Circular Fringes Research Articles

Absolute phase retrieval based on spatial ternary phase coding with circular fringe projection

Fringe projection profilometry (FPP) based on temporal phase unwrapping (TPU) plays a vital role due to its high precision and fast speed in three-dimensional (3D) measurements. However, a fundamental contradiction arises between the necessity for numerous auxiliary patterns and the goal of enhancing measurement efficiency, particularly when ensuring a big codeword space for fringe order. Thus, a spatial ternary phase coding (STPC) method is proposed, which achieves a big codeword space by projecting just one auxiliary coding pattern while ensuring a robust decoding. By equally dividing each fringe period into several segments, ternary coding of fringe order is accomplished using an arccosine operation to code three distinct phase values within each segment. Meanwhile, circular grating is adopted due to its excellent performance in phase sensitivity, which offers a higher measuring accuracy in 3D reconstruction. Accordingly, a robust TPU is achieved by combining the introduced phase unwrapping scheme and the circle center-cosine theorem (CCCT) strategy. Furthermore, high-efficiency measurements are attained through projecting three phase-shifting circular gratings and one STPC grating, rendering it suitable for real-time measurements. Experimental results with various static objects and real-time scene demonstrate the feasibility and validity of the proposed method.

Comparative analysis of circular and linear fringe projection profilometry: from calibration to 3D reconstruction

This study compares the accuracy of circular and linear fringe projection profilometry in the aspects of system calibration and 3D reconstruction. We introduce, what we believe to be, a novel calibration method and 3D reconstruction technique using circular and radial fringe patterns. Our approach is compared with the traditional linear phase-shifting method through several 2 × 2 experimental setups. Results indicate that our 3D reconstruction method surpasses the linear phase-shifting approach in performance, although calibration efficiency does not present a superior performance. Further analysis reveals that sensitivity and estimated phase error contribute to the relative underperformance in calibration. This paper offers insights into the potentials and limitations of circular fringe projection profilometry.

Phase-shifting determination and pattern recognition using a modified Sagnac interferometer with multiple reflections.

This work has implemented a diverse modification of the Sagnac interferometer to accommodate various measurement requirements, including phase shifting, pattern recognition, and a morphological analysis. These modifications were introduced to validate the adaptability and versatility of the system. To enable phase shifting using the multiple light reflection technique, a half-wave plate (HWP) was utilized with rotations at 0, π/8, π/4, and 3π/8radians, generating four interference patterns. It is possible to observe a distinct circular fringe width as the polarized light experiences diffraction at the interferometer's output as it travels through a circular aperture with various diameters ranging from 0.4 to 1mm. Further modifications were made to the setup by inserting a pure glass and a fluoride-doped tin oxide (FTO) transparent substrate into the common path. This modification aimed to detect and analyze a horizontal fringe pattern. Subsequently, the FTO substrate was replaced with a bee leg to facilitate morphology recognition. A deep learning-based image processing technique was employed to analyze the bee leg morphology. The experimental results showed that the proposed scheme succeeded in achieving the phase shift, measuring hole diameters with errors smaller than 1.6%, separating distinct transparent crystals, and acquiring the morphological view of a bee's leg. The method also has successfully achieved an accurate surface area and background segmentation with an accuracy over 87%. Overall, the outcomes demonstrated the potential of proposed interferometers for various applications, and the advantages of the optical sensors were highlighted, particularly in microscopic applications.

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.

Fabry-Perot Interferometer Used to Measure Very Low Static Pressure Measurements.

This paper describes the use of an optical instrument, the Fabry-Perot interferometer, adapted to measure very low pressures. The interferometer consists of two high-reflectance flat mirrors placed one in front of another. In addition, a metallic chamber contains air or a gas. In one of the faces of the chamber, a flexible thin silicone membrane is attached and, over it, one of the mirrors is glued. The other mirror rests in a fixed mechanical mounting. Light crosses both mirrors and, when it leaves them, forms an interference pattern consisting of concentric circular fringes. When the pressure is increased/decreased within the chamber, a displacement of the fringes is observed due to the movement of the glued mirror. By measuring the fringe displacement and knowing the pressure, a calibration plot can be made. Minimum pressure measurements of about tens of Pascals were achieved.

CF3DNet: A learning-based approach for single-shot 3D reconstruction from circular fringes

3D reconstruction by fringe projection is an ill-posed problem that is of significance in many practical applications. Conventional algorithms employ multiple steps like fringe analysis, phase unwrapping, and calibration to reconstruct the 3D profile. Recent deep learning (DL) methods circumvent this multi-step approach by directly mapping the deformations in linear fringes caused by objects to the absolute phase shifts. However, this mapping is not one-to-one for the discontinuous objects due to 2π periodicity of linear fringes, and hence it is impossible to reconstruct using a single deformed fringe. In this work, we propose Circular Fringe-to-3D reconstruction Network (CF3DNet), a learning-based approach for 3D reconstruction from circular fringes, that can establish a one-to-one mapping between the phase deformations and the absolute phase shifts. This is viable because the radially symmetric circular fringes can distinctly record the lateral shifts caused by objects, and the learning-based approach can uniquely map them to absolute phase shifts. Thus, the proposed CF3DNet can reconstruct discontinuous object profiles, which is infeasible with linear fringe-based methods. Further, to evade the tedious process of real data collection and to bring generalization, the model is trained with synthetically generated smooth, discontinuous and Computer-Aided Design (CAD) object profiles. The results demonstrate that the proposed method led to tenfold improved performance over linear fringe-based methods in 3D reconstruction of various synthetic objects, and accurately estimate the real object profile compared to the other state-of-art methods. The extensive analysis presented in this work shows that the proposed method can work with carrier frequencies as low as 10 cycles/row to reconstruct dynamic ranges up to 100 radians under high noise, paving a way for the development of a cost-effective fringe projection system.

Crystal-Momentum-Resolved Contributions to Harmonics in Laser-Driven Graphene

We investigate the crystal-momentum-resolved contributions to high-order harmonic generation in laser-driven graphene by semi-conductor Bloch equations in the velocity gauge. It is shown that each harmonic is generated by electrons with the specific initial crystal momentum. The higher harmonics are primarily contributed by the electrons of larger initial crystal momentum because they possess larger instantaneous energies during the intra-band motion. Particularly, we observe circular interference fringes in the crystal-momentum-resolved harmonics spectrum, which result from the inter-cycle interference of harmonic generation. These circular fringes will disappear if the inter-cycle interference is disrupted by the strong dephasing effect. Our findings can help to better analyze the mechanism of high harmonics in graphene.

Sub-Nanometer Acoustic Vibration Sensing Using a Tapered-Tip Optical Fiber Microcantilever.

We demonstrate a highly sensitive acoustic vibration sensor based on a tapered-tip optical fiber acting as a microcantilever. The tapered-tip fiber has a unique output profile that exhibits a circular fringe pattern, whose distribution is highly sensitive to the vibration of the fiber tip. A piezo transducer is used for the acoustic excitation of the fiber microcantilever, which results in a periodic bending of the tip and thereby a significant output power modulation. Using a multimode readout fiber connected to an electric spectrum analyzer, we measured the amplitude of these power modulations over the 10-50 kHz range and observed resonances over certain frequency ranges. Two types of tapered-tip fibers were fabricated with diameter values of 1.5 µm and 1.8 µm and their frequency responses were compared with a non-tapered fiber tip. Thanks to the resonance effect as well as the sensitive fringe pattern of the tapered-tip fibers, the limit of detection and the sensitivity of the fiber sensor were obtained as 0.1 nm and 15.7 V/nm, respectively, which were significantly better than the values obtained with the non-tapered fiber tip (i.e., 1.1 nm and 0.12 V/nm, respectively). The sensor is highly sensitive, easy to fabricate, low-cost, and can detect sub-nanometer displacements, which makes it a promising tool for vibration sensing, particularly in the photoacoustic sensing of greenhouse gases.

Estimation of TiO2 thin film shift by electronic speckle interferometry

Laser scattering from the thin film surface develops self-interference speckle images, which are treated by image processing. As a result, circular fringes are formed due to the phase differences of the two laser beams overlapping after the beams splitter in the Michelson interferometer. The speckle images are captured in double stages of the sample under test, one without loading and the other with loading. Subtracting speckle images provides the similarity of Young's fringes due to the common portions of the two images. Thus the carrier frequency is negated, and posterior spatial filtering contributes only to non-common parts such as fringes. This method shows the simple experimental setup with sufficient accuracy of fringe computation for the estimation of TiO2 thin film.

Circular fringe projection profilometry and 3D sensitivity analysis based on extended epipolar geometry

For fringe projection profilometry, the phase-shifting direction needs to fit the position layout of a camera and a projector, or the quality of 3D reconstruction will be poor. Two-direction fringe projection profilometry, i.e. scanning along both horizontal and vertical directions, achieves robust performance, while the number of patterns is doubled comparing with one-direction scanning. In this paper, based on the extended epipolar geometry, we propose circular fringe projection profilometry to overcome the direction issue without doubling the number of patterns. First, by introducing the farthest and nearest points to the epipole in projector space, we can robustly control the spatial frequency of circular fringes in patterns. Second, by analytically relating the circularly-shaped phase to 3D world coordinate, we naturally reduce the number of the equations from 6 down to 3 for the proposed algorithm, and compute 3D point clouds in real time. Finally, we propose 3D sensitivity to conveniently analyze the error of reconstruction point clouds. The experimental results show that, compared with two-direction fringe projection profilometry which is robust in accuracy, the proposed is of a little better quality while the number of the patterns is half.

Tapered tip optical fibers for measuring ultra-small refractive index changes with record high sensitivity.

Here we demonstrate an inexpensive, simple, and ultra-sensitive refractive index sensor based on a tapered tip optical fiber combined with a straightforward image analysis method. The output profile of this fiber exhibits circular fringe patterns whose intensity distribution dramatically changes even with ultra-small refractive index variations in the surrounding medium. The sensitivity of the fiber sensor is measured using different concentrations of saline solutions with a transmission setup consisting of a single wavelength light source, a cuvette, an objective lens, and a camera. By analyzing the areal changes in the center of the fringe patterns for each saline solution, we obtain an unprecedented sensitivity value of 24,160 dB/RIU (refractive index unit), which is the highest value reported so far among intensity-modulated fiber refractometers. The resolution of the sensor is calculated to be 6.9 ×10-9. Moreover, we measure the sensitivity of the fiber tip in the backreflection mode using salt-water solutions and obtained a sensitivity value of 620 dB/RIU. This sensor is ultra-sensitive, simple, easy to fabricate, and low-cost, which makes it a promising tool for on-site measurements and point-of-care applications.

An Improved Circular Fringe Fourier Transform Profilometry.

Circular fringe projection profilometry (CFPP), as a branch of carrier fringe projection profilometry, has attracted research interest in recent years. Circular fringe Fourier transform profilometry (CFFTP) has been used to measure out-of-plane objects quickly because the absolute phase can be obtained by employing fewer fringes. However, the existing CFFTP method needs to solve a quadratic equation to calculate the pixel displacement amount related to the height of the object, in which the root-seeking process may get into trouble due to the phase error and the non-uniform period of reference fringe. In this paper, an improved CFFTP method based on a non-telecentric model is presented. The calculation of displacement amount is performed by solving a linear equation instead of a quadratic equation after introducing an extra projection of circular fringe with circular center translation. In addition, Gerchberg iteration is employed to eliminate phase error of the region close to the circular center, and the plane calibration technique is used to eliminate system error by establishing a displacement-to-height look-up table. The mathematical model and theoretical analysis are presented. Simulations and experiments have demonstrated the effectiveness of the proposed method.

Optical fringe analysis using support vector machine

An optical fringe analysis is very important in many scientific fields, such as optical metrology, rheology, and Fourier optics. It is proposed to analyze optical fringe patterns, obtained by a diffraction grating and Michelson’s interferometer, by treating them as training datasets for a multiclass classification problem, which is solved by using a support vector machine (SVM). The relation between a decision boundary function obtained by SVM and various optical parameters in the lab coordinate system is derived. The information that is extracted by the SVM algorithm is (1) the orientation angle of straight-line fringes; (2) the location of a smeared diffraction spot; (3) upper bounds on the spot size of each diffracted order; (4) the ratio between laser beam propagation distances; (5) the ratio between the radii of circular fringes of a Michelson’s interferogram; and (6) the orientation of elliptic fringes.

Dynamic Fizeau interferometer with assistant defocused source.

We report and demonstrate a dynamic Fizeau interferometer (DFI) capable of high-precision measurements under severe vibration. The method involves two interferometric passes: a null-interference pass (NP) and a circular carrier-frequency pass (CP). The NP adopts an on-axis Fizeau design, preserving the common path of the reference and test beams. The CP uses an assistant defocused source to generate circular fringe patterns, assisting the NP in phase extraction and sign correction. The feasibility and performance of the proposed method are demonstrated experimentally. To the best of our knowledge, this is the first proposal of a general asynchronous phase-shifting DFI that can not only demodulate null-fringe patterns and correct phase-sign ambiguity but can also maintain the common-path advantages of the Fizeau interferometer.

An exact approach for radius of curvature measurement in a calibration test sphere

In this paper we present a technique to measure the radius of curvature of a test sphere based on the relation between acquired images of a circular cosine fringe pattern and size of virtual image formed on the calibration test surface. Radius of curvature is calculated with the exact equation proposed, using the parameters of the optical setup. Fringe pattern evaluation was performed by locating extrema indices. The mathematical formulation as well as the experimental setup and results are presented. After applying a linear fit algorithm to the data as a method of compensation, obtained results show an error within the tolerance established by the ISO 10343 specifications.

Calibrating large-FOV stereo digital image correlation system using phase targets and epipolar geometry

We propose an easy-to-implement and accurate calibration method for large-scale stereo-digital image correlation (stereo-DIC). First, the intrinsic parameters of each camera are separately calibrated using a regular-sized phase target. Using phase-shifted circular fringe patterns that feature the advantage of being insensitive to the defocus, the phase target can be placed in the close range of each camera. Then, scale-free extrinsic parameters are computed from the epipolar geometry, which can be easily retrieved from DIC registration of homologous point pairs in the stereo images of a test specimen surface. These intrinsic and scale-free extrinsic parameters are used as the initial guess for further optimizing the calibration results. Further, by measuring objects with known physical size, the scale information of the stereo-DIC system can be determined. The metrological performance of the proposed method is evaluated by large-scale shape reconstruction and deformation measurement tests. The obtained results demonstrate that the proposed method is a viable solution for the large-scale 3D deformation measurement tasks.

Bijlmermeer and Scampia: The potential of suburbs as centralities in sustainable and circular cities

In the sustainable paradigm of the circular economy, waste is a resource to be valued. Similarly, in the circular city, fringe areas and urban wastescapes are places full of potential. This study consider two modern suburbs in Amsterdam and Naples. The two cases aims to compare the characteristics, potential and problems of the two districts and the ongoing actions for sustainable or circular urban regeneration. The objective is to identify the potential of these two European suburbs in the wider process of development of fair and circular cities. Therefore, this paper aims to answer to the following research question: Based on the two case studies Bijlmermeer and Scampia, what is the potential for circular and socially sustainable urban transformation of the 1960s suburbs? Analysing the main changes that have taken place over the last ten years in these complex areas, the main drivers are identified with the historical value of modern architecture landmarks, the opportunities offered by big green public spaces in dense urbanized areas, the presence of active place-based communities and their position in nodal points of the metropolitan areas.

Single-shot circular fringe projection for the profiling of objects having surface discontinuities.

Fringe projection profilometry (FPP) is a widely used non-contact optical method for 3D profiling of objects. The commonly used linear fringe pattern in FPP has periodic intensity variations along the lateral direction. As a result, the linear fringe pattern used in FPP cannot uniquely represent the lateral shift induced by the objects having surface discontinuities. Thus, unambiguous surface profiling of objects, especially with surface discontinuities, using a single linear fringe image having a single fringe frequency, is unfeasible. This paper proposes using a radially symmetric circular fringe pattern as the structured light pattern for accurate unambiguous surface profiling of sudden height-discontinuous objects. To the best of our knowledge, this is the only method that can reconstruct discontinuous height profiles with the help of a single fringe image having a single frequency. The performance of the proposed algorithm is evaluated on several synthetic and real objects having smooth variations and discontinuities. Compared to the well-known fringe projection methods, the results depict that for a tolerable range of error, the proposed method can be applied for the reconstruction of objects with 4 times higher dynamic range and even at much lower fringe frequencies.

Two point-source interference: a study of the formation of concentric circular fringes

In this short paper, the hyperbola-based analysis of wave interference recently developed by Thomas is employed to study the pattern of concentric circular fringes captured on a distant screen, when its plane is oriented orthogonal to the line joining two coherent point-sources. Newton’s rings are another instance of concentric circular fringes, generated with an extended source and a plano-convex lens-glass plate combination. The underlying geometry of the latter scenario, as described in the literature, allows for the estimation of two important physical parameters viz. the wavelength of light and the refractive index of a liquid medium. It is demonstrated here that the geometrical arrangement of the former scenario can in principle, be utilized to reach the same ends as well. Additionally, the hallmark distinguishing features of both types of circular fringe patterns are qualitatively and quantitatively elucidated.

Multiple slit interference: a hyperbola based analysis

The textbook description of multiple slit interference employs a standard path difference formula that was originally calculated for the purpose of analyzing Young’s double slit experiment. It was arrived at on the basis of a pair of assumptions that help simplify the geometry of the slit-barrier-screen arrangement and the ensuing formalism needed to estimate the positions of interference fringes on the detection screen. According to the conventional approach, convergent rays of light that emanate from different slits can be treated as effectively parallel in the far field limit. Such a parallel ray approximation-based analysis was shown by Thomas (2019; 2020) to be redundant, paradoxical and limited in accuracy. In this paper, the task of reformulation that began previously with the elementary two-slit experiment is now extended to encompass the N-slit scenario, diffraction at a single-slit and the formation of circular fringes. In order to derive one of the key results (the generalized hyperbola theorem for N-point sources), a novel computational technique called analytical induction is introduced. The procedure involves a synthesis of concepts from Cartesian geometry and discrete mathematics, namely coordinate transformation by Euclidean translation and a method of proof called the principle of mathematical induction, respectively. The final goal attained are two distribution functions that succinctly and precisely describe the variation of intensity of interference fringes on the distant screen, when it is oriented parallel and perpendicular relative to the line joining the sources. A comparison is drawn between the predictions of the conventional and the new analyses by means of numerical-graphical simulation with MATLAB. The theoretical methods and results presented herein may have a significant bearing in areas of applied optics like interferometry and diffraction crystallography. On the pedagogic front, it is suggested that the current geometrical program being visually more intuitive, be incorporated into the standard curriculum of an advanced undergraduate/graduate level course in physical optics, to complement the conventional approach.