Interference imaging for aspheric surface testing

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.

Absolute measurement approach for crystal growth height based on a polarization-synchronized phase-shifting interferometer

The quality of the solid deuterium-deuterium (D-D) layer in the inertial confinement fusion (ICF) target plays a vital role in the success of fusion experiments. A good understanding of how the quality is affected by the unstable growth of D-D crystal is required. This article provides an approach of measuring D-D layer absolute height in real time by combining monitoring algorithms and a synchronous phase-shifting interferometer. In the approach taken, a real-time monitoring technology, in which an antivibration algorithm is added, is used to get an absolute height of monitoring zone, overcoming the inability to accurately detect the saltus step in the interferometric measurement. Meanwhile, the polarization-synchronized phase-shifting technology is propitious to retrieve the D-D height distribution in a whole interferogram. Consequently, the categorical altitude of the D-D layer in entire crystalline regions can be obtained. Simulation analysis together with experiments have proved that a non-contact, rapid, and high-precision measurement of the D-D crystal absolute height can be realized by using the interferometer and method proposed.

Enhanced calibration for freeform surface misalignments in non-null interferometers by convolutional neural network

Most tested surface calibration methods in interferometers, such as the direct coefficients removing method, the sensitive matrix (SM) method, and deep neural network (DNN) calibration method, rely on Zernike coefficients. However, due to the inherent rotationally non-symmetric aberrations in a non-null freeform surface interferometer, the interferograms are usually non-circular even if the surface apertures are circular. The Zernike coefficients based methods are inaccurate due to the non-orthogonality of Zernike polynomials in the non-circular area. A convolutional neural network (CNN)-based misalignment calibration method is proposed. Instead of Zernike coefficients, the well-trained CNN treats the interferogram directly to estimate the specific misalignments. Simulations and experiments are carried out to validate the high accuracy.

Model calibration by multi-null constraint for an optical freeform surface adaptive interferometer

Model calibration is performed for an adaptive freeform surface interferometer (AFI). In view of the non-unique null configuration in AFI, the multi-null constraint (MNC) calibration method is proposed to address error coupling in the null configuration modeling. The final figure error of the tested surface can be extracted together with the coupling parameters. The performance of the MNC method is evaluated in simulations and experiments. The higher accuracy is proved after the MNC calibration. This calibration is preparation for the subsequent system instrumentation.

ICF target DT-layer refractive index and thickness from iterative analysis

An iterative algorithm based on optical path difference (OPD) and ray deflection is proposed to obtain the DT (deuterium-tritium)-layer refractive index and thickness of the ICF (inertial confinement fusion) target simultaneously. Starting from an assumed initial value, the refractive index and thickness are solved back and forth until the iteration stopping criterion is reached. Simulations show that the relative retrieval error of the DT-layer refractive index is better than 0.05% after finite iterations, and that of the thickness is better than 0.1%. Experiments show that the target shell refractive index and thickness can be retrieved with a relative error below ±2%. The test uncertainties from experiments were also analyzed.

Aspheric optical surface profiling based on laser scanning and auto-collimation

Nowadays the utilization of aspheric lenses has become more and more popular, enabling highly increased degree of freedom for optical design and simultaneously improving the performance of optical systems. Fast and accurate surface profiling of these aspheric components is a real demand in characterization and optimization of the optical systems. In this paper, a novel and simple surface profiler instrument is designed and developed to fulfill the ever increasing need of testing the axially symmetric aspheric surface. The proposed instrument is implemented based on a unique mapping between the position and rotation angle of the reflective mirror in optical path and the coordinate of reflection point on the surface during rapid laser beam scanning. High accuracy of the proposed surface profiling method is ensured by a high-resolution grating guide rail, indexing plate, and position sensitive detector based on laser auto-collimation and beam center-fitting. Testing the meridian line of both convex and concave surfaces has been experimentally demonstrated using the developed instrument. In comparison to tested results from conventional image measuring instruments and coordinate measuring machines, coefficient of determination better than 0.999 99 and RMS less than 1.5 μm have been achieved, which validates the feasibility of this method. Analysis on the systematic error is beneficial to further improve its measurement accuracy. The presented instrument-essentially builds on the geometrical optics technique-provides a powerful tool to measure the aspheric surfaces quickly and accurately with stable structure and simple algorithm.

Retrace error reconstruction based on point characteristic function

Figure measuring interferometers generally work in the null condition, i.e., the reference rays share the same optical path with the test rays through the imaging system. In this case, except field distortion error, effect of other aberrations cancels out and doesn't result in measureable systematic error. However, for spatial carrier interferometry and non-null aspheric test cases, null condition cannot be achieved typically, and there is excessive measurement error that is referenced as retrace error. Previous studies about retrace error can be generally distinguished into two categories: one based on 4th-order aberration formalism, the other based on ray tracing through interferometer model. In this paper, point characteristic function (PCF) is used to analyze retrace error in a Fizeau interferometer working in high spatial carrier condition. We present the process of reconstructing retrace error with and without element error in detail. Our results are in contrast with those obtained by ray tracing through interferometer model. The small difference between them (less than 3%) shows that our method is effective.

Determination of aspheric vertex radius of curvature in non-null interferometry

Traditional spherical radius of curvature interferometry is not valid for an aspheric vertex radius of curvature (VROC) due to the obstacle in identifying null positions (cat's eye or confocal position). Simultaneous optimization for multiconfiguration of an interferometer model is proposed to retrieve the actual aspheric VROC from its biased nominal value. This procedure works out the contradiction between VROC deviation and positioning error and even surface figure error, independent of absolute positioning by cat's eye or confocal position. In this method, the aspheric VROC and surface figure can be measured simultaneously, which facilitates the test process remarkably in practical optical shop testing. Furthermore, the parent VROC of a hollow aspheric (annular surface) also can be determined in this method. The performance of the proposed method is validated by experiments.

Measurement improvement by high overlapping density subaperture stitching interferometry

The vibration-modulated subaperture stitching interferometer acquires the interferogram on the fly dynamically. With its highly improved measurement throughput, we applied the device for high overlapping density subaperture stitching interferometry to acquire hundreds of overlapping subapertures in a single phase stitching measurement. The averaging effect of the high overlapping density stitching interferometer is discussed. In the experiment, the proposed high overlapping density stitching interferometer is also proved to reduce measurement uncertainty and improve measurement quality effectively.

Reverse optimization reconstruction of aspheric figure error in a non-null interferometer

Aspheric non-null testing, as an alternative to the traditional null testing, achieves more flexible measurements. However, figure-error reconstruction in non-null tests has always been difficult due to the presence of retrace error. A novel method with reverse optimization is proposed for aspheric figure-error reconstruction in a non-null interferometer. It is a generalized and effective approach based on system modeling and polynomial fitting. An optimization function is set with polynomial coefficients of the desired figure error as variables and those of the detected experimental wavefront as optimization targets. Through the reverse optimization process with iterative ray tracing, the optimal solutions can be extracted and the desired figure error is reconstructed with a simple fitting procedure. Numerical simulations verifying the high accuracy of the proposed method are presented with error considerations. A set of experiments has also been carried out to demonstrate the validity and repeatability of this method.

Compensation of high-order misalignment aberrations in cylindrical interferometry

Interferometry with a null corrector can be used to test cylindrical surfaces. The requirement for accurate measurement is a null fringe pattern. When the tested cylindrical surface is not perfect or seriously misaligned, a nonzero fringe pattern might be obtained. As a result, high-order misalignment aberrations (e.g., coma and spherical aberration) are introduced into the measurement. The sources and types of high-order misalignment aberrations are analyzed by orthogonal Legendre polynomials. Based on the analysis, a mathematical model was proposed to estimate the high-order misalignment aberrations. Then a wavefront difference method was proposed to calibrate the coefficients of this model. With the calibrated coefficients, the high-order misalignment aberrations can be determined and separated from the measurement results. Several experiments were conducted to demonstrate the validity of the proposed method. Compared with the lower-order misalignment aberrations removal method, the proposed method can reduce the high-order misalignment aberrations by at least half, and highly accurate results can be achieved by the proposed method.

Optimal strategy for fabrication of large aperture aspheric surfaces

Aspheric surfaces are widely used because of their desirable characteristics. Such a surface can obtain nearly perfect imaging quality with fewer optical elements and reduce the size and mass of optical systems. Various machine systems have been developed based on modern deterministic polishing technologies for large aperture aspheric surfaces. Several factors affect the final precision of large aperture aspheric surfaces, such as the velocity limit of the machine and the path design. Excess velocity, which will be truncated automatically by the computer numerical control system, may cause the dwell time to deviate from the desired time. When a path designed on a two-dimensional surface map with equidistant pitch is projected onto an aspheric surface, the pitch changes as a result of the varied curvature of the aspheric surface. This may affect the removal map and cause some ripple errors. A multiregion distribution strategy, which includes velocity checking, is proposed in this study to avoid exceeding the velocity limits. The strategy can be used to modify local errors and edge effects. A three-dimensional spiral path generation method is also presented using an iterative method to ensure uniformity in the space length of the adjacent circle of the spiral path. This process can reduce the ripple error caused by the overlapping of tool paths. A polishing experiment was conducted, and the results proved the validity of the proposed strategies.

Practical and accurate method for aspheric misalignment aberrations calibration in non-null interferometric testing

Calibration for misalignment aberrations is one of the challenges in non-null interferometric aspheric testing. The high-order misalignment aberrations are especially difficult to distinguish from the rest of the testing system. The traditional calibration method removing the first four terms from the Zernike coefficients of the test wavefront is obviously inaccurate. Computer-aided alignment is considered to be an effective method; however, it is less practical due to its dependency on mechanical or manual adjustment, as are other common methods. A practical and accurate calibration method based on system modeling is proposed in this paper for misalignment aberrations' removal. In this work, actual misalignments, which are calculated from five selected low-order aberrations of the test wavefront in the experiment, are simulated in the model to predict all misalignment aberrations by ray tracing. These aberrations then are removed by a simple wavefront data subtraction. The method depends on neither a precise adjusting mechanism nor a troublesome manual adjustment. Experimental results showing feasibility and repeatability of the proposed method are presented.

Generalized data reduction approach for aspheric testing in a non-null interferometer

Data reduction in non-null tests is difficult due to the presence of retrace error. We propose a simple yet effective data reduction approach for aspheric testing in a non-null interferometer. The new approach gives figure error of the aspheric by just subtracting the theoretical wavefront and first-order errors from the real wavefront obtained in the non-null interferometer. Precise prediction of the theoretical wavefront can be achieved by accurate calibration of the partial compensation system. The approach can be considered a generalization of the traditional data processing method in null tests, and errors that may affect its accuracy are discussed. A set of experiments have been carried out to demonstrate its validity and feasibility.

Nonnull interferometer simulation for aspheric testing based on ray tracing

The nonnull interferometric method that employs a partial compensation system to compensate for the longitude aberration of the aspheric under test and a reverse optimization procedure to correct retrace errors is a useful technique for general aspheric testing. However, accurate system modeling and simulation are required to correct retrace errors and reconstruct fabrication error of the aspheric. Here, we propose a ray-tracing-based method to simulate the nonnull interferometer, which calculates the optical path difference by tracing rays through the reference path and the test path. To model a nonrotationally symmetric fabrication error, we mathematically represent it with a set of Zernike polynomials (i.e., Zernike deformation) and derive ray-tracing formulas for the deformed surface, which can also deal with misalignment situations (i.e., a surface with tilts and/or decenters) and thus facilitates system modeling extremely. Simulation results of systems with (relatively) large and small Zernike deformations and their comparisons with the lens design program Zemax have demonstrated the correctness and effectiveness of the method.

Practical methods for retrace error correction in nonnull aspheric testing

Nonnull test is often adopted for aspheric testing. But due to its violation of null condition, the testing rays will follow different paths from the reference and aberrations from the interferometer will not cancel out, leading to widely difference between the obtained surface figure and that of the real, which is called the Retrace-error accordingly. In this paper, retrace error of nonnull aspheric testing is analyzed in detail with conclusions that retrace error has much to do with the aperture, F number and surface shape error of the aspheric under test. Correcting methods are proposed according to the manner of the retrace errors. Both computer simulation and experimental results show that the proposed methods can correct the retrace error effectively. The analysis and proposed correction methods bring much to the application of nonnull aspheric testing.

Absolute interferometric test of aspheres by use of twin computer-generated holograms

A complete absolute interferometric test of axially symmetric aspheres is presented. The method is based on a specially designed computer-generated hologram (CGH) that reconstructs an aspherical wave as well as a spherical auxiliary wave. Since both phase functions have the same symmetry and their pattern is simultaneously encoded, we call this type of multiplex hologram a Twin-CGH. The spherical wave is used for calibration. The aberrations of the spherical auxiliary wave are measured absolutely with either a spherical mirror or an absolute test for Fresnel zone plates. Thus the two types of aberration inherent in the CGH can be identified and separated from each other. The errors of the spherical wave can be transferred to those of the aspherical wave. Two different methods thatuse Twin-CGHs for absolute testing of aspheric surfaces are described. Test procedures are explained, equations are derived, and experimental results are presented. A mutual comparison of the two results and a comparison with the established N-position rotation test are given.

Reflective grating interferometer: a folded reversal and shearing wave-front interferometer

The reflecting grating interferometer (RGI) is a folded and reversal wave-front interferometer sensitive only to asymmetrical aberrations such as third-order coma. The RGI can isolate and evaluate coma both in nearly collimated and in noncollimated beams. We propose a RGI with a different optical configuration that includes a lateral shearing in addition to folding and reversal operations. With lateral shear, the RGI also becomes sensitive to other terms of third-order aberrations such as defocusing, astigmatism, and spherical aberration. Optical path difference equations for interpreting interferograms and numerical simulations are presented to show how the interferometer works in the shearing configuration. Its potential applications are described and discussed.

Measurement and calibration of interferometric imaging aberrations

Phase-shifting interferometry is the standard method for testing figure error on optical surfaces. Instruments measuring spheres and flats are readily available, but the accurate measurement of aspheres requires null correction. One problem with the general (nonull) testing of aspheres is the loss of common path. Systematic errors are introduced into the measurement by the fringe imaging optics. The sources and types of error are reviewed, as well as their effect on a wave-front measurement. These nonnull errors are predicted generally, with third-order analytic expressions derived for a tilted or a defocused test surface. An interferometer is built to test the expressions. The imaging system is a single lens, nominally image telecentric. Measurements are performed on a test surface defocused from -5 to 5 mm. The resulting measurement bias is shown to be in good agreement with third-order aberration theory predictions.