Abstract

We report general characteristics of on-axis Z-scan transmittance for arbitrary circularly symmetric beams. Some experimental results are presented for a nearly top-hat-shaped beam and for a trimmed Airy beam whose electric field profile is the central portion of an Airy function inside its first zero. The sensitivity of Z-scan method with a trimmed Airy beam for measuring an induced index-of-refraction change is a factor of 1.5 greater than that of a Gaussian beam. Also, it is found that there are some advantages of experimental alignment and numerical convergence for a Z-scan measurement that uses a trimmed Airy beam over one that uses a top-hat beam.

© 1996 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 (1989).
    [CrossRef] [PubMed]
  2. 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 (1990).
    [CrossRef]
  3. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1991).
    [CrossRef]
  4. L. Yang, R. Dorsinville, Q. Z. Wang, P. X. Ye, and R. R. Alfano, “Excited-state nonlinearity in polythiophene thin films investigated by the Z-scan technique,” Opt. Lett. 17, 323 (1992).
    [CrossRef] [PubMed]
  5. B. A. Rockwell, W. P. Roach, M. E. Rogers, M. W. Mayo, and C. A. Toth, “Nonlinear refraction in vitreous humor,” Opt. Lett. 18, 1792 (1993).
    [CrossRef] [PubMed]
  6. W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613 (1993).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

1993 (2)

1992 (1)

1991 (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 (1990).
[CrossRef]

1989 (1)

Alfano, R. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Dorsinville, R.

Hagan, D. J.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1991).
[CrossRef]

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 (1990).
[CrossRef]

Mayo, M. W.

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613 (1993).
[CrossRef]

Roach, W. P.

Rockwell, B. A.

Rogers, M. E.

Said, A. A.

Sheik-Bahae, M.

Toth, C. A.

Van Stryland, E. W.

Wang, J.

Wang, Q. Z.

Wei, T. H.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1991).
[CrossRef]

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 (1990).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Yang, L.

Ye, P. X.

Young, J.

Zhao, W.

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613 (1993).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Measured irradiance profile of a nearly top-hat-shaped beam (open circles). The solid curve is a best-fit function to the experimental data.

Fig. 2
Fig. 2

Normalized on-axis Z-scan transmittance for the nearly top-hat beam with I0=1.3 GW/cm2. The solid curve is a numerical fit with ΔΦ=0.25.

Fig. 3
Fig. 3

Irradiance profile of the focused nearly top-hat beam measured 0.7 m from the lens.

Fig. 4
Fig. 4

Measured diffraction pattern for a circular aperture of 0.5-mm radius (open squares). The measurements were made 1.5 m from the aperture (input pinhole). The solid curve is the best fit to a square of the trimmed Airy function.

Fig. 5
Fig. 5

Normalized on-axis Z-scan transmittance for the trimmed Airy beam with I0=1.2 GW/cm2. The solid curve is the numerical fit with ΔΦ=0.23.

Equations (14)

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E(r=x2+y2, z)=Ei0λfgriJ02π rriλfexp-iπ zλrif22πridri.
E(r, z)=E0[fr(r, z)+ifi(r, z)],
Ee(r, z)=E(r, z)exp(-αL/2)(1+iFΔΦ-FΔΨ),
Ee(r, z)=E0 exp(-αL/2)[(fr-frFΔΨ-fiFΔΦ)+i(fi-fiFΔΨ+frFΔΦ)].
Ed(z=d)=1λdEeds.
Tz=1+2fidsfrFds-frdsfiFdsfrds2+fids2ΔΦ-2frdsfrFds+fidsfiFdsfrds2+fids2ΔΨ.
T(z)=1-2 fiFds frdsΔΦ-2 frFds frdsΔΨ,
g(ri)=1+0.3[(ri/a)2-(ri/a)4]|ri|a0|ri|>a,
E(r, z)=Ei0λf2πa201g(ρ)J0(vρ)exp(-iqρ2)ρdρ,
IIm f=limb 0b01 g(ρ)sin(qρ2)J0(vρ)ρdρvdv.
IIm f=limb b01g(ρ)sin(qρ2)J1(bρ)dρ.
g(ρ)sin(qρ2)=n=1dnρ2n,
In=limbb01ρ2nJ1(bρ)dρ.
In=limb s=1n-2bs-1 (n-1)!(n-s)!Js+1(b),

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