2.1 Planar PIV

Planar PIV (figure 2 b) was first used in the study of dusty plasmas in 1999 (Thomas Jr Reference Thomas1999). In this approach, a pair of images are acquired by a single camera that is oriented perpendicular to, and synchronized with, a pair of laser pulses that follow the same optical path and are expanded into a light sheet to illuminate a thin slice of the dust cloud. Each image pair (figure 3 a) is then divided into $n\times n$ subregions known as interrogation regions (figure 3 b) and a cross-correlation analysis, (2.1), is performed using similar subregions from the image pair (Keane & Adrian Reference Keane and Adrian1990, Reference Keane and Adrian1992). Overlapping interrogation regions are often employed to improve the spatial resolution and the quality of the measured displacements

(2.1) $$\begin{eqnarray}\displaystyle R(m,n) & = & \displaystyle R_{D}(m,n)+R_{F}(m,n)+R_{C}(m,n)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i,j}I(i,j,t_{1})I(i+m,j+n,t_{2}),\end{eqnarray}$$

where $R(m,n)$ is an element in the correlation plane, $R_{D}(m,n)$ , $R_{F}(m,n)$ and $R_{C}(m,n)$ represents that correlation between identical particles, the correlation between random particles and the convolution of background image intensity at location $(m,n)$ in the correlation plane, respectively and $I(i,j,t_{k})$ represents the image intensity at pixel location $(i,j)$ in the interrogation region at time $t_{k}$ , $k=1,2$ . While several algorithms have been developed to perform this correlation analysis (Meinhart, Wereley & Santiago Reference Meinhart, Wereley and Santiago2000; Piirto et al. Reference Piirto, Eloranta, Saarenrinne and Karvinen2005; Sciacchitano, Scarano & Wieneke Reference Sciacchitano, Scarano and Wieneke2012), the most common algorithm currently employed within the dusty plasma community is the standard fast Fourier transform cross-correlation function, as described in sections 3.4 and 5.4 of Raffel et al. (Reference Raffel, Willert, Wereley and Kompenhans2007).

Figure 3. (a) Representative image of a dust cloud with a $32\times 32$ pixel grid superimposed. (b) Representative interrogation regions showing a small fraction of the dust grains visible from the blue box seen in (a) and separated in time by ${\rm\Delta}t_{PIV}=t_{2}-t_{1}$ . (c) The correlation map that is constructed by performing the PIV analysis on the interrogation regions seen in (b). The resulting correlation map consists of three parts, the correlation between identical particles, $R_{D}$ , the correlation between random particles, $R_{F}$ , and the convolution of background image intensity, $R_{C}$ . Here, the average displacement is represented by the peak denoted by $R_{D}$ , while $R_{F}+R_{C}$ represents the noise in the measurement.

The resulting correlation plane (figure 3 c) represents a statistical measurement of the particle displacement and the peak in the correlation plane represents the most likely displacement of the particles located in the interrogation region (figure 3 b). This peak is then fit to find the average displacement of the particles in the interrogation region that was used in the application of the PIV algorithm. This fitting yields an uncertainty in the measured displacement that can be as small as 0.1 pixel. This process is repeated over the entire image, yielding an instantaneous measurement of the particle transport over the field of view of the camera. Often, this correlation analysis is repeated several times using the velocity field from the previous iteration to offset and deform the interrogation region used in the subsequent correlation analysis (Kim & Sung Reference Kim and Sung2006). This leads to fewer particles leaving the interrogation regions that are used in the final iteration (Kim & Sung Reference Kim and Sung2006) and a more accurate measurement of the particle displacement (Scarano & Riethmuller Reference Scarano and Riethmuller2000; Scarano Reference Scarano2002).

Because the correlation analysis is essentially a statistical particle-matching technique, it is important that there be several particles in each interrogation region. Additionally, the size of the interrogation region that is used depends on the value of ${\rm\Delta}t_{PIV}$ used when acquiring the data. In particular, there are three factors that should be met for the PIV algorithm to accurately capture the particle motion: the particles in each interrogation region should exhibit directed motion (i.e. the majority of the particles in an interrogation region should move in the same direction), the typical displacement of the particles in the interrogation region should be of the order of or less than $1/4$ the size of the interrogation region used in the PIV analysis and a minimal number of particles should leave the interrogation region over the duration of the measurement, ${\rm\Delta}t_{PIV}$ . Recent advances in PIV algorithms, including the use of non-square interrogation and search regions, allow for larger displacements to be measured. However, it is critical that directed motion be observed at the scale size of the interrogation region selected.

While the planar technique has proven to be a valuable diagnostic tool and has provided significant insight into a range of phenomena including particle transport (Thomas Jr Reference Thomas1999, Reference Thomas2001a , Reference Thomas2002b ; Thomas Jr et al. Reference Thomas, Amatucci, Compton and Christy2002a ; Aldewereld & Thomas Jr Reference Aldewereld and Thomas2007) the potential structure within a dust cloud (Thomas Jr Reference Thomas2002a ) and near the dust-cloud boundary (Thomas Jr Reference Thomas2001b ; Thomas Jr et al. Reference Thomas, Annaratone, Morfill and Rothermel2002b ; Thomas Jr Reference Thomas2003) and the structure of the dust acoustic wave, (Thomas Jr & Merlino Reference Thomas and Merlino2001), one important limitation is that this technique only measures the projection of the motion of the dust grains seen by the cameras onto the plane of the laser sheet. As a result, the motion perpendicular to the laser sheet is lost. While this is not a significant issue for two-dimensional dusty plasma systems, this can be significant for many of the three-dimensional dusty plasma systems which are often in a weakly coupled state and exhibit transport that is fully three-dimensional. For these systems, this projection of the observed motion onto the illuminated slice of the dust cloud represents a limitation of this technique, particularly when looking at the dust cloud–plasma boundary, wave structure or systems where there is significant motion in the direction perpendicular to the laser sheet that is used to illuminate the dust cloud (Thomas Jr et al. Reference Thomas, Williams and Silver2004).

2.2 Stereoscopic PIV

Stereoscopic PIV (figure 2 c) (Prasad Reference Prasad2000b ) was first used in the study of dusty plasmas in 2004 (Thomas Jr et al. Reference Thomas, Williams and Silver2004) and is an extension of the planar PIV technique. Here, two cameras are oriented obliquely to the plane of the laser sheet and each camera records a pair of images that are synchronized to the firing of a pair of lasers. The oblique viewing angles between each camera and the laser sheet results in non-uniform magnification across the field of view and the images being distorted (Prasad & Jensen Reference Prasad and Jensen1995). To account for this distortion, each camera is fitted with a tilt adapter between the camera lens and the imaging sensor. The field of view of each camera is adjusted using this tilt adapter until no distortion is observed and an empirical calibration is performed to determine a calibration function that describes the orientation of each camera relative to the laser sheet (Tsai Reference Tsai1986). Once the calibration process is complete, a PIV measurement can be made and a pair of two-dimensional velocity fields are computed using the image pairs acquired by each camera. The resulting pair of velocity fields that are generated represent the projection of the particle motion onto the illumination plane measured by each of the cameras. The true three-component velocity vector ( $v_{x},v_{y},v_{z}$ ) of a cluster of particles in a sub-region of the acquired images is reconstructed from this pair of two-dimensional velocity fields and the empirical calibration function.

This technique has provided significant insight into a range of phenomena involving the strength of the neutral drag force (Thomas Jr & Williams Reference Thomas and Williams2005), the spatial structure and thermal properties of the dust acoustic wave (Thomas Jr Reference Thomas2006, Reference Thomas2010; Fisher & Thomas Jr Reference Fisher and Thomas2010) and the thermal state of weakly coupled dusty plasma systems (Williams & Thomas Jr Reference Williams and Thomas2007; Fisher & Thomas Jr Reference Fisher and Thomas2011, Reference Fisher and Thomas2012, Reference Fisher and Thomas2013; Fisher et al. Reference Fisher, Avinash, Thomas, Merlino and Gupta2013). However, these measurements of the three-dimensional transport are restricted to a thin slice of the dusty plasma system and may not fully capture the true three-dimensional motion if particles leave the laser sheet during the duration of the measurement, ${\rm\Delta}t_{PIV}$ . Additionally, the reconstruction of the third velocity component requires spatial calibrations that are accurate to within a few pixels. As a result, these systems, when compared to the planar PIV systems, are much more sensitive to mechanical noise. It is noted, however, that is possible to use the image data acquired during an experiment to correct the mapping function, thereby accounting for small changes that might have occurred between the calibration process and the acquisition of data; including small errors in the initial calibration function or minor shifts in the relative orientation of the cameras and laser sheet due to mechanical vibrations. This self-calibration technique (Wieneke Reference Wieneke2005) identifies a small subset of particles in each of the acquired images and their triangulation in three-dimensional space over the measurement sequence to determine a statistical estimate of the local disparity vector that represents the most probable offset between the lines of sight for the different cameras and the laser sheet. This disparity vector can then be used to correct the calibration function, essentially eliminating this as a concern when using the stereoscopic approach.

2.3 Tomographic PIV

Tomographic PIV (figure 2 c) (Elsinga et al. Reference Elsinga, Scarano, Wieneke and van Oudheusden2006) was first used in the study of dusty plasmas in 2011 (Williams Reference Williams2011) and is an extension of the stereoscopic PIV technique. Here, a pair of laser pulses are expanded to illuminate a rectangular volume of the dusty plasma system; the size of the illumination volume is determined by the intensity of the laser. Three or more (typically, four) cameras are oriented obliquely to the illumination volume, are synchronized to the laser pulses and image the dust cloud when the lasers are fired. To account for the distortion from the oblique viewing angles between each camera and the illumination volume, each camera is fitted with a tilt adapter between the camera lens and the imaging sensor (Prasad & Jensen Reference Prasad and Jensen1995). The field of view of each camera is adjusted in the same way as described for the stereoscopic PIV system and an empirical calibration is performed to find a calibration function (Tsai Reference Tsai1986) that relates the orientation of each camera and the illumination volume. Once the calibration is complete, pairs of images are acquired from each of the cameras used and the particle distribution in the illuminated volume during each laser pulse is reconstructed from the set of images acquired by each of the cameras during a laser pulse as a three-dimensional intensity distribution. This reconstruction of the intensity distribution represents the particle locations in the illumination volume during the firing of each laser and is typically found through the use of multiple (typically five) iterations of a multiplicative algebraic reconstruction technique (MART) algorithm (Herman & Lent Reference Herman and Lent1976). The pair of three-dimensional intensity distributions that are reconstructed from the acquired image pairs are then decomposed into a series of $m\times n\times p$ (where $m$ , $n$ and $p$ are typically between 32 and 64 pixels) volumetric interrogation regions and the average displacement of the particles within each of these interrogation regions is found by performing a three-dimensional cross-correlation analysis (Wieneke Reference Wieneke2008).

This technique has been used to gain insight into the three-dimensional transport within a weakly coupled dusty plasma (Williams Reference Williams2011), the three-dimensional properties of the dust acoustic wave (Williams Reference Williams2012) and the thermal state of weakly coupled dusty plasma systems (Williams Reference Williams2011). A key advantage to this approach is that it provides a much more complete measurement of the transport properties over an extended volume. However, there are two limitations that should be noted.

First, the limited number of viewing angles (cameras) that are used in the reconstruction process results in the creation of non-existent particles in the reconstructed three-dimensional intensity distribution, an artefact known as ghost particles (Maas, Gruen & Papantoniou Reference Maas, Gruen and Papantoniou1993). As the number density increases, the ability of the reconstruction algorithm to uniquely identify the location of dust grains decreases, leading to an increase in the number of ghost particles. This effect can be limited by either increasing the number of cameras used or by limiting the number densities in the system being examined. In particular, the additional empty space that appears in the image due to the lower number densities helps to suppress this artefact. As a result, the number density for the typical four-camera system should be between 0.05 and 0.15 particles per pixel (Scarano Reference Scarano2013), which is lower than what can be used when using a planar or stereoscopic PIV system. For context, if the dust cloud was imaged using a tomographic PIV system consisting of four one-megapixel cameras and the dust cloud filled the field of view of the camera, there would be approximately 50 000 particles in the illuminated volume. That said, while the existence of these ghost particles is an unavoidable artefact of the reconstruction process, they are relatively easy to identify in the reconstructed image volume. Here, real particles appear as spherical objects and ghost particles appear as elongated objects in the direction perpendicular to the illumination volume (i.e. the $z$ -direction in figure 7). Further, one can quantify the fraction of ghost particles in the reconstructed volume by repeating the reconstruction process over a larger depth. Outside of the illuminated volume, one measures a baseline intensity coming only from ghost particles. Within the illumination volume, one measures an intensity with contributions from both real and ghost particles. The ratio of these intensities provides a measure of the fraction of ghost particles present (Scarano Reference Scarano2013). More important, the three-dimensional correlation analysis is reasonably robust to the presence of these ghost particles because the total intensity of the ghost particles in the reconstructed volume is smaller than that of real particles and this intensity is spread over a larger volume. As a result, the contributions of ghost particles in the volumetric correlation analysis is relatively small (Elsinga & Tokgoz Reference Elsinga and Tokgoz2014), resulting in them not contributing in a meaningful way in the three-dimensional cross correlation analysis provided that the ratio of of ghost particles to real particles in not too large.

Second, this approach is much more sensitive to errors in the calibration function, requiring errors of less than 0.1 pixel. At this maximum level of error in the calibration function, even the settling of individual lenses in the imaging optics is enough to ruin a valid calibration. However, as previously noted, recent advances in self-calibration (Wieneke Reference Wieneke2008) have significantly reduced this concern but it does require that the cameras be much more rigidly fixed than in the previously described approaches and that the calibration must be done as close in time to the acquisition of data as possible.

2.4 Time-resolved planar PIV

The time-resolved planar PIV technique involves acquiring image data using a conventional dusty plasma diagnostic system and then applying the PIV analysis technique to the acquired images. In this approach, a thin sheet of dusty plasma system is illuminated using a CW laser. Here, the exposure time and frame rate of the camera are used to determine when an image is acquired, while the exposure time of the camera serves as the duration of the illumination pulse, ${\rm\Delta}t_{laser}$ , in a dedicated PIV system. Once the image sequence has been acquired, appropriate image pairs are then collected and the planar PIV analysis technique is applied to these image pairs. This approach has been made possible by the availability of high frame-rate cameras and of several open-source PIV software packages (van der Graaf Reference van der Graaf2008; Vennemann Reference Vennemann2008; Taylor et al. Reference Taylor, Gurka, Kopp and Liberzon2010; Thielicke & Stamhuis Reference Thielicke and Stamhuis2014).

Since the analysis approach and orientation of the diagnostic system is identical to what is used in the planar PIV approach, this approach inherits all of the advantages and disadvantages that are associated with the planar PIV technique previously noted. However, this approach also has the advantage that additional specialized timing equipment is not necessary. Because of this, this approach has seen an increased amount of use in recent years and has been used to examine a range of transport, wave and thermal properties of dusty plasma systems. Additionally, in this approach the time between images does not need to be known a priori. This allows the researcher to determine the type of motion that they are interested in examining after the data has been acquired and allows for the examination of kinetic and fluid-like behaviour simultaneously. It should also be noted, however, that one does need to pay closer attention to quality of the acquired images. In particular, the longer exposure times of the camera (compared to the duration of the pulse laser, ${\rm\Delta}t_{laser}$ , used in a dedicated system) can lead to the particles moving during the image acquisition, potentially resulting in some amount of particle streaking in the acquired images. This streaking results in a broadening of the peak in the correlation plane that represents the average displacement of the particles in the interrogation region, resulting in larger uncertainties in the measured displacement.

4.1 Uncertainty quantification

In the application of the PIV technique, there are a number of potential sources of uncertainty, including noise in the acquired images (Huang, Dabiri & Gharib Reference Huang, Dabiri and Gharib1997), the number density of the particles in the acquired images, peak locking that occurs due to the use of small particles (Westerweel Reference Westerweel1997; Christensen Reference Christensen2004), particle motion that is perpendicular to the illumination plane (Nobach & Bodenschatz Reference Nobach and Bodenschatz2009) and the existence of velocity gradients (Meunier & Leweke Reference Meunier and Leweke2003; Westerweel Reference Westerweel2007). Many of these issues have been extensively examined by the fluid mechanics community through modelling the measurement process (Westerweel Reference Westerweel1997) or through the use of Monte Carlo simulations (Keane & Adrian Reference Keane and Adrian1992; Fincham & Delerce Reference Fincham and Delerce2000; Scarano & Riethmuller Reference Scarano and Riethmuller2000; Lecordier et al. Reference Lecordier, Demare, Vervisch, Réveillon and Trinité2001). The results of these efforts have shown that the error associated with the PIV technique is typically between 0.03 and 0.1 pixel (Raffel et al. Reference Raffel, Willert, Wereley and Kompenhans2007). However, it is also known that the Monte Carlo simulations can underestimate the errors that are observed in experiments (Stanislas et al. Reference Stanislas, Okamoto, Kähler and Westerweel2005). As a result, the fluid mechanics community has recently developed a number of techniques to assess the uncertainty in the measured velocity field using information that is available in the experimental data (i.e. the images used in the PIV analysis, the cross-correlation function and the velocity vectors). These methods include the uncertainty surface method (Timmins et al. Reference Timmins, Wilson, Smith and Vlachos2012), the particle disparity method (Sciacchitano, Wieneke & Scarano Reference Sciacchitano, Wieneke and Scarano2013), the peak ratio method (Charonko & Vlachos Reference Charonko and Vlachos2013) and the correlation statistics method (Wieneke Reference Wieneke2015). While each of these methods has strengths and limitations (Sciacchitano et al. Reference Sciacchitano, Wieneke and Scarano2013), it has also been shown that the uncertainty in the measured velocity field tends to vary across the measurement region and that the significance of each of the previously noted sources of uncertainty contributes different amounts to the total measurement uncertainty depending on the experimental conditions (Neal et al. Reference Neal, Sciacchitano, Smith and Scarano2015; Sciacchitano et al. Reference Sciacchitano, Neal, Smith, Warner, Vlachos, Wieneke and Scarano2015). Given that the application of PIV techniques to a dusty plasma does not meet the same conditions/criteria that are typically seen in a fluid experiment, it is necessary to perform a similar comparison on data from the dusty plasma community to provide a more accurate measure of the uncertainty that is present is the resulting velocity fields.

4.2 Time-resolved stereoscopic PIV

Advances in imaging technology, particularly the availability of higher frame-rate cameras, have made it possible to analyse data from a wide range of laboratory and microgravity experiments using the PIV technique. As a result, the planar PIV technique has been adapted and applied to a wide range of experimental data acquired using the conventional diagnostic system within the dusty plasma community. However, as previously noted, there are limitations to the planar PIV approach. Additionally, the dedicated systems that employ the more sophisticated PIV approaches (i.e. stereoscopic and tomographic PIV) that have been used in the study of dusty plasmas have limited temporal resolution ( ${\leqslant}15~\text{Hz}$ ). Given the three-dimensional nature of many dusty plasma systems and the need for better temporal resolution to better understand the transport and thermal properties of dusty plasmas, particularly those in a weakly coupled state, there is a need for a diagnostic system that can recover the three-dimensional motion with greater temporal resolution than currently exists. One way to address this need in the immediate term would be to leverage advances in high-speed imaging to develop a time-resolved stereoscopic PIV system using the work that has been done in adapting the planar PIV technique for use on data from the conventional diagnostic system within the dusty plasma community as a roadmap for the development of this new diagnostic approach.

4.3 Combining PIV and PTV analysis techniques

The PIV and PTV techniques have been the primary diagnostic techniques used in the study of dusty plasmas. While the most appropriate analysis technique is determined based on the number density and observed particle motion in the acquired images, there are cases where these techniques can be used simultaneously to gain greater insight (Williams et al. Reference Williams, Thomas, Couëdel, Ivlev, Zhdanov, Nosenko, Thomas and Morfill2012). This presents a new direction for the analysis of data from dusty plasma experiments, one where the PTV and PIV techniques are used in tandem to gain greater insight into the underlying physics. Additionally, it is possible that these diagnostic approaches could be merged to provide a single diagnostic tool for the study of dusty plasmas. One approach would be to perform an initial analysis using the PIV technique to find a robust estimate of the local velocity field. This local velocity field would be used in the PTV analysis to predict particle positions, reducing the number of possible particle pairs and the number of false measurements. A second approach would be to develop a single code that would determine which technique would be most appropriate based on the number density and particle motion in the acquired images before performing the analysis. Both of these approaches would require the development and benchmarking of new analysis codes but could provide a valuable diagnostic tool to the dusty plasma community.

4.4 Plenoptic PIV

Plenoptic PIV (Fahringer, Lynch & Thurow Reference Fahringer, Lynch and Thurow2015) is a developing technology that is based on light-field imaging. This imaging approach was first proposed in 1908 (Lippmann Reference Lippmann1908) and is based on the idea that the propagation of a light-ray is represented by a five-dimensional function, known as the light field, that describes the location of the light-ray in space, $(x,y,z)$ , the direction of the light-ray’s propagation, $({\it\theta},{\it\phi})$ and the wavelength of the light-ray, ${\it\lambda}$ . When applying this principle to PIV, it is often the case that a single wavelength is used when acquiring the image data and that the intensity of the light is constant. As a result, the light field reduces to a four variable function, $L(x,y,{\it\theta},{\it\phi})$ . In conventional imaging, a camera lens collects light over a range of angles, focusing the light onto the imaging sensor, effectively integrating the signal and resulting in only two of the relevant dimensions of the light field, $(x,y)$ , being measured. In the stereoscopic and tomographic PIV approaches previously described, the information that is gained by the use of additional cameras and the empirical calibration function relating the orientation of the cameras and illumination region are used to reconstruct the four-dimensional light field to recover the full three-dimensional motion.

A plenoptic, or light-field, camera measures the full light-field function by placing an array of short focal length micro lenses at the focal length of the camera lens in front of the imaging sensor. Each microlens determines the position $(x,y)$ of a light-ray collected by the main camera lens, and then projects the light-ray onto an array of pixels, allowing for the direction of the light-ray’s propagation, $({\it\theta},{\it\phi})$ , to be determined. The result is a complete measurement of the light field, $L(x,y,{\it\theta},{\it\phi})$ , and an image that can be refocused at any plane in the depth of field, allowing for a fully volumetric measurement with a single camera. When applied to dusty plasmas, it could be possible to fully reconstruct the three-dimensional intensity field that represents the illuminated volume and a volumetric cross-correlation can then be performed to find a complete three-dimensional transport of the dust component. A key advantage is that this would require only a single camera. While this technology is still being developed, promising results have been seen within the fluid mechanics community (Thurow et al. Reference Thurow, Fahringer, Johnson and Hellman2014; Deem et al. Reference Deem, Agentis, Nicolas, Cattafesta, Fahringer and Thurow2015). This approach represents a promising diagnostic technique that has the potential to provide a complete measurement of the particle transport in a way that is notably more simple than the techniques currently employed.