1. Introduction

High-speed impacts, explosive detonations, and other high-energy phenomena can transmit strong shocks to the free surface of a solid material. These shocks may cause the surface to fracture, spallate, and otherwise emit a spray of fine particulate. The current work specifically considers the very fine, ${\mathcal O}$(µm), and very fast, ${\mathcal O}$(km/s), particles emitted by a strongly shocked metal surface. Such particulate are generally referred to as ejecta by a community that has been investigating their dynamics since the 1950s [1]. In brief, ejecta originate from shock concentrations around surface imperfections and Richtmyer-Meshkov instabilities [2,3]. At a surface, a shock may liquify the metal, resulting in micro-sheets and micro-jets that eventually breakup into fine drops. Attempts to predict the associated dynamics have included development of theory [4–6], semi-empirical relations [7], molecular dynamic simulations [5,8–10], continuum simulations [5,11], and many others.

As reviewed in [1,12], several experimental diagnostics are commonly applied to the investigation of metal ejecta. Visible [13–16] and x-ray imaging [13,15,17] capture the overall dynamics and quantify line-of-sight mass. Asay-foil [18,19] and piezoelectric pins [20,21] quantify particle time of arrival and momentum transfer. Photonic Doppler Velocimetry (PDV) measures the jump-off velocity of the free surface and has been extended to quantify velocities in the ejecta cloud [22–27]. For further details on these diagnostics, the reader is referred to the above citations and many related works.

Buttler et al. [12,28] present recent experimental investigations of ejecta dynamics. As reviewed there, the transport and reaction of ejecta particles within gas-phase surroundings is of ongoing interest. Because these dynamics are strongly dependent on the particle size and velocity distributions, measurements and predictions of these quantities are crucial. Starting in the 1980s various multi-angle scattering (aka Mie scattering) diagnostics have been proposed to quantify the mean and sometimes higher order moments of the particle size distribution [29–32]. Such Mie scattering diagnostics utilize relatively inexpensive optical components, which can be replaced after each shot, and tend to have good sensitivity to small particles whose diameters are comparable to the probing wavelength. However, Mie scattering measurements cannot unambiguously quantify the functional form of the particle size distribution and do not quantify the joint distribution of particle size and velocity, which is of critical interest to ejecta transport models.

These shortcomings have inspired the development of microscopic imaging techniques for direct, in-situ measurements of ejecta particle sizes and their corresponding velocities [14–16,33–42]. To prevent motion blurring, exposure times must generally be very short, ${\mathcal O}$(ns or less). This has motivated the use of pulsed laser illumination for all prior imaging developments. In addition, to measure particles outside of the narrow depth-of-focus of traditional high-magnification imaging [14–16], many prior investigations have utilized pulsed laser holography [33–42]. McMillan and Whipkey [33] were likely the first to report a holographic configuration for study of metal ejecta. In their experiment, particles were illuminated by a 130 ps, 532 nm laser and the resulting diffraction patterns were optically magnified by 4× before being recorded on holographic film. Particle sizes down to 3 µm diameter were quantified and a double-exposure configuration was demonstrated for simultaneous measures of particle velocities [33]. Subsequently, starting in 1990s, several of the current co-authors began development of our own high-magnification holographic diagnostics [34–41]. Initial efforts quantified ejecta in a pulsed-power facility [34–37]. Experiments utilized a 100 ps, 532 nm holography laser and achieved a minimum particle resolution of 1.5 µm. Later, a custom, 5× magnification objective [38] was developed for UV holography using a 150 ps, 355 nm laser. In [39–41] this configuration was applied to explosively driven ejecta experiments where a minimum particle resolution of 500 nm was demonstrated. In addition, particle velocities have been quantified by double-exposure [39] and effects of varying helium environmental pressures have been investigated [39,40]. In [31,32] these holographic measurements are compared with mean sizes from the Mie scattering diagnostic and good agreement is reported. Finally, Li et al. [42] have also reported similar pulsed-holographic measurements capable of resolving ejecta particles as small as 4 µm diameter.

All of these prior experiments recorded their holograms onto photographic film [33–42]. Processing of such film holograms is labor-intensive, requiring chemical development and subsequent analog reconstruction. Due to these drawbacks, a vast majority of holographic techniques now utilize digital sensors and numerical reconstruction rather than film recording [43–48]. Unfortunately, the spatial resolution of digital sensors is also orders of magnitude less than film. As a result, in an ejecta holography configuration, direct replacement of film with a digital sensor is likely to significantly degrade resolution of the smallest particles.

Prior digital holographic measurements of high-speed or explosively generated particles [49–54], have focused on larger particles, ${\mathcal O}$(50 µm or greater) for which optical resolution is less challenging. The current work reports the first known attempt to perform digital holographic measurements of µm-scale ejecta particles traveling at km/s. Here, optical techniques are modified to improve the spatial resolution and advancements to digital processing methodologies are presented. Results demonstrate minimum resolvable particle diameters around 2 µm, which matches the size limits of many of the prior film-based techniques. At the same time, our digital holography is shown to have several advantages, including improved ability to quantify particle velocities and 3D positions.

The remainder of this work begins with a brief review of the principles and state-of-the-art of digital holographic microscopy (DHM). Next, the experimental configuration for DHM of ejecta is presented. Following this, typical holograms are shown, revealing challenges stemming from relatively large size dynamic ranges, dense particle fields, complex particle morphologies, and speckle noise. Data processing methodologies are then outlined with focus on proposed improvements to address these challenges, and simulations are used to study measurement uncertainties. Finally, results from three experimental realizations are analyzed and relations to ejecta formation and transport physics are discussed. The work concludes with a summary of findings and recommendations for future developments.

1.1 Principles of digital holographic microscopy (DHM) of 3D particle fields

The basic holographic configuration projects a laser beam through a particle field. The portion of the light which is scattered or diffracted by the particles is commonly referred to as the object wave. At a hologram plane, some distance away from the particle field, the object wave coherently interferes with a reference wave. The intensity of the resulting interference fringes is referred to as the hologram and is recorded using either photographic film or a digital sensor array. In analog holographic reconstruction, as in the prior works of [33–42], the developed film is illuminated by a conjugate of the reference wave, and diffraction of the transmitted light produces real images of the particles at their original three-dimensional (3D) positions. Today, digital holographic reconstruction is more common. In that case, the recorded hologram intensity is numerically multiplied by an estimate of the conjugate reference wave, and the diffraction equations are solved to reconstruct particle images throughout the 3D depth.

The literature contains many holographic configurations, which are summarized in several review articles [43–45] and books [46–48]. The current work investigates particles with mean sizes on the order of a few µm. Holographic investigation of such small particles requires large optical magnifications and is typically referred to as digital holographic microscopy (DHM). Among the various DHM configurations, the in-line DHM configuration is the most simple and common. In in-line DHM, a collimated laser beam is transmitted through the particle field, such that the scattered object wave propagates in-line with the transmitted reference wave. A lens is placed downstream of the particle field to magnify and image the signal onto a recording plane. This in-line DHM configuration can be constructed from common transmission microscopy hardware and has been widely applied for the study of µm or even nm-scale particles (e.g. [55]). When multiple holograms are recorded over short time intervals, in-line DHM can also be used to quantify particle motions.

Several techniques have been proposed to locate and measure particles from in-line holograms. When particle shapes are known a-priori (e.g., spherical particles), particle sizes and 3D locations can sometimes be inverted by comparing measured hologram intensities to simulations based on Mie or diffraction theory [56–60]. However, for many investigations, including the current work, particle morphologies are unknown and/or particle number densities are too high to permit successful application of inverse methods. Instead, particles are more commonly measured by refocusing the hologram along the optical depth direction and applying image processing methodologies for automated detection of the 3D locations and 2D particle morphologies.

In digital in-line holography, the numerically reconstructed complex amplitude, E, at optical depth, z, is given by,

Equation (1) is often evaluated using the angular spectrum method in the frequency domain as,

Equation (2) allows a digital hologram to be numerically refocused to visualize particle images at their original optical depths. Recent reviews in [44,45] summarize the many processing routines that have been proposed to automate the detection and quantification of particles from refocused images. Most typically, these methodologies calculate numerically refocused images throughout the range of expected particle depths and apply a metric to segment potential particle locations from background regions. Common focus metrics are those based on intensity minimization [62,63], maximization of 2D image gradients [64–66], combinations thereof [67–72], and others [73–75]. As necessary, particle localization can be iterated to refine and improve initial results [45,67–69]. Finally, when time resolution is available, 3D particle velocities and trajectories are quantifiable based on sequential matching [49,70,72,76,77] or multi-frame tracking [53,57,78–80].

2. Experimental methods

The prior works by co-authors Sorenson et al. [38–41] present a film-based analog holography configuration for the study of ejecta particulates from explosively shocked metals. The current work extends this facility by replacing film with digital recording and numerical reconstruction, and Fig. 1 schematically illustrates the experimental configuration. Subsections that follow provide additional details, emphasizing those aspects of the experimental methods which differ from the prior works in [38–41].

 

Fig. 1. Experimental configuration for two-frame, Digital Holographic Microscopy (DHM) of explosively generated ejecta.

Download Full Size | PDF

2.1 Ejecta package

The ejecta source is identical to that in [38–41]. Briefly, a strong shock is generated by the detonation of a high-explosive pellet (PBX 9501). This produces a ∼275 kbar pressure at the free surface of a Sn target. A vertical groove is machined into the free surface of the Sn target. Shock concentrations within this groove melt and accelerate a microsheet, which is ejected ahead of the free surface. In the duration of a few µs, this microsheet breaks apart, forming long microjets (aka tendrils) which themselves fragment into the µm-scale ejecta particulate of interest. As in [38–41] each ejecta package also included a PDV probe used to quantify the jump-off velocity of the Sn surface.

Explosive hazards are contained in a 50 cm diameter steel vessel with 13 mm thick walls. Line of sight optical access is provided through two fused silica viewports. To minimize optical propagation distances, the ejecta package is placed near to one of these viewports, as schematically illustrated in Fig. 1. (For further details on the ejecta package and mechanical configuration the interested reader is referred to Sorenson et al. [39], particularly figures 2.2.4 and 2.2.5 in that work.)

2.2 Holography configuration

As in the prior works of [38–41] a frequency tripled Nd:YAG laser provides collimated UV illumination at λ=355 nm, with a pulse duration of 150 ps. After traversing the particle field, diffraction patterns are first imaged by the custom, nine element long-distance microscope lens defined in Malone et al. [38]. This lens system achieves 5× optical magnification and was designed for film recording at the plane noted in Fig. 1. Design resolution is >400 lp/mm at the film plane, which corresponds to a minimum measurable particle diameter of around 500 nm [38–41].

Here, film is replaced with digital recording. This provides several advantages as illustrated throughout the current work. However, the significantly lower spatial resolution of digital sensors as compared to film is one distinct disadvantage. For example, the CCD used in this work (Imperx B6620, 6600 × 4400 pixels, 5.5 µm pixel pitch) can resolve 60 lp/mm. If this CCD were placed at the film plane, the smallest measurable particle diameter would be limited by the CCD resolution to around 3.4 µm.

To increase overall magnification and improve the ability to quantify the smallest particles, the additional magnifying relay lens pairs were added to the system as shown in Fig. 1. The combined system was optimized in Zemax with the goal of maximizing system magnification while matching the lens resolution at the image plane to that of the CCD. Due to cost and time constraints, the current relay lens design was limited to commercial off the shelf lenses. The final design has a predicted total magnification of 8.3×, and the smallest measurable particle diameter is expected to be around 2.0 µm.

In DHM, particle velocities are typically resolved using multiple time instances recorded on a single sensor. For example, two high-resolution images can be recorded on a single interline transfer CCD, which can achieve interframe times on the order of ∼150 ns or greater. However, in the current work, particles are traveling around 2 km/s and will displace ∼450 pixels in 150 ns. Such large pixel displacements would make unambiguous frame-to-frame particle matching challenging. Instead, in this work, two images with much shorter interframe times are recorded using two separate CCDs as shown in Fig. 1. The optical configuration is similar to our previous works in [49,50]. A single pulse is first split into two orthogonally polarized beams. One beam is delayed over a longer pathlength and subsequently recombined along the optical axis of the second beam. After passing through the 5× lens, these two beams are reseparated by a polarizing beam splitter and then individually relay imaged into two CCDs as shown in Fig. 1. Potential crosstalk due to stray reflections from one orthogonally polarized beam path to the other are reduced by polarizing filters on each CCD (Thorlabs WP50L-UB). In addition, to reduce sensitivity to broadband emission from the explosives or other sources, 355 ± 10 nm bandpass filters (Andover Corporation 355S10-50) were also located in front of each CCD.

3. Experimental results

3.1 System calibration

On three separate days during the test campaign, a metrology grade dot grid (Max Levy Autograph DA020) was placed at the object plane of the lens system. From the recorded images, the average pixel spacing at the object plane, Δx, was determined to be Δx₁= 664.8 + 0.2/-0.4 nm and Δx₂= 665.4 + 0.4/-0.5 nm for cameras 1 and 2, respectively, where the +/- values define the min and max values measured from the three separate calibrations. These values combined with the CCD’s manufacturer specified pixel pitch (5.5 µm) quantify the total system magnification as 8.273 + 0.005/-0.002 and 8.266 + 0.006/-0.005 for cameras 1 and 2, respectively. The small experimental variations indicate that uncertainty in the magnified pixel spacing is unlikely to significantly contribute to overall particle sizing uncertainty. Finally, the spatial variations of measured dot centroids were fit to fourth degree, two-dimensional polynomial transformations in MATLAB. These transformations are used to dewarp experimental hologram images to a cartesian grid. Overall distortion corrections are small, with a maximum value in object space of 30 µm at the edge of the field of view.

To study the resolution limit of the DHM configuration, a second, custom metrology grade reticle was placed near the expected optical depth of the ejecta particles (z≈1 mm). As described further in the next sub-section, recorded holograms were background normalized and dewarped. Following this, holograms were refocused via Eq. (2) to manually locate the best focus depth of small, opaque squares printed on the glass reticle. Figure 2 shows refocused and zoomed-in images of several such squares with side lengths ranging from 9 to 1 µm. White dotted lines in Fig. 2 show the known dimensions of each square. Finally, note that the displayed magnification of each sub-image in Fig. 2 increases with decreasing square size, such that discrete pixels are clearly visible at the smallest dimensions.

 

Fig. 2. Select refocused hologram images from a metrology grade calibration grid recorded at z = 1 mm. Each sub-image is zoomed into an individual opaque square with side length and approximate location given by the label and dotted lines, respectively.

Download Full Size | PDF

For the relatively large squares in Fig. 2, refocused images match the known dimensions. This provides qualitative confirmation of the accuracy of the current DHM configuration. In addition, Fig. 2 results also indicate that intensity and edge sharpness contrast degrade with decreasing particle size. At about 3 µm and below, particle contrast approaches that of the background, and at about 1 µm and below pixel resolution is ultimately insufficient. This indicates that the minimum measurable particle diameter is likely around 2 µm, consistent with the spatial resolution limitations of both the lens and CCD sensor as discussed in the previous section. Further study of particle size accuracy is also presented in section 5 with the aid of simulations.

Finally, the interframe time, Δt = 8.84 ± 0.04 ns, was measured with a 5 GHz silicone photodiode placed in the combined beam path. Uncertainty is estimated from variation in Δt measured over multiple laser pulses.

3.2 Hologram images

In an experiment, the ejecta package was first positioned with the free surface of the Sn target a lateral distance away from the centerline of the holography field of view. In addition, the groove was orientated to produce an ejecta sheet parallel to the object plane and at an optical depth, z, a few mm from the object plane. Next, the ejecta package was evacuated to less than ∼0.1 mbar, and the explosive pellet and initiator were inserted into the package. Following this, personnel were cleared for remote explosive operation. Immediately prior to a shot, ten hologram images were recorded of the empty field of view. Finally, the explosive shot was performed, using electronic delay generators to record the laser pulse at a fixed delay with respect to explosive initiation.

The top row of Fig. 3 shows an example hologram image corresponding to Shot A. (Further details on experimental conditions are given in section 6, Table 2). The Fig. 3 hologram has been normalized by the average of the ten background images and dewarped using the fourth-degree polynomial transformation described above. The left images show the full extent of the field of view, while the right images show a zoomed-in region corresponding to the dotted white box.

 

Fig. 3. An experimental hologram showing (a) the full field of view and (b) a zoomed-in region defined by the dotted box in (a).

Download Full Size | PDF

The Fig. 3 hologram shows many diffraction patterns created by the ejecta particles. This hologram is numerically refocused along the optical depth direction via Eq. (2), the results of which are visualized by the real valued amplitude,

For example, the bottom row of Fig. 3 shows the numerically refocused A at optical depth, z = 2.83 mm, where in several in-focus particles are observed.

Figure 3 results demonstrate the first digitally recorded and numerically refocused hologram images of explosively driven metal ejecta. As expected, particles are on the order of a few µm in diameter and many are non-spherical. As reviewed in section 1, the current authors [67–69] and several others [65,71,81], have reported prior experiments and data processing methodologies for automated measures of non-spherical particles. In comparison to those prior works, images in Fig. 3 appear to suffer from several significant noise sources. For one, the particle density in Fig. 3 is relatively high. This, along with the large optical magnification, likely contributes to the appearance of high-frequency noise caused by coherent interference between the scattered fields from multiple particles, sometimes referred to as speckle noise [82]. In addition, despite hologram normalization, some intensity variations are still apparent in the background regions of Fig. 3. These are likely attributed to inherent shot-to-shot spatial intensity variations in the high-energy ps-laser, which is exacerbated by the relatively high image magnifications.

In the following sections, automated data processing methods are proposed. Discussion emphasizes new methodologies, which were developed to address these noise sources and other challenges specific to the current experiment.

4. Data processing

4.1 Identification of initial 2D particle regions

As is typical in the literature [44,68,69,72], the first step in the automated detection of in-focus particle locations is the determination of a 2D “all in-focus image” from which initial estimates of 2D particle regions are segmented. One of the simplest and most common all in-focus images is the minimum amplitude projection [62],

For example, Fig. 4 shows A_min determined from 3001 uniformly spaced z-depths between 1.5 ≤ z ≤ 4.5 mm. Large dark regions in Fig. 4 correspond with the larger particles observed in the refocused plane shown in Fig. 3. Unfortunately, in Fig. 4 the contrast between small particles and background variations in A_min is also relatively weak. As a result, our attempts to segment A_min could not successfully identify true particle regions while avoiding significant false-positive detection of background variations.

 

Fig. 4. Minimum amplitude projection, A_min, showing the portion of the field of view corresponding to by the dotted box in Fig. 3.

Download Full Size | PDF

Figure 5(a) shows an x-z slice of the refocused amplitude where dotted lines indicate to the extent of the simulated particles. As expected, the refocused amplitude achieves a minimum near the true particle depth. However, Fig. 5(a) also shows significant variations in regions exterior to the particles. These variations are caused by diffraction in out-of-focus planes as well as contributions from the inherent twin-image of in-line holography. These effects lead to background variations in the projected A_min, as shown in Fig. 5(b). In the experimental results, the combination of many such closely spaced particles leads to the poor signal quality observed in Fig. 4.

 

Fig. 5. Reconstructions of a simulated hologram of two 10 µm diameter spheres separated by 40 µm in the x-direction and located at z = 2 mm. (a) x-z slice of the refocused amplitude showing a 300 µm z-extent centered at z = 2 mm and (b) the minimum projected amplitude, A_min. (c) x-z slice of the relative phase, ϕ, and (d) the inverse of the maximum projected absolute phase, ϕ_max.

Download Full Size | PDF

The digitally reconstructed E(x,y;z) contains information on the phase of the refocused lightfield in addition to the amplitude information considered thus far. This has motivated several authors to propose particle localization techniques that leverage phase information [58,74,75]. Here, we suggest a variant thereof based on the relative phase angle,

Figure 5(c) shows an x-z slice of ϕ for the simulation of the two closely spaced particles. Compared with the amplitude in Fig. 5(a), ϕ shows less apparent variations in regions exterior to the particles. This suggests that an effective all in focus image can be determined by the projection of the maximum absolute value of ϕ along the z-direction [58],

Figure 5(d) shows this ϕ_max projection for the current example. To aid visual comparison with A_min, Fig. 5(d) displays π-ϕ_max such that particles appear dark with respect to the light background. Qualitatively, the contrast between the particles and the background is significantly better for ϕ_max compared to A_min. This contrast is quantified by the signal to noise ratio,

Application to the experimental results is shown in Fig. 6(a). In this example, ϕ_max is calculated from 3001 uniformly spaced refocused planes between 1.5 ≤ z ≤ 4.5 mm. Compared with the A_min projection in Fig. 4, the ϕ_max projection in Fig. 6(a) shows significantly improved contrast especially for the smaller particles.

 

Fig. 6. (a) the inverse of the maximum projected absolute phase, ϕ_max, and (b) the binary thresholded image, ${\mathbf{\mathscr F}}$_t{π-ϕ_max}, showing the portion of the field of view corresponding to by the dotted box in Fig. 3.

Download Full Size | PDF

Finally, initial estimates of 2D particle regions are found by segmenting ϕ_max to determine a binary image of all connected regions for which π-ϕ_max is less than a threshold value, t. This work uses the nomenclature of Gao et al. [69] where a thresholding operation is denoted by $\mathbf{\mathbf{\mathscr F}}$_t{}. For example, Fig. 6(b) shows $\mathbf{\mathbf{\mathscr F}}$_t{π-ϕ_max} for a threshold value of t = 0.25. At this empirically determined t, the segmented ϕ_max appears to adequately identify true particle regions, while avoiding significant false positive detection.

4.2 3D particle localization

Methods described in the previous sub-section provide initial 2D estimates of regions that are likely to contain individual particles (Fig. 6(b)). This sub-section discusses the subsequent identification of optimal 3D particle positions and refocused particle morphologies. Techniques proposed here are similar to our previous works [67–69], and the interested reader is referred there for details.

For each connected region in Fig. 6(b) a sub-window is defined as twice the extent of the 2D bounding box. Optimal particle z-positions corresponding to each sub-window are found by refocusing the hologram via Eq. (2) to 3001 uniform spaced z-depths between 1.5 ≤ z ≤ 4.5 mm and applying the edge-sharpness based focus metric described in the next paragraph.

At each z-depth, the refocused amplitude, A(x,y;z), is found by Eq. (4), and a sharpness image is determined based on the Tenengrad operator,

Next, in-plane particle morphologies are determined using techniques from [68]. Briefly, for each sub-window, the hologram is refocused to the optimal depth, z_i, and the average edge sharpness is calculated as a function of segmentation threshold, t,

Finally, the optimum segmentation threshold is selected as the value of t where S_i(t) is maximum. This threshold is used to segment the refocused amplitude image to find the binary particle image and 2D centroid (x_i,y_i).

4.3 Interframe displacement measurements

The automated image processing routines in sections 4.1 and 4.2 are applied to both the camera 1 and camera 2 holograms. Next, particle x,y positions from camera 1 are translated and rotated to the camera 2 coordinate system using a best-fit non-reflective similarity transformation of matched points from dot grid images. This process, along with initial dewarping of the holograms to cartesian grids (described in section 3.1), is intended to transform all measured x,y,z positions to the common global cartesian coordinate system corresponding to camera 2. Finally, a standard nearest neighbor matching routine is applied to quantify the particle displacements, Δx,Δy,Δz, over the interframe time, Δt, from which velocity components are determined.

During the experiment, calibration images were recorded with the dot grid located at both the object plane and the film plane as labeled in Fig. 1. Coordinate transformations determined from the dot grid located at the object plane are ideally most accurate. However, in the current experiment, the object plane can only be physically accessed either before installing the ejecta package or after completion of an explosive shot. In between these times, significant mechanical adjustments are required, and the explosive shot itself inherently causes shock vibrations. These factors may lead to changes in alignment, which would reduce the accuracy of the transformation determined from dot grid images located at the object plane. In contrast, as schematically illustrated in Fig. 1, the film plane is outside of the containment vessel and is physically accessible immediately before the explosive shot. Transformations determined from film-plane images are, therefore, more likely to reflect the actual alignment during the shot. On the other hand, transformation from film plane images cannot account for any relative translation or rotation introduced by the main lens. Although such offsets are likely to be small, various alignment artifacts could compound to make the transformation determined from the film plane images slightly different than the true transformations in object space.

In analysis performed after the experiment, transformations measured at the object and film planes were found to be nearly identical for some shots while for others significant shifts were observed. This appears to confirm our hypothesis that precise control of alignment is difficult in the current explosively driven experimental configuration. In the end, results reported here are determined from the transformation, which produces mean ejecta velocities most consistent with the PDV measured free-surface velocity and our general expectations of the flow. In future work, the calibration processes could be improved with alignment fiducials that are integrated within the ejecta package, which would ensure that measured coordinate transformations exactly reflect alignment during each shot.

The calibration challenges described above were anticipated at the time of the experiment. In addition to this, after the experiment it was discovered that initial measures of particle x-velocities were significantly different than expected velocities, regardless of the choice of calibration plane. Furthermore, stationary objects in the field of view (such as dots on calibration grids and dust on optical surfaces) were found to produce non-zero Δx,Δy displacements, which linearly varied with respect to z-depth. After careful study of these issues, the most likely explanation was postulated to be small angular misalignments between the orthogonally polarized beams (Fig. 1), which would not be removed by the procedures described thus far.

To correct these issues, the apparent Δx,Δy displacements of several stationary objects as a function of z were used to quantify the angular misalignment of beam 1 with respect to beam 2 as θ_x= 0.3 ± 0.2 mrad and θ_y= -3.8 ± 0.3 mrad about the horizontal and vertical axes, respectively. Uncertainty is estimated from variation in θ_x and θ_y quantified from several stationary objects. The transformed camera 1 x,y particle positions are then corrected as

These small misalignments were not discovered during the test campaign, necessitating the post-test analysis and corrections discussed above. In future experiments, alignment procedures should be improved to reduce the need for such corrections. In addition, inclusion of fixed fiducials at various z-depths in the experimental package would again provide a straightforward means to detect and quantify any beam misalignment on a per-shot basis.

Finally, after applying the corrections described above, particle matching not only allows for accurate measures of velocity but also helps to identify erroneous particle regions, which are detected from the camera 1 and 2 holograms but are either not successfully matched or produce measured displacements that are statistical outliers. For the current results, the accepted particle displacement ranges of 12 ≤ Δx ≤ 25 µm, -5 ≤ Δy ≤ 5 µm, and 50 ≤ Δz ≤ 50 µm were empirically selected from study of displacement histograms. Note, the comparatively large range of acceptable Δz is necessary due to the higher positional uncertainty in the z-direction caused by the well-known depth-of-focus issue [43].

The top row of Fig. 7 shows an example result corresponding to the experimental hologram of Fig. 3. In Fig. 7 each particle image is refocused to it optimal depth, z_i, and cropped to the brighter rectangular areas shown. Detected particle regions are displayed with outlines, whose colors correspond with the measured 2D area equivalent diameter. White vectors show x- and y-components of measured velocities. Vector lengths are scaled by the velocity magnitude with a corresponding reference length shown in the lower right corner. The slightly darker background in Fig. 7 shows a refocused hologram image at the average measured particle depth. Figure 7(a) is the full extent of the hologram while Fig. 7(b) is zoomed in to highlight details.

 

Fig. 7. (top) Results after a single processing iteration and (bottom) final particle measurements after multi-iteration refinement. Column (a) shows the full field of view and (b) a zoomed-in region defined by the dotted box in (a).

Download Full Size | PDF

4.4 Multi-iteration refinement

As seen in the top row of Fig. 7, a single iteration of processing methodologies in sections 4.1 to 4.3 successfully quantifies most of the visible particles within the refocused measurement volume. Particles that are not successfully detected tend to be the smallest particles and those in close x,y proximity of other particles. To better quantify these particles, an iterative refinement process is proposed here.

First, the signal from all previously detected particle regions is scrubbed from the complex amplitude by adapting techniques from [83]. To start, the reference removed complex amplitude at z = 0 is propagated to the optical depth of the i^th particle,

Next, this refocused complex amplitude is multiplied by a binary mask, M, whose value is one for pixels interior to the detected particle region and zero elsewhere. As suggested in [83] this mask is “softened” via dilatation with a 2 pixel disk and low-pass filtering by a 11 × 11 Gaussian kernel with 2 pixel standard deviation. The masked complex amplitude is then propagated back to the hologram plane and subtracted from the original complex amplitude,

At this point, E’ is the complex amplitude at z = 0 wherein the signal due to the real particle image at z = z_i has been approximately removed. Similarly, the signal due to the twin image from the i^th particle at z = -z_i is scrubbed by substituting E’ for E(z = 0) and flipping the sign of z_i in Eqs. (15) and (16). Finally, this process is repeated for all detected particles.

Results are illustrated in Fig. 8, wherein the scrubbed complex amplitude has been refocused to the same optical depth and zoomed-in to the region shown in prior figures. Signal from the prior successfully detected particle regions is largely removed in Fig. 8, and signal contrast for remaining particles has been substantially improved. (Note, in [83] a multi-pass scrubbing process is suggested to further improve the suppression of detected particle regions. As shown in Fig. 8, a single pass is sufficient for the current measurements, while extinction to multi-pass scrubbing would be trivial if deemed necessary in future work.)

 

Fig. 8. The refocused amplitude where signals from previously detected particles has been removed. Field of view corresponds to by the dotted box in Fig. 3 and Fig. 7.

Download Full Size | PDF

After scrubbing the camera 1 and camera 2 complex amplitudes as described above, particle localization techniques are repeated starting with the identification of initial 2D regions in section 4.1 followed by the 3D methods in section 4.2. Newly detected particle regions are then combined with the successful detections from previous iterations and interframe matching is performed, as in section 4.3.

In the current work, this process is iterated twice accepting any particle with measured area equivalent diameter, d, greater than 5 µm. Following this, two additional iterations are performed using techniques tuned to detect small particles in the range of 2 < d ≤ 10 µm. For these final two iterations, the signal from most of the larger particles has been removed (e.g., Fig. 8), and we have found that comparatively simpler processing techniques [68,69] adequately detect the smaller particles. This is similar to [84] where it is suggested that different particle localization schemes should be optimized for detection of large vs small particles.

In these final two iterations, the minimum amplitude projection is first thresholded, $\mathbf{\mathbf{\mathscr F}}$_t{A_min}, at t = 0.3 to detect potential particle regions. Next, the computationally expensive process in section 4.2 to determine z_i via Eqs. (10) and (11) is replaced by the hybrid refinement technique of [68], which utilizes A_min and a similar maximum Tenengrad projection that are both precomputed during calculations in section 4.1. Finally, in-plane particle morphologies are again found via the process outlined by Eq. (12).

Results after completion of all four iterations are illustrated in the bottom row of Fig. 7. Overall, Fig. 7 results demonstrate successful quantification of nearly all distinguishable particles, along with measured velocity vectors that show the expected bulk flow in the positive x-direction with minimal outliers.

5. Measurement uncertainty

For experimental results, the true particle positions, sizes, and velocities are generally not known a-priori. Consequently, direct experimental quantification of measurement uncertainty is difficult. Instead, as is often done in the literature [58,66,68,69,79], this section quantifies the approximate uncertainty via analysis of simulated holograms wherein true particle properties are known. In brief, simulated 3D particle fields are constructed by randomly sampling particle positions, sizes, and displacements from distributions that approximate the experimental results. Specifically, spherical particle diameters are sampled from an exponential probability distribution whose single parameter matches the best-fit exponential of the experimentally observed size distribution corresponding to Fig. 7 results. An exponential distribution has monotonically increasing probability at decreasing diameter and was chosen to simulate the high probability of small particles below the detection size limit of the current system. To ensure physically realistic values, sampled diameters are constrained to within 50 nm < d < 50 µm. The total number of sampled particles was selected such that the number density of particles with d > 3 µm matches the experimentally observed value. Finally, sampled particle positions are constrained to ensure that no particle is located within the shadow of another particle along the z-direction.

Simulated holograms are constructed using the methodologies detailed in Supplement 1. For particles close to the wavelength of light, the scattered electric field is predicted by the Mie solution, while larger particles use an exact solution of Fresnel diffraction by an opaque disk, which is more computationally efficient compared to Mie scattering. Individual scattered electric fields are summed together at the simulated hologram plane along with a unit amplitude reference wave. As defined, simulations predict the coherent interference of multiple scattered fields at the hologram plane (sometimes referred to as speckle noise) as well as noise contributions due to particles below the detection limit of the system. Finally, simulated holograms are sampled at 3300 × 2200 pixels (half the experimental resolution) with pixel spacing matching the experimental value in the object plane (0.665 µm).

Both camera 1 and camera 2 holograms are simulated and processed identically to the experimental holograms utilizing the multi-step techniques in section 4. Figure 9 shows an example result. Like the experiments in Fig. 7, measured particle regions are shown by colored outlines. However, unlike the experimental results, true particle positions and sizes are also known and are shown by white outlines in Fig. 9. (Note, particle velocities are also measured in the simulations, but are not shown in Fig. 9). A false positive measurement appears in Fig. 9 as a measured colored outline without a corresponding true white outline. Similarly, a false negative measurement is a true white outline without a corresponding measured colored outline. Overall, Fig. 9 results show very few false positive measures, but several false negatives, especially for small diameter particles.

 

Fig. 9. Simulated particle measurements with a field a view matching the extent of the zoomed-in regions of the experimental results. See Fig. 7 for color scale.

Download Full Size | PDF

Simulations were performed for six randomly sampled instances resulting in over 6000 simulated particles. From processed results, such as Fig. 9, a successful particle detection is defined as a measured particle whose x-y centroid is within 4 µm of a true particle. For each successful detection, error in particle diameter and x, y-, and z-positions is defined as the measured value minus the true value, and Table 1 quantifies error statistics. For all quantities in Table 1, the mean and standard deviation of errors are generally small. This indicates that when particles are successfully detected, measured properties are reasonably accurate. Note, z-positional errors are higher compared to x and y, which is expected due to the depth-of-focus issue [43].

Table 1. Statistical error of measured particle properties based on hologram simulations. All values are in µm.

View Table | View all tables in this article

For many applications, including the current study of shock ejecta, particle velocities are a primary quantity of interest. As described in section 4, particle velocities are quantified by the measured x,y,z displacements over the known Δt. Additionally, for the current experimental results, camera 1 x- and y-positional measurements have also been corrected for small beam misalignments using Eqs. (13) and (14). Assuming independent variables, straightforward propagation of uncertainty provides a relation between uncertainties in the input variables, including values in Table 1 and others given in prior sections, to the uncertainty in the measured particle velocities. For a typical particle located at z = 2.8 mm, calculations estimate the one standard deviation uncertainties of measured x- and y-velocities around 100 m/s. However, due to the depth-of-focus issue [43], uncertainty in z-velocities is on the order of km/s. Because of this extremely high uncertainty and the relatively minor importance in comparison with the main flow velocities in the x-direction, the current work does not present measured trends in z-velocities.

Next, Fig. 10(a) compares the measured particle size histogram with the true value. Despite the accurate measurement of individual particle diameters quantified in Table 1, the measured size histogram in Fig. 10(a) is biased low, especially at small diameters. The cause of this bias can be understood in Fig. 10(b), which shows the false positive and false negative detection rates. As particle diameter decreases the number of pixels and contrast within the refocused particle interior decreases (see Fig. 2). As a result, automated detection of particles becomes more challenging, and as seen in Fig. 10(b) the rate of false negative detection increases with decreasing particle diameter, eventually reaching about 50% at the measurement cutoff diameter of 2 µm.

 

Fig. 10. (a) Actual and measured particle size distributions from simulated holograms and (b) the corresponding particle detection rates.

Download Full Size | PDF

If desired, the statistical results presented here could be extended to study measurement uncertainty in other quantities of interest such as surface or volumetrically weighted mean diameters, best fit distribution parameters, etc. In addition, with low false positive rates in Fig. 10, the difference between the true and measured size distribution could be viewed as a known measurement bias and a correction defined. Such efforts require additional study and will be left for future work. Instead, for the current work we simply note that the false negative bias in the size distribution is roughly constant for diameters greater than 4 µm, and measurement trends discussed in the next section will focus on this range.

Finally, it should be noted that simulations do not consider several additional sources of potential uncertainty such as lens distortions, effects of non-spherical particle morphologies, particle overlap in the z-direction, scattering off the free surface, and beam quality issues caused by shot-to-shot variations or dust on optics. Consequently, reported uncertainties are likely a lower bound of the true measurement uncertainties.

6. Results and discussion

Three experimental shots were investigated at the conditions listed in Table 2. For all cases, a 2 mm thick Sn sample initially had a single v-shaped groove machined in the vertical direction with a nominal groove depth of 40 µm. Shot A had a 120° groove opening angle, similar to much of our prior results in [38–41], while Shots B and C had a slightly wider groove opening angle. As explained in [39] the explosive package included a wire fiducial located 78 mm from the initial location of the Sn surface. In Shot A, this fiducial was placed slightly outside and to the right of the holographic field of view (FOV). Because this fiducial was not visible in the experimental image, the initial x-position of the Sn surface with respect to the center of the FOV in the third column in Table 2 is approximate. In the subsequent Shots B and C, the wire fiducial was located within the FOV, allowing for direct measurement of the initial x-position at the precision reported in Table 2. (Note the wire fiducial is not visible in the normalized results presented subsequently, because signal from the fixed wire is removed by normalization). Finally, for all three shots, the PDV measured jump off velocity of the Sn surface was within 2.01 ± 0.01 km/s.

Table 2. Experimental conditions.

View Table | View all tables in this article

The final column in Table 2 gives the experimental acquisition times with respect to the first motion of the Sn surface as quantified by PDV. Compared with much of our prior film results [38–41], acquisition times in Table 2 are relatively long. These long delay times were intentionally selected to quantify instances where the free surface is near to the FOV. At these locations, ejecta particle diameters are anticipated to be relatively large and more amenable to digital holographic recording, which, as described in section 2, is expected to have degraded ability to measure the smallest particles compared to film. Consequently, the experimental data reported here are at a location and time in the ejecta formation process that has yet to be extensively investigated.

Results presented in section 3 and 4 are for Shot A. Shots B and C were processed identically and results are visualized in Fig. 11. Shot B, in the left column of Fig. 11, was recorded at a time where the free surface was at x≈-7.3 mm, as determined by the product of the free surface velocity and the acquisition time. Shot C is nominally the same condition but recorded at a slightly later time, where the free surface is at x ≈ -5.3 mm. Because Shot C was recorded later, the FOV captures slower moving portions of the ejecta field compared with Shot B. For this reason, Shot C results in Fig. 11 show more elongated ligaments (sometimes referred to as tendrils or microjets). In Shot C, there are also relatively few particles measured in the left side of the image, near the free surface. At these locations, the hologram contains significant noise, likely due to scattering off the free surface. More work is needed to determine if particle densities are truly lower in these locations or if the lack of detection is driven by the higher noise.

 

Fig. 11. (top) Shot B and (bottom) Shot C results. Column (a) shows the full field of view and (b) a zoomed-in region defined by the dotted box in (a).

Download Full Size | PDF

Table 3 summarizes the measured particle statistics. The mean diameters in Table 3 are given by,

Table 3. Measured ejecta particle statistics.

View Table | View all tables in this article

Sorenson et al. [41] used film holography to quantify nominally the same conditions as Shot A. The mean diameters from the prior [41] data are D₁₀ = 9.8 µm and D₃₀ = 13.2 µm when only particles with d ≥ 2 µm are considered for consistency with the current resolution limits. These values are within 10% of the Shot A results in Table 3. This agreement increases confidence in both the current results and the prior film data in [41].

Particle size information is presented in more detail in Fig. 12, where probably densities of measured diameters are shown for all three shots along with the prior film results from [41]. In Fig. 12(a) probability densities are plotted with linear axes, and Fig. 12(b) plots the same data on a log-log scale to highlight differences. For data from the current experiment, the left most bin in Fig. 12 corresponds to diameters between 2 to 4 µm. As discussed in section 5, detection efficiencies in this range are relatively poor, and data in this bin were ignored during the normalization of probability densities. Finally, error bars in Fig. 12 show the 95% confidence bounds calculated using the methods in [85]. As such, error bars quantify the statistical convergence of each data point. If shots are repeated and assumed to be governed by the same mean statistics, then individual probability values will vary over the ranges shown simply due to the limited number of sample points in each bin. Differences that fall within the confidence ranges are, therefore, not statistically significant.

 

Fig. 12. Probability densities of measured particle diameters plotted on (a) linear and (b) log-log axes. Note, probability densities from the current work are normalized using diameters ≥ 4 µm. Error bars show the 95% confidence intervals. Solid black line in (b) is power-law relation with a -5.5 exponent.

Download Full Size | PDF

Probability density results for Shots A and B show no statistically significant differences and are generally consistent with the prior data of [41]. In addition, in [37,41], the probability density of the larger particles are shown to follow power-law relations with a best fit exponent of around -5.5. Such a relation is shown by the solid black line in Fig. 12(b), which has a -5.5 slope and appears to match the measured data at diameters >∼20 µm. We again reiterate that the current data are recorded significantly closer to the free surface compared to much of our prior film-based data, and additional experimental results will be needed to enable more quantitative comparisons to literature values.

Finally, we note that Shot C size distributions show some statistically significant differences compared to Shots A and B. Specifically Shot C quantifies higher probabilities at larger diameters. As already discussed, the elongated ligaments quantified in Shot C are likely to breakup further at later times. Therefore, particle size distributions of these near surface ejecta may evolve to become more like Shots A and B at later times. Again, more work is warranted to study these dynamics.

Next, we consider the particle x-velocities. In Fig. 13(a) measured velocities are plotted as a function of the relative x-position with respect to the free-surface location. This relative position is found by taking the x-position of each particle and subtracting the projected free-surface location at the measurement times (determined by assuming a constant 2.01 km/s free-surface velocity combined with the experimental conditions in Table 2). As expected, Fig. 13(a) shows that measured particle velocities are all faster than the free-surface velocity shown by the horizontal solid black line. The few outliers are likely due to measurement uncertainty discussed later. In addition, Fig. 13(a) also resolves the expected trend of increasing particle velocities with increasing distance from the free surface. This trend is quantified by the dotted line in Fig. 13(a), which is the best-fit for all Shot A, B, and C data combined. This dotted line intersects the free-surface velocity (solid black line) at approximately x = 0, which is consistent with expectations that ejecta velocities are primarily determined by the rapid dynamics at the free surface where the collapsing groove forms into initial ligaments and sheets.

 

Fig. 13. (a) Particle x-velocities plotted as a function of x-position with respect to the free surface location. The solid black line is the free-surface velocity while the dotted black line is the best fit to all Shot A, B, and C data combined. (b) The same particle x-velocities plotted as a function of the expected velocity. The solid black line is a one-to-one relation. Note, 250 randomly sampled data points are shown for each experimental shot.

Download Full Size | PDF

In many prior experiments, where direct measurements of particle velocities are unavailable, mean velocities are assumed to follow a simple ballistic model wherein particles are produced instantaneously when the shockwave impacts the initial surface. With this assumption and negligible in flight drag, mean ejecta velocity at a downstream location is found by dividing the measurement standoff distance (third column in Table 2) by the acquisition time (final column in Table 2). Figure 13(b) replots the measured x-velocities as function of this expected velocity. Data from the only other prior holographic measure of ejecta velocities by Sorenson et al. [39] are also shown for comparison. In Fig. 13(b) the solid black line shows a one-to-one relation between measured and expected velocities. In general, Fig. 13(b) shows that mean ejecta velocities are in good agreement with this simple model. Indeed, measured x-velocities are on average equal to 99%, 101%, and 99% of the expected velocities for Shots A, B, and C, respectively. Similarly, the average of the Sorenson et al. [39] data are also equal to 98% of the ballistic model. These observations all confirm that the ballistic model reasonably describes the mean ejecta velocity for experiments conducted in vacuum, as reported here.

In addition to the spatial velocity gradients noted in Fig. 13, there is also scatter about the linear trend. To study if this scatter contains any additional physically meaningful information, Fig. 14 shows probability densities of the measured x- and y-velocity components where the mean spatial variations in x-velocities given by the dotted line in Fig. 13(a) have been removed. Figure 14 results show that the remaining velocity variations are statistically similar for all shots and are statistically similar in the x- and y-directions. In fact, the standard deviations of all probability densities in Fig. 14 are within 90 ± 40 m/s. This magnitude is similar to the 100 m/s velocity uncertainties reported in section 5. Therefore, Fig. 14 indicates that current measurement precision is insufficient to quantify physically meaningful trends in velocity statistics beyond those reported in Fig. 13. Future experiments may want to reduce this velocity uncertainty to measure other quantities, such as the joint distribution of size and velocity, which could provide additional insight into breakup physics. Certainly, one should improve alignment procedures as discussed in section 4.3 and eliminate the need for the Eqs. (13) and (14). Doing so would reduce velocity uncertainties to about 30 m/s. Beyond that, increasing interframe time, Δt, to achieve larger interframe displacements is the most effective and straight forward means to reduce velocity uncertainty.

 

Fig. 14. Probability densities of (a) particle x-velocity after subtracting the spatial trends defined by the dotted lines in Fig. 13 and (b) particle y-velocities.

Download Full Size | PDF

Finally, Fig. 15 illustrates the 3D nature of the Shot A particle field. In this image, spheres are shown for each particle located at their measured x,y location and are colored by their measured z-position. The displayed diameter of each sphere is proportional to the measured area equivalent diameter. In Fig. 15, measured z-positions show a clear sinusoidal variation in the y-direction. In addition, horizontally aligned collections of large particles, which almost certainly originate from breakup of single elongated tendrils, tend to be located within a narrow range of z-positions. These variations are likely caused by a fluid dynamic instability acting in the z-direction. Prior to this work, the ability of holography to resolve these 3D flow structures has not been previously reported. Future work could leverage such information for improved study and validation of ejecta formation mechanisms.

 

Fig. 15. Three-dimensional (3D) variation of measured particle positions in Shot A.

Download Full Size | PDF

7. Conclusions

This work reports the first digitally recorded and numerically post-processed measurements of explosively generated ejecta particles whose diameters are ${\mathcal O}$(µm) and velocities are ${\mathcal O}$(km/s). This is achieved by upgrading the long-working distance, high-magnification, UV inline holographic configuration in [38–41] with CCD recording in place of film. Because the pixel pitch of digital sensors are orders of magnitude larger than gain sizes of holographic film, direct replacement of film with CCD recording would result in significant degradation of resolving power and an unacceptable increase in the minimum measurable particle diameter. To address this, the current work adds magnifying optics to the system in [38–41] and develops processing algorithms specifically tailored for the current pixel-limited resolution. In addition, velocity measures are achieved using two CCDs, which record orthogonally polarized pulses separated by an 8.84 ns optical time of flight delay. Uncertainties in measured particle size, 3D positions and velocities are studied using simulations. Finally, three experimental realizations are presented. Measurement size and velocity statistics are shown to be consistent with the literature expectations.

Important advancements and conclusions from the current work are:

In addition, this work also suggests many possible topics for additional development: