Three-dimensional encoding of a gas–liquid interface by means of color-coded glare points

The formation and characteristics of glare points in both the theory of geometrical optics (GO) and wave optics (Born et al. 1999) will be briefly recapitulated here in the context of droplets. Glare points are the bright spots on the surface of the droplet that result from the scattering of a wide beam of parallel light according to van de Hulst (1981). In GO, glare points are described as the exit points of light rays that are either reflected or refracted on the surface in the direction of scattering \(\theta \) (Kerker 1969), as can be seen in Fig. 1. This results in the emergence of different orders of glare points, defined by the chords traveled within the droplet p (Debye 1908), where \(p=0\) indicates the externally reflected light, \(p=1\) the transmitted and twice refracted light rays and \(p=2\) the refracted and internally reflected light rays.

As indicated in Fig. 1 by the parallel \(p=0\) (red) and \(p=3\) (green) rays, multiple orders of glare points can be visible simultaneously for the same scattering angle. Conversely there exist ranges of scattering angles at which certain orders of glare points are not visible at all, as illustrated by the \(p=1\) (orange) ray in Fig. 1. According to van de Hulst and Wang (1991) all relevant rays for the formation of glare points at a perfect sphere propagate within the scattering plane, defined by the direction of the incident light and the scatter direction. Furthermore, the GO theory of glare points is applicable for volumes with a plane of circular cross section and, therefore, to spheroids and cylinders as well. The description of glare points in GO is valid for size parameters \(x=(2\pi a)/\lambda \gg 1\), with 2a describing the droplet diameter and \(\lambda \) the wavelength of the incident light. This holds true for \(2a\approx 2\) mm and \(\lambda \) =  400–700 nm in the experiments conducted in the present work.

The required equations for the calculation of the glare point positions in GO will be summarized in the following. The angle between an incident light ray and the surface of the droplet \(\tau \) can be set in relation to the angle between the refracted ray and the surface \(\tau '\) according toconsidering Snell’s law (see e.g. Hecht 2017) with the refractive indices \(n_1\) of the surrounding gas and \(n_2\) inside the droplet.

Considering the possible cases of external reflection, \(p-1\) internal reflections and the refraction of a light ray, the deflection angle \(\theta '\) can be determined. Subsequently, the scattering angle \(\theta \) can be calculated to bewhere k is an integer, depending on the number of internal reflections ( van de Hulst 1981, pp.228ff) and \(q=\pm 1\), depending on the side of the projected droplet relative to the optical axis, which can be described by the variable \(w=\alpha l/a\). This dimensionless quantity w describes the position on the image plane relative to the droplets projection as function of the droplet diameter 2a, the distance between the lens and the droplet l, and the angle \(\alpha \) between the optical axis and the position on the droplet, as seen from the lens perspective (see Fig. 1). The position of the glare point in the dimensionless coordinate w follows asNote that Eqs. 2 and 3 only hold for a spherical droplet. Note, however, that a GO description for non-spherical droplets, which are symmetric around the optical axis was developed by Hovenac (1991).

The scattering angle is non-monotonically dependent on the incident angle for \(p \ge 2\). Therefore, \(\theta (\tau )\) possesses an extremum at the rainbow angle \(\theta _c\) (Walker 1976), which can be determined by \(\textrm{d}\theta /\textrm{d}w=0\) (van de Hulst 1981). The deflection of the incident light rays \(\theta '\) is minimal at the rainbow angle and increases toward both larger and smaller scattering angles. As such, the density of the light rays exiting the droplet reaches a maximum at \(\theta _c\), which results in the highest scattered intensity at the rainbow angle. The scattering angle \(\theta \) can either be maximal or minimal at \(\theta _c\), depending on the order of the rays p. A further consequence is that no scattered rays beyond the rainbow angle are possible.

Note that the two visible rainbows in nature can be explained accordingly—rays of the order \(p=2\) create the primary rainbow and \(p=3\) rays produce the weaker secondary rainbow above the primary rainbow. Higher orders of rainbows become increasingly dim as their incident angle increases such that less droplet surface is illuminated. Furthermore, the light is partly refracted at every reflection. Due to dispersion, the polychromatic light of the sun is scattered at slightly different angles around the rainbow angle, creating the well known color gradients. For \(p=2\) the rainbow angle is the minimal possible scattering angle and for \(p=3\) it is the maximum scattering angle, which results in inverted color gradients and also explains the appearance of Alexander’s dark band between the rainbows in which no light is scattered (Nussenzveig 1977).

The complete description of glare points for arbitrary-sized spheres is given in wave optics through Lorenz-Mie theory, which is the exact solution for the problem of light scattering on a spherical particle (Mie 1908). van de Hulst and Wang (1991) derive the equation for the amplitude of the glare points with the following assumptions. Firstly, a well defined scattering plane, that contains the direction of incidence and the direction of scattering is assumed. Consequently, the scattering plane excludes the \(\theta = 0^\circ \) and \(180^\circ \) scattering angles. Secondly, the authors assume that the lens both follows the thin lens equation (Hecht 2017) and is placed in the far field. The glare point equation accordingly follows asin which \(S_{\parallel \perp }(\theta )\) is the electrical field amplitude of the scattered light, \(\theta _0\) is the scattering angle and 2b is the diameter of the lens. \(S_{\parallel \perp }(\theta )\) represents either parallelly \(S_\parallel \) or perpendicularly \(S_\perp \) polarized light. Consequently, the size of glare points arising from a plane wave impinging on the droplet will depend on the size (and shape) of the receiving aperture. The expected amplitude of the scattered light following Eq. 4 for the combination of all glare point orders can be seen in Fig. 10. A comprehensive derivation of the scattering on spherical particles is provided in the treatise of van de Hulst (1981).

The present approach extends the shadowgraphy method by supplementing two lateral light sources to the standard shadowgraphy setup as shown in Fig. 2. These additional light sources are mounted on a custom-built rail system as can be seen in Fig. 2b in order to allow for the adjustment of the azimuth angle \(\theta \), while the elevation angle \(\Phi \) can be adjusted by a hinge. This mounting system facilitates the independent and continuously variable adjustment of both angles and, thereby, allows for the evaluation of the influence of the direction of illumination on the glare-point location and intensity on the gas–liquid interface. The angular adjustment uncertainty is estimated to be lower than \(\pm 2^\circ \). In order to ensure a reproducible droplet volume over multiple experiments and to avoid oscillations of the droplet from the introduction of momentum, the water droplet is produced by an automatic drop-application system. This system consists of a syringe with a cannula diameter of \(d_s = 0.1\) mm and a stepper motor with \(l=10\) mm linear displacement and a step angle of \(\alpha =7.5^\circ \), which is used as a linear actuator for the syringe. The resulting droplet diameter for distilled water is \(2a\approx 2\) mm at room temperature. The considered substrate has a hydrophobic surface created by a silicon oxide (SiOx) coating.

The two lateral light sources, as well as the backlight are high-power ILA_5150 LPSv3 LEDs with narrow-banded spectra and maxima in the visible spectrum at \(\sim 455\) nm (”blue”), \( \sim 521\) nm (”green”) and \(\sim 632\) nm (”red”) for the desired blue, green and red light sources, respectively. The light of the LEDs is captured either at 3,000 frames per second (fps) by a Photron Fastcam Mini WX RGB-Camera or at 7,500 fps by a Photron Nova R2, both equipped with a Schneider-Kreuznach Apo-Componon 4.0/60 enlarging lens, where the three-colors \(i=\) (”red”, ”green”, ”blue”) of illumination are expected to be mainly captured with the corresponding channels of the RGB camera chip.

The backlight produces a classical shadowgraphy image on the corresponding image channel, where only non-deflected light reaches the camera chip. In order to capture the shape of the droplet as accurately as possible a parallel light beam is required, leading to a better resolution of small features and optimizing contrast (Settles 2001). This is achieved by placing an optical collimator consisting of a pinhole aperture with a diameter of \(d_a = 4\) mm and a Spindler & Hoyer biconvex collimator lens with 300 mm focal length between the backlight and the droplet. The blue LED is used for the backlight, since the camera chip possesses the lowest relative sensitivity in this range of wavelengths and the highest response in the chip is expected from direct illumination of the backlight LED.

The lateral light sources are focused by Thorlabs plano-convex lenses with a broadband anti-reflective coating (reflectivity \( <0.5\%\)) and illuminate the droplet from an azimuthal angle \(\theta >90^{\circ }\) in respect to the blue LED. Consequently, only reflected light reaches the camera, resulting in the creation of \(p=0\) glare points (van de Hulst and Wang 1991). The illuminating light beams possess a cross section that is larger than the droplet and are approximately parallel, which implies that the beams can be represented as plane waves. Since the light is partially transmitted at the first interface between air and water, it is reflected again on the following interfaces in the path of the light ray, e.g. while leaving the droplet or at internal phase boundaries of air inclusion within the droplet, resulting in higher order glare points (\(p\ge 2\)).

Due to the smooth surface of the phase boundary, pure interface reflection is considered, which is captured by the camera as differently colored glare points on each side of the droplet and on any present air bubbles, thereby encoding additional volumetric information on the droplet in the remaining red and green image channels. Since the information is separated between the channels of the RGB-image, three individual grayscale images with accordingly different illumination conditions can be extracted from the RGB-image, each providing a unique representation of the droplet that can be used for the volumetric reconstruction of the droplet.

The spectral power distribution of the light sources do not perfectly match with the spectral sensitivity of the corresponding camera channels \(k=\)(R, G, B) as illustrated in Fig. 3. Therefore, each light source causes a perturbation of the images in the other channels, which consequently needs to be considered as follows.

The response of the kth color channel at pixel x on the camera chip \(I_k(x)\) can be calculated by spectral integration from the spectral power distribution (SPD) of the incoming light \(L(\lambda )\), the spectral transmittance of the color filter \(T_k(\lambda )\) for the kth color channel and the spectral responsivity \(s(\lambda )\) of the camera (Shafer 1982) asEquation 5 implies the assumption of a linear camera-chip behavior, i.e., the measured signal at a pixel’s location x is proportional to the intensity of the incident light at this position. This assumption generally holds true, however, with little error for CMOS cameras (Wang and Theuwissen 2017). The spectral transmittance \(T_k(\lambda )\) and the spectral responsivity of the camera \(s(\lambda )\) are often expressed in a combined form as the spectral sensitivity of the camera \(S_k(\lambda ) = T_k(\lambda ) s(\lambda )\) for the kth color channel. The term of the camera’s spectral sensitivity \(S_k(\lambda )\) also includes the wavelength-dependent transmittance of the objective lens \(T_o(\lambda )\), while the transmittance of the focusing lenses of the light sources can be assumed to be constant in the relevant wavelength-band, as seen in Fig. 3. Since spectral integration is a linear transformation, an integration over \(L(\lambda )\), which is the combined superposed spectral power distribution of all light sources i via \(L(\lambda ) = \sum _{i=1}^{n} \int _{\lambda } L_i(\lambda )\), results in the same camera response as the sum of an integration over all light sources \(L_i(\lambda )\) (Klinker et al. 1988). Therefore, Eq. 5 can be rewritten asConsidering that light transport from the two lateral light sources to the camera chip can occur either directly or by scattering on the droplet, thus creating glare points, the light transport coefficient \(l(\lambda ,x)\) at pixel position x on the camera sensor is introduced asIn order to consider the contributions of direct illumination, reflection at the droplet interface and transmission through the droplet, the light transport coefficient is expressed aswhere d(x) represents the direct light transport, \(r(\lambda ,x)\) the spectral reflectance and \(t(\lambda ,x)\) the transmission. Generally, spectral reflectance and transmission are dependent on direction and wavelength of the light incident on the surface and the geometrical shape of the illuminated object, whereas the participation from direct light only depends on the illumination angle. Furthermore, the spectral reflectance can be divided into interface (specular) reflection \(r_s(\lambda ,x)\) and body (diffuse) reflection \(r_d(\lambda ,x)\), according to the dichromatic reflection model (Shafer 1984), i.e.,For a water droplet, the diffuse reflection \(r_d(\lambda ,x)\) is neglected due to the smoothness of the surface, so that reflection can be assumed to be only specular reflection \(r_s(\lambda ,x)\) in the following.

The light transport by transmission \(t(\lambda ,x)\) is determined by the two-fold refraction at the droplet interface \(r_f(x,\lambda )\) and the spectral transmittance \(T_D(x,\lambda )\) of the droplet liquid. Following the Beer-Lambert law (Swinehart 1962), the absorption of distilled water in the visible spectrum is negligibly low for the considered distance traveled within the droplet (Hale and Querry 1973), even considering longer distances for higher order glare points. Therefore, the spectral transmittance can be considered to be \(T_D \approx 1\) in the range of the considered wavelengths (\(\lambda \approx 400-700\) nm) and its influence on the light transport trough transmission can be neglected.

Interface reflection and refraction are in general dependent on the refractive index n following the Fresnel equations (Hecht 2017). Since n is furthermore dependent on the wavelength of the incident light, i.e., \(n=n(\lambda )\), the intensity of both reflected and transmitted light on the interface likewise depend on the wavelength. Note that the light of the LEDs is unpolarised. Therefore, in the following the effects of polarisation on the reflection are not considered.

The neutral interface reflection (NIR) model is a good approximation for a water droplet. It states that the spectral power distributions (SPD) of the reflected and incident light are approximately equal (Lee et al. 1990). This assumption is valid for materials with insignificant variation of refractive index in the visible spectrum, which holds true for water. With the assumption of dominating specular reflectance and the NIR model the spectral reflectance r becomes independent from the wavelength, i.e., \(r(x) = r_s(x)\). Furthermore, for \(n \approx const.\) the refraction becomes independent of the wavelength \(r_f(x)\). Consequently, the light transport only depends on the geometry of the problem l(x) and Eq. 7 can be accordingly rewritten asSince both the SPD of the light sources and the spectral sensitivity of the camera are known, an integration across all relevant wavelengths for the camera sensor results in a transfer coefficientto quantify the transfer of light from each light source \(L_i\) to each channel of the camera chip \(I_k\).

For the presented measurement setup, consisting of three LEDs with different maximal wavelengths in the red, green and blue spectral range of the visible light \((L_r,L_g,L_b)\), and a camera chip with a red, a green and a blue channel \((I_R,I_G,I_B)\), the resulting transfer function is represented byFinally, the inverse of the transfer matrix \({\textbf{T}}\) provides a correction matrix \({\textbf{C}} = \mathbf {T^{-1}}\) for a determination of the image \({\textbf{l}}\) created as a result of the light transport from only a singular light sourcefrom the response of the three image channels \({\textbf{I}}\). Therefore, the analytical spectral correction (ASC) function 13 allows for a separation of the images created by each only one of the light sources, resulting in three independent images of the droplet.

The transfer coefficients in Eq. 12 and thus the correction matrix can also be obtained in-situ from the experimental apparatus by means of calibration images, in the following called in-situ color correction (ISC). ISC is particularly useful if the spectral characteristics of the optical components are unknown and, consequently, ASC cannot be applied. For this purpose three sets of calibration images have to be recorded in the presented setup through an RGB-camera, where consecutively only a single light source \(L_i\) is active per recording. Afterward the channels \(I_k\) of the RGB-images are split from each other to achieve nine grayscale images that contain the required information to calculate the transfer coefficients \(t_{k,i}\) of Eq. 12. An accurate estimate for the transfer coefficients can be obtained by calculating the maximum channel intensities and averaging over a large set of calibration images, thus reducing the influence of random noise. While ISC proved to be effective, additional measurement errors have to be considered in comparison with the above-introduced analytical method.

The images of a water droplet impinging onto a flat substrate recorded for the present work with the above-introduced RGB shadowgraphy setup and their processing are discussed in the following section. The required information for the ASC function 13 is calculated from the SPD of the LED lights and the spectral sensitivity of the camera with the values provided by the manufacturers; see Fig. 3.

Figure 4 shows an RGB frame of the droplet shortly after impact and the separated response of the uncorrected channels of the raw image. The comparison of the three separate image channels to the RGB image demonstrates that the additional information from reflective images of lateral light sources of different wavelength bands can be separated into the channels of an RGB image, which however contain some spurious crosstalk between the images. Saliently, the background of the green channel is considerably illuminated by the blue light source and the reflective images from the lateral light are apparent on the other channels, but with a lower intensity. The annotations in Fig. 4b–d highlight the most noticeable perturbations in the respective color of the causative light source.

To reduce the cross-talk effect, the above-discussed ASC function 13 has been applied to the RGB image of Fig. 4. The results are shown in Fig. 5 below the raw images to allow an immediate comparison of all spurious patches. Obviously, most disturbances in Fig. 4 have been successfully removed or at least reduced significantly. This is most saliently emphasized by the removal of the background light in the green channel and the transmitted light around the vertical center line resulting from the blue light source. The glare points from the red light source on the right side of the droplet in the green image are reduced as well. In the blue and red channels the glare points from the green light source on the left side of the droplet are likewise removed successfully. The full series of RGB frames shortly before and after the drop impact, corrected with the ISC function, is provided in Appendix 1 to further contrast the raw images (Fig. 19) to the corrected images (Fig. 20).

A quantitative accuracy evaluation of the image color correction has been conducted on images from the RGB imaging with only one activated light source. In this setup a complete color correction is expected to result in zero intensities for all but the considered image channels and only the channel corresponding to the active light source is expected to contain an image. The quality of the color correction is measured by the mean image intensity in the omitted image channels, i.e., the remaining disturbance after correction. Figure 6 shows the temporal evolution of mean image intensities for all channels in the first 50 images of the drop impact before (dashed lines) and after (solid lines) color correction.

As expected, each light source causes the highest response in the corresponding image channel and additionally a smaller response in each of the two other channels. The intensity of the disturbance in the omitted channels is significantly reduced by the color correction, as becomes obvious from the reduction of average intensity for the respective channel to near-zero values for most frames. This evaluation, therefore, confirms the accuracy of the applied analytical color-correction approach. While the correction of the blue light is nearly perfect, the disturbance in the other channels cannot be fully eliminated for particular frames recorded with only the lateral green or red light source. The corresponding frames are characterized by high overexposure, such that clipping effects lead to an effective overestimation of the other channels. One such event is shown in Fig. 7, which occurred in the 38th frame of an RGB shadowgraphy experiment with only the green lateral light source activated (cp. Fig. 6b, where the considered frame is marked with a black arrow).

To evaluate the accuracy of the recorded glare-point locations and intensities, the occurrence of the glare points in the experimental images is compared to the theoretically expected results as can be calculated from the Lorenz-Mie theory for scattering on a spherical particle with Eq. 4. In the following the glare-point position is defined by the location of the maximum glare-point intensity. First, the position of the glare points on the projected gas–liquid interface of the droplet, as well as their relative intensity is evaluated for the simple case of a spherical droplet illuminated at \(\Phi = 0^\circ \) elevation angle. The azimuth angle is varied from \(\theta = 95^\circ \) to \(145^\circ \) in increments of \(10^\circ \), which corresponds to the angular range for which back-scattering occurs and that was simultaneously possible with the presented experimental setup. A corresponding set of recordings is shown in Fig. 8, where only the red channel appears as gray scale images. To guide the unaided eye, the uncorrected image channel with e.g., spurious backlight is shown, even though the corrected image channels were processed and are discussed below.

Figure 9 shows the column-wise averaged intensity of the red image channel after applying the previously introduced ISC method, therefore revealing the distribution intensity of the glare points over the projected gas–liquid interface of the droplet. The column-wise intensity was calculated for five rows of the color-corrected image around the maximum horizontal expanse of the droplet and subsequently averaged over five frames. Note that the third (center) frame of this temporal average corresponds to Fig. 8. The reconstructed glare point signal allows for a comparison of the position of the glare points in the experiment to the position expected from the calculations by van de Hulst and Wang (1991) following Eq. 4, shown in Fig. 10.

The good agreement of the position of the glare-point peaks in the experimental images and the theoretically expected intensities allows for an interpretation and subsequent differentiation of the contributing orders in the observed glare-point signal from the experiment. In Fig. 9a only the zeroth order glare point (\(p=0\)) is clearly visible at around \(w=0.67\), which is in good agreement with the theoretically expected position. Toward higher scattering angles the zeroth order glare point moves toward the center of the droplet and reaches \(w=0.29\) at \(145^\circ \), as can be seen in Fig. 9f. The exact glare point positions in geometrical optics are calculated by Eq. 2 and subsequently Eq. 3 for further comparison. The results showed a good agreement with the experimentally determined positions.

In Fig. 9b additionally a third order glare-point peak (\(p=3\)) becomes visible at \(w=0.73\), despite the order-of-magnitude difference in intensity to the main peak. The comparison of the predicted intensity with the measured signal confirms the presence of a third order peak to the right of the strong zeroth order peak (cp. Fig. 10b). The intensity of the third order glare point increases toward higher scattering angles as it approaches its rainbow angle at \(129.5^\circ \). In Fig. 9c the third order glare point is clearly visible at \(w=0.8\). The theory predicts a second third order glare point at the \(w=1\) edge of the droplet that can be confirmed by a clearly visible intensity peak in the experiment. Both third order glare points are predicted to merge into a single, bright glare point at their rainbow angle, as can be seen in Fig. 18b. However, in Fig. 9d they can still be clearly differentiated at a scattering angle of \(125^\circ \) as evident from the experimental image and the corresponding intensity curve.

A second order glare point (\(p=2\)) first emerges in Fig. 9e as a large and bright glare point around \(w=-0.92\). Since the scattering angle lies close to the second order rainbow angle of \(137.7^\circ \), the intensity of the glare points is high. In Fig. 9f two second order glare points are visible at \(w\approx -0.61\) and close to the edge. As predicted by theory, the second order glare point, therefore, behaves in the opposite way as the third order glare point and splits into two peaks for scattering angles larger than the rainbow angle. The comparison of the measured and computed glare points close to the \(p=2\) rainbow angle (see Fig. 18c) reveals, that in fact the scattering angle in the experiments has been closer to \(137.7^\circ \) than \(135.0^\circ \), as the expected relative intensity peaks produce a better match. This elucidates that the intensity of the glare points close to their respective rainbow angles reacts sensible to small angular changes. However, the position of the glare points does not change significantly, therefore allowing for an accurate prediction of glare point occurrence by Eq. 4.

The first order glare point (\(p=1\)) was not visible in the range of evaluated azimuthal angles \(\theta \), as it appears only at scattering angles smaller than \(82.6^\circ \) according to geometrical optics and following Eq. 2. In a similar manner the intensity of the fourth and fifth order glare points is predicted to be too low to result in visible glare points, which is confirmed by the experimental results. Higher order rainbows accumulate at the \(w=\pm 1\) edges, resulting in intensity peaks from potentially multiple higher order glare points, complicating the interpretation of the edge peaks. Furthermore the higher order glare points become increasingly more sensible to deviations in the scattering angle, since the greater number of internal reflections amplify any deviation. Therefore, the interpretation of higher order glare points becomes less reliable.

According to van de Hulst and Wang (1991) additionally surface waves might contribute to the edge peaks as well. However sixth (\(p=6\)) and seventh order glare points (\(p=7\)) appear plausible following the descriptions of van de Hulst and Wang (1991). The intensity peak at the \(w=-1\) edge in Fig. 9d, e matches the theoretical position of the sixth order glare point, while in Fig. 9e, f the peaks at \(w=0.96\), respectively at \(w=0.98\) match the expected seventh order glare point. Additionally, the diffraction on the droplet is visible around the contour of the droplet in all images, contributing to all the glare point signal in Fig. 9 and in particular at the edges.

It should be noted that according to Walker (1976) the rainbow angle calculated in wave optics differs by \(\approx 0.15^\circ \) for \(p=2\) and \(\approx 0.21^\circ \) for \(p=3\) in comparison with geometrical optics. The resulting deviation in the \(p=3\) glare-point position was calculated to be \(\delta _w < 0.0005\), which is considered negligible. The angular difference for the \(p=2\) rainbow angle was even lower, which further confirms that the theoretical calculation provides a valuable and meaningful comparison with the experimentally observed glare points.

To further study the influence of the elevation angle of the lateral light sources on the visibility of glare points on the deforming droplet during impact, the lateral light sources were successively raised and simultaneously tilted in order to illuminate gas–liquid interface from different angles, as sketched in Fig. 2. The elevation angle was varied in a range of \(0^\circ \le \Phi \le 45^\circ \) with increments of \(15^\circ \) and the additional angle of \(\Phi =50^\circ \), while the azimuth angle was varied from \(\theta = 95^\circ \) to \(145^\circ \) with \(10^\circ \) increment. A drop impact experiment with the same kinematic boundary conditions was recorded for each possible combination of elevation and azimuth angles. The images from the experiments were post-processed with the previously introduced ISC in order to obtain the undisturbed glare point images. Figure 11 shows the development of the glare points for experiments performed at \(95^\circ \) azimuth angle in comparison between \(\Phi =0^\circ \) and \(\Phi =45^\circ \) elevation angles by the mean intensity of the color-corrected red and green image channels.

The comparison of Fig. 11a, b provides evidence that the glare-point intensity fluctuates significantly more at the lower elevation angle of \(\Phi =0^\circ \) with some frames reaching close to zero intensity, thus indicating that no glare points were visible in these frames. Such low intensity frames were already apparent in previous experiments at \(\Phi =0^\circ \) elevation angle in which only the lateral light sources were active, as can be seen in Fig. 6. At higher elevation angles, in contrast, significantly more frames contained glare points and the development of the intensity between the frames was more consistent as well. This effect increased toward \(45^\circ \), after which no further improvement was observed. Figure 12 shows the least luminous frame in the experiments at \(\Phi =45^\circ \) elevation angle (Fig. 12a) and in comparison a frame at \(\Phi =0^\circ \) (Fig. 12b) for a similarly shaped droplet.

As evident from the images at \(\Phi =45^\circ \) elevation angle even the least luminous frame contains clearly visible glare points although faint, as opposed to \(\Phi =0^\circ \) where no glare points are visible. This implies that glare points must be visible in all other frames, which was confirmed by visual inspection. At the chosen azimuth angle of \(\theta =95^\circ \), as expected from the results of Sect. 3.2, only zeroth order glare points appear independent from the elevation angle. Analogous results were obtained for both the intensity and consistency of the glare points at all other azimuthal angles tested in the experiments, including those that resulted in higher-order glare points. For elevation angles larger than \(\Phi =0^\circ \), however, no higher order glare points were observed after a certain deformation of the droplet due to the interaction of the light with the substrate after drop impingement.

Figures 4, 5 and 8 indicate that the zeroth order glare points appear only at locations on the surface of the droplet that align with the complementary angle \(\Phi _c = 90^\circ -\Phi \) to the direction of the incoming light as viewed from the image plane. In general, the phase boundary of the droplet takes any angle that ranges between the contact angle at the contact-line position and \(0^\circ \) degrees at the top of the droplet. Higher angles are possible due to the deformation of the droplet, as can be seen for example in Fig. 12, but do not necessarily occur. If the contact angle does not exceed the complementary angle to the direction of the incident light, accordingly, frames with no visible glare points might be produced. Consequently, those frames lack the desired additional three-dimensional information of the gas–liquid interface.

The completely horizontal setup with \(\Phi =0^\circ \) elevation angle reveals various frames without glare points for the stages of the drop impact while the contact angle drops below \(90^\circ \), which becomes obvious e.g., in Fig. 12. To guarantee the consistent occurrence of glare points, the complementary angle has to remain below the expected range of dynamically varying contact angles in the experiment. Therefore, an elevation angle of \(\Phi =90^\circ - \gamma _{r}\) is required, in which \(\gamma _{r}\) is the minimum receding contact angle observed in the experiments (cp. also Fig. 12). Thus, the optimal elevation angle lies within the range between \(0^\circ \) and the minimum receding contact angle \(\gamma _{r}\) and immediately relies on the wettability of the substrate. Conversely, an increase in the elevation angle leads to a decrease in the distance between the zeroth order glare points, which in turn results in a larger relative measurement error of the distance between the glare points. The theoretical optimum for the elevation angle, therefore, is the lowest angle at which glare points are still visible on all frames during impact. An equilibrium contact angle of \(\gamma = 80^\circ \) and a receding contact angle of \(\gamma _{r} = 66^\circ \) characterizes the considered water-droplet impact experiment onto a SiOx-substrate. For this configuration, an elevation angle of \(45^\circ \) was found to produce the most continuous glare point signal.

At elevation angles larger than \(0^\circ \), higher order glare points mostly disappear as the light that is refracted while entering the droplet interacts with the substrate surface instead of reflecting on the second gas–liquid interface of the droplet. Furthermore, the position of the higher order glare points and the light reflected from the substrate surface quickly become chaotic when the shape of the droplet deviates significantly from the spherical shape. Both of these effects render an interpretation of higher order glare points at elevation angles larger than \(0^\circ \) difficult, as the glare point positions become ambiguous. Combined with the aforementioned requirement for elevation angles \(\Phi >90^\circ - \theta _{rec}\) for a continuous glare point signal—and in particular for hydrophilic substrates—this leaves the zeroth order glare points as the best option for encoding the droplets shape. The restriction to the zeroth order glare point yields the additional benefit of a straight-forward description of the underlying physics and interpretation of the results.

The optimal azimuth angle consequently is the angle at which only the zeroth order glare point is visible. According to GO a scattering angle of \(\theta =96.5^\circ \) leads to the strongest intensity of the zeroth order glare point relative to all other glare points at a wavelength of \(\lambda = 632\) nm. At this angle the glare points of orders \(3n, n \in {\mathbb {N}}_0\) coalesce, with the rays forming an equilateral triangle in the droplet, as can be calculated by Eq. 2. The \(p=1\) and \(p=2\) glare points do not appear at \(\theta =96.5^\circ \), as the \(p=1\) rays are limited to \(\theta <82.6^\circ \), which is the angle of the edge ray. Likewise, the \(p=2\) rays are limited to \(\theta >137.7^\circ \), which is the rainbow angle.

Since higher order glare points quickly become less intense outside of a limited range close to their rainbow angle, these orders can be neglected as well. Therefore, \(\theta =96.5^\circ \) can be regarded as a scattering angle that only yields zeroth order glare points. The theoretical derivation above matches well with the experimental results at \(\theta =95^\circ \) azimuth angle (see Fig. 8), which was sufficiently close to the optimal azimuth angle to only produce visible zeroth order glare points. Furthermore, the comparison of the theoretically expected glare points for \(\theta =95.0^\circ \) and \(96.5^\circ \) (see Fig. 18a) reveals that the difference in the intensity distribution is negligibly small. This high tolerance for angular misalignment in turn facilitates a robust, straight-forward calibration of the experimental setup.

The mild influence of the wavelength-dependent refractive index \(n(\lambda )\) of water becomes obvious from the above discussion for red light (\(\lambda _R = 632\) nm) in comparison the the green light (\(\lambda _G = 521\) nm) of the second lateral light source. Accordingly, the green glare points are determined to occur at an optimal angle of \(\theta _G = 96.3^\circ \) by Eq. 2. This color-dependent \(0.2^\circ \) difference, however, can be considered negligible in comparison with the angular alignment accuracy of the measurement setup.

In the following section the informative value of the glare points regarding the three-dimensional shape of the droplet will be analyzed. For this purpose a render pipeline for the generation synthetic images was set up in the 3D rendering software Blender (Blender Online Community 2018) in combination with the ray-tracing library LuxCore (Bucciarelli et al. 2018), which allows to create realistic images of arbitrary droplet geometries that closely match the experimental images.

In order to evaluate the evolution of the glare-point images for non-axisymmetric droplets, ellipsoidal cap geometries with varying aspect ratios were produced by stretching the geometry of a gas–liquid interface extracted from a numerical simulation along one axis in the horizontal plane by factors of \(A=1.0\) to \(A=1.4\) with increments of 0.05. Finally the render setup, configured with an azimuth angle of \(\theta = 95^\circ \) and an elevation angle of \(\Phi = 45^\circ \) was used to render glare-point images from the droplet geometries at different rotation angles. The renderings created by the aforementioned render pipeline can be seen in Fig. 13 for the aspect ratio \(A=1.25\).

Figure 14a shows the geometrical description of the ellipsoidal droplet that has the aspect ratio\(A = \Vert \textbf{f}_1\Vert /\Vert \textbf{f}_2\Vert \) and is rotated by \(\varphi \) relative to the space-fixed frame of reference. Fig. 13a, e shows the geometry rotated by \(\varphi = 0^\circ \) and \(180^\circ \), so that the major axis of the ellipsoid is pointing toward the observer. In order to emphasize the differences between the rendered glare point images at different \(\varphi \) all images in Fig. 13 contain the contour of the \(\varphi = 0^\circ \) droplet and the positions of the glare points marked by white crosses. The relative position of the glare points in dependence of \(\varphi \) and A was measured relative to the center line of the projection, as indicated in Fig. 14b. The measured positions are plotted in Fig. 15a–c for the red and green zeroth order glare points, and the blue first order glare point, respectively. The aspect ratio A appears as family parameter in the diagrams and angle \(\varphi \) is the functional variable.

As can be seen in Fig. 13 the position of all three glare points moves relative to the shadowgraph contour in dependence of the rotational angle \(\varphi \) and the aspect ratio A. The positions of the red and green glare points appear as phase-shifted and mirrored functions, since both glare points are created through interface reflection (\(p=0\)). Their location, therefore, depends on the point on the droplets gas–liquid interface that matches the required angle for interface reflection \(\theta _S = (180^\circ -\theta )/2\), which is half the angle included between the camera and the respective light source. The relative position of the blue glare point shows a different dynamic, as it arises from the transmission through the droplet with correspondingly two refractions on the gas liquid interface (\(p=1\)).

With the assumption that the shape of the droplet can be described as an ellipsoidal cap with two planes of symmetry along the major and minor axis, two parameters—the rotational angle \(\varphi \) and the aspect ratio A—are sufficient to uniquely determine the three-dimensional shape and position of the droplet. As can be seen in Fig. 15, the relative position of the three (color-specific) glare points are three mutually independent families of functions, since their amplitudes as well as the positions of their maxima depend on the aspect ratio. For a given measurement setup that determines the position of the light sources and the volume of the droplet, \(\varphi \) and A can thus be unambiguously determined from the system of the two \(p=0\) glare-point positions. The relative glare-point positions were determined analytically (see dashed lines in Fig. 15) and show a good agreement with the results from the rendering and, therefore, provide further evidence on the applicability of the considered approach. The derivation of the underlying necessary Eqs. 21 and 22 can be found in the Appendix A. The two \(p=0\) glare points obviously encode the location of the droplets phase boundary of the in the object plane and the local orientation at two additional points outside the shadowgraph plane. For the presented setup the local orientation of the gas–liquid interface at the position of the lateral glare points is \(\theta _S = 42.5^\circ \) in the azimuth angle and \(\Phi = 45^\circ \) elevation angle.

The positive results of the color correction, as seen in Fig. 6 suggest that the proposed model for the light transport and the underlying assumptions are valid. After the color correction, the blue image channel shows only a pure shadowgraphy image without any specular reflection of the lateral light sources. This result demonstrates that the presented method is able to produce standard shadowgraphy images in a high quality with the addition of further illumination angles of the droplet’s interface that encode additional three-dimensional information of the droplets gas–liquid interface in the glare points as shown in Sect. 4.2.

The two evaluated methods (ASC and ISC) allow for an accurate reconstruction of the glare point and shadowgraph images by the correction of the mutual disturbance in the image channels, that resulted from polychromatic light and spectral sensitivity of the camera sensor. The analytical reconstruction (ASC) can reach a potentially higher reconstruction quality, if the spectral characteristics of all optical components can be determined accurately a-priori. If such information is missing or incomplete, then the in-situ method (ISC) can be used to obtain the correction matrix, which introduces additional measurement imperfections, such as e.g., image noise.

It should be noted that the proposed color-correction models neglect chromatic aberration and dispersion. Due to the dependence of the refractive index on the wavelength of the incident light the (polychromatic) transmitted light would be deflected at a slightly different angle. Therefore, all glare points of order \(p=1\) and higher are affected by dispersion. Due to a higher amount of interactions with the gas–liquid interface the dispersion becomes increasingly severe for higher order glare points. As seen in Fig. 10 first, second and third order glare points, each with an increasing magnitude of dispersion can be observed in the experiments. Higher order glare points might appear in the images as well, however, since these lie close to the edges of the projected droplet, their spatial extent is very small compared to the lower order glare points so that the effects of dispersion can be neglected. A significant level of dispersion could lead to a faulty color correction, since different areas of the glare points on the images would be composed by different wavelengths. This effect would consequently distort the pixel-wise reconstruction.

For the optimal angular position of the lateral lights determined in Sect. 4.1 only \(p=0\) glare points do appear, which are produced by pure interface reflection and thus remain unaffected by dispersion. In contrast, the \(p=1\) glare point of the backlight is affected by dispersion, since it is refracted by the gas–liquid while entering and again while leaving the droplet. However, the influence of dispersion can be assumed to be negligibly small due to a small change in the index of refraction for the narrow-banded spectrum of the lateral LED light sources. The maximum angular dispersion in the first order glare point, i.e.. the deviation in the angles of the refracted light, can be estimated to be lower than \(0.25^\circ \) for the given spectra of the light sources using Snell’s law (Hecht 2017).

The intensity of the light sources was found to have no significant influence on the results of the color correction. The intensity of the color-corrected images scaled linearly with the light source intensity, but otherwise the channels had undergone the same color correction for different input intensities, thus validating the assumption of the linearity of the camera sensor. While the assumption of a linear response of the chip to the incident light holds true for intensities within the dynamic range, it neglects clipping in the regions of glare points and other highlighted image regions. Since one or potentially multiple color channels can be clipped, changes in hue occur (Novak et al. 1990) and a proper calculation of the color correction is inhibited. The resulting loss of information for intensities above the dynamic range skews the correction function such that the mutual disturbance of the color channels cannot be fully removed in the overexposed frames, which has been discussed along selected frames in Fig. 6. Note that such frames can still be improved by the correction function. However, reduced light intensity mitigates this problem at the cost of likewise less pronounced glare points, especially on frames with larger areas of reflection. Therefore, the light source intensity needs to be fine tuned for conserving a maximum of information in dim and bright images.

The influence of the chosen elevation angle \(\Phi \) on the resulting continuous registration and homogeneous intensity of recorded glare points has been already discussed in Sect. 4.1 for convex gas–liquid interfaces. Non-convex droplet shapes, which occur during the dynamic deformation of the droplet during impact, comprise multiple glare points at each position on the droplet surface that has the orientation \(\Phi _c\) and \(\theta _S\). These glare points split and merge depending on the deformation of the gas–liquid interface as indicated e.g., in Fig. 4, where multiple glare points are visible at each capillary wave (see also Fig. 20 for the full time series). The appearance of \(p=0\) glare points on non-convex deformed droplets facilitates the encoding of additional three-dimensional information on a larger amount of points on the surface. However, a reconstruction of these droplet shapes will be more complicated in comparison with the spheroidal cap shape, for which an unambiguous encoding of the volumetric shape was confirmed in Sect. 4.2.

The overall good agreement of the glare-point positions in the experiment in dependence of the azimuthal angle \(\theta \) to the theoretically expected positions (see Fig. 8) for all tested azimuthal angles, confirms the assumption of a circular cross section for the undeformed droplet. However, small differences in the position of the glare points were observed that might originate either from a deviation from the spherical shape and/or from alignment errors in the experimental setup. On the one hand, small changes in the scattering angle due to alignment errors in the azimuth angle of the lateral light sources, can result in significant changes of both glare-point intensity and position, which particularly cumulates for higher order glare points due to a larger number of interactions with the gas–liquid interface. In particular, the intensity increases sharply if the azimuth angle approaches the rainbow angle of the respective order of scattering, as can be seen in Fig. 18. The reduction of the measurement system to \(p=0\) and \(p=1\) glare points, therefore, significantly limits this error. On the other hand, alignment errors in the elevation angle of the lateral light sources result in a shift of the glare-point position. For low elevation angles the resulting error becomes negligible for small angular deviations, as the horizontal movement of the glare points is small due its trigonometrical relation to the elevation angle. For large elevation angles, however, this error source has to be considered.

A third error source are deformations of the droplet from the spherical shape, resulting in an elliptical cross section in the scattering plane for the lateral light sources. As shown in Sect. 4.2 the aspect ratio of an ellipsoidal droplet has a significant influence on the position of the glare points such that the deformation of the droplet in any axis has to be considered. Irregular deformations of the droplet will inevitably introduce deviations in the position of the glare point, which can lead to reconstruction errors. The present experiments of a droplet impinging in the normal direction onto a flat substrate, however, revealed no significant irregular deformations. Instead a deformation in the vertical direction from the detachment of the droplet at the needle was observed that causes a vertical oscillation and results in a spheroidal shape of the droplet. Note that the aspherical shape of the droplet in this case can be determined analogously to Sect. 4.2 from the position of the glare points.