Abstract

We present a theoretical model of optical beam propagation in a nonlinear medium for determining the nonlinear refraction and nonlinear absorption accurately and simultaneously by a single Z-scan method. The model is based on the Gaussian decomposition method and symmetric analysis. In principle, accurate nonlinear refraction and nonlinear absorption can be obtained simultaneously by exact experimental data, no matter how much they are. The treatment procedure of Z-scan experimental data is demonstrated.

© 2004 Optical Society of America

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References

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  1. M. Sheik-Bahae, A. A. Said, and E. W. VanStryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  2. M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Vanstryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  3. P. B. Chapple, J. Straromlynska, 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]
  4. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser-beam propagation in InSb,” Opt. Lett. 4, 331–333 (1979).
    [CrossRef] [PubMed]
  5. R. L. Sutherland, Nonlinear Optics (Marcel Dekker, New York, 1996), Chap. 7.
  6. 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]
  7. 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, 2742–2747 (1998).
    [CrossRef]
  8. J. D. Gaskill, Linear System, Fourier Transforms, and Optics (Wiley, New York, 1978).
  9. G. Boudebs, F. Sanchez, J. Troles, and F. Smektala, “Nonlinear optical properties of chalcogenide glasses: comparison between Mach–Zehnder interferometry and Z-scan techniques,” Opt. Commun. 199, 425–433 (2001).
    [CrossRef]
  10. 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]
  11. X. Liu, S. Guo, H. Wang, and L. Hou, “Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption,” Opt. Commun. 197, 432–437 (2001).
    [CrossRef]
  12. J. A. Hermann, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1, 729–736 (1984).
    [CrossRef]

2001 (2)

G. Boudebs, F. Sanchez, J. Troles, and F. Smektala, “Nonlinear optical properties of chalcogenide glasses: comparison between Mach–Zehnder interferometry and Z-scan techniques,” Opt. Commun. 199, 425–433 (2001).
[CrossRef]

X. Liu, S. Guo, H. Wang, and L. Hou, “Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption,” Opt. Commun. 197, 432–437 (2001).
[CrossRef]

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]

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, 2742–2747 (1998).
[CrossRef]

1997 (1)

P. B. Chapple, J. Straromlynska, 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]

1990 (1)

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

1989 (1)

1984 (1)

1979 (1)

Boudebs, G.

G. Boudebs, F. Sanchez, J. Troles, and F. Smektala, “Nonlinear optical properties of chalcogenide glasses: comparison between Mach–Zehnder interferometry and Z-scan techniques,” Opt. Commun. 199, 425–433 (2001).
[CrossRef]

Chapple, P. B.

P. B. Chapple, J. Straromlynska, 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]

Guo, S.

X. Liu, S. Guo, H. Wang, and L. Hou, “Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption,” Opt. Commun. 197, 432–437 (2001).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Vanstryland, “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. Straromlynska, 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. A. Hermann, “Beam propagation and optical power limiting with nonlinear media,” J. Opt. Soc. Am. B 1, 729–736 (1984).
[CrossRef]

Hou, L.

X. Liu, S. Guo, H. Wang, and L. Hou, “Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption,” Opt. Commun. 197, 432–437 (2001).
[CrossRef]

Ji, W.

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]

Li, H. P.

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]

Liu, X.

X. Liu, S. Guo, H. Wang, and L. Hou, “Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption,” Opt. Commun. 197, 432–437 (2001).
[CrossRef]

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. Straromlynska, 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. Straromlynska, 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]

Miller, D. A. B.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Vanstryland, “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. VanStryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Samad, R. E.

Sanchez, F.

G. Boudebs, F. Sanchez, J. Troles, and F. Smektala, “Nonlinear optical properties of chalcogenide glasses: comparison between Mach–Zehnder interferometry and Z-scan techniques,” Opt. Commun. 199, 425–433 (2001).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Vanstryland, “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. VanStryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Smektala, F.

G. Boudebs, F. Sanchez, J. Troles, and F. Smektala, “Nonlinear optical properties of chalcogenide glasses: comparison between Mach–Zehnder interferometry and Z-scan techniques,” Opt. Commun. 199, 425–433 (2001).
[CrossRef]

Smith, S. D.

Straromlynska, J.

P. B. Chapple, J. Straromlynska, 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]

Tang, S. H.

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]

Troles, J.

G. Boudebs, F. Sanchez, J. Troles, and F. Smektala, “Nonlinear optical properties of chalcogenide glasses: comparison between Mach–Zehnder interferometry and Z-scan techniques,” Opt. Commun. 199, 425–433 (2001).
[CrossRef]

Vanstryland, E. W.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Vanstryland, “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. VanStryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Vieira Jr., N. D.

Wang, H.

X. Liu, S. Guo, H. Wang, and L. Hou, “Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption,” Opt. Commun. 197, 432–437 (2001).
[CrossRef]

Weaire, D.

Wei, T.

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

Wherrett, B. S.

Yin, M.

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]

Appl. Phys. B (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]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Vanstryland, “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. Straromlynska, 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 (2)

Opt. Commun. (3)

G. Boudebs, F. Sanchez, J. Troles, and F. Smektala, “Nonlinear optical properties of chalcogenide glasses: comparison between Mach–Zehnder interferometry and Z-scan techniques,” Opt. Commun. 199, 425–433 (2001).
[CrossRef]

X. Liu, S. Guo, H. Wang, and L. Hou, “Theoretical study on the closed-aperture Z-scan curves in the materials with nonlinear refraction and strong nonlinear absorption,” Opt. Commun. 197, 432–437 (2001).
[CrossRef]

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 (2)

J. D. Gaskill, Linear System, Fourier Transforms, and Optics (Wiley, New York, 1978).

R. L. Sutherland, Nonlinear Optics (Marcel Dekker, New York, 1996), Chap. 7.

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Figures (6)

Fig. 1
Fig. 1

Z-scan experimental setup. D2/D1 as a function of the medium’s position Z is measured. D2 and D1, detectors; BS, beam splitter; L, lens; NL, nonlinear medium.

Fig. 2
Fig. 2

(a) Group of curves of odd function To, where nonlinear phase shift ΔΦ0=1 and nonlinear absorption q0=0,0.3,0.6,0.9, respectively. (b) A group of curves of even function Te, where nonlinear phase shift ΔΦ0=1 and nonlinear absorption q0=0,0.3,0.6,0.9, respectively.

Fig. 3
Fig. 3

(a) Peak–valley difference ΔToP-V of To curves as a function of nonlinear absorption q0 for a fixed nonlinear phase shift ΔΦ0=1. (b) The dip of Te curves as a function of nonlinear absorption q0 for a fixed nonlinear phase shift ΔΦ0=1. (c) The peak’s position of To curves as a function of nonlinear absorption q0, where nonlinear phase shift ΔΦ0=1.

Fig. 4
Fig. 4

(a) Calculated nonlinear phase shift ΔΦ as a function of q0 for different numbers of summation items, where q0 is the real value of nonlinear absorption. (b) Calculated nonlinear absorption q as a function of q0 for different numbers of summation items, where q0 is the real value of nonlinear absorption. For (a) and (b), m=1, m=3, and m=15 denote the first-order approximation, the third-order approximation, and the fifteenth-order approximation, respectively. The real value of nonlinear phase shift ΔΦ0=1.

Fig. 5
Fig. 5

(a) Peak–valley difference as a function of nonlinear absorption q0 for different dimensionless aperture radius Ya, where nonlinear phase shift ΔΦ0=1. (b) The dip as a function of nonlinear absorption q0 for different dimensionless aperture radius Ya, where nonlinear phase shift ΔΦ0=1. (c) The peak’s position of To curves as a function of nonlinear absorption q0 for different dimensionless aperture radius Ya, where nonlinear phase shift ΔΦ0=1.

Fig. 6
Fig. 6

(a) Calculated nonlinear phase shift ΔΦ as a function of nonlinear absorption q0 for different numbers of summation items, where Ya=0.5 and the real nonlinear phase shift ΔΦ0=1. (b) Calculated nonlinear absorption q as a function of nonlinear absorption q0 for different numbers of summation items, where Ya=0.5 and the real nonlinear phase shift ΔΦ0=1. For (a) and (b), m=1, m=3, and m=15 denote the first-order approximation, the third-order approximation, and the fifteenth-order approximation, respectively.

Equations (47)

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n=n0+γI,
α=α0+βI,
dΔϕdz=kγI,
dIdz=-α0I-βI2,
E(z, r, t)=E0(t)w0w(z)×exp-r2w2(z)-ikr22R(z)exp-iϕ(z, t),
Ee(r, z)=E(r, z, t)exp(-α0L/2)×[1+q(z, r)](iΔΦ0/q0-1/2),
Ee=E(z, r)exp(-α0l/2)1ifm=0m=0q(z, r)mm!n=1miΔΦ0q0-1/2-n+1ifm1.
Ea(r, t)=E(z, r=0, t)exp(-α0L/2)m=0fm wm0wm×exp-r2wm2-ikr22Rm+iθm.
wm02=w2(z)/(2m+1),
dm=12kwm02,
Rm=d1-gg2+d2/dm2-1,
θm=tan-1d/dmg,
wm2=wm02(g2+d2/dm2),
fm=1m!iΔΦ01+z2/z02m1ifm=0n=1m1+i(n-1/2)ΔΦ0q0ifm1.
T(ra, z, ΔΦ0, q0)
=0ra|Ea(r, z, t, ΔΦ0, q0)|2rdr0ra|Ea(r, z, t, ΔΦ0=0, q0=0)|2rdr,
T(Ya, x, ΔΦ0, q0)=k=0Tk(Ya, x, ΔΦ0, q0)(ΔΦ0)k,
Tk(Ya, x, ΔΦ0, q0)=m=0,n=k-mkSmn(Ya, x, ΔΦ0, q0),
Smn(Ya, x, ΔΦ0, q0)
=Smnnr(x)Smnna(ΔΦ0, q0)Smnra(Ya, x),
Smnnr(x)=im-n(1+x2)m+nwm0wn0w02wmwnw002 exp(iθm-iθn),
Smnna(ΔΦ0, q0)=l=1m1+i(l-1/2)q0ΔΦ0×l=1n1-i(l-1/2)q0ΔΦ0,
Smnra=2[1-exp(-AmnYa2)]Amn[1-exp(-2Ya2)],
Amn=2(m+n+1)(1+x2)[x2-2i(m-n)x+(2m+1)(2n+1)][x2+(2m+1)2][x2+(2n+1)2],
T(Ya, x, ΔΦ0, q0)=To(Ya, x, ΔΦ0, q0)+Te(Ya, x, ΔΦ0, q0),
To(Ya, x, ΔΦ0, q0)
=[T(Ya, x, ΔΦ0, q0)-T(Ya, -x, ΔΦ0, q0)]2,
Te(Ya, x, ΔΦ0, q0)
=[T(Ya, x, ΔΦ0, q0)+T(Ya, -x, ΔΦ0, q0)]2.
To(x, ΔΦ0, q0)=4xΔΦ0(x2+1)(x2+9),
Te(x, ΔΦ0, q0)=1-(x2+3)q0(x2+1)(x2+9).
ΔToP-V=2To(Ya=0, xp, ΔΦ0, q0),
ΔTe=1-Te(Ya=0, x=0, ΔΦ0, q0).
FUN(1)=2To(Ya=0, xp,v, ΔΦ0, q0)-ΔToP-V,
FUN(2)=1-Te(Ya=0, x=0, ΔΦ0, q0)-ΔTe.
ΔΦ01=ΔToP-V/0.406,
q01=3ΔTe.
T(ra, z, ΔΦ0, q0)
=m=1n=1fmfn* wm0wn0wmwn exp(iθm-iθn)w002/w02Smnra,
fmfn* wm0wn0wmwn exp(iθm-iθn)w002/w02
=Smnnr(x)Smnna(ΔΦ0, q0)(ΔΦ0)(m+n),
Smnnr(x)=i(m-n)(1+x2)(m+n)wm0wn0w02wmwnw002 exp(iθm-iθn),
Smnna(ΔΦ0, q0)=l=1m1+il-1/2 q0ΔΦ0×l=1n1-il-1/2 q0ΔΦ0,
Smnra=0raexp-r2wm2-r2wn2-ikr22Rm+ikr22Rnrdr0raexp-2r2w02rdr.
Smnra=2[1-exp(-AmnYa2)]Amn[1-exp(-2Ya2)],
Amn=2(m+n+1)(1+x2)[x2-2i(m-n)x+(2m+1)(2n+1)][x2+(2m+1)2][x2+(2n+1)2].
k=0m=0,n=k-mk.

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