Recursive method for phase retrieval using transport of intensity and its applications.

Propagation of optical fields is governed by the Helmholtz equation or the paraxial wave equation. Transport of intensity is a noninterferometric method to find the phase of an object by recording optical intensities at different distances of propagation. The transport of intensity equation results from the imaginary part of the complex paraxial wave equation and is equivalent to the principle of conservation of energy. The real part of the paraxial wave equation yields the Eikonal equation in the presence of diffraction. The amplitude and phase of the optical field must therefore simultaneously satisfy both the real and imaginary parts of the paraxial wave equation during propagation. In this paper, we demonstrate, with illustrative examples, how to exploit this to retrieve the phase through recursive calculations of the phase and intensity. This is achieved using the transport of intensity equation, which is solved using standard techniques, and the real part of the paraxial wave equation, or the transport of phase equation, which is solved using a Gauss-Seidel iterative method. Examples include calculation of the imaged phase induced through self-phase modulation of a focused laser beam in a liquid and the imaged phase of light reflected from a surface, which yields the 3D surface profile.