Abbe Error Research Articles

Model for surface topography prediction in the ultra-precision grinding of silicon wafers considering volumetric errors

Ultra-precision grinding machine is widely used for thinning silicon wafers. The wafer surface quality is considerably affected by volumetric errors at the functional point of the ultra-precision grinding machine. Therefore, a volumetric error model is established based on the homogeneous transformation matrix (HTM) method. The model is, then, optimized by considering Abbe and Bryan errors induced volumetric errors during the error measurements process. A prediction model for the wafer surface topography is established by combining the manufacturing mechanism of the ultra-precision grinding and the optimized volumetric error model of the ultra-precision grinding machine. Volumetric errors are detected using an author’s group developed laser measurement system and a commercial spindle error analyzer. The model enables to predict the topography and total thickness variation (TTV) of the wafer surface. A group of grinding experiments are performed to verify the effectiveness of the proposed models.

Laser differential confocal axicon surface profile measurement method

Aiming to address the challenging problem of high-precision axicon profile measurement, a laser differential confocal axicon profile normal tracking (LDCNT) measurement method with high spatial resolution and anti-tilt capability is proposed in this paper. This method utilizes the principle of normal direction adaptive tracking for axicon profile measurements. It employs the Phase-Sensitive Detector (PSD) image plane spot position to calculate the normal angle and a high-precision rotary axis to drive the sensor normal to the measured surface to achieve the normal direction adaptive tracking measurement of the axicon profile and reduce the difficulty of pose adjustment of the measured axicon. Using the laser differential confocal focusing principle, the differential confocal signal is normalized to improve the axial resolution and avoid the Abbe error generated by the local inclination of the surface, thus realizing high-resolution and anti-tilt measurement of the axicon profile. Through the reference frame error decoupling axicon profile measurement and evaluation technique, spatial error compensation, three-dimensional profile reconstruction, and evaluation of the measurement data are performed to reduce the systematic measurement error and realize high-precision measurement and reconstruction of the axicon profile. Finally, the feasibility of the method is verified by measuring the profile of a standard cylinder and axicon at different axicon angles. The experimental analyses show that this method provides a new method for high-precision measurement of the axicon profile with Peak-to-Valley (PV) repeatability better than 40 nm (3σ) and Root-Mean-Square (RMS) repeatability better than 15 nm (3σ).

An on-machine measurement technique with sub-micron accuracy on a low-precision grinding machine tool

The on-machine measurement system can effectively improve the machining efficiency and accuracy of the optics in the grinding stage, but the common grinding machine has large geometric errors and ambient temperature errors, which are the key factors restricting the accuracy of on-machine measurement. In this paper, a high-precision error separation technique including geometric error separation and temperature error separation is proposed to achieve sub-micron level measurement accuracy in the grinding stage. The on-machine measurement system is built based on the color confocal sensor. A sensitive error model is established based on Abbe's principle and multi-body dynamics, and the geometric error separation with zero Abbe's error is achieved by a high-precision reference mirror. Then the temperature error influence model is established based on the scanning path, and the temperature error separation method based on the auxiliary profile line is proposed according to the temperature error model. Finally, it is verified through experiments that after error separation, the measurement accuracy of the on-machine measurement system is improved from tens of micron to sub-micron, which exceeds the measurement accuracy of ordinary coordinate measurement machines (CMMs), and can effectively improve the machining accuracy and efficiency in the grinding stage.

Full-Range Static Method of Calibration for Laser Tracker

This paper focuses on the challenge of the inability to accurately calibrate the static measurement of a laser tracker across the full scale. To address this issue, this paper proposes to add a hollow corner cube prism on a 50 m high-precision composite guide rail to achieve a double-range measurement of the laser tracker. Data analysis indicated that, in the 77 m identical-directional double-range measurement experiment, the maximum indication error of a single-beam laser interferometer was −29.5 μm, and that of a triple-beam laser interferometer was 14.6 μm, and the measurement indication error was obviously small when the Abbe error was eliminated. The single-point repeatability of the tracker was 0.9 μm. In the 50 m identical-directional verification experiment, the results of the direct measurement outperformed those of the double-range measurement, and the indication errors under standard conditions were −4.0 μm and −8.9 μm, respectively. Overall, the method used in the experiment satisfies the requirements of the laser tracker. In terms of the identical-directional measurement, the measurement uncertainty of the tracker indication error is U ≈ 1.0 μm + 0.2L (k = 2) L = (0~77 m). The proposed method also provides insights for length measurements using other high-precision measuring instruments.

A study on the real-time compensation system based on Abbe principle for range accuracy of submicron measuring instrument

Range accuracy is an important parameter to evaluate the performance of laser tracker. However, the different field-testing methods of laser tracker influenced the range accuracy. In this paper, we propose a universal scheme for correcting range accuracy of laser tracker based on Abbe principle. The reasons for the Abbe error have been analyzed and the theoretical calculation and simulation verification have been done. Finally, we built Abbe error compensation system by using the autocollimator to compensate the distance accuracy error of home-made laser tracker. The experimental results show that the ranging accuracy after compensation can be improved from 1.828 ppm to 0.496 ppm. The method proposed in this paper is also applicable to the range accuracy calibration of other submicron measuring instruments.

A Reflective-Type Heterodyne Grating Interferometer for Three-Degree-of-Freedom Subnanometer Measurement

A heterodyne grating interferometer (HTGI) able to simultaneously measure the three-axis translational displacements of ultra-precision motion stages with a sub-nanometer resolution is demonstrated. The HTGI is composed of two main parts, a reflective planar grating with a uniform micron-scale period in two orthogonal directions, and a reading head. Thanks to the reflective grating, the reading head design manages to make main optical paths common and located on the same side of this compact structure, potentially resulting in low Abbe errors and environmental variations. Relying on a laboratory-made dual-frequency laser source at 780 nm, two heterodyne interferometric measurement modules for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> -/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</i> -direction motions and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</i> -direction motion were realized based on first-order and zeroth-order diffraction, respectively. Experiments show that the proposed HTGI, in all three axes, can distinguish a 0.5 nm step or even better, perform an up to 2.5×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> 10linearity in a range of 80 μm, and hold a 5 nm stability within 300 s. These performances confirm that the proposed HTGI can greatly benefit nanoscience and technology, where multi-axis ultra-precision positioning is required, such as wafer stage in next-generation lithography machine, and so forth.

Development of the Heterodyne Laser Encoder System for the X-Y Positioning Stage.

This investigation develops a laser encoder system based on a heterodyne laser interferometer. For eliminating geometric errors, the optical structure of the proposed encoder system was carried out with the internal zero-point method. The designed structure can eliminate the geometric errors, including positioning error, straightness error, squareness error, and Abbe error of the positioning stage. The signal processing system is composed of commercial integrated circuits (ICs). The signal type of the proposed encoding system is a differential signal that is compatible with most motion control systems. The proposed encoder system is embedded in a two-dimensional positioning stage. By the experimental results of the positioning test in the measuring range of 27 mm × 27 mm, with a resolution of 15.8 nm, the maximum values of the positioning error and standard deviation are 12.64 nm and 126.4 nm, respectively, in the positioning experiments. The result shows that the proposed encoder system can fit the positioning requirements of the optoelectronic and semiconductor industries.

Two-dimensional abbe error analysis and modeling of CNC machine tool XY table unidirectional movement

Two-dimensional abbe error analysis and modeling of CNC machine tool XY table unidirectional movement

Development of a Rapid Optical Measurement System for Circular Workpieces with Irregular Tooth Contours after Broaching Process

During a manufacturing process, it is essential to quickly identify whether a tool needs to be replaced or adjusted, to ensure that production quality is not compromised. Therefore, the re-inspection of the product or first article inspection is an important process. Reducing the inspection time can reduce the time spent waiting for a product in the production line. This research aimed to design a system that can automatically and rapidly measure the dimensions of irregular tooth contours in the broaching process, to ensure cutting tools are replaced when necessary. This study developed an automatic machine for measuring the irregular tooth contours of large ring parts; the tooth root, tooth height, and tooth thickness of the workpiece are measured. The measurement diameter is approximately 200 mm, and the radial inspection accuracy is within ±20 μm; we aimed to reduce the detection time considerably. An optical micrometer and an automatic rotating platform were used in the measurement system. The workpieces to be measured were easy to install, and the eccentricity was automatically corrected by the system, thus saving time that would be taken to correct Abbe errors. This research successfully developed a rapid optical measurement system that can reduce the inspection time from 30 min to 60 s. Moreover, the maximum radial measurement error is −0.02 mm, which means that the measurement accuracy is within ±20 μm (total: 40 μm).

A New Computational Model of Step Gauge Calibration Based on the Synthesis Technology of Multi-Path Laser Interferometers

A step gauge is a commonly used length standard for international comparison, and its calibration accuracy is often used as a sign to measure a country’s length Calibration and Measurement Capability (CMC). Based on this, some developed countries and developing countries all over the world have been carrying out the research of precision calibration technology for step gauge. On the basis of summarizing the current situation of step gauge calibration technology in other countries, this paper presents a new computational model of step gauge calibration based on the Synthesis Technology of Multi-Path Laser Interferometers (SMLI) and an auto-collimator, which can synthesize the three laser light paths into the measured centerline of step gauge. It is very important to obtain a good measurement accuracy for the step gauge, conformed to the Abbe principle, no matter where it is installed on the CMM measurement platform. In this paper, the development of the mathematical model, the data collection algorithms, data analysis techniques, and measurement uncertainty budgets are discussed. Finally, the experimental measurement is carried out and the measurement accuracy is verified to be effective. The results show that this method can effectively avoid the influence of Abbe error in length measurement, and significantly enhance the calibration accuracy of the step gauge.

A Differential Measurement System for Surface Topography Based on a Modular Design

In this paper, a novel design of a surface topography measurement system is proposed, to address the challenge of accurate measurement in a relatively large area. This system was able to achieve nanometer-scale accuracy in a measurement range of 100 mm × 100 mm. The high accuracy in a relatively large area was achieved by implementing two concepts: (1) A static coordinate system was configured to minimize the Abbe errors. (2) A differential measurement configuration was developed by setting up a confocal sensor and a film interferometry module to separate the motion error. In order to accommodate the differential measurement probes from both sides of the central stage and ensure the system rigidity with balanced supports, separate linear guides were introduced in this system. Therefore, the motion Degree of Freedom (DoF) was analyzed in order to address the challenge of an over-constrained mechanism due to multiple kinematic pairs. An optimal configuration and a quick assembly process were proposed accordingly. The experimental results presented in this paper showed that the proposed modular measurement system was able to achieve 10 nm accuracy in measuring the surface roughness and 100 nm accuracy in measuring the step height in the range of 100 mm × 100 mm. In summary, the novel concept of this study is the build of a high-accuracy system with conventional mechanical components.

Optical Sensors for Multi-Axis Angle and Displacement Measurement Using Grating Reflectors

In dimensional metrology it is necessary to carry out multi-axis angle and displacement measurement for high-precision positioning. Although the state-of-the-art linear displacement sensors have sub-nanometric measurement resolution, it is not easy to suppress the increase of measurement uncertainty when being applied for multi-axis angle and displacement measurement due to the Abbe errors and the influences of sensor misalignment. In this review article, the state-of-the-art multi-axis optical sensors, such as the three-axis autocollimator, the three-axis planar encoder, and the six-degree-of-freedom planar encoder based on a planar scale grating are introduced. With the employment of grating reflectors, measurement of multi-axis translational and angular displacement can be carried out while employing a single laser beam. Fabrication methods of a large-area planar scale grating based on a single-point diamond cutting with the fast tool servo technique and the interference lithography are also presented, followed by the description of the evaluation method of the large-area planar scale grating based on the Fizeau interferometer.

Abbe error compensation for a micro/nano CMM with a coplanar stage

A micro/nano Coordinate Measuring Machine (CMM) with a stroke of 20 mm × 20 mm × 10 mm has been developed by our group. The “Pagoda structure” with Chinese style is adopted, which has fine mechanical stability. The base of the micro/nano CMM is made of granite to achieve high thermal stability. The stages are driven by piezoelectric motors with a resolution of 1 nm in three directions. The displacements of the stages in horizontal directions are detected by homemade Michelson interferometers. The optical axes of the two interferometers are consistent with their slide way of the coplanar stages respectively, which can reduce the Abbe errors in X and Y axes as much as possible. Semiconductor laser diodes are used as the light source of the interferometers due to low-cost and small size. A linear diffraction grating interferometer (LDGI) has also been developed to detect the displacement in Z direction. After subdivision, the measuring resolution can reach nanometer scale. The micro/nano CMM is compatible with different types of probes. A model of Abbe error and its compensation method are proposed in this paper. Experimental results show that the positioning errors were reduced from 435 nm to 91 nm and from 426 nm to 151 nm via Abbe error compensation method. The flatness of a 0-grade gauge block was tested by the micro/nano CMM, the repeatability of 44 nm (K=2) was achieved.

The Structure Design and Analysis of a Parallel Kinematic Mechanism Based on Abbe’s Principle

The purpose of this study is to design and analysis of a parallel kinematic mechanism based on Abbe’s principle can be applied to the 3D-coordinate measuring machine. In this study is the design and analysis of the parallel kinematic mechanism, which includes the mechanism design of linear drive rod mechanism and displacement sensing position coaxial, and the intersection of the three linear drive rod mechanisms are located at the contact probe. Therefore, the design of this study improves the displacement of the linear drive rod and Abbe error resulted from the angle variation of the displacement sensor. Furthermore, the mechanism performance has been verified experimentally. Through precision tests of the machine by a laser Doppler interferometer, the linear drive rod mechanism positioning precisions are within 1μm.

Laser encoder system for X-Y positioning stage

In this investigation, a novel opto-mechatronics design of linear Laser encoder system which can eliminate geometrical errors of X-Y positioning stage was proposed. The eliminated geometrical errors, which are squareness error, straightness error and Abbe error, can be reduced by this Laser encoder system. Those errors are induced by the mechanical components and assemblies. In this design, a pentaprism beam splitter was employed to divide the laser beam into two encoder axes which are perpendicular to each other. For this arrangement, a real zero point of the positioning stage was set inside the pentaprism beam splitter. Therefore, those specific geometrical errors can be minimized by the proposed system. In this research, the design of the opto-mechatronics structure and its theoretical simulations will be studied. The experimental results which were obtained in the ordinary environment revealed that the resolution of the proposed positioning stage is about 15.8 nm. In addition, the maximum standard deviation of static positioning error is about 50 nm. The Laser encoder system presented in this study is recommended to be used in the precision machinery and semiconductor industries for the precision positioning purpose.

Real-time displacement calculation and offline geometric calibration of the grating interferometer system for ultra-precision wafer stage measurement

For ultra-precision wafer stage measurement in a photolithography scanner, real-time six-degree-of-freedom displacement calculation and offline geometric calibration methods are proposed to solve the multiple nonlinearities caused by rotation-translation coupling, Abbe error, and cosine error of the grating interferometer system with non-ideal geometry. First, the interference signal phase-shift model related to the displacements of the center of mass of the wafer stage is established. Then, through double approximations, the displacement calculation process is efficiently reduced to polynomials of substitution variables of phase-shift to improve the real-time performance of computing. Finally, the non-ideal geometry of the nonlinear computing model caused by manufacturing and assembly errors is calibrated offline by solving overdetermined equations set up by redundant measurement to improve the calculation precision. The proposed methods are verified through ZEMAX simulation.

Development of a Compact Three-Degree-of-Freedom Laser Measurement System with Self-Wavelength Correction for Displacement Feedback of a Nanopositioning Stage

This paper presents a miniature three-degree-of-freedom laser measurement (3DOFLM) system for displacement feedback and error compensation of a nanopositioning stage. The 3DOFLM system is composed of a miniature Michelson interferometer (MMI) kit, a wavelength corrector kit, and a miniature autocollimator kit. A low-cost laser diode is employed as the laser source. The motion of the stage can cause an optical path difference in the MMI kit so as to produce interference fringes. The interference signals with a phase interval of 90° due to the phase control are detected by four photodetectors. The wavelength corrector kit, based on the grating diffraction principle and the autocollimation principle, provides real-time correction of the laser diode wavelength, which is the length unit of the MMI kit. The miniature autocollimator kit based on the autocollimation principle is employed to measure angular errors and compensate induced Abbe error of the moving table. The developed 3DOFLM system was constructed with dimensions of 80 mm (x) × 90 mm (y) × 20 mm (z) so that it could be embedded into the nanopositioning stage. A series of calibration and comparison experiments were carried out to test the performance of this system.

Measurement and compensation of machine tool geometry error based on Abbe principle

The error measurement of numerically controlled machine tools is fundamental to error compensation. The installation position of measurement equipment has a significant influence on the measurement accuracy. Accurate measurement should satisfy the requirement that five factors are on the same line, or the Abbe error will result in measurement data inaccuracy. In this study, the reasons for the Abbe offset are analysed. The measuring equipment and measured object are adjusted so that the Abbe arm length is reduced to zero. When the measurement system is limited by the object condition, the compensation methods for these errors are provided. The system error is significantly decreased by software error compensation according to the error model based on the Abbe principle. The experimental results demonstrate that the machine tool accuracy is improved significantly. Therefore, the proposed compensation method is effective in compensating for the geometric error of the machine tool.

Yaw error correction of ultra-precision stage for scanning beam interference lithography systems

The stage yaw error is a key factor affecting the phase distortion of gratings produced by scanning beam interference lithography system. In order to solve this problem, a coarse-fine dual-stage mechanism is proposed, in which an ultra-precision fine positioning stage with yaw error correction function is developed. To achieve nanoscale positioning and sub-microradian yaw motion accuracy, four Lorentz motors are used to drive the fine stage. The internal coupling factors and the mechanism of Lorentz motors motion control are analyzed. Besides, the Abbe error caused by the yaw error is investigated. Positioning and scanning experiments are conducted and the outcomes show that maximum yaw error is 0.33 μrad during constant velocity scanning, which completely meets the grating fabrication requirements.

Calibration and Compensation of Dynamic Abbe Errors of a Coordinate Measuring Machine

Error compensation technology is used for improving accuracy and reducing costs. Dynamic error compensation techniques of coordinate measuring machine (CMM) are still under study; the major problem is a lack of suitable models, which would be able to correctly and simply relate the dynamic errors with the structural and operational parameters. To avoid the complexity of local dynamic deformation measurement and modeling, a comprehensive calibration method is employed. Experimental research reveals specific qualities of dynamic Abbe errors; the results exceed the scope of ISO 10360-2 calibration method, showing the ISO 10360-2 dynamic error evaluation deficiencies. For calibrating the dynamic Abbe errors, the differential measurement method is presented based on the measurements of the internal and external dimensions. Referring probe tip radius correction, the dynamic Abbe errors compensation method is proposed for CMM end-users and is easy to use.