Retrieval of phase and three-dimensional topography using modified transport of intensity and phase equations with electrically programmable optical path lengths

Abstract. Transport of intensity (TI) is a well-known non-interferometric technique for phase retrieval. The TI and phase equations result from the Helmholtz equation and show the coupling of intensity and phase during optical propagation. TI is an alternative to digital holography, which requires a reference beam for a recording of the interference pattern. However, the conventional TI method has an experimental challenge in that mechanical displacement of the camera or object is needed to record the optical intensities at multiple defocused planes, which can cause errors from misalignments. This work expands on a modified TI technique that avoids mechanical displacements, instead invoking the use of electrooptic materials to create an optical phase difference and hence optical path length through the application of a bias voltage. We demonstrate the use of the modified TI equation (TIE) through simulation and experiment by selecting suitable objects and a biased nematic liquid crystal cell made from pentyl-4-cyanobiphenyl (5CB). The corresponding modified transport of phase equation is also derived and is used to enhance the accuracy of the modified TIE. After providing simulation results for imaged phase retrieval, we demonstrate the unwrapped image phase and hence height or profile extraction for an object with three-dimensional topography using this technique.