ON THE LASER DOPPLER VELOCIMETRY FOR SURFACE ROTATION MEASUREMENTS

The article aims to present the possible use of the simple modified optical training kit as a low-cost, simple setup for high precision surface speed measurements. A measurement capability evaluation of the optical Doppler kit used for velocimetry of a fast-rotating reflecting surface is presented. To get high repeatability measurements, we modified a fibre optic interferometry training kit. By using signal processing and statistics a precision under repeatability conditions of measurements was evaluated. Expressed by the standard deviations (3σ’s) the surface velocity measurements precision below 0.2 s is shown. The Cg’s capability indices were also evaluated. We postulate, the electric circuit stability of the measurement system power supply is essential for a signal noise reducing process for a wide range of metrology systems. It is crucial for precision measurements.


Introduction
The light-matter interaction can turn out as a complex network of the electromagnetic field. It can be physically and mathematically described nowadays quite widely and precisely. Because of the redefined in 2019 the SI units, based on the speed of light in a vacuum (c) [1,2], this beautiful part of nature now is commonly used in metrology for high precision, traceable measurements [3]. The light is already used in the manufacturing metrology for fast surface measurements in a various scale [4][5][6][7]. Except for high-accuracy interferometry [8], the velocity measurements of these fast-moving surfaces can also be determined precisely by using well known optical Doppler effect (non-relativistic) [9][10][11][12][13][14], as well as surface asperities. It can be considered even parallel in one measurement system, where surface topography and displacement are measured together, with comparable precision [15]. It is possible e.g. with interferometric surface texture measurement technique [16]. The Doppler radar technique has a wide application from technical science [17,18] to life science as well [19]. By using the Doppler effect principle, the vital signals are also measured in bio-radar systems for motion detection applications [20]. A groundbreaking in science was taking into account the Doppler effect by Albert Einstein in the relativity theory with time dilatation effect in 1905 [21]. So, the Doppler shift principle was, and is a decisive part of the wave theory, modern physics and modern technical science. An industrial laser Doppler velocimetry method has been developed for accurately measuring the velocity and length of moving surfaces already in 1983 [22]. Simultaneous measurement of the velocity and the displacement of the surface by a laser Doppler velocimeter is also well known [23]. The systems also have been miniaturized [24], but even nowadays, these systems are rather advanced, innovative and costly. In the paper, a simple, modified fibre optical training kit [25,26], for high precision velocimetry of a fast-rotating reflecting surface, is proposed. By using the isolation transformers, the laboratory electrical grid from the building grid noise was separated. A simple, stabilised, power supply for DC engine was prepared. We proved, electric noise reduction and stable power supply are essential for frequency measurements. They are especially important for high precision optical velocimetry using the Doppler kit, under repeatability conditions of measurements. The setup can be adapted as a non-lab, low-price measurement system, to measure the velocity of the moving surface up to 30 m s . Further investigations with various surfaces will begin presently. We aim to construct an optical measurement system based on lasers, for surface speed displacement and surface topography measurements with both nanometric precision, to be as a low-cost, fast, simple system, not necessarily high innovative, without an expensive, advanced optics. As the first part of the system, we propose to use our modified Doppler training kit. A second part might be based on scatterometric [27] or interferometric systems [16]. In the paper, the precision of the Doppler system measurements was evaluated by statistics.

The Doppler shift
Consider a source (S in fig. 1) emits an electromagnetic wave towards a reflecting surface (O in fig. 1). The reflected wave seems to come from S 1 . If surface moves at speed v ăă c, S' goes at speed 2v. It is related to the double change of the optical path length, caused by the surface displacement. The wave frequency detected by the receiver (R in fig. 1) is shifted. The effect is well known as the non-relativistic Doppler shift [28]. The laser speed gun works in this way.  [29] We used the optical setup to determine the tangential speed of the rotating disc by using the Doppler principle. The speed vectors relation scheme is shown in fig. 2. The vectors are oriented in a 3D space.  [25] Single-mode laser light (λ " 1550 nm) a has been reflected from a disc covered by retrodiffusing layer. The frequency of the incindent light is f 0 (see eq. 1) [25].
When the beam hits the rotating disc, the frequency shifts twice: at the wave reception by the disc, and the diffused wave emission by the same disc. For the same direction of the disc rotation as the direction of the light, the frequency detected by the disc is (see eq. 2 and fig. 2): where: f 0´f requency of the incident light, f 1´r eceived frequency by the rotating disc, v´speed of the disc, The reemitted frequency (went to collimator) f 2 is described by the eq. 3.

Setup
The optical layout of the setup is shown in fig. 3. The setup was laying on the granite isolation table (5 in fig. 4) for mechanical and acoustic noise reduction. The fibre interferometry and Doppler kit (see fig. 4) was additionally separated by an aluminium plate with four rubber feet (Sorbothane feet, I 45 mm). To keep a constant speed of the rotation (1600 rpm), surface to be measured (I 75 mm reflecting disc, see fig. 5) was mounted on a DC engine supplied by the self-built stabilised power supply (U " 3 V, I max " 850 mA) (see fig. 6).  A typical power source was used in the experiment, from the electrical grid of the building (V " 230 V, 50 Hz). To reduce the electric noise from many types of equipment in a whole building, the electrical grid of the laboratory was separated by the transformers (12 kVA; 0.4{0.4 kV; R « 4 MΩ). In this way, repeatable electric conditions for the experiment, not depending on the activities in the building, were prepared. The data were collected by using a scope (2 in fig. 4). Optical elements of the system were connected using optical fibres b and fibre APC connectors c , as mentioned in the fig. 3. The laser beam (P " 1 mW, λ " 1550 nm) is sent through the optical isolator (OI in fig. 3), to avoid damage of the laser resonator. Then, the fibre coupler (C in fig. 3) is split the beam into the two channels of the setup. One with the mirror as the reference (M in fig. 3) and the second as a measurement one with fibre collimator (Coll. in fig. 3). The mirror improves the carrier frequency. To reduce the power of the reference beam and makes equal two beams mixed into the one signal detected by the photodiode (D in fig. 3) d , the´5 dB optical attenuator (A in fig. 3) is placed in the reference channel, just before the mirror. The attenuator also avoids a photodiode saturation.

The measurement procedure
The scheme of the measurement procedure is shown in fig. 7. Before the measurements, the parameters are set: current of the laser diode (I " 25 mA, kept always the same), photodiode offset (V o f f set " 1´4 V), reflection angle (θ " 88.5, 78.75, 67.50˝) and the measurement place on the disc (radius R " 20, 25, 30 mm). Next, the DC motor is switched on, to achieve, after c.a. 30 seconds, a constant speed of the disc rotation. The signal is detected by the scope and is stored on the external USB stick for further evaluation. This procedure was repeated 30 times for the setup capability evaluation (see fig. 7).  The measurement procedure shown in fig. 7 corresponds to one set of parameters. Consider the parameters set as: I " 25 mA, V o f f set " 1 V, θ " 88.5˝, R " 30, and the disc started to rotate with constant speed 1600 rpm. After c.a. 30 s, the signal from the scope is stored on the USB stick. Next, the DC motor is switched off to stop the disc. For a measurement repeatability investigation, the above procedure was repeated 30 times. The data from the scope as the *dat files were imported in R programming language. The signal processing and statistics were evaluated by using the additional R package [30,31].

Signal processing
For Doppler shift evaluation, the fast Fourier transform was used. In fig. 8 example of the photodiode signal, registrered by the scope, is depicted. The sampling interval was always 10´8 s (8192 data points). The signal transform to the frequency domain is presented in fig. 9. The DC component clearly to be seen, comes from the photodiode offset (V o f f set " 1 V), and only negligibly affects the Doppler peak analysis.  Next steps are, to fit the filtered spectrum by using the Gaussian curve, and Doppler peak detection. The Doppler peak is found by using a 2nd derivative of the Gaussian curve (see fig. 12). In fig. 9 a high frequency noise is presented. The data were stored without using the separation transformers in the laboratory. The high frequency noise was reduced in the final setup with transformers (see fig. 13). The increased DC compomnent comes from the photodiode offset V o f f set " 4 V, used in the final experiment.

Disc speed evaluation
By using the Gaussian fit, the Doppler peak (see e.g. fig. 12) can be determined. For known frequency of the Doppler shift, the disc speed can be calculated from the eq. 6. Because of a lack of measurement standard, for measurement system capability evaluation, the tolerance limits from the speed fluctuations (v) due to the power voltage (U z ) were estimated. The constant (k) of the motor velocity is described by the eq. 7.
From eq. 7 and 8 a maximum error of the motor rotation speed can be described as (eq. 9): ∆ω s "ˇˇˇˇB ω s BU zˇ¨∆ U z " k¨∆U z " 484.85¨0.05 « 24.24 rrpms, and the angular speed, finally as (eq.10): The linear speed (v) for R " 0.03 rms is described by the eq. 11. In fig. 14 the linear relation from eq. 11 is presented. It was proved experimentally (see section 4, fig. 20). The Doppler shift varies as the cos function (see eq. 6 and fig. 15).
v " ω¨R « 5.02˘0.08 " m s ı .  From theoretical linear speed (eq. 11) of the disc in radius R " 0.03 rms, upper tolerance limit (UTL), lower tolerance limit (LTL) and the tolerance range T R (see eq. 12) were defined.
In the same way, linear speed tolerances for different points (radiuses R) on the disc (see eq. 13 were calculated.
T R"25 " 0.12 " m s ı , For the velocimetry system capability evaluation, well known in manufacturing metrology, the standard Measurement Systems Analysis (MSA) method [32,33] is proposed. The method is widely Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 October 2020 doi:10.20944/preprints202010.0192.v1 used [34,35]. In our case, a standard deviation parameter (3σ) and the C g metric (see eq. 14) were used for the precision estimation, under repeatability conditions of measurement. Measurements were repeated 30 times per series (total 18 series). To check the normal distribution of the data the Shapiro-Wilk test [36] was used. We used the open-source programming language R for statistical computing [31,37,38]. where:

Results
The 18 series of measurements were collected: The measurements were repeated of 30 times per series. We have started the investigation of the precision under repeatability conditions of measurements, with the standard offset photodiode voltage V o f f set " 1 V. During three series (#1´#3) for a different angle of the incident light beam (θ), quite high standard deviations (3σ's) for linear speed measurements were discovered, together with unacceptable low capability factors C g 's (see tab. 2). During the first three series, the setup was not capable at all for the precision laser Doppler velocimetry. The first step to improve it was to increase the offset voltage to V o f f set " 4 V (see tab. 3). Higher C g 's factors and lower 3σ's) were appeared. The important modification was, to use the electrical separation of the laboratory from the building electrical grid noise, by using the transformers (see sec. 2). Separation transformers are highly Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 9 October 2020 doi:10.20944/preprints202010.0192.v1 recommended for individual lab grounding electric circuit. A stable power supply in the lab, by using separation is essential, e.g. in highly sensitive quantum metrology [39][40][41], and we decided to do this as well. In this way, the noise signal was significantly reduced, and precision of the surface linear speed measurements improved (low 3σ's) with an acceptable C g 's factors (see tab. 4). By improving the signal with V o f f set adjusting and especially, by electric noise reduction with separation using the transformers, the C g 's factors were found in the correct level " 1, which means, the measurement system was capable enough for this purpose. In this way, we also proved, to achieve high precision, under repeatability conditions, of the surface linear speed measurement, some part of the information-rich metrology paradigm is good to be fulfilled [42]. It is important as well as for surface texture measurements. For frequency measurements, the electrical grid of the laboratory should be separated from the building grid noise.
The temperature in the laboratory during the measurements (20˘1 r˝Cs) has been monitored. The mechanical and acoustic noises were reduced by the setup construction. Other disturbing effects in the experiment, e.g. humidity, were negligible when the optical fibre setup is used.
All measurement results of the disc speed were normally distributed (see e.g. tab. 5 and fig. 16-19), it means, standard deviation parameters (σ's, in this proposal 3σ's) are related to the measurements uncertainties [43,44]. In fig. 17 and 19 examples of the quantile-quantile plots are presented. These probability plots are helpful to check if the normality assumption is plausible, and if not, how the assumption is violated and what data points contribute to the violation [45]. The normality is easy to see along the straight line, with the points in the confidence bands area [46].
As an extra check for the normality of the data distribution, the Shapiro-Wilk test was used (see e.g. tab. 5).  The disc speed (as average v) versus disc radius (R) graph for measured values (see fig. 20) shows full correspondence to the linear relation of the v " f pRq (see eq. 6 and compare fig. 20 with fig. 14). The disc speed (as average v) versus angle θ -?p Ý Ñ v , Ý Ñ c q graph for measured values shows correctly a constant line (see fig. 21). The slope of the line is related to the measurement errors evaluated.

Discussion
In the Industry 4.0 age, significant progress in surface metrology has been observed. Optical metrology is going to be a game-changer in that field. More research is needed to simplify the hardware and software of the systems, with well known and controlled electric parameters of the systems power supply, as well.
In the paper, a simple modification of the Doppler kit is presented. The laboratory electric grid separation from the building grid noise, and self-built DC engine stable power supply, improved fairly enough the setup to be used as the low-cost, high precision measurement system. For a fast surface Doppler-shift velocimetry, the precision under repeatability conditions of measurements has been evaluated. The precision of the linear speed measurements with the setup was above 0.2 m s . It was estimated as the standard deviations (3σ's ă 0.2), based on the normal distribution of the data. The capability factors C q 's, in final experiments, were in a close range of " 1, which means, the setup was capable enough for this metrological application, and modifications presented were essential. Here we postulate; not only the temperature, pressure, humidity or vibrations stabilisation/isolation are essential for high precision measurements, but the power supply stability of the system as well. Or even more, in fast, short-time measurements, where others effects are not disturbing so much. The electric circuit conditions of the power supply should be well known before the measurements and kept controlled well. Further investigations with various surfaces and measurement instruments will begin presently. The surface reflectivity, polarisation state of light, different object displacement, needs to be further investigated. The proposed setup is low-cost, fast, customisable, and by using optical fibres can be used for surface speed measurements, with a long-distance between the object and the unit. Using the optical fibres might be an advantage for real-time online measurements, where standard optical systems, can not be used due to the environmental conditions. The rise of new metrology tools has seen suppliers focus their attention only on the hardware devices, with everyone focusing mainly on speed, accuracy, portability. However, the one thing that makes all of these devices work and produce measurement results is the software. It seems that manufacturers seem to have created their own software that works only with their products. By accepting a manufacturer's software, users are immediately limited to what that brand can offer in the future.
In the article, the hardware modification is presented, but also the data evaluation as well. By using the open-source R programming language, the signal processing and statistics were given. With the open-source codes, the mathematics behind the data analysis steps are open to the public and can be inspected or even corrected by others. Some ongoing projects on the metrology background are quite promising [47].
After more improvements, the low-cost modified Doppler kit might be used in a complex system as a part responsible for precision surface displacement measurements [27].

Conclusion
A simple modification of the training Doppler kit by using separation transformers and self-built power supply has been presented. Signal improvements and data evaluation have been shown. An improvement of measurement precision using statistic was evaluated. For signal processing and data analysis, an open-source R programming language was proposed. Based on the well known Doppler radar technique, the proposed setup is promising as a low-cost, simple, fast, alternative system for high precision surface velocimetry.