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Miniaturized interferometric 3-D shape sensor using coherent fiber bundles
Optics and Lasers in Engineering 107 (2018) 364–369
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Miniaturized interferometric 3-D shape sensor using coherent fiber bundles Hao Zhang∗, Robert Kuschmierz, Jürgen Czarske Laboratory of Measurement and Sensor System Techniques, Technische Universität Dresden, Dresden 01062, Germany
a r t i c l e Keywords: Shape measurement Interferometry Coherent fiber bundle In-situ Image processing Speckle
i n f o
a b s t r a c t Mach-Zehnder interferometer based sensors for simultaneous distance and velocity measurement allow for absolute 3-D shape measurements of rotating workpieces for instance in cutting lathes. The achievable shape uncertainty is limited to around one micron due to the speckle effect and temperature drifts, however. In this paper, a laser Doppler distance sensor with phase evaluation (P-LDD sensor) with a camera based scattered light detection is investigated. A novel speckle separation technique and in-situ fringe distance calibration method are realized to reduce the measurement uncertainty. A coherent fiber bundle is employed to forward the scattered light towards the camera. This enables a compact and passive sensor head with keyhole access. Compared with a photo detector based sensor, the camera based setup allows to decrease the measurement uncertainty by the order of one magnitude. As a result, the total shape uncertainty of absolute 3-D shape measurements can be reduced to about 100 nm.
1. Introduction Absolute shape measurements of rotating workpieces are important for process monitoring and process control for instance at metal cutting lathes. Currently, as the state of the art, coordinate measurement machines (CMM) allow absolute shape measurements with submicron precision [1]. However, the measurement process of the workpiece is slow compared to the workpiece processing time in the lathe, due to the time required for setting up the measurement and due to the tactile nature of conventional CMMs. This also makes the measurement process susceptible to mounting tolerances. Furthermore, the measurement is usually performed ex-situ and after the processing. This means that an immediate control of the workpiece processing is not possible. For this reason, an absolute shape measurement is required inside of the lathe. With the advantages of non-contact and fast measurements, optical measurement techniques [2–9] enable high measurement rates and submicron distance uncertainties. Thus, they can be employed to measure the surface profile of the workpiece, either by scanning or by employing the inherent rotation of the workpiece inside the lathe. However, at high surface or scanning velocities the measurement uncertainty of these techniques increases, due to the decreasing averaging time and the speckle effect which occurs at rough surfaces. Moreover, all these techniques offer only one measurand, the distance. As a result, the absolute diameter of the workpiece is missing. One approach to measure the mean diameter of the workpiece is to measure the tangential velocity of the surface during rotation with a known frequency, additionally. Thus, a simultaneous measurement of
∗
position and tangential velocity at the same position enables absolute shape measurements. While lubricants and coolants might prohibit the application of optical sensors during the cutting process, due to light distortion, the approach allows for in-situ shape measurements directly after cutting or even in-process measurements for dry machining. Furthermore, the simultaneous velocity and distance measurement enables other applications at rotating machinery such as tip clearance and tip vibration measurements in turbo machinery, deformation measurements of high speed rotors [10] or spindle vibration measurements. The laser Doppler distance sensor with phase evaluation (P-LDD sensor) enables simultaneous velocity and distance measurements and is well equipped to measure at fast moving rough surfaces as well. Therefore, it enables an absolute, in-situ shape measurement inside of a lathe with key-hole access [11]. The measurement principle is based on the evaluation of the Doppler frequencies of the scattered light signals and thereby on speckles [12]. For the conventional setup, the scattered light is detected with photo detectors. Therefore, several speckles oscillating with equal Doppler frequency but random phases are superposed on the detector of only one pixel [13]. This results in an increased surface velocity uncertainty and distance uncertainty, and therefore in an increased uncertainty of the measured shape. Furthermore, temperature drifts result to a systematic uncertainty of the interference fringe distance, which increases the shape uncertainty further [14]. The aim of this article is to introduce a sensor to be employed in rotating machines for example for the in-situ, absolute 3-D shape measurement of rotating workpieces in cutting lathes. A setup with a camera based scattered light detection is proposed to evaluate the speckles individually and to achieve an in-situ fringe distance calibration, thereby
Corresponding author. E-mail address: [email protected] (H. Zhang).
https://doi.org/10.1016/j.optlaseng.2018.04.011 Received 1 November 2017; Received in revised form 12 April 2018; Accepted 17 April 2018 0143-8166/© 2018 Elsevier Ltd. All rights reserved.
H. Zhang et al.
Optics and Lasers in Engineering 107 (2018) 364–369
Note that, the contributors can be distinguished between uncertainties resulting from the sensor and uncertainties resulting from the machine, namely the spindle rotation speed and spindle vibrations. The spindle rotation speed can be measured precisely with angular encoders or by speckle correlation with the P-LDD sensor. It has a negligible influence towards the total shape uncertainty. Spindle vibration superposes the position signal and potentially increases the uncertainty of the angle dependent radius deviation. Its impact strongly depends on its amplitude and spectral properties and can be decreased by averaging, filtering or multi-step separation techniques [17]. In this paper we concentrate on reducing the uncertainties resulting from the sensor system. In order to analyze the measurement uncertainty, the scattered light signals are modeled. The detected signal of a single particle that crosses the measurement volume perpendicular to the interference fringes can be described by a sinusoidal modulation:
Fig. 1. Principle of P-LDD sensor: superposition of two interference fringe systems with constant and equal fringe spacing d, which are tilted towards each other by an angle 𝜓. The scattered light from each fringe system is detected with a photo detector.
𝑏(𝑡) = 𝐴𝑒
𝑠(𝑡) =
(1)
In order to measure the axial distance z simultaneously, two mutually tilted interference fringe systems with equal fringe distances and the tilting angle 𝜓 are superposed by wavelength multiplexing. This leads to a position dependent lateral offset between both fringe systems. Thus, z is calculated with the measured phase difference 𝜑 between both light signals by (2)
where the s is the slope of the calibration function 𝜑(z). By evaluating the surface distance and the surface velocity of a rotating workpiece, the absolute shape r(𝛼) can finally be evaluated by [11] 𝑣̄ − (𝑧̄ − 𝑧(𝛼)). 𝜔
cos (2𝜋𝑓D ⋅ (𝑡∕𝑓s − 𝑡a )),
𝑡 = 1, 2, … , 𝑁,
(4)
𝑏𝑘 (𝑡),
𝑡 = 1, 2, … , 𝑁,
(5)
which results in a distortion in the scattered light signal. The distortion depends on the speckle effect and therefore on the surface micro geometry. Thus, the measurement uncertainties caused by the speckle effect cannot be reduced by repetitive measurements of the same surface. In order to reduce the measurement uncertainty, the speckle signals should be processed and evaluated separately. In fluid measurements a temporal separation based on Hilbert transform [19], Empirical Mode Decomposition [20], or Wavelet transform [21] can be applied to separate consecutive burst signals. These approaches are not feasible for surface measurements, because several speckles occur simultaneously. Meanwhile, the total measurement uncertainty is also limited by the relative fringe distance uncertainty which is influenced by thermal effects [22]. The uncertainty of the fringe distance due to thermal effect should be eliminated by an in-situ fringe distance calibration method which enables the calibration at any moment when the temperature changes. For separating the individual speckle as well as to measure the fringe distance, we propose a matrix camera based approach, cf. Fig. 2. We employ one camera (Basler piA640-210gm, maximum frame rate of 210 Hz, resolution of 648 px × 488 px, and a photoelement size of 7.4 μm × 7.4 μm) for each fringe system. Two interference fringe systems are generated by two pairs of laser beams with wavelengths 656 nm and 686 nm, respectively. The light of the fringe systems is scattered at a metallic specimen with a plane, rough surface, which is mounted on a moving stage. The stage is moved laterally in steps of 1 μm. Both cameras are synchronized by a function generator. In the experiments of this paper, the velocity of the moving stage, the sampling frame rates of the cameras and the used areas of the CCDs amount to 200 μm/s, 200 Hz, and 200 px × 200 px, respectively. The measurement uncertainties result from 10 random segments on the specimen surface. Please note that the applied velocity is several magnitudes lower than the common surface velocity inside of cutting lathes and limited due to the framerate of the CCDs. However, camera speeds will further increase in the coming years and can already be increased by reducing the pixel number. In our experiments, a high camera resolution is employed to investigate the feasibility of the approach. A second prob-
As shown in Fig. 1, the P-LDD sensor is based on a laser surface velocimeter using interference fringe patterns [15,16]. It is easily described for single scattering particles passing the measurement volume with a certain velocity. The amplitude of the scattered light is then modulated with the Doppler frequency fD . Determining the fringe distance d by a calibration and measuring the Doppler frequency with the Fast Fourier Transform (FFT), the particle velocity v perpendicular to the fringe system can be calculated by
𝑟(𝛼) = 𝑅 + Δ𝑟(𝛼) =
𝐾 ∑ 𝑘=1
2. Principle and setup
𝑧 = 𝜑 ⋅ 𝑠−1 ,
2𝑓 2 ⋅(𝑡∕𝑓s −𝑡a )2 D 𝑡2 w
with the Doppler frequency fD and a gaussian envelope with the amplitude A. Whereas fs is the sampling frequency, ta describes the arrival time of the particle and tw the temporal 1/e2 width of the envelope which results from the geometrical width of the measurement volume and the particle velocity. For scattered light signals from rough surfaces the speckle effect has to be considered. Former investigations showed, that each speckle exhibits the same Doppler frequency fD but random amplitudes Ak and arrival times 𝑡a𝑘 . Thus, the detector signal from K speckles reads [18]
reducing the measurement uncertainty. In order to realize a compact and robust sensor for in-situ measurements, a coherent fiber bundle is used to transmit the scattered light signal from the sensor head to the cameras. The measurement principle of the P-LDD sensor and the measurement uncertainty budget are described in Section 2. Since the speckle and fringe distance related measurement uncertainties are dominant, the camera based methods and the respective image processing for the reduction of these measurement uncertainties are proposed in Section 3. Finally, the coherent fiber bundle based scattered light signal transmission is presented in Section 4.
𝑣 = 𝑓D ⋅ 𝑑.
−
(3)
The mean radius R of the workpiece can be estimated by the mean velocity 𝑣̄ of the surface and the angular velocity 𝜔, which is constant and known. The angle dependent deviation Δr(𝛼) of the workpiece radius from the mean radius is obtained from the difference of the mean value 𝑧̄ and the angle resolved distance z(𝛼). While the rotation of the workpiece in the lathe enables the angle resolved radius measurement (2-D shape), an additional feed forward of the sensor along the rotational axis achieves an absolute 3-D shape measurement, cf. Fig. 1. The dominant contributors to the shape uncertainty 𝜎 r and their resulting shape uncertainties have been derived experimentally and are listed in Table 1. 365
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Fig. 2. Camera based P-LDD sensor, (a) The scheme of the setup, (b) The experimental setup. Table 1 Measurement uncertainty budget for a 3-D shape measurement containing the most significant contributors. The numerical values are derived experimentally. The uncertainties are calculated for a representative 3-D shape measurement with stepwise scanning along the height of the workpiece with 𝑀 = 104 (102 measurement points per revolution × 102 scanning steps along the y direction), the mean radius of the workpiece 𝑅 = 20 mm and a temperature variation of Δ𝑇 = 1 K. Averaging length XL is the displacement of the workpiece surface, while sampling a time series of the scattered light signal for one measurement point. Description
Value
Relative uncertainty of Doppler frequency Distance uncertainty Relative uncertainty of fringe distance Relative uncertainty of angular velocity Spindle vibration Resulting 3-D shape uncertainty
√ 𝜎𝑓D ∕𝑓D = 4 × 10−3 ∕ 𝑋𝐿 ∕mm √ 𝜎𝑧 = 0.2 μm∕ 𝑋𝐿 ∕mm 𝜎𝑑 ∕𝑑 = 1.5 × 10−5 × Δ𝑇 ∕𝐾 𝜎𝜔 ∕𝜔 = 10−6 Dependent on spectral properties of vibration, signal processing and separation techniques 𝜎r
Resulting shape uncertainty in μm 0.8 0.2 0.2 0.02 1
lem resulting from using cameras opposed to single pixel detectors is the increase in recorded data and signal processing time. For checking the feasibility of the approach this processing speed was not optimized. We used a Windows PC with an Intel Core i7 processor and MATLAB. The processing time is currently 2.9 s/mm surface length. For real-time measurements this has to be increased by parallelizing the signal processing on FPGAs and faster data transfer for instance using frame grabbers. 3. Camera based measurement uncertainty reduction 3.1. Doppler frequency Fig. 3. (a) is the sketch of the speckle movement in the time series of the images shown in (b).
A speckle separation technique can be realized by using a line camera or a matrix camera. It achieves a reduction of the measurement uncer√ tainty of the Doppler frequency by 𝑁s , whereas Ns is the number of independent speckle signals that are detected simultaneously. Since Ns results from the number of camera pixels Np and the ratio between pixel area Ap and average speckle area As : [ ] 𝐴p 𝑁s = 𝑁p ⋅ min ;1 , (6) 𝐴s
speckles moving in a single camera row over time 𝑆(𝑥̃ , 𝑦̃ = 𝑦̃0 , 𝑡) and (c) ′ the sheared signal 𝑆 (𝑥̃ + 𝑡∕ tan(𝜃), 𝑦̃ = 𝑦̃0 , 𝑡) = 𝑆(𝑥̃ , 𝑦̃ = 𝑦̃0 , 𝑡). Thus, the amplitude spectra of each speckle can be calculated by ′ FFT on the sheared signals 𝑆 . Averaging the amplitude spectra of each speckle yields a seemingly undisturbed gaussian function, from which the Doppler frequency is calculated, cf. Fig. 5 (red curve). Compared to the photo detector based sensor, 𝜎𝑓D ∕𝑓D is decreased √ by a factor of 18 to 3 × 10−4 ∕ 𝑋𝐿 ∕mm where 𝑋𝐿 = 1 mm. Fig. 9(a)(red curve) shows 𝜎𝑓D ∕𝑓D decreases with the increasing number of detected speckle signals Ns . Thus, the contribution of 𝜎𝑓D ∕𝑓D towards the shape uncertainty is reduced to 60 nm.
the measurement uncertainty can be reduced by adjusting the speckle size to the pixel size by optimizing the numerical aperture. In this paper a matrix camera is employed to obtain a higher pixel number, meaning more speckles can be detected and evaluated simultaneously. In this case the speckles move across the camera in accordance with the surface movement, cf. Fig. 3. In order to allow for an easy frequency estimation by the FFT, we shear the speckle signal by the angle 𝜃 in the x̃ , t -plane by means of the affine transformation ⎡𝑥̃ + 𝑡∕ tan(𝜃)⎤ ⎡1 ⎢ ⎥ = ⎢0 𝑡 ⎢ ⎥ ⎢ 1 ⎣ ⎦ ⎣0
1∕ tan(𝜃) 1 0
0⎤⎡𝑥̃ ⎤ 0⎥⎢ 𝑡 ⎥, ⎥⎢ ⎥ 1⎦⎣1⎦
3.2. In-situ fringe distance calibration In order to reduce 𝜎 r further, 𝜎 d /d has to be reduced. Formally, the fringe distance of Mach-Zehnder velocimeter based sensors was calibrated by utilizing a rotating disc with a pin hole [23]. This needs to be done before measurements, which means that during measurements in a working lathe a calibration cannot be achieved with this method. In order to eliminate the impact of thermal effects, an in-situ calibration is necessary. This can be achieved by imaging the fringe system onto the camera and evaluating the fringe distance by a spatial FFT. However,
(7)
exploiting the fact that each speckle has the same velocity on the camera surface. 𝜃 is calculated by auto correlating the camera signal 𝑆(𝑥̃ , 𝑦̃ = 𝑦̃0 , 𝑡). Fig. 4(a) shows the measured speckle signal 𝑆(𝑥̃ , 𝑦̃, 𝑡 = 𝑡0 ), (b) the 366
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Fig. 4. (a) is the speckle signal 𝑆(𝑥̃ , 𝑦̃, 𝑡 = 𝑡0 ) captured by the matrix camera, as an example the speckle signal in 100th camera row (red rectangle) is shown in (b). ′ (c) is the sheared speckle signal 𝑆 in which the motions of the speckles are adjusted to vertical direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
averaging over 𝑥̃ direction, which corresponds to a line camera signal: 𝑆r (𝑦̃, 𝑡) =
𝑁c ∑
𝑆(𝑥̃ , 𝑦̃, 𝑡),
(9)
𝑥̃ =1
with Nc the number of camera columns. The phase difference 𝜑𝑘 = 2𝜋𝜏𝑘 ∕𝑓D can now be calculated for each line camera pixel pair by 𝐶(𝑦̃, 𝜏) =
𝑁 ∑
𝑆r,1 (𝑦̃, 𝑡)𝑆r,2 (𝑦̃, 𝑡 + 𝜏).
(10)
𝑡=1
To verify the approach, distance measurements on the same specimen are performed with the photo detector and the camera based scattered light detection. We can show, that the distance uncertainty is reduced by about the order of one magnitude to 33 nm, cf. Fig. 9(b)(red curve). In order to investigate the effect of the pixel number towards the measurement uncertainty, we vary the pixel number by software binning. As is depicted in Fig. 9(b)(red curve), the uncertainty decreases with the square root of the pixel number as it does for the Doppler frequency. This decrease is limited by the speckle size, only. Meaning that choosing pixels smaller than the speckle size, does not yield an advantage. With the decreased measurement uncertainties for fD , d and z the 3-D shape uncertainty 𝜎 r is reduced to 100 nm.
Fig. 5. Determined amplitude spectrum with the matrix camera based approach (red curve), where the amplitude spectra are determined for each speckle signal. For comparison, the formerly resulting amplitude spectra with the photo detector (integration over all camera pixels, blue curve) are shown. The uncertainties are given for scattered light signals with an averaging length of 𝑋𝐿 = 1 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
due to the speckle effect no fringe system is apparent for a single shot image, cf. Fig. 6(a). On the other hand, as a characteristic, the speckles are bright at the positions of light fringes and dark at the position of dark fringes. While the object is moving, the speckles from the object surface are distributed to random positions. Thus, the speckle effect can be eliminated by averaging the signal images over time and thereby obtaining the fringe pattern If : 𝐼f =
𝑁 1 ∑ 𝑠(𝑥̃ , 𝑦̃, 𝑡). 𝑁 𝑡=1
4. Miniaturization using coherent fiber bundle In order to realize in-situ measurements inside of a lathe, a small and robust sensor is required. Formally, this was achieved by employing multi-mode fiber coupled photo detectors [11]. Using multi-mode fibers is not possible here because the spatially resolved speckle images are required while multi-mode fibers distort these images due to modemixing [24]. Therefore, the possibility to use a coherent fiber bundle (CFB) is investigated. CFBs are used in endoscopy to transmit images coherently, meaning that the relative distance and angle between each pixel remains constant [25]. In experiments, one CFB (Fujikura FIGH-50-1100N) with 50,000 fiber cores and 4 μm core diameter is employed to forward the scattered light of both fringe systems towards two cameras cf. Fig. 7. The cameras are used to detect the image of the scattered light field at the distal end of the CFB. For comparisons, the experiments with the CFB are done under the same experimental condition as the experiments without the CFB, meaning with direct camera detection. The signal images detected with the CFB and without the CFB are compared at the same position of the specimen surface, cf. Fig. 8. The similarity of the images is given by a correlation coefficient 0.93. This means the speckle separation and the fringe distance calibration techniques of Section 2 can be employed in principle and should work on the signal images transmitted by the CFB. In order to investigate the influence of the number of camera pixels, the pixel number is varied by using software binning and the relative Doppler frequency uncertainty 𝜎𝑓D ∕𝑓D , the distance uncertainty 𝜎 z as well as the relative fringe distance uncertainty 𝜎 d /d are calculated. As
(8)
This has been demonstrated experimentally, cf. Fig. 6(a). As shown in Fig. 6(b), by means of the FFT, an in-situ and real-time calibration of the fringe distance d is realized. Thus, the influence of the temperature variation √ on d can be eliminated. Applying this approach 𝜎𝑑 ∕𝑑 = 5.9 × 10−4 ∕ 𝑋𝐿 ∕mm is determined cf. Fig. 9(c)(red curve). With 𝑋𝐿 = 1 mm and the total sample number 𝑀 = 104 as in the example in Table 1, 𝜎𝑑 ∕𝑑 ≈ 10−6 results. Thus, the contribution of 𝜎 d /d towards the shape uncertainty is reduced to 20 nm. 3.3. Distance Lastly the distance uncertainty is investigated. Conventionally with photo detector based detection, the distance was estimated by evaluating the phase difference of two time series of scattered light signals employing the cross-correlation function. Evaluating the phase difference 𝜑k for each speckle signal separately and subsequent averaging would lead to a maximum uncertainty reduction. However, the phase difference results in a shift of the speckle in 𝑥̃ direction on the camera as well due to the speckle movement. Thus, calculating 𝜑k is computationally expensive, since a 2-D cross correlation is required. In order to reduce computing time, the 3-D camera signal is reduced to a 2-D signal by 367
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Optics and Lasers in Engineering 107 (2018) 364–369
Fig. 6. (a) A single shot speckle image from a moving rough surface, (b) The fringe pattern If obtained by averaging the speckle images over time with 𝑋𝐿 = 1 mm, (c) The fringe frequency ff is evaluated by the FFT and averaged over 𝑦̃.
Fig. 7. Camera based P-LDD sensor with the coherent fiber bundle, (a) The scheme of the setup, (b) The experimental setup.
Fig. 8. (a) The signal image detected with CFB, (b) The signal image detected without CFB (direct camera detection). The correlation coefficient of the images is 0.93.
Fig. 9. Camera based measurement uncertainties with and without coherent fiber bundle scattered light detection. The relative Doppler frequency uncertainty (a) and distance uncertainty (b) decrease with square root of individually resolvable speckles. For a single photo detector with only one pixel, i.e. a single fiber core (green circle), the relative Doppler frequency uncertainties and the distance uncertainties are maximal and amount to 5.5 × 10−3 , 5.4 × 10−3 and 0.5 μm, 0.4 μm, relative to the experiments with CFB and without CFB. (c) As the result of the in-situ calibration, the relative uncertainty of the fringe distance is decreasing with the increase of averaging length. The reductions of the measurement uncertainties are consist with the theories (dotted lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Table 2 Experimental measurement uncertainties for 3-D shape measurements for 𝑋𝐿 = 1 mm. 𝜎𝑓D ∕𝑓D Photo Detector CCD CCD+CFB
𝜎𝑟 (𝜎𝑓D ∕𝑓D )
5.4 × 10 3 × 10−4 3.2 × 10−4 −3
0.8 μm 0.06 μm 0.064 μm
𝜎z 0.4 μm 0.033 μm 0.064 μm
𝜎 r (𝜎 z )
𝜎 d /d
0.4 μm 0.033 μm 0.064 μm
1.5 × 10 × Δ𝑇 ∕𝐾 5.9 × 10−4 1.8 × 10−3
shown in Fig. 9, the measurement uncertainties 𝜎𝑓D ∕𝑓D and 𝜎 z show the same behaviour regarding the pixel number as without the CFB and decrease with the number of individually resolved speckles, resulting a minimum uncertainty of 𝜎𝑓D ∕𝑓D = 3.2 × 10−4 and 𝜎𝑧 = 0.064 μm for an averaging length 𝑋𝐿 = 1 mm. Furthermore, the in-situ fringe distance calibration follows the √ same behaviour as without the CFB resulting in 𝜎𝑑 ∕𝑑 = 1.8 × 10−3 ∕ 𝑋𝐿 ∕mm. This shows, that CFBs are well suited to realize passive probes. As a result, with the coherent fiber bundle based signal transmission, the 3-D shape uncertainty 𝜎 r is reduced to 120 nm. However, the reductions of the measurement uncertainty are a little lower than with the direct camera detection, cf. Table 2. This has several reasons. First, the CFB uses a common cladding, thus optical cross talk occurs and distorts the transmitted images [26]. Second, each fiber core is multi-mode which results in an additional distortion. Lastly, the speckle size is not matched to the core diameter (diameter core: 4 μm, diameter speckle: 12 μm) due to the limited numerical aperture. Thus the number of independent signals is decreased. For visual comparisons, the measurement uncertainties with different detection approaches are listed in Table 2.
−5
𝜎 r (𝜎 d /d)
𝑁s,𝑓D
Ns, z
𝜎r
0.2 μm 0.02 μm 0.02 μm
1 3.9 × 104 3 × 104
1 32 28
1 μm 0.1 μm 0.12 μm
Acknowledgments The financial supports of the German Research Foundation (DFG) for the project Cz55/29-2 and the China Scholarship Council (CSC) for 201506690024 are gratefully acknowledged. References [1] Hocken RJ, Pereira PH. Coordinate measuring machines and systems. CRC Press; 2016. [2] Goh K, Phillips N, Bell R. The applicability of a laser triangulation probe to non-contacting inspection. Int J Prod Res 1986;24(6):1331–48. [3] Kempe A, Schlamp S, Rösgen T, Haffner K. Low-coherence interferometric tip-clearance probe. Opt Lett 2003;28(15):1323–5. [4] Sirat G, Paz F. Conoscopic probes are set to transform industrial metrology. Sensor Review 1998;18(2):108–10. [5] Lu S-H, Lee C-C. Measuring large step heights by variable synthetic wavelength interferometry. Meas Sci Technol 2002;13(9):1382. [6] Grün A, Kahmen H. Optical 3-d measurement techniques. Wichmann, Karlsruhe 1989. [7] Czarske J, Mobius J, Moldenhauer K, Ertmer W. External cavity laser sensor using synchronously-pumped laser diode for position measurements of rough surfaces. Electron Lett 2004;40(25):1584–6. [8] Schnell U, Gray S, Dändliker R. Dispersive white-light interferometry for absolute distance measurement with dielectric multilayer systems on the target. Opt Lett 1996;21(7):528–30. [9] Xiaoli D, Katuo S. High-accuracy absolute distance measurement by means of wavelength scanning heterodyne interferometry. Meas Sci Technol 1998;9(7):1031. [10] Kuschmierz R, Filippatos A, Günther P, Langkamp A, Hufenbach W, Czarske J, et al. In-process, non-destructive, dynamic testing of high-speed polymer composite rotors. Mech Syst Signal Process 2015;54:325–35. [11] Kuschmierz R, Davids A, Metschke S, Löffler F, Bosse H, Czarske J, et al. Optical, in situ, three-dimensional, absolute shape measurements in cnc metal working lathes. Int J Adv Manuf Technol 2016;84(9–12):2739–49. [12] Günther P, Kuschmierz R, Pfister T, Czarske JW. Displacement, distance, and shape measurements of fast-rotating rough objects by two mutually tilted interference fringe systems. JOSA A 2013;30(5):825–30. [13] Kuschmierz R, Koukourakis N, Fischer A, Czarske J. On the speckle number of interferometric velocity and distance measurements of moving rough surfaces. Opt Lett 2014;39(19):5622–5. [14] Schuster MS, Kuschmierz R, Czarske J. Measurement uncertainty of non-incremental, non-contact, in-situ shape measurements. tm-Technisches Messen 2017;84(6):401–10. [15] Truax BE, Demarest FC, Sommargren GE. Laser doppler velocimeter for velocity and length measurements of moving surfaces. Appl Opt 1984;23(1):67. [16] Matsubara K, Stork W, Wagner A, Drescher J, Mllerglaser KD. Simultaneous measurement of the velocity and the displacement of the moving rough surface by a laser doppler velocimeter. Appl Opt 1997;36(19):4516. [17] Whitehouse D. Some theoretical aspects of error separation techniques in surface metrology. J Phys E 1976;9(7):531. [18] Zhang H, Kuschmierz R, Czarske J, Fischer A. Camera-based speckle noise reduction for 3-d absolute shape measurements. Opt Express 2016;24(11):12130–41. [19] Nobach H. Analysis of dual-burst laser doppler signals. Meas Sci Technol 2001;13(1):33. [20] Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, et al. The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis. In: Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, 454. The Royal Society; 1998. p. 903–95. [21] Nobach H, Van Maanen H. Lda and pda signal analysis using wavelets. Exp Fluids 2001;30(6):613–25. [22] Berkovic G, Shafir E. Optical methods for distance and displacement measurements. Adv Opt Photonics 2012;4(4):441–71. [23] Kuschmierz R, Czarske J, Fischer A. Multiple wavelength interferometry for distance measurements of moving objects with nanometer uncertainty. Meas Sci Technol 2014;25(8):085202. [24] Yariv A. Three-dimensional pictorial transmission in optical fibers. Appl Phys Lett 1976;28(2):88–9. [25] Göbel W, Kerr JN, Nimmerjahn A, Helmchen F. Miniaturized two-photon microscope based on a flexible coherent fiber bundle and a gradient-index lens objective. Opt Lett 2004;29(21):2521–3. [26] Chen X, Reichenbach KL, Xu C. Experimental and theoretical analysis of core-to-core coupling on fiber bundle imaging. Opt Express 2008;16(26):21598–607.
5. Conclusions A novel camera based speckle separation method was presented for the speckle noise reduction and, thus, the systematic uncertainty reductions of Doppler frequency based velocity and distance sensors. Experiments were performed for validating the proposed camera based approach. By evaluating the Doppler frequency of each speckle separately, the Doppler frequency uncertainty decreases with the square root of independent speckles. Thus, a reduction by a factor of 18 for the relative Doppler frequency uncertainty was achieved in comparison to a photo detector based signal detection. Similarly, the distance uncertainty was reduced by the order of one magnitude as well. Furthermore, employing cameras allows for a novel in-situ calibration of the fringe distance, which can eliminate the impact of thermal effect on the measurement uncertainty. Thus, an absolute shape uncertainties of about 100 nm can be achieved for an in-situ shape measurement, for instance in turning lathes. In comparison, CMMs allow 3-D shape uncertainties of around 1 μm and do not support an in-situ measurement. A coherent fiber bundle has been employed in P-LDD sensor for the first time for signal transmission to demonstrate the capability of realizing a robust, all passive sensor head. As an outlook, the possibility to employ single mode fiber bundles will be investigated. The influence of coolants on the workpiece surface towards the measurement uncertainty will be investigated for applying the sensor to the near-dry machining. A high-speed, online signal processing using FPGAs is to be setup for evaluating the measurement data in real time. Furthermore, other applications such as turbo machinery or blade oscillation, mechanical vibration, car tire, wind turbine rotor, gear, and spindle play measurements can be achieved by using the P-LDD sensor as well. As a broad prospect, the camera based speckle separation technique and the in-situ fringe distance calibration method also can be applied in other optical sensors that are limited by the speckle effect or the thermal effect, and do not use matrix cameras yet, such as other interferometers or triangulation sensors.
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Miniaturized interferometric 3-D shape sensor using coherent fiber bundles Hao Zhang∗, Robert Kuschmierz, Jürgen Czarske Laboratory of Measurement and Sensor System Techniques, Technische Universität Dresden, Dresden 01062, Germany
a r t i c l e Keywords: Shape measurement Interferometry Coherent fiber bundle In-situ Image processing Speckle
i n f o
a b s t r a c t Mach-Zehnder interferometer based sensors for simultaneous distance and velocity measurement allow for absolute 3-D shape measurements of rotating workpieces for instance in cutting lathes. The achievable shape uncertainty is limited to around one micron due to the speckle effect and temperature drifts, however. In this paper, a laser Doppler distance sensor with phase evaluation (P-LDD sensor) with a camera based scattered light detection is investigated. A novel speckle separation technique and in-situ fringe distance calibration method are realized to reduce the measurement uncertainty. A coherent fiber bundle is employed to forward the scattered light towards the camera. This enables a compact and passive sensor head with keyhole access. Compared with a photo detector based sensor, the camera based setup allows to decrease the measurement uncertainty by the order of one magnitude. As a result, the total shape uncertainty of absolute 3-D shape measurements can be reduced to about 100 nm.
1. Introduction Absolute shape measurements of rotating workpieces are important for process monitoring and process control for instance at metal cutting lathes. Currently, as the state of the art, coordinate measurement machines (CMM) allow absolute shape measurements with submicron precision [1]. However, the measurement process of the workpiece is slow compared to the workpiece processing time in the lathe, due to the time required for setting up the measurement and due to the tactile nature of conventional CMMs. This also makes the measurement process susceptible to mounting tolerances. Furthermore, the measurement is usually performed ex-situ and after the processing. This means that an immediate control of the workpiece processing is not possible. For this reason, an absolute shape measurement is required inside of the lathe. With the advantages of non-contact and fast measurements, optical measurement techniques [2–9] enable high measurement rates and submicron distance uncertainties. Thus, they can be employed to measure the surface profile of the workpiece, either by scanning or by employing the inherent rotation of the workpiece inside the lathe. However, at high surface or scanning velocities the measurement uncertainty of these techniques increases, due to the decreasing averaging time and the speckle effect which occurs at rough surfaces. Moreover, all these techniques offer only one measurand, the distance. As a result, the absolute diameter of the workpiece is missing. One approach to measure the mean diameter of the workpiece is to measure the tangential velocity of the surface during rotation with a known frequency, additionally. Thus, a simultaneous measurement of
∗
position and tangential velocity at the same position enables absolute shape measurements. While lubricants and coolants might prohibit the application of optical sensors during the cutting process, due to light distortion, the approach allows for in-situ shape measurements directly after cutting or even in-process measurements for dry machining. Furthermore, the simultaneous velocity and distance measurement enables other applications at rotating machinery such as tip clearance and tip vibration measurements in turbo machinery, deformation measurements of high speed rotors [10] or spindle vibration measurements. The laser Doppler distance sensor with phase evaluation (P-LDD sensor) enables simultaneous velocity and distance measurements and is well equipped to measure at fast moving rough surfaces as well. Therefore, it enables an absolute, in-situ shape measurement inside of a lathe with key-hole access [11]. The measurement principle is based on the evaluation of the Doppler frequencies of the scattered light signals and thereby on speckles [12]. For the conventional setup, the scattered light is detected with photo detectors. Therefore, several speckles oscillating with equal Doppler frequency but random phases are superposed on the detector of only one pixel [13]. This results in an increased surface velocity uncertainty and distance uncertainty, and therefore in an increased uncertainty of the measured shape. Furthermore, temperature drifts result to a systematic uncertainty of the interference fringe distance, which increases the shape uncertainty further [14]. The aim of this article is to introduce a sensor to be employed in rotating machines for example for the in-situ, absolute 3-D shape measurement of rotating workpieces in cutting lathes. A setup with a camera based scattered light detection is proposed to evaluate the speckles individually and to achieve an in-situ fringe distance calibration, thereby
Corresponding author. E-mail address: [email protected] (H. Zhang).
https://doi.org/10.1016/j.optlaseng.2018.04.011 Received 1 November 2017; Received in revised form 12 April 2018; Accepted 17 April 2018 0143-8166/© 2018 Elsevier Ltd. All rights reserved.
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Optics and Lasers in Engineering 107 (2018) 364–369
Note that, the contributors can be distinguished between uncertainties resulting from the sensor and uncertainties resulting from the machine, namely the spindle rotation speed and spindle vibrations. The spindle rotation speed can be measured precisely with angular encoders or by speckle correlation with the P-LDD sensor. It has a negligible influence towards the total shape uncertainty. Spindle vibration superposes the position signal and potentially increases the uncertainty of the angle dependent radius deviation. Its impact strongly depends on its amplitude and spectral properties and can be decreased by averaging, filtering or multi-step separation techniques [17]. In this paper we concentrate on reducing the uncertainties resulting from the sensor system. In order to analyze the measurement uncertainty, the scattered light signals are modeled. The detected signal of a single particle that crosses the measurement volume perpendicular to the interference fringes can be described by a sinusoidal modulation:
Fig. 1. Principle of P-LDD sensor: superposition of two interference fringe systems with constant and equal fringe spacing d, which are tilted towards each other by an angle 𝜓. The scattered light from each fringe system is detected with a photo detector.
𝑏(𝑡) = 𝐴𝑒
𝑠(𝑡) =
(1)
In order to measure the axial distance z simultaneously, two mutually tilted interference fringe systems with equal fringe distances and the tilting angle 𝜓 are superposed by wavelength multiplexing. This leads to a position dependent lateral offset between both fringe systems. Thus, z is calculated with the measured phase difference 𝜑 between both light signals by (2)
where the s is the slope of the calibration function 𝜑(z). By evaluating the surface distance and the surface velocity of a rotating workpiece, the absolute shape r(𝛼) can finally be evaluated by [11] 𝑣̄ − (𝑧̄ − 𝑧(𝛼)). 𝜔
cos (2𝜋𝑓D ⋅ (𝑡∕𝑓s − 𝑡a )),
𝑡 = 1, 2, … , 𝑁,
(4)
𝑏𝑘 (𝑡),
𝑡 = 1, 2, … , 𝑁,
(5)
which results in a distortion in the scattered light signal. The distortion depends on the speckle effect and therefore on the surface micro geometry. Thus, the measurement uncertainties caused by the speckle effect cannot be reduced by repetitive measurements of the same surface. In order to reduce the measurement uncertainty, the speckle signals should be processed and evaluated separately. In fluid measurements a temporal separation based on Hilbert transform [19], Empirical Mode Decomposition [20], or Wavelet transform [21] can be applied to separate consecutive burst signals. These approaches are not feasible for surface measurements, because several speckles occur simultaneously. Meanwhile, the total measurement uncertainty is also limited by the relative fringe distance uncertainty which is influenced by thermal effects [22]. The uncertainty of the fringe distance due to thermal effect should be eliminated by an in-situ fringe distance calibration method which enables the calibration at any moment when the temperature changes. For separating the individual speckle as well as to measure the fringe distance, we propose a matrix camera based approach, cf. Fig. 2. We employ one camera (Basler piA640-210gm, maximum frame rate of 210 Hz, resolution of 648 px × 488 px, and a photoelement size of 7.4 μm × 7.4 μm) for each fringe system. Two interference fringe systems are generated by two pairs of laser beams with wavelengths 656 nm and 686 nm, respectively. The light of the fringe systems is scattered at a metallic specimen with a plane, rough surface, which is mounted on a moving stage. The stage is moved laterally in steps of 1 μm. Both cameras are synchronized by a function generator. In the experiments of this paper, the velocity of the moving stage, the sampling frame rates of the cameras and the used areas of the CCDs amount to 200 μm/s, 200 Hz, and 200 px × 200 px, respectively. The measurement uncertainties result from 10 random segments on the specimen surface. Please note that the applied velocity is several magnitudes lower than the common surface velocity inside of cutting lathes and limited due to the framerate of the CCDs. However, camera speeds will further increase in the coming years and can already be increased by reducing the pixel number. In our experiments, a high camera resolution is employed to investigate the feasibility of the approach. A second prob-
As shown in Fig. 1, the P-LDD sensor is based on a laser surface velocimeter using interference fringe patterns [15,16]. It is easily described for single scattering particles passing the measurement volume with a certain velocity. The amplitude of the scattered light is then modulated with the Doppler frequency fD . Determining the fringe distance d by a calibration and measuring the Doppler frequency with the Fast Fourier Transform (FFT), the particle velocity v perpendicular to the fringe system can be calculated by
𝑟(𝛼) = 𝑅 + Δ𝑟(𝛼) =
𝐾 ∑ 𝑘=1
2. Principle and setup
𝑧 = 𝜑 ⋅ 𝑠−1 ,
2𝑓 2 ⋅(𝑡∕𝑓s −𝑡a )2 D 𝑡2 w
with the Doppler frequency fD and a gaussian envelope with the amplitude A. Whereas fs is the sampling frequency, ta describes the arrival time of the particle and tw the temporal 1/e2 width of the envelope which results from the geometrical width of the measurement volume and the particle velocity. For scattered light signals from rough surfaces the speckle effect has to be considered. Former investigations showed, that each speckle exhibits the same Doppler frequency fD but random amplitudes Ak and arrival times 𝑡a𝑘 . Thus, the detector signal from K speckles reads [18]
reducing the measurement uncertainty. In order to realize a compact and robust sensor for in-situ measurements, a coherent fiber bundle is used to transmit the scattered light signal from the sensor head to the cameras. The measurement principle of the P-LDD sensor and the measurement uncertainty budget are described in Section 2. Since the speckle and fringe distance related measurement uncertainties are dominant, the camera based methods and the respective image processing for the reduction of these measurement uncertainties are proposed in Section 3. Finally, the coherent fiber bundle based scattered light signal transmission is presented in Section 4.
𝑣 = 𝑓D ⋅ 𝑑.
−
(3)
The mean radius R of the workpiece can be estimated by the mean velocity 𝑣̄ of the surface and the angular velocity 𝜔, which is constant and known. The angle dependent deviation Δr(𝛼) of the workpiece radius from the mean radius is obtained from the difference of the mean value 𝑧̄ and the angle resolved distance z(𝛼). While the rotation of the workpiece in the lathe enables the angle resolved radius measurement (2-D shape), an additional feed forward of the sensor along the rotational axis achieves an absolute 3-D shape measurement, cf. Fig. 1. The dominant contributors to the shape uncertainty 𝜎 r and their resulting shape uncertainties have been derived experimentally and are listed in Table 1. 365
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Fig. 2. Camera based P-LDD sensor, (a) The scheme of the setup, (b) The experimental setup. Table 1 Measurement uncertainty budget for a 3-D shape measurement containing the most significant contributors. The numerical values are derived experimentally. The uncertainties are calculated for a representative 3-D shape measurement with stepwise scanning along the height of the workpiece with 𝑀 = 104 (102 measurement points per revolution × 102 scanning steps along the y direction), the mean radius of the workpiece 𝑅 = 20 mm and a temperature variation of Δ𝑇 = 1 K. Averaging length XL is the displacement of the workpiece surface, while sampling a time series of the scattered light signal for one measurement point. Description
Value
Relative uncertainty of Doppler frequency Distance uncertainty Relative uncertainty of fringe distance Relative uncertainty of angular velocity Spindle vibration Resulting 3-D shape uncertainty
√ 𝜎𝑓D ∕𝑓D = 4 × 10−3 ∕ 𝑋𝐿 ∕mm √ 𝜎𝑧 = 0.2 μm∕ 𝑋𝐿 ∕mm 𝜎𝑑 ∕𝑑 = 1.5 × 10−5 × Δ𝑇 ∕𝐾 𝜎𝜔 ∕𝜔 = 10−6 Dependent on spectral properties of vibration, signal processing and separation techniques 𝜎r
Resulting shape uncertainty in μm 0.8 0.2 0.2 0.02 1
lem resulting from using cameras opposed to single pixel detectors is the increase in recorded data and signal processing time. For checking the feasibility of the approach this processing speed was not optimized. We used a Windows PC with an Intel Core i7 processor and MATLAB. The processing time is currently 2.9 s/mm surface length. For real-time measurements this has to be increased by parallelizing the signal processing on FPGAs and faster data transfer for instance using frame grabbers. 3. Camera based measurement uncertainty reduction 3.1. Doppler frequency Fig. 3. (a) is the sketch of the speckle movement in the time series of the images shown in (b).
A speckle separation technique can be realized by using a line camera or a matrix camera. It achieves a reduction of the measurement uncer√ tainty of the Doppler frequency by 𝑁s , whereas Ns is the number of independent speckle signals that are detected simultaneously. Since Ns results from the number of camera pixels Np and the ratio between pixel area Ap and average speckle area As : [ ] 𝐴p 𝑁s = 𝑁p ⋅ min ;1 , (6) 𝐴s
speckles moving in a single camera row over time 𝑆(𝑥̃ , 𝑦̃ = 𝑦̃0 , 𝑡) and (c) ′ the sheared signal 𝑆 (𝑥̃ + 𝑡∕ tan(𝜃), 𝑦̃ = 𝑦̃0 , 𝑡) = 𝑆(𝑥̃ , 𝑦̃ = 𝑦̃0 , 𝑡). Thus, the amplitude spectra of each speckle can be calculated by ′ FFT on the sheared signals 𝑆 . Averaging the amplitude spectra of each speckle yields a seemingly undisturbed gaussian function, from which the Doppler frequency is calculated, cf. Fig. 5 (red curve). Compared to the photo detector based sensor, 𝜎𝑓D ∕𝑓D is decreased √ by a factor of 18 to 3 × 10−4 ∕ 𝑋𝐿 ∕mm where 𝑋𝐿 = 1 mm. Fig. 9(a)(red curve) shows 𝜎𝑓D ∕𝑓D decreases with the increasing number of detected speckle signals Ns . Thus, the contribution of 𝜎𝑓D ∕𝑓D towards the shape uncertainty is reduced to 60 nm.
the measurement uncertainty can be reduced by adjusting the speckle size to the pixel size by optimizing the numerical aperture. In this paper a matrix camera is employed to obtain a higher pixel number, meaning more speckles can be detected and evaluated simultaneously. In this case the speckles move across the camera in accordance with the surface movement, cf. Fig. 3. In order to allow for an easy frequency estimation by the FFT, we shear the speckle signal by the angle 𝜃 in the x̃ , t -plane by means of the affine transformation ⎡𝑥̃ + 𝑡∕ tan(𝜃)⎤ ⎡1 ⎢ ⎥ = ⎢0 𝑡 ⎢ ⎥ ⎢ 1 ⎣ ⎦ ⎣0
1∕ tan(𝜃) 1 0
0⎤⎡𝑥̃ ⎤ 0⎥⎢ 𝑡 ⎥, ⎥⎢ ⎥ 1⎦⎣1⎦
3.2. In-situ fringe distance calibration In order to reduce 𝜎 r further, 𝜎 d /d has to be reduced. Formally, the fringe distance of Mach-Zehnder velocimeter based sensors was calibrated by utilizing a rotating disc with a pin hole [23]. This needs to be done before measurements, which means that during measurements in a working lathe a calibration cannot be achieved with this method. In order to eliminate the impact of thermal effects, an in-situ calibration is necessary. This can be achieved by imaging the fringe system onto the camera and evaluating the fringe distance by a spatial FFT. However,
(7)
exploiting the fact that each speckle has the same velocity on the camera surface. 𝜃 is calculated by auto correlating the camera signal 𝑆(𝑥̃ , 𝑦̃ = 𝑦̃0 , 𝑡). Fig. 4(a) shows the measured speckle signal 𝑆(𝑥̃ , 𝑦̃, 𝑡 = 𝑡0 ), (b) the 366
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Fig. 4. (a) is the speckle signal 𝑆(𝑥̃ , 𝑦̃, 𝑡 = 𝑡0 ) captured by the matrix camera, as an example the speckle signal in 100th camera row (red rectangle) is shown in (b). ′ (c) is the sheared speckle signal 𝑆 in which the motions of the speckles are adjusted to vertical direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
averaging over 𝑥̃ direction, which corresponds to a line camera signal: 𝑆r (𝑦̃, 𝑡) =
𝑁c ∑
𝑆(𝑥̃ , 𝑦̃, 𝑡),
(9)
𝑥̃ =1
with Nc the number of camera columns. The phase difference 𝜑𝑘 = 2𝜋𝜏𝑘 ∕𝑓D can now be calculated for each line camera pixel pair by 𝐶(𝑦̃, 𝜏) =
𝑁 ∑
𝑆r,1 (𝑦̃, 𝑡)𝑆r,2 (𝑦̃, 𝑡 + 𝜏).
(10)
𝑡=1
To verify the approach, distance measurements on the same specimen are performed with the photo detector and the camera based scattered light detection. We can show, that the distance uncertainty is reduced by about the order of one magnitude to 33 nm, cf. Fig. 9(b)(red curve). In order to investigate the effect of the pixel number towards the measurement uncertainty, we vary the pixel number by software binning. As is depicted in Fig. 9(b)(red curve), the uncertainty decreases with the square root of the pixel number as it does for the Doppler frequency. This decrease is limited by the speckle size, only. Meaning that choosing pixels smaller than the speckle size, does not yield an advantage. With the decreased measurement uncertainties for fD , d and z the 3-D shape uncertainty 𝜎 r is reduced to 100 nm.
Fig. 5. Determined amplitude spectrum with the matrix camera based approach (red curve), where the amplitude spectra are determined for each speckle signal. For comparison, the formerly resulting amplitude spectra with the photo detector (integration over all camera pixels, blue curve) are shown. The uncertainties are given for scattered light signals with an averaging length of 𝑋𝐿 = 1 mm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
due to the speckle effect no fringe system is apparent for a single shot image, cf. Fig. 6(a). On the other hand, as a characteristic, the speckles are bright at the positions of light fringes and dark at the position of dark fringes. While the object is moving, the speckles from the object surface are distributed to random positions. Thus, the speckle effect can be eliminated by averaging the signal images over time and thereby obtaining the fringe pattern If : 𝐼f =
𝑁 1 ∑ 𝑠(𝑥̃ , 𝑦̃, 𝑡). 𝑁 𝑡=1
4. Miniaturization using coherent fiber bundle In order to realize in-situ measurements inside of a lathe, a small and robust sensor is required. Formally, this was achieved by employing multi-mode fiber coupled photo detectors [11]. Using multi-mode fibers is not possible here because the spatially resolved speckle images are required while multi-mode fibers distort these images due to modemixing [24]. Therefore, the possibility to use a coherent fiber bundle (CFB) is investigated. CFBs are used in endoscopy to transmit images coherently, meaning that the relative distance and angle between each pixel remains constant [25]. In experiments, one CFB (Fujikura FIGH-50-1100N) with 50,000 fiber cores and 4 μm core diameter is employed to forward the scattered light of both fringe systems towards two cameras cf. Fig. 7. The cameras are used to detect the image of the scattered light field at the distal end of the CFB. For comparisons, the experiments with the CFB are done under the same experimental condition as the experiments without the CFB, meaning with direct camera detection. The signal images detected with the CFB and without the CFB are compared at the same position of the specimen surface, cf. Fig. 8. The similarity of the images is given by a correlation coefficient 0.93. This means the speckle separation and the fringe distance calibration techniques of Section 2 can be employed in principle and should work on the signal images transmitted by the CFB. In order to investigate the influence of the number of camera pixels, the pixel number is varied by using software binning and the relative Doppler frequency uncertainty 𝜎𝑓D ∕𝑓D , the distance uncertainty 𝜎 z as well as the relative fringe distance uncertainty 𝜎 d /d are calculated. As
(8)
This has been demonstrated experimentally, cf. Fig. 6(a). As shown in Fig. 6(b), by means of the FFT, an in-situ and real-time calibration of the fringe distance d is realized. Thus, the influence of the temperature variation √ on d can be eliminated. Applying this approach 𝜎𝑑 ∕𝑑 = 5.9 × 10−4 ∕ 𝑋𝐿 ∕mm is determined cf. Fig. 9(c)(red curve). With 𝑋𝐿 = 1 mm and the total sample number 𝑀 = 104 as in the example in Table 1, 𝜎𝑑 ∕𝑑 ≈ 10−6 results. Thus, the contribution of 𝜎 d /d towards the shape uncertainty is reduced to 20 nm. 3.3. Distance Lastly the distance uncertainty is investigated. Conventionally with photo detector based detection, the distance was estimated by evaluating the phase difference of two time series of scattered light signals employing the cross-correlation function. Evaluating the phase difference 𝜑k for each speckle signal separately and subsequent averaging would lead to a maximum uncertainty reduction. However, the phase difference results in a shift of the speckle in 𝑥̃ direction on the camera as well due to the speckle movement. Thus, calculating 𝜑k is computationally expensive, since a 2-D cross correlation is required. In order to reduce computing time, the 3-D camera signal is reduced to a 2-D signal by 367
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Fig. 6. (a) A single shot speckle image from a moving rough surface, (b) The fringe pattern If obtained by averaging the speckle images over time with 𝑋𝐿 = 1 mm, (c) The fringe frequency ff is evaluated by the FFT and averaged over 𝑦̃.
Fig. 7. Camera based P-LDD sensor with the coherent fiber bundle, (a) The scheme of the setup, (b) The experimental setup.
Fig. 8. (a) The signal image detected with CFB, (b) The signal image detected without CFB (direct camera detection). The correlation coefficient of the images is 0.93.
Fig. 9. Camera based measurement uncertainties with and without coherent fiber bundle scattered light detection. The relative Doppler frequency uncertainty (a) and distance uncertainty (b) decrease with square root of individually resolvable speckles. For a single photo detector with only one pixel, i.e. a single fiber core (green circle), the relative Doppler frequency uncertainties and the distance uncertainties are maximal and amount to 5.5 × 10−3 , 5.4 × 10−3 and 0.5 μm, 0.4 μm, relative to the experiments with CFB and without CFB. (c) As the result of the in-situ calibration, the relative uncertainty of the fringe distance is decreasing with the increase of averaging length. The reductions of the measurement uncertainties are consist with the theories (dotted lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Table 2 Experimental measurement uncertainties for 3-D shape measurements for 𝑋𝐿 = 1 mm. 𝜎𝑓D ∕𝑓D Photo Detector CCD CCD+CFB
𝜎𝑟 (𝜎𝑓D ∕𝑓D )
5.4 × 10 3 × 10−4 3.2 × 10−4 −3
0.8 μm 0.06 μm 0.064 μm
𝜎z 0.4 μm 0.033 μm 0.064 μm
𝜎 r (𝜎 z )
𝜎 d /d
0.4 μm 0.033 μm 0.064 μm
1.5 × 10 × Δ𝑇 ∕𝐾 5.9 × 10−4 1.8 × 10−3
shown in Fig. 9, the measurement uncertainties 𝜎𝑓D ∕𝑓D and 𝜎 z show the same behaviour regarding the pixel number as without the CFB and decrease with the number of individually resolved speckles, resulting a minimum uncertainty of 𝜎𝑓D ∕𝑓D = 3.2 × 10−4 and 𝜎𝑧 = 0.064 μm for an averaging length 𝑋𝐿 = 1 mm. Furthermore, the in-situ fringe distance calibration follows the √ same behaviour as without the CFB resulting in 𝜎𝑑 ∕𝑑 = 1.8 × 10−3 ∕ 𝑋𝐿 ∕mm. This shows, that CFBs are well suited to realize passive probes. As a result, with the coherent fiber bundle based signal transmission, the 3-D shape uncertainty 𝜎 r is reduced to 120 nm. However, the reductions of the measurement uncertainty are a little lower than with the direct camera detection, cf. Table 2. This has several reasons. First, the CFB uses a common cladding, thus optical cross talk occurs and distorts the transmitted images [26]. Second, each fiber core is multi-mode which results in an additional distortion. Lastly, the speckle size is not matched to the core diameter (diameter core: 4 μm, diameter speckle: 12 μm) due to the limited numerical aperture. Thus the number of independent signals is decreased. For visual comparisons, the measurement uncertainties with different detection approaches are listed in Table 2.
−5
𝜎 r (𝜎 d /d)
𝑁s,𝑓D
Ns, z
𝜎r
0.2 μm 0.02 μm 0.02 μm
1 3.9 × 104 3 × 104
1 32 28
1 μm 0.1 μm 0.12 μm
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5. Conclusions A novel camera based speckle separation method was presented for the speckle noise reduction and, thus, the systematic uncertainty reductions of Doppler frequency based velocity and distance sensors. Experiments were performed for validating the proposed camera based approach. By evaluating the Doppler frequency of each speckle separately, the Doppler frequency uncertainty decreases with the square root of independent speckles. Thus, a reduction by a factor of 18 for the relative Doppler frequency uncertainty was achieved in comparison to a photo detector based signal detection. Similarly, the distance uncertainty was reduced by the order of one magnitude as well. Furthermore, employing cameras allows for a novel in-situ calibration of the fringe distance, which can eliminate the impact of thermal effect on the measurement uncertainty. Thus, an absolute shape uncertainties of about 100 nm can be achieved for an in-situ shape measurement, for instance in turning lathes. In comparison, CMMs allow 3-D shape uncertainties of around 1 μm and do not support an in-situ measurement. A coherent fiber bundle has been employed in P-LDD sensor for the first time for signal transmission to demonstrate the capability of realizing a robust, all passive sensor head. As an outlook, the possibility to employ single mode fiber bundles will be investigated. The influence of coolants on the workpiece surface towards the measurement uncertainty will be investigated for applying the sensor to the near-dry machining. A high-speed, online signal processing using FPGAs is to be setup for evaluating the measurement data in real time. Furthermore, other applications such as turbo machinery or blade oscillation, mechanical vibration, car tire, wind turbine rotor, gear, and spindle play measurements can be achieved by using the P-LDD sensor as well. As a broad prospect, the camera based speckle separation technique and the in-situ fringe distance calibration method also can be applied in other optical sensors that are limited by the speckle effect or the thermal effect, and do not use matrix cameras yet, such as other interferometers or triangulation sensors.
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