### ASA 127th Meeting M.I.T. 1994 June 6-10

## 3aUW5. A perturbative Lanczos solution method for alternative wide-angle
parabolic equations.

**David Yevick
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**
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*Dept. of Elec. Eng., Queen's Univ., Kingston, ON K7L 3N6, Canada
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In this talk, the Lanczos procedure is applied to the high-order
alternative wide-angle propagation operator. This operator is obtained by
approximating [radical 1+(epsilon)+(mu)[radical , where (epsilon)=n[sup 2](Z)-1
and (mu)=k[sub 0][sup -2](cursive beta)/(cursive beta)z[sup 2], by the
expression [radical 1+(epsilon)+(mu)[radical =-1+[radical 1+(epsilon)[radical
+[radical 1+(mu)[radical
-1/8((epsilon)(mu)+(mu)(epsilon))+1/16[((epsilon)+(mu))[sup 3]-(epsilon)[sup
3]-(mu)[sup 3]]+... . The exponential of the first three terms on the
right-hand side of the above expression correspond to the Thomson--Chapman
propagator and can be evaluated with the standard split-step Fourier algorithm.
The remaining correction terms can be implemented either through standard Pade
approximants or with the Lanczos procedure. Unlike the Pade formalism, the
perturbative Lanczos scheme can be implemented equally efficiently in both two
and three spatial dimensions. The procedure further does not suffer from the
previously documented convergence difficulties associated with the Lanczos
evaluation of the full square-root operator [Hermansson et al., IEEE J. Light.
Technol. 10, 772--776 (1992)] and is therefore highly computationally
efficient. Finally, it is verified that the Lanczos method yields highly
accurate results for a leaky surface duct test case.