1. Introduction

Thin metal foils with typical thicknesses of 5 µm to several 100 µm play a key role in modern electronics, e.g., in the design of lithium-ion batteries and capacitors or in photovoltaic solar modules. Such foils are produced in specialized rolling mills equipped with inline measurement gauges. Figure 1 schematically depicts such a rolling mill and the rolling process [1]. The strip to be rolled (1) is transported with a defined strip tension from the decoiler (2) to the coiler (4). In the rolling gap formed by the work rolls (6), the strip thickness $d_1$ is reduced to the target thickness $d_2$. A specific rolling force P is set for this purpose. Because of the scattering of the thickness $d_1$, it is important to very accurately detect the strip thickness both before (3) and after rolling (3) and to trigger a fast response to dynamically control the rolling force $P$. Oil is partially removed by the strip drying systems (7) located between the work rolls and the thickness gauges. Typically, the final thickness is produced in several rolling passes.

 

Fig. 1. Schematic drawing of a rolling process; 1: metal strip; 2: decoiler; 3: thickness gauge; 4: coiler; 5: rolling direction; 6: work roll; 7: strip drying system; P: rolling force; d₁: initial thickness; d₂: target thickness.

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Increasingly tighter production tolerances necessitate the improvement of measurement accuracy into the sub-micrometer range for these processes. The hostile environment in a rolling mill and the need for unfailing reliability call for simple and robust setups. State-of-the-art inline thickness gauges are based on various measurement techniques [1]. Tactile gauges are the classic system for this purpose and have been used for decades in the metal rolling process. Wear and tear on sample and probe, however, are becoming increasingly problematic as foil thicknesses decrease [1]. As a consequence, non-contact measuring equipment is gaining more and more relevance in the field. Though very robust, pulsed eddy current measurements can only be applied for non-ferrous metals and are limited to a minimum strip thickness of approximately 20 µm [2]. X-ray absorption measurements are highly suitable for measuring the thickness of thin strips but require a reliable model of the alloy composition [3]. In addition, this technology necessitates enhanced work safety measures that in practice often inhibit its applicability. Optical methods like chromatic confocal sensors and triangulation sensors promise a good compromise, namely, contact-free measurement without the need for special safety measures or for consideration of the alloy composition. Nevertheless, the achievable uncertainty of triangulation-based gauges today is limited to 1 µm [1], while the nominal accuracy of chromatic confocal sensors, in the order of 0.5 µm, severely deteriorates due to the large thermal drift of about 2 µm/ K [4]. Furthermore, remnant oil films are known to lead to a systematic error of the strip thickness when optical techniques are used.

At first glance, it might also seem straightforward to apply interferometry for metal strip thickness measurements when targeting sub-micron uncertainties. The measurement environment, however, is extremely challenging. The position of the metal strip is not constant within the measuring range of the distance sensor, and the strip oscillates perpendicularly to the strip surface with frequencies of up to 250 Hz and an amplitude of several 100 µm. This all takes place at a transport speed of hundreds of meters per minute. Continuous measurement, as required for classic counting interferometry, is therefore not possible. A robust and reliable measurement method must further be able to deal with beam interruptions. Interferometric techniques capable of measuring absolute distances without the need for measurement continuity do exist [5–10]. In particular, multi-wavelength interferometry (MWLI) can be compactly and cost-efficiently realized using diode lasers and still maintain the high temporal resolution required for this measurement [11–13]. This study reports on the realization, verification and full uncertainty model of a thickness gauge concept that combines a triangulation and an MWLI measurement to measure the top and bottom sides of the strip [14]. This concept makes use of a modulation interferometry approach first suggested by Meiners-Hagen et al. in 2004 [15]. The triangulation sensor provides the coarse pre-value required by the method to achieve the necessary non-ambiguity range, allowing a measuring range of several millimeters to be realized while detecting features below 100 nm. Sections 2 and 3 describe the gauge’s measurement principle and its technical realization. Verification measurements against tactile measurements as well as real shop floor measurements are presented in section 4. Section 5 provides an analysis of the measurement uncertainty of the strip thickness measurement.

2. Measuring principle

When a distance is measured by two or more interferometers of different vacuum wavelengths $\lambda _i$ ($i = 1, 2, {\ldots }$), the additional information can be used to extend the non-ambiguity range of a single interferometer by several orders of magnitude [11,16,17]. In the case of two-wavelength interferometry, this new non-ambiguity range is given in air by half the synthetic wavelength

The geometric length $l$ can be reconstructed from the synthetic phase

The integer order $N_{\rm s}$ can be determined from a coarse pre-value $l_{\mathrm {pre}}$ [12] by

A simultaneous measurement with several wavelengths requires that the signals of each wavelength be separated for the analysis. If heterodyne detection schemes are used, well-separated carrier frequencies can be assigned to each wavelength. Such wavelength-dependent beat nodes can be generated by acousto- [18] or electro-optical [13,19,20] modulation. A direct modulation of the laser frequency itself allows a particularly cost-efficient and compact phase detection by lock-in processing. Assuming a laser source whose frequency $\nu _0$ is sinusoidally modulated with a modulation frequency $f_{\rm m}$ and a modulation index $\Delta \nu$, the frequency $\nu$ of the emitted light is given by $\nu =\nu _0+\Delta \nu \sin {\left (2\pi f_{\rm m}t\right )}$. For moderate modulation frequencies, the intensity detected by the receiver of an interferometer of path length difference $l$ is given by [21]

The free parameter in Eq. (7), the modulation index $\Delta \nu$, can be chosen such that the condition $J_2\left (\Delta \phi \right )\ =\ J_3\left (\Delta \phi \right )$ is fulfilled. The optical phase can then by retrieved according to [15]:

It should finally be noted that this so-called 2f-3f-measurement principle is, generally, notoriously sensitive to nonlinear effects [28]. To contain these effects, several measures were taken. Since the measurement range for the thickness sensor was well-defined within a few millimeters, the diode working point could be carefully selected in the linear response regime, and the modulation could be kept as small as possible. The measurement parameters could hence be optimized for a clean sinusoidal frequency modulation. In addition, a linear interferometer design was chosen that minimized the number of internal interfaces in the beam path (cf. Sec. 3.1). Remnant non-linearities were finally corrected by the implemented Heydemann correction [29,30].

3. Realization of the thickness gauge

3.1 Gauge setup and data processing

Figure 2 gives an overview of the design of the thickness gauge. Its main component is a C-shaped rigid invar frame (A) with two optical sensors (C) that measure the distance to the top and bottom surfaces of the test article, e.g., a running metal foil (B). The frame has a total height of 120 mm. It is temperature-controlled, maintaining a stability of 0.5 K. It can be moved horizontally and vertically via two linear slides with stepper motors. This allows the scanning of stationary targets, specifically a tray with four gauge blocks used for adjustments and as thickness reference. A separate control unit contains the light sources, beam splitters, and most of the electronics for detection and data reduction. It is connected to the frame via optical single mode fibers and an ethernet link. Two tunable distributed feedback (DFB) laser diodes (D1 and D2) with nominal wavelengths of 780 nm and 795 nm and nominal linewidth (FWHM) of 2 MHz are used as light sources (eagleyard Photonics EYP-DFB-0780/0795-00080-1500-BFW01-0002). Their wavelengths can be tuned without mode hops in a range of at least 1 nm by varying the diode temperature and current. Each diode is enclosed in a butterfly housing with fiber output that also includes a thermoelectric cooler and a thermistor for temperature control as well as an optical isolator (2) to prevent small mode hops induced by back-reflected light. The diode temperatures are kept constant with a short-term standard deviation of approx. 70 µK. The light from the two diodes is first combined into a single fiber and then split into four fiber outputs (3), each passing its light through a Y-coupler (ports 5.1 and 5.2) to a focusing lens (7). A silicon positive intrinsic negative (PIN) photodiode (bandwidth approx. 100 MHz) placed at the third port of the Y-coupler (5.3) detects the light that comes back from the interferometer.

 

Fig. 2. A: thickness gauge; B: metal strip; C: optical sensor. Detail C: schematic of the design of the interferometric distance sensor used in the thickness gauge. 1: DFB laser diodes D1 and D2; 2: optical isolator; 3: 2-4 fiber splitter; 4: single mode fiber; 5:Y-fiber splitter; 6: line detector; 7: focusing lens; 8: image lens; 9: controller; 10: triangulation readout FPGA, 11: metal strip; 12: photodiode; 13: clean air ventilation. Figure modified from [31] under CC BY-SA 4.0.

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Figure 3 shows the optical setup of a distance sensor. The fiber coming from the control unit forms at its exit an interferometer (1) as approximately 4 ${\%}$ of the light is reflected at the end of the fiber. The remaining light with a typical optical power of approx. 1 mW is focused onto the target via a lens. It is then partially reflected by the target and coupled back into the fiber by the focusing lens. As only a small fraction of the light impinging on the target re-enters the fiber, a good interference contrast can be achieved, especially on technical surfaces. To measure the distance via triangulation, a second lens (2) picks up the light diffusely scattered by the target at an angle under approx. 30° and projects it onto a linear image sensor (4) with 1024 pixels and a pixel readout clock of 40 MHz. This arrangement ensures that coarse and fine measurements are taken from the same spot on the strip. The sensors are mounted to the C-frame of the thickness gauge via three adjustable support points. Their angular orientation is ensured by parallel keys. For the mutual alignment of the horizontal position of the sensors, a very thin semi-transparent matt screen is placed within the system’s thickness measuring range. A camera with a macro lens detects the laser spots (diameter 50 - 60 µm) on the matt screen at different vertical positions. The sensors are then mutually aligned until optimal overlap of the laser spots is attained throughout the measuring range. Once this is done, the matt screen is removed and the interference signal from the two sensors maximized.

 

Fig. 3. Combination of fiber-based interferometer and triangulation sensor (left: schematic, right: photograph). 1: collimation lens; 2: triangulation imaging lens with wavelength filter, 3: mirror, 4: imaging sensor, 5: fiber tip, 6: fiber connector.

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The signal processing system is summarized in Fig. 4. For the analysis, a field programmable gate array (FPGA) continuously reads the image data from the triangulation sensor and calculates the distance to the target from the center of gravity of the light spot by applying a linearization polynomial permanently stored in flash memory. The exposure of the image sensor can be controlled via an electronic shutter in a range from 1.6 µs to 1 ms. This allows the exposure to be adapted to the properties of the target surface, enabling measurements on rough surfaces as well as on highly reflective surfaces like those of gauge blocks. The triangulation distance readings are transmitted to the control unit. Each reading contains a timestamp synchronized to a corresponding clock in the control unit with an absolute error of approx. $\pm$0.5 µs. The electronics in the control unit consist of a commercial system on module (SoM) complete with processing unit, RAM and flash memory, and a custom designed board for data acquisition and laser diode control. The processor used on the SoM is a Xilinx Zynq MPSoC, which combines an FPGA with several hard processor cores. The laser diodes are modulated with sine waves of 1.5 MHz and 1.2 MHz. The modulation waveforms are digitally generated in the FPGA with an output rate of 125 MHz and then amplified and superimposed onto the DC diode current via a bias-tee circuit. The modulation amplitudes can either be adjusted in the FPGA or by varying the gains of the amplifiers. The control unit further contains four photodiode receivers. Following transimpedance amplification, the voltage signal passes through a band pass filter that removes frequencies above approx. 40 MHz to avoid aliasing effects during digitization and also dampens those below approx. 2 MHz in order to partially suppress the strong signals from the modulation frequencies. Finally, the signal is amplified by a programmable gain factor and digitized with a sample rate of 125 MHz for further processing in the FPGA.

 

Fig. 4. Schematic of the simplified signal processing system.

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To measure the amplitude of the $i$-th harmonic of modulation frequency $f_{\rm m}$, the digitized input signal is multiplied (’mixed’) with a sine wave of frequency $i \times f_{\rm m}$ and a constant phase offset to the original modulation waveform. This effectively shifts the power spectrum in frequency space by $i\times f_{\rm m}$ down to around 0 Hz, where it can easily be isolated by lowpass filtering. The FPGA implements the lowpass filter in three stages using a combination of cascaded integrator comb (CIC) and finite impulse response (FIR) filters and reducing the sample rate after each stage. The final output rate is 100 kHz with a bandwidth limit of 35 kHz. The FPGA contains sixteen of these mixer/lowpass filter units, all of which are synchronized and work in parallel. Working with a single-channel lock-in detection scheme, the phase offset $\Delta \varphi$ between modulated signal and reference wave has to be individually minimized for each channel. Therefore, auxiliary measurements are performed on an oscillating target, tuning the phase offset so that the peak signal amplitude is maximized. Each sample set of sixteen lock-in amplitudes is assigned a timestamp from the same high-resolution clock to which the remote triangulation readout units are synchronized. Taking into account latencies in the FPGA and the exposure times of the triangulation sensors, all three data sets (top and bottom triangulation distances and interferometer data) are combined. The final processing step in the FPGA involves performing a Heydemann correction [29,30] that depends on the triangulation length pre-value on each pair of lock-in amplitudes and then calculating the corrected phase and amplitude values [31].

Phase unwrap and thickness calculation are performed on one of the dedicated real-time processors. First, the noise in the triangulation distance readings is reduced by removing obvious outliers and applying a moving average filter (window size $\pm 30$ samples). Each interferometric distance reading $l_{\rm I}^{\alpha }$ ($\alpha$ = top (t) or bottom (b)) is then calculated according to Eq. (4) by

Subtracting the sum of the two interferometric distances $l_{\rm I}^{\rm t}$ and $l_{\rm I}^{\rm b}$ from an adjustment constant $S$ yields the thickness $d$

The final processing step removes obvious outliers and applies a lowpass filter.

3.2 Adjustment procedure

The analysis described in Sec. 3.1 uses several parameters that must be determined beforehand. To that end, the gauge is equipped with a tray having four gauge blocks (0.3 mm, 0.5 mm, 0.7 mm, and 1.0 mm) of accuracy class K that can be scanned by moving the C-frame horizontally and vertically in an automated adjustment procedure consisting of three steps. First, a slow vertical scan is performed over the whole measuring range on one of the gauge blocks at a fixed horizontal position to set up the Heydemann correction in the FPGA. Using the corresponding triangulation lengths as independent reference, the raw lock-in amplitudes are divided into blocks of 10 µm travel with an overlap of 50 ${\%}$. For each block the local correction parameters are determined by fitting an ellipse to the data points. They are then interpolated with polynomials which are used to initialize the FPGA lookup tables.

Second, the offset parameter $G_0^{\alpha }$ is determined for each sensor $\alpha$ = top (t) or bottom (b). A series of horizontal scans on the gauge block is performed, varying the vertical gauge block positions in steps of 0.5 mm. During each scan on a fixed gauge block position, the distance between the sensor and the gauge block surface varies in addition, e.g., due to vibrations or imperfect mechanical guides. For each point $i$ of the scan, the fractional phase contribution according to the expression in Eq. (11) is first subtracted to obtain $\widetilde {N}_{{\rm s,}i}^{\alpha } = N_{{\rm s,}i}^{\alpha }+G_{0}^{\alpha }$ as an estimator for the sum of the integer order $N_{{\rm s,}i}^{\alpha }$ and the (constant) offset $G_{0}^{\alpha }$

As an initial approximation for the synthetic wavelength $\Lambda _{\rm s}^{\rm est}$, default values have been determined for the laser diode parameters by a wavemeter. To identify the common constant offset parameter $G_{0}^{\alpha }$, the impact of the noisy data from the triangulation length measurement value $l^{\alpha }_{{\rm T},i}$ has to be mitigated. To do so, an estimate integer value $K_i^{\alpha }$ is assigned to each point $i$ by the following recursion:

The final values of $G_0^{\alpha }$ are the averages over all scans $G_0^{\alpha }= \left \langle G_{0,m}^{\alpha } \right \rangle_m$.

The third adjustment step is a horizontal scan over all four gauge blocks. For each gauge block with known thickness ${d}_i^{\rm cal}$, a local calibration constant $S_i$ is calculated from the interferometric distances $l_{i}^{\rm t}$ and $l_{i}^{\rm b}$. The mean value is then used as the ’spacing’ $S$:

It should be noted that the algorithm leading to $G_0^{\alpha }$ determines only the fractional component of the offset between the full length detected by the triangular sensor and the interference sensor. Deviations by full integer orders are mapped into the magnitude of $S$ without loss of accuracy.

Using the parameters thus determined, the experimental thicknesses $d^{\rm exp}_{i}$ can be determined from the phase measurements from Eq. (10)–(12) for each gauge block $i$ ($i$=1,2,3,4). If one assumes all other inputs (phase measurement, fit parameters) to be correct, the deviations from the calibrated gauge block thicknesses $d^{\rm cal}_{i}$ can be attributed to the deviation of the refined synthetic wavelength $\Lambda _{\rm s}^{\rm ref}$ from the estimated value $\Lambda _{\rm s}^{\rm est}$ by

A refined estimate of the synthetic wavelength $\Lambda _{\rm s}^{\rm ref}$ can hence be derived from a linear regression over the data pairs $\left (d^{\rm exp}_i,d^{\rm exp}_i -d^{\rm cal}_{i}\right )$. As a change in $\Lambda _{\rm s}$ affects the parameters $G_0^{\alpha }$ and $S$, these are recalculated using the measurements from the second and third step discussed above. The condition $\Delta S = \max {(\left |S_i-S\right |)} < {50}\;\textrm{nm}$ can be used to perform a sanity check for the adjustment.

4. Measurement results

The performance of the thickness gauge was studied using a simulator to recreate the highly dynamic measurement conditions of a rolling mill. The two ends of a 4.5 m long and approximately 638 µm thick stainless-steel strip were first welded together. This ’endless’ strip was then mounted on two rolls capable of driving it at speeds of between 40 and 200 m/min. The strip oscillated approximately $\pm {0.25}\;\textrm{mm}$ in the vertical direction. The surface of this demo strip had a roughness $R_{\rm a}$ of 0.25 µm and $R_{\rm max}$ of 1.6 µm. The optical thickness gauge and a tactile reference gauge were positioned between the two rolls and aligned to take measurements at approximately the same distance from the strip edge. Figure 5 depicts exemplary data for a strip speed of 160 m/min. The thickness profiles are averaged over 25 revolutions using an autocorrelation algorithm. The qualitative features from the optical (Fig. 5(a)) and tactile (Fig. 5(b)) gauges agree well and correspond to a resolution of better than 100 nm. The noise in the tactile data is considerably greater than that seen in the optical measurement. The difference between the two averaged curves shows an overall offset. The position of the welding joint is clearly identifiable as disruption in the tactile data, while the optical sensor seems insensitive to the interface. This can be understood from the strip morphology. The front faces of the strip are welded together with an imperfect spatial match, leading to a step-like interface. The stylus of the tactile sensor was retracted due to this mechanical discontinuity. But the optical gauge is only sensitive to the thickness, not the vertical position of the strip within the eligible measurement range. As the thickness around the joint changes only within micrometers, the optical signal change at the joint is far less pronounced. This sensitivity to this small change is further reduced by the 10 ms integration over the moving strip. The observed deviation of interferometric and tactile sensor is well within the uncertainty of the tactile reference system, which is specified by the manufacturer as 1 µm (see also Sec. 5).

 

Fig. 5. Comparison between interferometric and tactile strip thickness gauge. Blue lines indicate the average profile, the pink area indicates the data scatter. a) Strip thickness measured by the interferometric thickness gauge at a strip velocity of 160 m/min. b) Thickness measured by a tactile gauge. c) Deviation between the two measurements. Figure modified from [31] under CC BY-SA 4.0.

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The deviations and observed ranges for six different speeds between 40 and 200 m/min are summarized in Fig. 6. The mean values of the strip thickness agree within $\Delta d_{\rm max} = {0.30}\;{\mathrm{\mu}\textrm{m} }$ for both techniques, the magnitude varying without a clear correlation to the speed. The interpretation of this deviation is not straightforward. For one thing, the measurement positions did not coincide perfectly. Also, the techniques are sensitive to different physical quantities. On a rough surface, a finite tactile probe will not be able to map the full depth, while the optical signal integrates over reflections from the complete surface. Thermal drifts within the two gauges can also not be ruled out. The range $R$ of the strip thickness is systematically larger for the tactile than for the interferometric gauge (cf. upper part of Fig. 6), which is attributed to the different sensitivities of contact and non-contact probe to surface protrusions and morphology oscillations.

 

Fig. 6. Comparison of tactile and interferometer measurements at different strip speeds. The upper graph indicates the mean range of the thickness values measured at identical spots for the 25 strip revolutions. The lower graph depicts the mean deviation between the thickness measurements of the interferometric and tactile gauges. Figure modified from [31] under CC BY-SA 4.0.

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Figure 7(b) shows the optical thickness gauge mounted in a real production environment in a rolling mill in Germany at a strip velocity of 200 m/min. The data set in Fig. 7(a) covers the first three minutes of the feedback-controlled milling process. The thickness gauge measurement value provided the control signal. After closing the control loop, a transient oscillation of the control system could be observed in the thickness signal. After approx. 60 s, the milling control system found the optimum working point. During the subsequent forty minutes of continuous production, the remnant deviation remained in the order of $\pm {0.5}\;{\mathrm{\mu}\textrm{m} }$ from the target thickness. The combination of triangulation, high-resolution MWLI, and FPGA-based real-time signal processing can thus provide a feedback signal capable of controlling the rolling process. Figure 7(c) shows a microscopy image of a sample of such a thin foil rolled in a different facility. The surface contains remnants of the rolling oil applied during the process. After the strip passes the drying system (Fig. 7), the oil no longer wets the entire surface, but forms droplets or accumulates in indentations. As shown in detail Sec. 5, the impact of the remnant oil on the optical thickness measurement remains limited. An initial first conclusion of this real-life verification is that the gauge design, the compact fiber-based interferometer design and the spatial separation of the optical source and electronic control from the production line provide sufficient robustness to withstand the challenges of this heavy industry environment over process-relevant time scales.

 

Fig. 7. Measurement under real-life production conditions. a) Thickness data over the first three minutes of production at a strip velocity of 200 m/min with initial transient oscillation of the milling control loop. b) Optical thickness gauge positioned in the rolling mill. c) Partial oil coverage of a different sample pictured by optical microscopy.

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5. Measurement uncertainty

The thickness gauge is intended for production process control applications in highly complex environments. In the following, a measurement uncertainty budget according to the Guide to the Expression of Uncertainty in Measurement (GUM) [32] is derived for typical application scenario parameters. Metal strips from stainless steel, aluminum and copper with a final thickness of 40 µm and a surface roughness $R_{\rm a}$ of between 0.05 and 0.30 µm are discussed. A coarse estimate of the strip thickness is always available from the parameter settings of the rolling mill. The strip is centered between both sensors at a distance of 30 mm to each, a position that may change by $\pm {1}\;\textrm{mm}$. Strip velocities of up to 200 m/min are considered. The remnant oil coverage downstream of the strip drying system is assumed below 500 mg/m², statistically distributed across the surface and covering up to 50 ${\%}$ of the surface. Clean air is constantly blown into the measurement volume to mitigate environmental changes and to avoid fumes (see Fig. 2 and [33]). In the service times in between two rolling passes, typically every hour, the gauge is readjusted using four calibrated reference steel gauge blocks at room temperature (with an uncertainty of 0.5 K). Under this condition, the following ambient condition parameter window can be assumed: ambient temperature $T = (20.0\pm 2.0)$ °C, ambient pressure $p = (1013.3\pm 5.0)$ hPa, relative humidity $rh = (80\pm 20)$ ${\%}$rh, and a carbon dioxide content of $x_c = (450\pm 450)$ ppm. The strip temperature does not deviate more than $\Delta T_{\rm strip} = {25}\;\textrm{K}$ from $T$.

The thickness gauge must be adjusted before each rolling pass by means of the four calibrated gauge blocks used as reference parts. When using accuracy class K gauge blocks, a specified standard measurement uncertainty (coverage factor $k$ = 1) of $u_{\rm ref} = {0.015}\;{\mathrm{\mu}\textrm{m} }$ can be assumed. The alignment procedure described in Sec. 3 ensures that the horizontal sensor displacement can be considered well below a tenth of the width of the optical beam and hence negligible. The angular orientation of sample surfaces to the measuring beam was aligned with a high-resolution spirit level (scale division: 0.2 mm/m = approx. 0.01°), leading to a residual deviation of $\Delta l_{\mathrm {mis}}=l (1/\cos {{0.01}^{\circ}}-1) \approx 1.5 \times 10^{-8} l$, which can as such likewise be neglected. The thermal expansion coefficient $\alpha _{T}^{\rm gb}$ of the gauge blocks is given by $\left (10.8 \pm 0.5 \right ) \times 10^{-6} /{K}$. With a maximum thickness of 1.0 mm, this corresponds to a gauge block temperature-induced uncertainty of $u_{\Delta T, {\rm gb}} = {0.0054}\;{\mathrm{\mu}\textrm{m} }$. The remnant systematic deviation of the measurement system after the adjustment, i.e., the bias standard uncertainty $u_{\rm bias}$, was studied by means of a measurement series with different calibrated gauge blocks. The observed deviation values were smaller than 0.060 µm. Assuming a rectangular distribution, the bias standard uncertainty can thus be estimated to be $u_{\rm bias}={0.06}\;{\mathrm{\mu}\textrm{m} }/\sqrt {3}={0.0346}\;{\mathrm{\mu}\textrm{m} }$. Similarly, the experimental instrumental standard deviation of the thickness gauge for a collaborative fixed sample can be estimated as $u_{\sigma, {\rm gb}}={0.032}\;{\mathrm{\mu}\textrm{m} }$.

The DFB laser sources are housed in the control unit and are thus spatially well separated from the harsh environment. The temperature sensor in the diode housing indicates a standard deviation for the temperature of below 70 µK. The stability of the current can be estimated at well below 0.10 mA. For the latter, a rectangular distribution is to be assumed, resulting in standard uncertainties of 70 µK and 0.057 74 mA. Assuming sensitivities of $\frac {\partial \lambda }{\partial T}\approx {0.06}\;\textrm{nm} / \textrm{K}$ and $\frac {\partial \lambda }{\partial I}\approx {0.003}\;\textrm{nm} / \textrm{mA}$, this corresponds to vacuum wavelength uncertainties $u_T(\lambda ) = 4.2\times 10^{-6}{}\;\textrm{nm}$ and $u_I(\lambda ) = 1.732 \times 10^{-4} {}\;\textrm{nm}$. Multiplied by the ratio of $\Lambda _{\rm s} /\lambda \approx 50$, this corresponds to an uncertainty of the synthetic wavelength $\Lambda _{\rm s}$ of 0.008 663 nm, or a relative uncertainty on the scale of $2.0716 \times 10^{-7}$ in between two adjustments of the synthetic wavelengths. The uncertainty of the measured thickness $d$ due to the short-term stability of the vacuum wavelengths of the DFB diodes can be estimated as $u_\lambda =2.0716\times {10}^{-7}\times {0.120}\;\textrm{m}={0.0249}\;{\mathrm{\mu}\textrm{m} }$.

For the measurement of the thickness, the synthetic wavelength in the traversed air enters the uncertainty analysis. In the uncertainty discussion, we assume a rectangular distribution for all environmental parameters. Ciddor’s equation [38] is used as a model for the phase index of refraction in moist air. Equation (2) describes the two-color group index of refraction $n_{\rm g}$. The uncertainty of $n_{\rm g}$ for the top (t) and bottom (b) sensor with path length $l^{\rm t,b}$ can be calculated by

A complete assessment of the measurement uncertainty of the rolling process requires the inclusion of further uncertainty contributions. Ambient light effects are suppressed by the modulation of the signal in the case of interferometric measurement, and by an optical filter in front of the triangulation sensor. To study the equipment variation uncertainty $u_{\rm EV}$ associated with the dynamical aspects of the measurement conditions, the data shown in Fig. 6 is used. The maximum observed mean range $\bar {\bar {R}}$ for all investigated strip velocities was smaller than 0.10 µm. According to [39], this observation can be used to derive the standard uncertainty of $u_{\rm EV}=\bar {\bar {R}}/3.735 ={0.0268}\;{\mathrm{\mu}\textrm{m} }$. The full free width $S$ of the C-frame (cf. Figure 2) of approx. 120 mm enters the analysis in accordance with Eq. (12). Using a thermal expansion coefficient of invar of $\alpha _{T}^{\rm C} = 0.5\times 10^{-6}\textrm{K}^{-1}$, the temperature uncertainty ${0.5}\;\textrm{K}/\sqrt {3}$ translates into an uncertainty of the strip thickness in the order of $u_{\Delta T, S} = {0.0520}\;{\mathrm{\mu}\textrm{m} }$. The possible temperature deviation $\Delta T_{\rm strip}$ implies a thickness standard uncertainty component of $u_{\Delta T, {\rm strip}} = {0.0058}\;{\mathrm{\mu}\textrm{m} }$ for a thermal expansion coefficient for steel of $10\times 10^{-6}/\textrm{K}^{-1}$.

The strip surface is not perfectly flat and clean after passing the work roll. The final average roughness $R_{\rm a}$ is usually determined in a rolling mill. It is related to the standard deviation $\sigma _{\rm Ra}$ of the Gaussian distribution $\rho (z)$ of the surface height deviations from the average height by $\sigma _{R_{\rm a}} \approx R_{\rm a} \sqrt {0.5 \pi }$ [40]. The surface roughness $R_{\rm a}$ of both surfaces contributes to the achievable uncertainty of the strip thickness measurement with $u_{R_{\rm a}} = \sqrt {2}\sigma _{\rm Ra}=\sqrt {\pi }R_{\rm a}$. The remnant oil on the surface (cf. Figure 7(c)) interacts with the probing light. To model this effect, the oil is assumed to fill the indentations from bottom to top, forming homogeneous ‘lakes’ with flat surfaces. Optical constants for biodiesel are used here as approximations for the optical constants of the rolling oil (Table 1). The distribution of the oil film thicknesses $t$ corresponds to the surface roughness distribution $\rho (z)$. The peak value $\mu$ depends on the oil coverage $\Theta$ by $\int _0^{\infty }\rho (\mu,\sigma _{R_{\rm a}}, t)dt = \Theta$. According to geometric optics, the optical path length increases by the difference of the group indices of refraction of oil and air. As such, the geometric model predicts a linear increase of the deviation with the film thickness (Fig. 8(a)). The air/metal and air/oil/metal interfaces, however, must be considered more thoroughly. When the optical wave is reflected at an air/metal interface, its phase changes in line with $\delta \phi _{\rm P}={\rm atan2}(-2n_{\rm air}\kappa _{\rm metal},n_{\rm air}^2-n_{\rm metal}^2-\kappa _{\rm metal}^2)$ [41], with $n$ denoting the refractive index and $\kappa$ the extinction coefficient. The phase jump $\delta \phi _{\mathrm {P}}$ implies a substrate-dependent length error $\delta l_{\rm P}$ of the measurement. Table 1 summarizes the associated length shifts for the different metals discussed here. Furthermore, fractions of the beam are reflected multiple times at the interfaces of the thin oil film. All these partial waves of different path lengths interfere, superposing to a reflected wavefront with a phase shift with respect to the directly reflected beam. Figure 8(a) shows the predictions of the multi-reflection interference (MRI) model calculated according to [42] using the optical constants given in Table 1. Sign and magnitude of the deviation $\delta l_{\rm oil}$ oscillate for the MWLI sensor. The probe beam averages over the surface. Moreover, it must be taken into account that the adjustment was performed using stainless steel gauge blocks. If the metal strip is made of another material, the different phase shift for the one-time reflection must be accordingly corrected for. The detected mean deviation from the correct value can be derived as

Figure 8(b) summarizes the complex result for the MRI model for a copper thin strip in dependence of oil coverage $\Theta$ and average surface roughness $R_{\rm a}$. The maximum deviations for the three metals discussed here are compiled in Fig. 8(c) together with the corresponding result for the linear model. While the latter leads to a linear increase of the systematic thickness deviation with coverage up to ${0.1}\;{\mathrm{\mu}\textrm{m} }$, the MRI model substantially reduces the maximum deviation ${\rm max}\left (\left |\delta \bar {l}_{\rm oil}\right |\right )$ to below 0.020 µm for remnant oil coverage below half coverage ($\Theta < 0.5$). The corresponding uncertainty contribution for both surfaces hence does not exceed $u_{\rm oil} = {0.0400}\;{\mathrm{\mu}\textrm{m} }$.

 

Fig. 8. Impact of the remnant oil film. a) Length shift $\delta l_{\rm oil}$ for multi-reflection interference (MRI) and geometric model and probability density $\rho _{\Theta }$ for film thickness $t$ at coverage $\Theta$. b) Average length shift $\delta \bar {l}_{\rm oil}$ as function of surface roughness $R_{\rm a}$ and coverage $\Theta$ for copper. c) Maximum length shift ${\rm max}\left (\left |\delta \bar {l}_{\rm oil}\right |\right )$ plotted for the different strip metals and the geometric model.

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Table 1. Optical constants $n$ and $\kappa$ for typical thin strip metals and biodiesel (as an approximation for rolling oil) and length shift $\delta l_{\rm P}^{\rm me}$ at the air/metal interface reflection.

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For completeness, the resolution of the gauge system is better than 0.01 µm. It is ultimately limited by the digital-to-analog (D/A) conversion at the interface to the mill controller. A value of 0.01 µm corresponds to a standard uncertainty of $u_{\rm res}={0.01}\;{\mathrm{\mu}\textrm{m} }/(2\sqrt {3}) = {0.0029}\;{\mathrm{\mu}\textrm{m} }$. All contributions to the measurement uncertainty are compiled in the measurement uncertainty budget in Table 2. As the $R_{\rm a}$ value depends on the actual rolling process and is more a property of the sample than of the probe, it is included as a parameter. The resulting combined uncertainty $u_{\rm comb}(d)$ is then given by $\sqrt {({0.15}\;{\mathrm{\mu}\textrm{m} })^2+\pi R_{\rm a}^2}$, or an expanded measurement uncertainty of $\sqrt {({0.30}\;{\mathrm{\mu}\textrm{m} })^2+4\pi R_{\rm a}^2}$ for a coverage factor $k = 2$. The constant portion of the uncertainty contribution is mainly determined by the uncertainty of the index of refraction. Experimental results, however, suggest that the air blown into the measurement volume mitigates this effect and that the uncertainty contribution might be overestimated. The comparison measurement between the interferometric and tactile gauges depicted in Fig. 6 corroborates the derived measurement uncertainty. According to Table 2, the combined uncertainty of the MWLI thickness measurement of the (oil-free) steel strip with $R_{\rm a} = {0.25}\;{\mathrm{\mu}\textrm{m} }$ amounts to $u_{\rm comb} = {0.47}\;{\mathrm{\mu}\textrm{m} }$. The accuracy of the tactile gauge is given by the manufacturer as approximately 1.0 µm for strip thicknesses below 1 mm. But even assuming a perfect accuracy of the tactile gauge, i.e., $u_{\rm tactile} = 0$, the degree of equivalence between the two measurements $E_n$ [43] remains, at 0.32, well below the critical $E_n$ value of 1.0 for the observed maximum deviation of $\Delta d_{\rm max} = {0.30}\;{\mathrm{\mu}\textrm{m} }$ between the two gauges.

Table 2. Measurement uncertainty budget for the metal strip thickness measurement.

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6. Conclusions

In this study, an interferometric metal foil thickness gauge was developed starting from the basic design concept and ending with the first successful performance tests under industrial production conditions. Two optical techniques were successfully combined: triangulation to establish the millimetric non-ambiguity range, MWLI to achieve the sub-micron uncertainty of the measurement. The highly efficient method of 2f-3f interferometry was successfully implemented by employing FGPA-controlled DFB diode lasers. This technology enabled the creation of a compact multi-wavelength double interferometer. Gauge blocks are regularly used to calibrate the synthetic wavelength as well as the system parameters for the absolute distance measurement. Once quality starting values have been determined for use as initial working parameters for the optical wavelengths, no further stabilization or complementary wavelength determination is needed. Based on multiple characterization experiments, the expanded measurement uncertainty of the complete strip thickness measurement process was evaluated according to the GUM as $\sqrt {({0.30}\;{\mathrm{\mu}\textrm{m} })^2+4\pi R_{\rm a}^2}$. This result is very well suited to the requirements of thin foil rolling and more than matches the best state-of-the-art strip thickness measurement techniques. It should also be stressed that for an average surface roughness $R_{\rm a}$ above 0.15 µm, more than 50 ${\%}$ of the uncertainty comes from the sample, not from the measuring instrument. The limited impact of remnant oil on the measurement result might at first seem counter-intuitive, but it is a direct consequence of the interferometric approach. The use of calibrated gauge blocks combined with the full measurement uncertainty budget provides metrology-grade traceability to the SI definition of the meter for this complex industrial measurement. The qualitative agreement between tactile and optical gauges is convincing, as both methods reproduce features with a vertical resolution of better than 0.1 µm. When comparing the strip thickness mean values, the observed deviations remain below 0.3 µm. It should be emphasized that this is well below the practical requirements in a rolling mill. Last but not least, the gauge successfully performed as expected in an actual rolling mill, proving that its design and measurement strategy are fit for real-world deployment.