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Abstract
Speckle patterns offer valuable insights into the surface characteristics or the characteristics of the light generating the speckle. One possible way to extract this information is via spectral speckle correlation (SSC). The cross-correlation between two speckle fields, generated at different wavelengths, can be used for example to determine the roughness of the illuminated surface. Taking defocused measurements of the surface or measuring on a tilted surface leads to a displacement between the speckle, which in turn affects the cross-correlation and leads to errors in the calculated roughness. In this work we present a model to determine the lateral speckle displacement for a change in wavelength in the case of subjective speckle and defocused, tilted objects. This model is therefore applicable to a wide range of applications and allows to estimate and correct for this speckle displacement. Experimental results show sub-pixel accuracy for object tilts up to ±7° and defocus distances up to ±25 mm.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Speckle patterns are formed due to multiple interference of scattered waves when coherent light, such as laser light, interacts with a rough surface [1]. These intensity patterns consist of distinct bright and dark spots. The random phase and intensity differences in the scattered waves, resulting from the reflection of the waves at various positions on the surface, are responsible for the generation of speckle. When speckles are observed directly (i.e., without an imaging system), they are referred to as objective speckles. Otherwise, a subjective speckle pattern is captured [2].
The properties of a speckle pattern are dependent on the optical setup, the properties of the light and the scattering surface generating the speckle. The speckle pattern sticks to the surface [2,3]. When the surface is moved, the speckle pattern moves accordingly like a rigid-body [4]. In [5] this is used to measure the in-plane and out-of-plane deformations. This also means, that a tilt of the surface will also tilt the speckle field. The three dimensional deformation of a surface can be measured using the speckle displacements before and after deformation and considering local speckle displacements [6]. A variation in the wavelength of the light leads to a linear alteration in the phase along the reflecting surface. The surface can be conceptualized as a collection of diffraction gratings with varying phases relative to one another [7]. The resulting speckle fields will therefore be scaled with respect to each other. This leads to individual speckles being displaced between two different wavelengths.
The surface roughness of a reflective surface, which is described here as the mean quadratic difference in surface height, has a considerable influence on the speckle pattern. As the roughness increases, the difference in phase between the various reflected light paths increases. These influence factors were investigated by Ruffing [8] who uses the correlation between two speckle patterns of different wavelengths (spectral speckle correlation, SSC) to measure the surface roughness. He also describes how to achieve this by using one wavelength and tilting the object surface (angular speckle correlation). The SSC is employed in [9] for determining local surface roughness in a defocused optical system. In this scenario, the displacement of speckle patterns of various wavelengths occurs. Hence, comprehending this displacement is crucial to grasp the impact of this phenomenon on measurements, allowing for enhanced measurement accuracy and larger measurement areas, as well as the design of suitable optical systems.
Figure 1 shows recorded speckle for an aligned sample surface which is displaced by −15 mm in axial direction out of the object plane. In this case the speckle in Fig. 1(b) are displaced downward respective to Fig. 1(a). To improve the visibility of the speckle displacement, the difference between Fig. 1(a) and Fig. 1(b) is shown in Fig. 1(c).
Fig. 1. Speckle images captured for an aligned surface with an axial displacement of −15 mm. (a) Speckle recorded using a wavelength of 633.3 nm. (b) Speckle recorded using a wavelength of 644.4 nm. (c) Difference between (a) and (b).
Hanson et al. investigated the displacement of the speckle in case of objective speckle and a tilted surface in order to measure the wavelength of the incident light [10]. Gibson et al. also explored this phenomenon and utilized it to determine the absolute angle of a surface [11]. The results of these works are valid for objective speckle close to the optical axis of the system and in the case of [10], a Fourier transform optical system is discussed. To the authors knowledge, there exists no work concerning the speckle displacements in the case of both subjective speckle and tilted, defocused objects.
This paper extends the model of [10], so the lateral speckle displacement for an arbitrary imaging system can be estimated.
2. Theory
2.1 Objective speckle displacement model
In the work of Hanson [10] the speckle displacement $\Delta p$ dependent on wavelength and parameters of the optical setup is described as
In this case, a camera is placed at a distance $l$ from a flat reflective surface, where the angle $\eta$ between the camera and the surface is defined by the angle between the normal vectors of both planes. A beam illuminates the surface under an angle $\zeta$ relative to the normal vector of this surface. $\bar {k}$ denotes the mean of the wavenumbers and $\Delta k$ is the difference $k_{\mathrm {max}}-k_{\mathrm {min}}$ in wavenumbers between the incident fields. This is illustrated in Fig. 2. This result is valid for collimated incident light and the center of the detector. An important result is that the speckle displacement is not dependent on the illuminated surface in this case.
Fig. 2. Objective speckle displacement model [10]. The incident light is colored blue, the observed light is colored green.
2.2 Subjective speckle displacement model
When an object is moved out of the object plane of the imaging system, the recorded speckle at different wavelengths will exhibit a displacement. Horváth et al. [12] showed that a defocused imaging system samples the speckle field at the object plane. This result was validated and employed by Heikkinen [13,14]. This simplifies the model for the case of subjective speckle. By calculating the lateral speckle displacement at the object plane and propagating this onto the image plane, one gets the speckle displacement for a complex optical system.
The application of this is illustrated in Fig. 3. The lateral speckle displacement is determined in two steps. First the speckle displacement for an objective speckle pattern between the object plane and the defocused, tilted object surface is calculated. Then this speckle displacement is imaged onto the camera sensor.
Fig. 3. Subjective speckle displacement model. The object is axially displaced from the object plane by $l_{\mathrm {O}}$ and tilted by $\theta$. Light incident on the surface is colored blue. Observed light is colored green. A ray bundle originating from the object plane and propagated into the image space is shown in gray lines. When the image plane is fixed, ray bundles have a focus in point $P$ on the object plane. This means that for a defocused, tilted object the rays through point $P$ originate from an ellipsoid area on the object surface. The area between points $P$ and $P_{\mathrm {O}}$ is shown in detail as inset on the top right corner of the image.
The optical system in Fig. 3 is simplified using a principal lens plane and a thin lens approximation. The object surface is moved away from the object plane in $z$-direction by an amount $l_{\mathrm {O}}$, the object displacement, and tilted be the angle $\theta$. Due to the tilt, the distance between the object and lens plane is increased by $l_{\mathrm {T}}$. The angle of incidence $\zeta$ is defined as in the objective speckle displacement. The angle of observation $\eta$ is different for each pixel. A point $P$ in the object plane is imaged onto a point $P\textrm '$ in the image plane. The distance between the optical axis and $P$ is denoted as $h$. The observation angle $\eta$ needs to be considered for each pixel separately. For a certain point on the object plane $\eta$ is given by
The distance $s$ is the distance between the principal plane of the lens and the object plane. This parameter is calculated using the lateral magnification $\beta \textrm '$ and the focal length $f$ of the optical system $s=f(1-\beta \textrm '^{-1})$. The observation angle is defined relative to the normal vector of the object plane. This tilt needs to be considered for the observation angle. For collimated light, the observation angle from Eq. (1) is then $\eta + \theta$.
To calculate the speckle displacement, we need the distance $l$ between the object and the object plane. This is the distance given by points $P$ and $P_{\mathrm {O}}$. $P_{\mathrm {O}}$ is found by extending the principal ray from the lens center through point $P$ onto the object. The solution for $l$ is derived using the inset in Fig. 3. The signum function of the object displacement $\textrm{sign} (l_{\mathrm {O}})$ is used to account for the displacement direction.
The speckle displacement for a certain pixel is then analogous to Eq. (1). The lateral magnification $\beta \textrm '$ of the optical system needs to be considered as well.
Thus, with knowledge of the respective parameters, the lateral speckle displacement for each pixel on the camera can be calculated. The required parameters are the pixel size, the magnification and the focal length of the optical system, the tilt of the inspected surface, the used wavelengths, the incident angle of the illumination and the object displacement. For the setup used in this paper, the parameters are given in Table 1.
Table 1. Parameters used in this work.
3. Experimental verification
3.1 Setup and experimental procedure
The experimental setup is illustrated in Fig. 4. A tunable continuous wave laser source based on an optical parametric oscillator (Hübner Photonics C-Wave VIS Low Power) is used as laser source to set a custom wavelength. This tunable laser is fiber coupled and covers wavelengths from 450 nm to 650 nm in the visible range.
Fig. 4. Experimental setup with on-axis illumination and defocused, tilted object. Light incident on the surface is colored blue. Observed light is colored green. The polarization state of the light is indicated by arrows to the right and above the beamsplitter.
A flat stainless-steel sample with a surface roughness 1.05 µm Ra is used as test sample. The sample is placed on a precision goniometer to control the tilt angle $\theta$. The rotation point of the goniometer is located 24.9 mm above the sample surface and 2 mm in $x$-direction. This leads to an additional displacement $l_{\mathrm {Offset}} = -24.9 (\sec (\theta )-1) - 2 \tan (\theta )$ of the object surface in mm, depending on the object tilt, which needs to be added to $l_{\mathrm {O}}$ before calculating the speckle displacement. The goniometer is fixed on a linear platform, which adjusts the distance $l_{\mathrm {O}}$ between the metal sheet and the object plane.
The linearly vertical polarized laser light is collimated by a lens. An adjustable aperture is used to ensure a uniform illumination profile by cutting off the outer part of the gaussian beam. We realize an on-axis illumination with circular polarized light using a polarizing beamsplitter (PBS) and a quarter-waveplate (QWP). The linear polarized light is reflected towards the sample in the first pass through the beamsplitter. The QWP changes the polarization twice. On the first pass through the quarter wave plate, the polarization is changed to circular. After diffuse reflection from the sample, the circular polarized light is changed to linearly horizontal polarized light on the second pass through the QWP. The linearly polarized light, now rotated by 90°, passes through the PBS towards the camera.
The sample is displaced in $z$-direction by an amount $l_{\mathrm {O}}$ and tilted with respect to the optical axis by the angle $\theta$. The illumination angle matches this tilt angle. The surface is imaged onto a CMOS camera (IO Industries Flare 48 MP monochrome) with a pixel size of 4.6 µm and a resolution of 7920 $\times$ 6004 pixels. To minimize chromatic aberrations in the optical setup, two achromatic lenses are used in the optical setup and positioned so that they operate close to their optimum working position. A manually adjustable aperture is placed between the lenses. This setup has a focal length of 76.5 mm and a lateral magnification of $\beta \textrm '=-0.95$.
The position and size of the aperture is critical for the properties of the specke field. As the numerical aperture of the system increases, more light will be incident on the detector. Additionally, the speckle size will decrease, which is beneficial for an accurate image registration with sub-pixel accuracy. On the other hand, the speckles need to be resolved by the detector. Therefore, the f-number $F/{\#}$ of the system is set to 13. This corresponds to an average speckle size of approximately two pixels. The illuminated area of the sample has a diameter of approximately 15 mm, which corresponds to an area of 3100 $\times$ 3100 px on the camera.
For this experiment three wavelengths were chosen: 633.3 nm, 637.8 nm and 644.4 nm. One image is recorded for each wavelength. Between the individual images, the wavelength of the laser was tuned. One set of all three wavelengths represents one measurement. For each measurement the captured speckle images are compared by using 2D cross-correlation to determine the speckle displacement between different wavelengths.
As previously mentioned, the displacement of speckles is influenced by the observation angle, which varies for each pixel in a camera. To calculate the displacement, we compare small areas of the different speckle patterns with each other. Consequently, the single images are divided into sub-images with dimensions of 100 $\times$ 100 pixels. We correlate the sub-images of different wavelengths to calculate the speckle-displacement in $x$- and $y$-direction. The displacement is determined by evaluating the distance of the maximum in the cross-correlation to the center. No displacement means the speckle patterns are identical and the peak is centered. To achieve sub-pixel accuracy we use the "single-step DFT approach" described in [15]. An implementation of this approach was published by Guizar in [16]: First the discrete Fourier transformation (DFT) $F(u, v)$ and $G(u, v)$ of the two sub-images $f(x, y)$ and $g(x, y)$ is calculated. $x$ and $y$ indicate the image point. Next the inverse DFT of the product $F(u, v)G^{*}(u, v)$ is calculated, to get an estimate about the peak location. A 1.5 $\times$ 1.5 pixel neighborhood (in units of the original pixels) around the estimate peak location is upsampled by a factor of 100. The inverse DFT of the product of the upsampled DFT $F$ with the complex conjugated upsampled DFT $G$ is used to get a correlation with sub-pixel accuracy.
In the following we use the $l^2$ norm of the lateral displacements $\Delta p_x$ in $x$-direction and $\Delta p_y$ in $y$-direction, denoted as $|\Delta p|_2$ and calculated as $({{\Delta p_x}^2 + {\Delta p_y}^2})^{1/2}$ to describe the results.
To set precise tilt angles of the object, the object surface is first aligned orthogonal to the optical axis of the system. This was done by evaluating the speckle displacement $|\Delta p|_2$ over the whole image. Each evaluated sub-image gives a $|\Delta p|_2$ according to the respective observation angle. The observation angle of an aligned surface is zero at the optical axis. With increasing observation angle the displacement increases. The object was therefore aligned such that the smallest speckle displacement is located at the optical axis. Figure 5 shows the displacement of the sub-images for a measurement with wavelengths 633.3 nm and 644.4 nm and an object displacement $l_{\mathrm {O}}$ of 10 mm. Each value in the two-dimensional map represents the displacement of the respective sub-images. The smallest speckle displacement $|\Delta p|_2$ is located at the image center and increases with distance to the optical axis or image center. Figure 5 therefore shows an aligned surface.
Fig. 5. Two-dimensional map of the speckle displacement $|\Delta p|_2$ for an object displacement $l_{\mathrm {O}}$ of 10 mm with an aligned surface (minimum in image center) and wavelengths of $\lambda _1=$ 633.3 nm and $\lambda _2=$ 644.4 nm. Regions where the displacement could not be calculated due to low light conditions are cropped.
Table 1 summarizes the parameters required to solve Eq. (10). Consider that due to the rotation point of the goniometer an offset $l_{\mathrm {Offset}}$ needs to be added to $l_{\mathrm {O}}$, as described previously.
4. Results and discussion
The results show that the model is capable to predict the speckle displacement with an accuracy which is typically a fraction of a pixel. With increasing distance from the optical axis, the deviation increases.
With increasing object displacement and increasing difference in wavelength the speckle displacement increases. In the absence of a surface tilt, the model describes the speckle displacement with equal accuracy for the whole image (Fig. 6) and less than 0.4 px deviation. For most measurements the deviation is even less than 0.2 px.
Fig. 6. Speckle displacement in $x$-direction through the optical axis of the system. Used wavelengths are 633.3 nm and 644.4 nm. The points are measurements while the solid line represents the speckle displacement calculated using Eq. (10). The calculated speckle displacement according to Eq. (10) is symmetric for positive and negative values of $l_{\mathrm {O}}$. Therefore $|\Delta p|_2$ is identical for $l_{\mathrm {O}} = -15 \, \mathrm {mm}$ and $l_{\mathrm {O}} = 15 \, \mathrm {mm}$ as well as $l_{\mathrm {O}} = -25 \, \mathrm {mm}$ and $l_{\mathrm {O}} = 25 \, \mathrm {mm}$.
The top graph in Fig. 6 shows the $l^2$ norm of the speckle displacement $(|\Delta p|_2)$ as measured and as calculated by the Eq. (10). The bottom graph shows the deviation between model and measurements, that is the $l^2$ norm of the difference between the measured and calculated speckle displacement ($|\Delta p_{\mathrm {measured}}-\Delta p_{\mathrm {calculated}}|_2= |\Delta p_{\mathrm {err}} |_2$). The speckle displacement is shown over the camera sensor in $x$-direction. The object displacement is varied for a surface with zero tilt.
The speckle displacement increases with observation angle and object tilt. Figure 6 shows that for pixel further away from the optical axis the speckle displacement increases and likewise decreases towards the optical axis. This is because the observation angle increases.
In order to visualize the distribution of speckle displacements, a polar coordinate representation can be employed when the object is aligned. Figure 7 displays the speckle displacements depicted as the radius relative to the image center. $|\Delta p_{\mathrm {err}} |_2$ is similar for different object displacements and increases with the radius. For $l_{\mathrm {O}}$ of −15 mm the deviation between model and measurement increases approximately by 0.1 px along the radius. The average value of $|\Delta p_{\mathrm {err}} |_2$ is lower than 0.2 px with a standard deviation lower than 0.1 px for all object displacements. The measured speckle displacements are 0.1 px larger than the model prediction for negative object displacements. For positive $l_{\mathrm {O}}$ the measured speckle displacements are 0.1 px smaller than the prediction.
Fig. 7. Speckle displacement over the radial coordinate. All sub-image displacements are shown. Origin of the polar coordinate system is the image center. Used wavelengths are 633.3 nm and 644.4 nm. The points are measurements while the solid line represents the speckle displacement calculated using Eq. (10). The calculated speckle displacement according to Eq. (10) is symmetric for positive and negative values of $l_{\mathrm {O}}$. Therefore $|\Delta p|_2$ is identical for $l_{\mathrm {O}} = -15 \, \mathrm {mm}$ and $l_{\mathrm {O}} = 15 \, \mathrm {mm}$ as well as $l_{\mathrm {O}} = -25 \, \mathrm {mm}$ and $l_{\mathrm {O}} = 25 \, \mathrm {mm}$.
In Fig. 8 the object is tilted by 5° around the $y$-axis. Figure 9 shows a fixed object displacement and increasing object tilt. At larger object tilt there is a deviation of the model from the measured displacement with increasing distance of the pixels from the optical axis, see Fig. 8. The deviation is larger for smaller object displacement. The model underestimates the speckle displacement at detector positions further away from the optical axis and small object displacement. With the tilt of the object, the position of zero observation angle moves away from the optical axis and camera center, see Fig. 9. In Fig. 9 is also shown that the deviation from the model increases, as the tilt of the object increases. For tilt angles including 7° the speckle displacement along the whole section matches the model with an accuracy of one pixel.
Fig. 8. Speckle displacement through optical axis in $x$-direction. The object is tilted by 5° around the $y$-axis. Used wavelengths are 633.3 nm and 644.4 nm. The points are measurements while the solid line represents the speckle displacement calculated using Eq. (10).
Fig. 9. Speckle displacement through optical axis in $x$-direction. The object is displaced by −15 mm. Used wavelengths are 633.3 nm and 644.4 nm. The points are measurements while the solid line represents the speckle displacement calculated using Eq. (10).
Figure 10 and Fig. 11 show the speckle displacement at the optical axis for different wavelength combinations. In Fig. 10 the surface is tilted by 5°, while the object displacement is changed. Figure 11 shows the speckle displacement for a fixed axial position of −15 mm and a variable surface tilt. For the image center in Fig. 10 the model estimates the speckle displacement with an accuracy of 0.2 px for object displacements up to −15 mm. This matches the results from Fig. 8 and Fig. 9. There is no significant trend of increasing deviation for larger object displacements. While the outer pixels in Fig. 8 and Fig. 9 show deviations larger than 0.6 px for 5° tilt, the deviations at the center pixel shown in Fig. 11 are lower than 0.6 px for tilt angles up to 10°. The difference increases slowly first and increases rapidly with angles larger than 7°. The work conducted by Fischer [17] indicates that there is a direct relationship between increasing surface tilt and a corresponding increase in measurement error.
Fig. 10. Speckle displacement at the optical axis for varying axial object displacement. The object is tilted by 5° around the $y$-axis. Used wavelengths are 633.3 nm and 644.4 nm. The points are measurements while the solid line represents the speckle displacement calculated using Eq. (10).
Fig. 11. Speckle displacement at the optical axis for varying object tilt around the $y$-axis. The object is displaced by −15 mm. Used wavelengths are 633.3 nm and 644.4 nm. The points are measurements while the solid line represents the speckle displacement calculated using Eq. (10).
Figure 12 shows the displacement separated into $x$- and $y$-components for 7° tilt and −15 mm object displacement. In this setup the object is tilted around the $y$-axis. In Fig. 12 the observation angle in $y$-direction is zero. For this reason, the respective measured speckle displacement is close to zero. The speckle displacement, given as $l^2$ norm of $x$- and $y$-component is mainly influenced by the $x$-component. This observation is true for the other results as well, as the observation angle in the $y$-direction remains consistently zero.
Fig. 12. Speckle displacement separated into $x$- and $y$-component at the optical axis. 7° tilt around the $y$-axis and −15 mm object displacement. Used wavelengths are 633.3 nm and 644.4 nm. The points are measurements while the solid line represents the speckle displacement calculated using Eq. (10).
Theory predicts the speckle displacement to change steadily with a change of parameters. The image registration approach used to determine the speckle displacement is said to achieve an accuracy of up to a hundredth of a pixel [15]. It is difficult to verify the accuracy here but comparing the absolute deviations from Eq. (10) to the registered displacements, an accuracy of about 0.1 px seems plausible (see Fig. 6, Fig. 7 and Fig. 12) which indicates a significant amount of noise.
Speckle displacement occurs not only in lateral direction, but also in axial direction. As the displacement increases, the speckle patterns are therefore less correlated, even if the lateral displacement is considered. For this reason, smaller displacements in general are calculated with a better reliability.
In this work we separate the images into sub-images and determine the speckle displacement for each of those sub-images. This gives an average displacement value for each sub-image and limits the lateral resolution of the displacement calculation. The sub-image size was chosen according to requirements of the SSC application in [9]. The reduction of the sub-image size is possible, but there is a threshold beyond which the reliability of the result is compromised. As can be seen in Fig. 5 there is a variance in the displacements, originating from the statistical nature of the speckle and the surface structure. Likewise increasing the sub-image size only benefits the accuracy up to a point because of the varying speckle displacement. Ideally, the speckle displacement should be calculated individually for each pixel. This can be achieved by moving the sub-image in pixel increments over the image.
Deviations between the model and the measurements become larger as the distance from the optical axis increases (see Fig. 7, Fig. 8 and Fig. 9). The objective spectral speckle model from Hanson et al. [10] is valid close to the optical axis. This indicates that the subjective speckle model derived in this paper has similar characteristics. The influence of the limiting aperture in the imaging system on the speckle displacement is not considered. In the model we associated the speckle displacement with one point on the object. In a defocused setup, the sampled speckles imaged onto one pixel of the camera sensor originate from a circular area, rather than a point. In the case of a tilted object, the surface contributing to one pixel lies within an ellipsoid. It is to be expected that this will influence the result. In addition, paraxial optics, which is only valid for small angles, is used here to derive the imaging of the objective speckles
An additional error source is the adjustment of the setup. The precision goniometer has a scale indicating 0.1° steps and is used to set the object tilt manually. The optical components are mounted on rails and slight misalignments are to be expected. Especially the position and alignment of the aperture is important, because this element determines the spatial frequencies contributing to the speckle displacement.
Figure 13 shows the correlation of the sub-images for two wavelengths. Here the surface is aligned and an object displacement of $l_{\mathrm {O}} = 10\mathrm {mm}$ is used. This correlation can be used to determine the roughness of the measured surface [8]. The measured surface is flat and exhibits a uniform rough surface, therefore we expect a uniform distribution of the correlation. In Fig. 13(a) the speckle displacement is not corrected. The correlation decreases with increasing distance to the optical axis, that is with increasing speckle displacement. This leads to an error in the calculation of the roughness in the SSC method described in [9]. In Fig. 13(b) the speckle displacement is corrected, which greatly improves the SSC.
Fig. 13. Correlation for speckle images recorded with this setup on a uniform surface with a roughness Ra of 1.05 µm, using wavelengths 633.3 nm and 644.4 nm. The surface is aligned and a displacement $l_{\mathrm {O}} =$ −10 mm is used. With increasing distance to the optical axis the speckle displacement increases and the correlation for respective sub-images decreases, as can be seen in (a). When the displacement is corrected, as in (b), the correlation is more uniform.
5. Conclusion
In this work, an extended model of [10] is theoretically derived and experimentally validated to describe the lateral displacement of subjective speckle patterns at different wavelengths of tilted and defocused objects. We make use of the concept that the optical system samples the objective speckle in the object plane. Thus, using the approximations of paraxial optics, the subjective speckle displacement can be calculated.
It is shown that this model, described by Eq. (10), can be used to calculate the speckle displacement in the case of focused and defocused objects as well as for tilted surfaces when the parameters in Table 1 are known. Using sub-images, the speckle displacement can be calculated with spatial resolution for the whole image sensor.
For aligned surfaces the speckle displacement is determined with an accuracy of 0.4 px pixel for axial object displacements up to $\pm$25 mm. With increasing tilt of the object, the accuracy deteriorates, especially in the corners of the image. At $\pm$7° tilt and at the edges of the image, the deviation is about one pixel. This deviation needs to be investigated further.
By using this model, the speckle displacement for arbitrary optical systems can be calculated. This approach enables the spatially resolved measurement of the roughness of technical objects with a large field of view and high precision using spectral speckle correlation.
Funding
Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung e.V. (SME 601 236).
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.
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