Transparent Parallel Plate Research Articles

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.

Multi-surface Phase-shifting Interferometry Using Harmonic Frequency Solution Based on the Total Least Squares

Wavelength-tuning phase-shifting interferometry plays a significant role in the measurements of transparent parallel plates. The difficulty of multi-surface measurements is that the harmonics with different frequencies and phases make the captured interference signal mixed. In multi-surface measurements, the demodulation of phases builds on the determination of the frequency of each signal, which is traditionally estimated from the prior information including the cavity length and the optical thickness of the measured plate. However, the estimation accuracy of this method is not satisfactory. In order to accurately obtain the signal frequencies and demodulate the initial phases of the measured surfaces simultaneously, this paper proposes a total least squares wavelength-tuning phase-shifting algorithm (TLSWPA). The developed TLSWPA includes two steps: Firstly, considering the influence of the Gaussian noise on the captured interferograms, the signal frequencies are obtained based on the total least squares principle with high accuracy; Secondly, introducing the Householder transformation, the least squares error based on the 2-Norm is used to demodulate the initial phase distribution for each signal. Comparison of the obtained results of the frequency solution and the accuracy of wavefront reconstruction shows that the developed TLSWPA allows obtaining results with an accuracy 2 orders of magnitude higher than the algorithm based on the fast Fourier transform and 1 order of magnitude higher than the advanced iterative algorithm. Experimental study on a borosilicate crown glass using a Fizeau wavelength-tuning interferometer further verifies its effectiveness.

Prediction of the settling velocity of the rod-shaped proppant in vertical fracture using artificial neural network

As a new technology proppant, rod-shaped proppant is receiving more and more attention because it can greatly improve the performance of hydraulic fracturing. With special shape, the rod-shaped proppant is subject to greater flow resistance, resulting in complex migration in fractures. Accurate prediction of the settling velocity of rod proppant in fracture is important for predicting the transport distance and bed height of rod proppant in hydraulic fractures to improve the fracture performance. In this study, high-speed photography and transparent parallel plates were used to simulate and record the settling process of proppant in vertical fractures. A total of 588 effective tests were conducted to reveal the influence of proppant shape, fracturing fluid properties, and the wall effect of fracture on the settling characteristics involving the range of proppant aspect ratio and Reynolds number are 1.5–5.0 and 0.001–172.54, respectively. It was found that the rod proppant settled vertically at the low Reynolds number and horizontally at the high Reynolds number. First, an artificial neural network model of settling orientation with the prediction accuracy of 100% is established as the basis for predicting settlement velocity. The topological structure of the neural network is further extended to establish the prediction model of settling velocity with mean relative error of 1.21%. The relative importance of each input factor to the settling velocity was analyzed by the connection weights algorithm, and the weight and bias of the neural network are provided for the application and generalization of the model. Finally, the neural network model is evaluated objectively by comparing with the mathematical model. This study is a successful exploration of artificial intelligence in particle migration and will provide guidance for improving the performance of rod proppant in hydraulic fracturing.

Multi-surface phase demodulation based on the selective weighted multi-step algorithms

The transparent parallel plate plays a practical role in the optical system. The accurate measurement of its surface topography is a great importance of quantifying and evaluating the performance of the transparent plate. Based on the phase shifting interferometry, a measurement method using triple iteration and selective phase demodulation algorithms is presented, which can realize multi-surface interferometry at arbitrary positions. Through the design of the three-dimensional data matrix used to store errors, its element pixels can be read out a wealth of key parameter information, thus guiding the design of the algorithm, realizing the unity of the actual use of the algorithm and the measurement scheme, achieving the multi-surface interference measurement with the minimum residual errors with arbitrary cavity length, and expanding the limitations of the traditional weighted multi-step algorithm.

Liquid penetration in metal wire mesh between parallel plates under normal gravity and microgravity conditions

The dynamics of liquid penetration in porous wire mesh are experimentally studied in this work by particularly considering the influence of gravity. Similar experimental setups are used on the ground and in a drop tower facility in order to compare the experimental results directly. Visualization of the liquid penetrating process is realized by using single-layered wire mesh which is put between a pair of transparent parallel plates. The dependence of wire mesh on the parallel plates is confirmed. The dimensional results show that liquid penetration in the wire mesh stops at an equilibrium height in normal gravity, while the liquid penetrates ceaselessly in microgravity following a parabolic relation between the liquid height and time. The fast rising early period of liquid rise in normal gravity reduces to a slow climbing period in microgravity. Different to that in normal gravity, liquid rises faster in the coarse wire mesh than in the fine wire mesh in microgravity. The effective contact angle of fluid in the parallel plates increases when it changes from normal gravity to microgravity, while it shows weak dependence on the gravity level as well as the pore radii in the wire meshes. Dimensionless analysis is performed to the results. Good consistency is obtained between the dimensionless experimental data and the theoretical solution, providing a generalized explanation to the dynamics of liquid penetration in metal wire mesh under both normal gravity and microgravity environments.

Interferometric profilometry of absolute optical thickness of transparent plate using wavelength tuning fringe analysis

The separation of the refractive index and absolute geometric thickness from the absolute optical thickness of a transparent parallel plate has been a long-standing issue. For the measurement of the separation, the absolute optical thickness should be measured with a nanoscale uncertainty. A combination of phase-shifting techniques and wavelength tuning was recently proposed to measure the absolute optical thickness of a transparent plate. However, as the phase-shifting technique is extremely sensitive, it is necessary to use the excess fraction method to eliminate the phase-shift error. In this study, the absolute optical thickness of an optical flat, with a thickness of 6 mm and diameter of 100 mm, was measured using wavelength-tuning fringe analysis without using the sensitive phase-shifting technique. Initially, the coarse value of the absolute optical thickness was determined by a wavelength-tuning Fourier analysis. The discontinuous errors that occurred during the wavelength tuning were compensated for and eliminated using an unwrapping correlation analysis, yielding a smooth thickness distribution. The experimental data indicate that the standard deviation of the thickness measurement was 7.071 nm, attributed to temperature variations in the laboratory. Finally, the absolute optical thickness was determined at a probability of 95 % by employing the Student’s t-test using 20 experimental data points.

Experimental study on the wall factor for spherical particles settling in parallel plates filled with power-law fluids

Hydraulic fracturing is a widespread stimulation technology in the petroleum industry. The retardation due to the fracture wall existence impacts the efficient placement of the proppant in fractures. Accurate prediction of the wall factor for proppant settling in fractures filled with Power-law fluids is important in predicting the transport distance and bed height of proppant in hydraulic fractures. However, few studies were conducted to analyze the wall factor of fracture on the settling proppant. In this study, a set of transparent parallel plates and a high-speed camera were used to record the settling behaviors of the proppant in the fracture. A total of 756 tests were carried out to analyze the effects of fluid properties, the dimensionless diameter, particle size, particle density, and the particle Reynolds number on the wall factor, involving the ranges of the dimensionless diameter and the particle Reynolds number are 0.04–0.9 and 0.0001–144.25, respectively. Results indicate that the wall factor is a function of both the dimensionless diameter and the particle Reynolds number. And a new model with 1.23% relative error was developed to predict the wall factor of the parallel plates on the spherical particles settling in the Power-law fluid. Finally, it was found that the wall effect vanishes when the dimensionless diameter is smaller than 0.057. The results of this study could help field engineers optimize proppant size and fracturing parameters.

Phase-measuring algorithm to suppress inner reflection of transparent parallel plate in wavelength tuning Fizeau interferometer

Optical thickness variation is a fundamental characteristic of transparent optical devices. When measuring this variation by using a wavelength-tuning Fizeau interferometer, the measurement accuracy depends on the phase-shifting algorithm being used. Therefore, the phase-shifting algorithm should compensate for any errors incurred during the measurement, including the phase-shift error, coupling errors, and bias modulation of intensity. Among these errors, however, the coupling errors between the higher harmonics resulting from the inner reflections of the transparent plate and phase-shift error have not previously been considered. This paper presents a derivation of a 19-sample phase-shifting algorithm that can compensate for the miscalibration and 1st-order nonlinearity of the phase shift, coupling errors, and bias modulation of intensity during wavelength tuning. The characteristics of the 19-sample algorithm were estimated with respect to the Fourier representation in the frequency domain. The phase error of measurements performed using the 19-sample algorithm was discussed and compared with that of measurements obtained using other conventional phase-shifting algorithms. Finally, the optical thickness variation of a fused silica parallel plate was obtained using a wavelength-tuning Fizeau interferometer and the 19-sample algorithm. The measurement accuracy was discussed by comparing the ripples in the crosstalk noise with those calculated using other phase-shifting algorithms. The experimental results indicated that the optical thickness variation measurement accuracy for the fused silica plate was approximately 2nm.

Scanning deflectometric profiler for measurement of transparent parallel plates.

An angle-based deflectometric surface profiler has been improved for the measurement of transparent parallel plates. In the developed system, the unwanted beam reflected from the back surface of the transparent parallel plate is removed by ensuring that the beam is obliquely incident to the measurement surface; this is realized by using a modified pentamirror unit comprising two mirrors installed at a predetermined angle to one another. The surface profile measurement of a transparent parallel plate with a repeatability of less than ±0.7 nm was successfully achieved. A measurement accuracy of around 3 nm was reached by comparing the developed system with other scanning deflectometric profiler systems for the measurement of a silicon bar mirror with a length of 300 mm.

A model of two-phase resin and void flow during composites processing

The reduction of porosity during composites manufacturing remains a critical design goal for successful processing. High porosity composite parts exhibit reduced mechanical properties and increased design risk. A primary source of porosity is the formation and transport of trapped air bubbles during polymer resin flow through fibrous porous media reinforcement. A model experiment has been developed in which a flow cell comprised of two transparent parallel plates is injected with a simulated resin and air bubbles to investigate two-phase flow. The flow cell is designed to simulate flow behavior exhibited during Resin Transfer Molding (RTM) or Out-of-Autoclave (OOA) prepreg composites processing. A flow visualization setup captures the two-phase resin flow and void migration through the flow cell. A computational model is formulated to describe the two phase flow and validated with experimental results. The model material and process parameters can be optimized to maximize the relative void velocity with respect to the resin, thus achieving effective void migration and overall porosity reduction during processing. A parametric scaling analysis is conducted to identify and quantify the sensitivity of material and process parameters on two phase resin and void transport in composites processing.

Two new methods for the determination of hydraulic fracture apertures in fractured-rock aquifers

Fracture apertures play a significant role in groundwater systems. For proper groundwater quantity and contamination management, fractures have to be properly characterised. However, due to their complexity, fracture characterisation is one of the main challenges for hydrogeologists all over the world. This is particularly important in South Africa, where aquifers are predominantly fractured. Two new methods have been developed to determine inclined and horizontal fracture apertures in fractured-rock aquifers. The first is a water-balance method, the slug-tracer (ST) test: . The slug-tracer (ST) test [b = (r2 / R2) ƒ¢ h] and the second is a tracer-detection method, comprising the NAPL entry pressure (NEP) test and the NAPL injection pressure (NIP) test: . The NAPL entry pressure (NEP) test [b = (r ƒI g h)] . The NAPL injection pressure (NIP) test [b = (ƒI g V/ 2 ƒI r)] and [b = (M g) / m] These mathematical formulations were developed from laboratory experimentation using transparent Perspex parallel plate physical models and 27 apertures of 0.008 mm to 6 mm, created by using aluminium foil and thickness gauges between 20 mm thick clamped Perspex plates. The ST test uses a slug to which is added NaCl as tracer (50 mg to 300 mg/.). An EC meter is used to detect breakthrough in the observation boreholes. The NEP test uses an NAPL (sunflower oil) hydraulic head and transducers to determine the entry pressure. Using these mathematical formulae, fracture apertures are then determined for horizontal and inclined apertures. The NIP test uses the entry pressure (by injection), recorded by transducers of an NAPL (sunflower oil) and its volume or mass to determine the fracture aperture for horizontal and inclined fractures. Results from smooth and rough (buffed to 10 x 20 ƒE) fracture surfaces gave accuracies of 96 to 98% for aperture determinations of 26 apertures from 0.04 to 6 mm.

Measurement of residual wedge angle of a high optical quality transparent parallel plate with a multipass optical configuration

A new technique of wedge angle measurement of a transparent nearly parallel plate (PP) is presented. In this technique, the single-pass angular deviation suffered by a pair of mutually parallel, laterally separated pencil beams is enhanced by using a multipass cavity formed with two right-angle prisms. The resultant angular deflection is converted to linear deviation by means of a Fourier transform lens (FTL), which focuses the pencil beams on to its focal plane. A two-dimensional CCD detector array placed at the focal plane of the FTL measures the linear shift of the intensity centroid (IC) of the resulting Young's fringes formed due to the superposition of the laterally separated pencil beams. The wedge angle is determined from the measured value of the linear shift of the IC due to the PP.

Dynamic optically induced planar terahertz quasioptics

Optical control of the propagation direction of a terahertz pulse inside an optically transparent parallel plate waveguide is demonstrated by patterned charge carrier photoexcitation of a silicon slab embedded within the waveguide. It is shown experimentally and through finite element simulations that photoexcitations with sufficient conductivity can induce a partial reflection, capable of steering the pulse inside the two-dimensional waveguide. A beamsplitter is demonstrated as proof of principle and is used to delay the arrival of the reflected terahertz pulse at the detector by several picoseconds by moving the excitation in the plane.

Measurement of residual wedge angle with a reversal shear interferometer

A technique for determination of residual wedge angle of high optical quality transparent parallel plate using a reversal shear interferometer-based optical system has been discussed. In this technique the parallel plate to be tested is used to introduce angular tilt in the preset two-beam interference fringes of a reversal shear interferometer. The parallelism is calculated from the angular tilt of the fringes. The technique is more sensitive compared to techniques using Fizeau interferometer and is most suitable for highly parallel transparent plate with very low residual wedge angle.

Factors affecting the performance of micromachined sensors based on Fabry–Perot interferometry

The noise and performance limitations of optical sensors that utilize Fabry–Perot interferometry detection are investigated. A Fabry–Perot interferometer consists of two partially transparent parallel plates with reflective inner surfaces. The plates form a cavity with an optical resonance that depends on the distance between them. At resonant wavelengths, the incident light energy is transmitted through the sensor and intensity peaks occur. The distance between the plates can be obtained by detecting the wavelength of the transmitted light. Various sensors can be based on Fabry–Perot interferometry, e.g. accelerometers, pressure sensors, and microphones. This paper considers factors affecting the performance of this type of sensor, including mechanical–thermal noise, contribution of noise in the detection system, and effects of reflectivity, surface roughness and parallelism of the mirrors. The presented experimental data support the results of the analysis.

Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer.

Wavelength-scanning interferometry permits the simultaneous measurement of variations in surface shape and optical thickness of a nearly parallel plate. Interference signals from both surfaces of the test plate can be separated in frequency space; however, these frequencies are shifted from the expected frequency by the refractive-index dispersion of the test plate and any nonlinearity that is due to wavelength scanning. Conventional Fourier analysis is sensitive to this detuning of the signal frequency and suffers from multiple-beam interference noise. We propose new wavelength-scanning algorithms that permit a large tolerance for dispersion of the test plate and nonlinearity of scanning. Two 19-sample algorithms that suppress multiple-interference noise up to the second order of the reflectance of the test plate are presented. Experimental results show that the variation in surface shape and optical thickness of a glass parallel plate of 250-mm diameter was measured with a resolution of 1-2 nm rms.

Simple technique for measurement of residual wedge angle of high optical quality transparent parallel plate

A general technique for determination of residual wedge angles of high optical quality parallel plates using a Fizeau interferometer is discussed. In this technique the parallel plate to be tested for degrees of parallelism is used to introduce angular tilt in the Fizeau fringes of an air cavity of a known wedge angle. The parallelism is calculated from the angular tilt of the fringes. The technique is most suitable for highly par- allel transparent plates with very low residual wedge angles. © 2003 So-

Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion.

Testing for flatness of an optical parallel plate in a Fizeau interferometer suffers from problems caused by multiple-beam interference noise. Each internal-reflection component can be separated from the signal by its modulation frequency in a wavelength-scanned interferometer; however, the frequency depends on the thickness and the refractive-index dispersion of the test plate and on the nonlinearity of the scanning source. With a new 19-sample wavelength-scanning algorithm we demonstrate the elimination of the reflection noise, the effect of the dispersion up to the second order of the reflectance of the test plate, and as the nonlinearity of the source. The algorithm permits large tolerance in the air-gap distance, thus making it somewhat independent of the thickness of the test plate. The minimum residual reflection noise with this algorithm for testing a glass plate is approximately lambda/600. Experimental results show that the front surface of the test plate was measured within 1 nm rms of its true shape over a 230-mm-diameter aperture.

Calculation of diffraction efficiencies of a grating made on a thick transparent plate

Mathematical formulas are derived for predicting diffraction efficiencies of a grating made on a thick transparent parallel plate (substrate) or cemented between two plates. The formulas are applicable in two situations: (1) when the coherence length of the light source is greater than the thickness of the grating but less than that of the substrate; (2) when the thickness of the substrate is not precisely known, therefore only the average values of the diffraction efficiencies can be expected. There is no limitation on the number of propagating diffraction orders that are present in the system. The derivation uses the concept of energy scattering matrix, therefore it is simpler than a previous method that yielded limited results.

Effects of coalescence and breakup on the steady-state morphology of an immiscible polymer blend in shear flow

The steady-state morphology of an immiscible polymer blend in shear flow has been investigated by optical microscopy techniques. The blend is composed by poly-isobutylene (PIB) and poly-dimethylsiloxane (PDMS) of comparable viscosity. Experiments were performed by means of a home-made transparent parallel plate device. The two plates can be independently counterrotated, so that sheared droplets of the dispersed phase can be kept fixed with respect to the microscope point of view, and observed for long times. The distribution of drops and their average size were measured directly during flow at different shear rates and for different blend compositions. It was found that the average drop size in steady-state conditions is a decreasing function of the applied shear rate, and does not depend on blend composition for volume fractions up to 10%. Experiments have proved that, in the shear rate range which could be investigated, the stationary morphology is controlled only by coalescence phenomena, droplet breakup playing no role in determining the size of the dispersed phase. More generally, it has been shown that the steady-state morphology is a function not only of the physical parameters of the blend and of the shear rate, but also of the initial conditions applied to the blend. The steady-state results reported in this paper constitute the first direct experimental confirmation of theoretical models which describe the mechanisms of shear-induced drop coalescence.