Retrace Error Research Articles

Dynamic Interferometry With Retrace Error Correction Using Fringe Projection For High-Gradient Fields

This paper introduces an innovative approach aimed at advancing the characterization of high-gradient flow fields through the integration of dynamic interferometry and fringe projection techniques. Traditional interferometry offers a precise tool sensitive to optical path changes, typically used to measure refractive index variations induced by factors such as density changes in fluid flow. Rays are usually considered as straight lines, and refraction can be neglected. However, in scenarios involving high-gradient fields, such as supersonic flow with shock waves, optical path changes due to ray bending become significant. In these high-gradient fields, the trajectory of the ray is altered, leading to longer optical paths (resulting in incorrect values) and geometric distortions of the image (resulting in inaccuracies in position). Although retrace error corrections using numerical models of refractive index distribution can mitigate some effects, complete compensation is often challenging. To address this, our proposed technique integrates fringe projection, which is highly sensitive to ray deflections caused by variations in the index of refraction within a transparent medium. The method involves projecting a precisely defined multiplexed cosine fringe pattern across the measured area. Modifications in the pattern indicate ray bending, providing data for interferometric corrections. Real-time interferometry is seamlessly combined with the fringe projection technique to compensate for optical path length changes and mitigate mapping errors induced by ray deflections. The dynamic nature of this combined approach enables the instantaneous capture of both interferometric and fringe-projection data, making it a suitable tool for real-time measurements. Experimental verification of the technique demonstrates its efficacy, with a comparative study against traditional interferometry showcasing a significant reduction in mapping errors.

Two-step retrace error calibration removing tilt ambiguity in coherence scanning interferometry.

In interferometry measurement, the retrace error often limits its high-precision metrology applications. Retrace error calibration with tilted flats can give a relation between the retrace error and the introduced tilt angles, but there is still an ambiguity between the introduced tilt angles and the tilt terms in the created retrace error. We propose a novel, to the best of our knowledge, two-step calibration method to resolve this tilt ambiguity. It involves additional measurements of spherical mirror(s) with known curvature(s). The experiment shows that the curvature deviation due to the tilt ambiguity can be significantly reduced after applying the proposed method.

Retrace error calibration for interferometric measurements using an unknown optical system.

In classical interferometric null test measurements, the measurement and reference beam path should be the same. A difference in the beam paths results in the so called retrace error. One very common approach to avoid retrace errors is to adapt the measurement wavefront to the reference wavefront with a computer generated hologram (CGH), which is costly and time consuming. A much more flexible approach is to do non nulltest measurement in combination with mathematical treatment of retrace errors. Most of such methods are based on iterative optimization or calibration of the nominal optical design of the interferometer. While this may be a convenient solution in the context of research, the more common use may be limited due to the need of the optical design of all interferometer components. In many cases, the optical designs of standard off the shelf optical assemblies are not available or disclosed by the manufacturer. This is especially true for transmission spheres of interferometers. We introduce the so called Black Box Model (BBM), used in the well known Tilted Wave Interferometry (TWI), as a mathematical model to account for retrace errors in interferometry. The Black Box Model is based on point characteristic functions which are adapted to the result and behavior of a real interferometer by calibration. With an extended calibration method, the need of a specific optical design of the interferometer is no longer necessary. Thus the method is attractive for a wide field of use in interferometry with standard off the shelf components.

Correcting retrace and system imaging errors to achieve nanometer accuracy in full aperture, single-shot Fizeau interferometry.

X-ray mirrors with single-digit nanometer height errors are required to preserve the quality of ultra-intense photon beams produced at synchrotron or free electron laser sources. To fabricate suitable X-ray mirrors, accurate metrology data is needed for deterministic polishing machines. Fizeau phase-shifting interferometers are optimized to achieve accurate results under nulled conditions. However, for curved or aspheric mirrors, a limited choice of reference optic often necessitates measurement under non-nulled conditions, which can introduce retrace error. Using experimental measurements of a multi-tilted calibration mirror, we have developed an empirical model of Fizeau retrace error, based on Zernike polynomial fitting. We demonstrate that the model is in good agreement with measurements of ultra-high quality, weakly-curved X-ray mirrors with sags of only a few tens of microns. Removing the predicted retrace error improves the measurement accuracy for full aperture, single shot, Fizeau interferometry to < 2 nm RMS.

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.

Circular Subaperture Stitching Interferometry Based on Polarization Grating and Virtual–Real Combination Interferometer

This paper presents a polarization grating based circular subaperture stitching interferometer. The system can be used for small F/# concave surface tests with a large F/# transmission sphere, where F/# is the ratio of focal length to aperture. A polarization grating was employed to deflect the incident beam for subaperture scanning by its axial rotation instead of a multi-axis motion-control system. Compared with the traditional subaperture stitching interferometric system, the system proposed in this paper is smaller in size and reduces the measurement error introduced by mechanical adjustment. Using a virtual interferometer model and a virtual-real combination algorithm to remove the retrace error, the full-aperture figure error can be directly obtained without the need for a complex stitching algorithm. The feasibility of the algorithm was verified, and the measurement error caused by the modeling error was analyzed by simulation. The capability of the polarization grating to scan subapertures was experimentally confirmed, and possible solutions to some engineering challenges were pointed out. The research in this paper has pioneering and guiding significance for the application of polarization grating in interferometry.

Iterative correction method of a retrace error in interferometry.

In interferometer measurements, the inconsistency of the optical range through which the reference and test lights pass introduces a retrace error in the phase measurement. In this study, we propose an iterative retrace error correction method in interferometry. A black-box model is established based on the linear and squared relationships between the retrace error and the tilt of the testing surface. The error correction phase is obtained using the least-squares method; thereafter, the global tilt is determined to iteratively correct the retrace error. The root mean square (RMS) of the residuals was > 3.2 × 10-5λ, >6.4 × 10-3λ, and >1.4 × 10-3λ in the simulation, experimentally computed retrace error correction in the planar measurement, and spherical measurement, respectively, proving that the retrace error can be effectively corrected.

Measurement and correction of lateral distortion in a Fizeau interferometer based on the self-calibration technique.

The lateral distortion of a surface measuring Fizeau interferometer may cause distorted image features in the lateral direction, as well as the surface form error in the axial direction (which is a source of the retrace error). Traditional method for lateral distortion measurement requires a high-accuracy calibration plate featuring a grid pattern. Such a calibration plate is not always available, especially when the required accuracy of the grid pattern comes to the order of sub-micrometer or even nanometer level. To remove the dependence on the plate accuracy, we propose a self-calibration method for the measurement and correction of lateral distortion in Fizeau interferometer. The self-calibration technique may separate the lateral distortion and the geometric error of the calibration plate. This method is verified using a 108-mm-aperture Fizeau interferometer. The experiments show that the form measurement error of a surface tilted at approximately 5° and 16° can be reduced from 92 nm to 43 nm and from 251 nm to 144 nm (peak-to-valley value), respectively, after the distortion correction.

Reduction of retrace error from broadband variable transmission spheres in Fizeau interferometry

We have previously presented broadband variable transmission sphere (BVTS) systems based on the Alvarez lens for use in Fizeau interferometry. A BVTS system may replace multiple conventional transmission spheres with one system composed of two plano-freeform optical elements. In this paper, we discuss retrace error as a source of aberrations in Fizeau interferometry measurements when using BVTS systems. We explore a design optimization method to decrease retrace error by optimizing freeform surfaces to reduce targeted Fringe Zernike wavefront aberrations. Significant reduction in peak-to-valley retrace error is demonstrated.

Error correction algorithms in non-null aspheric testing next generation sequencing data

Non-null tests are more versatile than null tests and could give fast, generic tests with sufficient performance. The non-null testing methods are able to measure aspheric surfaces with substantial deviations from a reference sphere. The ray will return along a different path and interfere with a separate reference ray in a non-null arrangement. The retrace error occurs because of the significant disparity between the aspheric and the reference. To correct retracing errors and replicate aspheric fabrication imperfections, complex systems modelling and simulations are required. We propose a ray-tracing-based technique for simulating the non-null interferometer that estimates the optical path differences by tracking lasers across the reference and testing paths. An error correction method is used by ray trace analysis to correct the retracing errors in non-null tests of aspheric surfaces. Experiments were conducted to demonstrate the effectiveness of this strategy. The result of the study shows that the proposed system successfully and effectively rectifies the retracing error compared to other practical findings and computer simulation methods. It is more versatile, faster, and provides adequate aspheric assessment performance.

Simulation and measurement of systematic errors of stitching interferometry for high precision X-ray mirrors with large radius of curvature.

To accurately measure the surface figure of curved mirrors with large radius of curvature (RoC) using stitching interferometry methods, three types of measurement errors are systematically studied, including retrace error, defocusing error within a single subaperture, and stitching angle error among different subapertures. It was found that part of the retrace error caused by the mismatch between the reference wavefront and reflected wavefront has little effect, while the overall retrace error, including the influence of the imperfect optical elements, will cause an error of 1-2 nm RMS within a single subaperture. Defocusing error will enlarge the error due to amplification of optical path error caused by the deviation of the position of the CCD. Because the error is mainly in the edge area, a slope threshold, which controls the maximum surface slope of each subaperture, can be optimized to reduce the effect of the defocus on stitching measurement error. Constant angle error among neighboring subapertures has the biggest accumulation effect on the final stitched figure. For the spherical mirror with RoC of 100 m of 80mm×40mm, the error of the one-dimensional residual profile is 4.67 nm PV, assuming a constant angle error of 2×10-7rad. For the elliptical mirror with RoC of 60-140 m, it is more than 15 nm peak-to-valley (PV). It is because the profile difference caused by constant angle error is closer to a circle, which can be mostly removed after subtraction of a best-fit sphere. Based on the above error analysis, the developed algorithm-based stitching method was used to measure an elliptical cylindrical mirror of 74mm×40mm with RoC of 60-140 m, and the result was compared with a slope measurement instrument from the Beijing Synchrotron Radiation Facility. After removing the best fitting ellipse profile, the one-dimensional difference between the two results is only 0.77 nm RMS, which demonstrated good measurement accuracy.

Effect of retrace error on stitching coherent scanning interferometry measurements of freeform optics.

We report on the effect of retrace error during measurement of freeform optics using a commercial coherence scanning interferometer (CSI), and its in-built stitching capabilities. It is shown that measuring segments of freeform optics under non-null conditions, results in artifacts on the measured zone, similar to the Seidel aberrations. An experimental approach is used to quantify the induced aberrations based on the local slopes of the surface. Simulation of surfaces containing different order aberrations is shown to have a significant effect on the measurement data. A correction method is proposed that uses experimental measurements to determine the required correction based on local slope and position in the aperture. These corrections reduce the measurement difference from a comparison measurement using a Fizeau interferometer.

Non-null interferometers for irregular surface measurement with system modeling

A design method of non-null interferometers is proposed in this paper for measuring irregular surfaces such as optical aspherics, freeform surfaces and droplet surfaces. With system modeling and raytracing, the improved reverse optimization reconstruction method is employed to correct the retrace error and obtain the accurate shape of the test part with all frequency information. To simplify the model in the raytracing program, an imaging lens images the desired interferogram at a certain plane on the detector and, therefore, only the test arm needs to be modeled. The imaging error of an interferogram, which can be reduced by the partial compensation system (PCS), is analyzed, and the design of the PCS is also demonstrated. Two cases with different optical layouts show the feasibility and the versatility of our method.

Dual-channel interferometer for vibration-resistant optical measurement

A dual-channel interferometer (DCI) is proposed for vibration-resistant optical measurement. The DCI employs a dual-channel interference optical path design, wherein an assistant channel is used to generate dense fringe patterns by introducing a spatial carrier frequency, and the coefficients of the vibration tilt phase plane are then extracted. The sparse fringe patterns of the main channel are combined with the coefficients of the tilt phase plane to calculate the measured phase distribution. This study reports the optical path and principles of the interference system. Simulations are performed using the proposed DCI phase extraction algorithm. To verify the DCI performance, phase measurements are carried out in a vibrating environment, and the results obtained using the proposed method are compared with those obtained using spatial phase-shifting interferometry (SPSI). The experimental results demonstrate that the DCI can effectively overcome the effect of vibration. Unlike spatial carrier frequency interferometry, although the spatial carrier frequency is also introduced in the assistant channel in the proposed DCI method, the dual-channel information extraction algorithm helps eliminate the calibration process of the retrace error before measurement, making this a calibration-free measurement. Moreover, the DCI utilizes environmental vibration to generate a ‘phase shift’, thus not only eliminating the need for a vibration isolation platform but also the need for a precise phase shifting device, with advantages such as structural simplicity, ease of operation, and low cost.

Improving the calibration of phase measuring deflectometry by a polynomial representation of the display shape

BackgroundPhase measuring deflectometry is a highly precise and full field metrology technique for specular surfaces based on the distortion of known reference patterns observed as a reflection at the surface under test. Typically, liquid crystal displays are employed to provide the required patterns. Due to a lack of research, these displays are used without sufficient calibration.MethodsIn this work, we present an enhanced calibration for phase measuring deflectometry, taking flatness deviations of the display surface into account. The display shape is modelled as a polynomial surface whose coefficients are determined by minimizing the retrace error in a global optimization procedure during calibration. This approach does not require any additional measurements or hardware. Improvements due to the enhanced calibration model are qualified experimentally using a flat and a spherical concave mirror.Results and conclusionThe model-based parameterization of the display surface yields significant improvement on both samples. The peak to valley (PV) of measured deviations on the plane mirror are reduced by 67% to 0.55 μm. Measuring the spherical sample without the display parameterization leads to a rather large shape deviation of 33.40 μm PV which is reduced by 94% to 1.98 μm. The viability of our approach confirms the dominant role of flatness deviations of the display surface as an error source in absolute shape measurement using phase measuring deflectometry.

Model-based multi-fringe interferometry using Zernike polynomials

In this paper, a general phase retrieval method is proposed, which is based on one single interferogram with a small amount of fringes (either tilt or power). Zernike polynomials are used to characterize the phase to be measured; the phase distribution is reconstructed by a non-linear least squares method. Experiments show that the proposed method can obtain satisfactory results compared to the standard phase-shifting interferometry technique. Additionally, the retrace errors of proposed method can be neglected because of the few fringes; it does not need any auxiliary phase shifting facilities (low cost) and it is easy to implement without the process of phase unwrapping.

部分补偿数字莫尔移相干涉的回程误差消除

A digital Moir phase-shifting interferometry with partial compensation lens was expounded to test the figure error of aspheric surfaces with high accuracy measurement. The real interferometer and the ideal interferometer model were established to obtain the real and ideal wavefront at the image plane of the interferometer. Then the image wavefront in the real interferometer related to the surface figure error of the aspheric surface under test was obtained by using the digital Moir phase-shifting technique. The error analysis of this measuring system was presented, and the reverse optimization procedure was applied to eliminate retrace error for the large figure error and reconstruct the test aspheric surface large figure error. Experimental results show that, compared to the profilometer, for the small figure error, the accuracy of the aspheric surface figure error measurement with the one-half method can achieve less than /20, both PV error and RMS error. For the large figure error, the reverse optimization method need to be used to obtain the accuracy of aspheric surface errors measurement of less than /5, both PV error and RMS error. Partial compensating digital Moir phase shifting interferometry for the test aspheric surface error based on reverse optimization procedure can effectively correct the retrace error, and reconstruct the large figure error of aspheric surfaces with high-accuracy.

Large radius of curvature measurement based on the evaluation of interferogram-quality metric in non-null interferometry

Non-null interferometry could use to measure the radius of curvature (ROC), we have presented a virtual quadratic Newton rings phase-shifting moiré-fringes measurement method for large ROC measurement (Yang et al., 2016). In this paper, we propose a large ROC measurement method based on the evaluation of the interferogram-quality metric by the non-null interferometer. With the multi-configuration model of the non-null interferometric system in ZEMAX, the retrace errors and the phase introduced by the test surface are reconstructed. The interferogram-quality metric is obtained by the normalized phase-shifted testing Newton rings with the spherical surface model in the non-null interferometric system. The radius curvature of the test spherical surface can be obtained until the minimum of the interferogram-quality metric is found. Simulations and experimental results are verified the feasibility of our proposed method. For a spherical mirror with a ROC of 41,400 mm, the measurement accuracy is better than 0.13%.

Circular carrier squeezing interferometry: Suppressing phase shift error in simultaneous phase-shifting point-diffraction interferometer

Circular carrier squeezing interferometry (CCSI) is proposed and applied to suppress phase shift error in simultaneous phase-shifting point-diffraction interferometer (SPSPDI). By introducing a defocus, four phase-shifting point-diffraction interferograms with circular carrier are acquired, and then converted into linear carrier interferograms by a coordinate transform. Rearranging the transformed interferograms into a spatial-temporal fringe (STF), so the error lobe will be separated from the phase lobe in the Fourier spectrum of the STF, and filtering the phase lobe to calculate the extended phase, when combined with the corresponding inverse coordinate transform, exactly retrieves the initial phase. Both simulations and experiments validate the ability of CCSI to suppress the ripple error generated by the phase shift error. Compared with carrier squeezing interferometry (CSI), CCSI is effective on some occasions in which a linear carrier is difficult to introduce, and with the added benefit of eliminating retrace error.

Characteristics of Linear Variable Differential Transformer(LVDT) Probe for Gauge Blocks Calibration

The LVDT probe is a very important component used in gauge blocks calibration via a mechanical comparative method. The probe is used to determine the central length difference between a reference gauge block and gauge block under-tested (UTC). Typically, an UTC and a reference gauge block have the same nominal length. However, some gauge blocks UTC are specially made for specific purpose where the reference gauge blocks with the same nominal length are not commercially available. Various reference gauge blocks are wrung in order to provide nominal length the same as that of the UTC. Wringing process is the troublesome step and causes larger measurement uncertainty. The lower accuracy is higher number of gauge block used to create reference gauge block, in order to improve accuracy of measurement, the LVDT probe was used at the longer range where the reference gauge block and the UTC don’t need to be the same nominal length. In this paper, characteristics of LVDT probe was investigated as it is related to the accuracy of the measurement result. Errors of LVDT probe came from non-linearity, calibration factor, retrace error, repeatability and maximum difference in length. A pair of gauge block, calculated by the Twyman-Green interferometer, length different range 5 μm to 230 μm was used in the study. Non-linearity of LVDT is evaluated by a simple linear regression model. The non-linearity of LVDT probe, calibration factor, retrace error, repeatability and maximum difference in length are 25 nm, 1.0003, 3 nm, 6 nm and 80 nm, respectively. Therefore, by using this technique, central length difference between the 2 gauge blocks up to 80 nm can be calibrated with the uncertainty due to non-linearity of 15 nm The experiment shows the large error of retrace closes to 0.02 μm at 0.09 μm. This can be determined the maximum difference in length to calibrate gauge blocks at difference nominal lengths. The measurement uncertainty of non-linearity is evaluated and it is close to 15 nm.