Intensity Equation Method Research Articles

Evolution of magnetization textures in Mn1.4PtSn under magnetic field

We have used Lorentz transmission electron microscopy (LTEM) and the transport of intensity equation (TIE) method to investigate the spin structures in the Mn1.4PtSn magnet, which exhibits D2d symmetry. The evolution of the spin structures under the external magnetic field was observed. In particular, a perpendicular field is able to modulate the striped domain widths and to induce the formation of anti-skyrmions and bubbles. While an in-plane magnetic field can control the state of these structures, the coexistence of different types of skyrmions suggests that the hysteresis properties of the magnet may affect the effectiveness of the field control.

Accurate decoding of data pages in an amplitude- and phase-modulated signal beam detected by the single-shot transport of intensity equation method with convolutional neural network-based classifiers

We propose an accurate method for the classification and decoding of data pages in complex-amplitude-modulated signal beams detected via the transport of the intensity equation (TIE) method using simple classifiers with a convolutional neural network (CNN) for holographic data storage (HDS). The classifiers allow a single-shot TIE method using two cameras to detect the complex-amplitude-modulated signal beam in HDS. Although the phase distribution detected using the single-shot TIE method tends to be superimposed with strong phase noise, we demonstrate experimentally that the CNN-based classifiers can classify/decode data pages in the complex-amplitude-modulated signal beam accurately without phase noise removal.

Correction to: Generation of amplitude- and phase-modulated signal beam with a phase-only spatial light modulator and its detection by the transport of intensity equation method for holographic data storage

Correction to: Generation of amplitude- and phase-modulated signal beam with a phase-only spatial light modulator and its detection by the transport of intensity equation method for holographic data storage

Generation of amplitude- and phase-modulated signal beam with a phase-only spatial light modulator and its detection by the transport of intensity equation method for holographic data storage

Generation of amplitude- and phase-modulated signal beam with a phase-only spatial light modulator and its detection by the transport of intensity equation method for holographic data storage

Recent Advances in Coherent Optical Communications for Short-Reach: Phase Retrieval Methods

Short-reach transmission systems traditionally utilize intensity modulation (IM) at the transmitter and direct detection (DD) at the receiver due to their cost-effectiveness, small footprint, and low power consumption. However, with the exponential increase in bandwidth demand, coherent optical communication systems have become necessary for long-haul distances, requiring application-specific integrated circuits (ASIC) and advanced digital signal processing (DSP) algorithms coupled with high-speed digital-to-analog and analog-to-digital converters to achieve Tbit/s speeds. As coherent technology matures, it will eventually become feasible for short-reach transmission. In this context, self-coherent systems have emerged as an intermediary solution, offering advantages over traditional IM/DD systems. While comprehensive review studies exist on self-coherent transceivers, they do not cover recent advances in phase retrieval methods for short-reach optical communications. This review article highlights recent developments in cost-effective self-coherent detection for short-reach systems through comparing the benefits of single sideband (SSB) transmission and Kramers-Kronig detection to carrier-assisted phase retrieval, the Gerchberg-Saxton (GS) algorithm, and the transport of intensity equation (TIE) method.

Binocular phase retrieval based on the transport of intensity equation under multiwavelength illumination

In view of the limitation that the transport of intensity equation method can only be used at a single wavelength, we propose a binocular phase retrieval algorithm based on multiwavelength illumination. First, we solve for the corresponding single-phase results at different wavelengths and then calculate the surface height of the object by combining the obtained synthetic wavelengths and synthetic phases, in order to reconstruct the initial phase results at different wavelengths. To solve the problem of amplification of traditional noise and surface profile noise in the phase diagram in the phase synthesis step, we introduce a step-by-step noise reduction method that can reduce the noise in the initial phase results to the level for a single wavelength and finally obtain the corresponding high-precision phase results at different wavelengths. The intensity images collected by the proposed dual-microscope system verify the correctness and effectiveness of the algorithm.

Solving the twin image problem in in-line holography by using multiple defocused intensity images reconstructed from a single hologram

Phase recovery from a hologram measurement is at the heart of coherent imaging techniques and online holography. In-line holography suffers from a blurred background problem that results from double images formed during hologram reconstruction. This so-called double image problem, inherent to the in-line geometry of the holographic experimental setup, limits the quality of the information (shape or refractive index of the object) obtained by reconstruction algorithms using a single hologram. We propose here a numerical reconstruction method combining the angular spectrum method (ASM) and the transport of intensity equation method using Gaussian process regression (GP-TIE) to solve this problem. In our approach, a single in-line hologram is used to recover the phase image free of any perturbation due to the presence of the double image.

Pebrine diagnosis using quantitative phase imaging and machine learning.

Pebrine is the most dreaded infectious disease of the silkworm and has devastated sericulture in Europe during the 18th century. Thereafter, if it is detected, the crop is burned to prevent further dissemination. The conventional microscopic examination of moth's body fluid is erroneous and it exacerbates on Metarhizium anisopliae (MA) contaminated test samples. This is due to the resemblance of pebrine and MA spores in the microscopic examination. Therefore, this study aims to demonstrate an efficient pebrine detection technique. In the proposed method, a motorised brightfield microscope is custom-made to acquire focused and defocused images of test spores. These images are used to produce quantitative phase images of the spores by the transport of intensity equation method. The phase images' histogram of oriented gradients feature is used by a machine learning classifier to categorise the spores. This system classified 92 pebrine and 185 MA spores with an accuracy of 97% at 0.04 seconds/spore. The duration taken for image acquisition is 2.5minutes per sample (10 fields of view covering an area of 302×260 μm2 ). The proposed method shows reliable results in pebrine diagnosis and would be an efficient alternative for current approaches.

Transport of intensity equation method and its applications.

A phase retrieval technique based on a transport of intensity equation (TIE) is one of the defocus series reconstruction techniques in microscopy. Since it does not require any dedicated devices like a biprism, and only three defocus images are enough to retrieve phase information, it has been applied to observe magnetic fields, magnetic domains, electrostatic potentials and strains. It is also used to improve image resolution by correcting spherical aberration. This technique is simple and easy to use, but some artifacts often appear in the retrieved phase map. One should pay careful attention to the experimental conditions and the algorithms and boundary conditions used to solve the TIE. This paper reviews the principle of the TIE method, the algorithms used to solve it and application results in materials science.

Transport of intensity equation method based on edge detection and duty ratio fusion

The non-interferometric phase recovery method which is based on the transport of intensity equation is an important phase acquisition technique in microscopy imaging. In the actual operational process of this technology, at least three intensity images are needed to solve the equation. In the image acquisition process, the selection of the focused image is very important, but it is usually determined by subjective methods, thus causing inaccurate positioning of the in-focus, which further affects the precision of phase recovery result. A phase recovery algorithm based on edge detection and duty ratio is proposed. Firstly, a series of ‘subjectively determined focused images’ collected by the microscope are edge-extracted to obtain corresponding edge information points. Then the real in-focus position is found by calculating the duty ratio, and the corresponding defocused images are selected according to the position of the focused image. Finally, the high-precision phase recovery result is obtained by using these images of the accurate position to solve the equation. The algorithm does not require complex operations such as hardware modifications and can be applied to both discrete and continuous sample distributions. The experimental results effectively prove that the algorithm can accurately find the in-focus position, which improves the accuracy of phase recovery compared with the traditional method that is subjectively determined in-focus image position.

Single-shot common-path transport of intensity equation method with Greek-ladder sieves

Phase recovery based on the transport of intensity equation (TIE) is a promising method for quantitative phase imaging, which is useful for the observation of dynamic objects. This study proposes a single-shot common-path TIE method with Greek-ladder sieves to reconstruct the phase distribution of small transmissive objects. A Greek-ladder sieve was designed to simultaneously capture bi-planar images at a single recording plane; thus, the phase distribution is quantitatively retrieved using a single-shot measurement of the intensity. A 120 µm diameter Greek-ladder sieve was designed to operate at a wavelength of 13.5 nm, and its feasibility was verified in the optical region experimentally. With the advantages of operating in real time and reducing the effects of vibration, this method can be potentially applied for various imaging and detection techniques, especially for biological imaging and wave front detection.

Detection method for the complex amplitude of a signal beam with intensity and phase modulation using the transport of intensity equation for holographic data storage.

Holographic data storage (HDS), in which both the amplitude and the phase of a signal beam are modulated, has been extensively studied with the goal of increasing its storage capacity. To detect such modulation during data retrieval, it is necessary to acquire the complex amplitude of the signal beam. In this study, we focus on the transport of intensity equation (TIE) method, which allows us to detect the phase distribution of the light wave quantitatively without using interferometry, contributing to miniaturization of the optical system and improvement of the vibration tolerance of HDS. We discuss the conditions of the modulation phase distribution of the signal beam required for accurate phase detection and propose a method to estimate and eliminate the noise that frequently appears in the phase distribution detected by the TIE method.

Hybrid method for accurate phase retrieval based on higher order transport of intensity equation and multiplane iteration

Because phase contains specific information of the sample, especially transparent specimens such as biological cells and tissues, it is of utmost significance to accurately extract the phase. Although the phase can be directly solved from multiplane intensities through the prevalent transport of intensity equation (TIE) method, too many approximations in the solving process ultimately reduce its recovery accuracy. In addition, an iterative idea as a classic but potential tool allows quantitatively reconstructing phase while it is prone to trap in local stagnation due to nonconvexity. In order to restore the phase quantitatively and precisely, we propose a hybrid method combining the initial phase value via higher order TIE for system error reduction, and a final recovery solution using multiplane iteration for phase accuracy enhancement. Simulation and experimental results show that the proposed method not only has good noise suppression ability but also high phase recovery accuracy, and thus can be used in fields such as biological observation and medical diagnosis in the future.

Higher Order Transport of Intensity Equation Methods: Comparisons and Their Hybrid Application for Noise Adaptive Phase Imaging

As a non-interferometric method, the transport of intensity equation (TIE) method suits for quantitative phase imaging with the commercial microscope platform, especially those higher order TIE approaches which can realize precise phase retrieval by eliminating undesired higher order derivatives are widely used. However, these approaches are mostly adopted separately without considering their phase retrieval precision in various noise levels. In this paper, we first compared these classical higher order TIE approaches through theoretical analysis, numerical simulations and experiments. Then, based on the quantitative comparisons mainly focusing on their phase retrieval accuracy and noise suppression capability, we determine the application scope corresponding to different noise levels of each higher order TIE approach. Finally, in order to deal with different noisy cases, we design the hybrid higher order TIE application, in which the specific higher order TIE approach is selected for phase retrieval according to the precise noise estimation, and the performance of the hybrid higher order TIE application is tested by both the numerical simulations and the experiments, proving it can perform high-quality phase imaging by balancing the trade-off between the phase retrieval accuracy and the noise influence. The paper not only provides a systematic reference for analysis and comparisons on different higher order TIE approaches, but also proposes the hybrid application for noise adaptive phase imaging, which can be a potential tool in biological observations and medical diagnostics.

Digital field of view correction combined dual-view transport of intensity equation method for real-time quantitative imaging

Dual-view transport of intensity equation (TIE) method is an ideal way for quantitative live cell imaging as it has advantages such as real-time imaging, multimode observations, compact setup, and large field of view (FoV). However, due to the image recorder installation error, the inevitable FoV mismatch between the captured under- and over-focus intensities reduces the accuracy in both amplitude and phase retrievals. Here, to eliminate this undesired FoV mismatch, the phase correlation-based digital FoV correction is adopted to recognize and compensate the rotation, scale, and translation between the under- and over-focus images. Both the numerical simulations as well as the experiments in standard sample detection and quantitative live cell imaging prove that the digital FoV correction combined dual-view TIE method can maintain the consistence of the dual FoVs, thus guaranteeing the high-accurate amplitude and phase computations, proving the proposed method is a promising quantitative live cell imaging tool in various applications such as biological observations and medical diagnostics.

Numerical tilting compensation in microscopy based on wavefront sensing using transport of intensity equation method

Wide-field microscopy is commonly used for sample observations in biological research and medical diagnosis. However, the tilting error induced by the oblique location of the image recorder or the sample, as well as the inclination of the optical path often deteriorates the imaging quality. In order to eliminate the tilting in microscopy, a numerical tilting compensation technique based on wavefront sensing using transport of intensity equation method is proposed in this paper. Both the provided numerical simulations and practical experiments prove that the proposed technique not only accurately determines the tilting angle with simple setup and procedures, but also compensates the tilting error for imaging quality improvement even in the large tilting cases. Considering its simple systems and operations, as well as image quality improvement capability, it is believed the proposed method can be applied for tilting compensation in the optical microscopy.

Rapid in-focus corrections on quantitative amplitude and phase imaging using transport of intensity equation method.

Transport of intensity equation (TIE) method can acquire sample phase distributions with high speed and accuracy, offering another perspective for cellular observations and measurements. However, caused by incorrect focal plane determination, blurs and halos are induced, decreasing resolution and accuracy in both retrieved amplitude and phase information. In order to obtain high-accurate sample details, we propose TIE based in-focus correction technique for quantitative amplitude and phase imaging, which can locate focal plane and then retrieve both in-focus intensity and phase distributions combining with numerical wavefront extraction and propagation as well as physical image recorder translation. Certified by both numerical simulations and practical measurements, it is believed the proposed method not only captures high-accurate in-focus sample information, but also provides a potential way for fast autofocusing in microscopic system.

Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method.

In order to realize high contrast imaging with portable devices for potential mobile healthcare, we demonstrate a hand-held smartphone based quantitative phase microscope using the transport of intensity equation method. With a cost-effective illumination source and compact microscope system, multi-focal images of samples can be captured by the smartphone's camera via manual focusing. Phase retrieval is performed using a self-developed Android application, which calculates sample phases from multi-plane intensities via solving the Poisson equation. We test the portable microscope using a random phase plate with known phases, and to further demonstrate its performance, a red blood cell smear, a Pap smear and monocot root and broad bean epidermis sections are also successfully imaged. Considering its advantages as an accurate, high-contrast, cost-effective and field-portable device, the smartphone based hand-held quantitative phase microscope is a promising tool which can be adopted in the future in remote healthcare and medical diagnosis.

Spatial-light-modulator-based adaptive optical system for the use of multiple phase retrieval methods.

We propose an adaptive optical setup using a spatial light modulator (SLM), which is suitable to perform different phase retrieval methods with varying optical features and without mechanical movement. By this approach, it is possible to test many different phase retrieval methods and their parameters (optical and algorithmic) using one stable setup and without hardware adaption. We show exemplary results for the well-known transport of intensity equation (TIE) method and a new iterative adaptive phase retrieval method, where the object phase is canceled by an inverse phase written into part of the SLM. The measurement results are compared to white light interferometric measurements.

In-focus quantitative intensity and phase imaging with the numerical focusing transport of intensity equation method

Microscopy combined with the transport of intensity equation is capable of retrieving both intensity and phase distributions of samples from both in-focus and defocus intensities. However, during measurements, the focal plane is often decided artificially and the improper choice may induce errors in quantitative intensity and phase retrieval. In order to obtain accurate in-focus information, quantitative intensity and phase imaging with the numerical focusing transport of intensity equation method combined with cellular duty ratio criterion and numerical wavefront propagation is introduced in this paper. Both numerical simulations and experimental measurements are provided proving this designed method can increase both retrieved in-focus intensity and phase accuracy and reduce dependence of focal plane determination in transport of intensity equation measurements. It is believed that the proposed method can be potentially applied in various fields as in-focus compensation for quantitative phase imaging and automatic focal plane determination, etc.