The principle of the self-calibration method for the evaluation of a planar scale grating having a constant pitch is extended to realize the evaluation of the pitch distribution of a planar scale grating having variable line spacings (VLSs) along the X- and Y-directions. In the conventional self-calibration method, the wavefronts in the zeroth-order diffracted beam and the first-order diffracted beams observed by a Fizeau interferometer arranged in the Littrow configuration were employed to evaluate the pitch deviation of a scale grating. The arithmetic operation with the wavefront data realizes the evaluation of the pitch deviation over a large area in a short time, while cancelling the influence of the out-of-flatness of a scale grating. Meanwhile, theoretical equations in the conventional self-calibration method cannot be directly applied to the evaluation of a VLS grating due to its unique properties of the pitch distribution. In this paper, major modifications are thus made to the conventional theoretical equations for deriving the pitch distribution of a VLS grating. To verify the performance of the newly proposed method, the pitch distribution of a VLS grating employed in a commercial planar absolute encoder is evaluated in experiments.
Major modifications are made to the setup and signal processing of the method of in-situ measurement of the pitch of a diffraction grating based on the angles of diffraction of the diffracted optical frequency comb laser emanated from the grating. In the method, the improvement of the uncertainty of in-situ pitch measurement can be expected since every mode in the diffracted optical frequency comb laser can be utilized. Instead of employing a Fabry-Pérot etalon for the separation of the neighboring modes in the group of the diffracted laser beams, the weight-of-mass method is introduced in the method to detect the light wavelength in the Littrow configuration. An attempt is also made to reduce the influence of the non-uniform spectrum of the optical comb laser employed in the setup through normalization operation. In addition, an optical alignment technique with the employment of a retroreflector is introduced for the precise alignment of optical components in the setup. Furthermore, a mathematical model of the pitch measurement by the proposed method is established, and theoretical analysis on the uncertainty of pitch measurement is carried out based on the guide to the expression of uncertainty in measurement (GUM).
A new method based on the interferometric pseudo-lateral-shearing method is proposed to evaluate the pitch variation of a large-scale planar variable-line-spacing (VLS) grating. In the method, wavefronts of the first-order diffracted beams from a planar VLS grating are measured by a commercial Fizeau form interferometer. By utilizing the differential wavefront of the first-order diffracted beam before and after the small lateral shift of the VLS grating, the pitch variation of the VLS grating can be evaluated. Meanwhile, additional positioning errors of the grating in the lateral shifting process could degrade the measurement accuracy of the pitch variation. To address the issue, the technique referred to as the reference plane technique is also introduced, where the least squares planes in the wavefronts of the first-order diffracted beams are employed to reduce the influences of the additional positioning errors of the VLS grating. The proposed method can also reduce the influence of the out-of-flatness of the reference flat in the Fizeau interferometer by taking the difference between the measured positive and negative diffracted wavefronts; namely, self-calibration can be accomplished. After the theoretical analysis and simulations, experiments are carried out with a large-scale VLS grating to verify the feasibility of the proposed methods. Furthermore, the evaluated VLS parameters are verified by comparing them with the readout signal of an absolute surface encoder employing the evaluated VLS grating as the scale for measurement.
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