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.
The amplitude and phase of the complex optical field in the Helmholtz equation obey a pair of coupled equations, arising from equating the real and imaginary parts. The imaginary part yields the transport of intensity equation (TIE), which can be used to derive the phase distribution at the observation plane. If a phase object is approximately imaged on the recording plane(s), TIE yields the phase without the need for phase unwrapping. In our experiment, the 3D image of a phase object and an amplitude object embedded in a phase object is recovered. The phase object is created by heating a liquid, comprising a solution of red dye in alcohol, using a focused 514 nm laser beam to the point where self-phase modulation of the beam is observed. The optical intensities are recorded at various planes during propagation of a low power 633 nm laser beam through the liquid. In the process of applying TIE to derive the phase at the observation plane, the real part of the complex equation is also examined as a cross-check of our calculations. For pure phase objects, it is shown that the real part of the complex equation is best satisfied around the image plane. Alternatively, it is proposed that this information can be used to determine the optimum image plane.