Abstract

The Helmholtz equation plays a central role in the description of propagation phenomena in optics and acoustics. The paraxial approximation to the Helmholtz equation, also known as the paraxial wave equation, has long been the instrument of choice for performing calculations because it is amenable to solution by accurate marching techniques. The generation of accurate solutions to the unapproximated Helmholtz equation by marching, on the other hand, requires the evaluation of a square root operator applied to some initial field. By using an orthogonalization procedure due to Lanczos one can generate a low-dimensional representation, valid over a sufficiently short propagation step, which accurately diagonalizes the square root operator.¹

© 1992 Optical Society of America