Proceedings PaperHelmholtz beam propagation by the method of Lanczos reduction
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The solution of the Helmholtz wave equation requires the application of an exponentiated square root operator to an initial field. This operation is greatly facilitated by the introduction of a representation in which the aforementioned operator is diagonal. The Lanczos method allows this diagonalization to be performed in a low dimensional space, e.g., of the order of 4-6, if one is interested in advancing the field over a limited propagation step of length Az. Although some boundary conditions may be ill-posed for the unapproximated Helmholtz equation, in the sense that certain plane wave components cannot propagate in the forward direction, the Lanczos method damps all of these components exponentially, thus guaranteeing the correctness of the solution.