2.1 Camera and illumination set-up

On the top of the vacuum chamber, we mount two video cameras: a high-speed scientific digital video camera which has a narrow field-of-view (FOV) and serves as the main data-taking device, and wide FOV camera used to survey the entirety of the dust cloud. The latter is generally not used for image analysis, so it can be a less expensive analogue camera. To avoid vibration, these cameras are not suspended in a cantilever configuration, but within a box-like support structure, as shown in figure 1. Preventing vibration is important because camera movement of just a few microns can lead to erroneous measurements of particle velocity. In addition to these top-view cameras, we also use a side-view camera that is positioned at the same height as the dust particles.

A typical scientific camera has a monochrome sensor with of the order of one megapixel resolution. Imaging rates of up to 1000 pictures per second are sometimes needed, although 200 frames  $\text{s}^{-1}$ , or even less, is more common. A typical example is the Phantom v5.2 shown in figure 1, which has 12 bits of greyscale information in each pixel. Such a camera has an adjustable shutter speed, although with a high frame rate it is common to operate the camera with the longest possible exposure time to achieve a sufficiently bright image. Movies are stored by the camera software as a sequence of bit-map images.

The camera lens is typically a ‘macro’ lens of the kind used for 35 mm film cameras. In our laboratory at Iowa we usually use a 105 mm focal length Nikon lens, as seen in figure 1. The face of the lens is typically positioned approximately 25 cm from the dust cloud; this long working distance is necessary because of the dimensions of the vacuum chamber. A filter is mounted on the entrance of the camera lens; this is typically a 10 nm bandpass filter chosen to admit the wavelength of the laser light used to illuminate the particles, while blocking most of the glow from the plasma.

To illuminate the particles, a laser beam is shaped into a fan or sheet of light. In figure 2 a steady-state laser beam, with a power of up to 100 mW, is deflected into a vertical axis where it is focused by a pair of spherical lenses L1 and L2, to achieve a Gaussian focal waist in the centre of the chamber. The beam then passes through a cylindrical lens L3 to disperse the beam into a fan shape. The lens can be rotated, and a subsequent mirror M2 can be tilted and translated to achieve a horizontal orientation for the sheet so that it coincides with the dust particle layer. As an alternative to the cylindrical lens shown in figure 2, it is also possible to raster a laser beam using a high-speed rotating polygonal mirror (Pieper, Goree & Quinn Reference Pieper, Goree and Quinn1996) or a galvanometer mirror that is tilted rapidly back and forth according to a repetitive waveform applied by a function generator (Feng et al. Reference Feng, Goree and Liu2008; Feng, Goree & Liu Reference Feng, Goree and Liu2012a ,Reference Feng, Goree and Liu b ).

Figure 2.

Laser illumination optics. A steady-state laser beam is deflected upward by mirror M1 and focused by a telescope consisting of spherical lenses L1 and L2. It is then dispersed into a laser sheet by a cylindrical lens L3 and deflected into a horizontal sheet by mirror M2. This horizontal sheet defines the illuminated volume in the chamber, and it is adjustable using translation stage TS.

Example top-view images are shown in figures 3 and 4, which are from the narrow FOV and wide FOV cameras, respectively. These images are from the flow experiment of Feng et al. (Reference Feng, Goree and Liu2012a ,Reference Feng, Goree and Liu b ). The narrow FOV camera is pointed near the centre of the particle cloud, where the interparticle distance is fairly uniform; this uniform portion of the dust cloud is ideal to mimic condensed matter with uniform conditions. The wide FOV camera is used to assess the overall cloud shape, and to identify problems such as rapidly moving out-of-plane particles. As seen in figure 4 with the wide FOV camera, the interparticle distance around the edge of the suspension is larger than in the centre; this variation is due to the radial confinement by the naturally present horizontal electric fields in the plasma environment.

Figure 3.

Narrow FOV of a 2-D dusty plasma suspension captured from a high-speed camera, Phantom v5.2, on the top the vacuum chamber during the flow experiment of Feng et al. (Reference Feng, Goree and Liu2012a ,Reference Feng, Goree and Liu b ). This image is used for particle identification in our data analysis.

Figure 4.

Wide FOV of a 2-D dusty plasma suspension captured from an analogue camera mounted on the top of the vacuum chamber. The corresponding wide FOV movie is used to monitor the whole dusty plasma suspension.

2.2 Camera and illumination adjustment

There are several factors to consider in choosing the intensity of the illuminating laser sheet. It should be bright enough so that all particles can be easily identified. At the same time, the illumination laser should not be too powerful as to push particles noticeably (Liu et al. Reference Liu, Goree, Nosenko and Boufendi2003)Footnote ¹ . After one experimental run, we need to check whether there are any saturated pixels (Feng, Goree & Liu Reference Feng, Goree and Liu2007) in the captured images. It is desirable to avoid large numbers of saturated pixels, because they will spoil the accuracy of the particle position measurement, as described later. In an image with ${\sim}10^{3}$ particles each occupying ${\sim}20$ pixels, it is acceptable to have a few saturated pixels, but certainly not a few thousand saturated pixels. In the latter case we can correct the problem either by diminishing the illumination power, shortening the exposure time or reducing the aperture of the camera.

The camera exposure time and frame rate are chosen according to how fast the particles move. In principle, before pixels are saturated, a longer exposure time means accumulating more useful information. However, when particles move quickly, a longer exposure time would result in sausage-shaped images of particles, as in figure 5(a). These sausage-shaped images should be avoided by using a shorter exposure time. Unfortunately, the shorter exposure time can result in an image that is too faint, or it can require using an illuminating laser beam that is too bright so that particles are pushed visibly; this problem arises most often with short exposure times such as $10^{-3}$  s. Another approach to avoiding sausage-shaped images is to raster the illuminating laser beam so that it strikes a given dust particle only briefly, instead of illuminating it constantly as when using a cylindrical lens to disperse the light.

Figure 5.

Non-ideal particle images: (a) a fast moving particle; (b) low illumination and focused too much; (c) defocused too much exhibiting a ring shape; (d) aperture distortions.

If the particle images are too sharply focused so that they fill only one or two pixels as in figure 5(b), it will be impossible to measure the particle’s position with subpixel accuracy. The camera lens should be adjusted to achieve a sufficiently large spot for each particle. For measuring particle positions, it is best if the image of a particle occupies at least 15–25 pixels to obtain good subpixel accuracy. Diffraction is often sufficient to blur a particle’s image, but if more blurring is needed one can defocus the lens slightly. However, if we defocus the lens too much, as in figure 5(c), the image of one dust particle becomes ring shaped, and this should be avoided. From our experience, the ideal image of one particle should have a circular symmetric shape, occupying 4 to 5 pixels in both $x$ and $y$ directions, as figure 6 shows. If all the particle images have a similar asymmetric shape, as in figure 5(d), there is a problem in the imaging set-up, for example an irregularly shaped aperture for the camera filter.

Figure 6.

An example of the ideal image of one particle. It has a nearly circular symmetric shape, occupying 4 to 5 pixels in both $x$ and $y$ directions.

3.1 Dark-field images

The experimenter generally records a dark-field movie along with the main movie, both using the same top-view camera. The main movie shows the particles. The dark-field movie is recorded without particles, but otherwise under exactly the same conditions as the main movie, with both the plasma and illuminating laser turned on. The dark-field images show only camera noise and the background light (for example due to plasma glow and inadvertent laser light reflections in the chamber). By recording of the order of $10^{3}$ frames of the dark field, and then averaging them pixel-by-pixel, one generates a dark-field image that is useful for subtraction from each frame of the main movie. This subtraction process can be performed using commonly available software such as ImageJ (Rasband Reference Rasband2016). Using this subtractive process, the images in the main movie are improved to eliminate unwanted background features, so that they are ready for the next step, which is often particle identification.

3.2 Particle identification

Standard image analysis software is used to identify the contiguous pixels that represent an individual particle in the bit-map image of a video frame. Although the particle might be only a few microns in diameter where it is suspended in the plasma, it will appear in the image as a larger spot, due to the effects of diffraction and possibly defocusing of the camera lens, as described above. A particle is identified by the presence of multiple adjacent pixels that are brighter than a specified ‘threshold’ level. One of the advantages of experiments with a monolayer dust cloud is that the images of two particles seldom overlap, so that this particle identification is unambiguous. On the other hand, with 3-D dust clouds, for example in microgravity experiments, sometimes two particles at different distances from the camera can appear to overlap.

Determining the $x$ – $y$ coordinates of each particle in an image is the central step for particle-level diagnostics of dusty plasmas. There are several methods available to do this, but due to the large number of particles that must be identified (typically $10^{3}$ particles in each movie frame and $10^{4}$ frames, for a total of the order of $10^{7}$ measurements), the measurement must not only be automated, it must also be efficient. The moment method (Feng et al. Reference Feng, Goree and Liu2007; Ivanov & Melzer Reference Ivanov and Melzer2007) is widely used in dusty plasma experiments to obtain particle positions because it is efficient, and when done properly, it offers great accuracy with errors much smaller than one pixel. In the moment method (Feng et al. Reference Feng, Goree and Liu2007; Ivanov & Melzer Reference Ivanov and Melzer2007), the particle position is calculated using

where $\boldsymbol{X}_{k}$ is the coordinate of the pixel $k$ belonging to the image of this particle. Our recommended choice for the weighting factor $w_{k}$ is the intensity of the pixel minus the threshold level (Feng et al. Reference Feng, Goree and Liu2007).

There are two sources of errors in calculating particle positions: random and systematic. Random errors can arise from noise in all pixels in experimental images, while systematic errors can arise from the image analysis method, due to the sampling of the imaged light by a rectangular array of pixels. A common systematic error is ‘pixel locking’, which means the computed particle positions tend to have some favourable locations, such as the edge of a pixel or the centre of a pixel. Based on a study that used synthetic images with known true particle positions, Feng et al. (Reference Feng, Goree and Liu2007) developed a recommended practice that minimizes both types of errors simultaneously. Several parts of this recommended practice are mentioned already above: the weighting factor choice for the moment method calculation and the adjustment of the camera lens and laser illumination.

Another part of our recommended practice is the choice of an algorithm for selecting the contiguous pixels that are included in the moment calculation. There is no question that the brightest pixels should be included, but there is considerable freedom in designing an algorithm when selecting which pixels that should be included at the edge of a particle’s image. In Feng et al. (Reference Feng, Goree and Liu2007) several algorithms were tested, and it was found that the one used in ImageJ offered the greatest suppression of pixel-locking errors. This is fortunate, because ImageJ is freely available from the National Institutes of Health (Rasband Reference Rasband2016).

While using the moment method with ImageJ, one key point is making the best choice of the threshold of intensity value to distinguish particles from background noise. The choice of this threshold is tricky (Feng et al. Reference Feng, Goree and Liu2007): it should be low enough to keep enough particle image information, and it should not be too low to wrongly identify noise as particles. In our method, we first subtract the averaged dark-field image from each experimental image pixel-by-pixel as described above and we then use ImageJ to identify particles for the whole movie, requiring as an option in ImageJ that the particle fill multiple pixels. After performing this identification for one value of the threshold, we repeat for various values, and then plot the total number of identified particles versus the corresponding threshold value. In this plot, there is a near-flat portion in the middle range of the threshold value, for example as shown in figure 16 of Feng et al. (Reference Feng, Goree and Liu2007). The desired choice of the threshold should be the lower value of the threshold in this near-flat portion. Further explanation of these steps of selecting the threshold are provided in § VII of Feng et al. (Reference Feng, Goree and Liu2007) with our experimental images.

After identifying particle positions, we test whether pixel locking is a significant problem. This is done using the $x$ – $y$ coordinates of the particles, in units of pixels. For a measurement of a particle, for example $(x,y)=(12.345,67.890)$ , the integer portions of $x$ is subtracted, and the same for $y$ , yielding the subpixel coordinates, for example $(0.345,0.890)$ . Note that in these subpixel coordinates, the possible values of $x$ range from 0 to 1, and likewise for $y$ . We repeat for a large number of particle measurements, and then we superimpose the results onto a single graph, which we call a subpixel map. Example subpixel maps from an experiment are shown in figure 7; these examples were prepared from the same data, but with a higher threshold value for figure 7(a) than for figure 7(b). Pixel locking is identified qualitatively by noting whether particles are assigned favoured positions such as the pixel’s middle. We routinely examine subpixel maps like these as one of our considerations when finalizing our choice of the threshold value.

Figure 7.

Examples of subpixel maps for calculated particle positions using a higher threshold (a) and a lower threshold (b). In both (a) and (b), approximately 700 particles have been identified within one frame of an experimental image. When the threshold is not suitable as in (a), the subpixel maps reveals pixel locking as a tendency of calculated particle positions to be concentrated at favoured positions such as the centre and edges of pixels. When the threshold is correctly chosen, the subpixel coordinates are much more uniformly distributed within the pixel, as in (b).

3.3 Particle tracking

From identified particle positions, we can track particle motion from one frame to the next, and then compute their velocities. Here we explain the practice that we follow.

To track a particle from one frame to the next, we use simple software, written by ourselves, to search a circular region in the second frame that is centred on the particle’s computed position in the first frame. This search circle has a radius that is a fraction (for example 10 %) of the average interparticle spacing. If exactly one particle is found in the search circle we assume that it is the same one as in the first frame and the tracking is deemed successful, but if there are zero or ${>}\!1$ particles in the search circle then the tracking was unsuccessful. The radius of the search circle can be chosen in a compromise to reduce zero-particle outcomes by adjusting the radius upward until there are a few incidents in the movie of ${>}\!1$ particles in the search circle.

For many purposes, such as computing particle velocities, it is only necessary to track particles for two consecutive frames. We call this two-frame threading. For other purposes, such as calculating the mean-square displacement of particles over a long time to obtain a diffusion coefficient, it is necessary to track the particles for multiple frames; in this practice of making long threads, a particle will be tracked for only a finite number of frames until it exits from the image or becomes indistinguishable. In experiments with monolayer particle clouds, these problems can be minimized by using a sufficiently high camera frame rate and by analysing only a central region of interest that does not include the edge of the camera image. In experiments with 3-D dust clouds, long threads in 3-D are often impractical due to particles moving in the direction perpendicular to the laser sheet.

The algorithm we use to compute the particle velocity is a simple subtraction of positions in two consecutive frames. This yields a particle velocity

where the index $k$ represents time and $t_{frame}$ is the time interval between the two position measurements. When computing quantities that involve both position and velocity, it is necessary to interpolate the particle positions so that they are recorded simultaneously with the velocity. There are other schemes for determining particle position, but they can all introduce different kinds of errors. We prefer the simple subtractive scheme mentioned above because it is efficient and the errors it generates are understood.

Errors in particle velocities mainly come from two sources (Feng, Goree & Liu Reference Feng, Goree and Liu2011): particle position and particle acceleration. We will discuss these separately. A key experimental parameter, the camera frame rate, actually affects these two kinds of errors in opposite ways, so that minimizing the total error in the particle velocity requires a compromise choice of the frame rate.

Particle position errors, as a source of error in velocity measurement, cause the most severe problems when the particle displacement is small and the frame rate is high, so that the displacement of a particle between two frames is less than the uncertainty of particle positions within a given frame. We can mention two examples of velocity errors due to particle position errors: the low velocity range in a velocity distribution functions can have unphysical oscillations as in figure 4(c) of Nosenko, Goree & Piel (Reference Nosenko, Goree and Piel2006), and calculated velocity autocorrelation function can have a weird shape, as in Feng et al. (Reference Feng, Goree and Liu2011). To minimize this problem, based on our simulation results, we have introduced a method of interlacing pairs of frames to calculate positions and velocities (Feng et al. Reference Feng, Goree and Liu2011): positions are calculated from the average of several consecutive frames, while velocities are calculated from frames with a larger time interval. Superimposing raw data of consecutive frames (Knapek et al. Reference Knapek, Samsonov, Zhdanov, Konopka and Morfill2007) is another method to reduce position and velocity calculation errors. It is interesting to note that a higher frame rate is not always desired for particle tracking in dusty plasma experiments.

Acceleration errors in the velocity determination arise from our definition of velocity in (3.2). This expression assumes that the motion between frames is always a straight line (Feng et al. Reference Feng, Goree and Liu2011). This assumption works well for a high frame rate since the time scale of dust particle’s motion is much longer than the time interval $t_{frame}$ between consecutive frames. However, some information is definitely lost by sampling the particle’s image at fixed time intervals. Causes of particle acceleration in a dusty plasma can include interparticle collisions due to the fluctuating electric fields as individual dust particles approach one another, and cyclotron motion if a strong magnetic field is used. In the latter case, the trajectory of a dust particle is mostly curved, so that our assumption of a straight line between frames may cause substantial systematic errors. In fluid dynamics, spline fit methods (Lüthi, Tsinober & Kinzelbach Reference Lüthi, Tsinober and Kinzelbach2005) have been introduced as an alternative to (3.2) to reduce the systematic error due to this straight line assumption between frames, although a spline fit is much more computationally expensive. Until now, in previous 2-D dusty plasma experiments, we have not noticed substantial systematic errors due to our velocity definition of this straight line assumption between frames.

Note that the time interval when computing velocity in (3.2) need not be the same as the time between frames in the recorded video. We can record data at a higher frame rate, such as 200 to 500 frames  $\text{s}^{-1}$ , while in later data analysis we can skip frames (Knapek et al. Reference Knapek, Samsonov, Zhdanov, Konopka and Morfill2007; Feng et al. Reference Feng, Goree and Liu2011) when using (3.2). Skipping frames in the analysis can be attractive to reduce the velocity error due to errors in particle position measurement, which is likely to be quite significant if the time interval $t_{frame}$ is so short that a particle’s displacement is less than 0.3 pixel (Feng et al. Reference Feng, Goree and Liu2007).

During data analysis, we choose the time interval between frames, $t_{frame}$ , according to two schemes that were presented in Feng et al. (Reference Feng, Goree and Liu2011). We summarize one of those schemes here. For velocity measurement, one chooses a value of $t_{frame}$ between two limits. The upper limit for $t_{frame}$ is significantly smaller than the time scale for the velocity to change significantly in the physical system, for example due to collisions. The lower limit for $t_{frame}$ is the accurate pixel resolution size (typically ${\sim}0.3$ pixel) divided by the typical speed, such as the thermal speed. In Feng et al. (Reference Feng, Goree and Liu2011), we performed systematic tests of various velocity-related physical quantities to obtain these rules for upper and lower limits.

Finally, we present an example, to illustrate the choice of the time interval for velocity measurement, using the physical quantities in the 2-D strongly coupled dusty plasma experiment of Feng, Goree & Liu (Reference Feng, Goree and Liu2010). In that experiment: the pixel resolution within an image corresponded to $30~{\rm\mu}\text{m}$ in the dust cloud, the thermal velocity was about $v=1.3~\text{mm}~\text{s}^{-1}$ and a characteristic time scale of a dust particle’s collisional motion was ${\it\omega}_{pd}^{-1}=1/30~\text{s}$ . According to the rules above, the time interval between frames for data analysis should have an upper limit of $t_{frame}<1/30~\text{s}$ and a lower limit of $t_{frame}>(0.3\times 30~{\rm\mu}\text{m})/1.3~\text{mm}~\text{s}^{-1}=0.0069~\text{s}$ . This range of time intervals between frames corresponds to an effective frame rate in the analysis of approximately 30–140 Hz for the experiment of Feng et al. (Reference Feng, Goree and Liu2010). In that experiment we operated the camera at a frame rate of 250 Hz, and in computing the velocity we skipped 1 frame when using (3.2) so that the effective frame rate was $1/t_{frame}=125~\text{Hz}$ .