A grating interferometer, called the “optical encoder,” is a commonly used tool for precise displacement measurements. In contrast to a laser interferometer, a grating interferometer is insensitive to the air refractive index and can be easily applied to multi-degree-of-freedom measurements, which has made it an extensively researched and widely used device. Classified based on the measuring principle and optical configuration, a grating interferometer experiences three distinct stages of development: homodyne, heterodyne, and spatially separated heterodyne. Compared with the former two, the spatially separated heterodyne grating interferometer could achieve a better resolution with a feature of eliminating periodic nonlinear errors. Meanwhile, numerous structures of grating interferometers with a high optical fold factor, a large measurement range, good usability, and multi-degree-of-freedom measurements have been investigated. The development of incremental displacement measuring grating interferometers achieved in recent years is summarized in detail, and studies on error analysis of a grating interferometer are briefly introduced.

Displacement measuring grating interferometer: a review

Phase generated carrier (PGC) demodulation schemes are widely used in fiber-optic interferometric sensors. All known PGC schemes use carrier signals to obtain in-phase and quadrature components from the interference signal. The unwanted phase delay between the reference carrier signal and the modulation signal in the interference signal might exist. This effect might lead to the incorrect measurement of the phase signal in the PGC scheme. In this paper, the original PGC demodulation scheme with the carrier phase delay compensation based on the basic PGC-Atan scheme is presented. The proposed scheme provides the real-time carrier phase delay compensation over the all possible phase delay range from 0° to 360° without using phase adjustments of the carrier signal. Theoretical analysis and mathematical modeling of the proposed scheme and the basic PGC-Atan demodulation scheme were performed to compare their performance under the same conditions. The proposed scheme, the basic PGC-Atan demodulation scheme, and the PGC synchronous carrier restoration scheme were implemented in the FPGA-based electronic processing circuit of the single interferometric fiber Bragg grating sensor. Experimental results showed that the proposed scheme provides carrier phase delay compensation without using phase adjustments of the carrier signal.

PGC-Atan Demodulation Scheme With the Carrier Phase Delay Compensation for Fiber-Optic Interferometric Sensors

Fiber-optic interferometric sensors (FOISs) are widely used in seismometers, hydrophones, and gyroscopes. The arctangent approach of phase-generated carrier (PGC-Arctan) demodulation algorithm is one of the key demodulation techniques in FOISs. The conventional PGC-Arctan demodulation algorithm requires the specific value of the phase modulation depth C to work properly. However, C will variate with laser wavelength, temperature, and humidity in the actual working environment, which leads to harmonic distortion and even demodulation failure. In this paper, a novel PGC demodulation algorithm called self-calibration PGC-Arctan (PGC-Arctan-SC) demodulation algorithm is presented. The proposed algorithm can jointly estimate the accurate C value by the elliptical parameters and C-related components while suppressing nonlinear distortion by ellipse fitting algorithm (EFA). Then C can be calibrated to the specific predefined optimal value by the closed-loop proportion integration differentiation (PID) module. The simulation results are consistent with theoretical analysis, and the all-digital PGC-Arctan-SC demodulation system is implemented on the embedded SoC. The experimental results show that C can be estimated and calibrated accurately in real time. The signal-to-noise and distortion ratio (SINAD) of the PGC-Arctan-SC demodulation output achieves 61.57 dB.

Real-time self-calibration PGC-Arctan demodulation algorithm in fiber-optic interferometric sensors.

Traditional phase generated carrier (PGC) schemes require that the carrier phase delay be kept zero and the phase modulation depth be maintained at a certain value; otherwise, serious nonlinear errors will be introduced into the phase demodulation of a sinusoidal phase modulation interferometer. Proposed herein is a robust carrier phase delay extraction method for nonlinear errors compensation of PGC demodulation that mitigates the effect of the interference phase shift. The carrier phase delay is directly extracted by adopting the first-, second-, and fourth-order in-phase and quadrature harmonic components generated by quadrature multi-harmonics frequency down conversion. Further, with the extracted phase delay, real-time compensations for the phase delay and phase demodulation depth are realized simultaneously in PGC demodulation. Theoretical analysis and derivation of the proposed method are given in detail. Simulation and displacement measurement experiments with different phase delays and phase modulation depths are performed to verify the effectiveness of the proposed method. The experimental results demonstrate that the proposed method is able to precisely extract the phase delay and efficiently eliminate the effects of the phase delay and modulation depth instability.

Extraction of Carrier Phase Delay for Nonlinear Errors Compensation of PGC Demodulation in an SPM Interferometer

This paper proposes a system utilizing a Renishaw XL80 positioning error measuring interferometer and sensitivity analysis design to measure six-degree-of-freedom (6 DOF) geometric errors of a machine tool’s linear guide. Each error is characterized by high independence with significantly reduced crosstalk, and error calculations are extremely fast and accurate. Initially, the real light path was simulated using Zemax. Then, Matlab’s skew ray tracing method was used to perform mathematical modeling and ray matching. Each error’s sensitivity to the sensor was then analyzed, and curve fitting was used to simplify and speed up the mathematical model computations. Finally, Solidworks was used to design the set of system modules, bringing the proposed system closer to a product. This system measured actual 6 DOF geometric errors of a machine tool’s linear guide, and a comparison is made with the Renishaw XL-80 interferometer measurements. The resulting pitch, yaw, horizontal straightness, and vertical straightness error deviation ranges are ±0.5 arcsec, ±3.6 arcsec, ±2.1 μm, and ±2.3 μm, respectively. The maximum repeatability deviations for the measured guide’s pitch, yaw, roll, horizontal straightness, vertical straightness, and positioning errors are 0.4 arcsec, 0.2 arcsec, 4.2 arcsec, 1.5 μm, 0.3 μm, and 3 μm, respectively.

Design of a Measurement System for Six-Degree-of-Freedom Geometric Errors of a Linear Guide of a Machine Tool

A precision PGC demodulation for homodyne interferometer modulated with a combined sinusoidal and triangular signal is proposed Using a triangular signal as additional modulation, a continuous phase-shifted interference signal for ellipse fitting is generated whether the measured object is in static or moving state The real-time ellipse fitting and correction of the AC amplitudes and DC offsets of the quadrature components in PGC demodulation can be realized The merit of this modulation is that it can eliminate thoroughly the periodic nonlinearity resulting from the influences of light intensity disturbance, the drift of modulation depth, the carrier phase delay, and non-ideal performance of the low pass filters in the conversional PGC demodulation The principle and realization of the signal processing with the combined modulation signal are described in detail The experiments of accuracy and rate evaluations of ellipse fitting, nanometer, and millimeter displacement measurements were performed to verify the feasibility of the proposed demodulation The experimental results show that the elliptical parameters of the quadrature components can be achieved precisely in real time and nanometer accuracy was realized in displacement measurements

Precision PGC demodulation for homodyne interferometer modulated with a combined sinusoidal and triangular signal.

In order to reduce the nonlinearity caused by an error of phase modulation depth, carrier phase delay and non-ideal performance of the low pass filters in the sinusoidal phase modulating interferometer (SPMI), a modified EOM-based SPMI is proposed in this paper to realize real-time normalization of the quadrature components for the arctangent approach of phase generated carrier (PGC-Arctan) demodulation. To verify the effectiveness of the real-time normalization technique, a fixed-phase-difference detection method is presented to evaluate the periodic nonlinearity in real time. The modified EOM-based SPMI is consisted of a monitor interferometer and a probe interferometer. The two interferometers share a reference corner cube, which is mounted on a slowly moving stage, thus periodic interference signals are generated for real-time normalization of the quadrature components in PGC demodulation. Subtracting the demodulated phase of the monitor interferometer from the phase of the probe interferometer, the phase to be measured can be obtained. The fixed-phase-difference detection method is realized by detecting an interference signal with two photodetectors, which are placed at an interval of quarter fringe, and the variation of the fixed-phase-difference can reflect the nonlinear error in PGC demodulation. Experiments of real-time normalization, nonlinear error evaluation of PGC demodulation, and displacement measurement were implemented to demonstrate the effectiveness of the proposed method. Experimental results show that the nonlinear error of phase demodulation reduced to less than ± 1° with real-time normalization, and nanometer displacement measurement is realized.

Real-time normalization and nonlinearity evaluation methods of the PGC-arctan demodulation in an EOM-based sinusoidal phase modulating interferometer.

A laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors is proposed for precision linear stage metrology. In this interferometer, the vertical straightness error and its position are measured by interference fringe counting, the yaw and pitch errors are obtained by measuring the spacing changes of interference fringe and the horizontal straightness and roll errors are determined by laser collimation. The merit of this interferometer is that four degrees of freedom motion errors are obtained by using laser interferometry with high accuracy. The optical configuration of the proposed interferometer is designed. The principle of the simultaneous measurement of six degrees of freedom errors including yaw, pitch, roll, two straightness errors and straightness error's position of measured linear stage is depicted in detail, and the compensation of crosstalk effects on straightness error and its position measurements is presented. At last, an experimental setup is constructed and several experiments are performed to demonstrate the feasibility of the proposed interferometer and the compensation method.

Laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors for precision linear stage metrology.

As the phase delay between the carrier component of the detected interference signal and the carrier has adverse effect for phase generated carrier (PGC) demodulation, it is essential to compensate the phase delay to improve the accuracy of precision displacement measurement in sinusoidal phase-modulation interferometer (SPMI). In this paper, a real-time phase delay compensation method is proposed by regulating a compensating phase introduced to the carrier to maximize the output of the low pass filter so as to make the carrier synchronize with the interference signal. The influence of phase delay for PGC demodulation is analyzed and the method for real-time phase delay compensation is described in detail. The simulation of the method was performed to verify the validity of the phase delay compensation algorithm. A SPMI using an EOM was constructed and several comparative experiments were carried out to demonstrate the feasibility of the proposed method. The experimental results show that the phase delay can be compensated accurately in real time, and nanometer accuracy is achieved for precision displacement measurement.

Shihua Zhang

Papers

Precision PGC demodulation for homodyne interferometer modulated with a combined sinusoidal and triangular signal.

Real-time normalization and nonlinearity evaluation methods of the PGC-arctan demodulation in an EOM-based sinusoidal phase modulating interferometer.

Laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors for precision linear stage metrology.

Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement.

A PGC-DCDM demodulation scheme insensitive to phase modulation depth and carrier phase delay in an EOM-based SPM interferometer