The finite-difference matrix for beam propagation: eigenvalues and eigenvectors

The partial differential equation for the three dimensional propagation of a light beam may be solved numerically by applying finite-difference techniques. We consider the matrix equation for the finite-difference, alternating direction implicit (ADI), numerical solution of the paraxial wave equation for the free-space propagation of light beams. The matrix is tridiagonal. It is also a Toeplitz matrix; Each diagonal descending from left to right is constant. Eigenvalues and eigenvectors are known for such matrices. The equation can be solved by making use of the orthogonality property of the eigenvectors.