Number Of Fringes Research Articles

High-accuracy fringe projection profilometry without phase unwrapping based on multi-view geometry constraints

The number of fringes and phase unwrapping in fringe projection profilometry result in two key factors. The first is to avoid the problems of excessive fringe patterns, and the second is phase ambiguity. This paper presents a three-dimensional (3D) measurement method without phase unwrapping. This method benefits from the geometric constraints and does not require additional images. Meanwhile, epipolar rectification is performed to calibrate the rotation matrix relationship between the new plane of the dual camera and the plane of the projector. Subsequently, using depth constraints, the point pairs with incorrect 3D positions are effectively eliminated, and the initial parallax map is obtained by establishing epipolar lines of the left and right matching points in the projector domain, obtaining the intersection points, and setting up the threshold for filtering. Finally, a function combining the modulation intensity and phase is proposed to refine the parallax map such that the 3D result is insensitive to phase error. The standard step block and standard ball were used to verify the validity of the proposed method, and the experimental results showed that the root mean square error of the method was 0.052 mm.

Modeling window smoothing effect hidden in fringe projection profilometry

The reconstruction using MEMS micro-vibration mirrors is the most promising. However, the influence mechanism of the line laser, which serves as the sole light source of MEMS-based systems, on reconstruction is unclear. To fill this theoretical gap, we propose the first interpretable laser influence model called the window smoothing model. This model reveals the window smoothing effect of laser beam scanning and quantifies its impact on modulation intensity through a decay factor. Given the absence of lens defocusing issues, the window smoothing effect becomes the primary factor affecting the contrast of fringe patterns. Therefore, a novel method using laser parameters to evaluate system accuracy is introduced. Leveraging these insights, we further propose an optimal fringe number calibration method, also based on laser parameters, to maximize the reconstruction accuracy of any given system. Finally, experimental results validate the effectiveness of the proposed method. As the first model tailored for MEMS systems, the window smoothing model is interpretable, highly accurate, and universal. Therefore, this model can provide new insights and a theoretical foundation for achieving high-performance MEMS-based reconstruction.

Two-plus-two fringe projection profilometry based on phase-shifted coding

In fringe projection profilometry based on temporal phase unwrapping, determining a fringe order map commonly requires a large number of fringes. To reduce the fringe number, this paper proposes a concise absolute phase retrieval algorithm just by projecting four fringes. The first two orthogonal fringes with relatively large frequency can collect reliable height information. The second two fringes are designed the same as the first two, but the only difference is that each 2π-phase of them is shifted by a unique amount, which can robustly label a large number of fringe orders. For decoding the fringes, we develop an average intensity one-time extraction algorithm, which allows for the rapid acquisition of the two pairs of alternating current components. From this, the wrapped phase containing height information and the stair-coded phase providing fringe orders can be directly extracted by arctangent operation in a point-to-point manner. Furthermore, we also develop a universal fringe order correction algorithm that can simultaneously correct the common errors and the misalignment between the wrapped phase and fringe orders. Experiment results demonstrate that this method achieves comparable accuracy and adaptability to the phase-coding method, while utilizing two fewer fringes.

Investigating the origin of the far-field reflection interference fringe (RIF) of microdroplets

We show that the reflection interference fringe (RIF) is formed on a screen far away from the microdroplets placed on a prism-based substrate, which have low contact angles and thin droplet heights, caused by the dual convex–concave profile of the droplet, not a pure convex profile. The geometric formulation shows that the interference fringes are caused by the optical path difference when the reflected rays from the upper convex profile at the droplet–air interface interfere with reflection from the lower concave profile at oblique angles lower than the critical angle. Analytic solutions are obtained for the droplet height and the contact angle out of the fringe number and the fringe radius in RIF from the geometric formulation. Furthermore, the ray tracing simulation is conducted using the custom-designed code. The geometric formulation and the ray tracing show excellent agreement with the experimental observation in the relation between the droplet height and the fringe number and the relation between the contact angle and the fringe radius. This study is remarkable as the droplet's dual profile cannot be easily observed with the existing techniques. However, the RIF technique can effectively verify the existence of a dual profile of the microdroplets in a simple setup. In this work, the RIF technique is successfully developed as a new optical diagnostic technique to determine the microdroplet features, such as the dual profile, the height, the contact angle, the inflection point, and the precursor film thickness, by simply measuring the RIF patterns on the far-field screen.

A systematic study on linear thermal expansion coefficient of metals based on interferometric measurement with Fresnel bimirror

AbstractLinear thermal expansion coefficient, which is vital for measuring the thermal expansion characteristics of metals, has been attracting considerable attention globally. Herein, a novel design based on Fresnel bimirror has been developed. In this design, when the upper end of the object to be measured comes in contact with a tilted double‐sided mirror, the temperature rises and intersection angle of the Fresnel bimirror decreases. Meanwhile, interference fringe spacing becomes narrower, while the number of fringes increases. An imaging system based on a digital microscope and smartphone is also incorporated in this design, which records the changes in the interference fringes. Then, using a self‐programmed software, the linear thermal expansion coefficients of Cu, Fe, and Al samples are determined at elevated temperatures as 17.85 ± 0.23 × 10−6/°C (), 11.8 ± 0.09 × 10−6/°C (), and 23.34 ±0.16 × 10−6/°C (), respectively, with a relative error of less than 1.6%. A cooling process is also designed, and the average value of the linear thermal expansion coefficient of metal samples during heating and cooling conditions is determined. The measurement results obtained via the finite‐method simulation demonstrate the feasibility and reliability of the system. Overall, this study provides a new idea for measuring the linear thermal expansion coefficient of metals.

Large-field and fast super-resolution microscopic imaging method based on laser interferometry

In recent years, structured illumination microscopy (SIM) has been drawing great attention for both technique development and application. However, conventional SIM, which uses a spatial light modulator (SLM) for fringe projection, often has a limited field of view. To meet the demand for high-throughput microscopic imaging in biomedicine research, a large-field super-resolution (SR) fluorescence microscopic imaging method based on laser interferometry was proposed. The method that combines a two-dimensional (2D) grating for fringe pattern projection and an SLM for selecting fringe orientation can break the limitation of fringe number limited by the digital projection devices. A spatial-domain reconstruction algorithm was developed to improve the computational speed of super-resolution imaging. Finally, an experimental platform for SIM microscopy was established. A large-field view of 1380 μm × 1035 μm under a 20×/NA0.75 objective is experimentally demonstrated, and an enhancement of 1.8-fold resolution is realized. The spatial-domain reconstruction algorithm can significantly improve the computational speed by approximately 10 times faster compared to the traditional frequency-domain algorithm.

Adaptive phase-shifting interferometry based on a phase-shifting digital holography algorithm.

Phase-shifting interferometry (PSI) requires accurate phase shifts between interferograms for realizing high-accuracy phase retrieval. This paper presents an adaptive PSI through synchronously capturing phase shifts measurement interferograms and phase measurement interferograms, in which the former is a series of spatial carrier frequency phase-shifting interferograms generated by an additional assembly and the phase shifts are calculated with the single-spectrum phase shifts measurement algorithm (SS-PSMA), the latter is employed for phase retrieval with an adaptive phase-shifting digital holography algorithm (PSDHA) based on complex amplitude recovery. In addition to exhibiting excellent reliability, high-accuracy phase retrieval (0.02 rad), and short calculation time (<25 ms), the proposed adaptive PSDHA is suitable for various interferograms with different fringe shapes and numbers. Importantly, both simulation analysis and experimental result demonstrate that this adaptive PSI based on PSDHA can effectively eliminate phase-shifting errors caused by phase shifter and external disturbance, ensuring high-accuracy phase shifts measurement and phase retrieval, meanwhile significantly reducing phase-shifting interferograms acquisition time and phase retrieval calculation time.

Error Sources of Interferometric Synthetic Aperture Radar Satellites

Interferometric synthetic aperture radar (InSAR) processing techniques have been widely used to derive surface deformation or retrieve terrain elevation. Over the development of the past few decades, most research has mainly focused on its application, new techniques for improved accuracy, or the investigation of a particular error source and its correction method. Therefore, a thorough discussion about each error source and its influence on InSAR-derived products is rarely addressed. Additionally, InSAR is a challenging topic for beginners to learn due to the intricate mathematics and the necessary signal processing knowledge required to grasp the core concepts. This results in the fact that existing papers about InSAR are easy to understand for those with a technical background but difficult for those without. To cope with the two issues, this paper aims to provide an organized, comprehensive, and easily understandable review of the InSAR error budget. In order to assist readers of various backgrounds in comprehending the concepts, we describe the error sources in plain language, use the most fundamental math, offer clear examples, and exhibit numerical and visual comparisons. In this paper, InSAR-related errors are categorized as intrinsic height errors or location-induced errors. Intrinsic height errors are further divided into two subcategories (i.e., systematic and random error). These errors can result in an incorrect number of phase fringes and introduce unwanted phase noise into the output interferograms, respectively. Location-induced errors are the projection errors caused by the slant-ranging attribute of the SAR systems and include foreshortening, layover, and shadow effects. The main focus of this work is on systematic and random error, as well as their effects on InSAR-derived topographic and deformation products. Furthermore, because the effects of systematic and random errors are greatly dependent on radar wavelengths, different bands are utilized for comparison, including L-band, S-band, C-band, and X-band scenarios. As examples, we used the parameters of the upcoming NISAR operation to represent L-band and S-band, ERS-1 and Sentinel-1 to represent C-band, and TerraSAR-X to represent X-band. This paper seeks to bridge this knowledge gap by presenting an approachable exploration of InSAR error sources and their implications. This robust and accessible analysis of the InSAR error budget is especially pertinent as more SAR data products are made available (e.g., NISAR, ICEYE, Capella, Umbra, etc.) and the SAR user-base continues to expand. Finally, a commentary is offered to explore the error sources that were not included in this work, as well as to present our thoughts and conclusions.

Structured Light in Vision Science Applications

The dichroic macular pigment in the eye acts as a natural radial polarization filter and enables humans to directly perceive polarization-related entoptic phenomena. Linearly polarized blue light induces a subtle bowtie-like pattern known as Haidinger’s brush in the central point of vision. The clarity and shape of this perceived pattern are directly linked to the health of the macula, rendering Haidinger’s brush a potential diagnostic marker in research on early-stage age-related macular degeneration (AMD) and central field visual dysfunction. However, due to the faint nature of this signal, integrating the perception of Haidinger’s brush into modern clinical methods remains a challenge. Here we review some advances in techniques to increase the strength of the perceived signal by employing polarization coupled orbital angular momentum states. We successfully achieved the creation of stimuli with higher numbers of azimuthal fringes, enabling the perception and discrimination of Pancharatnam-Berry phases, measuring the visual angle of entoptic phenomena, retinal imaging using structured light, and the creation of radially varying entoptic stimuli. Our current studies are focusing on applying the structured light methods that we developed to subjects that suffer from ocular diseases such as AMD.

Robust phase demodulation algorithm for single-frame interferogram

This paper presents a robust phase demodulation algorithm (PDA) for single-frame interferogram based on preprocessing and watershed segmentation transformation. Compared to existing methods, the PDA can be applied to closed fringe or non-closed fringe patterns with low number of fringes indiscriminately. The proposed PDA includes two main parts, one is the preprocessing stage based on the empirical mode decomposition and Hilbert spiral transformation for background and noise removal, as well as normalization, and the other one is the correction of the wrapped phase obtained by arccosine through the watershed segmentation transformation. Through numerical simulations, the robustness of PDA to high noise and different kinds of fringe patterns is proved. The experimental evaluation of the proposed algorithm focuses on surface shape measurement of optical components, and the comparative analysis of the first 36 Zernike coefficients demonstrates that the PDA has high accuracy and the phase is good accordance with that obtained by phase-shifting interferometry.

Interferometry analysis with fringe normalization and matrix Abel inversion for plasma diagnostics

In plasma diagnostics using interferometry, the phase shift caused by the plasma in the fringes is extracted to determine the plasma density. The common method to extract the phase shift from the fringes is the fast-Fourier-Transform (FFT), but this technique encounters challenges when dealing with insufficient fringe numbers, spatially varying fringe frequencies, or extremely sharp phase changes. These challenges result in errors and hinder the acquisition of precise phase measurements. To tackle this issue, we introduced the fringe normalization (FN) method. The simulations demonstrated that the FN method extracts accurate phase information, surpassing the capabilities of the FFT method. As a result, this advancement enables more precise plasma diagnostics by mitigating errors that arise during the phase data processing. Furthermore, we improved the code for the inverse matrix Abel inversion to convert phase information into density. The simulation employing this code showed that the developed code provides more accurate values in the analysis of plasmas with a sharp density profile, assisting in electron beam manipulation in laser-plasma acceleration.

Reconfigurable generation of chiral optical fields with multiple selective degrees of freedom.

Chiral optical fields caused by vortex beams possessing orbital angular momentum (OAM) can be used to fabricate helically structured materials and identify chiral molecules, in which the materials or molecules are associated with the character of the irradiated light. However, previously reported chiral optical fields can control only some of the parameters including the number of fringes, size, ellipticity, orientation, and local intensity distribution, which may hamper their applications. Thus, in this work, we propose both theoretically and experimentally an approach to fabricate chiral optical fields with five separately controllable degrees of freedom by overlapping two anisotropic vortices whose wavefronts have a nonlinear phase variation with the azimuthal angle. The local intensity distribution, number of fringes, size, orientation, and ellipticity of the chiral optical field can be dynamically controlled by adjusting the nonlinear coefficient, topological charges, axicon parameter, rotation angle, and stretching factor of the anisotropic vortices. Furthermore, the OAM density was investigated and proven to be continuously enhanced with the variation of the field's local intensity distribution, which gives the proposed approach the ability to continuously manipulate the OAM density of chiral optical fields. This work, supporting chiral optical fields by five separately controllable parameters, may make the applications of chiral optical fields in the fields of nanostructure fabrication and optical tweezers more flexible.

Simultaneous interferometric profilometry of surface and thickness of transparent plate using wavelength-modulation

Wavelength-modulation interferometers have been widely used to measure the geometric characteristics of transparent plates. However, several errors can be generated by multiple reflections of the laser beams and environmental uncertainties when the wavelength is modulated. In this research, two novel phase-calculation formulas with the same number of interference fringes (15) are proposed to measure the geometric characteristics of a transparent plate. First, a 15s-sample phase-calculation formula for measuring the surface shape of a transparent wedge plate was developed using a system of linear equations. Second, a 15t-sample phase-calculation formula for measuring the optical thickness irregularity of transparent parallel plate was developed by using the characteristic-polynomial theory. The two 15-sample formulas were visualized in the frequency domain and complex plane to confirm the properties of their error-compensation ability. Furthermore, the two formulas were compared with conventional phase-calculation formulas through the numerical analysis. Finally, the surface profile of the wedge plate and optical thickness irregularity of the parallel plate were measured via the two proposed formulas using wavelength-modulation Fizeau interferometer. The standard deviations of 15s-sample formula and the 15t-sample formula were confirmed as 4.3616 nm and 1.2875 nm, respectively. Using these formulas to measure the geometric properties of transparent plate on nanoscale can contribute to the production of high-performance semiconductor devices.

Analysis of Grating Lobe Effects on GEO DSC Distributed Antennas

Traditional single-antenna systems have inherent limitations in terms of antenna gain, anti-interference capability, and flexibility. To overcome these challenges, satellite-mounted distributed antenna systems disperse multiple antennas at different positions on the satellite to improve the reception quality and signal-to-noise ratio of satellite signals, enhancing the performance of the satellite communication system without additional bandwidth or transmission power. However, the dispersed locations and long distances between antennas on the satellite result in less compact spacing compared to terrestrial distributed antennas, leading to the generation of a significant number of grating lobes. The distributed satellite cluster (DSC) approach revolutionizes the traditional mode of satellite utilization, enabling close collaboration among distributed loads. In this study, we analyzed the impact of grating lobes produced by DSC distributed antennas in geostationary Earth orbit (GEO) and simulated the grating lobe patterns of two 1 m circular aperture satellite antenna arrays in GEO. The simulation results revealed that the relative position change of the satellites affected the width and number of interference fringes in a certain ground area, while change in the carrier phase led to the translation of the interference fringes. To mitigate the grating lobes, we employed a sparse array technique. The simulation results demonstrated that the sparse array effectively suppressed the grating lobes but at the expense of a decrease in the sidelobe level and beamwidth.

Deep learning-based binocular composite color fringe projection profilometry for fast 3D measurements

At present, deep learning plays a crucial role in structured light 3D reconstruction. Further, in the field of fringe projection profilometry, learning 3D features from fringes and performing 3D reconstruction are being studied by many researchers. This paper combines deep learning with binocular fringe projection, uses three channels to form a single-composite-color fringe pattern for three different frequencies as the input, and predicts the numerator and denominator required to solve the wrapped phase of the object. The wrapped phase is calculated using an arctangent function. The absolute phase is obtained by unwrapping the multi-frequency heterodyne method, and the absolute phase of the left and right cameras is matched to obtain a disparity map. The parameters obtained through camera calibration can restore the 3D shape of the object, which greatly reduces the number of fringes required; accuracy close to the phase of the training set is achieved. Finally, the experimental results demonstrate the feasibility of this approach.

Diffraction characteristics of optical vortices using annular radial grating with gradually changing period

Using a grating to measure the orbital angular momentum (OAM) of a vortex beam is a straightforward and economical approach. We devised an annular radial grating with a gradually changing period to characterise a vortex beam. By utilising the different off-axis positions of incidence, the spot distribution law for the far-field diffraction pattern can provide information regarding the topological charge magnitude and sign of the incident vortex beam. Our findings demonstrate that the fringe number and orientation of the diffracted spot can be employed to measure the vortex beam. Furthermore, we simulate the highest measurable OAM order, reaching +170. This measurement technique displays good tolerances for beam alignment and parameter selection. Our study offers a vital approach to the verification of high-order OAM beam generation and demultiplexing and wavefront correction in OAM multiplexing communication systems. These applications have significant value for large-capacity free-space optical communications and OAM holography.

A optomechanical design for Fabry-Perot Interferometer

Fabry-Perot Interferometer are particularly useful for high-resolution spectroscopy and other applications including astronomy and intra-cavity laser-line narrowing. Fabry-Perot etalon with diameters 1 mm square to 100 mm circular are commercially available and using a conglomerate of optomechanical components and lenses form a Fabry-Perot Interferometer (FPI) system. The etalon when illuminated with a known wavelength laser beam through fore-optics transmits the interference pattern onto a detector through back-optics and thus the image formed on the detector is a concentric circular interference fringe pattern. A sharp pattern with a maximum number of fringes is achieved by introducing a correct combination of the fore and back optomechanical system. Commercially available interferometer inclusive of the optics and the etalon are expensive and beyond the purchasing capacity of budget-conscious users. An alternative to these ready-made interferometer is to manufacture an indigenous optomechanical system and integrate it with the commercially available etalon that delivers a low cost, less weight, and a fairly simple FPI. This report discusses in detail one such system designed and developed in our institute laboratory. The important parameter finesse is calculated and its accuracy is found to be 85%. The developed system with further modifications, like the insertion of the field-of-view module, may serve as an alternative for the large size-based integrating sphere systems that are deployed in the field for observations.

Two-step differential phase-shifting phase retrieval using generative adversarial network

Phase-shifting interferometry (PSI) is a high-accuracy optical measurement technique which requires accurate phase-shift estimation or sufficient phase-shifting interferograms to ensure phase-retrieval accuracy. Therefore, obtaining the highest phase retrieval accuracy from fewer interferograms remains a challenge. Based on this, we propose a two-step differential phase-shifting phase retrieval method using generative adversarial network, which can achieve higher phase-retrieval accuracy from two arbitrary phase-shifting interferograms, compared to representative two-step arbitrary phase-shifting phase retrieval methods. A phase retrieval generative adversarial network (PR-GAN) which can be trained with artificial training data for phase retrieval from experimental interferograms is employed. Experimental results demonstrate that accurate phase results can be obtained, even when the phase shift is approximately π or the fringe number is relatively small.

Optical diffractometry by rough phase steps

Optical diffractometry (OD) using a phase step is an alternative for interferometry, further, has least sensitivity to environmental vibrations. Therefore, OD has found numerous interesting metrological and technological applications. OD utilizes a phase step to detect the influence of objects under measurement by the changes in the Fresnel diffraction pattern. Recently, we showed that such measurements do not require infinitively sharp phase steps, although fabrication of such sharp elements is also impossible. Here, we address the issue of smoothness of the phase step surfaces. So far, in all of the OD applications the surfaces of the incorporated phase steps are considered to be optically smooth and flat. However, practically, some amount of roughness and unflatness is unavoidable even in precise and careful fabrication process. We show that preserving the OD-diffraction-pattern characteristics of a phase step depends on the level of roughness in the surfaces of the phase step. We define number of detectable fringes and autocorrelation functions of the diffraction patterns as the measures for evaluating the similarity of the rough phase step diffractions to the ideal case. We derive the theoretical description and confirm the results with simulations and experiments.

Iterative Adaptive Spectroscopy of Short Signals.

We develop an iterative, adaptive frequency sensing protocol based on Ramsey interferometry of a two-level system. Our scheme allows one to estimate unknown frequencies with a high precision from short, finite signals consisting of only a small number of Ramsey fringes. It avoids several issues related to processing of decaying signals and reduces the experimental overhead related to sampling. High precision is achieved by enhancing the Ramsey sequence to prepare with high fidelity both the sensing and readout state and by using an iterative procedure built to mitigate systematic errors when estimating frequencies from Fourier transforms. A comparison with state-of-the-art dynamical decoupling techniques reveals a significant speedup of the frequency estimation without loss of precision.