Open Access
Add to BibSonomy
Abstract
With well-known speckle measurement techniques, the root mean square height as well as the autocorrelation length of isotropic surfaces can be determined quickly and over a large area of interest. Beyond that, the present article studies the speckle-based measurement of anisotropic surfaces. For this purpose, a measurement setup and evaluation algorithm are presented that enable the characterization of unidirectionally anisotropic surfaces machined by grinding. As a result, four measurands are obtained from one speckle image: the machining direction, the autocorrelation length perpendicular to the machining direction, as well as two root mean square roughness parameters parallel and perpendicular to the machining direction. The first two measurands are obtained from a two-dimensional fast Fourier transform of the diffraction pattern resulting from the unidirectional tool marks and the latter two by a bidirectional evaluation of the speckle contrast. In addition to measurements on physical reference samples, a spatial light modulator is used to create a large number of surface topographies with known model parameters in order to quantify the measurement uncertainty.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The surface texture is critical for the functionality and lifetime of a manufactured part. In industrial machining processes, surface parameters must be measured quickly and over a large area of interest requiring optical measurement approaches [1]. For isotropic surfaces, height parameters such as the root mean square height $S_{\textrm {q}}$, as well as spatial parameters such as the autocorrelation length $\Lambda$ are specified surface texture parameters in the international standard ISO 25178-2 [2]. Grinding is an important manufacturing process, accounting for about 20–25 % of the total expenditures on machining operations in industrialized nations [3], that produces anisotropic surfaces characterized by stochastically distributed unidirectional tool marks. Accordingly, we derive four measurands for the parametric characterization of this surface type: First, the machining direction in the surface plane that defines the direction of the $x$-axis. Second, the autocorrelation length $\Lambda _y$, which characterizes the lateral spatial appearance of the tool marks in the surface plane. $\Lambda _y$ is the lag value $l_y$ in the $y$-direction for which the surface height autocorrelation function
decays below $1/e$ of its zero lag value, i. e. $ACF(\Lambda _y)=ACF(0) \cdot 1/e$ [4]. $N_x$ and $N_y$ denote the number of surface height values $z(x,y)$ in $x$- and $y$-direction, respectively. And finally, the root mean square height1.1 State of the art of surface measurements
Tactile one-dimensional scanning stylus instruments are widely used for surface characterization [5]. Furthermore, faster optical measurements such as white-light interferometry (WLI) [6] and confocal microscopy [1] allow the measurement of the two-dimensional surface topography $z(x,y)$. However, the necessary scanning and the short working distance limit their potential for in-process measurements.
Laser speckle techniques allow for an even faster integrative measurement and a large working distance making them predestined for in-process quality inspection [7]. They yield single statistical roughness parameters for the illuminated and observed surface region, such as the root mean square surface roughness $S_{\textrm {q}}$ [8]. Already in the 70s the speckle contrast $C = \sigma (I) / \langle I \rangle$, i. e. the standard deviation $\sigma (I)$ divided by the mean $\langle I \rangle$ of the speckle intensity in an evaluation window, was used to determine the roughness parameter $S_{\textrm {q}}$ of isotropic surfaces [9,10]. With the advent of digital cameras a further measurement approach for $S_{\textrm {q}}$ was developed, which is based on evaluating the two-dimensional autocorrelation function of the speckle pattern [11,12].
Some authors reported measuring the bidirectional roughness parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$ for anisotropic surfaces, i. e. the root mean square roughness parallel and perpendicular to the machining direction. For this purpose, the two-dimensional correlation function [13,14], the gray-level co-occurence matrix [15,16], or the Gauss–Markov random field model [17] of the scattered light pattern were evaluated. However, in these experiments only few reference samples machined by grinding [13,17] or milling [14] were measured. For this reason, it was not yet experimentally investigated what measurement uncertainty is achievable when comparing the measurement results from a multitude of different random surface samples and in what range the parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$ can be determined from the speckle pattern. Note further that the tool marks on the manufactured workpiece from the studied anisotropic manufacturing processes form surface structures that are in one dimension significantly larger than the diffraction limit, which changes the structure of the speckle pattern [14]. A comprehensive study on how to extract additional information from these particular scattered light patterns that show both speckle characteristics and imaged surface structures is pending.
In addition to the measurement of the height parameters of a rough surface, such as $S_{\textrm {q}}$, the lateral spatial parameter correlation length $\Lambda$ of an isotropic surface was proven to be measurable with speckle techniques. The lateral correlation length was first determined from multiple measurements with different optical apertures [18–20]. An updated experimental setup was later employed by Cheng [21] to lower the measurement uncertainty of $\Lambda$ below one micrometer. However, the measuring rate was still very limited due to adjusting the aperture with a mechanical pinhole wheel. A more recent approach is to determine the roughness parameter and the correlation length from the autocorrelation of speckles in the deep Fresnel region [22,23]. Ground anisotropic surfaces in particular are described by the correlation length perpendicular to the machining direction $\Lambda _y$ as it characterizes the tool marks which influence both function and appearance of the surface finish [3]. However, the reported speckle measurement results are as of yet limited to isotropic samples as is the underlying theoretic description [24]. Additionally, because speckles in the deep Fresnel region near the surface were observed in [22,23], the working distance was below 1 cm. A measurement approach for both $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$, as well as $\Lambda _y$ of anisotropic surfaces that provides single-shot capability and a larger working distance to enable fast in-process measurements was not yet proposed. Thus the question remains, to what extent is it possible to determine the bidirectional roughness parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$, as well as the machining direction (defined as the direction of the $x$-axis) and the perpendicular autocorrelation length $\Lambda _y$ simultaneously from a single laser speckle pattern of anisotropic surfaces?
1.2 Aim and outline of the paper
The first aim of the article is to present a speckle measurement setup and evaluation algorithm for the parametric characterization of ground surfaces regarding the machining direction, the autocorrelation length $\Lambda _y$, as well as the root mean square height parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$. The machining direction and the autocorrelation length perpendicular to it are determined from a two-dimensional fast Fourier transform of the diffraction pattern resulting from the unidirectional tool marks. The roughness parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$ are obtained through a bidirectional evaluation of the speckle contrast. The second aim is to determine the measurement range and uncertainty of the experimental setup, which requires a large number of surface samples with known parameters that are difficult to obtain from manufacturing processes with a high variability such as grinding. Therefore, in order to fully characterize the measurement system, a phase-only spatial light modulator (SLM) is used to experimentally represent anisotropic surfaces according to given surface parameters, which bridges the gap between the theoretic analytical model on the one hand and the realization of a large number of physical reference samples with known statistical surface texture parameters for validation on the other hand. With the proposed measurement setup, surface parameters are varied to determine the achievable measurement range and measurement uncertainty.
First, the theory and measurement principle of the speckle analysis as well as the representation of anisotropic technical surfaces are explained in section 2. The experimental setup is presented in section 3. In section 4 the experimental results are shown and discussed. The final section 5 draws the conclusions and gives an outlook on further research aspects.
2. Measurement principle
2.1 Surface model
In the proposed experimental measurement principle, a surface topography is displayed as a phase map on an SLM to represent physical surface samples. The used surface topography is either determined from a reference sample with a WLI or simulated using a surface model. In this study, the surfaces are modelled with the moving average method [25,26]. Here, the surface topography
Figure 1(c) shows a surface topography simulated according to Eq. (5) that is made up of an isotropic component (Fig. 1(a)) and an anisotropic component (Fig. 1(b)). The isotropic model parameters are $M_{\textrm {i},R} = {0.05}\;\mathrm{\mu}\textrm{m}$, $M_{\textrm {i},\Lambda } = {1}\;\mathrm{\mu}\textrm{m}$, and the anisotropic model parameters in y-direction are $M_{\textrm {a},R} = {0.185}\;\mathrm{\mu}\textrm{m}$ and $M_{\textrm {a},\Lambda } = {4}\;\mathrm{\mu}\textrm{m}$. The surface topography in Fig. 1(d) is measured from a physical Rugotest N3 reference sample. Note that the surface topographies are displayed in a spatial resolution of 8 µm. Qualitative differences between the real and simulated surface topography remain, e. g., the tool marks are not infinitely long in $x$-direction and also not perfectly unidirectional. However, a sufficiently good qualitative match of the surface topography and the simulated speckle patterns between simulation and reference is achieved with closely matching quantitative surface texture parameters. Here, the model parameters of the simulated surface were selected so that the simulated surface exhibits comparable parameters to the physical reference sample, which has $S_{\textrm {q}} = {0.19}\;\mathrm{\mu}\textrm{m}$ and $\Lambda _y = {6}\;\mathrm{\mu}\textrm{m}$.
Fig. 1. Surface topography of (a) a simulated rough isotropic surface, (b) a simulated anisotropic surface, (c) the sum of the isotropic an anisotropic surfaces according to Eq. (5), and (d) a measured Rugotest reference sample.
2.2 Roughness measurement from speckle contrast
Various evaluation approaches to derive surface parameters from speckle patterns exist [27,28]. For a surface roughness smaller than the laser wavelength $\lambda$ a contrast-based evaluation is possible [8]. The speckle contrast of an isotropic surface over its roughness $S_{\textrm {q}}$ according to the theory from [24] is shown in Fig. 2. The curve results from the transition from specular reflection of laser light for very smooth surfaces to diffuse scattering for surfaces with $S_{\textrm {q}} > \lambda / 4$. In the transition region partially developed speckles occur. When the roughness is sufficiently large, diffuse scattering dominates, the contrast saturates at unity, and fully developed speckles occur. The shown theoretical curves demonstrate the cross-sensitivity of the speckle contrast with respect to the correlation length. This highlights a challenge in speckle surface metrology: the calibration of the measurement system depends on the surface type. However, the acquisition of additional parameters may allow to compensate for the cross-sensitivity.
Fig. 2. Speckle contrast over isotropic roughness $S_{\textrm {q}}$ for $\lambda = {638}\;\textrm{nm}$ and different correlation lengths $\Lambda$ calculated according to [24].
Our heuristically derived hypothesis, which needs to be validated, is that the roughness parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$ of anisotropic surfaces can be determined through a bidirectional contrast evaluation. The speckle contrast
2.3 Machining direction and correlation length measurement
In a lateral direction a surface topography is characterized by the correlation length $\Lambda$. For ground surfaces in particular, the parameter $\Lambda _y$ describes the shape and distribution of the tool marks. In Fig. 1 the prominent unidirectional tool marks in x-direction can be seen in the surface topographies. The tool marks form surface structures that are in one dimension significantly larger than the diffraction limited speckle size. Figure 3(a) shows the speckle pattern of the simulated surface from Fig. 1(c). The speckle pattern is simulated according to Goodman [8]. The tool marks of the ground surface act as an optical grating whose diffraction pattern is superimposed on the speckles that are caused by surface structures smaller than the lateral diffraction limit. These random surface height variations influence the speckle contrast along the tool marks in the machining direction (dashed red line in Fig. 3(a)). Because the diffraction pattern caused by the tool marks appears as a periodic, unidirectional element in the speckle pattern, it can be evaluated with a frequency analysis implemented through a two-dimensional fast Fourier transform. The resulting amplitude spectrum is shown in Fig. 3(b). Since the tool marks are stochastically distributed on the simulated surface, a large number of spatial frequencies perpendicular to the machining direction (dashed red line) appear in the spectrum. The spectrum has a global maximum at the zero-frequency origin and multiple local maxima in the $y$-direction (dashed red line in Fig. 3(b)). The center of mass of the spectrum $f_{\textrm {c}}$ is the weighted average of all spatial frequencies in the speckle pattern. Since the spatial frequencies perpendicular to the machining direction that characterize the tool marks are clearly distinguishable in the spectrum in Fig. 3(b), the machining direction can be determined from the angle $\beta$ to the ordinate (equal to $\beta$ in Fig. 3(a)). In addition, our heuristically derived hypothesis is that
For a specific surface height distribution a higher spatial frequency in the $y$-direction indicates that the tool marks are on average closer together and thus steeper, which results in a smaller correlation length $\Lambda _y$. This hypothesis is based on an approach theorized by Goodman [8] that links the autocorrelation function of the scattering surface to that of the light field leaving the scattering surface. In [8], the surface is assumed to be isotropic with Gaussian statistics and with a roughness $<\lambda /4$, i. e. in the regime of specular reflection as well as scattering. In addition, the generalized von Cittert–Zernike theorem is applied to link the autocorrelation function of the light field leaving the scattering surface to the average intensity distribution observed in the focal plane of a lens, which is sufficient for isotropic surfaces. However, in our approach for anisotropic surfaces the spatial amplitude spectrum is measured instead, based on the Wiener–Khinchin theorem that links the autocorrelation function to the power spectral density of a signal. Thus, by evaluating spatial frequencies in a certain direction we aim to determine the autocorrelation length of an anisotropic ground surface in that direction from the amplitude spectrum of its speckle pattern. However, the applicability of Eq. (7) to speckle patterns of ground surfaces and the valid measurement range for $\Lambda _y$ must be investigated experimentally subsequently.Fig. 3. (a) Simulated speckle pattern and (b) respective two-dimensional amplitude spectrum. A dashed red line indicates the machining direction.
3. Experimental setup
3.1 Measurement object
In order to represent simulated surface topographies in the experimental setup, the phase only spatial light modulator PLUTO-2.1-VIS-016 of HOLOEYE Photonics AG is the measurement object of the parametric characterization of anisotropic surfaces. Using the SLM, the spatially resolved phase of a laser wavefront can be modulated. In our measurement approach, the surface height profile is displayed on the SLM and the phase differences correspond to the path differences that result from a surface height variation. The SLM is operated in $4 \pi$ phase shift mode. At a laser wavelength of 638 nm, the $4 \pi$ maximum phase difference corresponds to a path difference of around 1.3 µm. We assume that surfaces with a roughness $S_{\textrm {q}} < {0.3}\;\mathrm{\mu}\textrm{m}$ are adequately represented, since 95 % of the height values are correctly represented for a normally distributed height profile. The resolution of the SLM is $1920 \times 1080$ with a pixel pitch of 8 µm, which results in an active area of ${15.4}\;\textrm{mm} \times {8.6}\;\textrm{mm}$. The pixel discrete representation with the spatial resolution of 8 µm smoothens the height profile compared to a real continuous surface. This limits the minimum correlation length of a surface that the SLM is able to represent correctly (see section 4.1.2). To validate both the surface model and its representation on the SLM, ground Rugotest surface standards were used as physical reference samples (see section 4.2). In order to analyze the surface topographies, they were measured using a WLI from GBS mbH with a lateral resolution of 0.94 µm.
3.2 Optical measurement system
Figure 4 shows a scheme of the measuring setup. A 638 nm continous-wave diode laser is expanded to a measuring spot diameter of about 5 mm, polarized, and collimated onto the SLM or reference sample surface. The angle of incidence is equal to the angle of reflection and they are $< {5}^{\circ}$ to reach the highest diffraction efficiency with the SLM. The resulting speckle pattern is viewed through a 4-f system with two $f = {150}\textrm{mm}$ lenses and a $d = {3}\;\textrm{mm}$ pinhole and imaged with a $2448\times 2048$ CMOS camera that has a pixel pitch of 3.45 µm. A 4-f imaging system is a standard configuration, and it was also chosen to allow easy modifications to the focal length and pinhole diameter and to keep the measurement system similar to that in the theoretic description of Goodman [8,24]. Furthermore, a non-zero incidence angle was chosen instead of a perpendicular surface illumination with a beam splitter in order to avoid unwanted interference effects.
3.3 Image processing
The monochromatic camera images are first cropped to a $1024 \times 1024$ pixel region of interest. The subsequent image processing and evaluation procedure is outlined in Fig. 5. In the first processing step the fast Fourier transfer MATLAB function fft2 is applied to the image. By windowing the amplitude spectrum for evaluation, both the very low and high frequency components are eliminated, i. e. high-pass and low-pass filters are applied. This improves the signal-to-noise ratio of the correlation length measurement. The frequency at the center of mass of the spectrum $f_{\textrm {c}}$ is then further evaluated. Its distance to the zero-frequency component was found to correlate with the correlation length $\Lambda _y$. The angle of this distance to the ordinate yields the machining direction. This angle is then used to rotate the speckle image so that the machining direction is aligned with the $x$-axis of the image. Afterwards, the speckle contrast in and perpendicular to the machining direction is determined according to Eq. (6) in a $500 \times 500$ pixel evaluation window. The evaluation window is homogeneously illuminated, which means that edge effects can be neglected. Within the measurement range, these contrast values correlate with the roughness parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$. With knowledge of the respective calibration curves, we are thus able to determine all four desired measurands from a single speckle image.
4. Results
4.1 SLM
In order to experimentally confirm the hypotheses stated in Eqs. (6) and (7), we first evaluate speckle patterns measured from the SLM surface, i. e. simulated surface topographies that are represented as a phase map on the SLM.
4.1.1 Roughness
Figure 6 shows the bidirectional speckle contrast measured from the SLM surface over the model parameters $M_{\textrm {i},R}$ and $M_{\textrm {a},R}$ of the simulated surface topography. Note that beyond the classical contrast evaluation shown in Fig. 2, we now measure the two contrast values $C_x$ and $C_y$ in and perpendicular to the machining direction. For each data point in the plot, 100 surface topographies with the same model parameters were randomly generated according to Eq. (5) and measured and the error bars show the standard deviation of the mean. In Fig. 6(a), the surface model parameter $M_{\textrm {i},R}$ is varied from 0.05 µm to 0.2 µm in order to selectively adjust the roughness parameter $R_{\textrm {q},x}$ of the surface. This linear increase is reflected in the speckle contrast in $x$-direction $C_x$. The other model parameters remain constant at $M_{\textrm {i},\Lambda } = {10}\;\mathrm{\mu}\textrm{m}$, $M_{\textrm {a},R} = {0.3}\;\mathrm{\mu}\textrm{m}$, and $M_{\textrm {a},\Lambda } = {20}\;\mathrm{\mu}\textrm{m}$. However, the speckle contrast in the y-direction $C_y$ also increases since $M_{\textrm {i},R}$, as an isotropic areal roughness parameter, also has an effect on $R_{\textrm {q},y}$.
Fig. 6. Speckle contrast $C_x$ in x- and $C_y$ in y-direction measured from the SLM surface. In (a) the isotropic roughness parameter $M_{\textrm {i},R}$ is varied and in (b) the anisopropic roughness parameter $M_{\textrm {a},R}$ is varied. Error bars show standard deviation of the mean of 100 random surfaces.
In Fig. 6(b), the parameter $M_{\textrm {a},R}$ of the surface model is varied from 0.1 µm to 0.5 µm. The other model parameters remain constant at $M_{\textrm {i},R} = {0.07}\;\mathrm{\mu}\textrm{m}$, $M_{\textrm {i},\Lambda } = {10}\;\mathrm{\mu}\textrm{m}$, and $M_{\textrm {a},\Lambda } = {20}\;\mathrm{\mu}\textrm{m}$. The speckle contrast $C_y$ evaluated perpendicular to the machining direction behaves in accordance with the theory from [8,24] for the transition from a partially developed to fully developed speckle pattern. The contrast saturates at a maximum value for the fully developed speckle pattern. In the theory this maximum value is unity, in the SLM measurements it stays below 0.8. In a fully developed speckle pattern, the diffusely scattered light should dominate the specularly reflected component. However, in the SLM setup the specular component does not disappear even for a large roughness close to the laser wavelength. The specular reflection increases the average intensity $\langle I \rangle$ of the speckle pattern, reducing the speckle contrast $C = \sigma (I) / \langle I \rangle$. Another reason for the upper limit of the speckle contrast is the $4 \pi \stackrel {\scriptscriptstyle \wedge }{=} {1.27}\;\mathrm{\mu}\textrm{m}$ phase range of the SLM, which limits the maximum surface roughness that can be represented. A surface roughness of ${0.5}\;\mathrm{\mu}\textrm{m}$ approaches this limit. Furthermore, a slight cross-sensitivity of the contrast in the x-direction with respect to the anisotropic height parameter $M_{\textrm {a},R}$ in the y-direction is evident in the diagram. This cross-sensitivity is possibly due to the large roughness range considered, i. e. the increasing ratio of $R_{\textrm {q},y}$ to $R_{\textrm {q},x}$. However, the cross-sensitivity is small compared to the roughness-sensitivity of the contrast values $C_x$ and $C_y$, which means that the bidirectional roughness evaluation is applicable throughout the entire considered measurement range.
In addition, the influence of the speckle noise on the speckle contrast, i. e. the uncertainty component caused by the statistically distributed speckle intensity pattern [29,30], is quantified by means of a measurement uncertainty analysis. For this purpose, the empirical measurement uncertainty of the speckle contrast is calculated according to the Type A evaluation from the Guide to the expression of uncertainty in measurement (GUM) [31]. The uncertainty of the roughness measurement is then calculated from the speckle contrast uncertainty by means of an uncertainty propagation using the measured calibration curve shown in Fig. 6. One hundred speckle images were evaluated for no speckle noise condition (of the same simulated surface) and with speckle noise, i. e. of 100 randomly generated surfaces with the same model parameters (shown in Fig. 6). In the no-speckle-noise measurement the uncertainty of $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$ are both 1 nm at a roughness of 0.1 µm. In the measurement with speckle noise the uncertainty of $R_{\textrm {q},x}$ is 1 nm and for $R_{\textrm {q},y}$ it is 1.7 nm. This shows that a measurement uncertainty in the nanometer range is possible for the bidirectional roughness evaluation. Furthermore, only the evaluation perpendicular to the machining direction appears to be significantly influenced by speckle noise.
4.1.2 Machining direction and correlation length
The machining direction and the spatial parameter $\Lambda _y$, i. e. the autocorrelation length perpendicular to the machining direction, are determined with a two-dimensional fast Fourier transform. As an example, Fig. 7(a) shows the measured speckle pattern of an SLM surface topography with the model parameters $M_{\textrm {i},R} = {0.07}\;\mathrm{\mu}\textrm{m}$, $M_{\textrm {i},\Lambda } = {10}\;\mathrm{\mu}\textrm{m}$, $M_{\textrm {a},R} = {0.3}\;\mathrm{\mu}\textrm{m}$, and $M_{\textrm {a},\Lambda } = {35}\;\mathrm{\mu}\textrm{m}$. In order to demonstrate the two-dimensional amplitude spectrum’s angular dependence of the machining direction, the surface topography is rotated by the angle $\beta = {45}^{\circ}$. The angle $\beta$ corresponds to the angle between the $\xi$-axis of the camera sensor and the $x$-axis of the surface topography, i. e. machining direction. The corresponding two-dimensional amplitude spectrum is shown in Fig. 7(b). As with the evaluation of the simulated speckle pattern in Fig. 3, the machining direction is extracted from the two-dimensional spectrum as the angle $\beta$ between the $\eta$- and the $y$-axes. For this purpose, the center of mass $f_{\textrm {c}}$ of the spectrum is calculated and the angle to the ordinate axis is $\beta$, which is in agreement with the initial rotation of the surface topography. The one-dimensional amplitude spectrum along the $y$-axis is shown in Fig. 7(c), visualizing the spatial frequencies of the tool marks. In this specific speckle pattern one frequency at $f_y = 73$ is dominant. However, since not all speckle patterns have only one dominant spatial frequency, the position of this local frequency maximum is not used as a measurand of the correlation length. The position of the center of mass $f_{\textrm {c}}$ is a much more robust measurand. Figure 7(d) shows $f_{\textrm {c}}$ plotted over the correlation length parameter $M_{\textrm {a},\Lambda }$. $M_{\textrm {a},\Lambda }$ was varied from 10 µm to 70 µm resulting in a linear increase of the lateral correlation length $\Lambda _y$. One hundred random surfaces were evaluated for each data point and the error bars show the standard deviation of the mean. The blue curve shows a nearly linear correlation of $f_{\textrm {c}}$ and $M_{\textrm {a},\Lambda }$ starting at $M_{\textrm {a},\Lambda } \approx {20}\;\mathrm{\mu}\textrm{m}$. Surface topographies with smaller correlation lengths cannot be represented correctly by the SLM because its spatial resolution is limited by the pixel size of 8 µm and the corresponding data points are grayed out in Fig. 7(d). In the surface simulation and on the SLM are almost no upper limits to the representable correlation length, however the simulated surfaces do not correspond to real ground surfaces after a certain point. Thus, from the results in Fig. 7(d) it follows that a measurement of $\Lambda _y$ is possible by evaluating $f_{\textrm {c}}$ in the measurement range indicated by the blue line, provided that the corresponding calibration curve is known.
Fig. 7. (a) Measured speckle pattern, (b) two-dimensional amplitude spectrum with center of mass $f_{\textrm {c}}$ marked red, (c) one-dimensional amplitude spectrum along the $y$-axis, i. e. perpendicular to the machining direction, with $f_{\textrm {c}}$ marked by a dotted red line, and (d) $f_{\textrm {c}}$ over the surface model parameter $M_{\textrm {a},\Lambda }$. Error bars show standard deviation of the mean of 100 random surfaces.
The influence of speckle noise was studied as well for the evaluation of the correlation length. Again, one hundred speckle images were evaluated for no speckle noise condition (of the same simulated surface) and with speckle noise, i. e. of 100 randomly generated surfaces with the same model parameters. The measurement uncertainty of $\Lambda _y$ corresponds in the considered range to the uncertainty of $M_{\textrm {a},\Lambda }$, which is calculated by means of an uncertainty propagation from the uncertainty of $f_{\textrm {c}}$ and its slope in Fig. 7(d). In the no-speckle-noise measurement the uncertainty of $\Lambda _y$ is negligible. In the measurement with speckle noise the uncertainty of $\Lambda _y$ is $\leq {0.77}\;\mathrm{\mu}\textrm{m}$ in the entire measurement range. Thus, speckle noise is the dominant component of the uncertainty budget. This is due to the fact that structures in the speckle pattern have to be detected, which vary stochastically in shape and size for different surfaces. If the same speckle pattern is measured several times, the measurement uncertainty is almost equal to 0, since the evaluation of the center of mass already filters out random noise in the measured spectrum. However, even with speckle noise a relative measurement uncertainty of $\Lambda _y$ well below 10 % is achievable.
4.2 Reference samples
Here, we evaluate speckle patterns measured from physical reference samples. The plane hand-polished Rugotest No. 5 reference samples of the increasing roughness classes N1 – N4 were measured in order to finally compare the findings from the emulated surface experiments with the SLM (on the basis of the surface model introduced in section 2.1) to physical samples. The hand-polished Rugotest samples were used instead of machine-ground ones because they were available in the desired roughness range and have a similar unidirectional, anisotropic surface structure. The roughness classes according to ISO 1302 have the following nominal mean roughness $R_{\textrm {a}}$ values: $\textrm {N}1={0.025}\;\mathrm{\mu}\textrm{m}$, $\textrm {N}2={0.05}\;\mathrm{\mu}\textrm{m}$, $\textrm {N}3={0.1}\;\mathrm{\mu}\textrm{m}$, and $\textrm {N}4={0.2}\;\mathrm{\mu}\textrm{m}$. However, these $R_{\textrm {a}}$ values without indication of the evaluation direction are not sufficient for the examination of anisotropic surfaces. Therefore, the surface topographies of the reference samples were recorded with a WLI and from this the parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$ in and perpendicular to the machining direction were evaluated. The results of these WLI measurements are shown in Fig. 8 (x-values).
Fig. 8. Bidirectional speckle contrast $C_x$ and $C_y$ measured from physical reference samples. The roughness values for $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$ on the abscissa were measured with a WLI. The error bars show the standard deviation of the mean over 5 measurements on different positions on the sample.
4.2.1 Roughness
Figure 8 shows the bidirectional speckle contrast evaluation of the four Rugotest reference samples N1 – N4. The transition from partially developed speckles (N2) to fully developed speckles (N3) is visible in the measured speckle contrast $C_y$. From sample N1 to N2 the contrast increases at a high rate and for samples N3 and N4 it approaches the upper saturation limit. The results of the contrast values $C_x$ demonstrate the advantage of the bidirectional evaluation. While the roughness $R_{\textrm {q},y}$ increases strongly from sample N3 to N4, the roughness $R_{\textrm {q},x}$ in the machining direction remains almost the same. This behavior is correctly represented in the contrast $C_x$, but would not be detectable from the standard areal speckle contrast evaluation.
The Rugotest results are qualitatively in agreement in comparison with the SLM measurements, but deviations occur in the absolute speckle contrast values. This is due to the differences of the surfaces displayed on the SLM compared to the physical surface topographies. The lateral spatial resolution of the SLM of 8 µm results in a different height distribution compared to the physical samples, which also changes the resulting speckle patterns and implies different calibration curves. Therefore, although the SLM setup cannot be used directly for calibration of the measuring system, it yields qualitatively valid results and is predestined for parameter studies and measurement uncertainty analysis, since a much larger number of surfaces with defined parameters can be measured than would be possible with physical reference samples.
4.2.2 Machining direction and correlation length
Analogous to Fig. 7, the machining direction of the physical samples is also correctly determined from the two-dimensional amplitude spectrum of the speckle pattern. Figure 9 shows the measured $f_{\textrm {c}}$ values of the Rugotest reference samples N1 – N4 over their $\Lambda _y$ values measured with the WLI. Additionally, the SLM measurement from Fig. 7(d) is shown in blue as a comparison. The results from samples N1 and N4 (red) are in qualitative agreement with the SLM results (blue). The higher $\Lambda _y$ values of N1 results in a decreased $f_{\textrm {c}}$ value at a rate similar to the blue line. In contrast, samples N2 and N3 (gray) do not agree with the hypothesis from Eq. (7). One reason for this are the small $\Lambda _y$ values of around 15 µm. For small $\Lambda _y$, the variance in the evaluation increases, as more spatial frequencies appear in the speckle pattern. As a result, samples N2 and N3 are outside of the measurement range and cannot be evaluated correctly with our current approach. Given this limitation, the reference measurement confirms the hypothesis stated in section 2.3 and is in agreement with the SLM measurements. However, due to the small number of samples, no definite statement is possible about the measuring range and the exact course of the calibration curve.
Fig. 9. Center of mass frequency $f_{\textrm {c}}$ over correlation length $f_{\textrm {c}}$ measured from Rugotest reference samples (red/gray) and over model parameter $M_{\textrm {a},\Lambda }$ measured from the SLM (blue).
5. Conclusions
We presented a speckle-based evaluation method for the parametric characterization of ground surfaces regarding the root mean square height parameters $R_{\textrm {q},x}$ and $R_{\textrm {q},y}$, as well as the machining direction and the autocorrelation length $\Lambda _y$. To enable a statistical study of the measurement capabilities, we realized a speckle measurement setup and introduced an SLM to represent anisotropic ground surfaces. The emulated surfaces follow from a parameterized surface model, which is introduced as the sum of an isotropic and an anisotropic Gaussian surface and consists of four parameters.
The roughness measurement is based on a bidirectional speckle contrast analysis that was demonstrated both for the SLM-based surface emulation as well as with physical reference samples. As a result, the roughness measurement range is around 0.05 µm to 0.4 µm and the measurement uncertainty is around 1 nm. The machining direction and the perpendicular autocorrelation length $\Lambda _y$ were measured using a two-dimensional fast Fourier transform. The spatial frequencies of the characteristic diffraction pattern caused by the unidirectional tool marks yield the orientation of the sample with respect to the machining direction. Additionally, in the correlation length range of 20 µm to 70 µm a linear relation of $\Lambda _y$ was found with the position of the center of mass of the two-dimensional amplitude spectrum. The measurement uncertainty of the correlation length is below 1 µm, with the measurement uncertainty budget dominated by speckle noise.
The SLM setup offers the possibility to test a variety of samples of desired surface parameters and thus to characterize the measurement technique and evaluation algorithms. The bidirectional evaluation of height and spatial surface texture parameters on ground anisotropic samples is fundamentally possible using speckle methods and our heuristically derived hypotheses from section 2 were experimentally validated. However, in order to enable a wide application of the measurement method and the determination of general calibration curves with the SLM-setup, the validity of the surface model for different machining parameters, the hardware limitations of the SLM and the fundamental limits of the speckle evaluation in particular for the measurement of the correlation length should be further investigated.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. Leach, Optical Measurement of Surface Topography (Springer-Verlag Berlin Heidelberg, 2011).
2. ISO 25178-2:2012, “Geometrical product specifications (GPS) – Surface texture: Areal part 2: Terms, definitions and surface texture parameters,” International Organization for Standardization (2012).
3. W. B. Rowe, Principles of Modern Grinding Technology (Elsevier Science, 2013), 2nd ed.
4. O. Taconet and V. Ciarletti, “Estimating soil roughness indices on a ridge-and-furrow surface using stereo photogrammetry,” Soil Tillage Res. 93(1), 64–76 (2007). [CrossRef]
5. T. Pfeifer, Production Metrology (Oldenbourg Wissenschaftsverlag, 2015).
6. G. Behrends, D. Stöbener, and A. Fischer, “Lateral scanning white-light interferometry on rotating objects,” Surf. Topogr.: Metrol. Prop. 8(3), 035006 (2020). [CrossRef]
7. A. Fischer and D. Stöbener, “In-process roughness quality inspection for metal sheet rolling,” CIRP Ann. 68(1), 523–526 (2019). [CrossRef]
8. J. W. Goodman, Speckle phenomena in optics: Theory and applications (Roberts, 2007).
9. H. M. Pedersen, “The roughness dependence of partially developed, monochromatic speckle patterns,” Opt. Commun. 12(2), 156–159 (1974). [CrossRef]
10. H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11(1), 35–38 (1974). [CrossRef]
11. P. Lehmann, “Surface-roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination,” Appl. Opt. 38(7), 1144–1152 (1999). [CrossRef]
12. S. Patzelt, D. Stöbener, and A. Fischer, “Laser light source limited uncertainty of speckle-based roughness measurements,” Appl. Opt. 58(23), 6436–6445 (2019). [CrossRef]
13. H. Hirabayashi, T. Matsuo, H. Ishizawa, H. Kanai, and T. Nishimatsu, “Surface Roughness Evaluation by Laser Speckle,” in 2006 SICE-ICASE international joint conference, (IEEE Service Center, 2006), pp. 5809–5812.
14. M. Shimizu, H. Sawano, H. Yoshioka, and H. Shinno, “Multi-dimensional assessment of nano/micro scale surface texture using laser speckle pattern analysis,” J. Adv. Mech. Des. Syst. Manuf. 9(1), JAMDSM0011 (2015). [CrossRef]
15. R.-S. Lu, G.-Y. Tian, D. Gledhill, and S. Ward, “Grinding surface roughness measurement based on the co-occurrence matrix of speckle pattern texture,” Appl. Opt. 45(35), 8839–8847 (2006). [CrossRef]
16. A. B. Pradana and P. Prajitno, “A Portable Surface Roughness Measurement System Using Laser Speckle Imaging Based on GLCM,” in 2019 6th International Conference on Instrumentation, Control, and Automation (ICA), (IEEE, 7/31/2019 - 8/2/2019), pp. 100–105.
17. L. Yang, R. Lu, and L. Lei, “Grinding surface roughness measurement based on Gauss-Markov random field model of laser speckle pattern texture,” in Fourth International Seminar on Modern Cutting and Measurement Engineering, J. Xin, L. Zhu, and Z. Wang, eds. (SPIE, 2010), SPIE Proceedings, p. 79971X.
18. H. Fujii, T. Asakura, and Y. Shindo, “Measurement of surface roughness properties by using image speckle contrast,” J. Opt. Soc. Am. 66(11), 1217 (1976). [CrossRef]
19. H. Kadono, T. Asakura, and N. Takai, “Roughness and correlation-length determination of rough-surface objects using the speckle contrast,” Appl. Phys. B: Photophys. Laser Chem. 44(3), 167–173 (1987). [CrossRef]
20. H. Kadono, T. Asakura, and N. Takai, “Roughness And Correlation-Length Measurements Of Rough Surface Objects Using The Speckle Contrast In The Diffraction Field,” in Statistical Optics, G. M. Morris, ed. (SPIE, 1988), SPIE Proceedings, p. 210.
21. C. Cheng, S. Teng, M. Liu, S. Gong, R. Li, and Z. Xu, “Extraction of height–height correlation function of random surfaces from the average intensity of image speckles,” Appl. Phys. Lett. 81(24), 4544–4546 (2002). [CrossRef]
22. G. Liang, X. Li, M. Zhang, Z. Li, C. Liu, and C. Cheng, “Experimental extraction of rough surface parameters from speckles in the deep Fresnel region with a scanning fibre-optic probe,” Eur. Phys. J. D 67(4), 77 (2013). [CrossRef]
23. C. Liu, Q. Dong, H. Li, Z. Li, X. Li, and C. Cheng, “Measurement of surface parameters from autocorrelation function of speckles in deep Fresnel region with microscopic imaging system,” Opt. Express 22(2), 1302–1312 (2014). [CrossRef]
24. J. C. Dainty, ed. Laser Speckle and Related Phenomena, vol. 9 of Topics in Applied Physics (Springer Berlin Heidelberg, 1975).
25. J. A. Ogilvy and J. R. Foster, “Rough surfaces: gaussian or exponential statistics?” J. Phys. D: Appl. Phys. 22(9), 1243–1251 (1989). [CrossRef]
26. A. K. Fung and M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2(12), 2274 (1985). [CrossRef]
27. P. Lehmann and G. Goch, “Comparison of Conventional Light Scattering and Speckle Techniques Concerning an In-Process Characterisation of Engineered Surfaces,” CIRP Ann. 49(1), 419–422 (2000). [CrossRef]
28. J. Stempin, A. Tausendfreund, D. Stöbener, and A. Fischer, “Roughness Measurements with Polychromatic Speckles on Tilted Surfaces,” Nanomanuf. Metrol. 4(4), 237–246 (2021). [CrossRef]
29. A. Fischer, “Fundamental uncertainty limit for speckle displacement measurements,” Appl. Opt. 56(25), 7013–7019 (2017). [CrossRef]
30. G. Alexe, A. Tausendfreund, D. Stöbener, L. Langstädtler, M. Herrmann, C. Schenck, and A. Fischer, “Uncertainty and Resolution of Speckle Photography on Micro Samples,” Nanomanuf. Metrol. 3(2), 91–104 (2020). [CrossRef]
31. JCGM 100:2008, “Evaluation of measurement data – Guide to the expression of uncertainty in measurement,” JCGM (2008).
