Data-based systematic error extraction and compensation methods based on wavelet transform in ultra-precision optical polishing.

Sub-aperture polishing is a key technique for fabricating ultra-precision optics. However, the existence of the polishing errors that are difficult to be compensated by physical modeling seriously affects the manufacturing accuracy and efficiency of optical components. To address this problem, a data-based systematic error extraction and compensation (DSEC) method was proposed to enhance the polishing accuracy on optics. To maximize the extraction quality in a small dataset condition, the wavelet transform is introduced into the extraction process, and the uncertainty of the piston term in the interferometer measurement is improved by L1-norm optimization. Furthermore, two typical error sources (loss of polishing fluid in the edge and the robot trajectory error) are used to verify the effectiveness of the proposed method; in experimental verification, the root mean square (RMS) of the surface figure of a ϕ85-mm mirror was decreased from 0.069λ to 0.017λ, and the RMS of the 610 × 440 mm mirrors was achieved at 0.019λ after the edge compensation, where the polishing accuracy can be improved by more than 4 times; additionally, the RMS of the surface figure with an effective aperture of 480 × 360 mm mirror was reached at 0.011λ after the trajectory error compensation, where the polishing accuracy can be improved by more than 2 times. The proposed DSEC model offers insights that will help achieve advancement in the sub-aperture polishing process.

Plug-and-play positioning error compensation model for ripple suppressing in industrial robot polishing.

The industrial robot-based polisher has wide applications in the field of optical manufacturing due to the advantages of low cost, high degrees of freedom, and high dynamic performance. However, the large positioning error of the industrial robot can lead to surface ripple and seriously restrict the system performance, but this error can only be inefficiently compensated for by measurement before each processing at present. To address this problem, we discovered the period-phase evolution law of the positioning error and established a double sine function compensation model. In the self-developed robotic polishing platform, the results show that the Z-axis error in the whole workspace after compensation can be reduced to ±0.06m m, which reaches the robot repetitive positioning error level; the Spearman correlation coefficients between the measurement and modeling errors are all above 0.88. In the practical polishing experiments, for both figuring and uniform polishing, the ripple error introduced by the positioning error is significantly suppressed by the proposed model under different conditions. Besides, the power spectral density (PSD) analysis has shown a significant suppression in the corresponding frequency error. This model gives an efficient plug-and-play compensation model for the robotic polisher, which provides possibilities for further improving robotic processing accuracy and efficiency.

Statistical perception of the chaotic fabrication error and the self-adaptive processing decision in ultra-precision optical polishing.

Subaperture polishing is a key technique for fabricating ultra-precision optics. However, the error source complexity in the polishing process creates large fabrication errors with chaotic characteristics that are difficult to predict using physical modelling. In this study, we first proved that the chaotic error is statistically predictable and developed a statistical chaotic-error perception (SCP) model. We confirmed that the coupling between the randomness characteristics of chaotic error (expectation and variance) and the polishing results follows an approximately linear relationship. Accordingly, the convolution fabrication formula based on the Preston equation was improved, and the form error evolution in each polishing cycle for various tools was quantitatively predicted. On this basis, a self-adaptive decision model that considers the chaotic-error influence was developed using the proposed mid- and low-spatial-frequency error criteria, which realises the automatic decision of the tool and processing parameters. An ultra-precision surface with equivalent accuracy can be stably realised via proper tool influence function (TIF) selection and modification, even for low-deterministic level tools. Experimental results indicated that the average prediction error in each convergence cycle was reduced to 6.14%. Without manual participation, the root mean square(RMS) of the surface figure of a ϕ100-mm flat mirror was converged to 1.788 nm with only robotic small-tool polishing, and that of a ϕ300-mm high-gradient ellipsoid mirror was converged to 0.008 λ. Additionally, the polishing efficiency was increased by 30% compared with that of manual polishing. The proposed SCP model offers insights that will help achieve advancement in the subaperture polishing process.

Wavefront-coded phase measuring deflectometry for the all-focused measurement.

Phase measuring deflectometry is a powerful measuring method for complex optical surfaces, which captures the reflected fringe images encoded on the screen under the premise of focusing the measured specular surface. Due to the limited depth of field of the camera, the captured images and the measured surface cannot be focused at the same time. To solve the position-angle uncertainty issue, in this Letter, the wavefront coding technology is used to modulate the imaging wavefront of the deflectometry, thereby making the measuring system insensitive to the defocus and other low-order aberration including astigmatism, field curvature, and so on. To obtain the accurate phase, the captured fringe images are deconvoluted using the modulated point spread function to reduce the phase error. Demonstrated with a highly curved spherical surface, the measurement accuracy can be improved by four times. Experiments demonstrate that the proposed method can successfully reconstruct the complex surfaces defocusing the captured images, which can greatly release the focusing requirement and improve measurement accuracy.

Iterative space-variant sphere-model deflectometry enabling designation-model-free measurement of the freeform surface.

Freeform optics, offering high degrees of design freeform to control light propagation, have already been widely applied in various photoelectric equipment. The form quality of those optics is crucial to their opto-electronics functionalities, which requires to be measured accurately. The deflectometry is a promising technology to test the complex freeform surfaces. In general, there is a designed surface model for the monoscopic deflectometry to estimate the positions of whole measured points to solve the issue of height-slope ambiguity. However, the unknown or inaccurate surface model can induce errors into the measured normal, thereby decreasing the measurement precision. In this paper, without relying on the known surface model, the proposed method iteratively optimizes a sphere model to describe the measured surface by changing the spherical radius. In order to reduce the global error, the space-variant spheres are optimized, respectively, to estimate the whole-aperture surface coordinate. With the help of the iteration surface reconstruction process, the optimal number of the space-variant spheres is achieved to meantime obtain the final reconstructed surface. Compared to the measurements by using the plane model, the form accuracy can be improved by three times. Experiments demonstrate that the proposed method can successfully reconstruct the complex surfaces without the need of a known surface model, which can greatly improve the measuring flexibility and measurement accuracy.

Correction of aberration-induced phase errors in phase measuring deflectometry.

Phase measuring deflectometry is a powerful measuring method of complex optical surfaces that captures the reflected fringe images associated with a displaying screen and calculates the normal vectors of the surface under test (SUT) accordingly. The captured images are usually set conjugate to the SUT, which in turn makes the screen defocused. As a result, the blurring effect caused by the defocus and aberrations of the off-axis catadioptric imaging system can severely degrade the phases solved from the blurred images. In order to correct the phase errors, the space-variant point spread functions (PSFs) are modeled using a skew-normal function. The phase bias is estimated by forward convolution between the captured images and the PSF models. Demonstrated with a highly curved aspheric surface, the measurement accuracy can be improved by three times.

Flexible one-shot geometric calibration for off-axis deflectometry.

Off-axis deflectometry is widely applied in the measurement of specular surfaces. However, the measuring accuracy depends on the reliability of geometrical calibration. Existing methods are inconvenient to be utilized due to their disadvantages of low efficiency and operational complexity. A simple geometrical calibration method is proposed by applying a flat mirror with markers, and only one image needs to be captured. A compensation process is introduced to correct the form error of the mirror. Experimental results show that the re-projection errors decrease from 0.319 pixels down to 0.12 pixels; thus the measuring efficiency and accuracy of optical surfaces can be greatly improved.

Extra-detection-free monoscopic deflectometry for the in situ measurement of freeform specular surfaces.

Extra detections of feature points conducted by a third-party instrument are indispensable in the monoscopic deflectometry to specify the surface position, but the measuring capability can be limited because most freeform surfaces do not contain any distinguishable and detectable feature points. In this Letter, the behaviors of the position error and form error of the measured surface are discriminated by reverse ray-tracing. A novel iterative optimization strategy is developed to position the workpiece for the in situ deflectometric measurement so that extra detections can be avoided. It is demonstrated that this method can achieve parallel measuring accuracy with conventional deflectometry, but the measuring efficiency and range of freeform surfaces can be greatly improved. A hollow off-axis parabolic mirror is measured, and its departure from the interferometric results is 139.85 nm root mean square.

Efficient phase retrieval of two-directional phase-shifting fringe patterns using geometric constraints of deflectometry.

In the phase measuring deflectometry, two groups of fringe patterns in orthogonal directions are usually applied to establish the correspondences between the pixel pairs on the screen and camera. Usually, 16 phase-shifting fringe patterns with different spatial frequencies are required in order to calculate the absolute phases in the conventional temporal phase unwrapping algorithms. This requirement makes the measurement inefficient and not robust against environmental noise. In this paper, an efficient phase retrieval strategy is developed, which requires only six fringe patterns. The modulating information in one-direction is obtained by first using four fringe patterns, and then it is applied to assist the phase calculation in the other direction, so that only two extra fringe patterns are needed. Subsequently the phases are unwrapped by using the geometric constraints of the software configurable optical test system without additional image acquisition. The measurement time is saved by 5/8, compared to the conventional methods. In this way, the influence of the low-frequency disturbances can be suppressed in the workshop condition. Experiments demonstrate that the proposed method can reliably retrieve the absolute phases, and it is of significance to improve the measuring efficiency and stability of in situ deflectometry.

Self-calibration of in situ monoscopic deflectometric measurement in precision optical manufacturing.

Accurate in situ measurement of complex optical surfaces is desired for precision manufacturing, and deflectometry is a promising measuring method. However, deflectometry's measurement accuracy depends heavily on the geometric calibration's reliability. Existing calibration methods suffer from the limitations of low efficiency, complicated operations, and expensive equipment. Therefore, a self-calibration method is proposed for the monoscopic deflectometry in order to determine the the camera and screen positions. A flat mirror without markers, rather than high-precision metrological instruments, is employed to assist with the self-calibration work, and a re-projection model is adopted in order to trace the rays from the screen to the camera. Numerical optimization is performed on the global parameters-including the camera intrinsic parameters, the screen intrinsic parameters, and the geometric transformations-in order to improve the geometric calibration's reliability. This self-calibration method only requires that four images be captured, hereby reducing the operating complexity and improving the measuring efficiency. Experiments demonstrate that the re-projection error can be significantly decreased, and also that the measuring accuracy of specular surfaces is comparable to high-precision interferometry.

3D shape measurement of discontinuous specular objects based on advanced PMD with bi-telecentric lens.

This paper presents an advanced phase measuring deflectometry (PMD) method based on a novel mathematical model to obtain three dimensional (3D) shape of discontinuous specular object using a bi-telecentric lens. The proposed method uses an LCD screen, a flat beam splitter, a camera with a bi-telecentric lens, and a translating stage. The LCD screen is used to display sinusoidal fringe patterns and can be moved by the stage to two different positions along the normal direction of a reference plane. The camera captures the deformed fringe patterns reflected by the measured specular surface. The splitter realizes the fringe patterns displaying and imaging from the same direction. Using the proposed advanced PMD method, the depth data can be directly calculated from absolute phase, instead of integrating gradient data. In order to calibrate the relative orientation of the LCD screen and the camera, an auxiliary plane mirror is used to reflect the pattern on the LCD screen three times. After the geometric calibration, 3D shape data of the measured specular objects are calculated from the phase differences between the reference plane and the reflected surface. The experimental results show that 3D shape of discontinuous specular object can be effectively and accurately measured from absolute phase data by the proposed advanced PMD method.

Generic precise augmented reality guiding system and its calibration method based on 3D virtual model.

Augmented reality system can be applied to provide precise guidance for various kinds of manual works. The adaptability and guiding accuracy of such systems are decided by the computational model and the corresponding calibration method. In this paper, a novel type of augmented reality guiding system and the corresponding designing scheme are proposed. Guided by external positioning equipment, the proposed system can achieve high relative indication accuracy in a large working space. Meanwhile, the proposed system is realized with a digital projector and the general back projection model is derived with geometry relationship between digitized 3D model and the projector in free space. The corresponding calibration method is also designed for the proposed system to obtain the parameters of projector. To validate the proposed back projection model, the coordinate data collected by a 3D positioning equipment is used to calculate and optimize the extrinsic parameters. The final projecting indication accuracy of the system is verified with subpixel pattern projecting technique.