## 2 Metrological Process and Subdivisional Errors in Optical Encoders  
Optical linear encoders use various types of optical-scanning principles (e.g., Talbot effect [46; 47], Lau effect [48; 49], Moire effect [50; 51; 52], generalized grating imaging [53; 54; 55; 56], interferometric [57; 58; 59]) to generate electric-output signals that are used to determine a precise position. Depending on subsequently used electronics, optical encoders can have different interfaces to ensure the reliable exchange of information [60]. One of the most widely used interfaces for incremental position measurement is two analog nearly sinusoidal voltage signals (Figure 1). Signals A and B are shifted by 90 electrical degrees and have an amplitude of typically 1 Vpp. One full period (360 electrical degrees)of these signals corresponds to a period of grating on the measuring scale. Using the arctangent algorithm, it is easy to determine the relative position inside the period:  
\[Xposition=\frac{p}{2\pi}arctan\bigg{(}\frac{S_{A}}{S_{B}}\bigg{)}, \tag{1}\]  
where _Xposition_, relative position between measuring scale and reading head; \(p\), the period of the main scale; and \(S_{A},S_{B}\), values of output signals A and B in a certain position, respectively.  
Such a process is known as interpolation, and it is directly related to the quality of the signals. The 1 Vpp interface is mainly used in applications where such an interpolation and analog-signal digitalization processes are executed by the subsequent electronics of the end-user equipment.  
Combining analog encoder signals into the \(X\) and \(Y\) axes of an oscilloscope, it is comfortable to observe and follow the quality and accuracy of the encoder. The ideal signals give a perfectly centered circular shape with a diameter corresponding to signal amplitude. Unfortunately, in real applications, encoder output signals are distorted due to imperfections in manufacturing, and assembly and optical-scanning operations, as well as negatively affected by variations of environmental conditions. Distortion of the signals causes SDE, which repeats in each period of the encoder grating. Variations of the signal background level (U\({}_{\text{A\_off}}\), U\({}_{\text{B\_off}}\)) are usually caused by imperfections or contaminants of the encoder measuring scale. Different signal offsets could also be related to the improper adjustment of electronic parts. Such a distortion leads to a decentered Lissajous curve, as shown in Figure 2a. Different peak-to-peak amplitudes (U\({}_{\text{A}}\), U\({}_{\text{B}}\)) of the A and B signals could appear due to uneven or inconsistent illumination of the photodetectors. This error leads to an elliptical Lissajous curve, as shown in Figure 2b. A phase shift from 90 electrical degrees made the curve oval, as shown in Figure 2c. The main reason for this type of error is the tilt between the gratings of the scanning reticle and the main measuring scale. All higher harmonics caused by optical effects and electronics make signals that are not perfectly sinusoidal. This type of error forms a noncircular Lissajous curve, as shown in Figure 2d.  
Figure 1: Definition of differential optical encoder signals. 1 Vpp interface.  
Figure 2: Lissajous curves of optical-encoder signals with relative subdivisional error (SDE): (**a**) offset error; (**b**) amplitude error; (**c**) phase-shift error; (**d**) signal-shape error.