CrossRef]
  10. S. Hughes, J. M. Burzer, G. Spruce, and B. S. Wherrett, “Fast Fourier transform techniques for efficient simulation of Z-scan measurements,” J. Opt. Soc. Am. B 12, 1888–1893 (1995).
    [CrossRef]
  11. D. I. Kvosh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
    [CrossRef]
  12. L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering (Wiley, New York, 1982), Chap. 4.

1999 (2)

P. B. Chapple, J. A. Hermann, and R. G. McDuff, “Power saturation effects in thick single-element optical limiters,” Opt. Quantum Electron. 31, 555–569 (1999).
[CrossRef]

D. I. Kvosh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[CrossRef]

1997 (1)

1995 (2)

1994 (2)

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and the thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

J.-G. Tian, W.-P. Zang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media by the variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

1993 (1)

1991 (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

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)

Burzer, J. M.

Chapple, P. B.

P. B. Chapple, J. A. Hermann, and R. G. McDuff, “Power saturation effects in thick single-element optical limiters,” Opt. Quantum Electron. 31, 555–569 (1999).
[CrossRef]

P. B. Chapple, J. Staromlynska, and R. G. McDuff, “Z-scan studies in the thin- and the thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
[CrossRef]

Hagan, D. J.

D. I. Kvosh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

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.

Hughes, S.

Kvosh, D. I.

McDuff, R. G.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

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 measurement,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

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 measurement,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Spruce, G.

Staromlynska, J.

Tian, J.-G.

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media,” Appl. Opt. 34, 4331–4336 (1995).
[CrossRef] [PubMed]

J.-G. Tian, W.-P. Zang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media by the variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

Van Stryland, E. W.

D. I. Kvosh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

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 measurement,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

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]

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Wherrett, B. S.

Yang, S.

Zang, W.-P.

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media,” Appl. Opt. 34, 4331–4336 (1995).
[CrossRef] [PubMed]

J.-G. Tian, W.-P. Zang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media by the variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

Zhang, C.-Z.

Zhang, G.-Y.

J.-G. Tian, W.-P. Zang, C.-Z. Zhang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media,” Appl. Opt. 34, 4331–4336 (1995).
[CrossRef] [PubMed]

J.-G. Tian, W.-P. Zang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media by the variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

Acta Phys. Sin. (1)

J.-G. Tian, W.-P. Zang, and G.-Y. Zhang, “Analysis of beam propagation through thick nonlinear media by the variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).

Appl. Opt. (2)

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. Opt. Soc. Am. B (4)

Opt. Eng. (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

P. B. Chapple, J. A. Hermann, and R. G. McDuff, “Power saturation effects in thick single-element optical limiters,” Opt. Quantum Electron. 31, 555–569 (1999).
[CrossRef]

Other (1)

L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering (Wiley, New York, 1982), Chap. 4.

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Figures (5)

Fig. 1
Fig. 1

Closed-aperture Z-scan curves for a thick medium with nonlinear phase shift ΔφR=0.5 and medium length l=6. a, Numerical simulation; b, our analytic solution; c, analytic solution given in Ref. 6; d, analytic solution given in Ref. 5.

Fig. 2
Fig. 2

Peak–valley normalized transmittance difference ΔTp-v as a function of nonlinear phase shift ΔφR. a, Numerical simulation; b, our analytic solution; c, analytic solution given in Ref. 6; d, analytic solution given in Ref. 5. Nonlinear phase shift ΔφR ranges from 0 to 1.0 and has medium length l=6.

Fig. 3
Fig. 3

Open-aperture Z-scan curves for several values of nonlinear refraction obtained by numerical simulation and our analytic solution. a, ΔφR=-0.5; b, ΔφR=0; c, ΔφR=0.5. These above calculations, QR=0.5 and l=6 were used. The solution given in Ref. 8 is represented by curve b.

Fig. 4
Fig. 4

Change in absorption dip Tmin as a function of nonlinear phase shift. a, Numerical simulation; b, our analytic solution; c, analytic solution given in Ref. 8, where ΔφR ranges from 0 to 1.0. QR=0.5 and l=6.

Fig. 5
Fig. 5

Closed-aperture Z-scan curves of a thick medium with nonlinear absorption and nonlinear refraction. a, l=2; b, l=6; c, l=10. In the above calculations, ΔφR=0.45 and QR=0.15 were used.

Equations (15)

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T=1+4Δφ0(t)x(x2+1)(x2+9)+O[Δφ(t)02],
T=i1n1+4Δφ0ixi(xi2+1)(xi2+9)expi=1n 4Δφ0ixi(xi2+1)(xi2+9),
T=[(x+l)2+1]×(x2+9)[(x+l)2+9]×(x2+1)ΔφR(t)/4,
Cφ(t)ΔφR(t)+tanh(l/3)×[ΔφR(t)]2/4+[ΔφR(t)]3/16.
T=[(x+l)2+1]×[(x2+9)][(x+l)2+9]×[(x2+1)]Cφ(t)/4.
xp,v=±(l2/2-10)+[(l2+10)2+108]1/261/2.
ΔTp-v=|2 sinh{C[ΔφR(t), l]×f(xp,v, l)}|,
f(xp,v, l)=[(|xp,v|+l)2+9]×(|xp,v|2+1)[(|xp,v|+l)2+1]×(|xp,v|2+9).
T=1q0(x, t) ln[1+q0(x, t)],
T=i=1n1q0i(xi, t) ln[1+q0i(xi, t)]1-12 i=1nq0i(x, t)+12 i=1nq0i(x, t)2+O12 i=1nq0i(x, t)3.
T=11+12 i=1nq0i(x, t)=11+12xx+lq0(x, t)dx.
T=11+½ QR(t)[tan-1(x+l)-tan-1(x)].
Cq(t)QR(t)×1+tanh(l/2)×3ΔφR(t)10+[ΔφR(t)]28.
T=11+1/2Cq(t)[tan-1(x+l)-tan-1(x)].
Tca=Tcr×Toa,

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