Abstract

An analytical expression capable of calculating the on-axis far-field electric field of a Gaussian beam carrying a phase with a Gaussian profile is obtained. This result is based on the Huygens–Fresnel principle and is not limited to small Gaussian phases. The analytical results are particularized to the z-scan case, agreeing with the Gaussian-decomposition results for small phases. For Gaussian phases with amplitude up to π, a small deviation of the constant peak-to-valley distance is found, but a significant displacement of the axis crossing point is predicted. The theoretical expression of the two-color z-scan is also obtained, and it agrees exactly with the expression obtained by the Gaussian-decomposition method, to first-order approximation. Also we apply our result to the case of a thick sample, verifying the range of coincidence between two different formalisms.

© 1998 Optical Society of America

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References

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1996

B. K. Rhee and J. S. Byun, J. Opt. Soc. Am. B 13, 2720 (1996).
[CrossRef]

L. C. Oliveira, T. Catunda, and S. C. Zilio, Jpn. J. Appl. Phys. 35, 2649 (1996).
[CrossRef]

1995

R. E. Bridges, G. L. Fischer, and R. W. Boyd, Opt. Lett. 20, 1821 (1995).
[CrossRef]

S. Hughes, J. M. Burzler, G. Spruce, and B. S. Wherrett, J. Opt. Soc. Am. B 12, 1888 (1995).
[CrossRef]

W. Nasalsky, Opt. Commun. 119, 218 (1995).
[CrossRef]

T.-H. Wei, T.-H. Huang, and H.-D. Lin, Appl. Phys. Lett. 67, 2266 (1995).
[CrossRef]

H. Ma and C. B. de Araujo, Appl. Phys. Lett. 66, 1581 (1995).
[CrossRef]

1994

1993

W. Zhao and P. Palffy-Muhoray, Appl. Phys. Lett. 63, 1613 (1993).
[CrossRef]

J. A. Hermann and R. G. McDuff, J. Opt. Soc. Am. B 10, 2056 (1993).
[CrossRef]

1992

1991

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, Opt. Eng. 30, 1228 (1991).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

1990

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

1989

R. Adair, L. L. Chase, and S. A. Payne, Phys. Rev. B 39, 3337 (1989).
[CrossRef]

1979

1966

Adair, R.

R. Adair, L. L. Chase, and S. A. Payne, Phys. Rev. B 39, 3337 (1989).
[CrossRef]

Boyd, R. W.

Bridges, R. E.

Burzler, J. M.

Byun, J. S.

Catunda, T.

L. C. Oliveira, T. Catunda, and S. C. Zilio, Jpn. J. Appl. Phys. 35, 2649 (1996).
[CrossRef]

Chase, L. L.

R. Adair, L. L. Chase, and S. A. Payne, Phys. Rev. B 39, 3337 (1989).
[CrossRef]

de Araujo, C. B.

H. Ma and C. B. de Araujo, Appl. Phys. Lett. 66, 1581 (1995).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

DeSalvo, R.

Fischer, G. L.

Gomes, A. S. L.

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

Hagan, D. J.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, Opt. Lett. 17, 258 (1992).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, Opt. Eng. 30, 1228 (1991).
[CrossRef]

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

Hermann, J. A.

Huang, T.-H.

T.-H. Wei, T.-H. Huang, and H.-D. Lin, Appl. Phys. Lett. 67, 2266 (1995).
[CrossRef]

Hughes, S.

Kogelnik, H.

Li, T.

Lin, H.-D.

T.-H. Wei, T.-H. Huang, and H.-D. Lin, Appl. Phys. Lett. 67, 2266 (1995).
[CrossRef]

Ma, H.

H. Ma and C. B. de Araujo, Appl. Phys. Lett. 66, 1581 (1995).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

McDuff, R. G.

Miller, D. A. B.

Nasalsky, W.

W. Nasalsky, Opt. Commun. 119, 218 (1995).
[CrossRef]

Oliveira, L. C.

L. C. Oliveira, T. Catunda, and S. C. Zilio, Jpn. J. Appl. Phys. 35, 2649 (1996).
[CrossRef]

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, Appl. Phys. Lett. 63, 1613 (1993).
[CrossRef]

Payne, S. A.

R. Adair, L. L. Chase, and S. A. Payne, Phys. Rev. B 39, 3337 (1989).
[CrossRef]

Rhee, B. K.

Said, A. A.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, Opt. Eng. 30, 1228 (1991).
[CrossRef]

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

Sheik-Bahae, M.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, Opt. Lett. 17, 258 (1992).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, Opt. Eng. 30, 1228 (1991).
[CrossRef]

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

Smith, S. D.

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, Opt. Eng. 30, 1228 (1991).
[CrossRef]

Spruce, G.

Tian, J.-G.

J.-G. Tian, W.-P. Zang, and G. Zhang, Opt. Commun. 107, 415 (1994).
[CrossRef]

Van Stryland, E. W.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, Opt. Lett. 17, 258 (1992).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, Opt. Eng. 30, 1228 (1991).
[CrossRef]

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

Wang, J.

Weaire, D.

Wei, T.

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

Wei, T.-H.

T.-H. Wei, T.-H. Huang, and H.-D. Lin, Appl. Phys. Lett. 67, 2266 (1995).
[CrossRef]

Wherrett, B. S.

Xia, T.

Zang, W.-P.

J.-G. Tian, W.-P. Zang, and G. Zhang, Opt. Commun. 107, 415 (1994).
[CrossRef]

Zhang, G.

J.-G. Tian, W.-P. Zang, and G. Zhang, Opt. Commun. 107, 415 (1994).
[CrossRef]

Zhao, W.

W. Zhao and P. Palffy-Muhoray, Appl. Phys. Lett. 63, 1613 (1993).
[CrossRef]

Zilio, S. C.

L. C. Oliveira, T. Catunda, and S. C. Zilio, Jpn. J. Appl. Phys. 35, 2649 (1996).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

W. Zhao and P. Palffy-Muhoray, Appl. Phys. Lett. 63, 1613 (1993).
[CrossRef]

T.-H. Wei, T.-H. Huang, and H.-D. Lin, Appl. Phys. Lett. 67, 2266 (1995).
[CrossRef]

H. Ma, A. S. L. Gomes, and C. B. de Araujo, Appl. Phys. Lett. 59, 2666 (1991).
[CrossRef]

H. Ma and C. B. de Araujo, Appl. Phys. Lett. 66, 1581 (1995).
[CrossRef]

IEEE J. Quantum Electron.

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

Int. J. Nonlinear Opt. Phys.

J. A. Hermann, Int. J. Nonlinear Opt. Phys. 1, 541 (1992).
[CrossRef]

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

L. C. Oliveira, T. Catunda, and S. C. Zilio, Jpn. J. Appl. Phys. 35, 2649 (1996).
[CrossRef]

Opt. Commun.

W. Nasalsky, Opt. Commun. 119, 218 (1995).
[CrossRef]

J.-G. Tian, W.-P. Zang, and G. Zhang, Opt. Commun. 107, 415 (1994).
[CrossRef]

Opt. Eng.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, Opt. Eng. 30, 1228 (1991).
[CrossRef]

Opt. Lett.

Phys. Rev. B

R. Adair, L. L. Chase, and S. A. Payne, Phys. Rev. B 39, 3337 (1989).
[CrossRef]

Other

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, UK, 1993), Chap. 4.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), Chap. 8.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, 10th ed. (Ntl. Bur. Stand., Gaithersburg, Md., 1972), Chap. 6.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, 1995), Chap. 10.

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Figures (8)

Fig. 1
Fig. 1

Parameters of a z-scan curve for a sample with a positive nonlinearity.

Fig. 2
Fig. 2

Scheme of the coordinates utilized in the calculation of the Fresnel integral.

Fig. 3
Fig. 3

Comparison between the previsions of GD theory (dashed curve) and TIGaF (solid curve) for small nonlinear phases. Shown are the curves for a nonlinear phase equal to π/100.

Fig. 4
Fig. 4

Comparison between the previsions of GD theory (dashed curves) and TIGaF (continuous curves) for great nonlinear phases. The graph depicts the results for ΔΦ0 equals π/4, π/2 and π.

Fig. 5
Fig. 5

Position of the point where the z-scan curve crosses the axis T(z)=1 as a function of the nonlinear phase.

Fig. 6
Fig. 6

Peak–valley distance as a function of the nonlinear phase; for phases up to 1.15π the error in Δz=1.72z0 is smaller than 1.5%.

Fig. 7
Fig. 7

(a) Comparison between the TIGaF (ΔTpv-Γ, solid curve) and GD (ΔTpv-GD dashed curve) previsions for the transmittance variation between peak and valley. (b) Percentage difference (ΔTpv-GD-ΔTpv-Γ)/ΔTpv-Γ.

Fig. 8
Fig. 8

TIGaF prevision for z-scan curves with large (⩾π) nonlinear phases. The curves represent the results for nonlinear phases equal to π, 2π, and 3π.

Equations (36)

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dIdz=-αI,
d(Δϕ)dz=δn(I)k,
Es(z, r)=E0 w0w(z) exp(-αL/2)exp-r2w(z)2-i kr22R(z)-ikzexp-iΔϕ0 exp-2r2w(z)2.
Δϕ0=kn2LeffI0(1+z2/z02)=ΔΦ0(1+z2/z02),
T(z)=1+4ΔΦ0x(x2+1)(x2+9)
ΔTpv=0.406|ΔΦ0|,
Δz=1.72z0.
W=ik2π exp(-ikr)r,
EC(z)=ik2π 02πdθ0dρρES(z, ρ) exp(-ikr)r.
r=d2+ρ2d+12 ρ2d.
α=1w(z)2+i k2 1R(z)+1d,
β=2w(z)2,
Λ=ikE0 w0w(z) exp(-αL/2-ikD) 1d,
EC=Λ0dρρ exp(-αρ2)exp[-iΔϕ0 exp(-βρ2)].
EC=Λ2β(iΔϕ0)μ+1 0iΔϕ0dννμ exp(-ν).
Γ(μ+1, a, b)=abdννμ exp(-ν).
γ(z)=μ(z)+1=12 iz0 z+z2+z02d+1.
EC(z, iΔϕ0)=i π2λ E0 exp(-αL/2-ikD) w02d×[1+(z/z0)2]1/2 Γ(γ, 0, iΔϕ0)(iΔϕ0)γ.
ICN(z, iΔϕ0)=γ(z) Γ(γ, 0, iΔϕ0)(iΔϕ0)γ2.
T(z, ΔΦ0)=γ(z) Γγ(z), 0, i ΔΦ01+(z/z0)2i ΔΦ01+(z/z0)2γ(z)2,
γ(z)=12 iz0 z+z2+z02D-z+1.
Γ(μ, 0, x)=0xdννμ-1e-ν=xμn=0(-1)n xnn!(μ+n)
T(1)(z)=1-iΔϕ0(z)1+1/γ(z).
T(1)(z)=1+4Δϕ0 x[1+z/d(1+1/x2)]9+x2[1+z/d(1+1/x2)]
T(z, ΔΦ0)=γ(z) Γγ(z), 0, i ΔΦ01+(z/zp0)2i ΔΦ01+(z/zp0)2γ(z) 2,
γ(z)=12 wp2w2 iz0 z+z2+z02D-z+1.
T(1)(z)=1+4a2bxΔΦ0(a+2b+ax2)2+4a2b2x2,
γ=(1+ζ02),
μ=(1-ζ0ζm/γ)2+(ζm/γ)2,
ζ0=z/zr,
ζm=L/zr,
u=[2 exp(2iy)-1]-1,
y=arctan(ζ0)+arctan(ζm-ζ0),
α=1γμr02 [1+i(ζ0-ζm)]+i k2d,
Δϕ0=b8 Re[i ln(u-1)],
β=b4Δϕ0γμr02 Re[i(1-u)].

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