Fringe Density Research Articles

Optimization-based approach for high-fidelity phase retrieval from sparse interferometric data: implications for industry and biological research

Optical profilometers provide a non-contact, non-destructive method for swiftly profiling 3D surfaces. White light interferometers, often used for this purpose, employ a 5-phase shifting technique for precise phase maps. However, capturing multiple frames introduces mechanical movement, which impedes imaging of dynamic objects. White light’s low spatial-temporal coherence mitigates speckles and spurious fringes while offering high axial resolution. Creating a high fringe density interferogram with low-coherence light is challenging. Introducing a tilt angle in the interferometer can increase the fringe density, which is still insufficient for phase map retrieval using the single-shot Fourier transform method. We propose an adaptive optimization framework to recover phase maps from single low fringe density interferograms. This method iteratively extracts reference beam information, eliminating mechanical movement and enhancing system stability while reducing costs and system bulkiness. The simulation and experimental results on a step-phase object (etched on silicon) and biological MG63 osteosarcoma cells validate the efficacy of a single-shot optimization scheme. For comparison, the phase maps of the same objects were obtained using the single-shot Fourier transform and multi-shot 5-phase shifted methods. The single-shot optimization technique shows efficient performance, yielding phase maps with reasonable accuracy, potentially replacing the 5-phase shifting technique in industrial and biological diagnostics.

Effect of Obstacle Shape on Natural Convection in Partially Open Square Enclosure

The present study reports a real-time experimental analysis of natural convection of air using a noninvasive technique (Mach-Zehnder interferometry). The experiments are conducted in a partially open square cavity with a centrally placed adiabatic obstacle and a heat source positioned at the center of the bottom horizontal wall. The left vertical wall has a partial opening occupying 50% of the wall height. The effect of obstacle shapes (square, circular, and triangular) and Rayleigh number on natural convection is evaluated. The interferograms obtained from the experiments give temperature field in the cavity. Air velocity at the opening is measured using an air velocity transducer. The experimental results are compared with numerical findings obtained using ANSYS Fluent 2020. The fringe density in the cavity enhances with increase in Rayleigh number. The temperature gradient, average Nusselt number, and average exit velocity of the flowing fluid are higher for triangular obstacle, whereas minimum for square obstacle. For Rayleigh numbers in the range of 107 to 108, an average increment in Nusselt number and average exit velocity is up to 15% and 16%, respectively. The maximum temperature in the enclosure is higher for square obstacle, whereas it is lower for triangular obstacle.

Radial vibration measurement of rotating shaft using constant density fringe pattern and line scan camera

The radial vibration signals of the rotor can provide abundant information about the health condition of the machine. In this paper, a simple vision-based measurement system is proposed to simultaneously measure two-dimensional displacements in radial directions for the rotating shaft, where the system consists of a constant density fringe pattern (CDFP), a line scan camera (LSC), and a lens. The CDFP should be installed around the surface of the rotating shaft to make the density of the fringe constant along the shaft axis, while the shaft axis is vertical to the optical axis of the LSC but not parallel to the line-array sensor of the LSC. Therefore, the density of the fringe imaged on the LSC is not constant because of the modulation of the circular surface of the shaft, and the distribution of the fringe density on the LSC is a U-shaped curve. Thus, the shaft centreline orbit can be tracked by the lowest point of the density distribution curve (DDC) of the fringe. Then, an efficient and accurate parameterized instantaneous frequency estimation method is employed to estimate the DDC of the fringe, because the variable density fringe can be regarded as an amplitude-modulated and frequency-modulated nonstationary signal whose instantaneous frequency function is equivalent to the DDC. Experimental results verify the feasibility and effectiveness of the proposed method by comparing it to the eddy current sensors.

High space–time bandwidth product imaging in low coherence quantitative phase microscopy

Current low coherence quantitative phase microscopy (LC-QPM) systems suffer from either reduced field of view (FoV) or reduced temporal resolution due to the short temporal coherence (TC) length of the light source. Here, we propose a hybrid, experimental and numerical approach to address this core problem associated with LC-QPM. We demonstrate high spatial resolution and high phase sensitivity in LC-QPM at high temporal resolution. High space–time bandwidth product is achieved by employing incoherent light source for sample illumination in QPM to increase the spatial resolution and single-shot Hilbert spiral transform (HST) based phase recovery algorithm to enhance the temporal resolution without sacrificing spatial resolution during the reconstruction steps. The high spatial phase sensitivity comes by default due to the use of incoherent light source in QPM which has low temporal coherence length and does not generate speckle noise and coherent noise. The spatial resolution achieved by the HST is slightly inferior to the temporal phase-shifting (TPS) method when tested on a specimen but surpasses that of the single-shot Fourier transform (FT) based phase recovery method. Contrary to HST method, FT method requires high density fringes for lossless phase recovery, which is difficult to achieve in LC-QPM over entire FoV. Consequently, integration of HST algorithm with LC-QPM system makes an attractive route. Here, we demonstrate scalable FoV and resolution in single-shot LC-QPM and experimentally corroborate it on a test object and on both live and fixed biological specimen such as MEF, U2OS and human red blood cells (RBCs). LC-QPM system with HST reconstruction offer high-speed single-shot QPM imaging at high phase sensitivity and high spatial resolution enabling us to study sub-cellular dynamic inside U2OS for extended duration (3 h) and observe high-speed (50 fps) dynamics of human RBCs. The experimental results validate the effectiveness of the present approach and will open new avenues in the domain of biomedical imaging in the future.

Incident angle optimization with multiple conflicting objectives based on BP-MOPSO in oblique laser interferometry

The oblique incident phase-shifting interferometric measurement method is an effective technique for assessing gear tooth form deviation. In the measurement process, the tooth flank incident angle is a crucial factor determining the quality of interferograms. Addressing the challenge in traditional optical path design, where the selection of the tooth flank incident angle encounters a trade-off among quality features such as measurable tooth area (MTA), interference fringe density (IFD) and interferogram compression ratio (ICR), this paper proposes an optimized method for the tooth flank incident angle. Firstly, a comprehensive evaluation model is established to calculate and analyze the MTA, IFD and ICR of the helical gear. Secondly, by integrating BP neural network and Multi-Objective Particle Swarm Optimization (MOPSO) algorithm, a tooth flank incident angle optimization method is devised to simultaneously optimize the measurement light incident angle γ and the rotation angle of the gear axis β. Finally, simulation analyses and physical experiments are employed to demonstrate the feasibility of the proposed method, compared to before optimization, the pseudocorrelation (PSD) value has increased by an average of 34%, and the residual points have decreased by an average of 80%, confirming its capability to improve quality interferogram and measurement accuracy.

Vision measurement system for tank settlement using Hanning-window changing density fringes

Vision measurement system for tank settlement using Hanning-window changing density fringes

AeroFeathers: Feathered airfoils inspired by the quiet flight of owls

Owls are well-known as the quietest flying birds. Recent studies show that they achieve noise-less flight because of their specialized feather design. Their feathers have three important components: a velvety surface texture, comb-like hooks on their leading edge, and jagged fibrous fringes on their trailing edges. Some researchers have attempted to imitate these design features within airfoil designs to reduce aeroacoustic noise emitted by fixed wing and vertical lift vehicles; however, such features cannot be easily fabricated using traditional manufacturing processes. In this student-led project funded by NASA’s University Student Research Challenge, we leverage our established success at additively manufacturing fibrous structures to introduce a new method to 3-D print velvety textures, fibrous fringes, and flexible edge serration using low-cost material extrusion-based printing methods. By altering the printer’s G-code, we create customized feathered airfoils with individual design parameters, including serration thickness and fringe density. We conduct acoustic tests in an anechoic chamber and aeroacoustic tests in an acoustic wind tunnel to determine the effect of each parameter on the overall acoustic signature of the airfoil. We analyze the data and explain the role of owl feather-like features on the noise reduction performance of propeller blades and scaled fixed-wing airfoils.

UN-PUNet for phase unwrapping from a single uneven and noisy ESPI phase pattern.

The wrapped phase patterns of objects with varying materials exhibit uneven gray values. Phase unwrapping is a tricky problem from a single wrapped phase pattern in electronic speckle pattern interferometry (ESPI) due to the gray unevenness and noise. In this paper, we propose a convolutional neural network (CNN) model named UN-PUNet for phase unwrapping from a single wrapped phase pattern with uneven grayscale and noise. UN-PUNet leverages the benefits of a dual-branch encoder structure, a multi-scale feature fusion structure, a convolutional block attention module, and skip connections. Additionally, we have created an abundant dataset for phase unwrapping with varying degrees of unevenness, fringe density, and noise levels. We also propose a mixed loss function MS_SSIM+L2. Employing the proposed dataset and loss function, we can successfully train the UN-PUNet, ultimately realizing effective and robust phase unwrapping from a single uneven and noisy wrapped phase pattern. We evaluate the performance of our method on both simulated and experimental ESPI wrapped phase patterns, comparing it with DLPU, VUR-Net, and PU-M-Net. The unwrapping performance is assessed quantitatively and qualitatively. Furthermore, we conduct ablation experiments to evaluate the impact of different loss functions and the attention module utilized in our method. The results demonstrate that our proposed method outperforms the compared methods, eliminating the need for pre-processing, post-processing procedures, and parameter fine-tuning. Moreover, our method effectively solves the phase unwrapping problem while preserving the structure and shape, eliminating speckle noise, and addressing uneven grayscale.

An InSAR Deformation Phase Retrieval Method Combined with Reference Phase in Mining Areas

The acquisition of precise deformation data, including the entirety of the subsidence basin resulting from subterranean mining operations, assumes critical significance in the context of surface impairment monitoring during the course of mining activities. In light of the constraints associated with InSAR technology when applied to the surveillance of expansive deformation gradient mining regions, an innovative approach is advanced herein for InSAR deformation phase retrieval. This approach integrates a reference phase, derivable through a variety of means, including pre-existing models or measurements. Initially, the reference deformation phase is subjected to subtraction from the wrapped InSAR deformation phase, culminating in the derivation of the wrapped phase indicative of the residual phase. Notably, it is posited that the fringe density characterizing the wrapped phase of the residual phase is theoretically diminished in comparison to that of the InSAR wrapped phase. This reduction in complexity in phase unwrapping ensues as a direct consequence. Subsequent to this, the phase retrieval process is effectuated through the summation of the reference phase and the unwrapped phase pertaining to the residual phase. The study harnesses Sentinel-1A and ALOS PALSAR-2 data, employing the PIM-predicted outcomes and GNSS-RTK monitoring outcomes as reference phases for the execution of phase retrieval experiments in two designated study areas. The computation of subsidence is subsequently realized through the combination of the displacement vector depression angle model and the retrieved phase, with the accuracy thereof corroborated through the utilization of leveling data. The experimental findings underscore the efficacy of the reference phase retrieval methodology in securing a more precise deformation phase characterization within expansive deformation gradient mining regions, thereby demonstrating the suitability of this methodological approach.

Freeform mirror validation by interferometric techniques using a spatial light modulator

The most widespread verification method for optical elements is interferometry but, in the case of freeform surfaces, a strong deviation of the slope along the surface can create areas in which the fringe density is too high for the interferometer to resolve them. The most desirable solution is to create a null or near null interferogram introducing compensating elements like a spatial light modulator (SLM) that provides the flexibility to accommodate the measurement of a wide range of free-form surfaces. This paper shows the process for a convex freeform mirror metrology. The method consists of inserting the SLM in the optical path to compensate the freeform component of the surface to be verified and to generate a null of aberrations in the interferometer. The system is previously modelled in an optical design software to calculate the required phase to be introduced in the SLM to generate the null. The arrangement of the SLM makes possible to keep its position fixed and use the same setup to measure a wide range of freeform surfaces, limited by the dynamic range of the SLM. For each specific surface, it is necessary to introduce suitable elements to compensate the base surface, reserving the SLM for the freeform component compensation. The method is illustrated with the verification of a convex freeform mirror whose freeform component is described by the astigmatism Zernike polynomial Z5.

Sexual dimorphism in a dioecious species with complex, specialist-pollinated flowers.

Pollinators with flower constancy and long nectar-feeding organs should favor less or no sexual dimorphism in the individual flowers of dioecious plants. This hypothesis is deduced because such pollinators can discriminate between intersexual flower size differences, and morphological differences between male and female flowers often diminish pollen transfer. We compared floral traits and pollinator behavior between male and female flowers in the hawkmoth-pollinated species, Trichosanthes cucumeroides. In field studies, we removed petal fringes on both sexes and observed pollinators to assess the role of elaborate petal fringes in pollinator attraction and pollination success for each flower sex. Female flowers had a similar front flower size and fringe extension as male flowers, supporting our hypothesis. In contrast, females allocated fewer resources to floral biomass. Additionally, they had smaller and narrower petal lobes, lower fringe density, shorter tubes with inferior nectar rewards, and lower display size than males, which is inconsistent with the hypothesis. Nocturnal hawkmoths prefer flowers with long fringe extensions. Fringe removal significantly decreased hawkmoth visitations to both female and male flowers but reduced success only in females. A literature survey indicated that female flowers of specialist-pollinated species are similar in size or larger than the males and thus tend to attract more pollinators compared with female flowers of generalist-pollinated species. Female flowers have evolved fringe extensions that are similar to those of male flowers, likely increasing pollinator attraction even slightly, and had less biomass in other floral parts and produced less nectar compared with male flowers. Our findings imply that female-biased resource limitation and flower-size sensitivity of pollinators together exert sex-specific selection of floral traits in T. cucumeroides.

Aspheric surface measurement by absolute wavelength scanning interferometry with model-based retrace error correction.

This paper presents a non-nulling absolute interferometric method for fast and full-area measurement of aspheric surfaces without the necessity of any mechanical movement. Several single frequency laser diodes with some degree of laser tunability are used to achieve an absolute interferometric measurement. The virtual interconnection of three different wavelengths makes it possible to accurately measure the geometrical path difference between the measured aspheric surface and the reference Fizeau surface independently for each pixel of the camera sensor. It is thus possible to measure even in undersampled areas of the high fringe density interferogram. After measuring the geometrical path difference, the retrace error associated with the non-nulling mode of the interferometer is compensated for using a calibrated numerical model (numerical twin) of the interferometer. A height map representing the normal deviation of the aspheric surface from its nominal shape is obtained. The principle of absolute interferometric measurement and numerical error compensation are described in this paper. The method was experimentally verified by measuring an aspheric surface with a measurement uncertainty of λ/20, and the results were in good agreement with the results of a single-point scanning interferometer.

An improved fourier transform method for measuring thermal boundary layer temperature field using fringe extrapolation

Using Gerchberg fringe extrapolation as the interferogram preprocessing, we improve the spatial carrier Fourier transform method (FTM) for measuring temperature field of the thermal boundary layer. Since the fringes abruptly disappear on the solid boundary, large ripples may occur in the retrieved phase/temperature if the original interferogram is directly analyzed by the FTM. Using the extrapolated interferogram, the edge effect can be shifted to the region near the virtual extrapolated boundary, so the ripples are significantly reduced near the physical solid boundary. In our interference experiment, the spatial resolution of the interferogram is 5.47 μm. Through adjusting the fringe density, a large enough spatial carrier frequency can be obtained to void frequency aliasing for high temperature gradient measurement. We first validate the proposed FTM through the measurement of a stagnant air layer between a hot upper plate and a cold lower plate. In comparison with temperatures measured by the thermocouple, the maximum difference is less than 0.3 °C for temperature measurement and 2% for local temperature gradient measurement. We then study the free convection of a hot vertical plate. Through the temperature field measurement, we obtain the local heat transfer coefficient and the Nusselt number. Compared to the similarity solution, the deviation of the local Nusselt number is 2%–6%, except the upper and lower edge regions. The proposed method can also be used to capture the transient temperature field for further studying the convection formation and evolution.

Optical counterpart of a Foucault pendulum.

The twin beam vortex interferometer with a phase-conjugating mirror in a rotating reference frame is analyzed. The circular motion of the interference pattern occurs due to the exchange of angular momenta between photons and the interferometer. Using the concept of the ideal phase-conjugating mirror, it is shown that the motion of the helical interference pattern of interacting vortex photons with topological charge ℓ may be used for detection of slow rotations. The higher density of interference fringes may improve sensitivity by a factor containing 2ℓ compared to conventional Michelson interferometry.

REARRANGEMENTS IN THE CONFORMATIONAL STRUCTURE OF POLYELECTROLYTES ON THE SURFACE OF A FLATTENED METAL NANOSPHEROID IN AN ALTERNATING ELECTRIC FIELD

A mathematical model has been presented for the formation of the conformational structure of chain units in a polyelectrolyte adsorbed on a flattened conducting charged nanospheroid polarized in an external electric field, which harmonically varies at a frequency much lower than the plasma frequency of the nanospheroid metal. Molecular dynamics has been employed to study the rearrangements in the conformational structure of uniformly charged polypeptides adsorbed on the surface of the oppositely charged flattened gold nanospheroid in an external alternating electric field, the strength vector of which varies along the rotation axis of the nanospheroid. One-dimensional density distributions along the rotation axis, as well as radial distributions, have been plotted for atoms of the polypeptides adsorbed on the nanospheroid surface. At a low temperature, a narrow ring-shaped polyelectrolyte fringe is formed in the equatorial region of the flattened metal nanospheroid, and the fringe density increases with the total charge of the nanospheroid and the number of charged units in polyelectrolyte macrochains. At a high temperature, the formed narrow macromolecular ring periodically shifts along the rotation axis of the nanospheroid with redirections of the polarizing electric field vector. The amplitude of the shifts increases with a decrease in the total charge of the nanospheroid and an increase in the fraction of charged units in a polyelectrolyte.

Optical path modulation of interference fringe density based on position errors analysis in oblique laser interferometry

When laser interferometry is applied to measure the form deviation of a helical tooth flank, dense interference fringes are almost inevitable because of the distorted shape of the helical tooth flank. The high-density fringe leads to poor processing accuracy of interferograms and is an important indicator of the interferogram quality. An optical path optimization method based on position errors analysis is proposed to modulate the fringe density. First, some factors that may affect fringe density in the optical system are modeled. Consequently, the incident angle of the measured light is proved to be the main factor affecting the fringe density and is the primary parameter that modulates the interference fringe density. Subsequently, an analytical model is established, and a fringe density modulation method based on the optimized incident angle is proposed for designing the optical path. Finally, simulated and actual experimental results are provided to verify the correctness of the analysis of the main factor affecting the fringe density and feasibility of the modulation method.

Theoretical Exposure Dose Modeling and Phase Modulation to Pattern a VLS Plane Grating with Variable-Period Scanning Beam Interference Lithography

Variable-period scanning beam interference lithography (VP-SBIL) can be used to fabricate varied-line-spacing (VLS) plane gratings. The exposure phase modulation method to pattern a VLS grating with a desired groove density must be carefully devised. In this paper, a mathematical model of the total exposure dose for VLS plane grating fabrication is established. With model-based numerical calculations, the phase modulation effects of the parameters, including the fringe locked phase, fringe density, and step size, are analyzed. The parameter combinations for the phase modulation are compared and chosen, and the optimal coordinate for phase compensation is selected. The calculation results show that the theoretical errors of the groove density coefficients can be controlled within 1e-8. The mathematical model can represent the deposited exposure dose for patterning VLS gratings during the lithography process, and the chosen parameters and proposed phase modulation method are appropriate for patterning VLS gratings with VP-SBIL.

Quantitative spatiotemporal density evolution of aluminum heated purely by monochromatic electrons

A spatially resolved air-wedge shearing interferometer and shadowgraph diagnostic provides measurements of electron density with a resolution of ∼40 μm. A ∼100-ns-long, monoenergetic electron bunch at 19.8 MeV and a current of 1.4 kA (8.5×1014 e−) heats 100-μm-thick aluminum (Al) foils in a 1-mm-spot to Te∼1 eV. A 5-ns-long, ∼60 mJ, frequency doubled Nd:YAG laser probes the dense Al plasma. Electron densities up to 1020cm−3 are resolved; the maximum resolvable density is limited by opacity, transmission, and spatial fringe density achievable with the detector. This diagnostic provides measurements of the total phase shift, transmission, and electron density. Several measurements at different time slices provide the ability to determine the velocity of the leading edge of the shadowgraph and compare it to the motion of different density shells. These measurements are also compared to radiation hydrodynamics simulations. A rough quantitative agreement is shown between the hydro simulations and the measurements; there are differences in the exact density distributions.

Single-shot phase retrieval for aspheric surface testing based on the transport of intensity equation and a prism-mirror module

Transport of intensity equation (TIE) relates the longitudinal intensity derivative of propagating light wave to the transverse phase gradient. The phase is retrieved by measuring the intensity of propagating light wave at two or more longitudinally displaced planes by moving the detector and then solving the TIE. We present a single-shot transport of intensity equation-based phase retrieval using a prism-mirror based module to capture the two defocused intensity images simultaneously. The results obtained by single-shot transport of intensity equation-based phase retrieval have been compared with interferometry and profilometer-based measurements. We highlight an important aspect of the TIE-based approach that this formulation decouples the intensity and the unwrapped phase of the unknown field. Intensity and phase are individually low-bandwidth functions when compared to the complex field associated with the unknown wavefront. When compared to interferometry, the TIE-based approach therefore inherently has many reduced sampling requirements. For a given array sensor, TIE can therefore measure large phase deviations for which an interferometer would produce very high density fringes that are impossible to sample as per the Nyquist criterion. The prism-mirror module shown in this work is easy to fabricate, align, and can be easily integrated with an experimental setup. The applications of the proposed method for the wavefront measurement of spherical and aspheric lenses are demonstrated.

Phase-shifting algorithms with known and unknown phase shifts: comparison and hybrid.

The phase-shifting interferometry has been intensively studied for more than half a century, and is still actively investigated and improved for more demanding precision measurement requirements. A proper phase-shifting algorithm (PSA) for phase extraction should consider various error sources including (i) the phase-shift errors, (ii) the intensity harmonics, (iii) the non-uniform phase-shift distributions and (iv) the random additive intensity noise. Consequently, a large pool of PSAs has been developed, including those with known phase shifts (abbreviated as kPSA) and those with unknown phase shifts (abbreviated as uPSA). While numerous evaluation works have been done for the kPSAs, there are very few for the uPSAs, making the overall picture of the PSAs unclear. Specifically, there is a lack of (i) fringe pattern parameters' restriction analysis for the uPSAs and (ii) performance comparison within the uPSAs and between the uPSAs and the kPSAs. Thus, for the first time, we comprehensively evaluated the pre-requisites and performance of four representative uPSAs, the advanced iterative algorithm, the general iterative algorithm (GIA), the algorithm based on the principal component analysis and the algorithm based on VU factorization, and then compare the uPSAs with twelve benchmarking kPSAs. From this comparison, the demand for proper selection of a kPSA, and the restriction and attractive performance of the uPSAs are clearly depicted. Due to the outstanding performance of the GIA, a hybrid kPSA-GIA is proposed to boost the performance of a kPSA and relieve the fringe density restriction of the GIA.