Optical testing using the transport-of-intensity equation

Flexible dynamic quantitative phase imaging based on division of focal plane polarization imaging technique

This paper proposes a flexible and accurate dynamic quantitative phase imaging (QPI) method using single-shot transport of intensity equation (TIE) phase retrieval achieved by division of focal plane (DoFP) polarization imaging technique. By exploiting the polarization property of the liquid crystal spatial light modulator (LC-SLM), two intensity images of different defocus distances contained in orthogonal polarization directions can be generated simultaneously. Then, with the help of the DoFP polarization imaging, these images can be captured with single exposure, enabling accurate dynamic QPI by solving the TIE. In addition, our approach gains great flexibility in defocus distance adjustment by adjusting the pattern loaded on the LC-SLM. Experiments on microlens array, phase plate, and living human gastric cancer cells demonstrate the accuracy, flexibility, and dynamic measurement performance for various objects. The proposed method provides a simple, flexible, and accurate approach for real-time QPI without sacrificing the field of view.

Optical module for single-shot quantitative phase imaging based on the transport of intensity equation with field of view multiplexing

We present a cost-effective, simple, and robust method that enables single-shot quantitative phase imaging (QPI) based on the transport of intensity equation (TIE) using an add-on optical module that can be assembled into the exit port of any regular microscope. The module integrates a beamsplitter (BS) cube (placed in a non-conventional way) for duplicating the output image onto the digital sensor (field of view - FOV - multiplexing), a Stokes lens (SL) for astigmatism compensation (introduced by the BS cube), and an optical quality glass plate over one of the FOV halves for defocusing generation (needed for single-shot TIE algorithm). Altogether, the system provides two laterally separated intensity images that are simultaneously recorded and slightly defocused one to each other, thus enabling accurate QPI by conventional TIE-based algorithms in a single snapshot. The proposed optical module is first calibrated for defining the configuration providing best QPI performance and, second, experimentally validated by using different phase samples (static and dynamic ones). The proposed configuration might be integrated in a compact three-dimensional (3D) printed module and coupled to any conventional microscope for QPI of dynamic transparent samples.

Two-dimensional Legendre polynomials as a basis for interpolation of data to optimize the solution of the irradiance transport equation analyzed as a boundary problem on surfaces testing

In this paper, we give a solution to the irradiance transport equation (ITE) using the two-dimensional (2D) Legendre polynomials (LPs) and an interpolator (i-LP) based on the LP. In the first place, we analyze the experimental data; subsequently, we proceed to fit the most probable 2D LPs' surface to the data in order to obtain the wavefront surface (W(x,y) of the elements under test) as a solution of the ITE differential equation associated with a boundary problem; and finally, we interpolate the resulting fitting. The interpolation is built from LP to increase the resolution and sharpness of the data. We apply the ITE to these results in order to obtain the wavefront as a nondeterministic solution that increases the resolution of the ITE as an optical test, and we compare our results regarding the obtained aberration surfaces (AS(x,y)).

Multiring pure-phase binary optical elements to extend depth of focus

Multiring pure-phase binary optical elements (BOEs) are widely used to extend the depth of focus (DOF) in many optical applications. Although researchers have designed various BOEs to extend the DOF, few theories and experiments have been reported to validate the performances of different N-ring pure-phase BOEs to realize the DOF as long as possible. In this paper, aberration theory is used to obtain the simple and straightforward initial phase, and a novel modified Gerchberg-Saxton algorithm is presented for generating N-ring 0-π-phase BOEs to optimally extend the DOF. Theoretical, numerical, and experimental results demonstrate that the DOF can be extended with increased N in the same NA.

Measurement of three-dimensional wavefronts using the Ichikawa-Lohmann-Takeda solution to the irradiance transport equation

In this paper, we use the irradiance transport equation and the Fourier transform-based experimental solution given by Ichikawa-Lohmann-Takeda. We analyze experimental factors such as the digital filter, the introduced error for the rotation and period of the Ronchi ruling, and a new method is demonstrated for the measurement of 3D wavefront information.

Fourier-based solving approach for the transport-of-intensity equation with reduced restrictions

The transport-of-intensity equation (TIE) has been proven as a standard approach for phase retrieval. Some high efficiency solving methods for the TIE, extensively used in many works, is based on a Fourier transform (FT). However, several assumptions have to be made to solve the TIE by these methods. A common assumption is that there are no zero values for the intensity distribution allowed. The two most widespread Fourier-based approaches have further restrictions. One of these requires the uniformity of the intensity distribution and the other assumes the parallelism of the intensity and phase gradients. In this paper, we present an approach, which does not need any of these assumptions and consequently extends the application domain of the TIE.

Phase retrieval based on transport of intensity and digital holography

We propose a technique in which intensity images are reconstructed from a digital hologram to provide inputs for the transport-of-intensity equation for unwrapped phase recovery. By doing this, we avoid shifting of the sample or the camera in the experiment, a method commonly employed while using the method of transport-of-intensity equation for phase retrieval. Computer simulations as well as experimental results have been demonstrated to verify the effectiveness of the proposed idea. The underlying numerical technique can also be viewed as an alternative to existing phase-unwrapping algorithms.

High-resolution transport-of-intensity quantitative phase microscopy with annular illumination

For quantitative phase imaging (QPI) based on transport-of-intensity equation (TIE), partially coherent illumination provides speckle-free imaging, compatibility with brightfield microscopy, and transverse resolution beyond coherent diffraction limit. Unfortunately, in a conventional microscope with circular illumination aperture, partial coherence tends to diminish the phase contrast, exacerbating the inherent noise-to-resolution tradeoff in TIE imaging, resulting in strong low-frequency artifacts and compromised imaging resolution. Here, we demonstrate how these issues can be effectively addressed by replacing the conventional circular illumination aperture with an annular one. The matched annular illumination not only strongly boosts the phase contrast for low spatial frequencies, but significantly improves the practical imaging resolution to near the incoherent diffraction limit. By incorporating high-numerical aperture (NA) illumination as well as high-NA objective, it is shown, for the first time, that TIE phase imaging can achieve a transverse resolution up to 208 nm, corresponding to an effective NA of 2.66. Time-lapse imaging of in vitro Hela cells revealing cellular morphology and subcellular dynamics during cells mitosis and apoptosis is exemplified. Given its capability for high-resolution QPI as well as the compatibility with widely available brightfield microscopy hardware, the proposed approach is expected to be adopted by the wider biology and medicine community.

Multimodal computational microscopy based on transport of intensity equation

Transport of intensity equation (TIE) is a powerful tool for phase retrieval and quantitative phase imaging, which requires intensity measurements only at axially closely spaced planes without a separate reference beam. It does not require coherent illumination and works well on conventional bright-field microscopes. The quantitative phase reconstructed by TIE gives valuable information that has been encoded in the complex wave field by passage through a sample of interest. Such information may provide tremendous flexibility to emulate various microscopy modalities computationally without requiring specialized hardware components. We develop a requisite theory to describe such a hybrid computational multimodal imaging system, which yields quantitative phase, Zernike phase contrast, differential interference contrast, and light field moment imaging, simultaneously. It makes the various observations for biomedical samples easy. Then we give the experimental demonstration of these ideas by time-lapse imaging of live HeLa cell mitosis. Experimental results verify that a tunable lens-based TIE system, combined with the appropriate postprocessing algorithm, can achieve a variety of promising imaging modalities in parallel with the quantitative phase images for the dynamic study of cellular processes.

Quantitative phase imaging using transport of intensity equation with multiple bandpass filters

A phase imaging based on the transport of intensity equation using multiple bandpass filters is proposed. The proposed method enables us to measure a phase distribution quantitatively from through-focus intensity images obtained by using a white light source and multiple bandpass filters. The technique improves the accuracy of a phase measurement by increasing the number of intensity images obtained at different defocused positions. The feasibility of the phase measurement and the improvement in the accuracy with the increasing the through-focus images are confirmed by numerical simulations and optical experiments.

Topological properties of the interaction between focusing regions kind cusped

We study here the cusped-cusped interaction between two kinds of Pearcey optical fields by analyzing its topological structure. We do it in two steps; the first one is an irradiance interaction that allows us to identify organization regions. The second one is an amplitude interaction, where it is shown that the interference fringes are organized around the irradiance distribution. The topological behavior of the optical field is analyzed identifying regions with different phase functions, one of them, corresponds with a catastrophe function which has associated a focusing region, the other region can be approximated by a quadratic function. The main consequence heritage from the phase structure is interference fringes emerge from focusing regions having similar features like a topological charges. We show computational and experimental results which are in very well agreement with the theoretical model.

Application of the transport of intensity equation in determination of nonlinear refractive index

We investigate the determination of nonlinear refractive index n(2), based on solving the transport of intensity equation (TIE) in conjunction with a pump-probe technique. As the pump and probe beams propagate through a sample, the pump-induced refractive index variations in the sample change the phase distribution of the probe beam. Using two recorded probe intensities in TIE, this phase change is derived, and so the nonlinear refractive index n(2) is obtained. The influence of some characteristics of the pump beam and noise on the accuracy of determining n(2) is also investigated. The simulation results show that the proposed method has a good capability for determining the nonlinear refractive index of materials.

Application of transport-of-intensity equation in fringe analysis

The transport-of-intensity equation (TIE) is applied in the reconstruction of two interfering wavefronts by analyzing the interference patterns and their derivatives along their common propagation directions. The TIE is extended from one wave to two waves and is then applied to calculate the phase of the interference field. Finally, the phase shift concept is applied to reconstruct the phase distribution of two waves. The consistency of the method is verified by simulation.

Boundary-artifact-free phase retrieval with the transport of intensity equation: fast solution with use of discrete cosine transform

The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. In this work, TIE phase retrieval is considered as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green's function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.

Enhanced deterministic phase retrieval using a partially developed speckle field

A technique for enhanced deterministic phase retrieval using a partially developed speckle field (PDSF) and a spatial light modulator (SLM) is demonstrated experimentally. A smooth test wavefront impinges on a phase diffuser, forming a PDSF that is directed to a 4f setup. Two defocused speckle intensity measurements are recorded at the output plane corresponding to axially-propagated representations of the PDSF in the input plane. The speckle intensity measurements are then used in a conventional transport of intensity equation (TIE) to reconstruct directly the test wavefront. The PDSF in our technique increases the dynamic range of the axial intensity derivative for smooth phase objects, resulting in a more robust solution to the TIE. The SLM setup enables a fast and accurate recording of speckle intensity. Experimental results are in good agreement with those obtained using the iterative phase retrieval and digital holographic methods of wavefront reconstruction.

Refractive index determination of transparent samples by noniterative phase retrieval

We present a simple method to determine the refractive indices of transparent specimens. The refractive index of an object under investigation is received by evaluating the optical path difference introduced by the object, while taking into account geometric parameters. The optical path difference that corresponds to the phase distribution is obtained by a noninterferometric, noniterative phase retrieval method based on Green's functions. It will be shown that this technique is a highly accurate and quantitative method for refractive index determination.

Topography retrieval using different solutions of the transport intensity equation

The topography of a phase plate is recovered from the phase reconstruction by solving the transport intensity equation (TIE). The TIE is solved using two different approaches: (a) the classical solution of solving the Poisson differential equation and (b) an algebraic approach with Zernike functions. In this paper we present and compare the topography reconstruction of a phase plate with these solution methods and justify why one solution is preferable over the other.

Non-interferometric, non-iterative phase retrieval by Green's functions

In this paper a non-interferometric, non-iterative method for phase retrieval by Green's functions is presented. The theory is based on the parabolic wave equation that describes propagation of light in the Fresnel approximation in homogeneous media. Green's first identity will be used to derive an algorithm for phase retrieval considering different boundary conditions. Finally it will be shown that a commonly used solution of the transport-of-intensity equation can be obtained as a special case of the more general Green's function formulation derived here.