## 3 Systematic Error Correction  
### Systematic Error Analysis  
We performed distance measurements repeatedly over a range of 0.5 to 2.5 m back and forth with 10 mm steps as illustrated in Figure 3. Each position value is represented by an average of 60 consecutive measurements taken over a period of 6 s. The averaged distance was compared with its counterpart value obtained simultaneously with a commercial incremental-type He-Ne laser interferometer, of which the discrepancy was plotted in Figure 3a. At each distance, with respect to the He-Ne laser interferometry, the measurement error of our interferometer was evaluated by processing eight individual data sets. The measurement error appears to be a sum of two separable terms; one is the non-repeatable random error and the other is the repeatable systematic error. In principle, the random error determines the measurement precision ultimately achievable, whereas the systematic error can be eliminated through correction at each position. Further, as depicted in Figure 3b, it is also  
Figure 2: Measurement repeatability. (**a**) Measurement fluctuation of a 1-m fixed distance traced over 1000 s along with its histogram. (**b**) Allan deviation from 0.1 to 100 s averaging.  
important to note that the repeatability of the systematic error can be either periodic (orange line) or non-periodic (purple line).  
In general, it is known that the periodic systematic error is caused by the signal leakage occurring in the interferometer optics or electronics, with the period appearing to be identical to the wavelength of the microwave signal used in distance measurement [24]. On the other hand, the non-periodic systematic error arises from the amplitude-to-phase error conversion due to the light intensity fluctuation as well as the imperfect RF electronics of nonlinear behaviors [25; 26]. For the 1 GHz microwave wavelength employed in this study, the periodic error is found to have an amplitude of 22.6 \(\upmu\)m with an actual period of 0.15 m. The periodic error is attributable to the cyclic cross talk of the measurement signal with unwanted signals. Yielding no position-dependency, the periodic error is generated by the RF circuit noise, as well as the partially-reflected laser beam from the interferometer optics for beam splitting and recombination. The effect of cross talk on the measurement error can be modelled as [24]:  
\[\cos\left(\frac{2\pi f}{c}2L\right)+\varepsilon_{\text{ crosstalk}}\cos(\theta_{\text{ crosstalk}})=\cos\left(\frac{2\pi f}{c}2L+\theta_{\text{ periodic error}}(L)\right) \tag{2}\]  
where \(f\) is the carrier frequency (here 1 GHz), \(c\) is the speed of light, \(L\) is the target distance, \(\varepsilon_{\text{ crosstalk}}\) is the cross talk ratio to the measurement signal, \(\theta_{\text{ crosstalk}}\) is the constant phase of the cross talk components and \(\theta_{\text{periodic error}}\) (\(L\)) is the periodic error induced by cross talk as a function of the target distance \(L\). In our case, the cross talk ratio \(\varepsilon_{\text{ crosstalk}}\) is found to be 0.1%, as illustrated in Figure 3c.  
The non-periodic systematic error shown in Figure 3a is found to be \(\pm\)300 \(\upmu\)m, with a position dependency with a strong correlation with the optical power actually received by the photodetector, as shown in Figure 3d. This confirms that the non-periodic error is attributable to the amplitude-to-phase conversion caused by the photodiode and RF components. However, the microwave phase is not reciprocal for increasing and decreasing the direction of the received optical power.  
Figure 3: Systematic measurement error. (**a**) Experimental data of measurement errors over a range from 0.5 to 2.5 m. (**b**) Periodic error vs. and non-periodic error. (**c**) Cyclic error amplitude vs. unwanted signal ratio (**d**) Photodiode output amplitude (proportional to the received optical power) in comparison to the non-cyclic error.