1. Introduction

Distance, shape and vibration measurements of rotating objects, e.g. to analyze dynamic rotation processes, are an important task in the field of process control and production measurement. Due to online monitoring, zero-error production becomes possible, which will increase the production efficiency. Additionally, a higher durability and reliability of the technical equipment is achievable. Hence, optical measurement techniques became more and more important in production metrology in the last years. Compared to tactile measurement techniques, which are commonly used up to the present, optical measurement techniques are fast and contactless, which is important for sensitive as well as fast moving surfaces, and easy to extend. Because of these advantages a lot of optical sensors are applied depending on the measurement problem, e.g. triangulation [1], low coherence interferometry [2], absolute distance interferometry (ADI) [3], conoscopic holography [4], mode-locking external cavity laser sensor [5], time-of-flight or laser Doppler techniques [6, 7].

Triangulation, a widely used technique for distance measurement, conoscopic holography and low coherence interferometry offer low measurement uncertainties down to the sub-micrometer range. Nevertheless, due to necessary scanning processes or because of using a position sensitive detector, the measurement rate is restricted to some kHz. Additionally, the measurement uncertainty increases with increasing object velocity due to the decreasing averaging time.

Laser Doppler vibrometers offer a resolution of some nanometers and a high measurement rate in the MHz range. However, they are restricted by their incremental measurement method, which can cause errors at rough surfaces if phase jumps higher than half the wavelength occur. In this case, the displacement can not be determined unambiguously.

The majority of optical measurement methods can determine only one measurand, e. g. distance or velocity, which is not sufficient to determine the absolute shape of a rotating object. The laser Doppler distance sensor with frequency evaluation we invented earlier [8, 9] overcomes this drawback by measuring the position and velocity of a moving object simultaneously. Based on the conventional laser Doppler velocimetry (LDV), two fan-shaped interference fringe systems with contrary fringe spacing gradients are superposed. By evaluating the quotient of the two resulting independent Doppler frequencies, the object position inside the measurement volume can be determined. With this frequency evaluating technique, simultaneous measurements of distance and velocity are possible allowing to determine the absolute shape of a rotating object by only one single sensor [8]. In addition, the position uncertainty is in principle independent of the object velocity in contrast to conventional distance sensors [9]. However, due to the measurement principle and the speckle effect at rough solid state surfaces, the distance uncertainty of the laser Doppler distance sensor σ_z,tot is limited to about 1% of the measurement range l_z, currently.

In order to achieve a higher accuracy concerning distance measurements, a novel laser Doppler distance sensor with phase evaluation was investigated, which is presented in this paper. This phase sensor is based on the superposition of two nearly identical interference fringe systems with approximately parallel fringes, which are slightly tilted towards each other. The two resulting scattered light signals from a passing measurement object exhibit a phase shift with respect to each other, which is proportional to the axial object position inside the measurement volume. The gradient of this phase shift can be varied without changing the size of the measurement volume significantly. Hence, the relative distance uncertainty σ_z,tot/l_z can be considerably reduced compared to the previous sensor setup with frequency evaluation by using a steep gradient of the phase difference function.

 

Fig. 1. Principle of the phase laser Doppler distance sensor. Superposition of interference fringe systems with approximately constant and equal fringe spacing d, which are tilted towards each other by an angle ?.

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This paper is structured as follows: section 2 contains a description of the principle of the novel phase sensor. In section 3, the experimental setup is described. Section 4 deals with the theoretical uncertainty estimation followed by experimental results in section 5. Finally, the most important results are summarized in section 6.

2. Sensor principle

Conventional laser Doppler velocimeters (LDV) are based on the evaluation of scattered light signals which are generated from measurement objects passing the interference fringe system in the intersection volume of two coherent laser beams. These scattered light signals exhibit an amplitude modulation with the Doppler frequency f. Thus, the measurement object velocity v can be calculated by [10, 11]:

where d is the mean fringe spacing due to the sensor setup. Conventionally, an average of the velocity over the measurement volume is obtained. In order to gain higher spatial resolution the two crossing laser beams have to be focused more strongly. However, this is accompanied with a higher velocity uncertainty due to increased wavefront curvature of the laser beams resulting in a stronger curvature of the fringe spacing function. Hence, with a conventional LDV, it is only possible to obtain either a high spatial resolution or a low velocity uncertainty [12]. In order to achieve both features simultaneously, a novel laser Doppler sensor with slightly tilted fringe systems and phase evaluation was invented for flow measurements [11].

Now, an adapted and modified sensor setup using phase evaluation is utilized for simultaneous distance and velocity measurements of moving solid state objects. Its measurement volume is formed by two interference fringe systems of different laser wavelengths. These interference fringe systems with approximately equal fringe spacing are superposed slightly tilted towards each other, see Fig. 1. A scattering object crossing this measurement volume results in two distinguishable scattered light signals. These two signals exhibit a phase difference which depends on the axial position z of the scattering object. Assuming plane wavefronts, the phase difference φ can be described as:

where s is the slope of the phase difference function φ(z) and φ ₀ the phase offset in the center of the measurement volume (z = 0). By evaluating this phase difference, the position z inside the measurement volume can be determined using the inverse function of Eq. (2). With the known working distance D ₀ between sensor front face and measurement volume, also the distance D = D ₀ + z of the measurement object with respect to the sensor can be determined. Consequently, only the position z is considered in the following.

Due to the known position z inside the measurement volume, the local fringe spacing can be taken into account allowing a more precise velocity determination compared to a conventional LDV. Thus, Eq. (1) can be transformed to:

In order to obtain a low measurement uncertainty, the calibration curve φ(z) should have a high slope s (cp. section 4). However, for an unambiguous determination of the position the calibration curve has to be bijective. Hence, the range of the phase difference inside the measurement volume usually has to be restricted to 2π. Therefore, an optimum slope s_opt exists, which is given by [11]:

where l_z is the length of the measurement volume in z-direction.

If a lower measurement uncertainty is needed, it is easily possible to increase the slope of the calibration function φ(z) by increasing the tilting angle ψ between the two interference fringe systems, see Fig. 1. With a higher slope of the calibration curve, a lower measurement uncertainty can be achieved if σ_φ remains constant, see Eq. (5). This possibility of reducing the measurement uncertainty without decreasing the size of the measurement volume significantly is a key advantage of the phase sensor compared to the laser Doppler distance sensor with frequency evaluation [8]. However, when using a higher slope than s_opt, the determination of the position is no longer unambiguous within the whole measurement volume. Hence, one additional information is needed, which can be obtained by a further interference fringe system employing a third discriminable laser wavelength. The three interference fringe systems have to be adjusted in such a way, that in addition to the steeper phase function φ ₁(z) one bijective phase function φ ₂(z) is obtained, see Fig. 2. Thus, it is possible to determine the object position first roughly via the calibration function φ ₂(z) and secondly very accurate via the steeper calibration function φ ₁(z). Consequently, a more precise position measurement is possible. Nevertheless, the requirement of a third laser wavelength demands a higher technical effort in practice.

3. Sensor setup

The experimental setup was arranged with an optical bench, see Fig. 3. Two laser diodes, a red one (660 nm) and an infrared one (785 nm), were used as light sources. The two laser beams were combined via a dichroic mirror and focused at a transmission phase grating, which acts as beam splitter. The first positive and negative diffraction orders were used as partial beams, all other orders were blocked by beam stops. A Keplerian telescope behind the grating, consisting of two achromatic lenses, focused the partial beams into the measurement volume, see Fig. 3. Since lenses with low chromatic abberation for the red and near-infrared wavelength were employed, a good overlap of the two interference fringe systems with an axial shift of only 25 μm was achieved. The length of the measurement volume in z-direction, i.e. the position measurement range, was about 800 μm. The lateral diameter of the measurement volume is equal to the spot size at a technical surface and was 60 μm with this setup.

 

Fig. 2. Principle of the high resolving sensor setup with one ambiguous and one unambiguous phase function.

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There are two possibilities of adjusting the relation between the phase difference φ of the two scattered light signals and the position z inside the measurement volume. Tilting the dichroic mirror results in a change of the angle between both interference fringe systems. Consequently, the slope s of the calibration function φ(z) can be tuned appropriately. The phase difference offset φ ₀ can be changed by tilting the glass plates inside the Keplerian telescope [11, 13].

The bi-chromatic scattered light from the measurement object was detected in backward direction. Collimated by the front lens, the scattered light was coupled out by a small mirror between the partial beams inside the Keplerian telescope. With a subsequent lens the light was coupled into a multimode fiber patch cable and guided to the detection unit. At this point, the bi-chromatic light was split into the two different wavelengths by a second dichroic mirror and focused onto two photo detectors. The electrical photo detector signals were sampled simultaneously by a 14-bit A/D converter card installed in a standard PC.

Further signal processing and evaluation was done by a MATLAB program. Thereby, the Doppler frequencies f _1,2 where calculated with a least square fit of the fast Fourier transformed photo detector signals. For phase estimation the cross-correlation function of the two photo detector signals was calculated. Via a cosine least square fit, the time shift of the maximum of the cross-correlation function was determined, which is proportional to the phase difference φ.

4. Theoretical measurement uncertainty

The statistical position uncertainty is defined as the standard deviation σ_z of repeated measurements at a fixed axial position of the measurement object. Using Eq. (2) the position uncertainty can be described by:

Assuming that the two scattered light signals have equal Doppler frequencies, the minimum standard deviation for the phase difference estimation is given by the Cramer-Rao Lower Bound (CRLB) [10]:

 

Fig. 3. Experimental setup of the phase laser Doppler distance sensor.

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where SNR is the signal-to-noise ratio and N the number of statistically independent sampling points. The minimum position uncertainty can be written now as:

Hence, the position uncertainty is independent of the object velocity, which is an important advantage of the laser Doppler distance sensor compared to other distance sensors.

To estimate the phase difference φ, the two Doppler frequencies f _1,2 have to be equal. Therefore, both interference fringe spacing functions d _1,2 have to be identical (d ₁(z) = d ₂(z)). Assuming ideal Gaussian laser beams with their beam waists located in the center of the measurement volume, the interference fringe spacing functions d _1,2(z) are given by [14]:

where θ_i denotes the half crossing angle of the partial laser beams and w _0i the beam waist radius for the respective laser wavelength i = 1,2. Since a diffraction grating is used for beam splitting with sin α ₁/λ ₁ = sinα ₂/λ ₂, the minimum fringe spacings at the center of the measurement volume, i.e. z = 0, are equal [13]. However, due to the different wavelengths, the fringe spacing curves differ outside the center of measurement range. Thus, in general, the two Doppler frequencies f _1,2 are not equal for z ≠ 0 resulting in a phase drift Δφ during the measurement time causing a systematic phase deviation [11, 13]

In order to obtain a phase drift Δφ = 0 the two modulation frequencies f _1,2 and thus the interference fringe spacings d _1,2 have to be equal. Therefore, the terms in the square brackets in Eq. (8) have to be equal for any position z. In approximation, this requirement is fulfilled when the beam waists of the two wavelengths w _01,2 match with [11, 13]:

In practice, a relative deviation between the two interference fringe spacings of 10^-4 … 10^-3 has been achieved by adjusting the sensor setup according to Eq. (10). Hence, it can bes assumed that d ₁(z) ≈ d ₂(z) ≈ d(z) and therefore the systematic phase drift described by Eq. (9) can be neglected.

Besides the position z, also the velocity of the measurement object can be determined. Based on Eq. (1) the relative measurement uncertainty for the velocity σ_v/v can be calculated by the Gaussian uncertainty propagation assuming the statistical independence of fringe spacing d and Doppler frequency f

With the relation σ_d = (∂d/∂z)σ_z and Eq. (5) the relative measurement uncertainty for the velocity can be written as:

Since the slopes of the interference fringe systems ∂d/∂z in the center of the measurement range are approximately zero, the second term inside the square root of Eq. (12) can be neglected and the relative measurement uncertainty for the velocity can be approximated by:

As a result, the relative velocity uncertainty is equal to the relative Doppler frequency uncertainty, which can be as small as the respective CRLB [15]. Hence, this sensor allows more precise velocity measurements than conventional LDVs.

5. Experimental results

In order to verify the theoretical estimations from section 4, measurements with test objects of steel and aluminum were carried out. These test surfaces had different but defined roughnesses with R_a-values (arithmetical mean deviation of the roughness profile) between 0.1 μm and 3.6 μm. The metal objects were mounted on a motor with stabilized rotation frequency, which was moved through the measurement volume in z-direction by a motorized translation stage. Thus, measurements at defined axial positions and with well known object velocities can be accomplished. At each position 25 individual measurements were carried out in order to obtain the statistical uncertainties. For each individual measurement, the Doppler frequencies f _1,2 and the phase difference φ were calculated from the whole time domain signals corresponding to an averaging over the 12 mm broad tip of the metal objects.

For the experiments, two different setups of the phase sensor were used. At first, the sensor was adjusted with a bijective phase function within the whole measurement range of l_z = 800 μm corresponding to a slope s_I = 0.45°/μm. Secondly, a sensor setup with a slope of s_II = 5.5°/μm was arranged corresponding to a bijective range of only 65 μm. The phase difference functions for the two setups are shown in Fig. 4. For the latter setup, the measurements were carried out only for the bijective range around z = 0 in order limit the technical effort. In future, also measurements within the whole measurement range of 800 μm will be investigated by using a third interference fringe system as explained in section 2.

Figures 5(a) and 5(b) show the measurement results of two test objects with a mean surface roughness of R_a = 0.2 μm and R_a = 3.6 μm, respectively. It can be seen that the measured position uncertainties σ_z(z) of the two different test objects are similar, see Fig. 5(a). In the center of the measurement range (z = 0) a minimum position uncertainty of σ_z = 0.98 μm was obtained. This value matches the theoretical uncertainty estimation of σ_ztheor =0.78 μm well, when using Eq. (7) with a SNR = 5 dB, a number of data points N = 18000 and a slope |s ^-1| =0.45°/μm, which are typical values occurring at these experiments. At the outer regions the statistical uncertainties increase due to the lower SNR and the lower numbers of fringes caused by the decreasing intersection area of the two partial laser beams. The mean value of the statistical position uncertainty over the whole measurement range is ⟨σ_zߩ ≥ 2.2 μm for both test objects.

 

Fig. 4. Measured phase difference functions of the two sensor setups: (a) bijective phase function with s_I - 0.45°/μm and (b) steep phase function with s_II - 5.5°/μm.

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However, due to the random surface structure at rough solid states, systematic deviations occur in addition to the statistical uncertainties, which are depicted in Fig. 5(b). Unlike the statistical uncertainties, these systematic deviations show no regular behavior and are a priori unknown. The functions Δz(z) are completely different from each other, which is due to the speckle pattern caused by coherent scattering at the random structures of the solid state surfaces. The maximum absolute values of the systematic deviations within the measurement range are Δz_max = 8 μm for R_a = 0.2 μm and Δz_max = 14 μm for R_a = 3.6 μm.

Using the Guide to the Expression of Uncertainty in Measurement (GUM), a total measurement uncertainty σ_z,tot can be calculated. Assuming that the probability density of the systematic deviations is uniformly distributed in the interval [-Δz_max, Δz_max], the Gaussian uncertainty propagation result in:

Thus, the total measurement uncertainties for the two test objects are σ_z,tot = 5.2 μm and σ_z,tot = 8.7 μm, respectively.

For the second setup, the statistical uncertainties and the systematic deviations were obtained in the same way. As an example, the measurement results for one test object with R_a = 0.2 μm are presented in Fig. 5(c) and 5(d). A minimum position uncertainty of σ_z = 140 nm and a corresponding mean value of ⟨σ_z⟩ = 280 nm were achieved over a measurement range of 65 μm. Due to the speckle effect at random surface structures the maximum systematic position deviation Δz_max = 1.5 μm is again significantly higher than the statistical uncertainties. Using Eq. (14), a total measurement uncertainty of σ_z,tot = 0.91 μm could be obtained with the steep phase function setup of the phase sensor.

Comparing the measurement results of the two sensor setups, it can be noticed that by increasing the slope s by a factor k = 12.2 from s_I = 0.45°/μm to s_II = 5.5 °/μm a reduction of the mean statistical position uncertainty by a factor k = 7.9 was obtained. Also the maximum systematic position deviation and thus the total measurement uncertainty are reduced by a factor k = 9.3. This is in good agreement with theory, see Eq. (2), since σ_z is indirectly proportional to the slope s of the phase difference function assuming that s_φ remains constant. In a subsequent experiment the influence of the slope s on the position uncertainty was investigated in more detail. Therefore, measurements with different slopes s were carried out. Figure 6(a) shows the measured statistical uncertainties in comparison with a 1/s- regression proving that theory and experimental results agree well. Also the systematic deviations behave in the same manner resulting in a 1/s- behavior also for the total measurement uncertainty, see Fig. 6(b). These results confirm that the measurement uncertainty can be reduced directly by increasing the slope s of the phase difference function as predicted. However, to expand this reduced uncertainty to the whole measurement range of 800 μm, a third interference fringe system has to be applied, see Fig. 2. Respective experiments with an extended setup are planned for the future.

 

Fig. 5. Measurement results of the position evaluation: (a) statistical uncertainties and (b) systematic deviations of the sensor setup with s_I = 0.45°/μm; (c) statistical uncertainties and (d) systematic deviations of the sensor setup with s_II = 5.5°/μm.

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Fig. 6. Experimentally obtained position uncertainties of the phase sensor depending on the slope s of the phase difference function. (a) statistical uncertainties σ_z and (b) total uncertainties σ_z,tot.

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Fig. 7. Total position uncertainty of the phase sensor σ_z,tot in dependence of R_a.

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In order to analyze the influence of the surface roughness on the measurement uncertainty, a measurement series with test objects exhibiting different but defined surface roughnesses were carried out with the phase sensor setup having a slope s_I = 0.45°/μm. Figure 7 shows the obtained total measurement uncertainties σ_z,tot within the whole measurement range of 800 μm according to Eq. (14), which are between 5.2 μm and 8.2 μm. No significant dependence on the surface roughness could be observed. In addition, for several test objects, measurements with different object velocities were accomplished. According to theory the object velocity should have no influence on the position uncertainty, see Eq. (7). The measurement results confirm this feature of the phase sensor, see Fig. 7.

For the purpose of detailed investigations of this special feature of the phase sensor, measurements on a rotating brass wheel with one tooth of 2 mm width and a radius of 40 mm were accomplished. At varying circumferential speed v, the tooth tip position was measured simultaneously by the phase sensor with a steep phase function (s_II = 5.5°/μm) and two commercial triangulation sensors manufactured by Micro-Epsilon, one with a measurement rate of 2.5 kHz (TS_2.5k) and the other one with 20 kHz (TS_20k). The latter one comprises an elliptical laser spot in order to reduce the influence of the speckle effect on the measurement result. Both triangulation sensors have a measurement range of 2 mm. Regarding Fig. 8, the position uncertainty of both triangulation sensors worsens with increasing velocity v of the brass tooth, which is due to the decreasing averaging time. The position uncertainty of the triangulation sensor TS_2.5k increases from σ_z = 2.1μm at v = 0.5 m/s to σ_z = 6.6μm at v = 7.5 m/s. Due to the higher measurement rate and the special shape of the laser spot, the measurement uncertainty of the triangulation sensor TS_20k is significantly lower and increases only from σ_z = 0.1 μm at v = 0.5 m/s to σ_z = 1.2 μm at v = 12.5 m/s. In contrast to the triangulation sensors, the position uncertainty σ_z of the phase sensor remains nearly constant at σ_z = 500 nm over the whole velocity range proving that its position uncertainty is indeed independent of the object velocity, see Fig. 8. Hence, the phase sensor is predestined for precise position measurements on fast moving objects. This feature opens up new applications such as process control of turbo machines, vacuum pumps and at grinding and turning.

 

Fig. 8. Measured statistical position uncertainties σ_z in dependence of the object velocity v for the phase sensor and the two triangulation sensors (TS).

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Fig. 9. Comparison of the relative uncertainties: (a) relative uncertainty of the frequency evaluation σ_f/f , (b) relative uncertainty of the velocity evaluation σ_v/v.

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In addition to the position determination inside the measurement volume, the velocity of the measurement object can be determined by Eq. (3). The measurement results presented in Fig. 9 confirm that the relative measurement uncertainty of the velocity depends mainly on the relative frequency uncertainty, cp. Eq. (13). Hence, the velocity can be determined very accurately without influence of the curvature of the fringe spacing functions d _1,2 resulting in a mean relative measurement uncertainty of σ_v/v = 3.5×10^-4. Due to the simultaneous measurement of axial position and tangential velocity, the diameter and thus the two-dimensional shape of rotating test objects can be calculated as described in [8, 16].

6. Conclusions

In this paper, we have presented a novel laser Doppler distance sensor using two tilted fringe systems and phase evaluation which allows simultaneous measurement of position and velocity of moving solid state objects. By using a high slope of the phase difference function, a total position uncertainty σ_z,tot < 1 μm within a measurement range of 65 μm was achieved. In the future, this measurement range can be extended without increasing the measurement uncertainty significantly by using a third interference fringe system. Since also the surface velocity is determined, the absolute radius and thus the two-dimensional shape of rotating objects can be calculated additionally. It was shown that the velocity can be measured with a relative uncertainty σ_v/v = 3.5 × 10^-4 within the whole measurement volume. Furthermore, it was demonstrated that the position uncertainty is independent of the object velocity, which is an important advantage compared to many other optical sensors, such as triangulation. Hence, interesting applications of the phase sensor are distance and shape measurements of fast rotating objects such as turbo machines and vacuum pumps or at grinding and turning.