An optimized transport-of-intensity solution for phase imaging

The transport-of-intensity equation (TIE) is often used to determine the phase and amplitude profile of a complex object by monitoring the intensities at different distances of propagation or around the image plane. TIE results from the imaginary part of the paraxial wave equation and is equivalent to the conservation of energy. The real part of the paraxial wave equation gives the eikonal equation in the presence of diffraction. Since propagation of the optical field between different planes is governed by the (paraxial) wave equation, both real and imaginary parts need to be satisfied at every propagation plane. In this work, the solution of the TIE is optimized by using the real part of the paraxial wave equation as a constraint. This technique is applied to the more exact determination of imaging the induced phase of a liquid heated by a focused laser beam, which has been previously computed using TIE only. Retrieval of imaged phase using the TIE is performed by using the constraint that naturally arises from the real part of the paraxial wave equation.