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Abstract
We show that the shape of a surface can be unambiguously determined from investigating the coherence function of a wave-field reflected by the surface and without the requirement of a reference wave. Spatio-temporal sampling facilitates the identification of temporal shifts of the coherence function that correspond to finite height differences of the surface. Evaluating these finite differences allows for the reconstruction of the surface using a numerical integration procedure. Spatial sampling of the coherence function is provided by a shear interferometer whereas temporal sampling is achieved by means of a Soleil-Babinet compensator. This low coherence profiling method allows to determine the shape of an object with sub-micrometer resolution and over a large unambiguity range, although it does not require any isolation against mechanical vibration. The approach therefore opens up a new avenue for precise, rugged optical metrology suitable for industrial in-line applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Interaction of light with matter is fundamental for optical metrology [1]. Light being scattered by, or transmitted through a specimen carries information about the structure and the properties of the object. This information is encoded in a propagating wave field [2] and can be investigated by means of dedicated sensors [3]. Examples include a wide variety of techniques for shape measurement from large objects using the recording of reflected intensity patterns [4,5] down to nano-sized objects using the investigation of polarization states [6]. In the state of the art, we also find established techniques that are based on the coherence state of a wave field, such as white light interferometry [7–9], optical coherence tomography [10–12] or spatial coherence profilometry [13,14]. In these techniques, light with limited coherence is superposed with a reference wave field after being scattered by a specimen under test, in order to determine its shape, volumetric structure or dispersion properties.
In this publication we further develop this concept towards a new paradigm. We investigate the hypothesis, whether sole knowledge of the coherence function of a reflected wave field is sufficient to determine the entire shape of an object, even if no reference wave field is present. This is expected to result in significant benefits for interferometric shape measurement, because the coherence function can be measured by differential methods, such as a Shack-Hartmann sensor [15] or a shear interferometer [16,17]. The common-path nature of a shear interferometer makes it insensitive against mechanical vibrations [18], thereby solving one of the primary problems of interferometric metrology: The need for mechanical stabilisation [19]. Furthermore, the absence of a reference wave field allows for decoupling of the illumination from the sensing unit, which enables a flexible and mobile system design, similar to a photographic camera. Finally, shear interferometric setups allow to measure relatively steep slopes in contrast to interferometric configurations based on a reference wave, because the fringe density can be adjusted by the magnitude of the shear.
In what follows, we will test the above hypothesis for the case of a reflective surface. Spatial sampling of the coherence function is provided by a shear interferometer whereas temporal sampling is achieved by means of a Soleil-Babinet compensator. Recently we have used a similar arrangement to derive step heights from the contrast of interference patterns measured by a shear interferometer [20]. Here we will further extent the approach of spatio-temporal sampling of the coherence function and we will show that we can determine the shape of a surface based on this concept and subsequent numerical integration. Because this technique establishes a direct relation between the shape of an object and the coherence function, which is often abbreviated by the Greek character $\Gamma$, we refer to it as $\Gamma$-profilometry. We will demonstrate experimental results using a step object and show that $\Gamma$-profilometry does not only represent a principally interesting concept but provides a new avenue for precise, rugged optical metrology.
2. Concept
In scalar wave theory, the time and space dependent fluctuations of a wave field are often expressed by means of a complex amplitude $U(\vec {x},t)$ [21]. Depending on its coherence properties, the light may exhibit a multitude of independent wave fields that correspond to various frequencies (wavelengths) and propagation directions. In this situation, it is very convenient to interpret the fluctuations of the complex amplitude as a statistical process and to describe the light by means of a correlation function, which is given by the mutual coherence function [22]
We will now explore, if the coherence function of light reflected by a surface allows to determine the shape of the surface, and how this information may be extracted. The basic relationship between the coherence function and the shape is illustrated in Fig. 1(a). It shows an object surface with a step profile as an example and light propagating along the vertical $z$-axis. For the sake of simplicity, we will restrict ourselves to cases in which the illumination is a plane wave. We discuss the coherence function for the two marked locations $\vec {x}_{1}$ and $\vec {x}_{2}$. Due to the surface step, the light reflected from the surface has to travel an additional distance of $2\Delta h$ to travel back to the height of $\vec {x}_{1}$, when compared to $\vec {x}_{2}$. Consequently it requires more time to reach $\vec {x}_{1}$ than $\vec {x}_{2}$, whereby the time difference can be expressed through the speed of light $c$ by $\tau _{h}(\vec {x}_{1}, \vec {x}_{2}) = 2\Delta h/c$. Hence, $\tau _{h}$ is directly related to the finite difference of the surface profile between $\vec {x}_{1}$ than $\vec {x}_{2}$.
Fig. 1. Concept of $\Gamma$-profilometry: Partially coherent light with average frequency $\nu$ and coherence time $\tau _{C}$ is reflected by an object surface. a) Because of the height difference $\Delta h$, reflected light requires more time to travel to point $\vec {x}_{1}$ than to $\vec {x}_{2}$. b) The resulting delay $\tau _{h}$ can be measured by sampling the coherence function along the temporal axis $\tau$. Knowing $\tau _{h}$ and the speed of light enables determination of $\Delta h$. Repeating the procedure at a large number of combinations $\vec {x}_{1}$ and $\vec {x}_{2}$ allows for numerical integration of the surface profile $h(\vec {x})$.
Since the coherence function depends on the time delay $\tau$, it should allow for the detection of $\tau _{h}$ by investigating shifts along the time axis. The idea is now to measure time differences for a large number of positions $\vec {x}_{1}$ and $\vec {x}_{2}$ across the object plane. The corresponding finite differences should then allow for determination of the surface shape by means of a numerical integration process. To determine $\tau _{h}$, we employ a light source with extended bandwidth, e.g. a light emitting diode (LED), that results in a limited temporal coherence, so that $\vert \Gamma \vert > 0$ only holds for a small interval along the $\tau$-axis. To clarify this approach, we take a closer look at the Cross Spectral Density $S_{xy}$ (CSD) [24], which is the temporal Fourier transform of the coherence function along the time delay $\tau$. The relation between both functions is thus given by
The spectral bandwidth of a typical LED is $\nu _{B}=13$ THz, which yields a coherence time of $\tau _{c}=67$ fs. With the speed of light $c~\approx ~300$ nm/fs, this translates into a coherence length of $l_{c} = 20.1$ µm along the propagation direction. Vice versa, a step height of $\Delta h = 150$ nm in the example of Fig. 1(a) will delay the light by $\tau _{h}=1$ fs and will therefore cause a corresponding shift of $\Gamma$. Since $\Gamma$ is confined (FWHM) in the interval $\tau _{c}$, we can evidently detect the shift $\tau _{h}$ by sampling of $\Gamma$ across the temporal domain. An illustration of $\Gamma (\tau )$ corresponding to LED light is seen from Fig. 1(b) along with the shift $\tau _{h}$. If we combine the temporal sampling with spatial sampling at numerous positions $\vec {x}_{n}$ and $\vec {x}_{m}$, it should be possible to recover the complete surface from the measured finite differences. In the following section, we will describe a measurement system which enables the spatio-temporal sampling of $\Gamma$ across its object plane and present experimental results.
3. Sensing device
Figure 2 shows the experimental setup that consists of two units. The dashed box on the left marks the temporal sampling unit, which makes use of a Soleil-Babinet (SB) compensator to provide temporal sampling of $\Gamma (\vec {x}_{1}, \vec {x}_{2}, \tau )$ along $\tau$. The box on the right indicates the spatial sampling unit, which allows for sampling of $\Gamma (\vec {x}_{1}, \vec {x}_{2}, \tau )$ across the spatial domain $\{ \vec {x} \}$ by means of a shear interferometer. The basic structure of the arrangement is a combination of two 4f-configurations with a common relay plane. The system creates two laterally separated images of the object across the sensor plane, thereby superposing the light originating from $\vec {x}_{1}$ and $\vec {x}_{2}$ at a point in the image plane. In the following, we will explain how the separation between $\vec {x}_{1}$ and $\vec {x}_{2}$, the so called shear, is controlled by means of the SLM, while the SB-compensator can be used to invoke a time delay $\tau$ between light coming from $\vec {x}_{1}$ and $\vec {x}_{2}$ to enable temporal sampling.
Fig. 2. Experimental setup: The arrangement consists of two $4f$-configurations, which are indicated by the grey areas. The first unit allows for temporal sampling of the coherence function by means of a Soleil-Babinet compensator. The second unit is a shear interferometer based on a liquid crystal spatial light modulator in the Fourier domain, which enables spatial sampling. The illumination comes from a fibre coupled LED Thorlabs M625F1. The CCD-Camera is an AVT Pike F145-B from Allied Vision Technologies with $2452 \times 2048$ pixels having a pixel pitch of $3.45$ µm. Further details are given in the text.
The sampling concept of the measurement system is based on birefringence. Both, the SB-compensator as well as the liquid crystal spatial light modulator (SLM) are birefringent elements. They are adjusted so that the orientations of their slow and fast axes coincide. The incoming illumination is randomly polarized, collimated light of a fibre coupled LED. The first polarizer is used to adjust the polarization towards $45^\circ$ relative to the birefringent axes of the compensator and the SLM. In this situation we can regard the light as being composed of two perpendicularly polarized but identical wave fields, with the respective field vectors being oriented towards the birefringent axes. In the following we will refer to them as the fast and the slow wave field.
The SB-compensator is used to insert a time delay $\tau$. As the slow axis of the SB-compensator and the slow axis of the following SLM-modulator are aligned, we can associate the fast wave field with position $\vec {x}_{1}$ and the slow wave field with $\vec {x}_{2}$. The 4f-configuration offers an important benefit for the imaging process. It ensures that the light reflected by the object is mostly collimated when passing the SB-compensator, thereby reducing spherical aberrations. In the spatial sampling unit, the SLM is located in the Fourier plane and generates a blazed grating [26]. Because of the birefringent nature of the SLM, only the slow wave field is affected by the blazed grating and therefore shifted in the sensor plane. The fast wave field will be simply reflected by the silicon back panel of the SLM. Due to this operation, the images of $\vec {x}_{1}$ and $\vec {x}_{2}$ are superposed at $P$. The polarizing analyzer in front of the camera is again set to $45^\circ$ with respect to the birefringent axes, to ensure interference between the slow and the fast light. For the comparably small shears we use, the diffraction efficiency of the SLM is about 90% for the first diffraction order, so that we face no significant loss in interference contrast. Higher orders appear due to the pixelation of the SLM, but the system is designed so that they arise outside of the area defined by the camera chip. Deviations caused by the cover glass of the SLM are very low frequent. Since the SLM is located in the Fourier domain, this only causes minimal blur in the camera domain which we found to be negligible. Because the spatial separation $\vec {s}$ between $\vec {x}_{1}$ and $\vec {x}_{2} = \vec {x}_{1}+\vec {s}$ can be adjusted by the period and the orientation of the blazed grating, this operation provides spatial sampling of $\Gamma$. The camera detects the intensity of the interference pattern, i.e. $I(\vec {x}) = \vert U(\vec {x})\vert ^2 + \vert U(\vec {x}+\vec {s})\vert ^2 + 2 \Re \{\Gamma (\vec {x},\vec {x}+\vec {s}, \tau )\}$, which directly depends on the real part $\Re \{\Gamma \}$ of the coherence function.
We should finally mention, that the method also requires a minimum spatial coherence defined by the magnitude of the shear. The LED in our setup is fibre coupled with the fibre having a diameter of $200$ µm. In our configuration the light in the object plane had a spatial coherence of approx. $500$ µm, which is the equivalent of approx. $150$ camera pixels in the image plane.
4. Measurement process and evaluation
To determine height differences $\Delta h(\vec {x}+\vec {s},\vec {x}) = h(\vec {x}+\vec {s}) - h(\vec {x})$ across the objects surfaces, we set a specific shear $\vec {s}$ using the SLM, and measured $461$ intensity distributions with the SB-compensator in different states. The compensator was adjusted in steps of $\Delta \tau =0.38$ fs, providing a temporal sampling in the range of $\pm 88$ fs. Hence, for any given position $\vec {x}$ in the object plane we yield a set of $461$ intensity values, which depend on the real part of the coherence function $\Re \{\Gamma \}$, as discussed in the previous section. A given height difference $\Delta h(\vec {x}+\vec {s}, \vec {x})$ causes a noticeable shift of $\Re \{\Gamma \}$, where the temporal sampling range of $\pm 88$ fs corresponds to detectable height differences in the range of $\pm 13.26$ µm.
From similar efforts in wave front sensing [27] we known that in order to unambiguously integrate a profile from finite differences, we require at least $3$ measurements with varying shears [2]. In our experiments, we performed $6$ measurements with the shears set to $\vec {s}_{1}=(7,0)$, $\vec {s}_{2}=(0,7)$, $\vec {s}_{3}=(23,0)$, $\vec {s}_{4}=(0,23)$, $\vec {s}_{5}=(73,0)$, and $\vec {s}_{6}=(0,73)$ in units of camera pixels in the sensor domain. From the $6$ sets of finite differences we integrated the profile $h(\vec {x})$ by minimizing the objective function
5. Experimental results
Figure 3 shows the sample and the procedure used to experimentally investigate the concept of $\Gamma$-profilometry. Both, the main requirement with respect to tip/tilt misalignment, and the main requirement with respect to slope and curvature of the object is that the reflected light has to pass through the entrance pupil of the imaging system. For the sake of simplicity, we used a specular diamond turned step height object with known shape, as shown in Fig. 3(a) that consists of concentric steps. The inner $4$ rings are separated by a step height of $5$ µm, followed by $4$ rings separated by step heights of $10$ µm. The investigated region of interest (ROI) is marked by the square and has a size of approx $6 \times 6$ mm$^2$. As an example, we show one of the $461$ intensity distributions recorded by the camera in Fig. 3(b), with the shear set to $\vec {s}_{5}=73$ pixels in horizontal direction and the SB-compensator in neutral position $\tau =0$. Due to the shearing process we yield a twin image. Every object feature appears twice but spatially separated by the shear. On the right, in Fig. 3(c), we have extracted the intensity variations along the temporal axis $\tau$ for the $2$ positions $P_{1}$ and $P_{2}$ marked in Fig. 3(b). It can be seen, that for position $P_{2}$, the peak of the coherence function is clearly shifted when compared to the situation at $P_{1}$. The reason is, that at $P_{2}$, images of two object points in different heights are superposed by the shear interferometer, so that the corresponding step height $\Delta h$ causes a time delay $\tau _{h}$ in the optical path. From the diagram, we see a time delay of $\tau = 33.64$ fs, which corresponds to a step of $\Delta h=5.05$ µm, as expected.
Fig. 3. The investigated sample and the experimental procedure to prove the concept of $\Gamma$-profilometry. a) The photo shows a diamond turned specular surface, which exhibits concentric steps of known height. The box marks the region of interest. b) An example of the recorded intensity corresponding to the horizontal shear $\vec {s}_{5}$ of $73$ camera pixels and $\tau =0$. c) The intensity variations along the temporal axis $\tau$ for the two positions $P_1$ (the central plateau of the object) and $P_2$ (across the first step) marked in b).
Figure 4(a) shows the measured profile, which has been obtained from the iterative integration process using measurements corresponding to all $6$ shears. As it can be seen, the shape of the step height normal has been precisely determined, which supports the fundamental hypothesis of this paper, that the shape of a surface can be determined from solely investigating the coherence function of a wave field reflected by it, for the case of reflective surfaces. As expected, the measured profile shows $4$ steps of $5$ µm followed by $4$ steps of $10$ µm. For comparison, the same area has been measured using a Keyence VK-X3000 white light interferometer. To achieve this, a $10X$ Mirau objective for the measurement was employed. To measure over a comparable area as the one presented in Fig. 4(a), the integrated stitching module was used, and $4\times 4$ patches were investigated. The result is shown in Fig. 4(b) and shows very good agreement with the one obtained from $\Gamma$-profilometry.
Fig. 4. a) Object shape obtained from measured finite differences by minimization of the objective function given in Eq. (5). b) Height profile of the sample surface in false color representation measured by means of a Keyence VK-X3000 white light interferometer. Both measurements agree very well.
To investigate statistical and systematic deviations, we measured the profile of a flat mirror with diameter of $1$" and a surface quality of $\lambda /20$. The resulting profile is shown in Fig. 5(a). We applied a $6$-th order fit, to approximate the systematic deviations which are shown in Fig. 5(b). The residual of the fit is seen in Fig. 5(c) and contains the statistical fluctuations. The systematic deviations are caused by the optical imaging system and the Soleil-Babinet-Compensator. The measurement shown in Fig. 5(b) can therefore be used in further measurements to compensate for them. The statistical fluctuations indicate a measurement uncertainty of approx. $\pm 35$ nm. The remaining uncertainty is mainly attributed to the comparably large coherence length of the LED light, so that in analogy to white light interferometry, the measurement uncertainty could be further improved by employing broad band light sources, e.g. a superluminescent diode (SLD).
Fig. 5. Measurement result for a flat mirror with surface quality $\lambda /20$: a) Measured profile, b) systematic deviations as obtained from a $6$-th order fit of the profile and c) difference between the profile and the systematic deviations (residual), showing the statistical fluctuations. The systematic deviations are caused by the optical imaging system, primarily by the Soleil-Babinet-Compensator, while the statistical fluctuations indicate a measurement uncertainty of $\sigma =\pm 35$ nm.
6. Conclusion
We have shown, that the shape of a surface profile can be determined by measuring the coherence function of light being reflected by it. The method we have presented here is based on spatio-temporal sampling of the coherence function of partially coherent light. From the coherence function we can identify temporal shifts that correspond to finite differences of the surface profile and allow recovering the shape subject to a numerical integration process. The corresponding experimental arrangement enables temporal as well as spatial sampling of the coherence function using a Soleil-Babinet compensator and a shear interferometer respectively. As a prove of principle, we have shown measurement results for a step height normal with known surface profile and we have investigated the measurement uncertainty by measuring a flat surface with known quality of $\lambda /20$.
The relation between the coherence function and the surface shape determined in this work has major implications towards the field of optical metrology. Similar methods, such as for example white light interferometry (WLI), rely on a separately guided reference wave field and are therefore significantly prone to environmental disturbances. In our results, even though no mechanical vibration compensation was required, we obtained measurement uncertainties of $\pm 35$,nm ($1\sigma$) over a large ambiguity range of $60$ µm. This figure is even likely to be improved when using broad band light sources. Therefore, measurement systems based on this principle may be employed in precise inspection of mechanical and electrical parts directly in an industrial production line.
Processing time is a decisive parameter for industrial applications. In our experiments we used a standard Intel Core i3 based office PC without any sophisticated hardware acceleration. On the software side we use MatLab for all our processing steps. On a basic system like this, it took us $10$ minutes for evaluating the step heights and the subsequent iterative surface reconstruction. However, on more sophisticated hardware with specialized FPGA or grafic card support and C++ programming, it should be possible to reduce the processing time towards much less than a minute.
In our investigations, we have proven the concept of $\Gamma$-profilometry by means of a reflective surface. Yet, the sole strictly limiting assumption is the spatial independence of the spectral characteristics of the reflected light. This is – to a good approximation within the wavelength regime used here – not only true for reflective surfaces, but also for many transmissive objects.
We have also successfully performed first investigations for the case of rough surfaces. However, the theory behind adaptation of Eq. (3) to rough surfaces is not straight forward and beyond the scope of this publication. The focus here is to demonstrate the basic principle of $\Gamma$-profilometry at the example of a simple object with known characteristics. Yet in the future, we will expand the scheme towards more general object structures, which requires adaptation of the method to speckle patterns and dispersion properties.
Annex
A. Relation to the Wiener-Khinchin theorem
We will start the discussion with light that is composed of a discrete set of plane waves with frequencies $\omega _{n}$, amplitudes $a_{n}(\vec {x})$ and wave vectors $\vec {k}_{n}$. In this case, we can describe the time dependent complex amplitude as
We seek to describe the situation in the object plane indicated by the dashed line in Fig. 1(a), which restricts $\vec {x}$ to this plane. Without loss of generality we can place the origin of our coordinate system also within the object plane. Since we have earlier assumed the illumination to be incident normal to the surface, we have then $\vec {k}_{n} \cdot \vec {x} = 0$ for all $\vec {x}$ and $\vec {k}_{n}$, and Eq. (6) simplifies to
To determine the coherence function for any two positions $\vec {x}_{1}$ and $\vec {x}_{2}$ and time delay $\tau$, we insert Eq. (7) into Eq. (1)
We will now consider the step $\Delta h$. The wave field at position $\vec {x}_{1}$ requires the additional time $\tau _{h}$ to reach the object plane, i.e. it is delayed by $-\tau _{h}$, so that
Although the sums under the integral produce a large number of cross terms, only those terms with the same frequency, i.e. $\omega _{n} = \omega _{m}$ will yield non vanishing contributions to the integration process, and we find
Up to now it was mathematically convenient to restrict ourselves to the discrete case. To extend the scheme to the more general case of a continuous spectrum, we will introduce the Cross Spectral Density $S_{xy}$, which for the discrete case can be written as
with $\Delta \omega = \omega _{n+1}-\omega _{n}$ being the spectral distance between neighboring frequencies. Please note, that $S_{xy}$ is indeed a frequency density, because it relates the $a_{n}$ to a spectral range $\Delta \omega$. Inserting this into Eq. (10) and using $\omega _{n} = n \cdot \Delta \omega$ yieldsWith this preparation, we can now reduce the spectral distance to infinitesimal small distances $\Delta \omega \rightarrow d\omega$, which creates the continuous variable $\omega = n \cdot d\omega$ and we yield
B. Sampling and shift detection
The temporal sampling was provided by a motorized micrometer screw which drove one of the calcite wedges of the SB-compensator. At each of the $461$ positions, the SLM was controlled to generate a sequence of $6$ shears (blazed phase gratings) and one image was recorded per shear. The result of the measurement was a set of $6$ stacks (one per shear) of $N=461$ recorded images.
Each stack was evaluated on a pixel basis, where we create one function $I_{xy}(n)$ per pixel position $(x,y)$, with $n=1{\ldots }N$ representing the sample number. From $I_{xy}$ we obtain the real part of the coherence function $R_{xy}(n)$ by subtracting the average $I_{xy,ave}=1/N \cdot \sum {I_{xy}(n)}$, i.e. $R_{xy}(n)=I_{xy}(n)-I_{xy,ave}$.
We measured the PSD $S_{xx}(\omega )$ of the LED and insert $\tau =n \cdot \Delta \tau$ in Eq. 3 to calculate shifted representations $\rho (n; \tau ')=\Re \{\Gamma (n; \tau ')\}$ of the real part of the coherence function across the sample positions, where $\tau '$ refers to the shift (represented by $\tau _{h}$ in Eq. 3). To find the correct shift $\tau _{h}$ in the measurements, we calculate the functional (scalar product)
and seek to estimate the optimum $\bar{\tau} \approx \tau _{h}$ which maximizes it, so thatC. Iterative integration process
In the following we will describe in detail how we numerically integrated the surface profile $h$ from the measured finite differences $\Delta h$. The numerical integration is based on the minimization of Eq. (5) with respect to $f(\vec {x})$. For the minimization we used the iterative steepest descent method, which follows the gradient $\nabla L$ to calculate the successor
where $k$ denotes the iteration index. The scalar constant $\alpha$ (step width) has to be found heuristically in each step, so that the current estimate $f^{(k+1)}$ minimizes Eq. (5). For convenience we also note the gradient here:Funding
Deutsche Forschungsgemeinschaft (265388903).
Acknowledgements
The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding this work within the frame of project $\Gamma$-Profilometrie, grant no. 265388903.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the presented results are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. T. Yoshizawa, Handbook of Optical Metrology (CRC Press, 2009).
2. U. Schnars, C. Falldorf, J. Watson, and W. Jüptner, Digital Holography and Wavefront Sensing (Springer-Verlag, 2015), 2nd ed.
3. J. Haus, Optical Sensors: Basics and Applications (John Wiley & Sons, 2010).
4. J. Burke, W. Li, A. Heimsath, C. von Kopylow, and R. B. Bergmann, “Qualifying parabolic mirrors with deflectometry,” J. Eur. Opt. Soc. - Rapid publications 8, 13014 (2013). [CrossRef]
5. M. Agour, C. Falldorf, and R. B. Bergmann, “Investigation of composite materials using SLM-based phase retrieval,” Opt. Lett. 38(13), 2203–2205 (2013). [CrossRef]
6. A. C. Diebold, A. Antonelli, and N. Keller, “Perspective: Optical measurement of feature dimensions and shapes by scatterometry,” APL Mater. 6(5), 058201 (2018). [CrossRef]
7. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31(7), 919–925 (1992). [CrossRef]
8. P. Lehmann, S. Tereschenko, and W. Xie, “Fundamental aspects of resolution and precision in vertical scanning white-light interferometry,” Surf. Topogr.: Metrol. Prop. 4(2), 024004 (2016). [CrossRef]
9. F. Wang, L. Liu, P. Yu, Z. Liu, H. Yu, Y. Wang, and W. J. Li, “Three-dimensional super-resolution morphology by near-field assisted white-light interferometry,” Sci. Rep. 6(1), 1–10 (2016). [CrossRef]
10. D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef]
11. A. G. Podoleanu, “Optical coherence tomography,” J. Microsc. 247(3), 209–219 (2012). [CrossRef]
12. V. Mazlin, P. Xiao, J. Scholler, K. Irsch, K. Grieve, M. Fink, and A. C. Boccara, “Real-time non-contact cellular imaging and angiography of human cornea and limbus with common-path full-field/SD OCT,” Nat. Commun. 11(1), 1868 (2020). [CrossRef]
13. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000). [CrossRef]
14. P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. 48(34), H40–H47 (2009). [CrossRef]
15. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. Sánchez-Soto, “Wavefront sensing reveals optical coherence,” Nat. Commun. 5(1), 3275 (2014). [CrossRef]
16. F. Pfeiffer, O. Bunk, S.-B. C A. Diaz, T. Weitkamp, C. David, V. J. D. Veen, I. Vartanyants, and I. K. Robinson, “Shearing interferometer for quantifying the coherence of hard x-ray beams,” Phys. Rev. Lett. 94(16), 164801 (2005). [CrossRef]
17. J.-H. Hagemann, C. Falldorf, G. Ehret, and R. B. Bergmann, “Form determination of optical surfaces by measuring the spatial coherence function using shearing interferometry,” Opt. Express 26(21), 27991–28001 (2018). [CrossRef]
18. P. Bon, N. Bourg, S. Lécart, S. Monneret, E. Fort, J. Wenger, and S. Lévêque-Fort, “Three-dimensional nanometre localization of nanoparticles to enhance super-resolution microscopy,” Nat. Commun. 6(1), 7764 (2015). [CrossRef]
19. S. Tereschenko, P. Lehmann, L. Zellmer, and A. Brueckner-Foit, “Passive vibration compensation in scanning white-light interferometry,” Appl. Opt. 55(23), 6172–6182 (2016). [CrossRef]
20. C. Falldorf, A. Hyra, A. Simic, and R. B. Bergmann, “Spatio-temporal sampling of the coherence function for step height measurements,” in Interferometry XIX, vol. 10749K. Creath, J. Burke, M. B. N. Morris, and A. D. Davies, eds., International Society for Optics and Photonics (SPIE, 2018), pp. 284–291.
21. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).
22. B. E. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, 2019), 3rd ed.
23. J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., 2015), 2nd ed.
24. M. J. Bastiaans, “Application of the wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3(8), 1227–1238 (1986). [CrossRef]
25. A. Keppens, W. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108(4), 043104 (2010). [CrossRef]
26. C. Falldorf, M. Agour, and R. B. Bergmann, “Digital holography and quantitative phase contrast imaging using computational shear interferometry,” Opt. Eng. 54(2), 024110 (2015). [CrossRef]
27. M. Servin, M. Cywiak, and A. Davila, “Lateral shearing interferometry: theoretical limits with practical consequences,” Opt. Express 15(26), 17805–17818 (2007). [CrossRef]
28. J. A. Snyman, Practical Mathematical Optimization (Springer, 2005).
