Abstract

The Z-scan analysis technique has been developed to permit characterization of nonlinear refractive indices and nonlinear absorption of non-Gaussian laser beams by a numerical approach. The new approach is based on a mode expansion of the electric field of the laser beam into Gaussian–Laguerre or Gaussian–Hermite modes. The individual modes are propagated within the nonlinear sample under the influence of the intensity-dependent phase shifts. The resulting electric field at the exit plane is expanded as a new sum of Gaussian–Laguerre or Gaussian–Hermite modes. From the final mode expansion the field distribution at the detector plane is calculated. The method also makes it possible to simulate a variety of complex optical limiting devices, such as stacks of samples with different material parameters and thick samples with a variation of material parameters along the z axis. A series of test simulations is presented and compared with experimental Z-scan data of solutions of substituted phthalocyanine molecules.

© 1998 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. J. A. Hermann, “Self-focusing effects and applications using thin nonlinear media,” Int. J. Nonlinear Opt. Phys. 1, 541–561 (1992).
    [CrossRef]
  4. W. Zhao, J. H. Kim, and P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 131–147 (1994).
    [CrossRef]
  5. P. B. Chapple and P. J. Wilson, “Z-scans with near-Gaussian laser beams,” J. Nonlinear Opt. Phys. Mater. 5, 419–436 (1996).
    [CrossRef]
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    [CrossRef]
  7. B. K. Rhee, J. S. Byun, and E. W. Van Stryland, “Z-scan using circularly symmetric beams,” J. Opt. Soc. Am. B 13, 2720–2723 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1997

1996

1994

D. J. Hagan, T. Xia, A. A. Said, and E. W. Van Stryland, “Tandem limiter optimization,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 179–190 (1994).
[CrossRef]

W. Zhao, J. H. Kim, and P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 131–147 (1994).
[CrossRef]

1992

1990

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, 760–769 (1990).
[CrossRef]

1989

1979

1966

Agnesi, A.

Byun, J. S.

Chapple, P. B.

P. B. Chapple and P. J. Wilson, “Z-scans with near-Gaussian laser beams,” J. Nonlinear Opt. Phys. Mater. 5, 419–436 (1996).
[CrossRef]

Gindre, D.

Hagan, D. J.

D. J. Hagan, T. Xia, A. A. Said, and E. W. Van Stryland, “Tandem limiter optimization,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 179–190 (1994).
[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, 760–769 (1990).
[CrossRef]

Hermann, J. A.

Kim, J. H.

W. Zhao, J. H. Kim, and P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 131–147 (1994).
[CrossRef]

Kogelnik, H.

Li, T.

Maillotte, H.

Marcano O., A.

Métin, D.

Mian, S. M.

Miller, D. A. B.

Palffy-Muhoray, P.

W. Zhao, J. H. Kim, and P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 131–147 (1994).
[CrossRef]

Reali, G. C.

Rhee, B. K.

Said, A. A.

D. J. Hagan, T. Xia, A. A. Said, and E. W. Van Stryland, “Tandem limiter optimization,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 179–190 (1994).
[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, 760–769 (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]

Sheik-Bahae, M.

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, 760–769 (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.

Taheri, B.

Tomaselli, A.

Van Stryland, E. W.

B. K. Rhee, J. S. Byun, and E. W. Van Stryland, “Z-scan using circularly symmetric beams,” J. Opt. Soc. Am. B 13, 2720–2723 (1996).
[CrossRef]

D. J. Hagan, T. Xia, A. A. Said, and E. W. Van Stryland, “Tandem limiter optimization,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 179–190 (1994).
[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, 760–769 (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]

Weaire, D.

Wei, T. H.

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, 760–769 (1990).
[CrossRef]

Wherrett, B. S.

Wicksted, J. P.

Wilson, P. J.

P. B. Chapple and P. J. Wilson, “Z-scans with near-Gaussian laser beams,” J. Nonlinear Opt. Phys. Mater. 5, 419–436 (1996).
[CrossRef]

Xia, T.

D. J. Hagan, T. Xia, A. A. Said, and E. W. Van Stryland, “Tandem limiter optimization,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 179–190 (1994).
[CrossRef]

Zhao, W.

W. Zhao, J. H. Kim, and P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 131–147 (1994).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

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, 760–769 (1990).
[CrossRef]

Int. J. Nonlinear Opt. Phys.

J. A. Hermann, “Self-focusing effects and applications using thin nonlinear media,” Int. J. Nonlinear Opt. Phys. 1, 541–561 (1992).
[CrossRef]

J. Nonlinear Opt. Phys. Mater.

P. B. Chapple and P. J. Wilson, “Z-scans with near-Gaussian laser beams,” J. Nonlinear Opt. Phys. Mater. 5, 419–436 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Proc. SPIE

W. Zhao, J. H. Kim, and P. Palffy-Muhoray, “Z-scan measurement on liquid crystals using top-hat beams,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 131–147 (1994).
[CrossRef]

D. J. Hagan, T. Xia, A. A. Said, and E. W. Van Stryland, “Tandem limiter optimization,” in Nonlinear Optical Materials for Switching and Limiting, M. J. Soileau, ed., Proc. SPIE 2229, 179–190 (1994).
[CrossRef]

Other

M. Lindgren, P.-O. Arntzen, and S. Svensson, “Spectral z-scans of organic dyes using an OPO laser,” Nonlinear Opt. 15, 85–88 (1996); further details are found in M. Lindgren and P.-O. Arntzen, “Laser beam quality characterization—testing the Sunlite OPO laser,” Rep. FOA-D—95–00134–3.1–SE (National Defence Research Establishment, Linköping, Sweden, 1995); A. Eriksson and M. Lindgren, “Gaussian–Hermite and Gaussian–Laguerre laser modes—simulation and characterization,” Rep. FOA-R—96–00255–3.1–SE (National Defense Research Establishment, Linköping, Sweden, 1996).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1988).

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Figures (8)

Fig. 1
Fig. 1

Schematic drawing of the experimental Z-scan setup. Details are given in Section 2. Sync., electric pulse from the laser that syncronizes the digital storage adaptor with the light pulse from the laser.

Fig. 2
Fig. 2

Simulated open-aperture Z scans with peak intensity I0=10 MW/cm2, Rayleigh length zR=3 cm, wavelength λ=500 nm, γ=10-6 cm2/MW, β=100 cm/GW, and a sample thickness of 2 mm (+). The solid curve is calculated with the formula of Sheik-Bahae et al.2 modified for a square temporal pulse. The dashed curve represents a Z scan with a Gaussian temporal pulse as given by Sheik-Bahae et al.2

Fig. 3
Fig. 3

(a) Simulated open-aperture Z scans with peak intensity I0=10 MW/cm2, Rayleigh length zR=3 cm, wavelength λ=500 nm, γ=10-6 cm2/MW, β=100 cm/GW, and a sample thickness of 2 mm. The solid curve is a simulation of a Z scan made with a beam with M2=2.3 (normalized amplitude weights for Gaussian–Laguerre expansion are A00, 0.74; A10, 0.53; A20, 0.37; A30, 0.18); for the dashed curve a Gaussian TEM00 mode was used. (b) Experimental open-aperture Z scan of a Pb phthalocyanine toluene solution (1.7 mM) in a 2-mm cell (dots). Fitting used the following parameters: I0=27 MW/cm2, Rayleigh length zR=1.7 cm, wavelength λ=523 nm, and β=680 cm/GW. β was introduced phenomenologically, and along with the Rayleigh length it was used as a free parameter in the fitting (solid curve).

Fig. 4
Fig. 4

zR=3 cm λ=500 nm, I0=10 MW/cm2, γ=10-6cm2/MW, β=0. An ordinary Z scan with a square pulse (solid curve) according to formula in Ref. 2. Calculations using the Gaussian TEM00 beam (+) and a beam with M2=2.3 (dashed curve); amplitude weights as given in Fig. 3.

Fig. 5
Fig. 5

Z scan of a tandem limiter configuration.

Fig. 6
Fig. 6

Sample thicknesses, 2 mm; distance between exit surfaces of the samples, 32 mm. Sample 1, β=25 cm/GW; γ=0; dye concentration, 0.25 mM. Sample 2 (detector side), β=50 cm/GW; γ=0; dye concentration, 0.5 mM. Beam, zR=1.5 cm; λ=523 nm, I0=220 MW/cm2; M2=2.3; amplitude weights as given in Fig. 3; solid curve, calculation.

Fig. 7
Fig. 7

Same as Fig. 6, but with samples 1 and 2 interchanged.

Fig. 8
Fig. 8

Simulations of the irradiance at the detector plane. Sample thicknesses, 2 mm; distance between exit surfaces of the samples, 32 mm. Sample 1, β=25 cm/GW, γ=0.25×10-6cm2/MW. Sample 2 (detector side), β=50 cm/GW, γ=0.510-6 cm2/MW. Beam, zR=1.5 cm, λ=523 nm, I0=220 MW/cm2, M2=2.3, amplitude weights as given in Fig. 3. Left, focus 1 m from entrance surface of sample 1; middle, focus 3 cm from entrance surface of sample 1; right, difference between the cases in the middle and on the left.The same scale is used for all three plots.

Equations (9)

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Ie=I(z, r, t)exp(-αL)1+βLeffI(z, r, t),
Δϕ(z, r, t)=kγβ ln[1+βLeffI(z, r, t)],
Leff=1-exp(-αL)α
E=p,lAplEpl,
I=c0n02 |E|2 =c0n02 p,lAplEpl2,
Epl(r, z, t)=E0 w0w(z) 2p!(1+δ0m)π(p+l)!1/2×2rw(z)lLpl2r2w(z)2×exp-r2w2(z)-i kr22R(z)ilθ×exp[-i(2p+l+1)ϕ(z, t)].
Ee=E(z, r, t)exp(-αL)1+βLeffI(z, r, t)1/2 exp[Δϕ(z, r, t)]=E(z, r, t)exp-αL2[1+βLeffI(z, r, t)](ikγ/β-1/2)=p,lAplEpl(z, r, t)exp-αL2×[1+βLeffI(z, r, t)](ikγ/β-1/2).
Ee=q,rBqrEqr,
Bqr=Eqr(x, y, z)Ee*(x, y, z)dxdy|Ee(x, y, z)|2dxdyx,yEqr(x, y, z)Ee*(x, y, z)x,y|Ee(x, y, z)|2.

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