Abstract
The trend to conduct volumetric particle tracking velocimetry (PTV) experiments with ever-increasing volumes, at a given particle density, poses increasing challenges on the design of such experiments in terms of the power of the laser source and the image analysis. These challenges, on one hand, require a reliable model to estimate the current signal from a pixel on a complementary metal-oxide semiconductor (CMOS) detector due to a Mie scattering particle. On the other hand, they require also a model for estimating the limiting factors upon the image resolution, where a large amount of particles within a three-dimensional (3D) volume are mapped into a two-dimensional (2D) image. Herein, we present a model that provides an analytical expression to estimate the current signal from a pixel of a CMOS detector due to a Mie scattering particle within an arbitrary large volume in a volumetric PTV experiments. We begin with a model for planar experiments and extend it into volumetric measurements. Our model considers the effect of the depth of field, particle density, Mie scattering signal and total Mie scattering loss, laser pulse energy, and other relevant optical parameters. Later, we investigate the consequence of the Rayleigh criterion upon the spatial resolution when it is applied to Mie particles within a volume of interest (VOI). Finally, we demonstrate how we applied our model to estimate the current signal and the limit upon the spatial resolution in three experiments carried out in our lab.
1 Introduction
The continual increase in computational power and the development of fast algorithms for particle tracking, such as shake the box (STB) [1], had been facilitating tracking of an increasing larger number of particles in tomographic particle imaging velocimetry (PIV) and PTV. Table 1 lists examples from the literature of several three dimensional-particle tracking velocimetry (3D-PTV) experiments using helium filled soap bubbles (HFSB), air-filled soap bubbles (AFSB), or Di-Ethyl-Hexyl-Sebacat (DEHS) particles. The values of the VOI, particle number N, particle diameter d, laser pulse energy Ep, and laser repetition rate frep are specified for each experiment. The last three rows in Table 1 are examples of 3D-PTV experiments (conducted in our lab) with small diameter DEHS particles, or with diameter AFSB particles. These experiments demonstrate that a high pulse-energy laser is required in order to obtain a signal in a relatively small VOI.
When one tries to work in an increasing VOI and with small tracer particles, one is confronted with the necessity to carefully optimize the light budget. A methodical study of 3D-PTV experiments shows that the current signal from a pixel of a CMOS camera depends on several factors: The initial laser pulse energy, the Fresnel losses due to guiding and shaping of the laser beam, scattering and absorption losses of the laser beam by the tracer particles and by the carrier fluid, the diameter of the tracer particle, the polar angle of the Mie scattering, the depth of field of the volume of interest, scattering and absorption losses of the Mie signal by the tracer particles and by the fluid; and the losses of the camera (transmission loss of the camera lens and the responsivity loss of the CMOS detector).
At the present, it is a common practice to study examples of existing experiments from the literature (c.f. Chapter 18 in Ref. [7]) and then conduct tests with existing equipment in order to estimate experimentally the attainable current signal and the spatial resolution for a given tracer and pulsed laser system in a given VOI. This type of experimental approach is inefficient in the design stage of 3D-PTV experiments. Since STB of volumetric PTV measurement are maturing (computationally) toward feasibility of tracking increasing amounts of particles in ever-increasing volumes, we recognize that there is a need to develop a sound and practical design method for volumetric experiments. In particular, it became evident to us that the dependence of the CMOS current signal and the spatial resolution upon the depth of field is not well understood. Thus, the main purpose of the present work is to develop an analytical (yet practical) model that can predict reliably the current signal from a CMOS pixel due to the imaged particles and the limit of the spatial resolution in 3D-PTV measurements.
2 Analytical Model for the Signal Level in Planar and Volumetric Particle Tracking Velocimetry
A simplified configuration of a 3D-PTV experiment is depicted in Fig. 1. A laser with pulse energy, Ep, irradiates a Mie particle, the particle scatters light into all directions according to Mie theory. Part of that scattered light passes through the camera lens and generates an image on the pixels of the CMOS detector. The image has an optical power Pi. If the image fills the area of one pixel, the current signal iS equals to the incident optical power times the responsivity of the CMOS detector at the wavelength λ of the laser, (the responsivity has the units of Ampere/Watts).
where the variables ra and rb are the radial components along the semiminor and semimajor axes of the elliptic cross section of the laser beam, respectively.
Now, we are ready to calculate the current signal by tracking the path of the laser pulse from its output, until a signal is generated in a pixel of the CMOS detector due to scattering by a Mie particle. Since planar PTV is a special case of volumetric PTV with a minute depth of field, it is simpler to consider first the signal level in a planar PTV experiment and then extend the model to the case of a 3D-PTV experiment.
2.1 The Current Signal in Planar Particle Tracking Velocimetry.
The following six factors should typically be considered:
First, the Gaussian laser beam is shaped by lenses into an elliptical Gaussian beam and it is guided by reflecting mirrors into the volume of interest V, as depicted in Fig. 1. If the volume of interest is located within a glass chamber that is filled with a liquid, the beam will suffer Fresnel losses also from the glass chamber (or any other dielectric media). All the optical surfaces (of lenses, mirrors, and the chamber) will have an accumulative optical power-loss, which we designate by L (a factor that is smaller than 1). These losses will reduce the initial laser intensity, I, to .
Second, when the laser beam propagates through a fluid (air or water, for example), it will suffer both scattering and absorption losses from the tracer particles and from the carrier fluid. The losses due to the tracer particles are characterized by scattering and absorption loss coefficients and , respectively. The losses due to the fluid are characterized by scattering and absorption coefficients and , respectively. For example, Mie particles with particle density n in air have a scattering loss coefficient , where σs is the total Mie scattering cross section of one particle (It accounts for scattering into a full solid angle). In the air (the fluid), the scattering loss coefficient by molecules (known as Rayleigh scattering) and the absorption loss coefficient (due to molecular absorption) are negligible in the range of the visible spectrum. On the other hand, in water, Mie particles cause for scattering and absorption losses; these losses are characterized by and . However, the water itself (the fluid) will show significant scattering losses due to density fluctuations and absorption losses, due to molecular absorption (see Chapter 43 in Ref. [9]). Let us assume that the laser beam will pass a distance zl through the seeded fluid until it reaches a particle that is located within the volume of interest. The laser intensity I after a path zl will be , where accounts separately for the absorption and scattering losses by the tracer particles and by the fluid.
where I is the intensity of the incoming laser and σ is the scattering cross section of a particle into a differential solid angle . The integration over will give the total Mie scattering cross section σs. We note that due to the symmetry of scattering from a sphere, the scattering cross section depends only on the polar angle θ. The polar Mie scattering can be calculated by using openly available resources, e.g., (see Footnote2).
where Df is the diameter of the lens. The intensity of Mie scattering into the camera lens will be approximately .
Fourth, the Mie scattering signal from a particle that is located at a distance zo from the camera lens will suffer scattering and absorption losses similar to those suffered by the incoming laser pulse (as explained in the second factor). Accordingly, the intensity of the signal will fall as .
Fifth, the camera lens usually consists of a combination of several types of lenses (to compensate for lens aberrations). If these lenses are not specifically coated by Anti-Reflection (AR) coatings, a considerable Fresnel loss will ensue. Let us designate the optical transmission of the camera lens at the laser wavelength, λ, by .
Sixth (and final), one needs to consider the ratio of the area of one pixel to the area of the image of one particle. In order to avoid peak locking [7], it is advised that the particle image will have an area of at least four pixels. Therefore, the amount of optical power incident onto one pixel will be about of the total power of the image. We designate this ratio by fill factor ratio (FFR).
Observe that particles that are located at the center of the elliptical laser beam (where ) will have the maximal laser intensity (and therefore the maximum signal). While particles that are located where will have the lowest laser intensity and therefore the lowest current signal.
2.2 The Current Signal in Volumetric Particle Tracking Velocimetry.
In 3D-PTV, we adjust the diameter Da of the camera aperture to obtain a depth of field that equals to the thickness of our rectangular cuboid VOI. By applying two minor and judicious changes to Eq. (5), we can obtain an expression that estimates the current signal, iS, for volumetric PTV with a large depth of field.
First, we need to identify the particles within the VOI that would have the faintest signal. We note that particles at the far face (the plane at ) of the VOI will have a smaller solid angle, Ωpar, compared with particles located at the object plane or at the plane (see Fig. 1). More specifically, considering the configuration of the camera and of the laser as is depicted in Fig. 1, we note that particles at the vertices of the far face of the VOI will have the smallest solid angle, Ωpar. The two vertices at the right hand with respect to the view seen by the camera will experience a laser beam with the lowest power due to scattering and absorption losses over the longest path. Furthermore, the polar Mie scattering by particles in these two vertices will be the weakest (generally, Mie scattering is maximal in forward scattering and falls down with increasing polar angle). Finally, these particles have the greatest distance to the camera lens and thus their Mie-signal will suffer the greatest scattering and absorption losses as it propagates toward the camera lens.
For simplicity, we will consider the signal from particles at the distance (the explicit expression for is found from (A2) and (A8) in the Appendix). Then, we will replace the lens diameter, Df, with the aperture diameter, Da, in Eq. (5). By doing so, we account for the signal from particles with almost the faintest Mie scattering signal. If we need to detect a signal from all particles within the rectangular cuboid VOI, we need to consider the signal from particles that are located at the vertices of the far face of the volume of interest, to the right with respect to the viewing camera.
Second, when the diameter of the camera aperture is decreased in order to match the required depth of field, Mie particles that are not in the object plane will appear blurred on the image plane. An ideal point-like light source at the object plane will appear in the image plane as a disk with diameter [11]. An ideal point-like light source at the far (or near) face of the VOI (relative to the camera view) will appear larger and blurred with diameter CoC (the Circle of Confusion, see explanation in the Appendix).
3 The Spatial Resolution in Planar and Volumetric Particle Tracking Velocimetry
To achieve the highest spatial resolution in a planar PTV experiment (or 3D-PTV), we would like to work with the smallest possible trace particles at the highest density. Therefore, let us consider two Mie scattering particles. What would be the image of such two adjacent point-like, light-source particles, which are located at the object plane?
Considering the wavelength of the illuminating laser and the distance of the camera lens from the particles, the scattered light that reaches the lens of the camera is as good as a coherent plane wave. Since our particles are point-like particles, we cannot use geometric optics to predict their image. The “image” of these two adjacent particles is a diffraction pattern.
According to diffraction theory, when a circular aperture (i.e., the lens or the camera aperture) is illuminated by a plane wave, one will observe a diffraction pattern that is called an Airy pattern (Ref. [11]). A single Airy pattern is made of a central disk (the Airy disk) that is surrounded by a series of dark and lighted concentric rings (Airy rings). When two adjacent point-like, light sources, are imaged by a lens with diameter Df, we will get two adjacent Airy patterns on the image plane.
The Rayleigh criterion sets a physical limit (due to diffraction) upon the optical resolution between two particles. is the minimal distance between two particles in the object plain that could be resolved in the image plane as two separate light points.
Apart from the physical limit set by the Rayleigh criterion, we identify four technical factors that may limit the spatial resolution :
Particles in the Front Overshadow Particles That are in the Rear: If the depth of field is equal to, or smaller than, the average distance between the particles, Dav, we will avoid overshadowing of particles. In planar PTV experiments, we tend to shape the laser beam to be thin (a laser sheet), hence we can mitigate overshadowing by reducing the thickness of the laser sheet. In volumetric PTV, the depth of field may take any value. Overshadowing will appear when . In that case, it is possible to mitigate overshadowing by using several cameras to view simultaneously a VOI from different angles. This allows for the algorithm of the image analysis [12] to identify particles that are overshadowed in one viewing camera but are distinguished when viewed by the other cameras.
Illumination by a Gaussian Elliptic Laser Beam: When a Gaussian elliptic laser beam is used to illuminate a rectangular cuboid VOI with height , thickness , and width W that equals to the Rayleigh range of a Gaussian beam, i.e., , the area of the field of view (FoV) with a proper laser intensity will be . Thus, we see that if we use a Gaussian elliptic laser beam, the area of the field of view (with proper light intensity) depends quadratically on its waist ωa. Typically, in order to avoid overshadowing, we would like that . Meaning that the more we attempt to improve the spatial resolution of our measurement, by decreasing the waist ωa to mitigate overshadowing, the smaller the area with proper light intensity we will get (within the field of view).
Signal Loss Due to Scattering and Absorption Losses: When a particle within the VOI generates a Mie signal, the signal needs to reach the camera lens at a distance zo (or in 3D-PTV). However, this signal will be rescattered and absorbed by other Mie particles and by the fluid as it propagates toward the camera lens. It was argued previously that the Mie scattering from a particle in the VOI is reduced according to , where n is the density of the Mie particles and σs is the total Mie scattering cross section (a similar factor will appear if the Mie particle has a significant absorption). To achieve the best spatial resolution, we will tend to increase the tracer density, n, toward the density that is limited by the Rayleigh spatial resolution. However, as we increase the tracer density n, the signal intensity will drop exponentially. Thus, we could face a situation where the density of the tracer will limit the optimal spatial resolution.
3.1 The Highest Spatial Resolution in Planar Particle Tracking Velocimetry.
where the subscript, R, reminds us that the limit is due to the Rayleigh Criterion. This is the best possible spatial resolution, limited by physical optics.
In order to reach the Rayleigh spatial resolution with a CMOS detector that has Npix pixels and an average particle distance as given by (10), we will need to reduce the dimensions of the VOI until the amount of particles left within the VOI is below . In addition, the image of the VOI needs to fill the whole area of the CMOS detector.
3.2 The Highest Spatial Resolution in Three-Dimensional-Particle Tracking Velocimetry.
This is the best possible spatial resolution for particles in a VOI. It is limited by physical optics and by the circle of confusion.
When we have a CMOS detector with Npix pixels, the area (in pixel number) that should be reserved to the image of each particle needs to account for the CoC: . The spatial resolution that is limited by the number of pixels of the CMOS detector will be . In addition, the image of the VOI needs to fill the whole area of the CMOS detector.
In order to reach the Rayleigh spatial resolution of Eq. (13) with a CMOS detector that has Npix pixels, we would need to reduce the dimensions of the VOI until the amount of particles left within the VOI matches the limit value due to the CMOS detector.
4 Experimental Examples
In this section, the practical design steps for a planar PTV, a small and a large volumetric PTV experiments are analyzed. We use the models and the relevant equations that were deduced in Secs. 2 and 3 along with experimental parameters which are specified in Tables 2–4.
Max. velocity | |
Temporal resolution | |
Spatial resolution | |
Laser: | Nanio from InnoLas |
Wavelength | |
Laser waist | |
Repetition rate | |
Pulse duration | |
Pulse energy | |
Particles: | DEHS |
Average diameter | |
Refractive index | 1.45 at |
CMOS camera(s): | 1 × Photron Nova S9 |
Image sensor dimensions | |
Pixels | 1024 × 1024 |
Pixel pitch | |
Camera frame rate | |
Responsivity | |
Optics | |
Beam guiding loss | L = 0.2 |
Signal transmission of the lens | |
Objective lens diameter |
Max. velocity | |
Temporal resolution | |
Spatial resolution | |
Laser: | Nanio from InnoLas |
Wavelength | |
Laser waist | |
Repetition rate | |
Pulse duration | |
Pulse energy | |
Particles: | DEHS |
Average diameter | |
Refractive index | 1.45 at |
CMOS camera(s): | 1 × Photron Nova S9 |
Image sensor dimensions | |
Pixels | 1024 × 1024 |
Pixel pitch | |
Camera frame rate | |
Responsivity | |
Optics | |
Beam guiding loss | L = 0.2 |
Signal transmission of the lens | |
Objective lens diameter |
Note: ? is to be calculated.
Max. velocity | |
Temporal resolution | |
Spatial resolution | |
Laser: | Blizz InnoLas |
Wavelength | |
Laser waist | |
Repetition rate | |
Pulse duration | |
Pulse energy | |
Particles: | DEHS |
Average diameter | |
Refractive index | 1.45 at |
CMOS camera(s): | 4 × Phantom V2640 |
Image sensor dimensions | |
Pixels | 2048 × 1952 |
Pixel pitch | |
Camera framerate | |
Responsivity | |
Optics | |
Beam guiding loss | L = 0.04 |
Signal transmission | |
Objective lens diameter |
Max. velocity | |
Temporal resolution | |
Spatial resolution | |
Laser: | Blizz InnoLas |
Wavelength | |
Laser waist | |
Repetition rate | |
Pulse duration | |
Pulse energy | |
Particles: | DEHS |
Average diameter | |
Refractive index | 1.45 at |
CMOS camera(s): | 4 × Phantom V2640 |
Image sensor dimensions | |
Pixels | 2048 × 1952 |
Pixel pitch | |
Camera framerate | |
Responsivity | |
Optics | |
Beam guiding loss | L = 0.04 |
Signal transmission | |
Objective lens diameter |
Note: ? is to be calculated.
Max. velocity | |
Temporal resolution | |
Spatial resolution | |
Laser: | Green elliptic |
Wavelength | |
Laser waist | |
Repetition rate | |
Pulse duration | |
Pulse energy | |
Particles: | Air-filled soap bubbles |
Average diameter | |
Refractive index | 1.33 at (soap membrane) |
CMOS camera(s): | 4 × Phantom V2640 |
Image sensor dimensions | |
Pixels | 2048 × 1952 |
Pixel pitch | |
Camera framerate | |
Responsivity | |
Optics | |
Beam guiding loss | L = 0.04 |
Signal transmission | |
Objective lens diameter |
Max. velocity | |
Temporal resolution | |
Spatial resolution | |
Laser: | Green elliptic |
Wavelength | |
Laser waist | |
Repetition rate | |
Pulse duration | |
Pulse energy | |
Particles: | Air-filled soap bubbles |
Average diameter | |
Refractive index | 1.33 at (soap membrane) |
CMOS camera(s): | 4 × Phantom V2640 |
Image sensor dimensions | |
Pixels | 2048 × 1952 |
Pixel pitch | |
Camera framerate | |
Responsivity | |
Optics | |
Beam guiding loss | L = 0.04 |
Signal transmission | |
Objective lens diameter |
Note: ? is to be calculated.
4.1 Planar Particle Imaging Velocimetry/Particle Tracking Velocimetry: .
This experiment concerns with measuring variations in the concentration of Mie-scattering particles [13]. We will demonstrate how the current signal and the spatial resolution are estimated by using the experimental parameters listed in Table 2.
Minimal Working Distance, zo: The aspect ratio of the FoV () corresponds to that of the CMOS detector (). In order to use the full area of the CMOS detector, we would like to work with a demagnification (Width of CMOS)/(Width of FoV) . We find the minimal working distance with a negligible lens aberration by requiring, (due to the paraxial ray approximation [11]). The height of the FoV is , hence . From the thin lens Eq. (A1) and the magnification we find . Thus, when the focal length , the working distance and our image at will have negligible lens aberrations. This estimation is for a single thin lens. If we use an objective lens that consists of several lenses to reduce spherical aberrations, the working distance could be made shorter and without lens aberrations. Adjusting the focal lens f and the working distance zo is a practical approach in the design of PTV experiments (lenses are cheaper than CMOS detectors, and adjusting a working distance is usually a practical option).
According to geometric optics, a diameter particle will have an image with a diameter of . This image diameter is smaller than the predicted diameter of a point-like light source using diffraction theory. Therefore, we cannot use geometric optic to predict the image diameter of the diameter particle with this magnification. Instead, we can calculate the diffraction image from a diameter particle that consists of a continuous distribution of point-like light sources. The diffraction image of one point-like light source would be an airy pattern. Its airy disk would have the diameter . By viewing the diameter particle as if it consists of continuous point-like light sources, we can approximate (instead of a rigorous complex integration) that the diameter of the image would be the sum of all Airy disks from each point. The total diameter would be about . The area of the image of one particle is smaller than the area of one pixel. With a proper particle density, the images of several particles could fit into one pixel with an area of . In such a case, the signal in one pixel would represent the number of particles in a voxel with a volume of . Hence, we will image the density of the particles (which is indeed the objective of this PIV experiment), rather than imaging the individual particle as in a PTV experiment.
Laser Repetition Rate, frep: For a maximum velocity within the object plane, the STB algorithm requires particle image displacement of about pixels. The time between pulses is given by . The laser repetition rate is determined by the required temporal resolution, .
Laser Pulse Width, τ: To avoid streaking, . The Nanio laser has a pulse duration of . Hence, this laser is suitable.
Let us consider a rarefied particle density so that only one particle is imaged into one pixel. For this to happen, there should be 1 particle in one voxel with dimensions of . This corresponds to a particle density of . In this case, the loss due to total Mie scattering and absorption by DEHS particles is negligible. The electron count from a pixel for is found by dividing iS with the electron charge Coulomb. For a pulse energy, we obtain 19 electron counts from an image of one particle. The Photron Nova S9 camera has an electron noise-count of about 20 per pixel. Hence, we will have SNR = 1 at laser pulse energy. If we will increase the particle density to above 10 particles per voxel, we will see a signal that corresponds to the density of the particles and our SNR will be workable.
The laser intensity at falls by about 87%. Since the Mie signal is proportional to the power density of the illumination source, the signal will fall exponentially, outward, from the center of the beam.
Spatial Resolutionand Particle Density n: According to the Rayleigh criterion, the minimum discernible distance between adjacent particles in the object plane is given by Eq. (8): . This corresponds to a distance of in the image plane. This imaged distance is smaller than the pixel size of the CMOS detector. Hence, with the present CMOS resolution (of ) and the given demagnification, we would not be able to reach the Rayleigh resolution. We could only distinguish between two particles with a distance larger than . The limit on the optical resolution is due to the pixel size (called the pixel resolution) and the demagnification. Also, we note that the thickness of the laser sheet is . If we work with an average distance between the particles of , we would experience some overshadowing (the laser sheet is 3.3 times larger than the average distance Dav between the particles). In order to mitigate overshadowing, we could reduce the laser waist to . However, then our field of view (with useful laser light) will be reduced due to the Rayleigh Range of a Gaussian laser beam. The new width will be . We see that although we speak of planar PTV, a more accurate understanding of planar PTV can be achieved in terms of volumetric PTV.
4.2 Volumetric Particle Tracking Velocimetry: .
This experiment is designed to measure the full instantaneous velocity gradient tensor at sub-Kolmogorov scales in a turbulent round jet. The measurements are carried out at a high spatial resolution in order to obtain the full dissipation rate tensor at Reynolds numbers ranging between . More details on the laboratory setup can be found in [6] and [14]. The parameters of this experiment are listed in Table 3.
Minimal Working Distance, zo: To avoid lens aberrations, the paraxial ray condition give a , thus . Due to experimental constraints, a minimum working distance of is required, which fulfills the aberration constraint. The volume of interest has a FoV of , which matches the CMOS image sensor area. Hence, the required magnification is M = 1. The thin lens equation with the required magnification, gives that for , the focal length of the lens should be and , which is an unusually large number for a CMOS camera (the sensor will need to be 1 meter away from the lens). The solution is to use a Galilean telescope. The Nikkor Telelphoto lens is basically an elaborated Galilean telescope (Ref. [11]). It has a Front Effective Focal Length . When it is continued with a Nikkor teleconverter 2 M lens, its . By turning the focusing cylinder knob, we tune the Back Effective Focal Lens BEFL so that the field of view is in focus at a distance of 240 mm from the edge of the Nikkor teleconverter 2 M lens.
Laser Repetition Rate, frep: For a maximum velocity of , the required time between pulses is given by . The corresponding laser repetition rate is thus
Laser Pulse Width, τ: To avoid streaking . The Blizz laser from InnoLas has a pulse duration of , which is suitable.
Let us assume that the density of the tracer particles is low and thus the total Mie scattering and the absorption by the tracer particles is negligible. We calculate the electron-count from a pixel that images particles, which are located at the center of the Gaussian beam where r = 0 (the radial distance from the center of the beam outward). We divide iS by the electron charge Coulomb to obtain 187 electron counts from an image of one particle per pulse energy. According to the manual of the V2640 camera, the read out noise per pixel in the standard mode is 7.2 electrons. Hence, the SNR = 26 for particles located at the center of the laser beam. Since the intensity of the cross section of the laser beam drops exponentially with r, the intensity of the laser at the waists will be about 8 times smaller. This means that the SNR for particles located at the faces of the VOI will be 8 times smaller.
Optical Resolution and Density: For and a depth of field , one can use Eq. (A9) to estimate that the circle of confusion, . and from Eq. (A8) we find that and . Using Eq. (8), we find that the Rayleigh distance: . The camera sensor has a pixel pitch of . The Rayleigh distance corresponds to a distance of two pixels.
As was explained in Sec. 3, the images of point-like light sources at and will have a diameter that is approximately the diameter of the airy disk plus the circle of confusion. The diameter of the Airy disk is and . Hence, the image diameter of a particle at and a particle at will be . The Rayleigh distance for particles at plane is . Hence, the best spatial resolution would be .
4.3 Volumetric Particle Tracking Velocimetry: .
In this experimental design, the idea is to globally measure the turbulent flow from a jet at Reynolds numbers ranging between ∼10.000 and 100.000. The diameter of the nozzle of the jet is 10 times smaller than the nozzle in the previous example. Therefore, one can cover a large portion of the flow-field with proper illumination. If we use a laser with pulse-energy as in the first example (a cross section of ), we cannot expect to have a detectable Mie-signal when the cross section of the VOI is . In fact, we would need a laser pulse that is 100 times more powerful in order to keep on the same power density as in the first example. However, if we will work with a shorter working distance zo and use air-filled soap bubbles (their Mie scattering signal is two orders of magnitude larger than the Mie scattering of diameter particles), the Mie signal might be detectable. We need to investigate if we can detect particles from other parts of the volume, in particular particles at the far face of the volume of interest.
Minimal Working Distance, zo: The spherical aberration of a simple lens are negligible when . Hence, the working distance zo should be larger than . As can be seen from Eq. (6), the current signal depends inversely on the second power of . To enhance the signal, it seems sensible to work with values of zo that are as small as possible. Therefore, we use a Nikkor lens (from Nikon Corporation) which has aberration-free imaging down to a distance of (this lens consists of 8 lenses that together decrease lens aberrations and facilitate an angle of view of ).
The dimensions of our CMOS sensor are and the dimensions of the FoV are . Thus, for optimal imaging of the field of view, the magnification should be (Width of CMOS)/(Width of the FoV). In order to maximize the signal, we consider first a working distance with M = 0.1. By using Eq. (A1) and the magnification, we find that for , the required working distance is . Since , for the aperture diameter is .
From Fig. 2, one can study the variations of and as a function of CoC for and . For a depth of field , the circle of confusion is (where and ).
Laser Repetition Rate, frep: For a maximum velocity of , the time between pulses is found from (note that ). Since , the laser pulse repetition rate should be above
Laser Pulse Width, τ: Since the ratio of the depth of field to the working distance is not negligible, one should use to estimate the pulse width . Our laser has and is therefore suitable.
Laser Pulse Energy, Ep: We assume an elliptic Gaussian laser beam with and that approximately fills the VOI. We recall that since the depth of field is not negligible, we need to calculate the signal level for a particle at the far edge of the depth of field (at ). We used Philip Laven's free-ware2 to calculate the differential Mie scattering at for and . The solid angle of this particle with the aperture is calculated from Eq. (3) to be . The image diameter of this particle on the image plane is , where the diameter of the particle due to diffraction () was used instead of dM+. Thus, the area of a blurred image of a particle will cover about 25 pixels(!) and the . This example emphasizes a situation where the FFR becomes a dominant limiting factor. For particles in the object plane, the FFR is 0.25. For particles at the far face of the volume of interest, the FFR is smaller by a factor of 6.25.
We assume that the losses due to total Mie scattering or absorption are negligibly small.
We divide iS by the electron charge Coulomb and obtain 0.014 electron counts in one pixel per one mJ of pulse energy. The read out noise in the Phantom camera is 7.2 electrons (in the standard mode). Hence, we would not be able to detect a signal with pulse energy. Our estimate is for a laser intensity at the center of the beam (at ). The laser intensity at the waist will be at least eight times smaller than its intensity at its center. This means that our estimation for the current signal from particles at the far face of the depth of field (at ) will have a signal 8 times smaller (the Mie signal is proportional to the intensity of the laser).
The above estimation shows that our current signal will be very poor. In order to improve the electron count, we should consider the following:
Change the angle of the Mie scattering toward forward scattering. For example: Mie scattering at is 18 times larger than Mie scattering at right angle,
Increase the pulse energy by an order of magnitude,
Decrease the volume of interest, in particular, by shortening of the depth of field so that the CoC decreases and the FFR increases.
Increase the particle diameter. A particle with diameter of will have 10 times more Mie signal.
Optical Resolution and Density: When and the camera aperture diameter is . Figure 2 shows that for a working distance and a depth of field , the far face of the VOI is at and the . The Rayleigh distance at is . The spatial resolution is the third power of this value. Due to the large depth of field, a point-like particle at has a circle of confusion , which compromises the current signal, the optical resolution and as a consequence, the spatial resolution .
5 Conclusions
This report presents an analytical model that estimates the current signal in a pixel of a CMOS camera in planar PTV and 3D-PTV experiments. Additionally we found expressions for estimating the spatial resolution in 2D and 3D-PTV experiments.
The key idea behind our approach is to identify the particles within the VOI which will have the faintest current signal and the worst optical resolution. The location of these particles depends on the configuration of the experiment. In an experimental configuration, as is shown in Fig. 1, the particles at the vertices of the far face will have the faintest signal and the worst optical resolution.
In typical 3D-PTV experiments, one uses 2–4 identical cameras to view the volume of interest with 2–4 corresponding viewing angles. To employ our model, one needs to use for each camera view: the appropriate polar Mie scattering, the specific scattering and absorption losses of the laser and the specific scattering and absorption losses of the signal.
Our analysis assumes the paraxial approximation of geometric optics, i.e., there are no spherical aberration to our images. However, when one is working with four cameras that look at the same VOI, it could happen that the field of view would have to be wide and then the effect of spherical aberration: astigmatism, coma and distortion (see Ref. [11]) will creep into the images. As a result, the current signal will fall and the image resolution will degrade. It is thus worth investing in high-quality lenses that compensate for spherical aberration at large viewing angles.
where, is the refractive index of water, θi is the angle of incidence of the laser beam, and θt is the angle of refraction, found from Snell law. The calculation of the current signal due to the reflection of a laser light from the surface of HFSB in a volumetric PTV experiment would result in a similar expression to Eq. (6). The same six optical consideration that were put forth in Sec. 2 would have to appear. However, the Mie scattering factor should be replaced by the Fresnel reflection, while considering the spherical surface of the HFSB and its effect on the angle of incident upon the Fresnel reflection. Additionally, the scattering cross section and absorption cross section by large HFSB need to be worked out (by using Fresnel reflection and transmission equations).
The presented model considers a Gaussian elliptic laser source as the light source, which is the most prevalent illumination source at present when using Mie particles. Light emitting diodes (LEDs) have been proven to be able to generate high power densities suitable for illuminating HFSB in large volumes [4]. LEDs have a wide spectral distribution and more complex beam profiles, such as, Lambertian, top-hat profile or a complex beam profile emanating from an array of LEDs. Therefore, a model for the current signal and the image resolution of HFSB within a VOI, which are illuminated by LEDs, could be of interest in future work.
Acknowledgment
H. A. and C. M. V. acknowledge financial support from the Poul Due Jensen Foundation: financial support from the Poul Due Jensen Foundation (Grundfos Foundation) for this research is gratefully acknowledged.
C. M. V., S. L. R. and Y. Z. acknowledge financial support from the European Research Council: this project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program.
The authors would like to thank Benny Edelsten and Jakob Skov Nielsen for proofreading the paper.
Funding Data
Poul Due Jensen Foundation (Grundfos Foundation) (No. 2018-03).
European Research Council (ERC) under the European Unions Horizon 2020 Research and Innovation Program (Grant No. 803419; Funder ID: 10.13039/501100000781).
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.
Nomenclature
- d =
particle diameter (μm)
- Da =
aperture diameter (mm)
- Dav =
averaged distance between particles (mm)
- Df =
lens diameter (mm)
- Ep =
pulse energy (J)
- frep =
laser repetition rate (Hz)
- is =
current signal (Ampere)
- I =
laser intensity (W/cm2)
- Is =
signal intensity (W/cm2)
- n =
particle density (cm)−3
- Pi =
optical power (W)
- zl =
distance between laser to VOI (mm)
- zo =
distance of a lens to the object plane (mm)
Appendix: Depth of Field and Circle of Confusion
The thin lens equation is essential for understanding the image generation of the Mie scattering particles in planar and volumetric PTV. Figure 3 illustrates the main concepts involved with a thin lens. Three planes are defined in perpendicular to the horizontal optical axis: The object plane (the plane where the object lies), the lens plane (the plane where the lens is positioned) and the image plane (the plane where the image is formed).
According to geometric optics, the finite-sized objects (seeding particles) consist of a distribution of point-like light sources that lie in the object plane. Each point source of the object emits light rays in all directions. The rays of each point source define a cone, where its apex is the point-like light source and its base is the area of the lens. Ideally, all the rays within that cone will be imaged into a single point on the image plane.
A digital camera consists of a CMOS detector, a lens and an aperture. In order to view an object at a distance zo away from the lens “in focus,” the CMOS detector must be at a distance zi away from the lens on the opposite side of the lens. Then all the point sources of the object will be mapped into image points in the image plane.
The point-like light source of the object at zo is imaged into a point-like light source at zi on the CMOS detector (the green rays). However, the other two point-like light sources, and , are imaged as blurred circles on the image plane zi, since their foci lie in front of and behind the image plane. The resulting blurred circles due to the lack of focus are known as the circle of confusion with diameter CoC.
The depth of focus is obtained from the difference . The depth of focus is thus the segment along the optical axis where a point source from the object plane is imaged into a circle that is smaller than the circle of confusion. Correspondingly, the depth of field, , is a segment along the optical axis where any point source within the depth of field will appear on the image plane as a circle of light with a diameter smaller than CoC.
where .