Abstract

We apply a finite-difference algorithm that combines the local one-dimensional approximation and the Crank-Nicolson algorithms to solve the three-dimensional nonlinear Schrödinger equation. This scheme is unconditionally stable and accurate to second order. Therefore it offers a simple and accurate means to study a two-dimensional Z scan for arbitrary beam shape and medium length. As an example, we analyze the characteristics of a Z scan by utilizing an elliptic Gaussian beam for a thick nonlinear medium. The effects of ellipticity and waist separation of the elliptic beam on the normalized transmittance of the closed-aperture and open-aperture Z scan are demonstrated.

© 2004 Optical Society of America

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References

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  1. M. Sheik-Bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
    [CrossRef] [PubMed]
  2. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, E. W. Van Stryland, “Sensitivity measurement of optical nonlinearities using a single beam,” IEEE. J. Quantum Electron. 26, 760–769 (1990).
    [CrossRef]
  3. W. Zhao, P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
    [CrossRef]
  4. T. Xia, D. J. Hagan, M. Sheik-Bahae, E. W. VanStryland.“Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317–319 (1994).
    [CrossRef] [PubMed]
  5. J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
    [CrossRef]
  6. J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik 98, 143–146 (1995).
  7. S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–863 (1996).
    [CrossRef]
  8. Y.-L. Huang, C.-K. Sun, “Z-scan measurement with an astigmatic Gaussian beam.” J. Opt. Soc. Am. B 17, 43–47 (2000).
    [CrossRef]
  9. J. H. Marburger, Prog “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  10. F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
    [CrossRef]
  11. S. M. Mian, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
    [CrossRef]
  12. J.-G. Tian, W.-P. Zang, C.-P. Zhang, “Analysis of beam propagation through thick nonlinear media by variational approach,” Acta Phys. Sin. 43, 1712–1717 (1994).
  13. J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “Analysis of beam propagation in thick nonlinear media,” Appl. Opt. 34, 4331–4336 (1995).
    [CrossRef] [PubMed]
  14. B. Gross, J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
    [CrossRef]
  15. W.-P. Zang, J.-G. Tian, Z.-B. Liu, W.-Y. Zhou, C.-P. Zhang, G.-Y. Zhang, “Variational analysis of z scan of thick medium with an elliptic Gaussian beam,” Appl. Opt. 42, 2219–2225 (2003).
    [CrossRef] [PubMed]
  16. J. M. Burzler, S. Hughes, B. S. Wherrett. “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
    [CrossRef]
  17. K. Kawano, T. Kitou. Introduction to Optical Waveguide Analysis (Wiley, New York, 2001), Chap. 5.
    [CrossRef]
  18. L. Lapidus, G. F. Pinder, Numerical Solution of Partial Differential Equationals in Science and Engineering (Wiley, New York, 1982), Chap. 4.
  19. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, London, 1986).
  20. W.-P. Zang, J.-G. Tian, Z.-B. Liu, W.-Y. Zhou, C.-P. Zhang, G.-Y. Zhang. “Comparison of the solutions from a novel variational method with numerical results for the study of beam propagation in a Kerr medium with nonlinear absorption,” Opt. Lett. 28, 722–724 (2003).
    [CrossRef] [PubMed]
  21. A. Yariv. Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.
  22. P. B. Chapple, J. Staromlynska, R. G. McDuff, “Z-scan studies in the thin- and thick-sample limits,” J. Opt. Soc. Am. B 11, 975–982 (1994).
    [CrossRef]

2003

2000

1996

J. M. Burzler, S. Hughes, B. S. Wherrett. “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–863 (1996).
[CrossRef]

1995

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

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik 98, 143–146 (1995).

S. M. Mian, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

1994

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

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

T. Xia, D. J. Hagan, M. Sheik-Bahae, E. W. VanStryland.“Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317–319 (1994).
[CrossRef] [PubMed]

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

1993

W. Zhao, P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

1992

B. Gross, J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

1990

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

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

1989

1975

J. H. Marburger, Prog “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Burzler, J. M.

J. M. Burzler, S. Hughes, B. S. Wherrett. “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

Chapple, P. B.

Cornolti, F.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, London, 1986).

Gross, B.

B. Gross, J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

Hagan, D.

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

Hagan, D. J.

Huang, Y.-L.

Hughes, S.

J. M. Burzler, S. Hughes, B. S. Wherrett. “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

Kawano, K.

K. Kawano, T. Kitou. Introduction to Optical Waveguide Analysis (Wiley, New York, 2001), Chap. 5.
[CrossRef]

Kitou, T.

K. Kawano, T. Kitou. Introduction to Optical Waveguide Analysis (Wiley, New York, 2001), Chap. 5.
[CrossRef]

Lapidus, L.

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

Liu, Z.-B.

Lucchesi, M.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Manassah, J. T.

B. Gross, J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

Marburger, J. H.

J. H. Marburger, Prog “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

McDuff, R. G.

Mian, S. M.

S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–863 (1996).
[CrossRef]

S. M. Mian, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

Palffy-Muhoray, P.

W. Zhao, P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

Pinder, G. F.

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

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, London, 1986).

Said, A. A.

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

M. Sheik-Bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sheik-Bahae, M.

Staromlynska, J.

Sun, C.-K.

Taheri, B.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, London, 1986).

Tian, J.-G.

W.-P. Zang, J.-G. Tian, Z.-B. Liu, W.-Y. Zhou, C.-P. Zhang, G.-Y. Zhang, “Variational analysis of z scan of thick medium with an elliptic Gaussian beam,” Appl. Opt. 42, 2219–2225 (2003).
[CrossRef] [PubMed]

W.-P. Zang, J.-G. Tian, Z.-B. Liu, W.-Y. Zhou, C.-P. Zhang, G.-Y. Zhang. “Comparison of the solutions from a novel variational method with numerical results for the study of beam propagation in a Kerr medium with nonlinear absorption,” Opt. Lett. 28, 722–724 (2003).
[CrossRef] [PubMed]

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik 98, 143–146 (1995).

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

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

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

Van Stryland, E. W.

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

M. Sheik-Bahae, A. A. Said, E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

VanStryland, E. W.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, London, 1986).

Wei, T. H.

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

Wherrett, B. S.

J. M. Burzler, S. Hughes, B. S. Wherrett. “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

Wicksted, J. P.

S. M. Mian, B. Taheri, J. P. Wicksted, “Effects of beam ellipticity on Z-scan measurements,” J. Opt. Soc. Am. B 13, 856–863 (1996).
[CrossRef]

S. M. Mian, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

Xia, T.

Yariv, A.

A. Yariv. Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

Zambon, B.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

Zang, W.-P.

W.-P. Zang, J.-G. Tian, Z.-B. Liu, W.-Y. Zhou, C.-P. Zhang, G.-Y. Zhang. “Comparison of the solutions from a novel variational method with numerical results for the study of beam propagation in a Kerr medium with nonlinear absorption,” Opt. Lett. 28, 722–724 (2003).
[CrossRef] [PubMed]

W.-P. Zang, J.-G. Tian, Z.-B. Liu, W.-Y. Zhou, C.-P. Zhang, G.-Y. Zhang, “Variational analysis of z scan of thick medium with an elliptic Gaussian beam,” Appl. Opt. 42, 2219–2225 (2003).
[CrossRef] [PubMed]

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

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik 98, 143–146 (1995).

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

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

Zhang, C.-P.

Zhang, G.-Y.

Zhao, W.

W. Zhao, P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

Zhou, W.-Y.

Acta Phys. Sin.

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

Appl. Opt.

Appl. Phys. B

J. M. Burzler, S. Hughes, B. S. Wherrett. “Split-step Fourier methods applied to model nonlinear refractive effects in optically thick media,” Appl. Phys. B 62, 389–397 (1996).
[CrossRef]

Appl. Phys. Lett.

W. Zhao, P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

IEEE. J. Quantum Electron.

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

J. Appl. Phys.

S. M. Mian, J. P. Wicksted, “Measurement of optical nonlinearities using an elliptic Gaussian beam,” J. Appl. Phys. 77, 5434–5436 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

F. Cornolti, M. Lucchesi, B. Zambon, “Elliptic Gaussian beam self-focusing in nonlinear media,” Opt. Commun. 75, 129–135 (1990).
[CrossRef]

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “Two modified Z-scan methods for determination of nonlinear-optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

Opt. Lett.

Optik

J.-G. Tian, W.-P. Zang, G.-Y. Zhang, “The modified Z-scan method with simplicity and enhanced sensitivity,” Optik 98, 143–146 (1995).

Phys. Lett. A

B. Gross, J. T. Manassah, “Numerical solution for the propagation of an elliptic Gaussian beam in a Kerr medium,” Phys. Lett. A 169, 371–378 (1992).
[CrossRef]

Prog. Quantum Electron.

J. H. Marburger, Prog “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Other

K. Kawano, T. Kitou. Introduction to Optical Waveguide Analysis (Wiley, New York, 2001), Chap. 5.
[CrossRef]

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

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, London, 1986).

A. Yariv. Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chap. 6.

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Figures (9)

Fig. 1
Fig. 1

Closed-aperture Z-scan curves with different medium lengths scaled by the Rayleigh length. Two-photon absorption coefficient α2r = 0, ellipticity e = 1, and astigmatism c = 0.

Fig. 2
Fig. 2

Closed-aperture Z-scan curves with different ellipticities (e = 1.0, 1.5, 2.0) and the same beam power. Two-photon absorption coefficient α2r = 0, medium length l = 4, and astigmatism c = 0.

Fig. 3
Fig. 3

Closed-aperture Z-scan curves with different ellipticities (e = 1.0, 1.5, 2.0) and the same beam intensity. Two-photon absorption coefficient α2r = 0, medium length l = 4, and astigmatism c = 0.

Fig. 4
Fig. 4

Closed-aperture Z-scan curves with different astigmatisms. Two-photon absorption coefficient α2r = 0, medium length l = 4, and ellipticity e = 1.

Fig. 5
Fig. 5

Closed-aperture Z-scan curves with different medium lengths scaled by the Rayleigh length. Two-photon absorption coefficient α2r - 5 cm/GW, ellipticity e = 1, and astigmatism c = 0.

Fig. 6
Fig. 6

Open-aperture Z-scan curves with different medium lengths scaled by the Rayleigh length. Ellipticity e = 1, and astigmatism c = 0.

Fig. 7
Fig. 7

Open-aperture Z-scan curves with different ellipticities (e = 1.0, 1.5, 2.0) and the same beam power. The medium length l = 4, and astigmatism c = 0.

Fig. 8
Fig. 8

Open-aperture Z-scan curves with different ellipticities (e = 1.0, 1.5, 2.0) and the same beam intensity. The medium length l = 4, and astigmatism c = 0.

Fig. 9
Fig. 9

Open-aperture Z-scan curves with different astigmatisms. The medium length l = 4, and ellipticity e = 1.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

2Ex2+2Ey2-2ik Ez-iki=1n αi|E|2i-1E+2n2k2n0 |E|2E=0,
ψz¯=a2ψx¯2+2ψy¯2+pψψ,
pψ=a2n2k2w0x2n0 |E0ψ|2-ikw0x2i=1nαi|E0ψ|2i-1,
12ψz¯=a 2ψx¯2+12 pψψ,
12ψz¯=a 2ψy¯2+12 pψψ.
12ψs,tr+1/2-ψs,trΔz¯/2=a2Δx¯2δx2ψs,tr+1/2+δx2ψs,tr+12 pψs,trψs,tr+1/2+ψs,tr,
12ψs,tr+1-ψs,tr+1/2Δz¯/2=a2Δy2δy2ψs,tr+1+δy2ψs,tr+1/2+12 pψs,tr+1/2ψs,tr+1+ψs,tr+1/2,
δx2ψs,tr=ψs+1,tr-2ψs,tr+ψs-1,tr,
δy2ψs,tr=ψs,t+1r-2ψs,tr+ψs,t-1r,
Xsr+1/2ψs,tr+1/2+Xs+1r+1/2ψs+1,tr+1/2+Xs-1r+1/2ψs-1,tr+1/2=Xsrψs,tr+Xs+1rψs+1,tr+Xs-1rψs-1,tr,
Ytr+1ψs,texr+1+Yt+1r+1ψs,t+1r+1+Yt-1r+1ψs,t-1r+1=Ytr+1/2ψs,tr+1/2+Yt+1r+1/2ψs,t+1r+1/2+Yt-1r+1/2ψs,t-1r+1/2,
Xs±1r+1/2=-ρx, Xsr+1/2=1+2ρx-12 pψs,trΔz¯,Xs±1r=ρx,Xsr=1-2ρx-12 pψs,trΔz¯, ρx=aΔz¯2Δx¯2;Ys±1r+1=-ρy, Ysr+1=1+2ρy-12 pψs,tr+1/2Δz¯,Ys±1r+1/2=ρy, Ysr+1/2=1-2ρy-12 pψs,tr+1/2Δz¯,ρy=aΔz¯2Δy¯2.
12ψ1z¯=a 2ψ1x¯2,
12ψ1z¯=a 2ψ1y¯2,
ψ2=ψ1 exppψ1Δz.
Ex, y, z, t=E0tw0xwxzw0ywyz1/2×exp-ikz-ηz×exp-x21wx2z+ik2Rxz-y21wy2z+ik2Ryz,
wx,y2z=wx0,y021+z-zx,y2z0x,0y2,
Rx,yz=z-zx,y1+z0x,0y2z-zx,y2,
ηz=12tan-1z-zxz0x+12tan-1z-zyz0y,
ψs,t0=EsΔx¯, tΔy¯, z¯=0/E0.
ψ-1,tr=ψ1,tr, ψs,-1r=ψs,1r,
ψN,tr=0, ψs,Nr=0.
ψN,tr=0, ψs,Nr=0, ψ-N,tr=0, ψs,-Nr=0.

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