A practical way to resolve ambiguities in wavefront reconstructions by the transport of intensity equation

On a universal solution to the transport-of-intensity equation

The transport-of-intensity equation (TIE) is one of the most well-known approaches for phase retrieval and quantitative phase imaging. It directly recovers the quantitative phase distribution of an optical field by through-focus intensity measurements in a non-interferometric, deterministic manner. Nevertheless, the accuracy and validity of state-of-the-art TIE solvers depend on restrictive pre-knowledge or assumptions, including appropriate boundary conditions, a well-defined closed region, and quasi-uniform in-focus intensity distribution, which, however, cannot be strictly satisfied simultaneously under practical experimental conditions. In this Letter, we propose a universal solution to TIE with the advantages of high accuracy, convergence guarantee, applicability to arbitrarily shaped regions, and simplified implementation and computation. With the "maximum intensity assumption," we first simplify TIE as a standard Poisson equation to get an initial guess of the solution. Then the initial solution is further refined iteratively by solving the same Poisson equation, and thus the instability associated with the division by zero/small intensity values and large intensity variations can be effectively bypassed. Simulations and experiments with arbitrary phase, arbitrary aperture shapes, and nonuniform intensity distributions verify the effectiveness and universality of the proposed method.

Gradient flipping algorithm: introducing non-convex constraints in wavefront reconstructions with the transport of intensity equation

The transport of intensity equation (TIE) is widely applied for recovering wave fronts from an intensity measurement and a measurement of its variation along the direction of propagation. In order to get around the problem of non-uniqueness and ill-conditionedness of the solution of the TIE in the very common case of unspecified boundary conditions or noisy data, additional constraints to the solution are necessary. Although from a numerical optimization point of view, convex constraint as imposed to by total variation minimization is preferable, we will show that in many cases non-convex constraints are necessary to overcome the low-frequency artifacts so typical for convex constraints. We will provide simulated and experimental examples that demonstrate the superiority of solutions to the TIE obtained by our recently introduced gradient flipping algorithm over a total variation constrained solution.

Recovering low spatial frequencies in wavefront sensing based on intensity measurements

The transport of intensity equation (TIE) offers a convenient method to retrieve the phase of a wave function from maps of the irradiance (images) recorded at different planes along the optic axis of an optical system. However, being a second-order partial differential equation, even for noise-free data a unique solution of the TIE requires boundary conditions to be specified which are generally not accessible experimentally, jeopardizing retrieval of the low-frequency information in particular. Here we introduce an iterative algorithm which forgoes the need for explicit boundary conditions and combines the well-known reciprocal space solution of the TIE with the charge-flipping algorithm that has originally been developed to solve the crystallographic phase problem in X-ray diffraction. Application of this algorithm to experimental data and comparison with conventionally used algorithms demonstrates an improved retrieval of the low spatial frequencies of the phase.

Singularimetry of local phase gradients using vortex lattices and in-line holography

We have developed a differential form of singularimetry, which utilizes phase vortices or intensity gradient singularities as topological fiducial markers in a structured illumination context. This approach analytically measures phase gradients imparted by refracting specimens, yielding quantitative information that is both local and deterministic. We have quantified our phase gradient experiments to demonstrate that lattices of wave field singularities can be used to detect subtle phase gradients imparted by a spherical specimen and fiber optic cylinders.