Abstract

We have developed a modified Arnoldi method that includes a complex square-root approximation, which excels at modeling the propagation of highly diverging beams in various media. Simulations of one transverse dimensional beam with an ultranarrow width and of cylindrical Gaussian beams with various divergence angles reveal the strength of this nonparaxial-beam propagation method.

© 2001 Optical Society of America

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References

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  7. Y. Y. Lu, SIAM J. Matrix Anal. 19, 833 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2000 (2)

H. El-Refaei, I. Betty, and D. Yevick, IEEE Photon. Technol. Lett. 12, 158 (2000).
[CrossRef]

Q. Luo and C. T. Law, Opt. Lett. 25, 869 (2000).
[CrossRef]

1998 (2)

R. B. Sidje, ACM Trans. Math. Software 24, 156 (1998).
[CrossRef]

Y. Y. Lu, SIAM J. Matrix Anal. 19, 833 (1998).
[CrossRef]

1997 (1)

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, J. Acoust. Soc. Am. 101, 760 (1997).
[CrossRef]

1994 (3)

D. Yevick, Opt. Quantum Electron. 26, S185 (1994).
[CrossRef]

Q. Ye, Math. Comput. 62, 179 (1994).
[CrossRef]

R. P. Ratowsky, J. A. Fleck, and M. D. Feit, Opt. Lett. 19, 1284 (1994).
[CrossRef] [PubMed]

1992 (2)

1991 (1)

Betty, I.

H. El-Refaei, I. Betty, and D. Yevick, IEEE Photon. Technol. Lett. 12, 158 (2000).
[CrossRef]

Brooke, G. H.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, J. Acoust. Soc. Am. 101, 760 (1997).
[CrossRef]

El-Refaei, H.

H. El-Refaei, I. Betty, and D. Yevick, IEEE Photon. Technol. Lett. 12, 158 (2000).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Law, C. T.

Lu, Y. Y.

Y. Y. Lu, SIAM J. Matrix Anal. 19, 833 (1998).
[CrossRef]

Luo, Q.

Milinazzo, F. A.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, J. Acoust. Soc. Am. 101, 760 (1997).
[CrossRef]

Ratowsky, R. P.

Saad, Y.

Y. Saad, SIAM J. Numer. Anal. 29, 209 (1992).
[CrossRef]

Sidje, R. B.

R. B. Sidje, ACM Trans. Math. Software 24, 156 (1998).
[CrossRef]

Ye, Q.

Q. Ye, Math. Comput. 62, 179 (1994).
[CrossRef]

Yevick, D.

H. El-Refaei, I. Betty, and D. Yevick, IEEE Photon. Technol. Lett. 12, 158 (2000).
[CrossRef]

D. Yevick, Opt. Quantum Electron. 26, S185 (1994).
[CrossRef]

Zala, C. A.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, J. Acoust. Soc. Am. 101, 760 (1997).
[CrossRef]

ACM Trans. Math. Software (1)

R. B. Sidje, ACM Trans. Math. Software 24, 156 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. El-Refaei, I. Betty, and D. Yevick, IEEE Photon. Technol. Lett. 12, 158 (2000).
[CrossRef]

J. Acoust. Soc. Am. (1)

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, J. Acoust. Soc. Am. 101, 760 (1997).
[CrossRef]

Math. Comput. (1)

Q. Ye, Math. Comput. 62, 179 (1994).
[CrossRef]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

D. Yevick, Opt. Quantum Electron. 26, S185 (1994).
[CrossRef]

SIAM J. Matrix Anal. (1)

Y. Y. Lu, SIAM J. Matrix Anal. 19, 833 (1998).
[CrossRef]

SIAM J. Numer. Anal. (1)

Y. Saad, SIAM J. Numer. Anal. 29, 209 (1992).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Overlap error ϵ versus step size for various ways to calculate square roots (exact, exact operation with eigenvalue decomposition; no preconditioner, approximated operation with the Padé expansion and no preconditioner; and with preconditioner, approximated operation with the Padé expansion and a preconditioner) for various orders of MAM with grid spacing Δx=λ/2 and N=1024.

Fig. 2
Fig. 2

Overlap error ϵ versus step size for the same conditions as for Fig.  1, except now with grid spacing Δx=λ/4 and N=2048.

Fig. 3
Fig. 3

Relative error ϵZ of paraxial propagation and nonparaxial calculations with the MAM (8th-order method with 512 grid points for the Padé approximation without a preconditioner) versus half divergence angle θ1/2 for Gaussian beams of various initial sizes propagating in free space for Z=0.1.

Fig. 4
Fig. 4

Relative error ϵZ versus half divergence angle θ1/2 under the same conditions as for Fig.  3, except that Z=5.

Equations (12)

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ψ/Z=j4k0z01+2k0z0-1H1/2-1ψ,
ψZ+ΔZ=exp(j4k0z01+2k0z0-1H1/2-1×ΔZ)ψZ.
expj4k0z0ΔZZ̲-I̲=S̲expj4k0z0ΔZD̲S̲-1.
q¯p=ψ¯Z/ψ¯Z,w¯p=Z̲q¯pp=1,
q¯p=w¯p-1-n=1p-1hn,p-1q¯n/hp,p-1,w¯p=Z̲q¯p-1-n=1p-1hn,p-1w¯n/hp,p-1p=2,,M+1,
hi,p=q¯i|w¯p,hp+1,p=w¯pi=1,,p,p=1,M+1,
Z̲=I̲+X̲C0I̲+n=1NpI̲+BnX̲-1AnX̲,
C0=expjα/21+n=1Npanexp-jα-11+bnexp-jα-1,
An=anexp-jα/21+bnexp-jα-1-2,
Bn=bnexp-jα1+bnexp-jα-1-1.
I̲+X̲=I̲+X̲appI̲+X̲.
ψLX,Z=m=-N/2+1N/2ΨkmexpjkmX×expj4k0z0βmZ,

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