Beam propagation in gradient refractive-index media

A generalized beam propagation method is described that uses the ABCD matrices to treat optical systems that have modest amounts of aberrations including gradient refractive-index elements. We can make calculations from any point in the near or far field to any other point by using appropriate numerical algorithms. The variation of the reduced length is discussed as a limitation to accuracy. The diffraction properties of a complex stigmatic system may be represented by those of an equivalent elementary system. This facilitates calculations using the standard diffraction operations for homogeneous media. The modified propagation technique replaces the large number of diffraction steps commonly used for the split-step solution of inhomogeneous media with one step for stigmatic media and in general no more than a few steps for aberrated media. Maxwell's fisheye lens is discussed in detail to show application of the method.